Quantum Fields and Fundamental Geometry

1. dcg@uakron.edu Quantum Fields and Fundamental Geometry Daniel Galehouse 17-19 February 2005

2. Introduction • Basic concept — fields and geometry • Quantum mechanics — interpretations • Gravitation — structure and interaction • Spin theory — eight dimensions • Ongoing studies — higher interactions and theoretical issues

3. What is Field theory?

4. 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.

5. 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?

6. 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

7. 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

8. Conformal Transformations Expansion plus rotation • Two dimensions • More dimensions • Conformal factor Curvilinear representation of the wave function: c c

9. Quantum Mechanics?

10. 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.

11. 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.

12. 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.

13. 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

14. 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.

15. 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

16. 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

17. How does geometry work?

18. 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

19. 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

20. 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.

21. Five dimensional quantum geometry • Fifth coordinate from proper time • Null displacements • Electromagnetic potential and wave function placed off-diagonal • Precise relationship with quantum fields

22. Geodetic currents • Electrodynamic-gravitational motion • Quantum scaling of coefficients • Accelerations from quantum forces • Probability current trajectories • Null displacements along trajectory

23. Quantum Field Equation • Gives the wave function, including • Diffraction and interference • Electromagnetic effects • Gravitational fields • Arbitrary coordinate systems • Geometrical mass corrections

24. 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

25. 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

26. 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

27. 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

28. 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

29. What is spin?

30. 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

31. 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.

32. 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

33. 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

34. 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

35. 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

36. Pluecker-Klein correspondence • General bilinear spinor combination • Six pair-wise combinations • Quadratic invariant for any spinors • Algebraic identity

37. Spinor invariants in five-space • Single spinor invariant • Known similarity transformation • Energy-momentum in classical limit • Extra physical quantities

38. 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

39. 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

40. What is next?

41. 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

42. 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

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