Quantum Fields and Fundamental Geometry

dcg@uakron.edu

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

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