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Simulating Nelsonian Quantum Field Theory

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Abstract

We describe the picture of physical processes suggested by Edward Nelson’s stochastic mechanics when generalized to quantum field theory regularized on a lattice, after an introductory review of his theory applied to the hydrogen atom. By performing numerical simulations of the relevant stochastic processes, we observe that Nelson’s theory provides a means of generating typical field configurations for any given quantum state. In particular, an intuitive picture is given of the field “beable”—to use a phrase of John Stewart Bell—corresponding to the Fock vacuum, and an explanation is suggested for how particle-like features can be exhibited by excited states. We then argue that the picture looks qualitatively similar when generalized to interacting scalar field theory. Lastly, we compare the Nelsonian framework to various other proposed ontologies for QFT, and remark upon their relative merits in light of the effective field theory paradigm. Links to animations of the corresponding beables are provided throughout.

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Notes

  1. Bell [1] suggested the particular term “beable” for such things (rather than, e.g., “being” or “beer”) to emphasize “the essentially tentative nature of any physical theory,” a sentiment we want to reiterate in this work. “In fact, ‘beable’ is short for ‘maybe-able’.”

  2. A few caveats: In [4], this problem is avoided by remarking that such eigenstate distributions, in realistic scenarios, arise from complex interactions, during which the wave function is not so simple. Second, once spin is included, the electron is no longer motionless in such states [5, 6].

  3. Not everybody regards this as unappealing; see [7] for various perspectives on the status of the wave function. See also [8, 9].

  4. Whether Nelson succeeded in deriving QM in its entirety has been argued in the negative by Wallstrom [14], but there is some disagreement [15]. Proposals to “complete” the derivation, assuming the validity of Wallstrom’s criticism, have also been made in recent years [16, 17].

  5. See [18, 29], and [30,31,32] for similar field-based proposals.

  6. What seems to have stuck around for longer was the related enterprise of Stochastic Quantization (SQ)—but this too has mostly fallen out of fashion. An important difference between SQ and Nelson should be stressed: the “time” of SQ is a fictitious “simulation time,” whereas the time for Nelson is a real, physical time. See [40] for a recent perspective on Nelson-Yasue QFT; and see [41] for an introduction to SQ.

  7. Although the derivative \(\dot{{\varvec{x}}} = \textrm{d}{\varvec{x}}/\textrm{d}t\) here is ill-defined, rigorous definitions exist via Wiener processes [41, 46], which are here swept aside to ease the presentation.

  8. Here \((Df)({\varvec{x}},t):= \lim _{\epsilon \rightarrow 0^+} \epsilon ^{-1}\textrm{E}[f({\varvec{x}}_{t+\epsilon },t+\epsilon )-f({\varvec{x}}_t,t)|{\varvec{x}}_t={\varvec{x}}]\). One can define a “backward” stochastic process associated with the forward process, i.e., a rule relating the current positions to prior positions in a probabilistic fashion, and an associated backward derivative \(D_*\); see [13, 19, 47].

  9. Yasue [48] has shown that Nelson’s Second Law follows from a stochastic variational principle, together with the demand of positive semidefinite mean kinetic energy.

  10. See [54] for an animation of this motion.

  11. Or, from an average along a single trajectory, since this state has a time-independent density. In this case one must properly deal with autocorrelations in the time series in order to obtain correct stochastic error estimates; see, e.g., [55]

  12. Hence there is no “sign problem,” as it is known in the lattice community [57], for Schrödinger picture expectation values. However, whether Nelson’s theory can be used to study problems whose wave functions are not known beforehand is an open question, and there is a debate about the ability to compute Heisenberg picture expectation values, like \(\langle \widehat{x}(t) \widehat{x}(0) \rangle \), in Nelson’s theory. See [58, 59] for the debate.

  13. We refer the reader to [29, 60, 61] as standard references on the Schrödinger picture.

  14. Strictly speaking, these are just ordinary functions of many variables \(\phi _{\varvec{x}}\), but we use the term “functional” by convention due to the similarity with continuum formulations of QFT [60].

  15. One may either compute the norms abstractly using the operator algebra, or by evaluating the functional integral directly, for which standard Gaussian integration formulas apply.

  16. The diffusion coefficient \(\nu \) in Nelsonian QFT is therefore independent of any particle mass, or the Klein-Gordon mass which appears in the dispersion relation, unlike in the nonrelativistic theory where \(\nu = \hbar /2m\).

  17. The square of \(\omega _{{\varvec{x}},{\varvec{y}}}\) as a matrix is (minus) the lattice Laplacian operator, plus a mass term.

  18. These states are similar to the Newton-Wigner states considered, for example, in [62], except for the particular momentum-dependence of coefficients. The states I use here are built from “positive-frequency” components of the field operator, \((\phi ^+_{\varvec{x}})^\dag \).

  19. Note that the normalization of \(\Psi _1\) is \((D_{0,0})^{-1/2}\), implying the overlaps \(\langle \Psi _1({\varvec{x}}_1)|\Psi _1({\varvec{x}}_2)\rangle = D_{0,0}^{-1} D_{{\varvec{x}}_1,{\varvec{x}}_2}\). Since \(D_{0,0}\) is UV-sensitive, while \(D_{{\varvec{x}},{\varvec{y}}}\) remains finite in the continuum limit, the overlap of normalized states is suppressed by the cutoff scale for non-coincident points.

  20. It’s nevertheless true that \(\langle \phi _{{\varvec{x}}} \rangle = 0\) in the state \(\Psi _1\), since the ground state is symmetric under \(\phi \rightarrow -\phi \). But whereas the configuration \(\phi _{\varvec{x}}= 0\) is typical in the ground state, it is not typical in \(\Psi _1\).

  21. See [63] for an animation of the field beable in this state.

  22. For example, regularizing the canonical commutators by defining \([\widehat{\phi }({\varvec{x}}), \widehat{\pi }({\varvec{y}})] = if_\Lambda ({\varvec{x}}-{\varvec{y}})\) (with sufficiently quickly decaying \(f_\Lambda \)) rather than \(i\delta ({\varvec{x}}-{\varvec{y}})\) yields a vacuum \(\Psi _0(\phi ) \propto \exp [-\frac{1}{2} \int \textrm{d}^d p \; \omega ({\varvec{p}}) \phi ({\varvec{p}}) \phi (-{\varvec{p}})/ f_\Lambda ({\varvec{p}})]\), which in turn yields a likeliest field configuration with \(\varphi ({\varvec{x}}_0)^2 \propto \int \textrm{d}^d p f_\Lambda ({\varvec{p}})/\omega ({\varvec{p}})\), and is finite if \(f_\Lambda ({\varvec{p}})\) decays more quickly than \(1/{\varvec{p}}^{d-1}\).

  23. See [64] for an animation of this field.

  24. This observation is not limited to just Bohm or Nelson’s theories: it applies to any interpretation which takes wave functionals and field beables seriously.

  25. See [65] for an example of a Bohmian field beable for a 2-particle time-dependent state.

  26. The coefficients have a structure analogous to the standard Feynman diagrams of QFT. The quartic term in \(R^{(1)}\) corresponds to the tree-level diagram for the 4-point function, while the quadratic term in \(R^{(1)}\) is the 1-loop correction to the 2-point function. The sextic term in \(R^{(2)}\) gives a tree-level 6-point function; the quartic term gives the 1-loop contribution to the 4-point function (both 1PI and non-1PI diagrams); the quadratic term gives the 2-loop contribution to the 2-point function. The superficial degree of divergence of the loops is the same as the corresponding standard Feynman diagrams (see [66] for a standard account). In this way we see how diagrams similar to those of standard QFT “contribute” to the vacuum wave functional.

  27. Equations of striking similarity to these appear in the context of functional RG, where \(\ln \Psi _0\) is replaced by an RG fixed point action, and \(\mathcal P_n\) is replaced by a scaling operator associated with that fixed point; see, e.g., [68]. A connection between RG and stochastic processes has been explored in [69, 70].

  28. There are also terms proportional to \(\phi _{\pm {\varvec{p}}}\) with undetermined coefficients, reflecting the degeneracy of the free states \(| 1,\pm {\varvec{p}} \rangle \). We have set these coefficients to zero.

  29. The extension to (pure) lattice gauge theories should be possible, without too much difficulty, by utilizing methods from the stochastic quantization of such theories [41, 53].

  30. See also [74] for a Lorentz covariant proposal.

  31. This is perhaps borrowed from the association of the divergences in classical electromagnetism with a point-like model of electrons. In QFT, the association is more difficult to make, since the ontology of QFT is not as clear to begin with.

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Acknowledgements

I would like to thank Maaneli Derakhshani, Ward Struyve, and Charles Sebens for discussions and feedback. This work has also benefited from comments received after presenting the material at the 2023 European Conference on Foundations of Physics in Bristol, UK, as well as the online Laws of Nature Series [79]. Lastly, I thank the reviewer for their valuable comments and suggestions for improving the manuscript. This work was supported in part by U.S. DOE Grant No. DE-FG02-95ER40907.

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Carosso, A. Simulating Nelsonian Quantum Field Theory. Found Phys 54, 30 (2024). https://doi.org/10.1007/s10701-024-00766-6

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