A 2026 paper by Camurati, Sobrero, Suassuna, and Paraguassú (J. Phys. A, arXiv:2602.00258) showed formally that, when an open quantum system is coupled to an environment through f(q)B^f(q)\hat{B} and the influence functional is truncated at Gaussian order and linearized in the quantum coherence coordinate y=q−q′y=q-q' (the strong-decoherence limit), the oscillatory Feynman path-integral description of the reduced density matrix turns into a Wiener-type stochastic path integral describing an underdamped Langevin equation with colored, generally non-Markovian noise coupled multiplicatively through f′(x)f'(x); the induced measure carries an explicit functional-determinant factor det⁡N−1\sqrt{\det N^{-1}} and Jacobian factors ∏t∣f′(xt)∣−1\prod_t|f'(x_t)|^{-1}, and reduces to the Onsager–Machlup weight in the high-temperature Caldeira–Leggett limit. The derivation is, however, formal and restricted: no parameter regimes or error estimates for the truncations are known; the induced measure has no rigorous mathematical construction for general multiplicative colored noise; the trajectory-level thermodynamics has so far been formulated only for additive coupling; the inverse problem (constructing a quantum influence functional realizing a prescribed classical Langevin equation) remains an explicitly phenomenological Gaussian construction; and the mapping is established only for one degree of freedom and Gaussian influence functionals. The problem is to determine when and in what precise sense this mapping holds and how far it reaches.

Background

An open quantum system—a system CC whose coordinates are denoted qq coupled to an environment QQ—has a reduced density matrix obtained by tracing out the environment. Feynman and Vernon (Ann. Phys. 24, 118, 1963) showed that this reduced dynamics can be written as a double path integral over a forward path q(t)q(t) and a backward path q′(t)q'(t), in which the environment enters only through a single object, the influence functional F[q,q′]\mathcal{F}[q,q'], multiplying the free forward–backward action phases. For a coupling of the form Hint∝f(q)B^H_{\rm int}\propto f(q)\hat{B} with B^\hat{B} an environment operator, expanding to second order in the interaction gives a Gaussian influence functional whose exponent splits into an imaginary (dissipation) part governed by a kernel D(t,t′)D(t,t') and a real (noise) part governed by a positive kernel N(t,t′)N(t,t'); at thermal equilibrium the two kernels are linked by a fluctuation–dissipation relation. Caldeira and Leggett's model of quantum Brownian motion (Physica A 121, 587, 1983) and the exact master equation of Hu, Paz, and Zhang (Phys. Rev. D 45, 2843, 1992) are the canonical instances, with NN a colored (time-correlated, hence non-Markovian) noise kernel.

Introduce the center coordinate x=(q+q′)/2x=(q+q')/2 and the difference coordinate y=q−q′y=q-q'. The variable yy measures the quantum coherence length—the off-diagonal spread of the density matrix—so decoherence suppresses integrals over yy. In the strong-decoherence limit, the coherence length lcl_c associated with yy is assumed small compared to the macroscopic scale ll on which ff and the potential VV vary, so the influence functional and the system action can be linearized in yy. Performing the Gaussian (Hubbard–Stratonovich) transformation that decouples the noise part introduces an auxiliary stochastic field ηt\eta_t with covariance N(t,t′)N(t,t'), weighted by a Wiener-type measure whose normalization involves the functional-determinant factor det⁡N−1\sqrt{\det N^{-1}}. The remaining integral over yy produces a functional Dirac delta that enforces a generalized Langevin equation for xx—the quantum-induced stochastic dynamics (QISD)—in which the noise enters multiplicatively (state-dependently) through f′(xt)ηtf'(x_t)\eta_t; isolating ηt\eta_t inside the delta generates Jacobian factors ∏t∣f′(xt)∣−1\prod_t|f'(x_t)|^{-1} in the measure.

On the classical side, diffusion processes have an equivalent path-integral representation, the Onsager–Machlup (Wiener) path integral, and classical stochastic dynamics can be reformulated as a field theory via the Martin–Siggia–Rose response formalism. The mathematical statuses of the two sides are, however, radically different: the real-time Feynman 'measure' is oscillatory and exists only as a formal or time-sliced object, never as a countably additive probability measure, whereas the Wiener measure of diffusion is a genuine probability measure. The claim that decoherence converts one into the other is therefore intrinsically a statement about a limit whose rigorous meaning is not automatic. For classical Langevin equations with multiplicative noise, the path integral itself is known to depend on the time-discretization prescription (Itô versus Stratonovich type), and building a discretization-covariant path-integral calculus for multiplicative noise is an active topic in its own right (Arenas and Barci, Phys. Rev. E 81, 051113, 2010; Cugliandolo, Lecomte, and van Wijland, J. Phys. A 52, 50LT01, 2019).

The QISD framework underlies current models of classical probes driven by quantum systems: optomechanical fluctuations of nanoparticles coupled to cavity light (Paraguassú, Abrahão, and Guerreiro, JOSA B 41, 2798, 2024), a classical mesoscopic oscillator driven by a two-level system (Sobrero et al., arXiv:2509.04216), graviton-noise phenomenology at gravitational-wave detectors (Parikh, Wilczek, and Zahariade, Phys. Rev. D 104, 046021, 2021), and noise in semiclassical gravity (Calzetta and Hu, Phys. Rev. D 49, 6636, 1994). In the Wigner representation (the phase-space quasi-probability obtained by Fourier transforming the density matrix in yy), the source paper shows that the reduced dynamics composes with a classical stochastic conditional probability—exact for linear system–environment couplings—and the high-temperature Caldeira–Leggett limit reproduces the Onsager–Machlup weight of underdamped Brownian motion, benchmarking the mapping. A follow-up by the same group (Paraguassú, Phys. Rev. E, 2026, arXiv:2602.03764) has since formulated a stochastic thermodynamics for the additive-coupling case, defining trajectory-level heat, work, and entropy production and a modified Second Law with a correction for non-equilibrium (e.g. squeezed) environment states. What remains unproven is the subject of the problem below: the validity domain of the approximations, the rigorous existence of the induced measure, and the extension of the mapping to multiplicative-coupling thermodynamics, to the inverse quantization in the multiplicative-noise regimes the source declares open, and to the regimes where the linearization in yy fails.

Problem Statement

Determine when and in what precise mathematical sense the strong-decoherence Feynman–Vernon construction defines a bona fide classical stochastic process, and extend the mapping beyond its current formal, Gaussian regime. Concretely, for a system particle of mass mm moving in a potential VV and coupled to an environment through f(q)B^f(q)\hat{B}, whose Gaussian-truncated influence functional in the linear-yy (strong-decoherence) approximation is

F[x,y]=exp⁡ ⁣(iℏ∫dt dt′ f′(xt) f(xt′) yt D(t,t′)−12∫dt dt′ yt [f′(xt) f′(xt′) N(t,t′)] yt′),\mathcal{F}[x,y]=\exp\!\left(\frac{i}{\hbar}\int dt\,dt'\, f'(x_t)\,f(x_{t'})\,y_t\,D_{(t,t')}-\frac{1}{2}\int dt\,dt'\,y_t\,\big[f'(x_t)\,f'(x_{t'})\,N_{(t,t')}\big]\,y_{t'}\right),

where xt=(qt+qt′)/2x_t=(q_t+q'_t)/2 and yt=qt−qt′y_t=q_t-q'_t denote the center and difference (coherence) coordinates of the forward and backward paths, DD is the dissipation kernel and NN the noise kernel (and f′f' denotes differentiation of the coupling function). The real noise quadratic form enters here with a negative sign, so that the integral over yy is suppressed and produces a functional Dirac delta enforcing the quantum-induced Langevin equation

mx¨t+V′(xt)=∫ds f′(xt) D(t,s) f(xs)+f′(xt) ηt,m\ddot{x}_t+V'(x_t)=\int ds\,f'(x_t)\,D_{(t,s)}\,f(x_s)+f'(x_t)\,\eta_t ,

and the Wiener-type measure over η\eta is normalized by the functional-determinant factor det⁡N−1\sqrt{\det N^{-1}}. With these objects fixed, resolve all five of the following jointly linked facets, which together constitute a solution:

  1. Validity: establish the parameter regimes — in terms of the coherence-length ratio lc/ll_c/l, the coupling strength, the temperature, the bath spectral density, and the anharmonicity of VV and ff — in which the strong-decoherence limit, the linearization in yy, and the Gaussian truncation hold, with quantitative error estimates for the neglected higher-order yy corrections and non-Gaussian environmental cumulants.

  2. Rigor: for general multiplicative colored noise, either construct the induced path-integral measure — giving a mathematically well-defined meaning of the functional determinant det⁡N−1\det N^{-1}, the Jacobian factors ∏t∣f′(xt)∣−1\prod_t|f'(x_t)|^{-1}, the continuum limit of the time-sliced measure, and the discretization prescription that preserves covariance — together with positivity and normalization of the resulting stochastic process, or characterize precisely where such a positive, normalized measure fails to exist, in particular for non-Markovian kernels N(t,t′)N(t,t') and at zeros of f′f'.

  3. Thermodynamics: formulate the stochastic energetics of the quantum-induced trajectories in the regimes not covered by the existing additive-coupling treatment, assigning trajectory-level heat, work, and entropy production when the noise enters multiplicatively as f′(xt)ηtf'(x_t)\eta_t, and derive the corresponding fluctuation relations and second-law statements.

  4. Inverse problem: extend the source's inverse-quantization scheme — presently an explicitly phenomenological construction that realizes prescribed Langevin equations of the form mx¨+V′(x)+∫0tdt′ g(xt′) D(t,t′)=f(xt) ξ(t)m\ddot{x}+V'(x)+\int_0^t dt'\,g(x_{t'})\,D(t,t')=f(x_t)\,\xi(t) with state-dependent noise and dissipation, for Gaussian noise ξ\xi with ⟨ξ(t)ξ(t′)⟩=N(t,t′)\langle\xi(t)\xi(t')\rangle=N(t,t') — in the direction the source itself declares open, namely effective Langevin descriptions for more complex systems in regimes characterized by multiplicative noise. Given such a prescribed classical Langevin equation, construct a quantum influence functional, or a microscopic open-quantum-system model, that realizes it, or establish that no such realization exists (for instance from positivity or complete-positivity constraints on influence functionals).

  5. Extensions: determine whether a Wiener-type probabilistic description survives in regimes where the linearization in yy fails (highly anharmonic potentials, low temperatures with large coherences), or must be replaced by other objects.

Facets 1 and 2 establish the validity and the mathematical meaning of the mapping; facets 3, 4, and 5 extend the same mapping to the regimes that the source work and its follow-up explicitly left open; together they answer whether the emergent classical stochastic description of a decohering open quantum system is a controlled approximation backed by a well-defined mathematical object, and how far it reaches.

References

  1. Antonio Camurati, Felipe Sobrero, Bruno Suassuna, Pedro V. Paraguassú, 'From Feynman–Vernon to Wiener stochastic path integral', Journal of Physics A: Mathematical and Theoretical 59, 245304 (2026), DOI: 10.1088/1751-8121/ae748c; arXiv:2602.00258, https://doi.org/10.1088/1751-8121/ae748c
  2. Pedro V. Paraguassú, 'Stochastic thermodynamics of quantum-induced stochastic dynamics', Physical Review E (2026), DOI: 10.1103/tgsz-1mm9; arXiv:2602.03764, https://doi.org/10.1103/tgsz-1mm9
  3. Daniel Valente, Maurício Matos, Thiago Werlang, 'Thermoforesis from generalized Caldeira-Leggett models', Brazilian Journal of Physics 56, 213 (2026), DOI: 10.1007/s13538-026-02131-6; arXiv:2603.26980, https://doi.org/10.1007/s13538-026-02131-6
  4. Pedro V. Paraguassú, Luca Abrahão, Thiago Guerreiro, 'Quantum-induced stochastic optomechanical dynamics', Journal of the Optical Society of America B 41, 2798–2810 (2024), DOI: 10.1364/JOSAB.532994; arXiv:2401.16511, https://doi.org/10.1364/JOSAB.532994
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  6. Pedro V. Paraguassú, Thiago Guerreiro, 'Apparent violations of the second law in the quantum-classical dynamics of interacting levitated nanoparticles', Physical Review E 112, 024134 (2025), DOI: 10.1103/5s83-qzzk, https://doi.org/10.1103/5s83-qzzk
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  12. Leticia F. Cugliandolo, Vivien Lecomte, Frédéric van Wijland, 'Building a path-integral calculus: a covariant discretization approach', Journal of Physics A: Mathematical and Theoretical 52, 50LT01 (2019), DOI: 10.1088/1751-8121/ab3ad5, https://doi.org/10.1088/1751-8121/ab3ad5
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