Abstract
A Lorentz-covariant system of wave equations is formulated for a quantum-mechanical two-body system in one space dimension, comprised of one electron and one photon. Manifest Lorentz covariance is achieved using Dirac’s formalism of multi-time wave functions, i.e., wave functions \(\Psi ^{{(2)}}(\mathbf {x}_{\text{ ph }},\mathbf {x}_{\text{ el }})\) where \(\mathbf {x}_{\text{ el }},\mathbf {x}_{\text{ ph }}\) are the generic spacetime events of the electron and photon, respectively. Their interaction is implemented via a Lorentz-invariant no-crossing-of-paths boundary condition at the coincidence submanifold \(\{\mathbf {x}_{\text{ el }}=\mathbf {x}_{\text{ ph }}\}\), compatible with particle current conservation. The corresponding initial-boundary-value problem is proved to be well-posed. Electron and photon trajectories are shown to exist globally in a hypersurface Bohm–Dirac theory, for typical particle initial conditions. Also presented are the results of some numerical experiments which illustrate Compton scattering as well as a new phenomenon: photon capture and release by the electron.
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Notes
This may require suitable conservation laws, see Sect. 4.
We are only interested in Lorentz transformations with identical action on the photonic and electronic variables.
“Almost everywhere” without any measure specified refers to the Lebesgue measure, here and henceforth.
References
Aharonov, Y., Albert, D., Vaidman, L.: How the result of measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. Phys. Rev. Lett. 60, 1351–1354 (1988)
Alazzawi, S., Dybalski, W.: Compton scattering in the Buchholz-Roberts framework of relativistic QED. Lett. Math. Phys. 107, 81–106 (2017)
Berndl, K., Dürr, D., Goldstein, S., Peruzzi, G., Zanghì, N.: On the global existence of Bohmian mechanics. Commun. Math. Phys. 173, 647–673 (1995)
Bohm, D.: Quantum Theory. Dover Publication, Mineola (1989)
Bohm, D.: A suggested interpretation of the quantum theory in terms of “hidden” variables. Part I. Phys. Rev. 85:166–179 (1952). Part II ibid., 180–193 (1952)
Born, M.: Zur Quantenmechanik der Stossvorgänge. Z. Phys. 37, 863–867 (1926)
Born, M.: Quantenmechanik der Stossvorgänge. Z. Phys. 38, 803–827 (1926)
de Broglie, L.V.P.R.: La nouvelle dynamique des quanta, in “Cinquième Conseil de Physique Solvay” (Bruxelles 1927), ed. J. Bordet, (Gauthier-Villars, Paris, 1928); English transl.: “The new dynamics of quanta”, p.374-406 in: G. Bacciagaluppi and A. Valentini, “Quantum Theory at the Crossroads. Cambridge University Press, Cambridge, 2009
Bricmont, J.: Making Sense of Quantum Mechanics. Springer, Berlin (2016)
Buchholz, D., Roberts, J.E.: New light on infrared problems: sectors, statistics, symmetries, and spectrum. Commun. Math. Phys. 330, 935–972 (2014)
Chen, T., Fröhlich, J., Pizzo, A.: Infraparticle scattering states in non-relativistic QED: I. The Bloch-Nordsieck paradigm. Commun. Math. Phys. 294, 761–825 (2010)
Chen, T., Fröhlich, J., Pizzo, A.: Infraparticle scattering states in non-relativistic QED: II. Mass shell properties. J. Math. Phys. 50, 012103–012134 (2009)
Deckert, D.-A., Nickel, L.: Consistency of multi-time Dirac equations with general interaction potentials. J. Math. Phys. 57(7), 072301 (2016)
Deckert, D.-A., Nickel, L.: Multi-time dynamics of the Dirac-Fock-Podolsky model of QED. J. Math. Phys. 60, 072301 (2019)
Dirac, P.A.M.: Relativistic Quantum Mechanics. Proc. R. Soc. Lond. A 136, 453–464 (1932)
Dürr, D., Goldstein, S., Münch-Berndl, K., Zanghì, N.: Hypersurface Bohm-Dirac models. Phys. Rev. A 60, 2729–2736 (1999)
Dürr, D., Teufel, S.: Bohmian Mechanics. Springer, Berlin (2009)
Dürr, D., Goldstein, S., Norsen, T., Struyve, W., Zanghì, N.: Can Bohmian mechanics be made relativistic? Proc. R. Soc. A 470(2162), 20130699 (2014)
Einstein, A.: Zum gegenwärtigen Stand des Strahlungsproblems. Phys. Zeitschr. 10, 185–193 (1909)
Einstein, A.: Über die Entwicklung unserer Anschauungen über das Wesen und die Konstitution der Strahlung. Verh. Deutsch. Phys. Ges. 7, 482–500 (1909)
Fröhlich, J., Griesemer, M., Schlein, B.: Asymptotic completeness for Compton scattering. Commun. Math. Phys. 252, 415–476 (2004)
Garabedian, P.R.: Partial Differential Equations. AMS Chelsea Publication, New York City (1998)
Guerra, F.: Introduction to Nelson stochastic mechanics as a model for quantum mechanics. In: Garola, C., Rossi, A. (eds.) The Foundations of Quantum Mechanics, pp. 339–355. Kluwer, Amsterdam (1995)
Jost, R.: Das Märchen vom elfenbeinernen Turm. Springer, Wien (2002)
Kiessling, M.K.-H., Tahvildar-Zadeh, A.S.: On the quantum mechanics of a single photon. J. Math. Phys. 59, 112302 (2018)
Kocsis, S., Braverman, B., Ravets, S., Stevens, M.J., Mirin, R.P., Shalm, L.K., Steinberg, A.M.: Observing the average trajectories of single photons in a two-slit interferometer. Science 332, 1170–1173 (2011). https://doi.org/10.1126/science.1202218
Lienert, M.: A relativistically interacting exactly solvable multi-time model for two massless Dirac particles in 1+1 dimensions. J. Math. Phys. 56, 042301 (2015)
Lienert, M., Nickel, L.: A simple explicitly solvable interacting relativistic \(N\)-particle model. J. Phys. A Math. Theor. 48(32), 325301 (2015)
Lienert, M., Nickel, L.: Multi-time formulation of particle creation and annihilation via interior-boundary conditions. Rev. Math. Phys. 32, 2050004 (2020)
Lienert, M., Petrat, S., Tumulka, R.: Multi-time wave functions. J. Phys. Conf. Ser. 880(1), 012006 (2017)
Lienert, M., Tumulka, R.: Born’s rule for arbitrary Cauchy surfaces (2017). arXiv:1706.07074
Penrose, R.: The Emperor’s New Mind, 2nd edn. Oxford University Press, New York (1990)
Petrat, S., Tumulka, R.: Multi-time Schrödinger equations cannot contain interaction potentials. J. Math. Phys. 55, 032302 (2014)
Petrat, S., Tumulka, R.: Multi-time wave functions for quantum field theory. Ann. Phys. 345, 17–54 (2014)
Riesz, M.: Sur certaines notions fondamentales en théorie quantique relativiste, In “Dixieme Congres Math. des Pays Scandinaves,” 123–148, Copenhagen (1946)
Schweber, S.: QED and the Men Who Made It: Dyson, Feynman, Schwinger, and Tomonaga. Princeton University Press, Princeton (1994)
Teufel, S., Tumulka, R.: Simple Proof for Global Existence of Bohmian Trajectories. Commun. Math. Phys. 258, 349–365 (2005)
Teufel, S., Tumulka, R.: Hamiltonians without ultraviolet divergence for quantum field theories (2015). arXiv:1505.04847
Teufel, S., Tumulka, R.: Avoiding ultraviolet divergence by means of interior-boundary conditions. In: Finster, F., Kleiner, J., Röken, C., Tolksdorf, J. (eds.) Quantum Mathematical Physics. Birkháuser, Cham (2016)
Thaller, B.: The Dirac Equation. Springer, Berlin (1992)
Weinberg, S.: The Quantum Theory of Fields, vol. I. Cambridge University Press, Cambridge (1995)
Acknowledgements
We would like to thank Hans Jauslin, Stefan Teufel and Roderich Tumulka for helpful discussions. Thanks to Lawrence Frolov and Samuel Leigh for giving our paper a careful reading and for their helpful suggestions.
Funding
Thanks go also to the referees for their comments. 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.
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Appendices
Appendix
A. Solution formulas for the Klein–Gordon equation
We recall that the general solution to the Cauchy problem for the Klein–Gordon equation
is
where \(J_\nu \) is the Bessel function of index \(\nu \).
We also need the solution formula for the Goursat (i.e., characteristic initial data) problem for the 1-dimensional Klein–Gordon equation:
which is (see e.g., [22], eq. (4.85)):
(Note that \(\zeta (0)=\xi (0)\) is necessary for continuity.) Using integration by parts, the above can also be written as
B. Existence and uniqueness for the electron–photon trajectories
For completeness, we include the theorem about the global existence and uniqueness of Bomian trajectories by Teufel and Tumulka [37] which we heavily use in Sect. 7.
The notation is as follows: The configuration space \({\mathcal {Q}}\) is taken to be either \({\mathcal {Q}} = {\mathbb {R}}^d\) or \({\mathcal {Q}} = {\mathbb {R}}^d \backslash \bigcup _{l=1}^m S_l\) where each \(S_l\) is an admissible set. A set \(S \subset {\mathbb {R}}^d\) is said to be admissible if there is a \(\delta > 0\) such that the distance function \(q \mapsto \text {dist}(q,S)\) is differentiable on the open set \((S + \delta ) \backslash S\) where \(S + \delta = \{ q \in {\mathbb {R}}^d : \mathrm{dist}(q,S) < \delta \}\). Let \({{\mathcal {N}}}_t\), \(Q_q(t)\), and \(\varphi _t\) be defined as before. This allows us to introduce the notion of equivariance as follows: Let \(\rho _t\) be the distribution of \(Q_q(t)\) if q has distribution \(\mu _0\), i.e., \(\rho _t = \mu _0 \circ \varphi _t^{-1}\). One then says that the family of measures \(\mu _t\) is equivariant on a time interval I if \(\rho _t = \mu _t\) for all \(t \in I\). (Intuitively, this means that the measures \(\mu _t\) capture the time-evolved distribution of trajectories correctly.)
Theorem B.1
([37, Theorem 1]) Let \({\mathcal {Q}} \subset {\mathbb {R}}^d\) be a configuration space as defined above and let \(j=(\rho , J)\) be a current with \(\rho : [0,\infty ) \times {\mathcal {Q}} \rightarrow [0,\infty )\), \(J : [0,\infty ) \times {\mathcal {Q}} \rightarrow {\mathbb {R}}^d\) where:
-
(i)
\(\rho \) and J are continuously differentiable,
-
(ii)
\(\partial _t \rho + \mathrm{div} \, J = 0\),
-
(iii)
\(\rho > 0\) whenever \(\rho \ne 0\),
-
(iv)
\(\int _{\mathcal {Q}} dq \, \rho (t,q) = 1\) for all \(t \ge 0\).
Let \(T>0\) and let \({\mathcal {B}}({\mathcal {Q}})\) denote the set of all bounded Borel sets in \({\mathcal {Q}}\). Suppose that
and, if \({\mathcal {Q}}={\mathbb {R}}^d \backslash \bigcup _{l=1}^m S_l\), in addition that for every \(l =1,\ldots ,m\):
Here, \(\mathrm{dist}(q,S_l)\) is the Euclidean distance of q from \(S_l\) and \(e_l(q) = -\nabla _q \mathrm{dist}(q,S_l)\) is the radial unit vector toward \(S_l\) at \(q \in {\mathcal {Q}}\).
Then for almost every \(q \in {\mathcal {Q}}\) relative to the measure \(\mu _0(dq) = \rho (0,q) dq\), the solution of (7.2) starting at \(Q(0) = q\) exists at least up to time T, and the family of measures \(\mu _t(dq) = \rho (t,q) dq\) is equivariant on [0, T]. In particular, if (B.1), (B.2) and, if appropriate, (B.3) are true for every \(T>0\), then for \(\mu _0\)-almost every \(q \in {\mathcal {Q}}\) the solution of (7.2) starting at q exists for all times \(t \ge 0\).
Remark The conditions (B.1), (B.2) and (B.3) have the following intuitive meanings. If (B.1) holds, \(\mu _0\)-almost no trajectory approaches a node for \(0 \le t \le T\). If (B.2) is satisfied, \(\mu _0\)-almost no trajectory escapes to infinity for \(0 \le t \le T\). If (B.3) holds, \(\mu _0\)-almost no trajectory approaches the critical set \(\bigcup _{l=1}^m S_l\) for \(0 \le t \le T\).
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Kiessling, M.KH., Lienert, M. & Tahvildar-Zadeh, A.S. A Lorentz-covariant interacting electron–photon system in one space dimension. Lett Math Phys 110, 3153–3195 (2020). https://doi.org/10.1007/s11005-020-01331-8
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DOI: https://doi.org/10.1007/s11005-020-01331-8