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Multi-Time Wave Functions Versus Multiple Timelike Dimensions

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Abstract

Multi-time wave functions are wave functions for multi-particle quantum systems that involve several time variables (one per particle). In this paper we contrast them with solutions of wave equations on a space–time with multiple timelike dimensions, i.e., on a pseudo-Riemannian manifold whose metric has signature such as \({+}{+}{-}{-}\) or \({+}{+}{-}{-}{-}{-}{-}{-}\), instead of \({+}{-}{-}{-}\). Despite the superficial similarity, the two behave very differently: whereas wave equations in multiple timelike dimensions are typically mathematically ill-posed and presumably unphysical, relevant Schrödinger equations for multi-time wave functions possess for every initial datum a unique solution on the spacelike configurations and form a natural covariant representation of quantum states.

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Notes

  1. An initial-value problem for a PDE is called well-posed (in the sense of Hadamard) [10] if for every initial datum (from an appropriate function space) a solution exists for all times, is unique, and depends continuously on the initial datum.

References

  1. Bera, P.K.: Review of [22]. Math. Rev. 3165768 (2015)

  2. Bloch, F.: Die physikalische Bedeutung mehrerer Zeiten in der Quantenelektrodynamik. Phys. Z. Sowjetunion 5, 301–305 (1934)

    MATH  Google Scholar 

  3. Craig, W., Weinstein, S.: On determinism and well-posedness in multiple time dimensions. Proc. R. Soc. A 465, 3023–3046. http://arxiv.org/abs/0812.0210 (2009)

  4. Crater, H.W., Van Alstine, P.: Two-body Dirac equations. Ann. Phys. 148, 57–94 (1983)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Dirac, P.A.M.: Relativistic quantum mechanics. Proc. R. Soc. Lond. A 136, 453–464 (1932)

    Article  ADS  MATH  Google Scholar 

  6. Dirac, P.A.M., Fock, V.A., Podolsky, B.: On quantum electrodynamics. Phys. Z. Sowjetunion 2(6), 468–479 (1932). Reprinted in Schwinger, J.: Selected Papers on Quantum Electrodynamics. Dover, New York (1958)

    MATH  Google Scholar 

  7. Droz-Vincent, P.: Relativistic wave equations for a system of two particles with spin \(\tfrac{1}{2}\). Lettere al Nuovo Cimento 30, 375–378 (1981)

    Article  MathSciNet  Google Scholar 

  8. Droz-Vincent, P.: Second quantization of directly interacting particles. In: Llosa, J. (ed.) Relativistic Action at a Distance: Classical and Quantum Aspects, pp. 81–101. Springer, Berlin (1982)

    Google Scholar 

  9. Droz-Vincent, P.: Relativistic quantum mechanics with non conserved number of particles. J. Geom. Phys. 2(1), 101–119 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  10. Evans, L.C.: Partial Differential Equations, 2nd edn. American Mathematical Society, Providence (2010)

    MATH  Google Scholar 

  11. Lampart, J., Schmidt, J., Teufel, S., Tumulka, R.: Particle creation at a point source by means of interior-boundary conditions. Preprint. https://arxiv.org/abs/1703.04476 (2017)

  12. Lienert, M.: A relativistically interacting exactly solvable multi-time model for two mass-less Dirac particles in 1+1 dimensions. J. Math. Phys. 56, 042301. http://arxiv.org/abs/1411.2833 (2015)

  13. Lienert, M.: On the question of current conservation for the Two-Body Dirac equations of constraint theory. J. Phys. A 48, 325302. http://arxiv.org/abs/1501.07027 (2015)

  14. Lienert, M.: Lorentz invariant quantum dynamics in the multi-time formalism. PhD Thesis, Mathematics Institute, Ludwig-Maximilians University, Munich (2015)

  15. Lienert, M., Nickel, L.: A simple explicitly solvable interacting relativistic \(N\)-particle model. J. Phys. A 48, 325301. http://arxiv.org/abs/1502.00917 (2015)

  16. Lienert, M., Petrat, S., Tumulka, R.: Multi-time wave functions. J. Phys. Conf. Ser. 880, 012006. http://arxiv.org/abs/1702.05282 (2017)

  17. Lienert, M., Tumulka, R.: Born’s rule on arbitrary Cauchy surfaces. http://arxiv.org/abs/1706.07074 (2015)

  18. Moshinsky, M., Laurrabaquio, G.L.: Relativistic interactions by means of boundary conditions: the Breit–Wigner formula. J. Math. Phys. 32, 3519–3528 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  19. Nickel, L., Deckert, D.-A.: Consistency of multi-time Dirac equations with general interaction potentials. J. Math. Phys. 57, 072301. http://arxiv.org/abs/1603.02538 (2016)

  20. Petrat, S., Tumulka, R.: Multi-time Schrödinger equations cannot contain interaction potentials. J. Math. Phys. 55, 032302. http://arxiv.org/abs/1308.1065 (2014)

  21. Petrat, S., Tumulka, R.: Multi-time wave functions for quantum field theory. Ann. Phys. 345, 17–54. http://arxiv.org/abs/1309.0802 (2014)

  22. Petrat, S., Tumulka, R.: Multi-time equations, classical and quantum. Proc. R. Soc. A 470(2164), 20130632. http://arxiv.org/abs/1309.1103 (2014)

  23. Petrat, S., Tumulka, R.: Multi-time formulation of pair creation. J. Phys. A 47, 112001. http://arxiv.org/abs/1401.6093 (2014)

  24. Piceno, E., Rosado, A., Sadurní, E.: Fundamental constraints on two-time physics. Eur. Phys. J. Plus 131, 352. http://arxiv.org/abs/1512.05345 (2016)

  25. Schweber, S.: An Introduction To Relativistic Quantum Field Theory. Row, Peterson and Company, Evanston (1961)

    MATH  Google Scholar 

  26. Sparling, G.A.J.: Germ of a synthesis: space–time is spinorial, extra dimensions are time-like. Proc. R. Soc. A 463, 1665–1679 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Tegmark, M.: On the dimensionality of space–time. Classical Quantum Gravity 14, L69–L75. http://arxiv.org/abs/gr-qc/9702052 (1997)

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Acknowledgements

We thank an anonymous referee for her or his comments on a previous version of this note. This project has received funding from the European Union’s Framework for Research and Innovation Horizon 2020 (2014–2020) under the Marie Skłodowska–Curie Grant Agreement No. 705295. S.P. gratefully acknowledges support from the German Academic Exchange Service (DAAD).

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Authors and Affiliations

  1. Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ, 08854-8019, USA

    Matthias Lienert

  2. Department of Physics, Princeton University, Jadwin Hall, Washington Road, Princeton, NJ, 08544-0708, USA

    Sören Petrat

  3. Fachbereich Mathematik, Eberhard-Karls-Universität, Auf der Morgenstelle 10, 72076, Tübingen, Germany

    Roderich Tumulka

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  1. Matthias Lienert
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Correspondence to Matthias Lienert.

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Lienert, M., Petrat, S. & Tumulka, R. Multi-Time Wave Functions Versus Multiple Timelike Dimensions. Found Phys 47, 1582–1590 (2017). https://doi.org/10.1007/s10701-017-0120-5

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