Can Quantum Field Theory be Recovered from Time-Symmetric Stochastic Mechanics? Part II: Prospects for a Trajectory Interpretation

Quantum Physics [Submitted on 30 Mar 2026] Can Quantum Field Theory be Recovered from Time-Symmetric Stochastic Mechanics? Part II: Prospects for a Trajectory Interpretation Simon Friederich, Mritunjay Tyagi In a companion paper we derived a unique time-reversal-invariant stochastic generalization of the Liouville equation and showed that it coincides with the evolution equation for the Husimi Q-function in a broad class of bosonic quantum field theories. Here we investigate the prospects for interpreting that evolution equation in terms of underlying stochastic trajectories. Drawing on Drummond's time-symmetric stochastic action formalism, we show that the traceless diffusion Fokker-Planck equation defines a natural measure over stochastic trajectories conditional on mixed-time boundary conditions. However, we identify a significant gap: it has not been established that every Q-function can be represented as a weighted average of these conditional probabilities over boundary values. The trajectory interpretation holds for ensembles with fixed boundary conditions but does not straightforwardly extend to arbitrary quantum states. Despite this limitation, we show that Drummond's trajectory dynamics are fundamentally non-Markovian -- a natural consequence of combining stochasticity with time-reversal invariance. This non-Markovianity places the dynamics outside the scope of the ontological models framework and thereby explains why the major no-go theorems for hidden-variable theories do not rule out the approach. These results clarify both the achievements and the remaining challenges in the project of understanding quantum field theory as the statistical mechanics of time-symmetric stochastic processes. Comments: 13 pages, 2 figures Subjects: Quantum Physics (quant-ph); History and Philosophy of Physics (physics.hist-ph) Cite as: arXiv:2603.28983 [quant-ph]   (or arXiv:2603.28983v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2603.28983 Focus to learn more Submission history From: Simon Friederich [view email] [v1] Mon, 30 Mar 2026 20:31:35 UTC (31 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-03 Change to browse by: physics physics.hist-ph References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)