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The Klein-Gordon Equation Revisited

The Klein-Gordon Equation Revisited. Ken Wharton Associate Professor Department of Physics San Jos é State University San José, CA; USA. PIAF-1 February 1-3, 2008 Sydney, Australia. The PIAF Connection.

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The Klein-Gordon Equation Revisited

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  1. The Klein-Gordon Equation Revisited Ken Wharton Associate Professor Department of Physics San José State University San José, CA; USA PIAF-1 February 1-3, 2008 Sydney, Australia

  2. The PIAF Connection • I will outline a foundational research program that naturally links to very different work by at least three PIAF participants: • Robert Spekkens • Huw Price • Lucien Hardy • If successful, this program should also interest: • PI’s quantum gravity experts • Australia’s retrocausal experts • Bayesians (?)

  3. The Big Picture(for neutral, spinless fields) C General Relativity Klein-Gordon in curved space Quantum Gravity ?????? Quantum Field Theory B Klein-Gordon Equation Special Relativity A Non-Relativistic Limit Schrödinger Equation Quantum Mechanics ClassicalQuantum

  4. Relativistic Particle Klein-Gordon Equation (KGE) Schrödinger’s starting point:The Klein-Gordon Equation (KGE) deBroglie Waves: Advantages: Time-Symmetric, Relativistically Covariant Problem: No consistent, spatially meaningful interpretation General solutions to KGE:

  5. Then where What happens in the non-relativistic limit? One does NOT get the Schrödinger equation!(1st and 2nd order differential equations aren’t equivalent in ANY limit.) Schrödinger’s critical assumption: By dropping half of the allowed parameters, Schrödinger reduced the KGE to a 1st order differential equation (in t).

  6. The critical assumption, in detail The Klein-Gordon Equation The Schrödinger Eqn. (V=0) • Halves number of free parameters in the solution. • (no longer need  and d/dt to solve; just ) • Introduces an explicit time-asymmetry. • Arbitrary way to halve solutions (Asin+Bcos, A+iB, etc.) • This particular halving fails in curved spacetime! • (Perhaps why QM has never been reconciled with GR)

  7. It’s long past time to revisit the KGE! The fathers of quantum mechanics never meant to devise a relativistically-correct theory... and yet we’re still using their basic formalism as a starting point 80 years later. If relativistic thinking was irrelevant, then extrapolating to SR or GR would be simple. The fact that it isn’t easy strongly implies that SR+GR have foundational relevance to QM. Ambitious Research Goal: Use KGE to re-derive QM probabilities (associated with preparation-measurement pairs) without dropping b(k). (i.e. learn how to quantize a second-order differential equation)

  8. The KGE’s “Extra” Free Parameters The Klein-Gordon equation has a solution with exactly twice the free parameters of the Schrödinger Equation solution  But 80 years of experiments say we can’t learn any more information (at one time) than can be encoded by . (Deeply connected with the Uncertainty Principle) Therefore, if we start with the KGE as the master Equation, one gets the axiomatic foundation of Spekkens’s toy model! (We can only know half the total information in .) However, this means one can never get enough information to solve the KGE as an initial boundary condition problem.

  9. Initial Boundary Conditions vs. CPT Both quantum field theory and relativity are CPT symmetric; should reduce to a time-symmetric non-relativistic picture. But QM is explicitly time-asymmetric. (The T-asymmetry in the Schrödinger Eqn is fixed “by hand”, the collapse is not.) Connection with Huw Price’s work: Asymmetries appear because boundary conditions are imposed asymmetrically. To replace the T-asymmetric “collapse” with a CPT-symmetric picture, maybe we shouldn’t be looking for initial boundary conditions in the first place!

  10. A B B Upon reaching B, the rest of the wave “collapses”?! Boundary conditions are often implemented time-asymmetrically Atom A emits a photon, and it is later absorbed by atom B: Very symmetric. Using only initial boundary conditions leads to a strange picture: No time-symmetry in this picture! A symmetric picture requires two-time boundary conditions.

  11. CPT and KGE: Natural Partners Larry Schulman has attempted to impose two-time boundary conditions on the Schrödinger equation. Leads to an overconstrained equation; non-exact solutions. “Time’s arrows and quantum measurement”, L.S. Schulman, Cambridge Univ. Press (1997) But the Klein-Gordon equation requires a 2nd boundary condition to determine the “extra” free parameters… … it can’t go at the beginning, and physical time- symmetry implies it’s far more natural to put it at the end! A Novel Proposal: Keep the full Klein-Gordon equation. Impose half the boundaries at one time, and half at another time.

  12. t=t0 (x,t) Time t=0 Mapping two-time boundaries to QM Mathematical boundary conditions correspond to external physical constraints (i.e. measurements). Final measurement (procedure + results); allows retrodiction. Initial measurement (preparation) can’t specify a unique wavefunction. (Would need both If the boundary conditions correspond to measurements, the “collapse” becomes the continuous effect of a future boundary.

  13. 45o “+” 0o 45o 0o ? Two-Boundary FAQs Huw Price’s picture of a photon passing through 2 polarizers Doesn’t this violate our intuitive notion of causality? Yes -- perhaps a benefit in disguise. (Intuition is biased against time-symmetry) Does this permit causal paradoxes? It’s impossible to retrieve any future-information without changing the boundary conditions. Where does probability fit in?

  14. Discrete Probability Weights The 2-boundary problem is solvable, but cannot predict. Furthermore, once you retrodict the solution , what sense is there to extract an outcome probability from ? Bayesian answer: “Probability is assigned to propositions, not wavefunctions!” Fact: Some pairs of boundaries are more likely to occur together than other pairs of boundaries. If relative weights for each pair are known, one can generate probabilities for any time-biased proposition.

  15. yes no yes no Student “A” 90% 5% 5% 0% time yes yes no no yes no yes no Student “D” 20% 35% 35% 10% time yes yes no no A Classical Example Last semester, did a given student come to class for two consecutive lectures? Recovered probabilities: If A and D came to previous class, A had a 94.7% attendance probability, while D had 36.4%.

  16. Implementation Questions This research program comes down to 2 main issues: • What mathematical boundary condition corresponds to • a given physical measurement/constraint? • Map to existing measurement theory? • Construct GR-friendly measurement theory? • What is the discrete probability weight that • corresponds to any complete solution? • Demand exact correspondence to QM in NR-limit? • - Use known results as a guide, not a rule?

  17. Standard theory: Boundary conditions are eigenfunctions of an operator. (in position space, ) Problem #1: fails for the KGE! Propagates in k direction Propagates in -k direction eigenvalues of both terms are , which does not correspond to physical momentum of the wave Tentative solution: Use only time-even operators Recent Results (arXiv:0706.4075)

  18. Initial Boundary Condition = F(r) Final Boundary Condition = G(r) t = 0 t = to IBC: Fourier-expand F(r) and G(r) FBC: Plug into and solve for coefficients a and b determine ; we know F(k) from initial boundary, G(k) from final boundary, and , but… First Attempt: Two-time Boundary Conditions

  19. Problem #2;  is a function of k, so for any value of to, there will always be values of k where , and the coefficient denominators go to zero! Next problem: infinite poles Import the solution from quantum field theory: give the mass a tiny imaginary component. New KGE: Then calculate probability and take limit as

  20. The “retrodicted” wavefunction • No Collapse ( automatically conforms to the final boundary condition) • Not pre-dictable: need measurement result G(r) • (ExplainsEPR/Bell w/o faster-than-light influences) In other words, this is a “hidden variable” model that violates Bell’s inequality, because the parameters a(k) and b(k) depend on future events.

  21. Charge density of KGE: FBC t=to  t=0 x IBC Covariant Probability Weight (not well-defined in curved space) ct  Here  is a unit four-vector, perpendicular to the boundary condition’s 3D hypersurface (inward pointing). Covariant generalization on arbitrary closed boundary:

  22. Known non-relativistic limit: Discrete Probability Postulate (Wmax-Wmin)2 = P Given by square of range of W: W has a range because we don’t know the relative phase between F and G, and we don’t know the exact value of to Given: 1) Non-relativistic limit 2) Additional time-energy constraint (but not quite!)

  23. Four postulates: 3 good, 1 bad • 1) Start with the Klein-Gordon Equation. • (Not the Schrödinger Equation!) • 2) Constrain with a closed boundary condition in 4-D. • (Deal with infinities using m2 => m2-i) • 3) Weight the probability with All of these postulates are easily extendible to a general relativity framework (curved space), except… • 4) The boundary condition corresponds to the eigenstate • from ordinary quantum measurement theory.

  24. hypersurface boundary condition  time time space space A spacetime view leads to a new perspective of measurements Standard View: Spacetime View: Measurement  Spatial boundary conditions Preparation The preparation and spatial boundaries give , from which one calculates the measurement probabilities. Partial information on a hypersurface constrains the solution . More solutions lead to a larger weightP.

  25. System Lab+System Physical interactions determine shape and content of boundary conditions Time Space Further insight can be found in recent papers: R. Oeckl: “General Boundary Quantum Field Theory”: arXiv.org/hep-th/0509122 L. Hardy: “Non-Fixed Causal Structure”: arXiv.org/gr-qc/0608043

  26. Clues to a GR-friendly measurement theory • Momentum is not fundamental for fields in GR: • The stress energy tensor, T, is fundamental. • On a closed 3-surface (with dual ), one can extract: • Energy density everywhere on surface: T0  • Momentum density everywhere on surface: Ti  • These appear to roughly map to the info in (x,t). • On a space-like 3-surface, one can integrate the above values to get total energy, angular momentum, etc...

  27. A B (x,t) The missing piece of the puzzle... Quantization! • Without eigenfunction rule, all possible boundary conditions become reasonable. • (T00(x) need not be localized;  is a scalar field) • Possible paths forward: • Find probability weight that effectively selects for eigenfunctions. • New GR-friendly axiom: No paradoxes allowed. A,B are space-like separated, but can have a causal effect via 

  28. Conclusions • Relativity and CPT symmetry must inform quantum foundations research, even in the non-relativistic limit. • Both foundations and quantum gravity could benefit from a new interpretation of the Klein-Gordon Equation and a spacetime picture of measurement/boundaries. This is a hugely ambitious research program... ...but PIAF is the group with the abilities and research inclinations best suited to carrying it out.

  29. Acknowledgements Thank you to: - Huw Price, Guido Bacciagaluppi, Centre for Time - Jerry Finkelstein, Lawrence Berkeley Laboratory - Eric Cavalcanti, Griffith University, Australia - Philip Goyal, Perimeter Institute, Canada More information can be found in these papers: K.B. Wharton, “Time-symmetric quantum mechanics”, Foundations of Physics, v.37 p.159 (2007) K.B. Wharton, “A novel interpretation of the Klein-Gordon Equation,” arXiv:0706.4075 [quant-ph] Email: wharton@science.sjsu.edu

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