Quantum Fluctuations From a Local-causal Information Dynamics - Agung Budiyono

8/13/2019 Quantum Fluctuations From a Local-causal Information Dynamics - Agung Budiyono

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arXiv:1312.3366v1

[quant-ph]11Dec2013

Quantum fluctuations from a local-causal information dynamics

Agung Budiyono

Jalan Emas 772 Growong Lor RT 04 RW 02 Juwana,

Pati, 59185 Jawa Tengah, Indonesia

(Dated: December 13, 2013)

Abstract

We shall show that the abstract and formal rules which govern the quantum kinematic and

dynamics can be derived from a law of change of the information content or the degree of uncer-

tainty that the system has a certain configuration in a microscopic time scale, which is singled

out uniquely, up to a free parameter, by imposing the condition of Macroscopic Classicality and

the principle of Locality. Unlike standard quantum mechanics, however, the system always has

a definite configuration all the time as in classical mechanics, following a continuous trajectory

fluctuating randomly in time. Moreover, we shall show that the average of the relevant physical

quantities over the distribution of the configuration is equal to the quantum mechanical average of

the corresponding quantum mechanical Hermitian operators over a quantum state.

PACS numbers: 03.65.Ta; 03.65.Ud; 05.20.Gg

Keywords: Reconstruction of quantum mechanics; Physical origin of quantum fluctuations; Informationdynamics; Principle of Locality; Macroscopic Classicality

Electronic address: [email protected]

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I. MOTIVATION

The violation of Bell inequality by quantum mechanics is widely believed to lead to a

bizarre conclusion that quantum mechanics allows the statistical results of a pair of mea-

surement events spacelike separated from each other to have a stronger correlation than that

is allowed by any local-causal theory [13]. The nonlocal correlation has been claimed to

be verified in numerous experiments[416], in spite of the fact that no experiment hitherto

conducted is free from loopholes[17,18]. Such a nonlocality prima facie contradicts the spirit

of the special theory of relativity which presumes a finite maximum velocity of interaction

given by the velocity of light in vacuum. Further careful investigation however showed that

the quantum mechanical nonlocal correlation can not be exploited by one party to influence

the statistical results of measurement performed by the other distantly separated party, thusprohibits signaling, in accord with the assertion of the special theory of relativity [1923].

The co-existence of nonlocal correlation and no-signaling in quantum mechanics has in-

spired some authors to ask if quantum mechanics can be derived from a certain balance

between some kind of nonlocality and the principle of no-signaling [2428]. While it is

shown that the constraints put by the nonlocal correlation and no-signaling are notsuffi-

ciently strong to single out quantum mechanics [2628], it has renewed an interest in an

approach to clarify the meaning of quantum mechanics by deriving its formal mathematical

structures and numerous abstract postulates from a set of conceptually simple and phys-

ically transparent axioms. In such a program, one attempts to directly answer the most

tantalizing foundational question: why the quantum? [29]. One of the advantages of

the program to reconstruct quantum mechanics is that it might provide physical insights for

possible natural extensions of quantum mechanics either by modifying the axioms or varying

the free parameters that are left unfixed by the axioms. Extension of quantum mechanics

is not only necessary to set up precision tests against quantum mechanics, but might turn

out to be the necessary step to solve some of the foundational problems of the latter.

A lot of works along this line has been reported recently by regarding information as the

basic ingredient of Natural phenomena[3047] : all things physical is information-theoretic

in origin thus It from Bit [29]. In those works, one searches for a set of basic features of

information processingwhich can be promoted as axioms to reconstruct quantum mechanics.

Such an approach is partly motivated by the advancement of quantum information science

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[48]: the fact that quantum mechanics allows information processing tasks that can not be

performed at least as efficiently within classical mechanics suggests an intimate relationship

between the the foundation of quantum mechanics and the basic features of information

processing. This approach thus plays with information within operational-instrumentalist

theoretical framework in which the notion of preparation and measurement play central

role. Another different way to reconstruct quantum mechanics is to assume that quantum

fluctuations is objectively real thus should be properly modeled by a stochastic processes.

A lot of efforts have been made within this realist theoretical framework to derive the

Schrodinger equation from a stochastic processes[4962]. The greatest challenge of such an

approach is how to explain the nonlocal correlation predicted by quantum mechanics.

In the present paper, we shall follow the above second point of view. We shall first

propose a statistical model of stochastic deviation from classical mechanics in microscopic

regime based on a stochastic fluctuations of infinitesimal stationary action. We shall then

show that the abstract and strange [63] rules of quantization of classical systems can

be derived from a specific law of infinitesimal change of the information content or the

degree of uncertainty that the system hasa certain configuration along an infinitesimally

short path, induced by the stochastic fluctuations of the infinitesimal stationary action.

This law for the dynamics of information is shown to be singled out uniquely, up to a free

parameter, by imposing the condition of Macroscopic Classicality and the principle of Local-Causality. Note that here, as will be detailed later, information is used to quantify an actual

degree of of uncertainty referring directly to an event regardless of measurement. It is then

imperative to ask: how to explain the violation of Bell inequality in experiments? Putting

the problem aside, we will show that the local-causal statistical model thus developed leads

to the derivation the linear Schrodinger equation with Borns statistical interpretation of

wave function and quantum mechanical uncertainty relation, two of the cornerstones of

standard quantum mechanics.

We shall thus argue that quantization is physical and Planck constant acquires physical

interpretation as a statistical average of the stochastic deviation from classical mechanics in

a microscopic time scale. Two concrete examples of the application of the statistical model

will be given. In the main text, we shall apply the model to quantize a system of particles

subjected to external potentials and in the appendix we shall apply the model to quantize

a classical mechanical model of measurement of physical quantities. Both reproduce the

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prediction of the standard canonical quantization. Unlike the latter, however, the system

always has a definite configuration all the time as in classical mechanics, fluctuating ran-

domly with time. We shall also show that the average of the relevant physical quantities

over the distribution of the configuration is equal to the quantum mechanical average of

quantum mechanical Hermitian operators corresponding to the physical quantities over a

wave function.

II. A STATISTICAL MODEL OF MICROSCOPIC RANDOMNESS AND THE

DYNAMICS OF UNCERTAINTY OBEYING THE PRINCIPLE OF LOCALITY

A. A class of statistical models of microscopic stochastic deviation from classical

mechanics

There is a wealth of empirical evidences that phenomena in microscopic regime involve a

universal stochastic element. Yet, unlike the Brownian motion, hitherto there is no consensus

on the nature and origin of its randomness. Moreover, the prediction of quantum mechanics

on the AB (Aharonov-Bohm) effect [64] and its experimental verification [65] suggest that

the randomness in microscopic regime can not be adequately described by introducing some

kinds of conventional random forces as in Brownian motion. The force has to act at a

distance.

To discuss the universal randomness in microscopic regime, let us consider the following

class of statistical models. Letqdenotes the configuration of the system and tis time param-

eterizing the evolution of the system. Let us assume that the Lagrangian is parameterized

by a random variable fluctuating in a microscopic time scale dt, whose origin is not our

present concern: L= L(q, q; ), where q .= dq/dt. Let us then consider two infinitesimally

close spacetime points (q; t) and (q+ dq; t+ dt) such that is constant. Let us assume

that fixing , the principle of stationary action is valid to select a path, denoted byJ(),that connects the two points. One must then solve a variational problem (Ldt) = 0 with

fixed end points. This leads to the existence of a function, the Hamiltons principal function

denoted byA(q; t, ), whose differential along the segment of trajectory is given by [66], for

a fixed ,

dA= Ldt = p dq Hdt, (1)

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where p(q) = L/q is the classical momentum and H(q, p) .= p q(p) L(q, q(p)) is the

classical Hamiltonian. The above relation implies the following Hamilton-Jacobi equation:

p= qA,

H(q, p) =tA. (2)

Varying, the principle of stationary action will therefore pick up various different paths

J(), all connecting the same two infinitesimally close spacetime points, each might havedifferent values of infinitesimal stationary actiondA(). dA() thus is randomly fluctuating

due to the random fluctuations of. The system starting with a configuration qat time t

may therefore take various different paths randomly to end up with a configuration q+dq

at time t+dt. We have thus a stochastic processes driven by the random fluctuations of

in a microscopic time scale. Hence a complete description of a single event is impossible.

Instead, one has to rely on a statistical approach.

One can see that the randomness enters into the dynamics in a microscopic time scale

in a fundamentally different way from that of the Brownian motion. In the model, it is

the infinitesimal stationary action that is randomly fluctuating in a microscopic time scale.

By contrast, the randomness in the Brownian motion is induced by some random forces.

We have thus assumed that the Lagrangian schema based on energies is more fundamental

than the Newtonian schema based on forces. We expect that this will lead to a local-causal

explanation of the AB effect. To see another implication of such a difference, let us consider

a compound composed of two interacting subsystems. Within the formalism of Brownian

motion, it is then possible to introduce a joint-probability for two random forces each act-

ing locally to a subsystem. By contrast, since action is evaluated in configuration space

rather than in ordinary space, then in the statistical model based on a random fluctua-

tions of infinitesimal stationary action, one can notdefine a joint-probability density for the

fluctuations of infinitesimal stationary action of each subsystem.We have thus a class of stochastic models which differ fundamentally from the conven-

tional Brownian motion. In the following subsections, we shall select one of them by imposing

the constraints that the statistical model has a smooth classical limit in macroscopic regime

and respects the principle of Locality demanded by the theory of relativity.

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B. The dynamics of information or uncertainty with a smooth Macroscopic

Classicality

To develop a statistical description of the stochastic processes, let us denote the joint-

probability density that at time t the configuration of the system is qand a random value of

is realized as (q, ; t). We would like to find an equation which describes how changes

along an infinitesimally short trajectoryJ(). To do this, instead of working directly with, we shall below consider a quantity defined as

I(q; t, ) .= ln(q, ; t). (3)

This quantity is introduced by Shannon as a measure of information content or the degree

of uncertainty of an event. Within the context of the stochastic model under study, fixing

, it is the information content or the degree of uncertainty that the configuration of the

system isqfor the following intuitive reasons: i) it is vanishing if the system definitely has

a configuration qso that (q) = 1; ii) it is increasing monotonically as the probability that

the system has a configuration qis decreasing and iii) it is additive for independent events.

Let us first note that the information content or the degree of uncertainty defined above

is objective referring directly to the configuration, thus the factual state, of the system. It

isnot

the information that one obtains by performing some measurements over the systemof interest. Hence, we shall in the present paper work with information within a realist

rather than instrumentalist-operational theoretical model. The latter approach is however

followed by most works in the reconstruction of quantum mechanics based on information

theory, which is apparently motivated by the central role of measurement in the formalism

of standard quantum mechanics. Moreover, let us note that the information quantified by

I(q) refers to a single event that the system has a particular configuration q, rather than

the whole possible events of the system distributed according to (q). The information with

regard to the whole possible events is usually quantified by the average ofI(q) given by the

Gibbs-Shannon entropy which is central in information theory[67].

The interpretation of I(q) as the amount of information or degree of uncertainty that

the system has a configuration qmay also be argued within the concept of microcanonical

ensemble as follows. First, given the parameters of the system, let N denotes the total

number of the microstates accessible by the system. Let us assume that the system may be in

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one of the microstates equally probably. Let us then assume that qis a macroscopic coarse-

grained variable of the microstates. LetNq denotes the number of microstates compatible

with q. The probability that the system has a configuration q is thus given by =Nq/N.

One therefore has ln N = I(q) + ln Nq. Interpreting ln Nas the amount of information or

uncertainty that the system lies in one of theNpossible microstates, and ln Nqas the amount

of information or uncertainty that the system lies in one of the Nq microstates compatible

with q, then it is natural to interpret I(q) as the amount of information or uncertainty that

the system has a configuration q.

To avoid confusion, below we shall only use the term uncertainty to refer to information

content or degree of uncertainty of an event. Let us proceed to again consider two infinites-

imally close spacetime points (q; t) and (q+ dq; t+dt) such that is constant, connected

by an infinitesimally short path J(). Let us then assume that as the configuration evolvesalong J(), the uncertainty that the system has a configuration qalso changes according tothe following balance equation:

dI(q; t, ) = d ln(q, ; t) =(q; t, )dt, (4)

where is a function ofq, and t. Our main goal in the present section is then to find a

unique functional form ofand express it in terms of the physical properties of the system,

by imposing a set of conceptually simple and physically transparent axioms.First, it is instructive to impose the condition of Macroscopic Classicality which demands

that in a physical regime corresponding to macroscopic world, one should regain the clas-

sical mechanics. Since the deviation from classical mechanics, as assumed in the previous

subsection, is due to the fluctuations of infinitesimal stationary action induced by the fluc-

tuations of, then in the classical limit of macroscopic regime, such fluctuations must be

ignorable. In the macroscopic regime, one must therefore regain the dynamics of ensemble

of classical trajectories driven by the deterministic flow of classical velocity field. The in-

finitesimal change of the uncertainty must in this case solely be given by the flux due to the

deterministic classical velocity field. Notice then that the uncertainty should increase if the

velocity divergence along the infinitesimally short trajectory J() is positive and vice versa.On the other hand, since the system under consideration is closed, then probability has to

be conserved. These two conditions combined suggest that in the macroscopic regime whose

mathematical formulation will be clarified later, on the right hand side of Eq. (4) must

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reduce to

q vc= c, (5)

where vc is the classical velocity field which is related to the classical Hamiltonian and

the Hamiltons principal function via the kinematic part of the Hamilton equation and theHamilton-Jacobi equation of (2) as

vc=H

p

p=qA

. (6)

Indeed, inserting Eq. (5) into Eq. (4), dividing both sides bydt and taking the limitdt 0,one obtains the continuity equation

t +q

vc

= 0, (7)

which guarantees the conservation of probability.

Hence the demand of Macroscopic Classicality suggests that the right hand side of Eq.

(4) should be given by the following terms:

dI(q; t, ) = (q; t, )dt+(q; t, ),

with .=q v, (8)

where v is a velocity field which in the classical limit of macroscopic regime must approach

vc

v vc, (9)

and is a function ofq,and t which must be vanishing in the classical limit

(q; t, ) 0. (10)

(q; t, ) may thus be regarded as the rate of production of uncertainty along the random

pathJ() due to the fluctuations of in microscopic regime.From the discussion above, especially the demand thatmust be vanishing in the classical

limit, it is then natural to assume that is a function of a quantity that measures the

deviation from classical mechanics in microscopic regime due to the fluctuations of. To

identify such a quantity, let us first assume that is the simplest random variable with two

possible values, a binary random variable. Without losing generality let us assume that the

two possible values ofdiffer from each other only by their signs, namely one is the opposite

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of the other, =||. Suppose that both realizations of lead to the same path so thatdA() =dA(). Since the stationary action principle is valid for both values of, thensuch a model must recover the classical mechanics. Hence, the non-classical behavior must

be measured by the difference ofdA() at

|

|,dA()

dA(

).

Now let us proceed to assume that may take continuous values. Let us assume that

even in this case the difference of the values ofdA at,

Z(q; t, ) .=dA(q; t, ) dA(q; t, ) = Z(q; t, ), (11)

measures the non-classical behavior of the stochastic processes, namely the larger the dif-

ference, the stronger is the deviation from classical mechanics. (q; t, ) should therefore be

a function ofZ(q; t, ):

= Z(q; t, ). (12)For later purpose, let us introduce a new stochastic quantity S(q; t, ) so that the differ-

ential along the segment of pathJ() is given by

dS(q; t, ) =dA(q; t, ) +dA(q; t, )

2 =dS(q; t, ). (13)

SubtractingdA(q; t, ) from both sides, one gets

dS(q; t, ) dA(q; t, ) =dA(q; t,

)

dA(q; t, )

2

= Z(q; t, )/2. (14)

of Eq. (12) may thus be written as a function ofdS() dA()

=

dS() dA(). (15)Note that the assumed universality of the law of physics demands that the functional form

of must be independent from the details of the system of interest: the number of particles,

masses, etc. It must only depend on dS dA.Let us then express the condition of macroscopic classicality of Eqs. (9) and (10) in term

ofSdefined above. Let us first assume that the sign ofis fluctuating randomly in a time

scaledt. Let us then denote the time scale for the fluctuations of|| as, and assume thatit is much larger than dt:

dt. (16)

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Within a time interval of length , the magnitude of is thus effectively constant while its

sign fluctuates randomly. In order for the stochastic system to have a smooth classical limit

for all time, then it is necessary that the classical mechanics is recovered in a time interval

of length during which the magnitude of is effectively constant while its sign fluctuates

randomly. As discussed above, for this binary random variable, the classicality is regained

when dA() =dA(). In this case, one also has dS() =dA() by the virtue of Eq. (14),so that due to Eq. (1),Ssatisfies the Hamilton-Jacobi equation of (2). Taking into account

this fact, first, the condition of Macroscopic Classicality of Eq. (10) should be rewritten as

limdSdA

(dS dA) = 0. (17)

Moreover, the condition of Macroscopic Classicality of Eq. (9) is attained by assuming that

v in Eq. (8) is related to Sas follows

v=H

p

p=qS

. (18)

One can see that in the limit dS dAone hasqS qAso thatv vc as expected. Letus emphasize that the above condition is sufficient to recover the classical mechanics only

within the time interval of length in which|| is constant. While it is also a necessarycondition to recover the classical dynamics for the whole time, it is not sufficient. One needs

to have more conditions to recover classical mechanics for the whole time. This problem willbe discussed later.

C. An infinitesimal change of uncertainty respecting the principle of Locality

Let us then proceed to show that imposing the principle of Locality will pick up uniquely,

up to a free parameter, the functional form of (dS dA). To do this, let us consider acompound of two subsystems, say two particles whose configuration are denoted respectively

byq1 and q2. Let us assume that they are spacelike separated from each other so that due

to the principle of Locality, there is no mechanical interaction between the two particles.

The total Lagrangian is thus decomposable as L(q1, q2, q1, q2) = L1(q1, q1) + L2(q2, q2) and

accordingly, dA(q1, q2) anddS(q1, q2) are also decomposable: dA(q1, q2) =dA1(q1) + dA2(q2)

and dS(q1, q2) =dS1(q1) +dS2(q2). can thus be written as

(dS dA) = (dS1 dA1) + (dS2 dA2). (19)10


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Further, in this case, the classical Hamiltonian is also decomposable H(q1, q2, p1, p2) =

H1(q1, p1) +H2(q2, p2), where pi, i = 1, 2, is the momentum of iparticle. Putting thisinto Eq. (18) and recalling that dS is decomposable, then defined in Eq. (8) is also

decomposable

(q1, q2) = 1(q1) +2(q2). (20)

The change of the uncertainty that the compound system has a configuration q = (q1, q2)

moving along the pathJ() thus reads, by the virtue of Eq. (19) and (20),

dI(q1, q2) =

1+2

dt+

(dS1 dA1) + (dS2 dA2)

. (21)

On the other hand, since the two subsystems are spacelike separated from each other,

the principle of Locality demands that the change of the uncertainty that the first (second)

subsystem has a configurationq1 (q2), when the compound system moves along an infinitesi-

mally short trajectory J(), must be independent from what happens with the second (first)subsystem. Otherwise, the uncertainty that one subsystem has a certain configuration can

be influenced by the state of the other distantly separated subsystem by varying the control

parameters of the latter despite of no interaction. Hence, the change of the uncertainty that

each subsystem has a certain configuration must only depend on the corresponding single

particle Lagrangian. One therefore has the following pair of decoupled relations:

dI1(q1) = 1dt+(dS1 dA1),dI2(q2) =2dt+(dS2 dA2), (22)

where dIi =d(lni), and i(qi), i= 1, 2, is the probability density for the configurationof the iparticle. dIi is thus the change of the uncertainty that the isubsystem has aconfiguration qi. Let us note again that the assumed universality of the law of physics

demands that the functional form of for the whole compound system on the right hand

side of Eq. (21) must be the same as those for each subsystem on the right hand side of Eq.

(22).

Let us first assume that the probability distribution of the configuration of the com-

pound system is separable: (q1, q2) = 1(q1)2(q2). In this case, the total change of the

uncertainty that the compound system has a configuration q = (q1, q2) as the configura-

tion evolves alongJ() is then decomposable as dI(q1, q2) = dI1(q1) +dI2(q2). Now let us

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consider a general case when the distribution of the configuration of the two spacelike sepa-

rated subsystems are correlated. One thus has (q1, q2) = 12(q1|q2)2(q2), where 12(q1|q2)is the conditional probability that the configuration of the first subsystem is q1 when the

configuration of the second subsystem is q2. As the configuration of the compound system

evolves along an infinitesimally short pathJ(), the total change of the uncertainty thatthe compound system has a configurationq= (q1, q2) is then

dI(q1, q2) =dI12(q1|q2) +dI2(q2), (23)

where dI12= d ln 12 is the infinitesimal change of the uncertainty that the configurationof the first subsystem is q1 when the configuration of the second subsystem is q2. The

principle of Locality however demands that, since the two subsystems are spacelike separated

from each other, the infinitesimal change of the uncertainty that the first subsystem has aconfiguration q1 must be independent from the configuration of the second subsystem q2.

One must thus have dI12(q1|q2) =dI1(q1). Inserting into Eq. (23), one therefore concludesthat in general the total infinitesimal change of the uncertainty that the two non-interacting

subsystems have a configuration q= (q1, q2) is decomposable as

dI(q1, q2) =dI1(q1) +dI2(q2). (24)

Finally inserting Eqs. (21) and (22) into Eq. (24), in general(dSdA) must also satisfythe following decomposability condition:

(dS1 dA1) + (dS2 dA2)

= (dS1 dA1) +(dS2 dA2). (25)

The above functional equation together with the necessary condition for Macroscopic Clas-

sicality of Eq. (17) can then be solved to give the following linear solution:

(dS dA) = (; t)(dS dA), (26)

where is a real-valued function independent from dS dA, yet might depend on t and,hence is randomly fluctuating within a microscopic time scale. Let us emphasize that Eq.

(26) now applies for general cases, not only for a compound of non-interacting subsystems.

For the reason that will be clear later, let us introduce a new non-vanishing random vari-

able(; t) = 2/. The change of the uncertainty that the system has a certain configuration

along an infinitesimally short pathJ() of Eq. (4) is thus given by

dI= d ln =dt +2

(dS dA). (27)

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Further, since is fixed during the time interval dt, one can expand all the differentials as

dF =tF dt+qF dq to have the following pair of coupled partial differential equations:

qln = 2

qSp(q)

,

tln = 2H(q, p) +tS +(S), (28)

where we have made use of Eq. (1). The spatial and temporal changes of the uncertainty

that the system has a certain configuration are thus related to the momentum and energy

of the system, respectively. Let us emphasize that the above pair of equations are valid only

for a time interval in which is constant.

For later convenient let us write the pair of coupled equations of Eq. (28) as follows:

p(q) = qS+ 2

q

,

H(q, p) = tS+ 2

t

+

2(S), (29)

where the momentum and energy are put on the left hand side. The above pair of relations

must notbe interpreted that the momentum or velocity and energy of the system are deter-

minedcausallyby the change of the uncertainty, which is physically absurd. Rather both the

momentum and energy provide the source of change of the uncertainty that the system has

a certain configuration along an infinitesimally short trajectory J() as shown explicitly byEq. (27). Further, it is evident that in the formal limit 0 whose physical meaning willbe clarified in the next subsection, Eq. (29) reduces back to the Hamilton-Jacobi equation

of (2). In this sense, Eq. (29) can be regarded as a generalization of the Hamilton-Jacobi

equation. Unlike the Hamilton-Jacobi equation in which we have a single unknown function

A, however, to calculate the velocity or momentum and energy, one now needs a pair of

unknown functions Sand .

D. A stochastic processes with a transition probability given by an exponential

distribution of deviation from infinitesimal stationary action

We have started from a stochastic processes in which the system with a configuration

qat time t can take one of many possible random pathsJ() selected by the principle ofstationary action with different random values of, to end up with a configurationq+ dqat

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timet+dt. We then derived a law of infinitesimal change of the uncertainty of an event along

an infinitesimally short path by imposing the condition of Macroscopic Classicality and the

principle of Locality. It is then tempting to investigate if the law of change of uncertainty

given by Eq. (27) completely determines the stochastic processes. To see this, fixing , let

{(q+ dq; t+dt), (q; t)}J() denotes the conditional joint-probability density that theconfiguration of the system is q at time t, tracing the trajectoryJ() and end up with aconfiguration q+dqat time d+dt. Using this quantity, the change of probability density

d due to the transport along the pathJ() is given by

d(q, ; t) = {(q+dq; t+dt), (q; t)}J() (q, ; t). (30)

Inserting into Eq. (27) one therefore has

{(q+dq; t+dt), (q; t)}J()

=

1 (S)dt 2

(dS dA)(q, ; t). (31)Let us then consider the case when |(dSdA)/| 1. Equation (31) can then be written

approximately as

{(q+dq; t+dt), (q; t)}

J()

e

2(dS()dA())(S)dt

(q, ; t). (32)

The above relation can obviously be read within the conventional probability theory as

follows: the joint-probability density that the system initially at (q; t) traces the segment

of trajectoryJ() and end up at (q+dq; t+dt), {(q+dq; t+dt), (q; t)}J(), is equalto the probability that the configuration of the system is qat time t, (q, ; t), multiplied

by a transition probability between the two infinitesimally close spacetime points via the

segment of trajectoryJ() given by

P((q+dq; t+dt)|{J(), (q; t)}) e 2 (dS()dA())/Z, (33)

whereZ= exp((S)dt).Some notes are instructive. First, to guarantee the normalizability of the above transition

probability, then the exponent (dS()dA())/() must be non-negative for any spacetimepoint (q, t). This demands that dS() dA() must always have the same sign as (). Onthe other hand, from Eq. (14), one can see thatdS() dA() changes its sign as flips its

14


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sign. Hence, to guarantee the non-negativity of (dS() dA())/(), must also changeits sign as flips its sign. This allows us to assume that the sign of is always the same

as that of. The time scale for the fluctuations of the sign of must therefore be the same

as the time scale for the fluctuations of given by dt. Moreover, the second term on the

right hand side of Eq. (27), , namely the rate of production of uncertainty due to the

fluctuations of infinitesimal stationary action is always non-negative.

It is also evident that for the distribution of Eq. (33) to make sense mathematically, the

time scale for the fluctuations of||, denoted by , must be much larger than that of||.One thus has

dt. (34)In other words,

|

|fluctuates much faster than

|

|, yet both and always have the same

sign fluctuating randomly in the time scale dt. Hence, within a time interval of length

during which|| is effectively constant, one may assume that dS() dA() is randomlyfluctuating due to the fluctuations of || distributed according to the exponential law of Eq.(33) characterized by||.

Next, there is no a priori reason on how the sign of the values of dS dA should bedistributed. Following the principle of indifference (principle of insufficient reason) [68], it is

then advisable to assume that the sign ofdS dA is distributed equally probably. Further,since the sign ofdS() dA() changes as flips its sign, then the sign ofmust also befluctuating randomly with equal probability so that the probability density of the value of

at any given time, denoted below by PH(), must satisfy the following unbiased condition:

PH() = PH(). (35)

Since the sign of is always the same as that ofthen the probability distribution function

of must also satisfy the same unbiased condition. Further, since PH() =

dq(q, ; t),

then Eq. (35) demands the following symmetry relation:

(q, ; t) = (q, ; t). (36)

One also has, from Eq. (13), the following symmetry relations for the spatiotemporal gra-

dient ofS(q, ; t):

qS(q; t, ) = qS(q; t, ),tS(q; t, ) = tS(q; t, ), (37)

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which together with Eq. (36) will play important role later.

Recall that the pair of relations in Eqs. (28) or (29) are valid when is fixed. However,

since as discussed above, the second term on the right hand side of Eq. (27) is insensitive

to the sign of which is always the same as the sign of , then the pair of equations in

(29) are valid in a microscopic time interval of length during which the magnitude of

is constant while its sign may change randomly. To have an evolution for a finite time

interval t > , one can proceed along the following approximation. First one divides the

time into a series of microscopic time intervals of length : t [(k 1), k),k = 1, 2, . . . ,and attributes to each interval a random value of (t) = k according to the probability

distributionPHk(k) =PHk(k). Hence, during the interval [(k 1), k), the magnitudeof(t) =k is kept constant while its sign changes randomly in an infinitesimal time scale

dt. One then applies the pair of relations in Eqs. (28) or (29) during each interval of time

with fixed|(t)| = |k|, consecutively.SincedA is just the infinitesimal stationary action along the short pathJ(),|dS dA|

may be regarded as the deviation from infinitesimal stationary action, the distribution of

which is given by Eq. (33). Such an exponential distribution was firstly suggested heuristi-

cally in Ref. [69] to model a microscopic stochastic deviation from classical mechanics. An

application of the statistical model to model quantum measurement is given recently in Ref.

[70]. For a fixed value of|| which is valid during a time interval of length , one can seefrom Eq. (33) that the average deviation from infinitesimal stationary action is given by

|dS dA| = ||/2. (38)

It is then evident that in the regime where the average deviation is much smaller than the

infinitesimal stationary action itself, namely|dA/| 1, or formally in the limit|| 0,Eq. (33) reduces to

(dS

dA), (39)

so that dS()dA(). Such a regime thus must be identified as the macroscopic regime.This fact suggests that || must have a very small microscopic value. In this regard, the pairof equations in (29) may be regarded as a generalization of the Hamilton-Jacobi equation of

(2) due to the exponential distribution of deviation from infinitesimal stationary action of

Eq. (33). Let us also note that since || in general may depend on time, then the transitionprobability is in general notstationary except when =Q all the time, whereQ is a

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constant. We shall in the next section and appendix consider a stationary case of empirical

interest whenQ = .One can also see that the decomposability of the infinitesimal change of the uncertainty

for a pair of spacelike separated subsystems given by Eq. (24), which is demanded by the

principle of Local-Causality, implies directly the separability of the transition probability of

Eq. (33) for the non-interacting subsystems. Namely, for non-interacting two subsystems

such that dA and dSare decomposable as dA(q1, q2) =dA1(q1) +dA2(q2) and dS(q1, q2) =

dS1(q1) + dS2(q2), respectively, so that is also decomposable(q1, q2) =1(q1) + 2(q2), the

transition probability of Eq. (33) is separable as

PS(dS1+dS|dA1+dA2) =PS(dS1|dA1)PS(dS2|dA2). (40)

Hence, the transition probability which determines the stochastic behavior of one subsystem

is independent from that of the other subsystem as intuitively expected for spacelike sepa-

rated non-interacting subsystems. Otherwise, the dynamics and statistics of one subsystem

is influenced by the other subsystem, which contradicts the principle of Locality. See also

Ref. [71] for a different approach to single out Eq. (33) by imposing the principle of Locality

without directly employing the concept of information.

III. QUANTIZATION

A. The Schrodinger equation and Borns statistical interpretation of wave function

Let us apply the above general formalism to stochastically modify a classical system of

a single particle subjected to external potentials so that the classical Hamiltonian takes the

following general form:

H(q, p) =gij(q)

2

(pi

ai)(pj

aj) +V, (41)

where ai(q), i = x, y, z and V(q) are vector and scalar potentials respectively, the metric

gij(q) may depend on the position of the particle, and summation over repeated indices

are assumed. The application to many particles system with different kind of classical

Hamiltonians can be done in the same way by following exactly all the steps that we are

going to take below. In the appendix we shall give another example of the application of the

statistical model to a system of two interacting particles which is commonly used to model

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the measurement of physical quantities, the detail elaboration of which is reported recently

in Ref. [70].

Let us first consider a time interval of length during which the magnitude of is

effectively constant while its sign fluctuates randomly together with the random fluctuations

of the sign of in a time scale dt. Let us then divide it into a series of microscopic time

intervals of length , [(k 1), k), k = 1, 2, . . . and attribute to each interval a randomvalue of(t) =k according to the probability distribution PHk(k) = PHk(k). Hence, ineach interval, the magnitude of(t) = k is constant while its sign changes randomly and

the pair of equations in (29) with fixed|k| apply.Let us first consider a microscopic time interval [(k 1), k). Within this interval of

time, using Eq. (41) to express qin term ofp via the (kinematic part of the) usual Hamilton

equation q= H/p, one has, by the virtue of the upper equation of (29)

qi() =gij

qjS() +

2

qj()

() aj

. (42)

Assuming the conservation of probability one thus obtains the following continuity equation:

0 =t +q (q)=t +qi

gij(qjS aj)

+

2qi(g

ijqj). (43)

On the other hand, from Eq. (41),(S) defined in Eqs. (8) and (18) is given by

(S) =qigij(qjS aj). (44)

Using the above form of(S), the lower equation of (29) thus becomes

H(q, p) = tS+ 2

t

+

2qig

ij(qjS aj). (45)

Plugging the upper equation of (29) into the left hand side of Eq. (45) and using Eq. (41)

one has, after arrangement

tS+g ij

2 (qiS ai)(qjS aj) +V

2

2

gij

qiqjR

R +qig

ijqjR

R

+

2

t +qi

gij(qjS aj)

+

2qi(g

ijqj)

= 0, (46)

where we have defined R .=

and used the identity:

1

4

qi

qj

=

1

2

qiqj

qiqjR

R . (47)

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Inserting Eq. (43), the last line of Eq. (46) vanishes to give

tS+g ij

2 (qiS ai)(qjS aj) +V

2

2 gij

qiqjR

R

+qigij

qjR

R = 0. (48)We have thus a pair of coupled equations (43) and (48) which are parameterized by. Recall

that the above pair of equations are valid in a microscopic time interval of length during

which the magnitude ofis constant while its sign changes randomly with equal probability.

Moreover, recall also that the sign of is always the same as the sign of. Keeping this in

mind, averaging Eq. (43) for the cases, thus is also over, one has, by the virtue ofEqs. (36) and (37),

t +qigij(qjS aj)= 0. (49)Similarly, averaging Eq. (48) over the cases will not change anything. We have thusfinally a pair of coupled equations (48) and (49) which are now parameterized by a constant

||, valid during a microscopic time interval of length characterized by a constant||.Next, since|| is non-vanishing, one can define the following complex-valued function:

.=

exp

iS

||

. (50)

Using , recalling the assumption that|| is constant during the time interval of interestof length , the pair of Eqs. (48) and (49) can then be recast into the following modified

Schrodinger equation:

i||t =12

(i||qi ai)gij(q)(i||qj aj) +V. (51)

Notice that the above equation is valid only for a microscopic time interval [(n 1), n)during which the magnitude of=n is constant. For finite time interval t > , one must

then apply Eq. (51) consecutively to each time intervals of length with different randomvalues of|n|,n = 1, 2, 3, . . . .

Let us then consider a specific case when|| is given by the reduced Planck constant for all the time, namely = , so that the exponential distribution of the deviation frominfinitesimal stationary action of Eq. (33) is stationary and the average deviation is given

by

/2. (52)

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Let us further assume that the fluctuations of|| around its average is sufficiently narrow.In this case, one may approximate (q, ||; t) and S(q; t, |n|) by the corresponding zerothorder terms of their Taylor expansion around the average of||, respectively denoted byQ(q; t) and SQ(q; t). The zeroth order approximation of Eq. (51) therefore reads

itQ(q; t) = HQ(q; t),

Q(q; t) .=

Q(q; t)eiSQ(q;t), (53)

where His just the quantum Hamiltonian given by

H=1

2(pi ai)gij(q)(pj aj) +V, (54)

with pi.

=iqi is the quantum mechanical Hermitian momentum operator. Unlike Eq.

(51), Eq. (53) is now deterministic and is parameterized by . Moreover, from Eq. (53), onecan see that the Borns statistical interpretation of wave function is valid by construction

Q(q; t) = |Q(q; t)|2. (55)

Hence, for the specific type of classical Hamiltonian with a quadratic momentum given

in Eq. (41), one regains the results of canonical quantization. The rules of canonical

quantization thus effectively arise from a statistical modification of classical mechanics in

microscopic regime based on a specific law of change of the uncertainty that the system has a

certain configuration of Eq. (27) chosen uniquely by imposing the condition of Macroscopic

Classicality and the principle of Local-Causality. Unlike the canonical quantization which is

formal-mathematical with obscure physical meaning, the statistical model of quantization

is thus physical. As shown above, Planck constant acquires a physical interpretation as

the average deviation from classical mechanics in a microscopic time scale. Moreover, unlike

canonical quantization which in general leads to an infinite number of possible quantum

Hamiltonians if gij in Eq. (41) depends on q due to operators ordering ambiguity, the

statistical model gives a unique quantum Hamiltonian with a specific ordering of operatorsof Eq. (54) where gij(q) is sandwiched by p a= iq a.

B. Configuration as beable and quantum mechanical average

Recall that in standard quantum mechanics, the state of the system is assumed to be de-

termined completely by specifying the wave function. The wave function is thus regarded as

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fundamental. Statements about position and momentum are then relegated operationallyto

certain acts of measurement over the state of the system represented by the wave function.

The canonical uncertainty relation between the statistical results of position and momentum

measurement is usually mentioned to support the argumentation that it is in general impos-

sible to attribute a pair of definite values of position and momentum to a system, nor such an

attribution is useful. By contrast, in the statistical model of quantization developed in the

present paper, one assumes the objective ontology of particles with a definite configuration

for all the time as in classical mechanics. Hence, configuration of the system is regarded as

the beable of the theory in Bells sense [72]. The wave function, on the other hand, is

argued to be emergent artificial convenient mathematical tool to describe the dynamics and

statistics of ensemble of trajectories. The objective ontology of the trajectories guarantees

a conceptually smooth classical correspondence.

For general types of classical Hamiltonian, the velocities of the particles are then obtained

by first solving p(q) on the left hand side of the upper equation of (29) in term of qto have

q() =H

p

p=qS()+

()2

q()

()

. (56)Hence, the velocities of the particles are fluctuating randomly due to the fluctuations of.

The configuration of the system therefore follows a continuous and randomly fluctuating

trajectory. Since, as argued in the previous section, in the limit || 0 which is interpretedphysically to correspond to classical regime, one has dS dA, then one can see that in thislimit, Eq. (56) reduces to the classical relation q= H

p|p=qA. We have thus a formally and

conceptually smooth classical correspondence.

Note again that fixing ||, Eq. (56) is valid only within a time interval of length duringwhich the sign ofis fluctuating randomly. It is then natural to define an effective velocity

as

q(||) .=

q() + q()2

. (57)

For the type of classical Hamiltonian given by Eq. (41), Eq. (56) reduces to Eq. (42). Let

us now consider the case when the classical Hamiltonian is given by Eq. (41) withgij = 1/m

and ai = 0, describing a particle of mass m subjected to an external scalar potential V(q).

Let us again consider a specific case when = so that the average of the deviation frominfinitesimal stationary action distributed according to Eq. (33) is given by /2, and as

shown in the previous subsection, one regains the prediction of canonical quantization. In

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this case, the zeroth order approximation of Eq. (42) then reads

q=qSQ

m

2m

qQQ

, (58)

where the

signs change randomly with equal probability. Moreover, calculating the

effective velocity one gets, recalling that the sign of is the same as that of, and taking

into account Eqs. (36) and (37),

q= qSQ/m, (59)which, unlike Eqs. (42) and (58), is now deterministic due to the deterministic time evolution

ofSQ given by the Schrodinger equation of (53). Hence, q in general fluctuates randomly

around qSQ/m except when the particle happens to lie at the extremum points ofQ so

that the second term of Eq. (58) vanishes and one has q= qSQ/m.

One thus expects that the actual trajectory of the particle is in general fluctuating ran-

domly around the integration over time of the effective velocityq=qSQ/m. The latter isjust the Bohmian trajectory of the particle in pilot-wave theory [73]. Hence, we have a physi-

cal picture that the actual trajectory is fluctuating randomly around the Bohmian trajectory

while the latter moves as if it is guided by the wave function evolving deterministically ac-

cording to the Schrodinger equation. Yet, unlike the pilot-wave theory, the wave function

in the statistical model is notphysically real but an artificial mathematical construct, and

the Schrodinger equation and the guidance relation of Eqs. (53) and (59) are derived from

first principle rather than ad-hoc-ly postulated as in pilot-wave theory. Recall that the fun-

damental assumption in pilot-wave theory that the wave function is a physical field, living

in configuration space rather than in ordinary space, is known to lead to a conceptual diffi-

culty, and furthermore implies rigid nonlocality in direct conflict with the special theory of

relativity. By contrast, the present statistical model is developed based on the principle of

Locality. In this sense, the upper equation in (29) can not be regarded as a causal-dynamical

guidance relation as in pilot-wave theory, but a kinematical relation. Moreover, unlike the

pilot-wave theory which is deterministic and relegates the microscopic randomness to our

ignorance of the initial condition, the statistical model is strictly stochastic.

It is then imperative to calculate the statistical averages of the relevant physical quantities

over all possible configuration distributed according to (q, ; t). It is natural to ask how

such statistical averages are related to quantum mechanical averages. For this purpose, we

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should again assume that = with equal probability in case of which the statisticalmodel reproduces the results of canonical quantization as the zeroth order approximation.

First, the average of any function of the configuration O(q) at any time is given by

O(q) .

= dqdO(q) = dqdO(q)

dqQO(q)Q.

= Q|O(q)|Q, (60)

where we have again counted only the zeroth order terms. Numerically, it is thus equal to

the quantum mechanical average of a quantum mechanical observable O(q) =O(q) when

the state of the system is given by the wave function Q. In particular, for the cases

O(q) =qandO(q) = (qq)2, the left hand side of Eq. (60) are the average and standarddeviation of the fluctuations ofq, which are numerically equal to the quantum mechanical

average of position operator and its standard deviation over the state Q given by the right

hand side.

Let us further calculate the average of the actual value of momentum p at a given time.

One directly gets, from the upper equation in (29)

p =

dqd

qS+

2

q

=

dqd(qS) =

dqd(i||q)

dqQ(iq)Q .= Q|p|Q, (61)where in the second equality we have used Eqs. (36) and (37) and taken into account the

fact that the sign of is the same as that of, and in the last approximate equality we have

imposed = and counted only the zeroth order terms. Again, it is numerically given bythe quantum mechanical average of momentum operator p= iq over the state Q.

Let us proceed to calculate the average of a quantity which is a function of momentum

up to second degree. Such a quantity can always be put into a quadratic form. It is thus

sufficient to calculate the average of a quantity of the type O(q, p) =a(p f(q))2, where ais constant andf(q) is a vector-valued function. One directly gets

O(q, p) =

a

qS+

2

q

f(q)

2=

a2

4

q

2+ a(qS f(q))2

=

dqda

i||q f(q)2 Q|ap f(q)2|Q. (62)

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25/36

over all possible configuration of the system are equal to the quantum mechanical average

of the corresponding quantum observables represented by some Hermitian operators O over

a wave function Q representing the corresponding quantum mechanical state:

O .= dqdO(q, p) Q|O|Q, (67)where means that we have considered the case when= and counted only the zerothorder terms. It is then tempting to guess that the above conclusion applies for all quantities

of a function of position and momentum O(q, p). One can however show that this is not

the case. For example, averaging O = p3, regardless of its physical meaning and proceeding

as before, one can check thatp3 =Q|p3|Q. Let us further emphasize however thatwhileQ|O|Q in the standard quantum mechanics refers to the average of the outcomesof measurement over an ensemble of identically prepared state represented by Q,Oin the statistical model refers to the objective properties of the ensemble independent of

measurement.

C. The principle of Locality, linearity of quantum dynamics and uncertainty re-

lation

Notice that as argued in the previous section, the unique form of the law of infinitesimalchange of the degree of uncertainty that the system has a certain configuration along a short

path given by Eq. (27) is singled out by imposing the condition of Macroscopic Classicality

and the principle of Locality. In particular, the principle of Locality is decisive in selecting

the linear form of the second term on the right hand side of Eq. (27) which describes the

production of information or uncertainty due to the fluctuation of infinitesimal stationary

action. Since the stochastic processes based on such a change of the information leads to

the derivation of the linearSchrodinger equation, one may thus argue that the principle of

Locality expressed in Eq. (25) is a necessary condition for the linearity of the Schrodinger

equation. To support this argumentation, let us mention that a nonlinear extension of

quantum dynamics[74] may lead to signaling [75,76] thus violating the principle of Locality.

The importance of the linearity of the Schrodinger equation can not be over emphasized.

It gives the mathematical basis for the superposition principle which plays very crucial

roles for the explanation of particle interference in double slits experiment, quantum entan-

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glement, and also in quantum measurement. The linearity of the Schrodinger equation is

also responsible for another important characteristic trait of quantum mechanics that it is

impossible to copy an unknown quantum state, the no-cloning theory [7779].

Let us proceed to show that the principle of Locality also plays a very important role in the

derivation of the quantum mechanical uncertainty relation. First, as argued in the previous

subsection, the actual trajectory of the system in configuration space is fluctuating randomly

around the Bohmian trajectory. It is then imperative to ask how the fluctuations around the

Bohmian trajectory is distributed. First, from the normalization of ,

dqd(q, ) = 1,

and the assumption that |q= 0 which is valid for arbitrary value of, one has

1 =

dqd =

dqd(q q0)q

= dqd{(q q0)}q, (68)where q0 is an arbitrary real number and the integration over spatial coordinate is taken

fromq= toq= . Applying the Schwartz inequality one gets dqd(q q0)2

dqd

q

2 1. (69)

Substituting Eq. (42), considering the case when g ij = 1/mand ai= 0, one directly obtains

dqd(q q0)2

dqd(mq qS)2

2

4. (70)

Let us again consider a specific case when =. The zeroth order approximation ofthe above inequality therefore reads [80]

dq(q q0)2Q(q)

dq(mq qSQ)2(q) 2

4. (71)

One then sees that the width of the fluctuations ofmqaroundqSQis bounded from below

by the inverse of the width of the distribution ofqand vise-versa, in a similar fashion as the

standard quantum mechanical uncertainty relation.

On the other hand, from Eqs. (60) and (63), one has

(q q)2(p p)2= Q|(q q)2|QQ|(p p)2|Q

2

4, (72)

where the inequality is due to [q,p] = i. Hence, the width of the distribution of the

actual momentum is bounded from below by the inverse of the width of the distribution of

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the actual position in the ensemble, in exactly the same manner as the standard quantum

mechanical uncertainty relation. Let us emphasize again that unlike the former which is

objective referring to no measurement, the latter is referring to the statistical results of

measurement of position and momentum over an ensemble of identically prepared system.

The above uncertainty relation is related to the uncertainty relation of Eq. (71) via the

fact that

(p p)2

2

q

2+ (qSQ qSQ)2

2

q

2=

(mq qSQ)2

, (73)

where in the first approximate equality we have used Eqs. (58) and (61) and taking into

account Eqs. (36) and (37) and the fact that the sign ofis always the same as the sign of,

and the last equality is due again to Eq. (58). Multiplying both sides with (qq0)2, takingq0 =q and imposing Eq. (71), one obtains Eq. (72). Hence, the quantum mechanicaluncertainty relation can be derived starting from the upper equation in Eq. (29). Since,

as argued in the previous section, the pair of equations in (29) are derived by imposing

the principle of Locality, then one may also conclude that the latter is necessary for the

derivation of quantum mechanical uncertainty relation.

IV. CONCLUSION AND REMARKS

We first developed a stochastic processes for a microscopic stochastic deviation from

classical mechanics in which the randomness is modeled by a stochastic fluctuations of the

infinitesimal stationary action, thus is physically different from that of the Brownian motion

based on random forces. Such a stochastic processes leads to a production of the uncertainty

that the system has a certain configuration in a microscopic time scale, which is assumed

to be vanishing in the classical limit of macroscopic regime. We then showed that imposingthe principle of Locality, which requires the infinitesimal change of the uncertainty that a

subsystem has a certain configuration to be independent from the configuration of the other

spacelike separated subsystem, will select a unique law of infinitesimal change of uncertainty,

up to a free parameter. We then further showed that such a law of infinitesimal change of

uncertainty determines a stochastic processes with a transition probability between two

infinitesimally close spacetime points along a randomly chosen path that is given by an

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exponential distribution of deviation from infinitesimal stationary action.

Given a classical Hamiltonian, we showed that the statistical model leads to the deriva-

tion of Schrodinger equation with Borns statistical interpretation of wave function and

uncertainty relation, two of the cornerstones of standard quantum mechanics. Unlike the

canonical quantization, however, in the statistical model, the system always has a definite

configuration all the time as in classical mechanics, fluctuating randomly along a continuous

trajectory. We have also shown, for a system of spin-less particles, that the average of the

relevant physical quantities over the distribution of the configuration is numerically equal

to the quantum mechanical average of the corresponding quantum mechanical Hermitian

operators over a quantum state represented by a wave function. Since the principle of Lo-

cality is derived from our conception of spacetime structure, then one may conclude that

the dynamics and kinematics of quantum mechanics is intimately related to the former.

Some problems are left for future study. It is first imperative to ask how such a local-

causal statistical model would explain the violation of Bells inequalities predicted by the

quantum mechanics and verified in numerous experiments which is widely believed to give

strong evidences that Nature is nonlocal? This is a crucial problem needed an explanation

within the statistical model. Recall that Bells inequalities are derived by assuming 1) the

separability of probability of outcomes in a pair of spacelike joint-measurements (Bells lo-

cality assumption) and 2) the so-called measurement independence or experimental free-will,that the distribution of the hidden variables underlying the measurement outcomes is inde-

pendent from the setting parameters of the apparatus chosen freely by the observer [8184].

It is tempting to guess that the objective locality of the statistical model implies the Bells

locality assumption so that the model must somehow violate measurement independence. It

is therefore instructive to study the above two fundamental hypothesis within the statistical

model by first applying the model to develop quantum measurement in realistic physical

systems and derive the Borns rule.

Next, it is also tempting to ask why canonical quantization corresponds to a specific

case when|| in Eq. (27), the free parameter of the statistical model, is given by sothat the average deviation from infinitesimal stationary action distributed according to the

exponential law of Eq. (33) is given by /2. Is the relation|| = exact? Or whetherNature allows for a small fluctuations of|| around ? Recall also that the Schrodingerequation is derived as the zeroth order approximation of the statistical model. It is then

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imperative to study the higher orders corrections. These last two cases may thus provide

precision tests against quantum mechanics.

Acknowledgments

Appendix A: Quantization of classical mechanical model for the measurement of

angular momentum

To show the robustness of the statistical model, let us give an application of the model to a

system of two interacting particles. Let us assume that the interaction classical Hamiltonian

is given by

HI=glz1p2, with lz1 =x1py1 y1px1. (A1)Here g is an interaction coupling, and lz1 is the zpart angular momentum of the firstparticle. Let us further assume that the interaction is impulsive (g is sufficiently strong) so

that the single particle Hamiltonians, of the type given by Eq. (41), are ignorable.

The interaction Hamiltonian above can be used as a classical mechanical model of mea-

surement of the angular momentum lz1 of the first particle by regarding the position of the

second particle as the pointer of the apparatus of measurement. To see this, first, in such

a model lz1 is conserved: lz1 ={lz1, HI}= 0 where{, } is the usual Poisson bracket. The

interaction Hamiltonian of Eq. (A1) thus correlates the value of lz1 with the momentum

of the apparatus p2 while keeping the value oflz1 unchanged. On the other hand, one also

has q2 ={q2, HI} = glz1, which, noting the fact that lz1 is a constant of motion, can beintegrated to give

q2(tM) =q2(0) +glz1tM, (A2)

where tM is the time span of the interaction. The value of lz1 prior to the interaction can

thus in principle be inferred from the observation of the initial and final values ofq2. Inthis way, the measurement of the physical quantity lz1 of the first particle is reduced to the

measurement of the position of the second particle q2. In the model, q2(t) therefore plays

the role of the pointer of the apparatus of measurement.

Now let us apply the statistical model discussed in the main text to stochastically modify

the above classical mechanical model of measurement. We shall repeat all the steps and

manipulations performed in the main text. What we need to do is to put the Hamiltonian

29


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of Eq. (A1) in place of Eq. (41). Again, let us first consider a time interval of the scale

in which the absolute value of is effectively constant while its sign fluctuates randomly

together with the random fluctuations of the sign of. Let us then divide it into a series of

microscopic time intervals of length , [(k

1), k), k = 1, 2, . . . and attribute to each

interval a random value of (t) = k according to the probability distribution PHk(k) =

PHk(k) so that in each interval, the magnitude of is constant while its sign changesrandomly. During each time interval [(k 1), k), the pair of equations in Eq. (29), eachwith constant value of|k|, thus apply.

Let us first consider a microscopic time interval [(k 1), k). Using the form of HIgiven by Eq. (A1) to express q in term ofp via q = H/p, the upper equation of (29)

becomes

x1= gy1

q2S+

2

q2

, y1= gx1

q2S+

2

q2

,

q2= g

x1

y1S+

2

y1

y1

x1S+

2

x1

, (A3)

and z1 = 0. Again, assuming that the probability is conserved, one gets, after a simple

calculation, the following continuity equation:

0 =t +q (q)

=t gy1x1(q2S) +gx1y1(q2S) +gx1q2(y1S)gy1q2(x1S) g(y1x1q2 x1y1q2). (A4)

On the other hand, from Eq. (A1),(S) defined in Eqs. (8) and (18) is given by

(S) = 2g(x1q2y1S y1q2x1S). (A5)

Substituting this into the lower equation of (29), one then obtains

HI(q, p) = tS+ 2

t

+g(x1y1q2S y1x1q2S). (A6)

Inserting the upper equation of (29) into the left hand side of the above equation, and using

30


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Eq. (A1), one has, after arrangement

tS+g

x1y1S y1x1S

q2S g2

x1y1q2R

R

y1x1q2RR

+

2t gy1x1(q2S)

+gx1y1(q2S) +gx1q2(y1S) gy1q2(x1S)g(y1x1q2 x1y1q2)

= 0, (A7)

where R =

and we have again used the identity of Eq. (47). Substituting Eq. (A4), the

last term of Eq. (A7) in the bracket vanishes to give

tS+g

x1y1S y1x1S

q2S

g2

x1

y1q2R

R y1x1q2R

R = 0. (A8)

One thus has a pair of coupled equations (A4) and (A8) which are parameterized by .

This pair of equations are valid in a microscopic time interval of length during which the

magnitude ofis constant while its sign changes randomly with equal probability. Averaging

Eq. (A4) over the cases, recalling that the sign of is the same as that of, one has, bythe virtue of Eqs. (36) and (37),

t gy1x1(q2SQ) +gx1y1(q2SQ)

+gx1q2(y1SQ) gy1q2(x1SQ) = 0. (A9)On the other hand, averaging Eq. (A8) over the cases does not change anything. Wefinally have a pair of Eqs. (A8) and (A9) which are now parameterized by|| valid for amicroscopic time interval of duration characterized by a constant||.

Next, using defined in Eq. (50), and recalling that || is constant during the microscopictime interval of interest with length , the pair of Eqs. (A8) and (A9) can then be recast

into the following compact form:

i||t = 22

HI. (A10)

Here HIis a differential operator defined as

HI.

=glz1 p2, (A11)

where pi =iqi , i= 1, 2, are the quantum mechanical momentum operator referring tothe iparticle and lz1 .= x1py1 y1px1 is the zpart of the quantum mechanical angular

31


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momentum operator pertaining to the first particle; all are Hermitian. Recall that Eq.

(A10) is valid only for a microscopic time interval [(k 1), k) during which|| =|k|is constant. For finite time interval, t > one must then apply Eq. (A10) to each time

intervals, each is parameterized by a random value of

|k

|,k = 1, 2, . . . , consecutively.

Let us again consider a specific case when || = . Moreover, let us further assumethat the fluctuations of|| around its average is sufficiently narrow so that (q, ||; t) andS(q; t, ||) can be approximated by the corresponding zeroth order terms, given respectivelybyQ(q; t) andSQ(q; t). In this case, Eq. (A10) can be approximated as

itQ(q; t) = HIQ(q; t), (A12)

where Qis defined as in Eq. (53). Unlike Eq. (A10), Eq. (A12) is now deterministic and is

parameterized by the reduced Planck constant

. Equation (A12) together with Eq. (A11)is just the Schrodinger equation for the von Neumann model of measurement of angular

momentum operator lz1 of the first particle using the second particle as the apparatus.

Without giving the detail, let us mention that the same results will be obtained for the

measurement of position and linear momentum. In these cases, the interaction classical

Hamiltonian will be given by

HI=gO1(q1, p1)p2, (A13)

where O1

= q1

and O1

=p1, respectively. Proceeding as before, repeating all the steps and

manipulations performed above, one will arrive at the Schrodinger equation with a quantum

Hamiltonian given by

HI=g O1p2, (A14)

where O1= q1 = q1 and O1 = p1 = iq respectively for the measurement of position andmomentum. The measurement of energy should be reduced to the measurement of position,

momentum and angular momentum.

It is then instructive to discuss whether such a model of quantum measurement can

reproduce the prediction of quantum mechanics. It is also interesting to ask how the model

addresses the measurement problem. The above two problems are discussed in detail in Ref.

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Quantum Fluctuations From a Local-causal Information Dynamics - Agung Budiyono