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A Lorentz-covariant interacting electron–photon system in one space dimension

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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

  1. This may require suitable conservation laws, see Sect. 4.

  2. We are only interested in Lorentz transformations with identical action on the photonic and electronic variables.

  3. “Almost everywhere” without any measure specified refers to the Lebesgue measure, here and henceforth.

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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

$$\begin{aligned} w_{tt} - w_{ss} + w = 0,\qquad w(0,s) = f(s),\qquad w_t(0,s) = g(s), \end{aligned}$$
(A.1)

is

$$\begin{aligned} w(t,s)= & {} \frac{1}{2}\{ f(s-t)+f(s+t) \}- \frac{t}{2}\int _{s-t}^{s+t} \frac{J_1(\sqrt{t^2 - (s-\sigma )^2})}{\sqrt{t^2 - (s-\sigma )^2}} f(\sigma ) \mathrm{d}\sigma \nonumber \\&+ \frac{1}{2}\int _{s-t}^{s+t} J_0(\sqrt{t^2 - (s-\sigma )^2}) g(\sigma ) \mathrm{d}\sigma , \end{aligned}$$
(A.2)

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:

$$\begin{aligned} U_{tt} - U_{ss} + U = 0 \text{ for } |s|<t,\qquad U(b,b) = \zeta (b),\qquad U(c,-c) = \xi (c),\nonumber \\ \end{aligned}$$
(A.3)

which is (see e.g., [22], eq. (4.85)):

$$\begin{aligned} U(t,s)= & {} \frac{1}{2}(\zeta (0)+\xi (0))J_0(\sqrt{t^2-s^2})\nonumber \\&+ \int _0^{(t+s)/2} \zeta '(b) J_0(\sqrt{(t-s)(t+s-2b)}) \mathrm{d}b \nonumber \\&+ \int _0^{(t-s)/2} \xi '(c) J_0(\sqrt{(t+s)(t-s-2c)}) \mathrm{d}c \end{aligned}$$
(A.4)

(Note that \(\zeta (0)=\xi (0)\) is necessary for continuity.) Using integration by parts, the above can also be written as

$$\begin{aligned} U(t,s)= & {} \zeta \left( \frac{t+s}{2}\right) + \xi \left( \frac{t-s}{2}\right) - \frac{1}{2}(\zeta (0)+\xi (0))J_0(\sqrt{t^2-s^2}) \nonumber \\&- (t-s) \int _0^{(t+s)/2} \zeta (b) \frac{J_1(\sqrt{(t-s)(t+s-2b)})}{\sqrt{(t-s)(t+s-2b)}} \mathrm{d}b\nonumber \\&- (t+s) \int _0^{(t-s)/2} \xi (c) \frac{J_1(\sqrt{(t+s)(t-s-2c)})}{\sqrt{(t+s)(t-s-2c)}} \mathrm{d}c. \end{aligned}$$
(A.5)

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:

  1. (i)

    \(\rho \) and J are continuously differentiable,

  2. (ii)

    \(\partial _t \rho + \mathrm{div} \, J = 0\),

  3. (iii)

    \(\rho > 0\) whenever \(\rho \ne 0\),

  4. (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

$$\begin{aligned} \forall B \in {\mathcal {B}}({\mathcal {Q}}): ~~~ \int _0^T dt \int _{\varphi _t(B) \backslash \{ \diamondsuit \}} dq \left| \left( \frac{\partial }{\partial t} + \frac{J}{\rho } \cdot \nabla _q \right) \rho (t,q) \right| < \infty , \end{aligned}$$
(B.1)
$$\begin{aligned} \forall B \in {\mathcal {B}}({\mathcal {Q}}): ~~~ \int _0^T dt \int _{\varphi _t(B) \backslash \{ \diamondsuit \}} dq \left| J(t,q) \cdot \frac{q}{|q|} \right| < \infty , \end{aligned}$$
(B.2)

and, if \({\mathcal {Q}}={\mathbb {R}}^d \backslash \bigcup _{l=1}^m S_l\), in addition that for every \(l =1,\ldots ,m\):

$$\begin{aligned} \exists \delta > 0: \forall B \in {\mathcal {B}}({\mathcal {Q}}): ~~~ \int _0^T dt \int _{\varphi _t(B) \backslash \{ \diamondsuit \}} dq ~ \mathbb {1} (q\in (S_l +\delta )) \frac{|J(t,q) \cdot e_l(q)|}{\mathrm{dist}(q,S_l)} < \infty . \nonumber \\ \end{aligned}$$
(B.3)

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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