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dcg@uakron.edu. 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
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dcg@uakron.edu 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 • Ongoing studies — higher interactions and theoretical issues
Quantum field concepts • Point Classical Particles and countability • Particle fields in classical physics • Experimental point particles and wave particles 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?
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 2:03 2:02
Conformal Transformations Expansion plus rotation • Two dimensions • More dimensions • Conformal factor Curvilinear representation of the wave function: c c
Quantum Mechanics?
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 sequence of refinements • 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.
Radiation Forces • 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 emitter B hυ A
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 time detector detector source
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
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. Q P 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.
Five dimensional quantum geometry • Fifth coordinate from proper time • Null displacements • Electromagnetic potential and wave function placed off-diagonal • Precise relationship with quantum fields
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 Electrodynamics Gravitation Quantum gravitational waves Classical gravitational waves Quantum electrodynamics Classical electrodynamics Feynman, Schwinger Tomonaga Ashtekar,. . . Wheeler, Feynman Kilmister Davies Time symmetric classical gravitational waves Time symmetric quantum gravitational waves Time symmetric quantum electrodynamics Time symmetric classical electrodynamics Hoyle, Narlikar
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 masssymmetrically
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 arbitraryorientation
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 G.R. E.D. Q.M. Spin Weak Strong Q.C.D Standard Model G.R. E.D. Q.M. Spin Weak Strong Q.E.D 5-D Theory ? 8-D Theory
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
Summary • 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