Upload
keaton-oneal
View
24
Download
0
Tags:
Embed Size (px)
DESCRIPTION
[email protected]. Quantum Fields and Fundamental Geometry. Daniel Galehouse. 17-19 February 2005. Introduction. Basic concept — fields and geometry Quantum mechanics — interpretations Gravitation — structure and interaction Spin theory — eight dimensions - PowerPoint PPT Presentation
Citation preview
Quantum Fields and Fundamental Geometry
VII International Symposium : Meeting the Unknown
Industrial Physics Engineering, Monterrey Campus, Monterrey, Mexico
Daniel Galehouse17-19 February 2005
● Basic concept — fields and geometry● Quantum mechanics — interpretations● Gravitation — structure and interaction ● Spin theory — eight dimensions● Ongoing studies — higher interactions and theoretical
issues
Introduction
What is Field theory?
Quantum field concepts
● Point Classical Particles and countability● Particle fields in classical physics● Experimental point particles and wave particles
The justification of the constructs, which represent “reality” for us, liesalone in their quality of making intelligible what is sensorially given . . .
-- A. Einstein
Quantum field:
A description of physical objects based on countable wave fields.
What is Quantization?
● Is there a way to be sure that classical physics is right?● Is there a verifiable starting point?● Study values of 0<β<1. ● Is the process mathematically justified?
quantbox.pdf
Essential quantum terms from geometry
● Quantum terms can appear without quantization● Intrinsic Quantization:
– Weyl theories — gauge invariance + general covariance
– Kaluza and Klein theories — intrinsically quantum
– Implicit for curvilinear formalism
● All quantum terms can come from geometry
Twin paradox and accelerated motion
● Twin paradox of general relativity
● Requires a curvilinear theory
● Equivalence implies the same problem for quantum motion
● Any failure of Lorentz invariance requires a curvilinear theory
● Special relativity fails for and real interaction.
0:00
0:00
2:02
2:03
g g.
Conformal Transformations
Expansion plus rotation• Two dimensions• More dimensions• Conformal factor
Curvilinear representation of the wave function:
gg
QuantumMechanics?
Quantum Measurements
● A source emits particles which are diffracted by a screen and detected.
● An explicit model of the detector models the basis of measurement.● Wave particles are captured on target nuclei remaining as localized.● Radiation is emitted as the capture occurs.● Radiation details match the transition of the wave particle.
● A particle traverses several slits in order, and is deflected at each● The implied selection of the initial trajectory is refined at each step● The argument for point like character fails.● Radiation is emitted at each refinement.● Information is carried away by the radiation.
A sequence of refinements
Radiation Forces
EI
0
● For one antenna, the field is E ~ I0 and the power is P ~ E2 ~ I02
● For two antennas, the total field is E ~ 2I0 and the power is P ~ 4E2 ~ 4I02
● Double the expected energy from input excitation voltage to tower● Increased force of radiation reaction to first tower from second.
Radiation symmetry
● Emitter and absorber one system● Time symmetric interaction● Forces of emission equivalent to
absorption● Time reversal exchanges emitter
and absorber● Interaction of universe assumed
fundamentally symmetrical.● Advanced forces essential to state
change of emitterA
B
hυ
Entanglements
● Two wave particles interact● Covariant interactions are light-like.● Near field forces are symmetric● Far field forces taken symmetric● Absorption and emission symmetrical● Complexity of connections implies
space-like forces indirectly.
Delayed Correlations
● Two photon emitter● No stable intermediate● Both photons required to force final
state transition.● “Double” radiation reaction forces
required● Polarization correlation also
required● Detected correlations present for
any timesource
detector
detector
Determinism
● Cat in box with spontaneous trigger.● Can cat be in a superposition state?● Statistics depend on distant
absorbers ● Determinism requires a closed system● Box not perfectly closed in quantum
statistical sense● Universe is a determined system● Evolution is determined if box isolates
from the distant absorber
hυ
How does geometry work?
Gravitational fields
● Universal field assumption for point particles– Motion described by one field or metric
● Individual field assumption for quantum particles
– Interactions must be separated on overlap.
– Each quantum wave particle must have
separate electromagnetic, gravitational and
quantum fields.
P Q
P Q
Geometrical Quantum Theory
● Use a separate tensor for each particle● Essential quantum terms appear
automatically● Electromagnetic interactions ● Gravitational interactions● Quantum effects● All invariants come from the Riemann tensor● Electron and neutrino spin
Some common difficulties in field theory
● Avoid double quantization.● Justify from experiment, never classical theory.● General relativity contains essential quantum terms .
and cannot be actively quantized.● Quantization of a classical theory may or may not work.● A quantum theory that is only Lorentz covariant (such as
Q.E.D.)
is an approximation and cannot be written in closed form.
● Use geometrical quantization.
● Fifth coordinate from proper time● Null displacements● Electromagnetic potential and wave function placed off-
diagonal● Precise relationship with quantum fields
Five dimensional quantum geometry
Geodetic currents
● Electrodynamic-gravitational motion
– Quantum scaling of coefficients
– Accelerations from quantum forces
– Probability current trajectories
– Null displacements along trajectory
Quantum Field Equation
● Gives the wave function, including– Diffraction and interference
– Electromagnetic effects
– Gravitational fields
– Arbitrary coordinate systems
– Geometrical mass corrections
Positrons and electrons
• e-p pairs are connected at the point of origination
• They may start with an acute angle or they may curve around
• The sharp angular representation is common but studies following the perspective of G.R. are smooth
• Five dimensional terms suggest a connection of the spaces following the Riemannian theory
• Experimental tests are difficult• Calculations may be affected in
some detail
Mass corrections
● Energy density correction● Integral to in 5-d theory● Part of 5-covariance● Simple of mass theory● Electron correction beyond
measurement● Neutrino correction may be within
range● Numerical factors for more
dimensions
Quantum gravitational source terms
● Source currents from five dimensional conformal effects.– Quantum relativistic corrections
– Essential quantum gravitational effects
– Densities for electromagnetic sources
– Constants and interactions
Black holes?
● Quantum-gravitational corrections may bring the horizon into the star surface
● Quantum information may persist● Gravitational pair production● Pressure term may affect cosmological constant
Field quantization
Quantumelectrodynamics
Classicalelectrodynamics
Time symmetricquantum
electrodynamics
Time symmetricclassicalelectrodynamics
Classicalgravitationalwaves
Quantumgravitationalwaves
Time symmetricquantumgravitational waves
Time symmetricclassicalgravitational waves
Wheeler, Feynman
Feynman, SchwingerTomonaga
Davies
Hoyle,Narlikar
Ashtekar,. . .
Kilmister
Electrodynamics Gravitation
What is spin?
Dirac Equation in 5-symmetric form
● Dirac equation converts to symmetric form suitable for five dimensions● A similarity transformation is used to include the mass symmetrically
Spin Matrices and Geometry
● Standard gamma matrices relate to general metric● Fifth anti-commuting Dirac matrix completes the set for five
dimensions.● Dotted values for observers' space● Un-dotted values for particle space.
Eight dimensional spinor basis.
● Eight real coordinates are combined into four complex pairs● Standard spinor metric is used● Transformation to the five dimensional space depends on gamma
matrices● Spinor type Lorentz transformations● Delta parametrizes local frame orientation
Spinor space curvature invariant
● Zero curvature scalar corresponds to eight dimensional D'Alembertian● Local conformal parameter equal to the two thirds power of the wave
function● Conformal transformations are sufficient● All spaces taken conformally flat
Spin from the gradient of a scalar
● 8-Gradient of scalar wave function space gives Dirac spinor● Standard transformation properties follow from local coordinate
relation.● Characteristic equation becomes first order● Use chain rule to get differential equation in five space
Spinor wave by differentiation
● Scalar plane wave in five dimensional form● Spinor differentiation gives related Dirac wave function● General solutions are locally of the Dirac form● Parameterization is in five dimensional spinor basis with arbitrary
orientation
Pluecker-Klein correspondence
● General bilinear spinor combination
● Six pair-wise combinations● Quadratic invariant for any spinors● Algebraic identity
Spinor invariants in five-space
● Single spinor invariant● Known similarity transformation● Energy-momentum in classical limit● Extra physical quantities
Lepton mass
● Mass is generated from two of the six quantities in the sum
● Mass zero quantities constrain allowable spinor wave functions
● Positive or negative helicities required● Neutrinos and electrons satisfy same equation
Types of field theory
Standard Model Q.C.D
Q.E.D
5-D Theory
8-D Theory ?
G.R. E.D. Q.M. Spin Weak Strong
What is next?
Ongoing studies and physical implications
● General mass theory● Propagating mass and rest mass● Inertia, gravity and the Higgs● Geometries for weak and strong interactions● Curvilinear description of elementary particles● Particle transmutation● Regularization requirements● Renormalization● Theory of the vacuum● Black holes
● Basic concepts
– Fields, quantization, geometry, waves, conformal transformations
● Quantum mechanics
– Refinements, entanglements, measurements, radiation, correlations, cats
● Gravitation
– Metrics, geodesy, wave equations, source equations, five dimensions
● Spin theory
– Matrices, Dirac equation, eight dimensions, waves, invariants, lepton mass
● Ongoing studies
– Field quantization, applications, conflicts to study
Summary
Back to work