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A Time-Symmetric Soliton Dynamics à la de Broglie

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Abstract

In this work we develop a time-symmetric soliton theory for quantum particles inspired from works by de Broglie and Bohm. We consider explicitly a non-linear Klein–Gordon theory leading to monopolar oscillating solitons. We show that the theory is able to reproduce the main results of the pilot-wave interpretation for non interacting particles in a external electromagnetic field. In this regime, using the time symmetry of the theory, we are also able to explain quantum entanglement between several solitons and we reproduce the famous pilot-wave nonlocality associated with the de Broglie-Bohm theory.

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Notes

  1. We use the Minkowski metric \(\eta _{\mu \nu }\) with signature \(+,-,-,-\) and the convention \(\hbar =1\), \(c=1\).

  2. In [22] we obtained the expression for \({\mathcal {M}}_{u}(x)\) (Eq. 66) without expliciting the first order correction \(O(\xi )\). The details of the calculations show that we have \({\mathcal {M}}_{u}(x)\simeq {\mathcal {M}}_{\Psi} (z(\tau ))+{\varvec{\xi }}\cdot {\varvec{\nabla }}{\mathcal {M}}_{\Psi} (z(\tau ))+O(\xi ^{2})\). A careful analysis shows that we have \(\partial _{\mu} {\mathcal {M}}_{u}(z(\tau ))=\partial _{\mu} {\mathcal {M}}_{\Psi} (z(\tau ))\).

  3. We have the fluid conservation: \(v_{u}\partial \ln {(f^{2}{\mathcal {M}}_{u}\delta ^{3}\sigma _{0})}:=\frac{d}{d\tau }\ln {(f^{2}{\mathcal {M}}_{u}\delta ^{3}\sigma _{0})}=0\).

  4. Ehrenfest theorem has been also applied by Bialynicki-Birula and Mycielski [5] in the context of classical Gausson dynamics driven by external electromagnetic forces. The theorem has also been used by Durt and coworkers [10, 33] in the particular context of the nonrelativistic Schrödinger–Newton equations. Our own independent results are based on the relativistic NLKG an is valid for a large range of equations.

  5. This invariance allows us to circumvent the conclusions of the Hobart–Derrick theorem [17, 31, 34] which usually precludes the existence of static and stable solitons in 3D space. In “Appendix 2” we give an elementary proof of this result.

  6. We stress that de Broglie rejected the non singular wave \(u(t,r)=e^{-i\omega _{0} t}\frac{\sin {(\omega _{0} r)}}{r}\) solution of the homogeneous wave equation. This alternative solution was later rediscovered by many authors including Mackinnon [36] and Barut [3]. At the experimental level such non dispersive de Broglie waves have been recently created [32] using optical methods inspired from diffraction-free beams works [23]. This is also related to the work of Fink concerning time reversal mirrors in acoustics and optics [2, 29] that plays with causality. This could perhaps play a role in order to develop a physical wave analog of our time symmetric soliton.

  7. We have \(G^{(0)}_{sym,\omega }(R)=\frac{1}{2}[G^{(0)}_{ret,\omega }(R)+G^{(0)}_{adv,\omega }(R)]\) and \(G^{(0)}_{ret/adv,\omega }(R)=\frac{e^{\pm i\omega R}}{4\pi R}\) are the retarded and advanced Green functions respectively.

  8. We have also \(K^{(0)}_{sym}(x,x')=\int _{-\infty }^{+\infty }G^{(0)}_{sym,\omega }(R)e^{-i\omega (t-t')}\frac{d\omega }{2\pi }\) with \(G^{(0)}_{sym,\omega }(R)\) given by Eq. 34.

  9. We mention that F. Fer in 1957 developed a method for analyzing the motion of singularities in the context of the DSP [27, 28]. Moreover, his approach using only retarded Green’s functions missed the time-symmetry needed to recover the wave particle duality considered here.

  10. More precisely we have \(\sigma _{\mp} =r\left( 1+\frac{\xi \ddot{z}}{2} \mp \frac{r\xi z^{(3)}}{6}-\frac{r^{2}(\ddot{z})^{2}}{24}+\frac{3(\xi \ddot{z})^{2}}{8}\right) +O(r^{4})\) with \(z^{(3)}:=\frac{d^{3} z(\tau )}{d\tau ^{3}}\).

  11. A proof is obtained by using the Fourier transform \(K(x,x')=\int \frac{d^{4}k}{(2\pi )^{4}}e^{ik(x-x')}G_{k}\). Equation 49 reads thus \(G_{k}=G_{k}^{(0)}+G_{k}^{(0)}(e^{2}A^{2}-2eAk)G_{k}\), i.e., \(G_{k}=\frac{G_{k}^{(0)}}{1-(e^{2}A^{2}-2eAk)G_{k}^{(0)}}\). Using \(G_{k}^{(0)}=-1/k^{2}\) we deduce \(G_{k}=-1/(k-eA)^{2}\) and after using the inverse Fourier transform we obtain \(K_{sym}(x,x')=K^{(0)}_{sym}(x,x')e^{-ieA(z)(x-x')}\).

  12. To prove this rather general statement a qualitative argument could go like this: Considering the LKG equation \(D^{2}\Psi =-\omega _{0}^{2}\Psi \) in a electrostatic potential \(V({\textbf{x}})\) the first Born order scattering amplitude for an incident plane wave \(\Psi _{0}({\textbf{x}})=e^{ik\textbf{n}_{0}\cdot {\textbf{x}}}\) (with \(k^{2}=\omega ^{2}-\omega _{0}^{2}\)) reads \(\Psi _{s}\simeq -2\omega \int d^{3}{\textbf{x}}''G_\omega ^{(0)}({\textbf{x}},{\textbf{x}}'')eV({\textbf{x}}'')\Psi _{0}({\textbf{x}}'')\simeq -2\omega \frac{e^{ikr}}{4\pi r} e\hat{V}_{\textbf{q}}\) where we neglected the quadratic term \(e^{2}V^{2}\), \(G_\omega ^{(0)}({\textbf{x}},{\textbf{x}}'')=\frac{e^{ikR|{\textbf{x}}-{\textbf{x}}''|}}{4\pi |{\textbf{x}}-{\textbf{x}}''|}\) [computed here for a retarded wave], and where \(\hat{V}_{\textbf{q}}\) is the Fourier transform of the potential at the wave wavevector \(\textbf{q}=k(\textbf{n}_{s}-\textbf{n}_{0})\). In a Coulomb field for example we have \(\Psi _{s}\simeq -\frac{2\omega \alpha }{\textbf{q}^{2}}\frac{e^{ikr}}{ r}\). The same calculation done for a plane wave solution of the linearized equation for u \(D^{2}u=0\) leads to the same expression with \(\omega ^{2}\) replacing \(k^{2}\). Therefore the scattered field \(u_{s}\) is smaller than \(\Psi _{s}\) by a coefficient \(\frac{u_{s}}{\psi _{s}}\simeq k^{2}/\omega ^{2}=v^{2}\) where v is the particle velocity. In general \(v^{2}\ll 1\) and \(u_{s}\) is negligible.

  13. We have \(D_{i}=\partial _{i}+eA(x_{i})\), \(\partial _{i}:=\frac{\partial }{\partial x_{i}}\) and the polar form \(\Psi _{N}(x_{1},\ldots ,x_{i},\ldots ,x_{N})=a_{N}(x_{1},\ldots ,x_{i},\ldots ,x_{N})e^{iS_{N}(x_{1},\ldots ,x_{i},\ldots ,x_{N})}\).

  14. This is also true in the recent important experimental/theoretical work done with hydrodynamical analogs by Bush, Vervoort and coworkers [38, 40, 50]. In these experiments there is a form of superdeterminism driven by stationnary Faraday waves moving along a liquid surface. Yet, the development was for the moment limited to time-independent configurations like in the first Aspect’s experiments. Time varying settings would require a stronger conspiracy that can be generated by a time symmetric causality. The problem is thus to generate the right amount of conspiracy in order to reproduce quantum predictions and not look too magical or too contrived. This is exactly the solution offered in the present work with non-linear waves involving time symmetry.

  15. The error is small in the integration since \(U(f^{2})-f^{2}N(f^{2})\sim 1/r^{6}\) at large distances.

  16. This description made in the regime \(\omega _{0}T\gg 1\) is of course an approximation that neglects the transient effects associated with the discontinuities at A and B contributing to the energy balance.

  17. Of course the problem is absent if we limit the present model to neutral solitons with \(e=0\).

  18. Note that in order to have \(I_{p}< \infty \) we must have \(m>\frac{3}{2(p+1)}\) so that globally \(m> max\left[ \frac{1}{2},\frac{3}{2(p+1)}\right] \) [45].

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    Aurélien Drezet

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Drezet, A. A Time-Symmetric Soliton Dynamics à la de Broglie. Found Phys 53, 72 (2023). https://doi.org/10.1007/s10701-023-00711-z

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