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Energy and Momentum Conservation in
Bohm’s Model for Quantum Mechanics
Bryan Hall
Ph.D. Thesis
University of Western Sydney
2004
ii
Abstract
Bohm’s model for quantum mechanics is examined and a well-known drawback of the
model is considered, namely the fact that the model does not conserve energy and
momentum. It is shown that the Lagrangian formalism and the use of energy-momentum
tensors provide a way of addressing this non-conservation aspect once the model is
considered from the point of view of an interacting particle-field system. The full
mathematical formulation that is then presented demonstrates that conservation can be
reintroduced without disrupting the present agreement of Bohm’s model with experiment.
iii
Acknowledgement
I would most sincerely like to thank Dr Rod Sutherland for his continual guidance and
suggestions, as well as his careful checking of the text of this thesis.
iv
This is to certify that the work embodied in this thesis is original and has not been
submitted for a higher degree at any other institution.
Bryan Hall
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Contents
Chapter 1: Introduction.................................................................................................... 1
Chapter 2: Interpretations of Quantum Mechanics & the Measurement Problem ... 4
2.1 Historical Context................................................................................................... 4
2.2 Mathematical Structure and Statistical Interpretation ............................................ 5
2.3 The Correspondence Principle................................................................................ 6
2.4 The Copenhagen Interpretation .............................................................................. 6
2.5 Hidden Variable Theories....................................................................................... 9
Chapter 3: Bohm's Model............................................................................................... 12
3.1 Summary of Bohm's Model.................................................................................. 12
3.1.1 Equation of Continuity................................................................................. 13
3.1.2 Hamiltonian - Energy Considerations .......................................................... 15
3.1.3 Potential Gradient and Force in Bohm’s Model........................................... 16
3.2 Velocity as a Function of Position - Bohm’s Equation of Motion ....................... 17
3.3 Bohm’s Model and Conventional Quantum Mechanics....................................... 19
3.4 Energy and Momentum Not Conserved ............................................................... 20
3.4.1 Restoring Conservation ................................................................................ 22
3.5 Extensions to Bohm’s Model ............................................................................... 23
3.5.1 Holland's Generalisation .............................................................................. 24
3.5.2 Deoto and Ghiradi's Generalisation ............................................................. 25
3.5.3 Sutherland's Generalisation.......................................................................... 26
Chapter 4: Lagrangian Formalism................................................................................ 28
4.1 Lagrangian Formalism for Particle Motion .......................................................... 28
4.2 Lagrangian Formalism for Fields ......................................................................... 30
vi
4.3 Noether’s Theorem & Conservation..................................................................... 33
4.4 Overall Lagrangian for a Particle & Field in Interaction...................................... 33
4.5 Squires attempted Lagrangian Formulation of Bohmian Mechanics ................... 35
Chapter 5: A Lagrangian Formulation of Bohms Model ............................................ 38
5.1 Proposed Lagrangian Density............................................................................... 38
5.2 Derivation of Bohm’s Equation of Motion from the Lagrangian Density ........... 41
5.3 Field Equation Deriving from the Proposed Lagrangian Density ........................ 42
5.4 Consistency of the Derived Field Equation with Experiment .............................. 47
Chapter 6: Energy-Momentum Tensors ....................................................................... 49
6.1 Basic Theory......................................................................................................... 49
6.2 Energy-Momentum Tensor for a Scalar Field...................................................... 50
6.3 Energy and Momentum for a Scalar Field interacting with a Particle ................. 51
6.3.1 Energy and Momentum Conservation Equations......................................... 51
6.3.2 Introduction of Tµµµµννννparticle ............................................................................... 53
6.3.3 Global Equations .......................................................................................... 54
6.4 Tentative Application to Bohm's Model............................................................... 55
Chapter 7: Relativistic Treatment ................................................................................. 60
7.1 De Broglie’s Model .............................................................................................. 61
7.2 Lagrangian Density for de Broglie’s Model......................................................... 63
7.3 Equation of Motion for the Particle...................................................................... 66
7.4 Field Equation....................................................................................................... 66
7.5 Energy-Momentum Tensor for the Particle.......................................................... 68
7.6 Noether’s Theorem adapted to the Present Case .................................................. 69
7.7 Summary of Equations describing Overall Conservation .................................... 72
7.8 Energy-Momentum Tensors Tµνfield and Tµν
interaction ............................................. 73
vii
7.9 Divergence and Conservation............................................................................... 75
7.9.1 Divergence of Tµµµµννννfield .................................................................................... 75
7.9.2 Divergence of Tµµµµννννparticle ................................................................................. 76
7.9.3 Divergence of Tµµµµννννinteraction ............................................................................. 76
7.9.4 Divergence of Tµµµµννννtotal .................................................................................... 77
Chapter 8: Non-Relativistic Limit ................................................................................. 78
8.1 Non-Relativistic Energy-Momentum Tensor for the Particle .............................. 78
8.1.1 Physical Interpretation of Tµµµµννννparticle ............................................................... 78
8.1.2 Rules for obtaining the Non-Relativistic Limit ............................................ 83
8.1.3 Derivation of Tµµµµννννparticle .................................................................................. 85
8.2 Non-Relativistic Energy-Momentum Tensor for the Field .................................. 88
8.2.1 Non-Relativistic Tijfield.................................................................................. 91
8.2.2 Non-Relativistic Ti0field ................................................................................. 91
8.2.3 Non-Relativistic T0ifield ................................................................................. 91
8.2.4 Non-Relativistic T00field................................................................................. 92
8.2.5 Overall Non-Relativistic Result for Tµµµµννννfield .................................................. 92
8.3 Non-Relativistic Energy-Momentum Tensor – Interaction Component .............. 93
8.3.1 Non-Relativistic Tijinteraction ........................................................................... 95
8.3.2 Non-Relativistic Ti0interaction........................................................................... 96
8.3.3 Non-Relativistic T0iinteraction........................................................................... 96
8.3.4 Non-Relativistic T00interaction .......................................................................... 96
8.3.5 Overall Non-Relativistic Result for Tµµµµννννinteraction ............................................ 97
8.4 Divergence and Conservation............................................................................... 97
8.4.1 Divergence of Tµµµµννννfield .................................................................................... 98
8.4.2 Divergence of Tµµµµννννparticle ................................................................................. 99
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8.4.3 Divergence of Tµµµµννννinteraction ........................................................................... 101
8.4.4 Divergence of Tµµµµννννtotal .................................................................................. 103
8.5 Simplifications in the Bohmian Case ................................................................. 104
Chapter 9: Discussion and Conclusions ...................................................................... 107
Appendix 1: Non Locality............................................................................................. 109
A1.1 The EPR Paradox............................................................................................. 109
A1.2 Bells Theorem.................................................................................................. 111
A1.3 Counterfactual Definiteness............................................................................. 112
A1.4 Bohm's Model and Non-locality...................................................................... 113
A1.5 Kochen and Specker’s Proof ........................................................................... 113
Appendix 2: Velocity Expression corresponding to the Modified Schrodinger
Equation ......................................................................................................................... 115
Appendix 3: Rate of Change of a Particle's Energy in a Scalar Field...................... 117
Appendix 4: Schrodinger Energy-Momentum Tensor .............................................. 118
Appendix 5: Conservation Difficulty with the Schrodinger Energy-Momentum
Tensor ............................................................................................................................. 120
Appendix 6: Viability of a Scalar Potential Description with de Broglie’s Relativistic
Model .............................................................................................................................. 122
Appendix 7: Relativistic Equation of Motion ............................................................. 123
A7.1 Derivation from the Relativistic Lagrangian Density...................................... 123
A7.2 Consistency of the Equation of Motion with the Identity uµuµ = c2 ................ 125
Appendix 8: Modified Klein-Gordon Equation ......................................................... 127
Bibliography .................................................................................................................. 130
ix
Please Note:
The symbol for the quantity “h bar” (= Planck’s constant divided by 2π) appears as h
in the printing of this thesis (i.e., as a letter h with a small gap in its vertical stroke).
_Chapter_1.doc
1
Chapter 1: Introduction
This thesis looks at a particular interpretation of the formalism of quantum mechanics,
viz., the model proposed by David Bohm. The aim is not to argue for or against this
model, since the whole interpretation question for quantum mechanics is an area of much
controversy. Rather, the aim is to resolve a precisely defined physical and mathematical
problem that has been highlighted by several authors as being a possible deficiency of the
model. It is demonstrated here that this feature of Bohm’s model, namely that it does not
conserve energy and momentum, can be successfully eliminated if desired.
Advocates of Bohm’s model can, of course, claim that it is already both empirically
adequate and logically consistent without introducing such conservation. Nevertheless,
there seems to be a general view, shared by supporters of the model, that the possibility
of restoring energy and momentum conservation remains an interesting and aesthetically
appealing idea.
The structure of the thesis is as follows:
Chapter 2 provides a general discussion of the development of quantum mechanics and
the problem of its interpretation. It considers the Copenhagen interpretation, the
Measurement Problem and the possibility of hidden variables.
Chapter 3 summarizes the basic structure of Bohm’s model for quantum mechanics. It
describes the model’s derivation from the equation of continuity and compares the
modern minimalist version of the model with Bohm’s original version. Expressions for
Bohm’s “quantum potential” are derived in preparation for later use in the thesis. The fact
_Chapter_1.doc
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that Bohm's model does not conserve energy and momentum is then highlighted, this
aspect of the model being the main focus of subsequent chapters. Finally, possible
extensions to Bohm’s model that have been suggested by other authors are discussed.
In chapter 4, the Lagrangian formalism is outlined in preparation for applying it to
Bohm’s model. The eventual aim is to introduce energy and momentum conservation via
Noether’s theorem. Examples of a Lagrangian for particle motion and Lagrangian
densities for free field evolution are first discussed, followed by sample Lagrangian
densities for a particle and field in interaction. These expressions serve as possible
analogies and guides towards a Lagrangian density for Bohm’s model. Finally, an earlier
attempt at a Lagrangian formalism for Bohm’s model, proposed by Squires, is
summarized and discussed.
In chapter 5, a Lagrangian density suitable for Bohm’s model is introduced. It is then
demonstrated that this expression yields the usual equation of motion for the Bohmian
particle. Such a Lagrangian formulation characterizes Bohm’s model as an interacting
particle-field system and pursuing this approach necessarily causes some modification to
the Schrodinger equation. It is shown, however, that the particular modification
introduced by the Lagrangian density proposed here does not compromise the
Schrodinger equation’s standard, experimentally-verified predictions.
Chapter 6 summarizes the general theory of energy and momentum conservation for
particle-field systems in terms of the divergence of energy-momentum tensors. It then
tentatively considers the application of this formalism to Bohm’s model and highlights
some difficulties that arise.
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Chapter 7 proceeds to resolve these difficulties encountered in the non-relativistic theory
by instead formulating a relativistic treatment, using a Klein-Gordon version of Bohm’s
model published by de Broglie. The mathematical proof of Noether’s theorem is then re-
derived from first principles for this particular situation. The previous problems are
thereby eliminated, with the intention then being to proceed by taking the non-relativistic
limit. In preparation for this step, separate expressions are obtained for the energy-
momentum tensors of the field, particle and interaction, with the overall divergence being
shown to be zero as required.
Chapter 8 takes the non-relativistic limit of the formulation in the previous chapter.
Particular attention is paid to the appropriate expression for the energy-momentum tensor
of the particle, so that certain subtleties can be addressed concerning rest energy and the
symmetry of the tensors. Three rules are thereby identified which allow the non-
relativistic limits for the field and interaction expressions to be obtained easily. The
overall divergence is then confirmed to be zero for the non-relativistic case, showing that
energy and momentum conservation have been successfully introduced into Bohm’s
model.
Finally, chapter 9 summarizes all the steps that have been taken in developing the
argument and the problems encountered, including some comments on the strengths and
weaknesses of the formulation.
_Chapter_2.doc
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Chapter 2: Interpretations of Quantum Mechanics and theMeasurement Problem
2.1 Historical Context
Quantum physics grew from attempts to understand the behaviour of infinitesimally
small sub-atomic entities. As outlined by Heisenberg, in his 1932 Noble Prize address1,
the basic postulates of the quantum theory arose from the fact that atomic systems are
capable of assuming only discrete stationary states, and therefore of undergoing only
discrete energy changes.
Initially the program of quantum mechanics involved attempting to model observable
phenomena such as the electromagnetic emission and absorption spectrum of atoms.
Classical physics had dealt with “objective” processes occurring in space and time by
specifying some initial conditions and modelling the time evolution of such processes. In
addressing the quantum problem, Heisenberg observed that, according to the program of
classical physics, it ought to be possible to calculate the exact path of electrons “orbiting”
atomic nuclei from the measured properties of the emitted and absorbed radiation.
However, the program of producing a causal model in which the frequency spectrum is
directly related to the path of an electron “orbiting” around an atom met with very
considerable difficulties. Heisenberg’s ultimate solution to the problem was to develop
the theory of Matrix Mechanics2, in which any concept which could not be
experimentally verified was excluded. Heisenberg observed that by abandoning notions
which were not experimentally testable, contradictions between experiment and theory
1 Heisenberg W., Nobel Prize in Physics Address: A General History of the Development of Quantum
Mechanics, 1932. Published by Elsevier Publishing Co, with the permission of the Nobel Foundation. Cited
from The World of Physics Vol. 2, pp. 353-367. Simon and Schuster, New York (1987).
_Chapter_2.doc
5
could be avoided. Consequently, Heisenberg argued that classical concepts such as the
electron trajectory (position & momentum), which remains unobservable, should be
abandoned at the quantum level. Heisenberg emphasised that the existence of entities
which are in-principle unobservable cannot be objectively established and belief in their
existence is therefore a matter of personal choice.
Soon afterwards, Schrodinger produced his “Wave Mechanics”, in which a quantum
mechanical description of a system is presented in terms of a characteristic function
known as the wave function. Following the publication of his original paper3,
Schrodinger initially advanced the view that entities such as electrons and photons were,
in fact, waves. A wave model, which interpreted the Schrodinger wave function as
describing the spatial extent of real physical waves, seems well suited to explaining
quantum interference. However, there are a number of difficulties with erecting a wave-
based quantum theory to describe individual electrons which can be counted by Geiger
counters and observed as spots on photographic plates.
2.2 Mathematical Structure and Statistical Interpretation
The mathematical structures of the Heisenberg and Schrodinger formulations of quantum
mechanics are well understood and their formal equivalence was established very early
on by Schrodinger and Dirac. Consistent with the original formulations, the general
Hilbert Space representation was developed.
2 Heisenberg W., Z. Physik Vol. 33, p. 879 (1925).
3 Schrodinger, E., Ann. Physik Vol. 79, pp. 361 and 489 (1925); Vol. 80, p. 437 (1926); Vol. 81, p. 109
(1926)
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In deducing the correct statistical meaning for the normalised Schrodinger wave function,
Max Born provided the central, experimentally verified tenet of non-relativistic quantum
mechanics. Born's postulate requires that the volume integral of the square of the
Schrodinger wave function's modulus give the probability of finding the particle in that
volume. In a similar manner, the statistical distribution of measurement results for any
other observable quantity may be determined by switching the wave function to the
representation corresponding to that observable. The desired distribution is then given by
the squared modulus of the transformed wave function. In this scheme, physical
quantities are incorporated as representation-dependent, self-adjoint4 mathematical
operators. The point must be made emphatically that, in terms of Born's Interpretation,
the Schrodinger wave function, or state function, describes the statistical behaviour of an
aggregated collection. This quantum mechanical statistical algorithm need not constrain
individual ensemble members.
2.3 The Correspondence Principle
The Correspondence Principle requires that, under appropriate limiting circumstances
(usually expressed as lim
_Chapter_2.doc
7
existing reality”. Schrodinger5 eventually conceded that the quantum mechanical
formalism developed provides only a statistical algorithm for making predictions about
measurement results and does not provide any clear picture of entities existing between
measurement events. It therefore gives no insight into the nature of any possible
underlying reality and fails to support a “principle of causality” in any form.
Adherents to the "Copenhagen interpretation" of quantum mechanics assert that a
“complete description of reality” is in fact provided by Born's experimentally verified
statistical hypothesis (above) and that models describing the time evolution of individual
entities between observations are neither useful nor possible. Niels Bohr summarises the
Copenhagen view well claiming that “in quantum mechanics we are not dealing with an
arbitrary renunciation of a more detailed analysis of atomic phenomena, but with a
recognition that such an analysis is in principle excluded.6”
In an effort to challenge the Copenhagen interpretation, which proposed that quantum
theory provided a complete description of individual quantum entities rather than a
statistical algorithm for determining the behaviour of quantum ensembles, Einstein and
others developed a variety of objections to various peculiarities inherent in the
Copenhagen viewpoint. Schrodinger’s Cat and the non-locality following from the EPR
paradox rank as the most famous of these challenges. In contrast with Bohr, Einstein
5 Schrodinger E., Science & Humanism; Physics in Our Time, Cambridge 1951. Cited from Newnam J.R.,
Causality and Wave Mechanics in The World of Mathematics Vol II, pp. 1056-1068. George Allen &
Unwin Ltd (1960).6 Goldstein S., Quantum Theory Without Observers - Part One, in Physics Today, March 1998, pp. 42-46.
_Chapter_2.doc
8
asserted that the wave function provided a description of only quantum ensembles and
not of individual quantum entities7.
Toulmin8 observes that much of the unfocussed and unresolved controversy concerning
the interpretation of quantum mechanics has its roots in the fact that Einstein and his
supporters have refused to accept the change in standards of “what needs explaining”
which has been made with the development of the Copenhagen interpretation of quantum
mechanics. In Einstein’s view, these changes require one to restrict the horizon of
scientific endeavour in an unjustifiable way. Einstein’s opponents, on the other hand,
claim that his objections show only that he has not properly understood the theory.
Toulmin does not deal with the substance of the dispute but draws significant attention to
the language in which the dispute is carried on. The dispute is couched in terms of the
question, “Is a quantum mechanical description of a physical system complete or not?”
Toulmin argues that this way of posing the problem confuses the issue, giving it too sharp
an appearance of opposition. A complete or exhaustive description of a physical system is
one from which one can, using the currently accepted laws of nature, infer all properties
of the system for which it is a physicist’s ambition to account. Where two physicists do
not share a common standard of what does and does not need to be explained, there is no
hope on their agreeing that the corresponding description can be called complete. The
use of the word complete, with its implicit reference to particular criteria of
completeness, may serve to conceal rather than reveal the point at issue. A similar moral
holds more generally where, in the absence of any explanation, the term “reality” is
7 Einstein A. and Franklin J., Physics and Reality (1936). Cited from Dewitt B.S. and Graham N.R.,
Resource Letter IQM-1 on the Interpretation of Quantum Mechanics, American Journal of Physics. Vol. 39
pp. 724-738 (1971). See especially pp. 730 & 731.
_Chapter_2.doc
9
frequently used. Heisenberg highlighted this when asked directly the question: “Is there a
fundamental level of reality?” He responded as follows:
“This is just the point; I do not know what the words fundamental reality mean. They are
taken from our daily life situation where they have a good meaning, but when we use
such terms we are usually extrapolating from our daily lives into an area very remote
from it, where we cannot expect the words to have a meaning. This is perhaps one of the
fundamental difficulties of philosophy: that our thinking hangs in the language. Anyway,
we are forced to use the words so far as we can; we try to extend their use to the utmost,
and then we get into situations in which they have no meaning.”9
2.5 Hidden Variable Theories
In spite of the Copenhagen interpretation, there have been extensive efforts to introduce
theories providing a deeper description of nature. Principally these theories have taken
the form of “hidden variable” theories in which certain properties of individual quantum
entities always pre-exist before an act of measurement. One motivation for the hidden
variables program is that the Copenhagen interpretation of the Schrodinger equation is
unable to account in a satisfying way for the process of measurement wherein a
discontinuous transition from a spread-out state to a definite experimental result occurs.
Schrodinger wave functions evolve continuously and smoothly through time and after a
particle and an apparatus interact they are described by a single, overall wave function
from then on. This transition to a correlated state should result in the state of the
8 Toulmin S., The Philosophy of Science, pp. 118-9. Hutchison and Company, London. (Sixth Impression
1962).9 Buckley P. and Peat F.D., A Question of Physics; Conversations in Physics and Biology, pp. 3-16.
Routledge and Kegan Paul, London and Henley (1979). See especially p. 9.
_Chapter_2.doc
10
apparatus becoming less definite, rather than the particle’s state becoming more definite.
However, at the macroscopic level, definite measurement results are always obtained.
The Copenhagen “analysis” of the measurement process simply invokes Von Neumann's
Projection Postulate, which asserts that the state vector evolves according to the
Schrodinger equation while the system is isolated, but changes discontinuously during
measurement to an eigenstate of the observable that is measured10. Because of the
apparent necessity that the postulate apply only for "measurement" interactions, not for
"non-measurement" interactions, there has been much controversy concerning this
infamous Measurement Problem and the Copenhagen interpretation in general since they
were first proposed.
Hidden variables programs frequently take their motivation from other areas of physics
such as the classical theory of gases, which is understood as a macroscopic
approximation arising statistically from the aggregated behaviour of a large number of
microscopic gas molecules. On the other hand, advocates of the Copenhagen
interpretation have attempted to produce "impossibility proofs" intended to demonstrate
the incompatibility of hidden variables theories with quantum mechanics. Von Neumann
claimed to present a proof that hidden variables theories were not possible, but the proof
failed since it made the incorrect assumption that an algebraic rule which must hold in the
mean for non-commuting observables must also hold for the individual hidden values11.
Since the formalism of quantum mechanics does not necessarily imply the Copenhagen
interpretation, the possibility of constructing different models that are observationally
10
Ballentine L.E., Resource Letter IQM-2: Foundations of Quantum Mechanics since the Bell Inequalities.
American Journal of Physics Vol. 55, pp. 785-792 (1987).
_Chapter_2.doc
11
equivalent to conventional quantum mechanics remains open12. Although certain types of
hidden variables models can be ruled out, it is not possible to invalidate all hidden
variables models. Writing in Physics Today (1998), Goldstein13 claims that the Bohr-
Einstein debate has actually been resolved in favour of Einstein since a number of
observer-free formulations of quantum mechanics, in which the process of measurement
can be analysed in terms of more fundamental concepts, have been produced. Examples
of observer-free formulations include: Decoherent Histories, Spontaneous Localisation
and Pilot Wave theories (including Bohm’s Model).
11
Von Neumann J., Mathematical Foundations of Quantum Mechanics, Princeton U. P., New Jersey
(1955). Also, Bell J.S., Rev. Mod. Phys. Vol.38, p. 447 (1966).12
Cushing J.T., Quantum Mechanics: Historical Contingency & the Copenhagen Hegemony. p. 42.
University of Chicago Press (1994).13
Goldstein S., Quantum Theory Without Observers - Part One. Physics Today, March 1998, pp. 42-46.
_Chapter_3.doc
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Chapter 3: Bohm's Model
3.1 Summary of Bohm's Model
For non-relativistic quantum mechanics, David Bohm has explicitly constructed a scheme
which supports a continuously evolving underlying “particle trajectory” and yields results
entirely consistent with experimental evidence1. Even if suitable for no other purpose, the
Bohm model has demonstrated that an unqualified refutation of hidden variables theories
is, in fact, not possible. This model also refutes certain other claims, such as that we must
necessarily abandon realism, determinism, analyzability, etc.
The mathematical structure of the Bohmian model arises from combining the
Schrodinger Equation, the Equation of Continuity and the requirement of Conservation of
Probability in a fairly straightforward manner. Writing the wavefunction in the form:
ψ(x,t) = R(x,t) exp (iS(x,t)
h) [3-1]
Bohm's non-relativistic model requires three basic physical assumptions:
1. An electron or other quantum entity is a particle (represented by a position coordinate
x that is a well-defined, continuous function of time).
2. The particle's velocity is given at all times by v = ∇∇∇∇S/m.
3. P(x;t) = R2 is the probability distribution for particle positions in a statistical
ensemble of similar systems.
1 Bohm. D., A Suggested Interpretation of the Quantum Theory in Terms of Hidden Variables, Physical
Review Vol. 85, pp. 166-179 and 180- 193 (1952).
Bohm D. and Hiley B.J., Measurement Understood Through the Quantum Potential Approach,
Foundations of Physics Vol. 14, pp. 254-274 (1984).
_Chapter_3.doc
13
3.1.1 Equation of Continuity
The Schrodinger Equation2 and its complex conjugate can be written as:
– h
2
2m∇2 ψ + V(x) ψ = ih
∂ψ∂t
[3-2a]
– h
2
2m∇2 ψ* + V(x) ψ* = – ih
∂ψ*
∂t[3-2b]
and the classical equation of continuity for fluids takes the form3:
∇ . j +∂ρ∂t
= 0 ; j = ρ v [3-3]
Here, j(x,t) is the fluid flux, or mass of fluid passing through a defined unit cross-section
per unit time. The flux is obtained by multiplying the flow velocity v(x,t) by the fluid's
local density ρ(x,t) within the cross-section.
Evaluating Ψ* x (SE) - Ψ x (SE)*, where SE denotes the Schrodinger equation, the
following expression may be obtained4:
–h
2
2m∇( ψ*∇ψ – ψ ∇ψ* ) – ih
∂(ψ*ψ)
∂t= 0 [3-4]
Using the R,S polar notation for the complex function Ψ, this equation reduces to:
∇ [R
2 ∇Sm ] +
∂ R2
∂t= 0 [3-5]
Comparing equations [3-3] and [3-5], Bohm’s Model develops from making the obvious
associations:
ρ = R2 [3-6]
v = ∇∇∇∇S/m [3-7]
The second of these equations can be rewritten as a momentum expression:
2 See, e.g., p. 95 in Saxon D.S., Elementary Quantum Mechanics. Holden Day Publishers, San Francisco,
California. (1968).3 See, e.g., p.121 in Messiah A., Quantum Mechanics. Vol. 1. North - Holland Publishing Company
Amsterdam (1964).
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14
p = ∇∇∇∇S [3-8]
The minimalist version of Bohm's model requires only the three basic physical
assumptions numbered above. However, the original presentation of Bohm's model,
which may be distinguished as the "de Broglie-Bohm model," included a derived
quantum potential Q, outlined in the next section. It was later realised that Bohm's model
did not actually require presentation of the quantum potential to reach agreement with
experimental results. For this reason the inclusion of the quantum potential Q is not
actually necessary. Durr, Goldstein and Zanghi have stated that, from their perspective,
the artificiality suggested by the quantum potential is the price one pays for attempting to
cast the non-classical Bohmian theory into a classical mould5. They use the name
"Bohmian mechanics" for the minimalist version of the theory which does not contain the
quantum potential in its formulation.
In Bohm's model, the use of statistics via P(x,t) = R2(x,t) is a consequence only of our
ignorance of the particles exact position rather than being inherent in the conceptual
structure of the model. The wave function ψ plays two conceptually distinct roles in that
it determines both the influence of the environment on the particle's position6 and the
probability density P(x,t) 7.
4 See, e.g., pp. 25-27 in Schiff L.I., Quantum Mechanics, 3
rd Edition. McGraw Hill Book Company (1968).
5 Cushing J.T., Quantum Mechanics: Historical Contingency & the Copenhagen Hegemony. p. 45.
University of Chicago Press (1994). (See also other references cited therein.)6 More fundamentally, the wave function generates the vector field on configuration space defining the
equation of motion of the particle.7 Durr D., Goldstein S. and Zanghi N., Quantum mechanics, Randomness, and Deterministic Reality.
Physics Letters A. Vol. 172, pp. 6-12 (1992). See also footnote 5 above.
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15
3.1.2 Hamiltonian - Energy Considerations
By evaluating Ψ* × (SE) + Ψ × (SE)*, it is possible to produce an equation containing
terms similar to the classical Hamiltonian, a function which expresses the system's energy
in terms of momentum p, position x and possibly the time t. The relevant classical
equation is:
Kinetic Energy + Potential Energy = Total Energy [3-9]
which can be written as8:
p2
2m+ PE = E [3-10]
In the case of Bohm’s model, the corresponding equation is9:
[∇S]
2
2m+ V –
h2
2m∇2
RR
= – ∂S∂t
[3-11]
Given the previous association p = ∇∇∇∇S for momentum, the Schrodinger equation can now
be reinterpreted, within Bohm’s model, as representing a classical particle having
potential energy and total energy given, respectively, by10:
PE = V –h2
2m∇ 2R
R[3-12]
E = – ∂S∂t
[3-13]
The potential consists of a classical component V plus a quantum component, usually
represented by the letter Q:
Q = –h
2
2m∇2
RR
[3-14]
8 Here, p
2 is taken to mean |p|
2 which is simply p . p (similarly [∇S]
2 = |∇S|
2 = ∇∇∇∇S . ∇∇∇∇S).
9 It is assumed in this thesis that V(x) is real.
10 Bohm. D., A Suggested Interpretation of the Quantum Theory in Terms of Hidden Variables, Part1,
Physical Review Vol. 85, pp. 166-179 (1952).
_Chapter_3.doc
16
3.1.3 Potential Gradient and Force in Bohm’s Model
It can be shown11 that the "quantum mechanical force" required to produce the
accelerations described implicitly by the velocity relationship v = ∇∇∇∇S/m is equal to minus
the gradient of the potential given in [3-14]. The derivation is as follows (i and j have the
values 1,2 and 3 here, xi and xi are related via xi = −xi and a summation is implied over
repeated indices):
Fi= d
dt(mvi) [3-15a]
= m (dx j
dt
∂vi
∂x j+
dt
dt
∂vi
∂t) [3-15b]
= ( v j∂
∂x j+
∂∂t
) mvi [3-15c]
Substituting in the expression mvi = − ∂S/∂xi from equation [3-7], we obtain:
Fi= – ( – 1
m∂S
∂x j
∂
∂x j+
∂∂t
)∂S
∂xi
[3-16a]
= – ( – 1m
∂S
∂x j
∂2S
∂x j∂xi
+∂2
S
∂t∂xi
) [3-16b]
= –∂
∂xi
( – 12m
∂S
∂x j
∂S
∂x j+
∂S
∂t) [3-16c]
and using the relationship xi = − xi, this equation can be written in the form:
Fi= –
∂
∂xi( 1
2m
∂S
∂x j
∂S
∂x j–
∂S
∂t) [3-16d]
i.e.,
F = – ∇∇∇∇ ( –[∇S]
2
2m–
∂S
∂t) [3-16e]
Employing equation [3-11]:
11 Belinfante F.J., A Survey of Hidden Variable Theories, p. 185. Pergamon, Oxford (1973).
_Chapter_3.doc
17
[∇S]
2
2m+ V –
h2
2m∇2
RR
= – ∂S∂t
the force equation [3-16e] then becomes:
F = – ∇∇∇∇ [ V –h
2
2m∇2
RR
] [3-17]
Hence, referring to equations [3-12] and [3- 14], we can write:
F = – ∇∇∇∇ PE [3-18a]
= – ∇∇∇∇ (V + Q) [3-18b]
3.2 Velocity as a Function of Position - Bohm’s Equation of Motion
According to the minimalist version of Bohm’s model (which views [3-18b] as
superfluous), a Bohmian particle traces out a smooth trajectory and its velocity evolves in
a continuous manner as determined by ∇∇∇∇S. There is a clear contrast between this
Bohmian mechanics and classical mechanics. In classical mechanics, the Newtonian
equation of motion involves the second derivative of the particle's position coordinate
with respect to time:
md
2x
dt2 = – ∇V [3-19]
whereas Bohm's equation of motion only involves the first derivative:
m dxdt
= ∇S [3-20]
This has the following consequences. Solving the Newtonian equation in order to
determine the particle's trajectory x(t) requires performing two integrals with respect to
time, whereas solving Bohm's equation to obtain x(t) requires only one time integral. It
follows that two unknown constants of integration arise in the Newtonian case, but only
one in the Bohmian case. Physically, this means that, in attempting to determine a
particle's trajectory uniquely in this way, we need to specify both the initial position and
_Chapter_3.doc
18
the initial velocity in the Newtonian case but only the initial position in the Bohmian
case.
Contrasting the two different mechanics further, Bohm's model has been described as
"Aristotelian"12. This refers to the ancient "common sense" viewpoint, attributed to
Aristotle, that all objects will eventually come to rest unless kept moving by a force. In
the subsequent physics of Newton, on the other hand, a moving object keeps on moving
uniformly in a straight line unless acted upon by a net force. It has been argued by other
authors (see footnote 10) that the above two equations of motion can be considered to
exhibit this distinction in the following sense. Looking at the Newtonian equation,
suppose the external influence is "switched off", which in this case means setting the
potential V equal to zero. The particle's acceleration then becomes zero, but its velocity is
not affected. If moving beforehand, the particle keeps moving in a uniform manner. In
contrasting this result with Bohm's model, we will assume that [3-20] (in conjunction
with the Schrodinger equation [3-2a]) is taken as providing a fundamental
characterisation of Bohmian mechanics and that any other equations of the model are
treated as secondary. We then suppose that the external influence can somehow be
"switched off" in the Bohmian case, which this time means deleting the wavefunction
accompanying the Bohmian particle so that R and S become zero (and the particle is left
on its own). Setting S to zero in equation [3-20], we see that now it is the velocity that
becomes zero and the particle jerks immediately to a halt.
12
Durr D., Goldstein S. & Zanghi N., Quantum Equilibrium and the Origin of Absolute Uncertainty,
Journal of Statistical Physics, Vol. 67, pp. 843-907 (1992).
Also Valentini A., Pilot Wave Theory, p. 47 in Bohmian Mechanics and Quantum Theory: An Appraisal,
Edited by Cushing J.T., Fine A. and Goldstein S. Kluwer Academic Publishers, Dordrecht (1996).
_Chapter_3.doc
19
3.3 Bohm’s Model and Conventional Quantum Mechanics
Bohm has shown that, for a statistical ensemble of particles, the additional postulate p =
∇∇∇∇S, together with Born's statistical law P(x) = R2, provides exact agreement with
conventional non-relativistic quantum mechanics for all possible experimental
circumstances13. This precise agreement means that Bohm’s model cannot be
experimentally distinguished from the conventional theory. Bohm’s scheme is
mathematically deterministic in the sense that the equation p = ∇∇∇∇S uniquely determines a
particle's future trajectory once the initial position is specified. However, as with classical
mechanics, since it is not possible to measure or prepare the initial position with infinite
precision, complete "predictability" cannot be achieved.
Bohm’s model copes reasonably well with the Measurement Problem by postulating the
existence of hidden variables which uniquely determine measurement outcomes
(observations) as part of the measurement process. The variables (actually just the
particle positions) are distributed such that the usual probabilities are obtained. Bohm’s
model also provides a comprehensible physical mechanism whereby the correct post-
measurement statistical distributions for all quantum mechanical observables can be
deduced14 from the postulated pre-measurement position distribution |Ψ(x)|2.
De Broglie15 emphasized that the measurement process must allow us to distinguish
between the different states un and that typically this means separating the different states
(or something they interact with) in space. In simple cases, the outcome of the separation
13 Bohm. D., A Suggested Interpretation of the Quantum Theory in Terms of Hidden Variables, Part 1:
Physical Review Vol. 85, pp. 166-179 (1952).14
Bohm. D., A Suggested Interpretation of the Quantum Theory in Terms of Hidden Variables, Part 2:
Physical Review Vol. 85, pp. 180- 193 (1952).15
de Broglie L., Non-Linear Wave Mechanics. Elsevier, Amsterdam (1960).
_Chapter_3.doc
20
stage of the measurement is that the wavefunction Ψ(x) evolves into a collection of
spatially non-overlapping wave packets c1u1(x) + c2u2(x) + … As the packets gradually
become spatially distinct, the particle (which is assumed to be travelling along a definite
trajectory within the wavefunction) flows continuously and smoothly into one of them.
The process of measurement is therefore completed simply by determining in which
packet the particle is finally located. [Sentence deleted here.]
The non-local aspects of Bohm’s model are discussed in some detail in Appendix 1.
3.4 Energy and Momentum Not Conserved
Bohm’s model proposes for quantum mechanics an underlying reality consisting of
particles possessing continuous and smooth trajectories which are guided by a field
whose properties are defined by the associated wavefunction Ψ. As shown earlier (see
equations [3-11] to [3-13]), the Schrodinger equation can be manipulated to yield an
equation containing terms that resemble a classical Hamiltonian:
[∇S]2
2m+ V –
h2
2m∇2
RR
= – ∂S∂t
thereby pointing to the following expressions for potential energy PE and total energy E:
PE = V –h
2
2m∇2
RR
E = – ∂S∂t
(E = KE + PE)
Now, from [3-13] it follows that the total energy E of the particle is not constant, i.e., not
conserved, except in the special case where the wavefunction's phase S depends linearly
on the time t. (Similar considerations apply for momentum.) Classically, one would
explain this lack of conservation by arguing that the particle is exchanging energy with
the field with which it is interacting (i.e., the particle considered on its own is not a closed
_Chapter_3.doc
21
system). Here, the field is presumably the Schrodinger wave function. An “energy-
momentum tensor” (which classically describes the energy and momentum content of a
field) can be constructed for the Schrodinger field. Using this, however, the field’s total
energy turns out to be separately conserved without involving the particle16. Hence the
total energy of the particle-field system is not conserved either. This is in conflict with
the situation everywhere else in physics.
A number of authors have suggested that this seems unsatisfactory17 and that the absence
of "action and reaction" between the guiding wave and the particle in Bohm’s theory
represent a deficiency in the model. Holland18, writing in The Quantum Theory of
Motion, has summarised the situation as follows:
“One might expect the conservation laws would apply to the total field plus particle
system in interaction, as in classical electrodynamics. The reason they do not is that the
particle does not react back on the wave; the field satisfies its own conservation laws...
From the standpoint of general theoretical principles this feature of the causal
interpretation may appear as unsatisfactory, calling for a development of the theory to
include a more symmetrical relation between wave and particle. At present we have no
idea how a source term for the ψ-field could be consistently introduced into the
dynamical equations in such a way that it does not disturb the empirically well-verified
predictions of quantum theory...”
16
Holland P.R., The Quantum Theory of Motion, Section 3.9.2, Cambridge University Press (1995).
17 Cushing J.T., Quantum Mechanics, Historical Contingency and the Copenhagen Hegemony, p. 45.
University of Chicago Press (1994).18
Holland P.R., The Quantum Theory of Motion, p. 120. Cambridge University Press (1995).
_Chapter_3.doc
22
Anandan and Brown19 have expressed similar reservations by asserting that the Bohm
model fails to provide a satisfactory account of the nature of particle trajectories because
its violation of the action-reaction principle prevents it being dynamically complete.
3.4.1 Restoring Conservation
The non-conservation of energy and momentum in Bohm’s model can be traced to the
fact that the model attempts to erect a particle interpretation using the standard
Schrodinger equation, a field equation not containing any reference to the particle's
position. This equation does not describe any influence of the particle on the field.
Consequently, Bohm’s quantum potential, which derives directly from the Schrodinger
equation, appears to act unilaterally in the sense that the quantum potential acts on the
particle (determining its trajectory) but the particle does not react back to change the
magnitude of the field. This energy non-conservation deficiency in Bohm’s model can be
addressed by adding a source term to the Schrodinger equation which permits appropriate
interaction between the particle and the field and in so doing reinstates the necessary
conservation requirements. The problem with such a source term is, of course, that it is
likely to interfere with the Schrodinger equation's highly successful agreement with
experiment. In order for a model to be viable it is therefore necessary that the source term
added be so constructed that the equation's empirically well-verified predictions remain
intact. An aim of the subsequent chapters is to consider such a way in which the
conservation principles can be incorporated within single-particle Bohmian mechanics.
19
Anandan J. and Brown H.R., On the Reality of Space-Time Geometry and the Wave-function,
Foundations of Physics, Vol. 25, pp. 349-360 (1995).
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23
3.5 Extensions to Bohm’s Model
Bohm’s original model was constructed as a provisional point of view in an effort to
provide new insight into quantum theory and suggest new possibilities for conceptual
understanding. In particular, it aimed to show that the Copenhagen interpretation was not
essential. A number of generalisations of Bohm's model have been shown to be possible.
In each case, however, the assumption of a particle trajectory existing independently of
measurement is central to the model. Consequently, Von Neumann's “Projection
Postulate” is not required and the process of measurement can be understood
satisfactorily. Bohm himself considered stochastic generalisations of his model20, in
which the quantity v = ∇∇∇∇S/m becomes only the average velocity in a stochastic process
and in which P = R2 is the limiting distribution after allowing a sufficient period to
establish a random diffusion. Subsequently, alternative generalisations have been
developed as follows (these will be discussed further below):
• Holland exploited an additional angular degree of freedom that is already implicit in
the Schrodinger equation.
• Deotto and Ghiradi added a term to the equation of continuity which maintains the
required zero divergence. (Their models were not presented as serious proposals, but
to make a point about nonuniqueness.)
• Sutherland relaxed the requirement p = ∇∇∇∇S and considered a class of models
20 Bohm D. and Hiley B.J., Measurement Understood Through the Quantum Potential Approach,
Foundations of Physics, Vol. 14, pp. 255-274 (1984). See also Bohm D., Proof that Probability Density
Approaches |ψ|2 in the Causal Interpretation of Quantum Theory, Physical Review 89, pp. 458-466 (1953),
and Bohm D. and Vigier J.P., Model of the Causal Interpretation of Quantum Theory in Terms of a Fluid
with Irregular Fluctuations, Physical Review, Vol. 96, pp. 208-216 (1954).
_Chapter_3.doc
24
described by joint probability distributions and satisfying the phase space continuity
equation.
• Bohm’s model has also been extended to include spin, perhaps most effectively by
Bell21.
3.5.1 Holland's Generalisation
In formulating extensions to Bohm’s Model, Holland22 has identified two important
matters for consideration:
(i) Is the representation unique? Can we develop valid trajectory theories in
representations other than the position representation described above? If so, how are the
laws of motion in the various representations connected?
(ii) Within a specific representation, is the law of motion unique?
Holland’s paper made the point that, in the absence of a canonical transformation theory23
for the particle position and momentum variables in the de Broglie-Bohm theory, no
general conclusions can be drawn as to connections between descriptions of motion in
different representations. Beyond pointing this out and observing that the theory must be
reconciled with results in position space, since all our measurements are finally made in
the position representation, Holland’s paper did not address point (i) in any significant
way.
21
Bell J.S., Speakable and Unspeakable in Quantum Mechanics, Paper 4: Introduction to the Hidden-
Variable Question, Cambridge University Press (1987).22
Holland P.R., New Trajectory Interpretation of Quantum Mechanics, Foundations of Physics, Vol. 28,
pp. 881-911 (1998).23
A Canonical Transformation theory for particle position and momentum variables in the de Broglie-
Bohm model would provide a standard form for expressing a change in the values for the position variables
directly in terms of a change in the values for the momentum variables and vice versa.
_Chapter_3.doc
25
With respect to the second point, Holland demonstrated that other deterministic trajectory
interpretations can be produced by exploiting an internal angular degree of freedom in the
Schrodinger equation. Holland's' argument develops from the observation that, without
disturbing the density of the particles (given by ρ = |ψ|2), a vector field having zero
divergence (∇∇∇∇.A = 0) can be added to the continuity equation in [3-3].
∇ . j +∂ρ∂t
= 0 ; j = ρ v
(See also Deotto and Ghiradi below.) The addition of the divergenceless vector field
permits the introduction of a variety of physically natural constraints to describe
trajectories other than those specified by Bohm's equation of motion (p = ∇∇∇∇S). Holland
argues that the Schrodinger equation tacitly involves a degree of freedom which is
manifest when expressing the Schrodinger equation as a differential equation in an
extended configuration space. Agreement with Bohm’s model p = ∇∇∇∇S is achieved when
the new model is “averaged over the internal freedom.” Under such circumstances, the
predictions for Holland's formulation are indistinguishable from both Bohm's model and
the standard Schrodinger formulation of quantum mechanics.
3.5.2 Deotto and Ghiradi's Generalisation
Deotto and Ghiradi24 have presented a paper whose purpose was to investigate whether
the Bohmian program of assuming that particles have definite trajectories leads
unavoidably, when some general requirements of symmetry are taken into account, to
Bohmian Mechanics. They concluded that there are infinitely many non-equivalent (from
the point of view of trajectories) Bohmian models reproducing the predictions of
24
Deotto E. and Ghirardi G.C., Bohmian Mechanics Revisited, Foundations of Physics Vol. 28, pp. 1-30
(1998).
_Chapter_3.doc
26
quantum mechanics (because there are infinitely many terms with zero divergence that
can be added to the Schrodinger current density).
3.5.3 Sutherland's Generalisation
Sutherland25 has presented a non-relativistic, single-particle generalisation of Bohm’s
model based on the observation that the restriction P(x) = ψ2
is essential to Bohm’s
theory of measurement, whereas p = ∇∇∇∇S is not. Sutherland's generalisation therefore
relinquishes the momentum relationship and allows a spread of momentum values at each
position. He points out that the equation of continuity, which ensures compatibility with
continuous trajectories, remains valid provided the less restrictive relationship <p>x =
∇∇∇∇S(x) is satisfied, where the notation <p>x stands for the mean value of momentum p at
position x. Having thus characterised a class of suitable models, Sutherland then
constructs a particular generalisation of the de Broglie-Bohm model by choosing a
specific joint distribution P(x,p) for the particle's position and momentum. He then
formulates an underlying dynamics for the motion of the particles such that the ensemble
continues to conform to the chosen distribution through time. In his generalisation of the
de Broglie-Bohm model, the expression obtained for dp/dt shows that, as in the original
model, the particles can follow smooth trajectories (i.e., trajectories containing no
discontinuous changes in velocity).
Sutherland's paper has some relevance to the present work, as follows: A Lagrangian
density expression will be introduced here in a subsequent chapter with the aim of
reinstating conservation of energy. In terms of the quantum potential Q (with the classical
25
Sutherland R.I., Phase Space Generalisation of the de Broglie-Bohm Model, Foundations of Physics Vol.
27, pp. 845-863 (1997).
_Chapter_3.doc
27
potential V ignored for simplicity), this Lagrangian expression leads to an equation of
motion of the "Newtonian" form (as shown in equation [3-19]):
md
2x
dt2 = – ∇Q
rather than of the "Aristotelian" form [3-20]:
m dxdt
= ∇S
This then means that the quantum potential [3-14]:
Q = –h
2
2m∇2
RR
becomes relevant again, despite the arguments in the literature that this potential should
be discarded from Bohm's model as superfluous. This apparent dilemma is, however,
brought into better perspective by Sutherland's work, which essentially presents a whole
class of models, all of which are in agreement with the predictions of conventional
quantum mechanics. Bohm's model is then seen to be just one model in this class and, in
fact, the only one involving an Aristotelian equation of motion. This therefore shows that
the Aristotelian form is not an essential feature of a trajectory model for quantum
mechanics and thereby makes the proposed reintroduction of the equation md
2x
dt2 = – ∇Q
quite reasonable.
_Chapter_4.doc
28
Chapter 4: Lagrangian Formalism
Lagrange’s general formulation of mechanics in terms of variational principles shows
that conservation of energy arises as a direct consequence of temporal symmetry - the
invariance of physical laws under a time translation. Similarly, conservation of
momentum arises from spatial symmetry – the invariance of physical laws under a spatial
translation (ie, the freedom to choose the origin of our coordinate system arbitrarily)1.
While Bohm’s model has the correct temporal and spatial symmetry, it does not contain
the corresponding conservation laws. (This possibility is permitted by the loophole that
the model is not derivable from a Lagrangian.) Consequently, since the model does not
possess the usual linkage between symmetry and conservation, this feature should be
examined critically as a possible deficiency.2.
4.1 Lagrangian Formalism for Particle Motion
The Lagrangian formalism provides a general formulation of the laws governing the
behaviour of mechanical systems3. In the case of a single particle, the action S is a
functional of the entire trajectory, which may be described by the parameterisation [x0(t),
v(t)], where v ≡ dx0/dt. The subscript "0" is inserted here to distinguish the point x0 at
1 Tsung-Dao L., Particle Physics and Introduction to Field Theory, in The World of Physics. Weaver J.H.
Published by Simon and Schuster, New York (1987).2 Annandan and Brown view the situation as follows: "It is well known that the dynamics of particles and
fields, in classical and quantum physics, may be described by the action principle. The space-time
translational invariance of the action of the system under consideration implies that the energy and
momentum of the system are conserved. This means that the different components of the system satisfy the
action-reaction (AR) principle. But if the action does not have translational invariance, then we would say,
rather than give up energy-momentum conservation, that there is some external influence on the system, so
that the internal components of the system do satisfy the AR principle. This suggests that the AR principle
is more fundamental than any other law of physics, as if it is a condition for the reality and being of entities
in a physical theory." Annandan J. & Brown H.R., On the Reality of Space-Time Geometry and the
Wavefunction, Foundations of Physics, Vol. 25, pp. 349-360 (1995).3 See, e.g., Landau L.D. & Lifshitz E.M., Course of Theoretical Physics. Vol. 1: Mechanics, 2
nd Edition.
Pergamon Press, Oxford, London. (1969).
_Chapter_4.doc
29
which the particle is located at time t from arbitrary spatial points x at which field values
φ(x) are to be specified. The quantity v(t) represents the particle's velocity. The action S
is defined to be the time integral of the difference between the particle's kinetic and
potential energies over the trajectory between the end points of the motion under
consideration4
Action = S = {KE[v(t)] – PE[x0(t), v(t)]} dtt 1
t2
[4-1]
The “principle of least action” provides a global description for the time evolution of the
system by asserting that an object's trajectory between specified points over a certain time
interval is that for which the action is minimised5. The integrand of the action function is
called the Lagrangian L[x0(t), v(t)]:
L = KE − PE [4-2]
It is a function of the particle's position and velocity.
One of the principal advantages of the Lagrangian formulation of mechanics arises from
the fact that the Lagrangian is required to satisfy a number of symmetries, such as, for
example translational and rotational invariance and Lorentz and Gauge invariance. While
the Lagrangian describing a physical system is not unique, these restrictions and the
physical properties of the system under consideration frequently serve to identify an
appropriate Lagrangian from which the differential form of the system's equations of
motion may be determined. In other words, the Lagrangian for a system can often be
4 See, e.g., Feynman R.P, Leighton R.B. and Sands M., Ch. 19, The Principle of Least Action, in The
Feynman Lectures on Physics, Vol. 2. Addison-Wesley, Reading Massachusetts (1994). This reference
provides a very readable introduction to the material under consideration.5 Strictly speaking, the Lagrangian formulation develops under the assumption that the action takes on a
stationary value for the correct trajectory.
_Chapter_4.doc
30
guessed by imposing certain physical restrictions, which include the homogeneity of
space and time.
The equations of motion may always be obtained from the Lagrangian by determining the
conditions under which the action takes on a stationary value. This leads to the general
result, for determining equations of particle motion, known as Lagrange's equations (i =
1, 2, 3):
ddt
∂L∂vi = ∂L
∂x0i
[4-3]
where vi is defined by:
v i ≡≡≡≡dx0
i
dt[4-4]
These equations determine the particle's trajectory.
For a single particle in a real scalar field φ(x), the equation of motion describing the
particle's trajectory can be derived from the following Lagrangian:
L particle =mv2
2– qφ(x0) [4-5]
4.2 Lagrangian Formalism for Fields
The Lagrangian formulation also provides a general description of the time evolution of
fields. For a field defined by the function
_Chapter_4.doc
31
The principle of least action asserts that the field develops in such a way that its action
integral over a specified time be minimised6. By performing variations in the field δφ(x,t)
while imposing this requirement, a differential equation describing the field's time
evolution may be deduced. When φ is a real scalar valued function, the general solution
for the differential field equation is (µ = 0,1,2,3):
∂∂xµ
∂
_Chapter_4.doc
32
A free, massless scalar field may be described by the following relativistically invariant
Lagrangian density7:
_Chapter_4.doc
33
4.3 Noether’s Theorem & Conservation
Another advantage of using the Lagrangian formulation is that it provides a direct method
for determining the energy-momentum tensor for a field. Consequently, it permits a
complete specification of both the distribution and time rate of change of energy and
momentum density throughout the field. Furthermore, Noether’s theorem9 states that
overall energy and momentum conservation will hold in any physical system that can be
described by a Lagrangian having no explicit dependence on the space or time
coordinates.
4.4 Overall Lagrangian for a Particle & Field in Interaction
Our aim is to construct a Lagrangian formulation of Bohm's model, with a single
Lagrangian density describing both the Bohmian particle and the field with which it
interacts. The latter is a scalar field defined in terms of the wave-function ψ(x). As a
guide to obtaining such a Lagrangian density expression, we will examine the known
cases of a classical particle interacting with either a scalar or vector field. In the scalar
case, the following two Lagrangians (already introduced above) must be combined
together in a consistent manner:
L particle =mv2
2– qφ(x0) [4-5]
L massless scalarfield =
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34
For convenience in the present development, only the field is being treated
relativistically. The particle motion is described by the non-relativistic equation arising
from the Lagrangian [4-5]. Using a delta function, equation [4-5] can be written
equivalently as:
L particle = [mv2
2– qφ(x) ] δ(x – x0(t)) d
3x
– ∞
∞
[4-16a]
≡ [mv2
2– qφ(x) ] ρ d
3x
– ∞
∞
[4-16b]
where
ρ ≡ δ(x – x0(t)) [4-17]
can be thought of as the distribution of the particle through space.
Equation [4-16b] provides the link for combining [4-5] and [4-15b] in a unified way to
obtain an overall particle-field Lagrangian density. The combined Lagrangian is:
L system =
_Chapter_4.doc
35
Similarly the Lagrangian density for an electromagnetic field in interaction with a particle
is10:
_Chapter_4.doc
36
where λ is a Lagrange multiplier, k is an arbitrary parameter and the separate terms
making up this overall action are given explicitly by:
A s = dt dx [ ih2
(ψ*∂ tψ – ψ∂ tψ*) –
h2
2m(∇∇∇∇ψ*)(∇∇∇∇ψ) – V ψ*ψ ] [4-23]
A λ = dt λλλλ . [ v + ih2m
dx (∇∇∇∇ψψ –
∇∇∇∇ψ*
ψ*) δ(x – x0) ] [4-24]
AN = dt [mv2
2– V(x0) ] [4-25]
Variation with respect to λλλλ, ψ, and x0, respectively, yields the equations:
v = – 1m Re (
ih ∇∇∇∇ψψ )
x = x0
[4-26]
ih ∂ tψ = Hψ – ih2m ψ*
λλλλ . ∇∇∇∇δ(x – x0) [4-27]
∂ tλλλλ = ih2m
∇∇∇∇ [λλλλ . (∇∇∇∇ψψ –
∇∇∇∇ψ*
ψ*)]x = x0
– k(mv + ∇V) [4-28]
Equation [4-27] is the Schrodinger equation with the addition of an extra term. This can
be viewed as a source term and suggests that the particle might somehow be regarded as
the source of the wave function, which produces the quantum force. The existence of the
parameter k allows us to assign the magnitude of the source term arbitrarily.
Squires stated that "work is in progress on these equations". No further developments in
his approach are known at this time. There are two points of concern with his
formulation. One is that the condition v = ∇∇∇∇S/m, which arises from the unmodified
Schrodinger equation and which remains part of Squires' model (see [4-26]), is actually
not consistent with the modified Schrodinger equation [4-27] he introduces, assuming
11
Squires E.J., Some Comments on the de Broglie-Bohm Picture by an Admiring Spectator, pp. 125-38 in
Waves and Particles in Light and Matter, Edited by van der Merwe A. and Garuccio A. Plenum Press, New
York and London (1994).
_Chapter_4.doc
37
that a conserved probability density ψ*ψ is required. The other is that it is not clear how
to obtain an energy-momentum tensor expression corresponding to Squires' Lagrangian,
as would be needed to formulate energy and momentum conservation explicitly for the
system.
As stated above, Squires' approach relates only to the minimalist version of Bohm's
model characterised by the equation dx/dt = ∇∇∇∇S/m, without the introduction of the
quantum potential Q. The view to be pursued in our subsequent development here is that
the equations involving this potential are the more appropriate ones to use in attempting
to reinstate conservation via a Lagrangian approach.
_Chapter_5.doc
38
Chapter 5: A Lagrangian Formulation of Bohms Model
The Lagrangian formalism will now be used to construct a modified version of Bohm’s
model which addresses the non-conservation deficiencies characteristic of that model.
We will limit ourselves to the single particle case. The first step is to propose an overall
Lagrangian density for the particle-field system and show that it yields Bohm's equation
of motion for the particle, plus a field equation consistent with the Schrodinger equation.
5.1 Proposed Lagrangian Density
The development of a Lagrangian density for describing the Bohmian system of a particle
and Schrodinger field in interaction will proceed as outlined in the previous chapter. It
will be assumed that the Lagrangian density consists of distinct "free-field", "particle"
and "interaction" components. In line with the previous development, it is also assumed
that:
(i) the terms of the free-field component are the familiar ones1 for generating the
Schrodinger equation,
(ii) the "particle" component has its standard form ½mv2ρ,
(iii) the "interaction" term is the usual one for a scalar field (see previous chapter).
Consequently, the proposed Lagrangian density for describing the Bohmian system of a
particle and field in interaction is:
1 See, e.g., p. 18 in Greiner W., Relativistic Quantum Mechanics – Wave Equations, 2nd Ed. Springer,
Berlin (1994).
_Chapter_5.doc
39
_Chapter_5.doc
40
The quantity ρ has the form of a delta function representing the density distribution of the
particle through space. It depends entirely upon the particle's trajectory:
x0(t) = [x01(t), x0
2(t), x03(t)] [5-2]
and expands as:
ρ = δ[x – x0(t)] [5-3a]
= δ[x1 – x01(t)] δ[x2 – x0
2(t)] δ[x3 – x03(t)] [5-3b]
The spatial dependence of the last two terms in the Lagrangian density should be
carefully observed. After evaluating the full spatial integral ∫
_Chapter_5.doc
41
factor k into the source term of the field equation, as will be discussed later in this
chapter.
The Lagrangian density [5-1] is translationally and rotationally invariant. Since it is not
an explicit function of the coordinates, one can conclude from Noether’s Theorem that
the system's energy and momentum will be conserved overall.
It will now be confirmed that variation of the particle's world line in [5-1] yields Bohm’s
equation of motion.
5.2 Derivation of Bohm’s Equation of Motion from the Lagrangian Density
From Chapter 3, Bohm’s equation of motion [3-18b] may be equivalently written as:
dp i
dt=
∂Q
∂xi[5-4]
where Q is the quantum potential and where the external potential V has been neglected
for simplicity. To obtain this equation from the proposed Lagrangian density [5-1], it is
necessary to insert the Lagrangian:
L =
_Chapter_5.doc
42
which is the same as equation [5-4]. Hence it is apparent that the proposed Lagrangian
density yields Bohm's equation of motion as required.
Note that this result would remain valid if, as proposed earlier, the Lagrangian density [5-
1] were generalised by multiplying the particle and interaction terms by an arbitrary
constant k.
5.3 Field Equation Deriving from the Proposed Lagrangian Density
The field equation arising from our Lagrangian density can be obtained by applying
Lagrange’s equations for fields [4-7]. In performing this task, the interaction component
of the Lagrangian Density in [5-1] is found to contribute a source term to the usual
Schrodinger equation, as expected. Once this new term has been derived, its
compatibility with the experimentally verified quantum mechanical predictions, as
described by the Schrodinger equation, will be addressed.
The relevant form of Lagrange's equations is:
∂µ∂
_Chapter_5.doc
43
∂µ
∂∂(∂µψ*)
[h
2
2m(∂ j ψ∗
) (∂ j ψ) + ih2
(ψ∗ ∂ t ψ – ψ ∂ t ψ∗) ]
–∂
∂ψ*[
h2
2m(∂ j ψ∗
) (∂ j ψ) + ih2
(ψ∗ ∂ t ψ – ψ ∂ t ψ∗) ]
=h
2
2m∂ j∂
jψ –ih
2∂ tψ –
ih
2∂ tψ
= –h
2
2m∇2ψ – ih∂ tψ
[5-10]
Furthermore, the particle term m2
ρ v j v j of the Lagrangian density is not a function of Ψ*
and so the differentiations with respect to Ψ* and ∂µψ* in [5-9] eliminate this term. The
interaction term, however, contains the potential Q, which is a function of both Ψ* and
∂µψ* when written out explicitly. It may therefore be concluded that [5-9] reduces to the
following field equation:
–h
2
2m∇2ψ – ih∂ tψ – [∂µ
∂∂(∂µψ*)
– ∂∂ψ*
] ρ Q = 0 [5-11]
In accordance with convention, this field equation will be written with the free-field
terms on the left and the source term on the right:
–h
2
2m∇2ψ – ih∂ tψ = [∂µ
∂∂(∂µψ*)
– ∂∂ψ*
] ρ Q [5-12]
The Lagrangian density [5-1] has thus yielded a modified Schrodinger equation. To
obtain further insight into this equation, the source term needs to be written out in detail
in terms of the wave function and its derivatives. In order to proceed towards this goal, it
is necessary first to express the quantum potential Q as a function of ψ. As described
earlier, this potential is usually written in the form [3-14]:
Q = –h
2
2m∇2
RR
However, from section 3.1.3 (especially equation [3-16d]) it is clear that Q can also be
expressed in the equivalent form:
_Chapter_5.doc
44
Q = 12m
∂ jS ∂ jS – ∂ tS [5-13]
This latter expression will actually be the more appropriate one for our present purpose,
since the derivation of the former expression requires one to assume the standard
Schrodinger equation, which we are in the process of modifying here. It is true that
expression [5-13] is itself derived from the velocity given by the Schrodinger current
density, but this is a somewhat weaker assumption. In any case, it will be shown in
Appendix 2 that the usual Schrodinger velocity expression remains unmodified by the
present considerations, thereby confirming the consistency of choosing expression [5-13].
Now, the potential Q stated in equation [5-13] may be written in a form more directly
amenable to analysis in terms of the new field equation [5-12] by making use of the
definition ψ = ReiS/
_Chapter_5.doc
45
Part B:
– ∂∂ψ* { ρ ( –
h2
8m[∂ jψψ –
∂ jψ*
ψ*] [
∂ jψψ –
∂ jψ*
ψ*] + ih
2[∂ tψψ –
∂ tψ*
ψ*] ) } [5-17b]
Beginning with Part A, we have:
[5-17a] =
∂µ { –h
2
8m( [ 0 – ∂
∂(∂µψ*)
(∂ jψ*
ψ* ) ] [∂ jψψ –
∂ jψ*
ψ* ] + [∂ jψψ –
∂ jψ*
ψ* ] [ 0 – ∂∂(∂µψ
*)(∂ jψ*
ψ* ) ] ) ρ
+ ih2
[ 0 – ∂∂(∂µψ
*)(∂ tψ*
ψ* ) ] ρ }
[5-18]
where we have used the familiar identities:
∂xν/∂xµ ≡ gµν [5-19a]
∂xν/∂xµ ≡ δµν [5-19b]
∂t/∂xµ ≡ δµt [5-19c]
Hence, continuing on by using the tensor rules:
∂µ gµν = ∂ν [5-19d]
∂µ δµν = ∂ν [5-19e]
∂µ δµt = ∂t [5-19f]
we obtain:
[5-18] = { –h
2
8m∂ j ( – [ 1
ψ* ] [∂ jψψ –
∂ jψ*
ψ* ] – [∂ jψψ –
∂ jψ*
ψ* ] 1ψ* ) ρ – ih
2∂ t
1ψ* ρ }
=h
2
4m∂ j [ 1
ψ*] [
∂ jψψ –
∂ jψ*
ψ*] ρ – ih
2( –
∂ tψ*
ψ* 2ρ +
∂ tρψ*
)
=h
2
4m( [ –
∂ j ψ*
ψ* 2 ] [∂ jψψ –
∂ jψ*
ψ* ] ρ + [ 1ψ* ] ∂ j [
∂ jψψ –
∂ jψ*
ψ* ] ρ ) + ih2
(∂ tψ*
ψ* 2 ρ –∂ tρψ* )
[5-20a]
= ∂µ { –h
2
8m( – [
δ jµ
ψ* ] [∂ jψψ –
∂ jψ*
ψ* ] + [∂ jψψ –
∂ jψ*
ψ* ] [ –gµj
ψ* ] ) ρ + ih2
[ –δt
µ
ψ* ] ρ }
_Chapter_5.doc
46
This is our result for Part A. Turning to Part B, we have:
[5-17b] = ∂∂ψ*
{h
2
8m[
∂ jψψ –
∂ jψ*
ψ*] [
∂ jψψ –
∂ jψ*
ψ*] ρ – ih
2[
∂ tψψ –
∂ tψ*
ψ*] ρ }
=h
2
8m( [ 0 – ∂
∂ψ*(∂ jψ*
ψ*) ] [
∂ jψψ –
∂ jψ*
ψ*] + [
∂ jψψ –
∂ jψ*
ψ*] [ 0 – ∂
∂ψ*(∂ jψ*
ψ*) ] ) ρ
– ih2
[ – ∂∂ψ*
(∂ tψ*
ψ*) ] ρ
=h
2
8m( [
∂ jψ*
ψ* 2] [
∂ jψψ –
∂ jψ*
ψ*] + [
∂ jψψ –
∂ jψ*
ψ*] [
∂ jψ*
ψ* 2] ) ρ – ih
2[
∂ tψ*
ψ* 2] ρ
=h
2
4m[
∂ jψ*
ψ* 2 ] [∂ jψψ –
∂ jψ*
ψ* ] ρ – ih2
[∂ tψ*
ψ* 2 ] ρ [5-20b]
Now, adding together parts A and B (i.e., equations [5-20a] and [5-20b]) allows some
cancellation, so that the following expression is found for the source term to go on the
right hand side of the new field equation [5-12]:
[∂µ∂
∂(∂µψ*)
– ∂∂ψ* ] ρ Q =
h2
4m( [ 1
ψ* ] ∂ j [∂ jψψ –
∂ jψ*
ψ* ] ρ ) – ih2
∂ tρψ*
[5-21]
Finally, using the identity [5-14], this source term can be written more simply as:
[∂µ∂
∂(∂µψ*)
– ∂∂ψ* ] ρ Q = ih
2ψ* [ ∂ j (ρ∂ j
Sm ) – ∂ t ρ ]
= – ih2ψ* [∇.(ρ∇S
m ) + ∂ tρ ][5-22]
Summing up, the modified Schrodinger field equation that follows from the Lagrangian
density [5-1] is:
–h
2
2m∇2ψ – ih∂ tψ = – ih
2ψ*{∇.(ρ∇S
m ) + ∂ tρ } [5-23]
where ρ(x−x0) is the delta function defined in [5-3] and S(x) is the phase of the wave
function as usual.
_Chapter_5.doc
47
It may be helpful to finish by pointing out that this equation can, if desired, be written out
fully in terms of ψ simply by replacing S with the expression:
S = h2i
ln(ψψ* ) [5-24]
5.4 Consistency of the Derived Field Equation with Experiment
It will now be considered to what extent the new equation is compatible with the wealth
of experimental evidence supporting the standard Schrodinger equation. As mentioned
earlier in this chapter, an additional constant factor k could have been included in the
particle and interaction terms of the Lagrangian density [5-1]. If this had been done, the
constant k would then have appeared in the source term on the right of [5-23]. Since this
constant is arbitrary, its value could be assumed very small and so the difference between
the predictions of the standard and modified Schrodinger equations could then be asserted
to be too tiny to detect experimentally. This would be a satisfactory way of reconciling
the new source term with the known facts.
However, an alternative and more intriguing possibility also exists, as will now be
discussed. The essential point to note is that the terms in the curly brackets on the right of
[5-23] resemble those of a continuity equation [3-3]. In particular, once the restriction v =
∇∇∇∇S/m is reimposed, this bracket equalling zero becomes the condition for conservation of
the “matter” making up the particle (since ρ is the matter density). Since such
conservation can be assumed in the non-relativistic domain, the new field equation [5-23]
reduces to the standard Schrodinger equation [3-2a] within the latter's realm of
applicability. In other words, the experimental predictions are unchanged as long as there
is no particle creation or annihilation.
_Chapter_5.doc
48
This is a surprising and thought-provoking result. It encourages us to pursue the present
approach further. It also raises the question as to what extent, if any, new predictions are
likely to arise in the relativistic domain. An experimentally distinguishable generalisation
of quantum mechanics would be interesting. On the other hand, reinterpreting ρ as charge
density, rather than matter density, may allow the relativistic predictions to remain
unchanged as well. In any case, we know from Noether's theorem that energy and
momentum conservation will be achieved either way.
_Chapter_6.doc
49
Chapter 6: Energy-Momentum Tensors
This chapter outlines the general theory of energy and momentum conservation in terms
of energy-momentum tensors, then applies these ideas to the case of Bohm's model. We
will focus in particular on the case of a scalar field interacting with a particle. The rate of
change of energy and momentum is described in terms of tensor divergence equations.
6.1 Basic Theory
With fields, we are concerned with densities, such as charge, probability, energy and
momentum density. The treatment of the distribution of energy and momentum within
the field proceeds in the same way as for the more familiar cases of charge and
probability. Conservation of these latter quantities is described by a continuity equation
involving both a density ρ and a current density ji = ρvi (i = 1,2,3):
∂ i ji+ ∂ t ρ = 0 [6-1]
The densities that characterise a field's energy and momentum content are summarised in
the form of the energy-momentum tensor Tµν. The various terms in this quantity
correspond to energy and momentum densities and energy and momentum currents. In
particular, the momentum density component in the ith direction (for example, ρvi in the
case of a fluid having a mass density ρ and no internal stresses) will have a current
component in the jth direction (ρvivj for a stressless fluid). Thus we are led to a
description involving two indices i and j. In the relativistic case, the indices can
separately have any of the values µ,ν = 0, 1, 2, or 3 and so the energy momentum tensor
Tµν consists of 16 components. In analogy to the continuity equation [6-1], energy and
momentum conservation is described by the following set of 4 equations:
∂ j Tµj+ ∂0 Tµ0
= 0 [6-2]
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50
The equation corresponding to µ = 0 contains the terms T0j and T00 and describes
conservation of energy. The 3 equations corresponding to µ = i = 1, 2, 3 contain the terms
Tij and Ti0 and describe conservation of each component of momentum. In the relativistic
case, it can be shown that conservation of angular momentum requires Tµν to be
symmetric in µ and ν and, as a consequence, the number of independent components is
reduced from 16 to 10. These components have the following interpretation (ignoring any
factors of c):
• T00 = energy density [6-3a]
• Ti0 = T0i = three components of momentum density (equivalent to energy current)
[6-3b]
• Tij = Tji = six components of momentum current [6-3c]
Equations [6-2] can be written more elegantly as:
∂ν Tµν
= 0 [6-4]
6.2 Energy-Momentum Tensor for a Scalar Field
It can be shown1 that the energy-momentum tensor for a real, free scalar field φ described
by a Lagrangian density
_Chapter_6.doc
51
Using the Lagrangian density [4-10] for a real, massless scalar field:
_Chapter_6.doc
52
equations, we will evaluate the 4-divergence of the field's energy-momentum tensor [6-
7]:
∂ν Tfieldµν = ∂ν [ (∂µφ) (∂νφ) – ½ gµν (∂λφ) (∂λφ) ] [6-9a]
= (∂ν∂µφ) (∂νφ) + (∂µφ) ( ∂ν∂
νφ) – 12
[ (∂µ∂λφ) (∂λφ) + (∂λφ) (∂µ∂λφ) ] [6-9b]
= (∂λ∂µφ) (∂λφ) + (∂µφ) ( ∂ν∂
νφ) – 12
[ (∂µ∂λφ) (∂λφ) + (∂µ∂λφ) (∂λφ) ] [6-9c]
= (∂µ∂λφ) (∂λφ) + (∂µφ) ( ∂ν∂νφ) – (∂µ∂λφ) (∂λφ) [6-9d]
= (∂µφ) ( ∂ν∂νφ) [6-9e]
Now, the field equation which follows from the Lagrangian density [6-8] above is
equation [4-20]:
∂µ∂µφ = – q ρ
which is simply the free-field equation with a source term added. Inserting this field
equation into [6-9e] yields the tensor divergence equation:
∂ν Tfield
µν= – q ρ ∂µφ [6-10]
Returning again to the Lagrangian density [6-8], the particle equation of motion it yields
via the integral equations [4-16] and Lagrange's equations [4-3] is the usual one for a
particle in a scalar field:
dp i
dt= q ∂ iφ [6-11]
Also, from this equation for the rate of change of the particle's momentum, it is
straightforward to derive an analogous one for the particle's energy (see Appendix 3):
dE
dt= q
∂φ
∂t[6-12]
We are now in a position to write down the equations we are seeking. Inserting [6-11]
into the right hand side of [6-10], we obtain:
∂ν Tfieldiν = – ρ
dp i
dt(i = 1,2,3) [6-13]
_Chapter_6.doc
53
Similarly, inserting [6-12] into [6-10], we obtain:
c ∂ν Tfield0ν = – ρ
dE
dt[6-14]
These are the two desired equations. They link the local changes in the field's momentum
and energy to those of the particle, in accordance with the requirement of conservation.
Equations [6-13] and [6-14] also hold for other classical cases, such as a particle
interacting with an electromagnetic field (see the Lagrangian density [4-21] earlier). In
developing our Lagrangian approach to Bohm's model, it will be necessary for something
similar to hold in the case of a Bohmian particle interacting with a Schrodinger field.
6.3.2 Introduction of Tµνparticle
In the case of a particle interacting with a scalar field, conservation of momentum and
energy can also be expressed by introducing an energy-momentum tensor for the particle
and writing the following set of divergence equations (µ,ν = 0,1,2,3):
∂ν(Tµνfield + Tµν
particle) = 0 [6-15]
For a relativistic particle, Tµνparticle has the form2:
Tµνparticle = ρ0muµuν [6-16]
where ρ0, m and uµ are rest density, rest mass and 4-velocity, respectively. This
expression for Tµνparticle will be utilised in a later chapter. The set of equations [6-15] can
be shown3 to be equivalent to the relativistic versions of [6-13] and [6-14] provided
expression [6-16] is chosen for Tµνparticle.
From [6-15], the conservation of the three components of momentum (i = 1,2,3) will be
described by the equations
∂ν(Tiνfield + Tiν
particle) = 0 [6-17a]
2 See, e.g., Sec. 10-1 in Adler R., Bazin M. and Schiffer M., Introduction to General Relativity, 2
nd Ed.
McGraw-Hill Kogakusha, Tokyo (1975).
_Chapter_6.doc
54
and conservation of energy will be described by
∂ν(T0νfield + T0ν
particle) = 0 [6-17b]
6.3.3 Global Equations
Equations [6-17] involve momentum and energy densities and ensure conservation
"locally" at each point in space. On the other hand, the conservation of the total values of
momentum and energy will be described by the following "global" equations (i = 1,2,3):
ddt
[ p fieldi + pparticle
i ] = 0 [6-18a]
ddt
[ E field + Eparticle ] = 0 [6-18b]
Equations [6-18a] and [6-18b] can be derived from the "local" versions [6-17a] and [6-
17b] by integrating over all space:
∂νTfield
iν d3x
– ∞
∞
+ ∂νTparticle
iν d3x
– ∞
∞
= 0 [6-19a]
∂νTfield
0ν d3x
– ∞
∞
+ ∂νTparticle
0ν d3x
– ∞
∞
= 0 [6-19b]
The densities of momentum and energy will thereby be converted to total values. The
equivalence of equations [6-19a] and [6-19b] to equations [6-18a] and [6-18b] will now
be demonstrated. For both Tµνfield and Tµν
particle, the required integral over space can be
written out in detail as follows:
∂νTµν d
3x
– ∞
∞
= (∂0Tµ0 + ∂1T
µ1 + ∂2Tµ2 + ∂3T
µ3) d3x
– ∞
∞
[6-20]
Under the reasonable assumption that the energy-momentum tensor falls off to zero at
plus and minus infinity (in any spatial direction), the last three terms of [6-20] will be
zero and so only the term containing the time derivative survives:
∂νTµν d
3x
– ∞
∞
= ∂0 Tµ0d
3x
– ∞
∞
= 1c
d
dtTµ0
d3x
– ∞
∞
[6-21]
3 Felsager B., Geometry, Particles and Fields, Sec. 1-6, Springer-Verlag, NY (1998).
_Chapter_6.doc
55
In the last equality, the partial derivative has been replaced by the total derivative
because, after the spatial integration d3x has been performed, only time dependence
remains. With the aid of [6-21], equations [6-19a] and [6-19b] can be written as:
d
dtTfield
i0d
3x +
d
dtTparticle
i0d
3x = 0
– ∞
∞
– ∞
∞
[6-22a]
d
dtTfield
00d
3x +
d
dtTparticle
00d
3x = 0
– ∞
∞
– ∞
∞
[6-22b]
Now, Ti0 and T00 can be identified from equations [6-3a] and [6-3b] earlier as momentum
density and energy density, respectively. Therefore these equations reduce to the global
equations [6-18a] and [6-18b] as required:
ddt
[ p fieldi + pparticle
i ] = 0
ddt
[ E field + Eparticle ] = 0
6.4 Tentative Application to Bohm's Model
Having summarised the relevant theoretical formalism, we will now attempt to employ it
to introduce conservation of energy and momentum into Bohm's model. In doing so, it
will be found that some difficulties arise. Fortunately these can all be overcome by a
deeper and more careful analysis. We will look briefly here at the problems that are
encountered as a pointer towards an appropriate course of action to follow in the next
chapter.
As discussed in chapter 5, our proposed Lagrangian density for Bohm's model is:
_Chapter_6.doc
56
By analogy with the classical cases of a particle in a scalar or vector field, we tentatively
expect that equation [6-15] will continue to hold:
∂ν(Tµνfield + Tµν
particle) = 0
in the case of the system described by [5-1]. Equation [6-15] describes transfers of energy
and momentum between the field and the particle, in accordance with the requirements of
conservation. As a first step towards establishing whether this equation remains valid in
our Bohmian case, we will derive the free-field energy-momentum tensor corresponding
to [5-1].
Because our Lagrangian density is non-relativistic, it does not possess the sort of
symmetry between space and time that is characteristic of relativistic Lagrangians. It is
therefore necessary to obtain separate expressions for Tij, Ti0, T0i and T00 (i,j = 1,2,3),
rather than just a single Tµν expression (µ,ν = 0,1,2,3). This lengthens the derivation
somewhat. The desired expressions are found from the free-field part of the Lagrangian
density [5-1] by applying the formula:
Tfieldµν = [ ∂µφ ∂
∂(∂νφ)+ ∂µφ* ∂
∂(∂νφ*)
– gµν ]
_Chapter_6.doc
57
Looking at these expressions, our first difficulty is apparent. The energy-momentum
tensor is not symmetric, since we have:
Ti0 ≠ T0i [6-25]
whereas a symmetric tensor had been expected from the relativistic discussion earlier in
this chapter. Techniques exist to symmetrise an energy-momentum tensor4. However, as
later analysis will show, the present lack of symmetry should not simply be removed in
this way. Instead, its significance should and will be examined carefully. This matter will
be resolved in the next chapter.
Leaving this point and continuing on, we want to see whether the energy-momentum
tensor above yields conservation by satisfying equations [6-13] and [6-14]. Of these two
equations, it will be sufficient to discuss [6-13]:
∂ν Tfieldiν = – ρ
dp i
dt(i = 1,2,3)
For our present purpose, the divergence on the left of this equation needs to be split into
separate space and time components, so that we have:
∂ j Tfieldij + ∂ t Tfield
i0 = – ρdp i
dt[6-26]
To check whether the energy-momentum tensor summarized in equations [6-24] is
consistent with this conservation condition, expressions [6-24a] and [6-24b] will be
inserted into the left hand side of [6-26]. This is done in Appendix 5. For the usual non-
relativistic situation of a single particle with no creation or annihilation, the following
result is obtained:
∂ j Tfield
ij+ ∂ t Tfield
i0= 0 [6-27]
Hence, unlike the scalar and vector field cases discussed in the previous chapter, the
divergence of the field's energy-momentum tensor is zero here even when there is field-
_Chapter_6.doc
58
particle interaction. This result is not consistent with equation [6-26] and forbids energy
and momentum transfer between the field and the particle. Since we appear to need an
equation like [6-26] to hold, we are faced with a second difficulty.
Of course, this zero divergence of the Schrodinger energy-momentum tensor is well
known and is one reason why people have concluded that Bohm's model is not
compatible with energy and momentum conservation5. Nevertheless, Noether's theorem
assures us that the desired conservation must exist for the Lagrangian density we have
chosen. A closer examination of Noether's theorem will be needed to resolve this
problem. However, some insight into the course to be followed can be gained by
considering another well-known case, viz., an electromagnetic field and a Dirac spinor
field in interaction. Before the interaction between these two fields begins, the
divergences of the tensors Tµνelectromag and Tµν
Dirac are, of course, separately zero:
∂ν(Tµνelectromag) = 0 [6-28]
∂ν(TµνDirac) = 0 [6-29]
With the onset of the interaction, the expressions for Tµνelectromag and Tµν
Dirac do not
change (i.e., they each still look the same), but their individual divergences are no longer
zero6. Now, from our experience with the classical cases of a particle in a scalar or vector
field, one might expect the following overall condition to hold:
∂ν(Tµνelectromag + Tµν
Dirac) = 0
by analogy with [6-15]. However, this is not the case. The correct overall divergence
equation contains an extra term, as follows7:
∂ν(Tµνelectromag + Tµν
Dirac + Tµνinteraction) = 0 [6-30]
4 See, e.g., Ch. 3, Sec. 4 in Barut A., Electrodynamics and Classical Theory of Fields and Particles.
Macmillan, N.Y. (1964).5 See p. 115 in Holland P.R., The Quantum Theory of Motion. Cambridge University Press (1995).
_Chapter_6.doc
59
This example demonstrates that the appearance of an additional term Tµνinteraction may
sometimes be needed to achieve conservation. In so doing it suggests a way in which our
second difficulty may be tackled.
Pursuing this possible approach, it seems at first sight that a suitable extra term
Tµνinteraction could be obtained simply by applying the square bracket in equation [6-23] to
the interaction part of [5-1] to construct the tentative expression:
Tinteractionµν = [ ∂µφ ∂
∂(∂νφ)+ ∂µφ* ∂
∂(∂νφ*)
– gµν ]
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60
Chapter 7: Relativistic Treatment
In the previous chapters, we have been using a mixture of relativistic and non-relativistic
formalism. For instance, we have treated particles non-relativistically while describing
the classical scalar field relativistically. This complication has arisen because the focus of
our attention is the non-relativistic Bohm model, whereas the classical field examples we
have drawn from textbooks as guiding illustrations all obey Lorentz covariant equations.
Furthermore, discussions of energy-momentum tensors in books are nearly always
formulated relativistically. For example, the only expression used for Tµνparticle is
ρ0muµuν. Such presentations have a certain elegance whereas, as seen from equations [6-
24a] to [6-24d] earlier, developing a non-relativistic treatment for Tµν is messy and more
tedious because of the need to keep track of separate expressions for Tij, Ti0, T0i and T00.
The situation becomes more critical in attempting to construct a particular formulation of
Noether's theorem that will satisfy our present needs. A non-relativistic approach is more
difficult and becomes unclear, whereas a relativistic one is found to be comparatively
straightforward. For this reason, we will adopt a policy in the present chapter of
presenting a fully relativistic treatment. The non-relativistic results that we will
eventually need can then be obtained at the end by taking the non-relativistic limit.
To pursue this plan, it will be necessary to make temporary use of a relativistic version of
Bohm's model before taking the limit. A suitable model for this purpose has, in fact, been
formulated by Louis de Broglie1. While his model has certain contentious features
compared with Bohm's original model, these features will not have any bearing on the
present discussion because they do not affect the validity of our treatment and they vanish
in the non-relativistic limit.
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61
In the present chapter, it will be shown that a version of de Broglie's model incorporating
energy and momentum conservation can be constructed in a straightforward way.
7.1 De Broglie’s Model
In keeping with Bohm’s approach, de Broglie presumes that a particle has a definite
trajectory at all times. However, while Bohm’s model is based on the Schrodinger
equation, de Broglie’s formulation involves the Klein-Gordon equation instead:
h2
2m[∂µ∂
µφ + (mch
)2φ] = 0 [7-1]
where φ is the Klein-Gordon wavefunction. (The dimensional factor h2
2m has been
included here for ease of comparison with equations [7-8] and [7-17] later.) From chapter
3 (and using the notation ψ = Re iS/h), the basic postulate of Bohm’s model is equation [3-
8] for the particle’s momentum p:
p = ∇∇∇∇S [7-2]
Using the analogous notation φ = Re iS/h in the Klein-Gordon case, the basic equation of de
Broglie’s relativistic model is:
pµ = – ∂µS [7-3]
where now pµ is the particle’s 4-momentum. Equations [7-2] and [7-3] are sufficient for
the minimalist versions (see chapter 3) of Bohm’s and de Broglie’s models, respectively.
If one wishes to go further and introduce a “quantum potential” Q into each model (as is
convenient for our purposes), the appropriate expressions are as follows. From equation
[3-14], the potential for Bohm’s non-relativistic model is the familiar expression:
Q = –h
2
2m∇2
RR
and, as pointed out in equation [5-13], this potential can be written in the equivalent
form:
1 de Broglie L., Nonlinear Wave Mechanics. Elsevier, Amsterdam (1960).
_Chapter_7.doc
62
Q = 12m
∂ jS ∂ jS – ∂ tS
In comparison with this last expression, the appropriate expression for de Broglie’s
relativistic model is:
Q = c (∂µS) (∂µS) – mc2 [7-4]
The corresponding equations of motion for the particle are then
dp
dt= – ∇∇∇∇Q
in the non-relativistic case and
dpµ
dτ= ∂µQ [7-5]
in the relativistic case (τ being the proper time).
There are three questionable features of de Broglie’s relativistic model compared with
Bohm’s non-relativistic one:
1. De Broglie’s model is based on the Klein-Gordon equation, whereas it might have
been more appropriate to have a relativistic model corresponding to the Dirac
equation. On the positive side, however, we are interested only in the non-
relativistic limit and this limit is more easily derived in the Klein-Gordon case.
2. De Broglie bases his model on the Klein-Gordon equation’s current density,
which leads him to the following probability density for the particle’s position at
any time:
P(x,t) = ih2mc2 [ φ* ∂ tφ – φ ∂ tφ
*]
= – 1mc2
R2 ∂ tS
[7-6]
This expression has the disadvantage of not being positive definite (unlike the
simple expression R2 in Bohm’s model) and so requires the dubious notion of
negative probabilities. De Broglie attempts to explain this result physically in
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63
terms of the particle’s world line turning backwards in time. Fortunately this
controversial point need not be considered further here because the probability
density [7-6] reduces back to the positive expression R2 in the non-relativistic
limit.
3. In order for equations [7-3] and [7-5] to be compatible, de Broglie found it
necessary to introduce a “variable” rest mass2:
M = 1c (∂µS) (∂µ
S)
= m +Q
c2
[7-7]
i.e., the rest mass is a function of the wavefunction φ. Again, this rather
unwelcome feature is no problem from our point of view because expression [7-7]
can be shown to reduce back to the usual, constant mass m in the non-relativistic
limit.
In Appendix 6 it is confirmed that equation [7-5] is consistent with [7-3] once [7-7] is
assumed. In other words, we can assume in formulating our relativistic Lagrangian
density that the particle’s motion is still governed by a scalar potential.
7.2 Lagrangian Density for de Broglie’s Model
By analogy with the Lagrangian density introduced in chapter 5 for Bohm’s model, a
similar expression will be proposed here for de Broglie’s relativistic case. We will begin
by simply stating the proposed expression, then discuss in detail the forms chosen for the
various terms. Our relativistic Lagrangian density is:
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64
where:
• φ is the Klein-Gordon wave-function,
• ρ0 is the rest density distribution of the particle through space3 (ρ0 will be a delta
function),
• m is the constant rest mass usually associated with the particle (not de Broglie’s
variable rest mass M),
• uµ is the particle's 4-velocity (uµ ≡ dxµ/dτ, where τ = proper time),
• Q is the scalar potential,
• c is the speed of light.
As with the Lagrangian densities considered in chapters 4 and 5, our expression here
consists of a “free-field” component, a “particle” component and an “interaction”
component. The free-field terms are the standard ones from which the Klein-Gordon
equation may be derived4. The particle term and interaction terms are also standard
expressions5. The form of this Lagrangian density is manifestly Lorentz invariant. The
various constant factors in its terms ensure that it has the required dimensions of energy
density.
It is to be understood here that, in deriving the equation of motion for the particle, one
must employ the well-known technique of replacing the proper time with an arbitrary
parameter while performing the variation process6. The interaction term in [7-8] is similar
in appearance to the non-relativistic one in [5-1], except for the additional factor uµu
µ
c ,
which ensures parameterisation independence of the action. This factor also ensures that
3 i.e., the matter density in the particle’s instantaneous rest frame.
4 See, e.g., p. 14 in Greiner W., Relativistic Quantum Mechanics – Wave Equations, 2
nd Ed. Springer,
Berlin (1997).5 See, e.g., p. 289 in Anderson J.L., Principles of Relativity Physics, Academic Press, N.Y. (1967).
6 See, e.g., Sec. 7-9 in Goldstein H., Classical Mechanics, 2
nd Ed. Addison-Wesley, Massachusetts (1980).
_Chapter_7.doc
65
the equation of motion7 obtained for the particle from this Lagrangian density is
compatible with the familiar relativistic identity uµuµ = c2 that the 4-velocity must satisfy
once the proper time is reinstated in place of the arbitrary parameter of variation. This
point will be demonstrated in Appendix 7.
The identity uµuµ = c2 does, of course, allow us to rewrite the particle and interaction
terms in the simpler forms
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66
ρ0 = cu0 δ(x – x0) [7-13]
This is the required expression for the particle’s rest density distribution. It will be
needed in the next section for the derivation of the particle’s equation of motion.
Finally it should be noted that, in analogy to the non-relativistic case discussed in section
5.1, a more general relativistic Lagrangian density than [7-8] could be used in which the
particle and interaction terms are multiplied by an arbitrary constant k. Again, this would
leave the resulting equation of motion for the particle unchanged and would multiply the
source term of the resulting field equation by k.
7.3 Equation of Motion for the Particle
In Appendix 7 it is confirmed that our proposed relativistic Lagrangian density yields the
correct equation of motion [7-5]. Note that, as with the non-relativistic Lagrangian
density in chapter 5, we are effectively treating the particle’s velocity as an independent
variable here and temporarily suspending the de Broglie-Bohm restriction Muµ = – ∂µS.
This restriction can be restored at the end without any inconsistency once the Lagrangian
formalism has yielded the required equations for energy and momentum conservation.
7.4 Field Equation
The field equation corresponding to the Lagrangian density [7-8] will now be considered.
In analogy with the modified Schrodinger equation in chapter 5, this will be found to take
the form of the Klein-Gordon equation with an extra term added. From equation [5-8], the
appropriate form of Lagrange's equation for our needs is
∂µ∂
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67
Now, there is no need to insert the field terms of [7-8] into equation [7-14], because that
would simply yield the standard Klein-Gordon equation, as these terms have been
designed to do9. Furthermore, inserting the particle term of [7-8] would simply yield zero,
since this term is not a function of φ. Therefore, anything extra to be added to the Klein-
Gordon equation will come purely from the interaction term of the Lagrangian density.
This term is
∂
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68
In analogy with the modified Schrodinger equation in chapter 5, the expression ∂µ(ρ0uµ)
on the right of [7-18] is seen to resemble the form of a continuity equation. This
expression equalling zero is, in fact, the condition for conservation of the “matter”
making up the particle. In the low energy, single particle case (i.e., in the absence of
creation and annihilation), this expression is zero and our new field equation simply
reduces back to the standard Klein-Gordon equation.
7.5 Energy-Momentum Tensor for the Particle
The remainder of this chapter will be concerned with demonstrating conservation of
energy and momentum for the relativistic model under consideration here. This will be
achieved by considering the energy-momentum tensors corresponding to the various
terms in the Lagrangian density [7-8]. The main result will be derived in the next section.
As a preliminary step, we will briefly focus on the energy-momentum tensor for the
particle. The expression for this tensor has already been given in equation [6-16].
Allowing for the variable rest mass M in de Broglie’s model, the particle’s energy-
momentum tensor has the form:
Tparticle
µν= ρ0Muµuν [7-19]
where uµ is the particle’s 4-velocity and the rest density ρ0 is defined in [7-13]. It is a
standard result10 that the divergence of this tensor is related to the rate of change of the
particle’s 4-momentum as follows:
∂νTparticleµν
= ρ0
dpµ
dτ[7-20]
Combining this with the equation of motion [7-5]:
dpµ
dτ= ∂µQ
we then obtain the relationship:
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69
∂νTparticle
µν= ρ0 ∂µ
Q [7-21]
This result will be needed in the following section.
7.6 Noether’s Theorem adapted to the Present Case
A formulation of Noether’s theorem designed specifically to serve our particular needs
will now be developed from first principles. As discussed at the end of the previous
chapter, this way of proceeding is necessary because of difficulties that arise in
attempting a more routine approach.
In most textbook examples of classical particle-field interactions, the interaction term of
the Lagrangian density does not involve derivatives of the field. This can be shown to
have the consequence that the overall energy-momentum tensor for that Lagrangian
density consists simply of Tµνfield plus Tµν
particle, with no additional terms Tµνinteraction. For
our more complex interaction term [7-15], this simple situation no longer holds. To find
the more general expression for the overall Tµν that is applicable to our case, we will
return to Noether's theorem and derive the required expression.
Our Lagrangian density is an explicit function of the field, its first derivatives and the
particle’s rest density, rest mass and 4-velocity:
_Chapter_7.doc
70
Noether’s theorem states that the system’s energy and momentum will be conserved
provided
_Chapter_7.doc
71
∂µ∂
_Chapter_7.doc
72
∂ν (∂µφ) ∂
_Chapter_7.doc
73
7.7 Summary of Equations describing Overall Conservation
Summarising the results of the previous section, conservation of energy and momentum
for the Lagrangian density [7-8] is described by the condition:
∂ν Ttotal
µν= 0 [7-30]
where the overall energy-momentum tensor can be written component-wise as:
Ttotal
µν= Tfield
µν+ Tparticle
µν+ Tinteraction
µν [7-31]
and the individual tensor components have been determined to be:
Tfieldµν = [ ∂µφ ∂
∂(∂νφ)+ ∂µφ* ∂
∂(∂νφ*)
– gµν ]
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74
It therefore remains for us to find an explicit expression only for Tµνinteraction. To do this,
we need to evaluate [7-34] for our particular interaction term [7-15]:
_Chapter_7.doc
75
This is the desired term needed to complete the system’s overall energy-momentum
tensor. (It should be kept in mind that it is possible to write out this expression in full
using φ and φ* only, instead of using S as an abbreviation.)
In summary, gathering [7-33], [7-35] and [7-39] together, the overall energy-momentum
tensor for the system described by our relativistic Lagrangian density is made up of the
following parts:
Tfieldµν
=h
2
2m{(∂µφ)(∂νφ*
) + (∂µφ*)(∂νφ) – gµν [(∂λφ
*)(∂λφ ) – (mc
h)
2φ*φ]} [7-40]
Tparticle
µν= ρ0Muµuν [7-41]
Tinteraction
µν=
– c (∂µS) (∂
νS) ρ0
(∂αS) (∂αS)
[7-42]
7.9 Divergence and Conservation
The final task in this chapter is to check explicitly that the divergence of the overall
energy-momentum tensor for the particle-field system is zero and thereby confirm that
energy and momentum are conserved. Towards this end, the divergences of Tµνfield,
Tµνparticle and Tµν
interaction will be evaluated separately.
7.9.1 Divergence of Tµνfield
Taking the divergence of expression [7-40], we obtain
∂νTfieldµν =
h2
2m{(∂ν∂
µφ)(∂νφ*) + (∂µφ)(∂ν∂
νφ*) + (∂ν∂
µφ*)(∂νφ ) + (∂µφ*
)(∂ν∂νφ)
– ∂µ[(∂λφ
*)(∂λφ ) – (mc
h)
2φ*φ]}
=h
2
2m{(∂λ∂
µφ)(∂λφ*) + (∂µφ)(∂ν∂
νφ*) + (∂λ∂
µφ*)(∂λφ ) + (∂µφ*
)(∂ν∂νφ)
– (∂µ∂λφ*)(∂λφ ) – (∂λφ
*)(∂µ∂λφ ) + ∂µ
[(mch
)2
φ*φ]}
=h
2
2m{(∂µφ)(∂ν∂
νφ*) + (∂µφ*
)(∂ν∂νφ) + (mc
h)
2(φ* ∂µφ + φ ∂µφ*
)}
[7-43]
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76
This can be simplified further by using the field equation corresponding to our
Lagrangian density, i.e., by using the extended Klein-Gordon equation [7-17]:
h2
2m[∂µ∂
µφ + (mch
)2φ] = ihc
2
1
φ*∂µ
ρ0 ∂µS
(∂λS)(∂λS)
which can be written more conveniently in the form:
∂ν∂νφ = – (mc
h)
2φ + imc
h
1
φ*∂ν
ρ0 ∂νS
(∂λS)(∂λS)
[7-44]
Inserting [7-44] and its complex conjugate into [7-43], we obtain
∂νTfieldµν
=h
2
2m{(∂µφ) [ – (mc
h)
2φ*
– imch
1
φ∂ν
ρ0 ∂νS
(∂λS)(∂λS)
]
+ (∂µφ*) [ – (mc
h)
2φ + imc
h
1
φ*∂ν
ρ0 ∂νS
(∂λS)(∂λS)
]
+ (mch
)2(φ* ∂µφ + φ ∂µφ*
)}
= – ihc2
[∂µφ
φ–
∂µφ*
φ*] ∂ν
ρ0 ∂νS
(∂λS)(∂λS)
and using [7-38]:
∂µS = – ih2
{∂µφ
φ–
∂µφ*
φ*}
the divergence of Tµνfield is seen to reduce to
∂νTfieldµν
= c (∂µS) ∂ν
ρ0 ∂νS
(∂λS)(∂λS)
[7-45]
7.9.2 Divergence of Tµνparticle
The divergence of Tµνparticle for the particular case of our Lagrangian density has already
been stated earlier. From [7-21], it is:
∂νTparticle
µν= ρ0 ∂µ
Q [7-46]
7.9.3 Divergence of Tµνinteraction
Taking the divergence of expression [7-42], we obtain
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77
∂ν Tinteraction
µν= – c (∂µ
S) ∂ν
(∂νS) ρ0
(∂λS)(∂λS)
– c(∂ν
S) ρ0
(∂λS)(∂λS)
∂ν(∂µS)
= – c (∂µS) ∂ν
(∂νS) ρ0
(∂λS)(∂λS)
– cρ0
(∂λS)(∂λS)
(∂νS)∂µ
(∂νS)
= – c (∂µS) ∂ν
(∂νS) ρ0
(∂λS)(∂λS)
– ρ0∂µ
(c (∂νS)(∂νS)
and using expression [7-4] for the quantum potential:
Q = c (∂µS) (∂µS) – mc2
the divergence of Tµνinteraction then becomes
∂ν Tinteraction
µν= – c (∂µ
S) ∂ν
(∂νS) ρ0
(∂λS)(∂λS)
– ρ0 ∂µQ [7-47]
7.9.4 Divergence of Tµνtotal
From equation [7-31] we have:
Ttotal
µν= Tfield
µν+ Tparticle
µν+ Tinteraction
µν
The divergence of this overall energy-momentum tensor can now be obtained by
combining [7-45], [7-46] and [7-47] to obtain:
∂ν Ttotalµν
= ∂ν Tfieldµν
+ ∂ν Tparticleµν
+ ∂ν Tinteractionµν
= c (∂µS) ∂ν
ρ0 ∂νS
(∂λS)(∂λS)
+ ρ0 ∂µQ + – c (∂µ
S) ∂ν
(∂νS) ρ0
(∂λS)(∂λS)
– ρ0 ∂µQ
which cancels to:
∂ν Ttotalµν = 0
This is the desired result for energy and momentum conservation. (The divergence
calculation above also serves as a useful double-check on our derivations of Tµνfield,
Tµνparticle and Tµν
interaction.)
Therefore, from the viewpoint of conservation, a satisfactory relativistic model has been
achieved.
_Chapter 8
78
Chapter 8: Non-Relativistic Limit
In the previous chapter, a relativistic version of Bohm’s model incorporating energy and
momentum conservation has been successfully formulated. The task now is to take the
non-relativistic limit of that formalism. This will provide us with a mathematical
description which incorporates conservation into Bohm’s original model.
In the relativistic case, the symmetry between space and time made it sufficient to
consider a single tensor expression Tµν (µ,ν = 0,1,2,3). However, in dealing with the non-
relativistic limit, we must obtain separate expressions for each of the tensor components
Tij, Ti0, T0i and T00 (i,j = 1,2,3). Separate expressions must then be evaluated for the
divergence of the Tij and Ti0 together in the first instance and T0i and T00 together in the
second instance. This lengthens the analysis somewhat.
In taking the non-relativistic approximation, it will also be found that some subtleties
have to be taken into account. These will be illustrated by focussing our attention initially
on the energy-momentum tensor of the particle.
8.1 Non-Relativistic Energy-Momentum Tensor for the Particle
8.1.1 Physical Interpretation of Tµνparticle
From equation [7-19], the relativistic expression for the particle’s energy-momentum
tensor is:
Tparticle
µν= ρ0Muµuν [8-1]
_Chapter 8
79
In what follows, it needs to be kept in mind that the 4-velocity uµ is defined to be dx0µ/dτ,
where x0µ(τ) is the particle’s position at proper time τ. Note that uµ is a function of τ
whereas, in the non-relativistic limit, the 3-velocity vi = dx0i/dt is a function of the
ordinary time t. Now, at first sight it would seem to be straightforward to take the non-
relativistic limit of expression [8-1]. The rest density ρ0 will become simply the density
ρ:
ρ0 → ρ [8-2]
and de Broglie’s variable mass M will reduce to the constant mass m:
M → m [8-3]
Furthermore, since the proper time τ will become simply the ordinary time t:
τ → t [8-4]
the 4-velocity uµ will reduce to 3-velocity for µ = 1,2,3:
dx0i
dτ→
dx0i
dt(i=1,2,3) [8-5]
and will reduce to the constant c for µ = 0:
dx00
dτ→ c (since x0 ≡ ct) [8-6]
Using the above limits then leads to the following result (with the expressions for Tij, Ti0,
T0i and T00 written out separately):
Tijparticle = ρmvivj (vi ≡
dx0i
dt) [8-7a]
Ti0particle = T0i = ρmvic [8-7b]
T00particle = ρmc2 [8-7c]
Examination of these expressions, however, raises two problems. First, factors of c are
still present, even though the expressions are meant to be non-relativistic. Second,
_Chapter 8
80
looking at the energy density term T00 in [8-7c], we see that the particle’s rest energy mc2
has been retained in taking the above limit, while its kinetic energy has been lost. This is
not in keeping with the standard non-relativistic notion that the rest energy is a constant
which plays no role and can be ignored (even though it is usually larger than the kinetic
energy). To resolve these matters, a more careful analysis will now be given which
focuses on the physical interpretation of the initial expression [8-1].
As stated in [6-3a] to [6-3c], the various terms in Tµνparticle describe densities and currents
of both momentum and energy. In particular, using the relationship:
ρ0 uv = ρ cu0
uv (from [7 –12])
= ρ dτdt
dx0ν
dτ
= ρdx0
ν
dt
[8-8]
plus the 4-momentum definition:
pµ ≡ Muµ [8-9]
the relativistic expression [8-1] can be written as:
Tparticleµν
= pµ ρdx0
ν
dt[8-10]
Then, writing out the spatial and temporal components of this separately, we have:
Tparticle
ij= p i ρ v j [8-11a]
Tparticle
i0 = p i ρ c [8-11b]
Tparticle
0i= p0 ρ vi [8-11c]
Tparticle
00= p0 ρ c [8-11d]
_Chapter 8
81
Now, ρ in these expressions indicates a density and ρvi indicates a current. Hence, noting
that pi is the particle’s relativistic 3-momentum and p0c is its relativistic energy, we can
identify piρ as the momentum density of the particle, piρvj as momentum current, etc.,
and obtain:
Tijparticle ≡ momentum current [8-12a]
Ti0particle ≡ momentum density × c [8-12b]
T0iparticle ≡ energy current ÷ c [8-12c]
T00particle = energy density [8-12d]
Note that, despite the fact that the terms Ti0 and T0i refer to two different physical
quantities, viz. momentum density and energy current, the tensor is nevertheless
symmetric: Ti0 = T0i. This is because, in the relativistic domain, momentum density and
energy current are equal apart from a constant factor.
At this point we will consider the special case of a free particle with constant momentum
and energy, so that the following divergence equation holds:
∂ν Tparticle
µν= 0 [8-13]
Breaking this up into separate spatial and temporal terms, we have:
∂ j Tparticle
ij + ∂0 Tparticle
i0 = 0 [8-14a]
∂ j Tparticle
0j + ∂0 Tparticle
00 = 0 [8-14b]
Now, inserting expressions [8-12] into equations [8-14], we note that all factors of c
cancel and we obtain:
∂j(momentum current) + ∂t(momentum density) = 0 [8-15a]
∂j(energy current) + ∂t(energy density) = 0 [8-15b]
_Chapter 8
82
These two equations can be recognized as equations of continuity describing momentum
and energy conservation. Examining [8-12] and [8-15] highlights the basic physical
meaning of the energy-momentum tensor and its divergence. This essence should be
maintained in going to the non-relativistic limit. In other words, the non-relativistic
formalism for a free particle:
∂ j Tparticle
ij+ ∂ t Tparticle
i0 = 0 [8-16a]
∂ j Tparticle
0j+ ∂ t Tparticle
00= 0 [8-16b]
should still have the interpretation presented in equations [8-15]. This fact will be used to
obtain the correct non-relativistic form for Tµνparticle.
Now, the non-relativistic momentum and energy of a particle are mv and E, respectively.
(Here, E is the particle’s total non-relativistic energy, i.e., the sum of the kinetic and
potential energies1). Incorporating this extra detail into equations [8-15], we obtain:
∂j(mvi ρvj) + ∂t(mvi ρ) = 0 [8-17a]
∂j(E ρvj) + ∂t(E ρ) = 0 [8-17b]
Comparing [8-16] with [8-17] then yields the results:
Tijparticle = mvi ρvj [8-18a]
Ti0particle = mvi ρ [8-18b]
T0iparticle = E ρvi [8-18c]
T00particle = E ρ [8-18d]
These expressions do not suffer from the two problems mentioned earlier, i.e., there are
no longer any factors of c present2 and the rest energy mc2 has been eliminated in favour
1 The reason for specifying the total energy here, rather than just the kinetic energy, will become clear in
the next section.
_Chapter 8
83
of kinetic plus potential energy. On the other hand, the symmetry Ti0 = T0i has been lost.
It is important to note the source of this non-symmetry. Essentially it is due to the fact
that neglecting rest energy ends the similarity between momentum density and energy
current that had existed in the relativistic domain. Now that we have a physical
explanation for the lack of symmetry, the objection raised towards the end in chapter 6
has lost its efficacy3 and expressions [8-18] can be adopted as the appropriate non-
relativistic form for Tµνparticle.
8.1.2 Rules for obtaining the Non-Relativistic Limit
A systematic procedure for obtaining [8-18] can be summarized by the three rules set out
below. This will be helpful later in considering the cases of Tµνfield and Tµν
interaction.
Starting with the relativistic expressions for Tij, Ti0, T0i and T00, the rules are as follows:
1. Remove terms containing mc2 from T0i and T00. (To keep the overall
divergence zero, it may also be necessary to remove any term whose divergence
would previously have cancelled with that of a deleted mc2 term.)
2. Divide Ti0 by c and multiply T0i by c (to remove redundant factors of c from
these two expressions).
3. Take the non-relativistic limit c → ∞.
2 Note that the various factors of c appearing in the relativistic case are needed to ensure that all the
components of Tµν
have the same dimensions (i.e., units of energy density) and thereby ensure that time
and space remain on an equal footing. This symmetry between time and space components is not necessary
in the non-relativistic case.3 It was also mentioned near the beginning of chapter 6 that symmetry of the energy-momentum tensor is
required for conservation of angular momentum in the relativistic case. For the non-relativistic realm, it
turns out that angular momentum conservation is related to the symmetry of a different tensor, namely the
“mass-momentum” tensor. This alternative tensor continues to satisfy Ti0
= T0i
, but its divergence describes
conservation of mass and momentum (instead of energy and momentum). The non-relativistic mass-
momentum tensor for a particle has the form: Tij = ρmv
iv
j, T
i0 = T
0i = ρmv
i, T
00 = ρm.
_Chapter 8
84
The motivation for Rule 1 has already been discussed. Rule 2 can be obtained by
returning to [7-20]:
∂νTparticleµν
= ρ0
dpµ
dτ
and breaking this relativistic equation up into separate spatial and temporal terms:
∂ j Tparticleij
+ ∂0 Tparticlei0
= ρ0
dp i
dτ[8-19a]
∂ j Tparticle
0j+ ∂0 Tparticle
00=
ρ0
cdEdτ
[8-19b]
Noting the factors of c contained in the derivatives ∂0 ≡ ∂/∂(ct), these equations can then
be written as
∂ j Tparticle
ij+ ∂ t ( 1
c Tparticle
i0) = ρ 0
dp i
dτ[8-20a]
∂ j (c Tparticle
0j) + ∂ t Tparticle
00= ρ0
dEdτ
[8-20b]
Now, in contrast to this relativistic case, the non-relativistic version should be:
∂ j Tparticleij
+ ∂ t Tparticlei0
= ρdp i
dt[8-21a]
∂ j Tparticle
0j+ ∂ t Tparticle
00= ρ dE
dt[8-21b]
Comparing equations [8-20] with [8-21] term by term, it is then seen that Rule 2 is
necessary for the correct non-relativistic limit to be obtained.
The three rules above can be summarized in equation form as follows:
T non –rel
ij= lim
c → ∞T rel
ij [8-22a]
T non –rel
i0 = limc → ∞
(T rel
i0 ÷ c) [8-22b]
Tnon–rel
0i= lim
c → ∞{ [ Trel
i0– (mc2 terms) ] × c } [8-22c]
Tnon–rel
00= lim
c → ∞[ Trel
00– (mc2 terms) ] [8-22d]
_Chapter 8
85
A more formal derivation of the non-relativistic form for Tµνparticle is given in the next
section using the rules stated above.
8.1.3 Derivation of Tµνparticle
The non-relativistic expressions [8-18] will be derived here from the corresponding
relativistic expression [8-1]:
Tparticle
µν= ρ 0Mu µu ν
= ρ 0 Mdx 0
µ
dτ
dx 0ν
dτ
[8-23]
by using the three rules formulated in the previous section. We begin by using equation
[8-8]:
ρ0 uµ = ρdx0
µ
dt
to rewrite expression [8-23] as:
Tparticleµν = ρ M
dx0µ
dt
dx0ν
dτ
= ρ Mdx0
µ
dt
dx0ν
dtdtdτ
[8-24]
Now, in formulating the non-relativistic limit, the following binomial expansion will be
useful4:
dtdτ
= 1
1 – v 2
c 2
= 1 + 12
v 2
c 2 + 38
v 4
c 4 + ...
[8-25]
Inserting this into [8-24] and presenting the expressions for Tij, Ti0, T0i and T00 separately,
we have:
_Chapter 8
86
Tparticle
ij= ρ M vi v j [ 1 + 1
2v2
c2 + ... ]
Tparticle
i0= ρ M vi c [ 1 + 1
2v2
c2 + ... ]
Tparticle
0i= ρ M c vi [ 1 + 1
2v2
c2 + ... ]
Tparticle
00= ρ M c2 [ 1 + 1
2v2
c2 + ... ]
which will be written more conveniently here in the form:
Tparticle
ij= ρ M vi v j [ 1 + 1
2v2
c2 + ... ] [8-26a]
Tparticle
i0= ρ M vi c [ 1 + 1
2v2
c2 + ... ] [8-26b]
Tparticle
0i= ρ vi
c [ Mc2 + 12
Mv2 + ... ] [8-26c]
Tparticle
00= ρ [ Mc2 + 1
2Mv2 + ... ] [8-26d]
To proceed further, we refer back to [7-7]:
M = m +Q
c2
which allows us to rewrite our equations as:
Tparticle
ij= ρ M vi v j [ 1 + 1
2v2
c2 + ... ]
Tparticle
i0= ρ M vi c [ 1 + 1
2v2
c2 + ... ]
Tparticle
0i= ρ vi
c [ mc2 + Q + 12
Mv2 + ... ]
Tparticle
00= ρ [ mc2 + Q + 1
2Mv2 + ... ]
Now, applying Rule 1, i.e., deleting terms containing the rest energy mc2 from the energy
current expression T0i and the energy density expression T00, we obtain:
Tparticle
ij= ρ M vi v j [ 1 + 1
2v2
c2 + ... ]
4 See, e.g., pp. 67 and 85 in Rindler W., Special Relativity, 2
nd Ed., Oliver and Boyd, Edinburgh (1969).
_Chapter 8
87
Tparticle
i0= ρ M vi c [ 1 + 1
2v2
c2 + ... ]
Tparticle
0i= ρ vi
c [ Q + 12
Mv2 + ... ]
Tparticle
00= ρ [ Q + 1
2Mv2 + ... ]
(The deletion of mc2 from T0iparticle can be shown to balance its deletion from T00
particle, so
that the overall divergence remains zero. This will be verified later when the total
divergence is evaluated in detail.) Proceeding on, we divide Ti0 by c and multiply T0i by c
in accordance with Rule 2, which yields:
Tparticle
ij= ρ M vi v j [ 1 + 1
2v2
c2 + ... ]
Tparticle
i0= ρ M vi [ 1 + 1
2v2
c2 + ... ]
Tparticle
0i= ρ vi [ Q + 1
2Mv2 + ... ]
Tparticle
00= ρ [ Q + 1
2Mv2 + ... ]
Finally, we take the non-relativistic limit c → ∞ in accordance with Rule 3. This also
requires using the result [8-3] plus the following known limit5:
(de Broglie’s relativistic Q) → (Bohm’s non-relativistic Q) [8-27]
The expressions resulting from this step are:
Tparticle
ij= ρmviv j [8-28a]
Tparticle
i0= ρmvi [8-28b]
Tparticle
0i= ρvi [1
2mv2 + Q] [8-28c]
Tparticle
00= ρ [1
2mv2 + Q] [8-28d]
i.e.,
5 See p. 121 in: L. de Broglie, Nonlinear Wave Mechanics, Elsevier, Amsterdam (1960).
_Chapter 8
88
Tparticle
ij= ρmviv j
Tparticle
i0 = ρmvi
Tparticle
0i = ρviE
Tparticle
00= ρE
where E ≡ ½mv2 + Q. These equations are then seen to be expressions [8-18] as required.
Note that the reason for the lack of symmetry of Tµνparticle in the non-relativistic case can
be seen clearly by looking at equations [8-26]. In applying our rules to the components
Ti0 and T0i, we keep the first order term but drop the second order one in Ti0, whereas in
contrast we keep the second order term but drop the first order one in T0i. Not
surprisingly, this reversal results in the two expressions becoming different.
We are now in a position to find the appropriate non-relativistic expressions for Tµνfield
and Tµνinteraction.
8.2 Non-Relativistic Energy-Momentum Tensor for the Field
A possible form for the Schrodinger Tµνfield has already been derived in Appendix 4 using
the standard formula [6-23]:
Tfieldµν = [ ∂µψ ∂
∂(∂νψ)+ ∂µψ* ∂
∂(∂νψ*)
– gµν ]
_Chapter 8
89
To take this limit for comparison, we need to know the relationship between the Klein-
Gordon wavefunction φ and the Schrodinger wavefunction ψ. This relationship, which is
a standard formula of quantum mechanics6, is as follows:
φ = ψ e– ih
mcx0(x
0 = ct) [8-30]
Inserting [8-30] in [8-29] yields
Tfieldµν
=h
2
2m{ (∂µ
[ψe– ih
mcx0]) (∂ν
[ψ*eih
mcx0]) + (∂µ
[ψ*eih
mcx0]) (∂ν
[ψe– ih
mcx0])
– gµν (∂λ[ψ*e
ih
mcx0]) (∂λ
[ψe– ih
mcx0]) + gµν (mc
h)
2ψ*ψ }
and employing the identities [5-19]:
∂xν/∂xµ ≡ gµν
∂xν/∂xµ ≡ δνµ
we then obtain:
Tfieldµν
=h
2
2m{ (∂µψ) (∂νψ*) + ψψ*(– imc
h) g0µ ( imc
h) g0ν
+ (∂µψ) ψ* ( imch
) g0ν + ψ (– imch
) g0µ (∂νψ*)
+ (∂µψ*) (∂νψ) + ψ*ψ ( imch
) g0µ (– imch
) g0ν
+ (∂µψ*) ψ (– imch
) g0ν + ψ* ( imch
) g0µ (∂νψ)
– gµν [ (∂λψ*) (∂λψ) + ψ*ψ ( imc
h) δλ
0(– imc
h) g0λ
+ (∂λψ*) ψ (– imc
h) g0λ + ψ* ( imc
h) δλ
0(∂λψ) ]
+ gµν (mch
)2 ψ*ψ }
=h
2
2m{ (∂µψ) (∂νψ*) + (∂µψ*) (∂νψ) + 2 (mc
h)
2g0µ g0ν ψψ*
+ imch
[ g0µ (ψ* ∂νψ – ψ ∂νψ*) + g0ν (ψ* ∂µψ – ψ ∂µψ*)
– gµν [ (∂λψ*) (∂λψ) + imc
h(ψ* ∂0ψ – ψ ∂0ψ*) ] }
6 See, e.g., p. 7 in Greiner W., Relativistic Quantum Mechanics – Wave Equations, 2
nd Ed., Springer, Berlin
(1997).
_Chapter 8
90
=h
2
2m[ (∂µψ) (∂νψ*) + (∂µψ*) (∂νψ) ] + mc2 g0µ g0ν ψψ*
+ ihc2
[ g0µ (ψ* ∂νψ – ψ ∂νψ*) + g0ν (ψ* ∂µψ – ψ ∂µψ*) ]
– gµν [h
2
2m(∂λψ
*) (∂λψ) + ihc2
(ψ* ∂0ψ – ψ ∂0ψ*) ]
[8-31]
To proceed towards the non-relativistic approximation, we now apply Rule 1 and delete the
mc2 term from this expression. In doing so, it is necessary to keep the overall divergence of
the energy-momentum tensor zero. This means it is also necessary here to remove the term
that would previously have cancelled with the deleted mc2 term. It is not difficult to identify
this term, as follows. The divergence (∂ν) of the mc2 term would have given a result of the
form:
mc2 g0µ ∂0(ψψ*)
i.e., a term containing the factor g0µ. This could cancel only with another term containing
g0µ. Hence, looking at [8-31], we deduce that it is the term:
(ihc/2) g0µ (ψ* ∂νψ − ψ ∂νψ*)
that should also be deleted. (This conclusion will be verified more rigorously later by
calculating the divergence in full.) On the above basis, [8-31] reduces to:
Tfieldµν =
h2
2m[ (∂µψ) (∂νψ*) + (∂µψ*) (∂νψ) ] + ihc
2g0ν (ψ* ∂µψ – ψ ∂µψ*)
– gµν [h
2
2m(∂λψ
*) (∂λψ) + ihc2
(ψ* ∂0ψ – ψ ∂0ψ*) ]
[8-32]
Continuing on, the relativistic limit is obtained by taking the speed of light to be essentially
infinite. In taking this limit, we will need to consider the Tij, Ti0, T0i and T00 cases
separately.
_Chapter 8
91
8.2.1 Non-Relativistic Tijfield
For this case, [8-32] yields
Tfield
ij =h
2
2m[ (∂ iψ) (∂ jψ*) + (∂ iψ*) (∂ jψ) ] + 0
– g ij [h
2
2m{ (∂kψ
*) (∂kψ) + (∂0ψ*) (∂0ψ) } + ihc
2(ψ* ∂0ψ – ψ ∂0ψ*) ]
(where k = 1,2,3). Switching from x0 to ct, this becomes
Tfield
ij=
h2
2m[ (∂ iψ) (∂ jψ*) + (∂ iψ*) (∂ jψ) ]
– g ij [h
2
2m{ (∂kψ
*) (∂kψ) + 1c2 (∂ tψ
*) (∂ tψ) } + ih2
(ψ* ∂ tψ – ψ ∂ tψ*) ]
and taking the limit c → ∞, our Schrodinger expression for Tij is found to be
Tfieldij
=h
2
2m[ (∂
iψ) (∂jψ*) + (∂
iψ*) (∂jψ) ] – g ij [
h2
2m(∂kψ
*) (∂kψ) + ih
2(ψ* ∂ tψ – ψ ∂ tψ
*) ]
[8-33]
8.2.2 Non-Relativistic Ti0field
Inserting µ = i, ν = 0 into [8-32], we obtain
Tfieldi0 =
h2
2m[ (∂ iψ) (∂0ψ*) + (∂ iψ*) (∂0ψ) ] + ihc
2(ψ* ∂ iψ – ψ ∂ iψ*) – 0
=h
2
2mc[ (∂ iψ) (∂ tψ
*) + (∂ iψ*) (∂ tψ) ] + ihc2
(ψ* ∂ iψ – ψ ∂ iψ*)
Dividing through by c in accordance with Rule 2, this becomes:
Tfield
i0=
h2
2mc2 [ (∂ iψ) (∂ tψ*) + (∂ iψ*) (∂ tψ) ] + ih
2(ψ* ∂ iψ – ψ ∂ iψ*)
and taking the limit c → ∞, the Schrodinger expression for Ti0 is then found to be:
Tfield
i0= ih
2(ψ* ∂ iψ – ψ ∂ iψ*) [8-34]
8.2.3 Non-Relativistic T0ifield
Inserting µ = 0, ν = i into [8-32], we obtain
_Chapter 8
92
Tfield
0i=
h2
2m[ (∂0ψ) (∂ iψ*) + (∂0ψ*) (∂ iψ) ] + 0 – 0
=h
2
2mc[ (∂ tψ) (∂ iψ*) + (∂ tψ
*) (∂ iψ) ]
Multiplying through by c in accordance with Rule 2, then gives:
Tfield0i
=h
2
2m[ (∂ iψ) (∂ tψ
*) + (∂ iψ*) (∂ tψ) ] [8-35]
This is our Schrodinger expression for T0ifield, since there are no factors of c remaining to
require the limit c → ∞ to be taken.
8.2.4 Non-Relativistic T00field
Inserting µ = 0 and ν = 0 into [8-32], we obtain:
Tfield00
=h
2
2m[ (∂
0ψ) (∂0ψ*) + (∂
0ψ*) (∂0ψ) ] + ihc
2(ψ* ∂0ψ – ψ ∂0ψ*)
–h
2
2m(∂λψ
*) (∂λψ) – ihc2
(ψ* ∂0ψ – ψ ∂0ψ*)
= h2
m (∂0ψ) (∂
0ψ*) –h
2
2m(∂λψ
*) (∂λψ)
=h
2
mc2 (∂ tψ) (∂ tψ*) –
h2
2m{ (∂kψ
*) (∂kψ) +
1
c2 (∂ tψ) (∂ tψ*) }
and taking the limit c → ∞, the Schrodinger expression for T00 is found to be:
Tfield00
= –h
2
2m(∂kψ
*) (∂kψ) [8-36]
8.2.5 Overall Non-Relativistic Result for Tµν
field
Gathering together expressions [8-33] to [8-36], our non-relativistic form for Tµνfield is:
Tfieldij
=h
2
2m[ (∂
iψ) (∂jψ*) + (∂
iψ*) (∂jψ) ] – g ij [
h2
2m(∂kψ
*) (∂kψ) + ih
2(ψ* ∂ tψ – ψ ∂ tψ
*) ]
[8-37a]
Tfield
i0= ih
2(ψ* ∂ iψ – ψ ∂ iψ*) [8-37b]
Tfield0i
=h
2
2m[ (∂
iψ) (∂ tψ*) + (∂
iψ*) (∂ tψ) ] [8-37c]
Tfield00
= –h
2
2m(∂kψ
*) (∂kψ) [8-37d]
_Chapter 8
93
Comparison with Appendix 4 then shows that the two different derivations have yielded the
same result.
8.3 Non-Relativistic Energy-Momentum Tensor – Interaction Component
The non-relativistic form for Tµνinteraction will now be derived. From equation [7-42], the
relativistic expression is
Tinteraction
µν= –
c (∂µS) (∂
νS) ρ0
(∂αS) (∂αS)
[8-38]
This expression can be written in terms of the Klein-Gordon wavefunction φ and its
complex conjugate φ*, instead of in terms of the phase S, by using [7-38]:
∂µS = – ih2
{∂µφ
φ–
∂µφ*
φ*} [8-39]
It will be more convenient, however, to proceed by first re-expressing [8-39] in terms of the
Schrodinger wavefunction ψ, using the relationship [8-30] that connects φ and ψ:
φ = ψ e– ih
mcx0(x
0 = ct)
Inserting this relationship into [8-39] yields
∂µS = – ih2
{∂µ[ψe– i
hmcx0
]
ψe– ih
mcx0–
∂µ[ψ*e
ih
mcx0]
ψ*eih
mcx0}
and employing the identities [5-19], this becomes:
∂µS = – ih2
{∂µψ
ψ– imc
hδµ
0–
∂µψ*
ψ*– imc
hδµ
0}
= ih2
(∂µψ
*
ψ*–
∂µψ
ψ) – mc δµ
0
This result will now be inserted into equation [8-38] so that we obtain Tµνinteraction expressed
directly in terms of ψ:
_Chapter 8
94
Tinteraction
µν= – c
[ ih2
(∂
µψ*
ψ*–
∂µψ
ψ) – mc g0µ ] [ ih
2(∂
νψ*
ψ*–
∂νψ
ψ) – mc g0ν ] ρ0
[ ih2
(∂αψ*
ψ*–
∂αψ
ψ) – mc δα
0] [ ih
2(∂
αψ*
ψ*–
∂αψ
ψ) – mc g0α ]
[8-40]
We will now focus briefly on just the denominator of this expression, which can be
rewritten as follows:
[ ih2
(∂αψ*
ψ*–
∂αψ
ψ) – mc δα
0] [ ih
2(∂αψ*
ψ*–
∂αψ
ψ) – mc g0α ]
= –h
2
4(∂αψ*
ψ*–
∂αψ
ψ) (
∂αψ*
ψ*–
∂αψ
ψ) – ihmc (
∂0ψ*
ψ*–
∂0ψ
ψ) + m2c2
= mc – ( h2mc
)2
(∂αψ*
ψ*–
∂αψ
ψ) (
∂αψ*
ψ*–
∂αψ
ψ) – ih
mc (∂0ψ*
ψ*–
∂0ψ
ψ) + 1
Since the square root in this result will appear frequently in the rest of the present section,
we will represent it using the letter K as follows:
K ≡ 1
– ( h2mc
)2
(∂αψ*
ψ*–
∂αψ
ψ) (
∂αψ*
ψ*–
∂αψ
ψ) – ih
mc (∂0ψ*
ψ*–
∂0ψ
ψ) + 1
[8-41]
Note that the non-relativistic limit of K is simply:
K → 1 [8-42]
Returning to [8-40], the expression for Tµνinteraction can now be written more simply as:
Tinteraction
µν= – K
m [ ih2
(∂µψ*
ψ*–
∂µψ
ψ) – mc g0µ ] [ ih
2(∂νψ*
ψ*–
∂νψ
ψ) – mc g0ν ] ρ0
= – Km [ –
h2
4(∂µψ*
ψ*–
∂µψ
ψ) (
∂νψ*
ψ*–
∂νψ
ψ) – mc ih
2(∂νψ*
ψ*–
∂νψ
ψ) g0µ
– mc ih2
(∂µψ*
ψ*–
∂µψ
ψ) g0ν + m2c2 g0µ g0ν ] ρ0
= – K [ –h
2
4m(∂µψ*
ψ*–
∂µψ
ψ) (
∂νψ*
ψ*–
∂νψ
ψ) – ihc
2(∂νψ*
ψ*–
∂νψ
ψ) g0µ
– ihc2
(∂µψ*
ψ*–
∂µψ
ψ) g0ν + mc2 g0µ g0ν ] ρ0
[8-43]
_Chapter 8
95
We now apply Rule 1 and delete the mc2 term from this expression. As in the case of Tµνfield
earlier, it is also necessary here to remove the term that would previously have cancelled
with the deleted mc2 term (in order to keep the overall divergence of the energy-momentum
tensor zero). In order to identify this term, we note that the divergence (∂ν) of the mc2 term
would have given a result of the form:
− mc2 g0µ ∂0(Kρ0)
i.e., a term containing the factor g0µ. This can cancel only with another term containing g0µ.
Hence, looking at [8-43], we must also delete the term:
– ihc2
(∂νψ*
ψ*–
∂νψ
ψ) g0µ
(Again, this conclusion will be confirmed later when the full divergence is calculated.)
With these two deletions, [8-43] reduces to:
Tinteraction
µν= – K [ –
h2
4m(∂µψ*
ψ*–
∂µψ
ψ) (
∂νψ*
ψ*–
∂νψ
ψ) – ihc
2(∂µψ*
ψ*–
∂µψ
ψ) g0ν ] ρ0
[8-44]
The relativistic approximation will now be obtained by taking the limit c → ∞. In taking
this limit, we will need to consider the Tij, Ti0, T0i and T00 cases separately.
8.3.1 Non-Relativistic Tijinteraction
For this case, [8-44] yields:
Tinteractionij
= – K [ –h
2
4m(∂ iψ*
ψ*–
∂ iψ
ψ) (
∂ jψ*
ψ*–
∂ jψ
ψ) – 0 ] ρ0
and taking the non-relativistic limit via [8-2] and [8-42], we obtain:
Tinteractionij
=h
2
4m(∂ iψ*
ψ*–
∂ iψ
ψ) (
∂ jψ*
ψ*–
∂ jψ
ψ) ρ [8-45]
_Chapter 8
96
8.3.2 Non-Relativistic Ti0interaction
Inserting µ = i, ν = 0 into [8-44], we have:
Tinteractioni0 = – K [ –
h2
4m(∂ iψ*
ψ*–
∂ iψ
ψ) (
∂0ψ*
ψ*–
∂0ψ
ψ) – ihc
2(∂ iψ*
ψ*–
∂ iψ
ψ) ] ρ0
= – K [ –h
2
4mc(∂ iψ*
ψ*–
∂ iψ
ψ) (
∂ tψ*
ψ*–
∂ tψ
ψ) – ihc
2(∂ iψ*
ψ*–
∂ iψ
ψ) ] ρ0
Dividing through by c in accordance with Rule 2, this becomes:
Tinteractioni0
= – K [ –h
2
4mc2 (∂ iψ*
ψ*–
∂ iψ
ψ) (
∂ tψ*
ψ*–
∂ tψ
ψ) – ih
2(∂ iψ*
ψ*–
∂ iψ
ψ) ] ρ0
and taking the limit c → ∞ then yields:
Tinteractioni0
= ih2
(∂ iψ*
ψ*–
∂ iψ
ψ) ρ [8-46]
8.3.3 Non-Relativistic T0iinteraction
Inserting µ = 0, ν = i into [8-44], we obtain:
Tinteraction0i
= – K [ –h
2
4m(∂0ψ*
ψ*–
∂0ψ
ψ) (
∂ iψ*
ψ*–
∂ iψ
ψ) – 0 ] ρ0
= – K [ –h
2
4mc(∂ tψ
*
ψ*–
∂ tψ
ψ) (
∂ iψ*
ψ*–
∂ iψ
ψ) ] ρ0
Multiplying through by c in accordance with Rule 1 then gives:
Tinteraction0i
= – K [ –h
2
4m(∂ tψ
*
ψ*–
∂ tψ
ψ) (
∂ iψ*
ψ*–
∂ iψ
ψ) ] ρ0
and taking the non-relativistic limit via [8-2] and [8-42] then yields:
Tinteraction0i
=h
2
4m(∂ tψ
*
ψ*–
∂ tψ
ψ) (
∂ iψ*
ψ*–
∂ iψ
ψ) ] ρ [8-47]
8.3.4 Non-Relativistic T00interaction
Inserting µ = 0 and ν = 0 into [8-44], we obtain:
_Chapter 8
97
Tinteraction00 = – K [ –
h2
4m(∂0ψ*
ψ*–
∂0ψ
ψ) (
∂0ψ*
ψ*–
∂0ψ
ψ) – ihc
2(∂0ψ*
ψ*–
∂0ψ
ψ) ] ρ0
= – K [ –h
2
4mc2(∂ tψ
*
ψ*–
∂ tψ
ψ) (
∂ tψ*
ψ*–
∂ tψ
ψ) – ih
2(∂ tψ
*
ψ*–
∂ tψ
ψ) ] ρ0
and taking the non-relativistic limit c → ∞, together with [8-2] and [8-42], then yields:
Tinteraction00
= ih2
(∂ tψ
*
ψ*–
∂ tψ
ψ) ] ρ [8-48]
8.3.5 Overall Non-Relativistic Result for Tµν
interaction
Gathering together expressions [8-45] to [8-48], our non-relativistic form for Tµνinteraction is:
Tinteractionij =
h2
4m(∂ iψ*
ψ*–
∂ iψ
ψ) (
∂ jψ*
ψ*–
∂ jψ
ψ) ρ [8-49a]
Tinteractioni0
= ih2
(∂ iψ*
ψ*–
∂ iψ
ψ) ρ [8-49b]
Tinteraction0i
=h
2
4m(∂ tψ
*
ψ*–
∂ tψ
ψ) (
∂ iψ*
ψ*–
∂ iψ
ψ) ] ρ [8-49c]
Tinteraction00 = ih
2(∂ tψ
*
ψ*–
∂ tψ
ψ) ] ρ [8-49d]
8.4 Divergence and Conservation
The final task in this chapter is to check explicitly that the overall divergence of the non-
relativistic energy-momentum tensor for the particle-field system is zero and thereby
confirm that energy and momentum are conserved. Towards this end, the divergences of
Tµνfield , T
µνparticle and Tµν
interaction will be evaluated separately.
_Chapter 8
98
8.4.1 Divergence of Tµνfield
There are two distinct parts to the divergence of the non-relativistic Tµνfield , corresponding
to the cases µ = i ( = 1,2,3) and µ = 0, respectively. For the first of these, we have (using
expressions [8-37]):
∂ j Tfieldij
+ ∂ t Tfieldi0
= ∂ j {h
2
2m[ (∂ iψ) (∂ jψ*) + (∂ iψ*) (∂ jψ) ]
– g ij [h
2
2m(∂kψ
*) (∂kψ) + ih2
(ψ* ∂ tψ – ψ ∂ tψ*) ] }
+ ∂ t { ih2
(ψ* ∂ iψ – ψ ∂ iψ*) }
=h
2
2m[ (∂ j∂
iψ) (∂ jψ*) + (∂ iψ) (∂ j∂jψ*) + (∂ j∂
iψ*) (∂ jψ) + (∂ iψ*) (∂ j∂jψ) ]
–h
2
2m[ (∂ i∂kψ
*) (∂kψ) + (∂kψ*) (∂ i∂kψ) ]
– ih2
[ (∂ iψ*) (∂ tψ) + ψ* (∂ i∂ tψ) – (∂ iψ) (∂ tψ*) – ψ (∂ i∂ tψ
*) ]
+ ih2
[ (∂ tψ*) (∂ iψ) + ψ* (∂ t∂
iψ) – (∂ tψ) (∂ iψ*) – ψ (∂ t∂iψ*) ]
=h
2
2m[ (∂ iψ) (∂ j∂
jψ*) + (∂ iψ*) (∂ j∂jψ ) ] + ih
2[ (∂ iψ) (∂ tψ
*) – (∂ iψ*) (∂ tψ) ]
= (∂iψ*) [
h2
2m(∂ j∂
jψ) – ih2
(∂ tψ) ] + (∂iψ) [
h2
2m(∂ j∂
jψ*) + ih2
(∂ tψ*) ]
[8-50]
This can be simplified further by using the field equation corresponding to our
Lagrangian density, i.e., by using the extended Schrodinger equation [5-21]:
h2
2m∂ j∂
jψ – ih∂ tψ = – ih2ψ*
{∇.(ρ∇Sm ) + ∂ tρ } [8-51]
Inserting [8-51] and its complex conjugate into [8-50], yields:
∂ j Tfield
ij+ ∂ t Tfield
i0= – ih
2ψ* (∂iψ*) { ∇.(ρ∇S
m ) + ∂ tρ } + ih2ψ
(∂iψ) { ∇.(ρ∇S
m ) + ∂ tρ }
= ih2
[ 1ψ (∂ iψ) – 1
ψ*(∂ iψ*) ] { ∇.(ρ∇S
m ) + ∂ tρ }
and using the identity [5-14]:
∂ jS = h2i
[∂ jψψ –
∂ jψ*
ψ*]
_Chapter 8
99
we then obtain:
∂ j Tfieldij
+ ∂ t Tfieldi0
= – (∂iS) { ∇.(ρ∇S
m ) + ∂ tρ }
= (∂ iS) { ∂ j(ρ
∂ jS
m ) – ∂ tρ } [8-52]
We now turn to the second part of the divergence of Tµνfield (corresponding to µ = 0). Using
expressions [8-37], we have:
∂ j Tfield0j
+ ∂ t Tfield00
= ∂ j {h
2
2m[ (∂ jψ) (∂ tψ
*) + (∂ jψ*) (∂ tψ) ] } + ∂ t [ –h
2
2m(∂kψ
*) (∂kψ) ]
=h
2
2m[ (∂ j∂
jψ) (∂ tψ*) + (∂ jψ) (∂ j∂ tψ
*) + (∂ j∂jψ*) (∂ tψ) + (∂ jψ*) (∂ j∂ tψ) ]
–h
2
2m[ (∂ t∂kψ
*) (∂kψ) + (∂kψ*) (∂ t∂
kψ) ]
=h
2
2m[ (∂ j∂
jψ) (∂ tψ*) + (∂ j∂
jψ*) (∂ tψ) ]
Using the field equation [8-51] and its complex conjugate, this can be re-expressed as:
∂ j Tfield
0j+ ∂ t Tfield
00= [ ih∂ tψ – ih
2ψ* { ∇.(ρ∇Sm ) + ∂ tρ } ] ∂ tψ
*
+ [ – ih∂ tψ* + ih
2ψ{ ∇.(ρ∇S
m ) + ∂ tρ }] ∂ tψ
= ih2
[ 1ψ (∂ tψ) – 1
ψ*(∂ tψ
*) ] { ∇.(ρ∇Sm ) + ∂ tρ }
and, using the identity [5-14] again, we obtain:
∂ j Tfield0j
+ ∂ t Tfield00
= – (∂ tS) { ∇.(ρ∇Sm ) + ∂ tρ }
= (∂ tS) { ∂ j(ρ∂ j
Sm ) – ∂ tρ } [8-53]
8.4.2 Divergence of Tµνparticle
As with Tµνfield, there are two distinct parts to the divergence of the non-relativistic
Tµνparticle, corresponding to the cases µ = 1,2,3 and µ = 0, respectively. In evaluating these
_Chapter 8
100
two parts, it is necessary to keep in mind the following functional dependencies in the non-
relativistic domain:
x0 = x0(t) ≠ x0(x) [8-54a]
v = v(t) ≠ v(x) [8-54b]
ρ = ρ[x − x0(t)] = ρ[x0(t) − x] [8-54c]
Using equations [8-18], the first part of the divergence is:
∂ jTparticle
ij + ∂ tTparticle
i0 = ∂ j (mvi ρv j) + ∂ t (mvi ρ)
= mviv j ∂ jρ + m ∂ t(viρ) [since v
i ≠ v
i(x)]
= mviv j ∂ρ∂x j
+ mρ ∂ tvi + mvi ∂ tρ
= mviv j (–∂ρ∂x0
j) + mρ
∂vi
∂t+ mvi ∂ρ
∂x0j
∂x0j
∂t
= – mviv j ∂ρ∂x0
j+ mρ
dvi
dt+ mvi ∂ρ
∂x0j
v j
= ρ mdvi
dt
and referring back to [5-4], this can then be written as:
∂ jTparticle
ij+ ∂ tTparticle
i0= ρ ∂ i
Q [8-55]
where Q is Bohm’s non-relativistic quantum potential.
The second part of the divergence is:
∂ jTparticle
0j+ ∂ tTparticle
00= ∂ j (E ρv j) + ∂ t (E ρ)
= ρv j ∂ jE + Ev j ∂ jρ + ρ ∂ tE + E ∂ tρ
= ρv j ∂ jE + Ev j ∂ρ∂x j
+ ρ ∂ tE + E∂ρ∂x0
j
∂x0j
∂t
= ρv j ∂ jE + Ev j (–∂ρ∂x0
j) + ρ ∂ tE + E
∂ρ∂x0
j
dx0j
dt
_Chapter 8
101
= ρv j ∂ jE – Ev j ∂ρ∂x0
j+ ρ ∂ tE + E
∂ρ∂x0
jv j
= ρv j ∂ jE + ρ ∂ tE
= ρv j ∂ j(½mv2 + Q) + ρ ∂ t(½mv2 + Q)
= ρv j ∂ jQ + ρ ∂ t(– ½mvjvj) + ρ ∂ tQ
= ρv j ∂ jQ + ρ dvi
dt∂
∂vi (– ½mvjvj) + ρ ∂ tQ
= ρv j ∂ jQ – ρmdvi
dt½(g ijv
j+ vjδ i
j) + ρ ∂ tQ
= ρv j ∂ jQ – ρvi mdvi
dt+ ρ ∂ tQ
Using [5-4], this can then be written as:
∂ jTparticle
0j+ ∂ tTparticle
00= ρv j ∂ jQ – ρvi ∂ i
Q + ρ ∂ tQ
= ρ ∂ tQ [8-56]
8.4.3 Divergence of Tµνinteraction
From [8-49], our non-relativistic form for Tµνinteraction is:
Tinteractionij
=h
2
4m(∂ iψ*
ψ*–
∂ iψ
ψ) (
∂ jψ*
ψ*–
∂ jψ
ψ) ρ
Tinteractioni0
= ih2
(∂ iψ*
ψ*–
∂ iψ
ψ) ρ
Tinteraction0i
=h
2
4m(∂ tψ
*
ψ*–
∂ tψ
ψ) (
∂ iψ*
ψ*–
∂ iψ
ψ) ] ρ
Tinteraction00 = ih
2(∂ tψ
*
ψ*–
∂ tψ
ψ) ] ρ
and, using the identity [5-14], these expressions can be written more simply as:
Tinteractionij
= –(∂ i
S) (∂ jS)
m ρ [8-57a]
Tinteraction
i0= (∂ i
S) ρ [8-57b]
_Chapter 8
102
Tinteraction0i
= –(∂ tS) (∂
iS)
m ρ [8-57c]
Tinteraction
00= (∂ tS) ρ [8-57d]
As with Tµνfield and Tµν
particle, there are two distinct parts to the divergence of the non-
relativistic Tµνinteraction. Employing expressions [8-57], the first part is:
∂ jTinteractionij
+ ∂ tTinteractioni0
= ∂ j [ –(∂ i
S) (∂ jS)
m ρ ] + ∂ t [ (∂ iS) ρ ]
= – (∂iS) { ∂ j(ρ
∂ jS
m ) – ∂ tρ } –(∂
jS) ∂ j(∂
iS)
m ρ + ρ ∂ t(∂iS)
= – (∂iS) { ∂ j(ρ
∂ jS
m ) – ∂ tρ } –(∂
jS) ∂ i
(∂ jS)m ρ + ρ ∂ i
(∂ tS)
= – (∂ iS) { ∂ j(ρ
∂ jS
m ) – ∂ tρ } – ρ ∂ i[
(∂ jS) (∂ jS)
2m– (∂ tS) ]
and, using [5-13], this can be written in terms of the quantum potential Q as:
∂ jTinteractionij
+ ∂ tTinteractioni0
= – (∂ iS) { ∂ j(ρ
∂ jS
m ) – ∂ tρ } – ρ ∂ iQ [8-58]
Similarly, the second part of the divergence is:
∂ jTinteraction0j
+ ∂ tTinteraction00
= ∂ j [ –(∂ tS) (∂
jS)
m ρ ] + ∂ t [ (∂ tS) ρ ]
= – (∂ tS) { ∂ j(ρ∂ j
Sm ) – ∂ tρ } –
(∂ jS) ∂ j(∂ tS)
m ρ + ρ ∂ t(∂ tS)
= – (∂ tS) { ∂ j(ρ∂ j
Sm ) – ∂ tρ } –
(∂ jS) ∂ t(∂ jS)
m ρ + ρ ∂ t(∂ tS)
= – (∂ tS) { ∂ j(ρ∂ j
Sm ) – ∂ tρ } – ρ ∂ t [
(∂ jS) (∂ jS)
2m– (∂ tS) ]
and using [5-13] again, this can be expressed more simply in terms of the potential Q, as
follows:
∂ jTinteraction0j
+ ∂ tTinteraction00
= – (∂ tS) { ∂ j(ρ∂ j
Sm ) – ∂ tρ } – ρ ∂ tQ [8-59]
_Chapter 8
103
8.4.4 Divergence of Tµνtotal
From equation [7-31] we have
Ttotal
µν= Tfield
µν+ Tparticle
µν+ Tinteraction
µν
The divergence of this overall energy-momentum tensor can now be obtained by
combining the various results obtained above. As usual, the divergence will be written in
two parts. First, using [8-52], [8-55] and [8-58], we have:
∂ jTtotal
ij+ ∂ tTtotal
i0= ∂ jTfield
ij+ ∂ tTfield
i0+ ∂ jTparticle
ij+ ∂ tTparticle
i0+ ∂ jTinteraction
ij+ ∂ tTinteraction
i0
= (∂iS) {∂ j(ρ
∂ jS
m ) – ∂ tρ} + ρ ∂ iQ + – (∂
iS) {∂ j(ρ
∂ jS
m ) – ∂ tρ} – ρ ∂ iQ
which cancels to:
∂ jTtotal
ij+ ∂ tTtotal
i0= 0 [8-60]
Second, using [8-53], [8-56] and [8-59], we have:
∂ jTtotal
0j+ ∂ tTtotal
00= ∂ jTfield
0j+ ∂ tTfield
00+ ∂ jTparticle
0j+ ∂ tTparticle
00+ ∂ jTinteraction
0j+ ∂ tTinteraction
00
= (∂ tS) {∂ j(ρ∂ j
Sm ) – ∂ tρ} + ρ ∂ tQ + – (∂ tS) {∂ j(ρ
∂ jS
m ) – ∂ tρ} – ρ ∂ tQ
which cancels to:
∂ jTtotal
0j + ∂ tTtotal
00 = 0 [8-61]
Equations [8-60] and [8-61] are the desired results for energy and momentum
conservation. (The divergence calculations above also serve as a useful double-check on
our derivations of the non-relativistic expressions for Tµνfield, T
µνparticle and Tµν
interaction.)
Therefore, from the viewpoint of conservation, a satisfactory non-relativistic model has
been achieved.
_Chapter 8
104
8.5 Simplifications in the Bohmian Case
Some additional discussion is now needed to highlight the simplifications which occur in
the above equations in the main case of interest. It will be helpful here to restate the three
divergence results of Sec. 8.4 for further examination:
∂ j Tfield0j
+ ∂ t Tfield00
= (∂ tS) { ∂ j(ρ∂ j
Sm ) – ∂ tρ } [8-53]
∂ jTparticle
ij+ ∂ tTparticle
i0= ρ ∂ i
Q [8-55]
∂ jTinteractionij
+ ∂ tTinteractioni0
= – (∂iS) { ∂ j(ρ
∂ jS
m ) – ∂ tρ } – ρ ∂ iQ [8-58]
In developing our Lagrangian formulation, it was necessary to suspend the Bohmian
restriction p = ∇∇∇∇S on the velocity of the particle. This meant we were actually
considering a wide class of models, all of which satisfy the conservation laws for energy
and momentum, but most of which need not be in agreement with the predictions of
quantum theory. These models all satisfy the three divergence equations above. Note that,
in general, the three different divergences (for Tµνparticle, Tµν
field and Tµνinteraction) are all
non-zero so that, for example, energy and momentum are being exchanged between the
particle and the field.
Eventually, however, it is necessary to restore the restriction p = ∇∇∇∇S in order to return to
Bohm’s model and agreement with experiment. This limits us to one particular model
within the class considered. Since the whole class of models satisfies the energy and
momentum conservation laws, the special model now singled out will do so as well. (The
restriction is just an extra constraint which does not conflict with the earlier
considerations in any way.) However, the assumption of no creation or annihilation of
_Chapter 8
105
particles, in conjunction with p = ∇∇∇∇S, simplifies the above divergence equations
significantly, so that they become:
∂ j Tfield
0j+ ∂ t Tfield
00= 0 [8-62]
∂ jTparticle
ij+ ∂ tTparticle
i0= ρ ∂ i
Q [8-63]
∂ jTinteraction
ij+ ∂ tTinteraction
i0= – ρ ∂ i
Q [8-64]
In other words, in the special case of the Bohmian model singled out, the general
relationship:
∂ν Tfield
µν+ ∂ν Tinteraction
µν+ ∂ν Tparticle
µν= 0 [8-65]
reduces to the two separate relationships:
∂ν Tfield
µν= 0 [8-66]
and
∂ν Tinteraction
µν+ ∂ν Tparticle
µν= 0 [8-67]
so that the formalism becomes somewhat less elegant in the Bohmian case7. This break-
up into equations [8-66] and [8-67] is a necessary consequence of having a source-free
wave equation. It tends, however, to disguise the fact that conservation is present, with
the relationship [8-66] being particularly misleading in this regard. The apparent
difficulty posed by this equation, as highlighted in the discussions of Sec. 6.4, has
nevertheless been resolved by the necessary existence of the term Tµνinteraction, to which
we have been led by examining Noether’s theorem.
7 An additional simplification is that the independent expressions for T
µνparticle and T
µνinteraction become
connected by Tparticle
µν= – Tinteraction
µν
_Chapter 8
106
In conclusion, although the equations become rather simple in the one special case with
which we are most concerned, this should not serve as a distraction from the successful
reintroduction of energy and momentum conservation and the necessity of reaching it via
the more general Lagrangian formulation employed.
_Chapter_9.doc
107
Chapter 9: Discussion and Conclusions
In this thesis, it has been demonstrated that the well-known Bohmian model for quantum
mechanics can be made compatible with the laws of conservation of energy and
momentum. This has been achieved by constructing a Lagrangian formulation of the
model, so that the required conservation is then assured by Noether’s theorem. Although
this conservation is then known to be present in general terms, extracting a detailed
description of it in terms of energy-momentum tensors was found to be not at all
straightforward. First, it was necessary to realize that the usual energy-momentum tensors
Tµνfield and Tµν
particle that appear in such a formulation had, in this case, to be augmented
by a third tensor Tµνinteraction. Furthermore, attempting to obtain a specific expression for
Tµνinteraction by re-deriving Noether’s proof from first principles was found to lead to
difficulties and ambiguities in the non-relativistic case.
The chosen way forward was to formulate the details of a relativistic Lagrangian model
and then take the non-relativistic limit to obtain the appropriate formalism corresponding
to Bohm’s model. Although the relativistic case proved to be straightforward, the taking
of the non-relativistic limit was also found to involve subtleties and to require care.
During this procedure, it was necessary to scrutinize the physical interpretation of the
various expressions that arose and to clarify the physical meaning of symmetric and non-
symmetric energy-momentum tensors. The resulting formalism was then found to have
the desired properties for demonstrating conservation.
The construction of a Lagrangian formulation of Bohm’s model necessarily leads to some
modification of the relevant field equation, i.e., of the Schrodinger equation. This has
_Chapter_9.doc
108
resulted in perhaps the most intriguing aspect of the present approach, namely that the
modified Schrodinger equation is found to be automatically of a special form that reduces
back to the standard Schrodinger equation1 in the usual, non-relativistic case of no
creation or annihilation of particles, thereby maintaining compatibility with all the
relevant experimental evidence.
Concerning this restriction to conserved particle number in the non-relativistic case, it
may be observed that the divergences of the various parts of the energy-momentum
tensor become somewhat trivial under this assumption. However, this does not affect the
basic result that energy and momentum conservation have been successfully introduced
into Bohm’s model. Of course, the fact also remains that Bohmian mechanics is not a
widely accepted interpretation of quantum mechanics. Nevertheless, the aim here has
simply been to answer the question of whether this model can be compatible with the
usual conservation laws (this question having been published in several places), not to
argue for the superiority of the model in other ways.
Finally it should also be noted that, from a metaphysical point of view, the present work
establishes that the laws of energy and momentum conservation are quite compatible with
attempts to formulate an interpretation of quantum mechanics incorporating realism.
1 once the Bohmian constraint v = ∇∇∇∇S/m is applied.
Bibliography
130
130
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Appendix 1 (Ch3)
109
Appendix 1: Non Locality
(An appendix to Chapter 3, Section 3.3)
Since it is known that any hidden variable interpretation must incorporate "non-local"
behaviour, we will look at the non-locality question in some detail and then examine
briefly how Bohm's model deals with it.
A1.1 The EPR Paradox
In 1935, Einstein, Podolsky and Rosen published an objection to the Copenhagen
interpretation of quantum mechanics1 in an article which has come to be known as the
"EPR paper”. This paper demonstrated that the completeness of quantum mechanics,
as interpreted by the reigning Copenhagen interpretation, could not be reconciled with
the assumption of locality. The assumption of locality requires that, for any two
particles, the result obtained by performing a measurement on one particle is
independent of the type of measurement (if any) performed on the other particle when
the two measurements events are separated by a space-like interval in space-time. By
producing this demonstration, Einstein and his supporters presented a clear choice
between locality and the assumption that quantum mechanics provided a "complete"
or "sufficient" description of individual quantum entities.
It was the intention of the original EPR program to take locality as given and show
that quantum mechanics could not be a complete theory in terms of describing
individual quantum entities. This was achieved by introducing two essential
definitions and then considering the case of two spatially separated quantum entities
having correlated states. The definitions presented were as follows:
1 Einstein A., Podolsky B. & Rosen N., Can a Quantum Mechanical Description of Reality be
Considered Complete? Physics Review. Vol. 47, p. 777 (1935).
Appendix 1 (Ch3)
110
Definition 1: A necessary condition for a complete theory is that “every element of
physical reality must have a counterpart in the physical theory”.
Definition 2: A sufficient condition for identifying an element of reality is, “If,
without in anyway disturbing a system, we can predict with certainty (i.e., with
probability equal to unity) the value of a physical quantity, then there must exist an
element of reality corresponding to this physical quantity.”
Despite Toulmins comments (reported in Chapter 2) and Born's later assertion that
“the concept of reality is too much connected with emotions to allow a generally
acceptable definition2”, within the context of the definitions used in the EPR paper,
the EPR argument remains valid. In presenting EPR, it was intended that a resolution
to the difficulties established would be obtained by admitting the existence of
additional quantities consistent with quantum mechanics but restoring locality. The
proposed introduction of such hidden values was clearly substantially at variance with
the Copenhagen program. Ballentine3 has concisely summarised Einstein’s
conclusions, and the contribution of the EPR Paper in general, as follows:
“The following two statements are incompatible:
(1) The state vector provides a complete and exhaustive description of an individual
system;
(2) The real physical conditions of spatially separated (non-interacting) objects are
independent.
Of course, one is logically free to accept either one of these statements (or neither).
Einstein clearly accepted the second while Bohr apparently favoured the first. The
importance of the EPR argument is that it proved for the first time that assuming the
2 Born M., Natural Philosophy of Cause and Chance. Oxford University Press. London (1951).
3 Ballentine L.E., The Statistical Interpretation of Quantum Mechanics, Reviews of Modern Physics
Vol. 42, p. 363 (1970).
Appendix 1 (Ch3)
111
first statement above demands rejection of the second, and vice-versa, a fact that was
not at all obvious before 1935, and which may not be universally realised today.”
A1.2 Bells Theorem
In the sometimes unsatisfactory debate that followed the presentation of the 1935 EPR
paper, a very significant mile-stone was the later appearance of "Bell's theorem."
Starting from the EPR argument, Bell went further and demonstrated that no hidden
variable theory which leads to the predictions given by the quantum mechanical
algorithm can be compatible with locality4. Bell achieved this by producing an
inequality which constrains the distribution of measurement results possible for
coincident events in EPR experiments assuming local hidden variables5. In a recent
paper, Cramer6 explains that Bell's inequality deals with the way in which the
coincidence rate R(θθθθ) of an EPR experiment changes as θθθθ starts from zero and
becomes progressively larger. Bell proved mathematically that for all local hidden-
variable theories the rate R(θθθθ) of coincident events in EPR type experiments must
decrease linearly (or less rapidly) as θθθθ increases, i.e., the fastest possible decrease in
R(θθθθ) is proportional to θθθθ. On the other hand quantum mechanics predicts that the
coincidence rate is proportional to cos2θθθθ, so that for small θθθθ it will decrease roughly as
θθθθ2 (since cos2
θθθθ is approximately equal to 1−θθθθ2 for small θθθθ). Therefore, quantum
mechanics and locality require quantitatively different predictions about EPR
measurements.
4 In fact, Bell’s argument has been realised to be even more general than this, as will be discussed
below.5 Bell J.S., Physics Vol.1, p.195 (1964).
6 Cramer J., Quantum Nonlocality and the Possibility of Superluminal Effects. Published in the
Proceedings of the NASA Breakthrough Propulsion Physics Workshop, Cleveland, OH, (August 12-14,
1997). (Also available from http://www.npl.washington.edu/npl/int_rep/qm_nl.html).
Appendix 1 (Ch3)
112
In response to Bells Theorem, experiments using spatially separated but correlated
quantum entities (mainly pairs of photons) have been undertaken7. These experiments
have verified the standard quantum mechanical predictions and, in so doing, have
ruled out local hidden variable theories. It is generally agreed that the non-local
quantum correlations observed are not in direct contradiction with relativity since they
do not permit information transfer between space-like separated points (in particular,
they do not permit faster-than-light signalling).
Unfortunately, the first experimental results from EPR experiments were frequently
incorrectly interpreted as demonstrating the inadmissibility of hidden variable
theories. However, since Bell's theorem assumes only a local hidden variable theory,
the possibility of non-local hidden variable theories satisfying both Bell's theorem and
the experimentally verifiable predictions of quantum mechanics remains open. In
other words, from the viewpoint of hidden variables advocates, Bells theorem and the
subsequent experiments mentioned above have simply demonstrated that locality, not
hidden variable models, conflicts with experiment. Bohm’s theory is an example of a
non-local hidden variable theory that is consistent with the experimentally testable
predictions of quantum mechanics.
A1.3 Counterfactual Definiteness
Later it became clear that the argument was, in fact, even more general and quantum
mechanics cannot even be compatible with both locality and Stapps' assumption of
counterfactual definiteness, where the latter is defined as follows:
7 Freedman and Clauser demonstrated a 6σ violation of Bell's inequality in 1972. Freedman S.J. &
Clauser J.F., Physical Review Letters, Vol. 28, pp. 938-941 (1972).
Aspect et al demonstrated a 46σ violation of Bell's inequality in 1982. Aspect A , Dalibard J. & Roger
G., Physical Review Letters Vol. 49, pp. 91 & 1804 (1982).
Appendix 1 (Ch3)
113
"For each particle on which a measurement is performed, a definite value would have
been found if a different spin component had been measured on it instead (although
we cannot know what the specific value would have been) and, furthermore, the
complete set of such values (measured and unmeasured together) can be meaningfully
discussed".8
A1.4 Bohm's Model and Non-locality
Bohm's model deals with the required non-locality as follows: Since a correlated pair
of particles is described by a single, overall wavefunction, a measurement on one of
the particles must have an effect on the wavefunction description of the other particle.
Then, by the nature of Bohm's model, this also necessarily (and instantaneously)
affects the second particle's hidden position and momentum. Thus Bohm's theory
automatically incorporates an explicit description of the non-locality implied by Bell's
theorem. It does this, however, at the expense of a conflict with the principle of
relativity, albeit a hidden one. Hardy9 has argued that such a hidden conflict with the
equality of all reference frames may be a necessary feature of any hidden variable
model for quantum mechanics.10
A1.5 Kochen and Specker’s Proof
Kochen and Specker presented a proof11 which showed that any hidden variable
theory must also be "contextual," viz, the value obtained by a measurement must
sometimes depend on what other observable happens to be measured at the same time
(i.e., the value obtained depends on the "context"). In other words, the observable
8 Stapp H., p. 637-652 in Symposium on the Foundations of Modern Physics, Edited by Lahti P. &
Mittelstaedt P. World Scientific Publishing Co. (1985).9 Hardy L., Physical Review Letters Vol. 68, p. 2981 (1992); Hardy L. & Squires E.J., Physics Letters
Vol. A168, p. 169 (1992).10
However, the hyperspace Bohm-Dirac model in Durr D., Goldstein S., Munch-Berndl K., et al,
Physical Review A, Vol. 60, pp. 2729-2736 (1999), can be considered a counterexample to this claim.
Appendix 1 (Ch3)
114
values can't all just be pre-existing and waiting to be measured. If observables do have
values before they are measured, then measurements do not in general yield those
values. This is indeed the case for Bohm's model, since the measurement outcomes
obtained don't exist prior to the measurement. Rather, the measured values are created
during the measurement process, i.e., during the gradual spatial separation of the wave
function into non-overlapping wave packets. In the case of momentum, for example,
the measured value replaces the pre-existing value during this time, whilst for other
observables (such as spin in Bell's extension of Bohm's model12) there may be no pre-
existing value at all beforehand.
The Kochen and Specker proof (and those of other people, such as Gleason13) was
rather complicated and Mermin14 has pointed out that Bell's theorem essentially
proves the same thing more simply (in addition to its implications about non-locality).
11
Kochen S. and Specker E.P., Journal of Mathematics and Mechanics Vol. 17, p. 59 (1967).12
Bell J.S., Paper 4 in Speakable and Unspeakable in Quantum Mechanics. Cambridge University
Press (1987).13
Gleason A.M., Journal of Mathematics and Mechanics Vol. 6, p. 885 (1957).14
Mermin N.D., Physical Review Letters Vol. 65, pp. 3373-3376 (1990); Reviews of Modern Physics
Vol. 65, pp. 803-815 (1993).
Appendix_2_(Ch5).doc
115
Appendix 2: Velocity Expression corresponding to the
Modified Schrodinger Equation
(An appendix to Chapter 5, Section 5.3)
It will be shown here that the velocity expression v = ∇∇∇∇S/m corresponding to the current
density of the standard Schrodinger equation remains unchanged in going to the modified
Schrodinger equation [5-23]:
–h
2
2m∇
2ψ – ih∂ tψ = – ih
2ψ*{∇.(ρ∇S
m ) + ∂ tρ } [A2-1]
We will follow the usual steps involved in deriving the Schrodinger continuity equation.
The conjugate equation to [A2-1] is:
–h
2
2m∇2ψ* + ih∂ tψ
* = ih2ψ
{∇.(ρ∇Sm ) + ∂ tρ } [A2-2]
Multiplying [A2-1] by ψ* and [A2-2] by ψ then subtracting the two results, we have
ψ*( –h
2
2m∇
2ψ – ih∂ tψ) – ψ ( –
h2
2m∇
2ψ* + ih∂ tψ
*)
= – ih2
{∇.(ρ∇Sm ) + ∂ tρ } – ih
2{∇.(ρ∇S
m ) + ∂ tρ }
which simplifies to:
ψ* h2im
∇2ψ – ψ h2im
.∇2ψ* + ψ*∂ tψ + ψ∂ tψ* = ∇.(ρ∇S
m ) + ∂ tρ
i.e.,
h2im
∇.(ψ*∇ψ – ψ∇ψ*) + ∂ t(ψ*ψ) = ∇.(
ρ∇Sm ) + ∂ tρ [A2-3]
Expressing the wave function ψ in the form:
ψ = R exp ( iSh
)
equation [A2-3] can be written as:
∇.(R2∇Sm ) + ∂ t(R
2) = ∇.(ρ∇Sm ) + ∂ tρ
Appendix_2_(Ch5).doc
116
which rearranges to:
∇.[(R2– ρ)
∇Sm ] + ∂ t(R
2– ρ) = 0
This can be recognized as a continuity equation containing a current density (R2
– ρ) and
a flow velocity ∇Sm . Now, although the current density expression is different from the
usual Schrodinger one, the velocity expression is the same. It has therefore been shown
that our modified Schrodinger equation yields the same velocity expression as does the
standard Schrodinger equation.
Appendix_3_(Ch6).doc
117
Appendix 3: Rate of Change of a Particle's Energyin a Scalar Field
(An appendix to Chapter 6, Section 6.3.1)
Equation [6-12] will be derived here. Starting from the fact that total energy equals
kinetic energy plus potential energy, we can write:
dE particle
dt= d
dt(KE particle + PE particle )
= ddt
(p 2
2m+ qφ)
= 12m
dp i
dtd
dp i(–p jp
j) + q(dx0
i
dt
∂ φ(x0)
∂x0i
+∂ φ(x0)
∂t)
= – 12m
dp i
dt(g ijp
j + p jδ i
j) + q(
p i
m ∂ iφ +∂φ
∂ t)
= – 1m
dp i
dtp i + q(
p i
m ∂ iφ +∂φ
∂ t)
Therefore, using equation [6-11], we have:
dE particle
dt= – 1
m (q∂ iφ)p i + q(p i
m ∂ iφ +∂φ
∂ t)
and cancelling yields:
dE particle
dt= q
∂φ
∂ t
which is equation [6-12].
Appendix_4_(Ch6).doc
118
Appendix 4: Schrodinger Energy-Momentum Tensor
(An appendix to Chapter 6, Section 6.4)
To derive the various parts of the energy-momentum tensor corresponding to the free-
field portion of our Lagrangian density [5-1] (i.e., corresponding to the standard
Schrodinger equation), we will apply the formula [6-23]:
Tfieldµν = [ ∂µψ ∂
∂(∂νψ)+ ∂µψ* ∂
∂(∂νψ*)
– gµν ]
Appendix_4_(Ch6).doc
119
(iii)
Tfield
0i = [ ∂ tψ ∂∂(∂ iψ)
+ ∂ tψ* ∂∂(∂ iψ
*)– g0i ]
Appendix_5_(Ch6).doc
120
Appendix 5: Conservation Difficulty with the SchrodingerEnergy-Momentum Tensor
(An appendix to Chapter 6, Section 6.4)
It will be shown here that the energy-momentum tensor in equations [6-24] is not
consistent with the conservation condition [6-26].
Inserting expressions [6-24a] and [6-24b]:
Tfieldij
=h
2
2m{ (∂
iψ) (∂
jψ
∗) + (∂
iψ
*) (∂
jψ) – g ij (∂kψ
∗) (∂
kψ) }
– g ij i h2
(ψ∗
∂ tψ – ψ ∂ tψ∗)
Tfield
i0= i h
2{ ψ
∗∂
iψ – ψ ∂
iψ
*}
into the left hand side of [6-26]:
∂ j Tfieldij
+ ∂ t Tfieldi0
= – ρdp i
dt
we obtain:
∂ j Tfieldij
+ ∂ t Tfieldi0
= ∂ j [h
2
2m{ (∂
iψ) (∂
jψ
∗) + (∂
iψ
*) (∂
jψ) – g ij (∂kψ
∗) (∂
kψ) }
– g ij i h2
(ψ∗
∂ tψ – ψ ∂ tψ∗) ] + ∂ t [ i h
2{ ψ
∗∂
iψ – ψ ∂
iψ
*} ]
=h
2
2m{ (∂ j∂
iψ) (∂
jψ
∗) + (∂
iψ) (∂ j∂
jψ
∗)
+ (∂ j∂iψ
*) (∂
jψ) + (∂
iψ
*) (∂ j∂
jψ)
– (∂i∂kψ
∗) (∂
kψ) – (∂kψ
∗) (∂
i∂
kψ) }
+ i h2
{ – (∂iψ
∗) (∂ tψ) – ψ
∗∂
i∂ t ψ)
+ (∂iψ) ∂ tψ
∗) + ψ ∂
i∂ tψ
∗
+ (∂ tψ∗) (∂
iψ) + ψ
∗∂ t∂
iψ
– (∂ tψ) (∂iψ
*) + ψ ∂ t∂
iψ
*}
which cancels to:
∂ j Tfieldij
+ ∂ t Tfieldi0
=h
2
2m{ (∂ iψ) (∂ j∂
jψ∗) + (∂ iψ
*) (∂ j∂
jψ) }
+ ih { – (∂ iψ∗) (∂ tψ) + (∂ iψ) ∂ tψ
∗) }
= (∂ iψ*) {
h2
2m∂ j∂
jψ – ih∂ tψ } + (∂ iψ) {h
2
2m∂ j∂
jψ*
+ ih∂ tψ*
}
= (∂ iψ*) { –
h2
2m∇2ψ – ih∂ tψ } + (∂ iψ) { –
h2
2m∇2ψ
*+ ih∂ tψ
*}
[A5-1]
Appendix_5_(Ch6).doc
121
Hence, using the modified Schrodinger equation [5-23]:
–h
2
2m∇2ψ – ih∂ tψ = – ih
2ψ*{∇.(ρ∇S
m ) + ∂ tρ }
and its complex conjugate:
–h
2
2m∇
2ψ* + ih∂ tψ
* = ih2ψ
{∇.(ρ∇Sm ) + ∂ tρ }
equation [A5-1] becomes:
∂ j Tfield
ij+ ∂ t Tfield
i0= (∂
iψ
*) { – ih
2ψ* {∇.(ρ∇Sm ) + ∂ tρ } + (∂
iψ) { ih
2ψ{∇.(ρ∇S
m ) + ∂ tρ } }
This can be written more compactly as:
∂ j Tfield
ij+ ∂ t Tfield
i0= ih
2(
∂ iψψ –
∂ iψ*
ψ*) {∇.(ρ∇S
m ) + ∂ tρ }
and using the identity [5-14]:
∂jS = h
2i[∂
jψ
ψ –∂
jψ*
ψ* ]
we finally have:
∂ j Tfieldij
+ ∂ t Tfieldi0
= – (∂iS) {∇.(ρ∇S
m ) + ∂ tρ } [A5-2]
Now, identifying ∇Sm as the velocity of the Bohmian particle, the curly bracket is
recognized as corresponding to the continuity equation describing the conservation of the
matter making up this particle. This allows us to demonstrate that [A5-2] is not consistent
with [6-26] by considering the usual, non-relativistic case of no particle creation or
annihilation. In this case the curly bracket is zero and so equation [A5-2] reduces simply
to
∂ j Tfield
ij+ ∂ t Tfield
i0= 0
This result prohibits exchanges of energy and momentum between the field and particle
and so is not compatible with equation [6-26].
Appendix_6_(Ch7).doc
122
Appendix 6: Viability of a Scalar Potential Description withde Broglie’s Relativistic Model
(An appendix to Chapter 7, Section 7.1)
In this appendix it will be shown that the basic definition [7-3]:
pµ = – ∂µS [A6-1]
of de Broglie’s model is compatible with the idea of motion under a scalar potential once we
impose the condition that the particle has a variable mass given by [7-7]:
M = 1c (∂µS) (∂
µS) [A6-2]
Taking the τ derivative of equation [A6-1], we obtain:
dpµ
dτ= –
d(∂µS)
dτ
= –dxν
dτ
∂(∂µS)
∂xν
= – uν ∂ν∂µS
= –pν
M∂µ∂νS
= 1M
(∂νS)∂µ(∂νS)
Hence, inserting the definition [A6-2] for M, we have:
dpµ
dτ=
c
(∂λS) (∂λS)
(∂νS)∂µ(∂νS)
= ∂µ [ c (∂νS) (∂νS) ]
and using the definition [7-4] for the quantum potential:
Q = c (∂µS) (∂µS) – mc2
we then arrive at the following equation:
dpµ
dτ= ∂µQ
In other words, the particle’s equation of motion is then the relativistic version of "rate of change
of momentum equals gradient of scalar potential", as required.
Appendix_7_(Ch7).doc
123
Appendix 7: Relativistic Equation of Motion
(An appendix to Chapter 7, Sections 7.2 and 7.3)
A7.1 Derivation from the Relativistic Lagrangian Density
It will be shown here that the Lagrangian density [7-8] yields the correct equation of
motion [7-5] for the particle. The action function corresponding to this L will have the
form:
action =
= 1c
[A7-2]
Rather than going right back to first principles, it is simpler to perform our derivation via
a relativistic version of Lagrange’s equation. The appropriate generalisation of the non-
relativistic equation [4-3] is1:
ddτ
∂L∂uµ = ∂L
∂x0µ
[A7-3]
This equation highlights a further consideration. We actually need a Lagrangian L to
insert into this equation, not a Lagrangian density L. Now, the partial action in [A7-2] is
related to the required Lagrangian L via2:
action
Appendix_7_(Ch7).doc
124
both the action and L being Lorentz scalar invariants. Using the 4-velocity definition:
u0 =dx0
dτ
expression [A7-2] can also be written in the form:
action =u0
c (
Appendix_7_(Ch7).doc
125
we obtain the equation of motion:
d(Muµ)
dτ=
∂Q
∂x0µ
or, equivalently:
dpµ
dτ= ∂µQ
in agreement with the expected result [7-5].
A7.2 Consistency of the Equation of Motion with the Identity uµµµµuµµµµ = c2
In deriving the equation of motion for the particle, the restriction uµuµ = c2 is temporarily
suspended until after the variation process has been performed3. We will now carry out a
standard check that the resultant equation of motion is then consistent with the identity
uµuµ = c2 without any unwanted restrictions arising. For this purpose, it is most convenient
to use the form shown in equation [A7-6]. Introducing uµ on both sides of [A7-6], we
obtain
uµ ddτ
[ (m +Q
c2) uµ ] = uµ ∂Q
∂x0µ
which can be written as:
uµuµddτ
(m +Q
c2) + (m +Q
c2) uµduµ
dτ=
dQ
dτ
i.e.,
uµuµddτ
(Q
c2) + (m +Q
c2) ½d(uµuµ)
dτ=
dQ
dτ
Using the identity uµuµ = c2 we then have:
c2 ddτ
(Q
c2) + (m +Q
c2) ½d(c2)
dτ=
dQ
dτ
i.e.,
dQ
dτ+ 0 =
dQ
dτ[A7-7]
Appendix_7_(Ch7).doc
126
and the fact that an identity has been obtained without imposing any extra assumption
establishes the desired degree of consistency.
The need for us to do the above check can be seen by considering the discussion prior to
equations [7-9] and [7-10]. If one chooses not to include the factor of uµuµ in the
interaction term of the Lagrangian density [7-8], it is easily shown that the following
equation of motion is obtained for the particle instead:
d(muµ)
dτ=
∂Q
∂x0µ [A7-8]
Repeating the above consistency check by introducing uµ on both sides of this new
equation, it is then found that the strong condition:
dQ
dτ= 0
is deduced instead of the simple identity [A7-7]. Hence choosing the alternative equation
of motion [A7-8] and its corresponding Lagrangian density would lead to an
unacceptable restriction on the form of the potential Q.
3 See, e.g., p. 329 in Goldstein H., Classical Mechanics, 2
nd Ed. Addison-Wesley, Massachusetts (1980).
Appendix_8_(Ch7).doc
127
Appendix 8: Modified Klein-Gordon Equation
(An appendix to Chapter 7, Section 7.4)
The extra term that is added to the Klein-Gordon equation by the interaction part of our
relativistic Lagrangian density will be deduced here by inserting [7-15]:�
��*)
– ∂
*
Appendix_8_(Ch7).doc
128
Applying the identity:
∂(∂µφ*)
∂(∂νφ*)
≡ gµν
we then have:
∂µ
∂
Appendix_8_(Ch7).doc
129
This is the extra term to be added to the Klein-Gordon equation. (Note that, if desired,
this expression can be written in terms of φ and φ*, instead of S, by employing [A8-3].)
In analogy to the non-relativistic analysis in Appendix 2, it is easily shown that the
modified Klein-Gordon equation still yields the same expression for the particle’s
velocity as does the standard Klein-Gordon equation. This means that all the formalism
earlier in chapter 7 remains valid, despite the extra term derived above.
