We present an extension of Nelson's stochastic quantum mechanics to finite temperature. Utilizing the formulation of Thermo Field Dynamics (TFD), we can show that Ito's stochastic equations for tilde and non-tilde particle positions reproduce the TFD-type Schrödinger equation which is equivalent to the Liouville-von Neumann equation. In our formalism, the drift terms in the Ito's stochastic equation have the temperature dependence and the thermal fluctuation is induced through the correlation of the non-tilde and tilde particles. We show that our formalism satisfy the position-momentum uncertainty relation at finite temperature.
A serious objection made by Wallstrom against quantum interpretations based flow variables, in particular Nelsonian stochastics, is their empirical inequivalence with quantum theory: They are unable to obtain a quantization condition for flows around zeros which is automatically fulfilled by the wave function. It is found that the quantization condition follows from a simple additional postulate: The Laplace operator of the density has to be finite and positive at zeros of the density. This postulate is a quite natural consequence of subquantum theories, as far as they conform to a simple prin…
We present a class of nonlinear Schroedinger equations (NLSEs) describing, in the mean field approximation, systems of interacting particles. This class of NLSEs is obtained generalizing expediently the approach proposed in Ref. [G.K. Phys. Rev. A 55, 941 (1997)], where a classical system obeying to an exclusion-inclusion principle is quantized using the Nelson stochastic quantization. The new class of NLSEs is obtained starting from the most general nonlinear classical kinetics compatible with a constant diffusion coefficient D=\hbar/2m. Finally, in the case of s-stationary states, we propose…
The generalized Fenyes--Nelson model of quantum mechanics is applied to the free scalar field. The resulting Markov field is equivalent to the Euclidean Markov field with the times scaled by a common factor which depends on the diffusion parameter. This result is consistent between Guerra's earlier work on stochastic quantization of scalar fields. It suggests a deep connection between Euclidean field theory and the stochastic interpretation of quantum mechanics. The question of Lorentz covariance is also discussed.
I examine whether Nelson's stochastic formulation of Schrödinger equation could be derived from a phase space process through a colored noise smoothing. If this conjecture is true, it would yield a local stochastic hidden variable theory. I discuss how this does not necessarily contradict Bell type theorems as general local stochastic theories can violate local causality assumptions. I also discuss the generalization to quantization of fields and speculate about the gravitational origins of noise.
Wallstrom's criticism of existing formulations of stochastic mechanics is that they fail to derive the empirical predictions of orthodox quantum mechanics because they require an ad hoc quantization condition on the postulated velocity potential, \emph{S}, in order to derive Schrödinger wave functions. We propose an answer to this criticism by modifying the Nelson-Yasue formulation of non-relativistic stochastic mechanics for a spinless particle with the following hypothesis: a spinless Nelson-Yasue particle of rest mass $m$ continuously undergoes a driven steady-state oscillation of `zitterbe…
The well-known hydrodynamical representation of the Schrödinger equation is reformulated by extending the idea of Nelson-Yasue's stochastic variational method. The fluid flow is composed by the two stochastic processes from the past and the future, which are unified naturally by the principle of maximum entropy. We show that this formulation is easily applicable to the quantization of scalar fields.
We embed Nelson's stochastic quantization in the Schwartz-Meyer second order geometry framework. The result is a non-perturbative theory of quantum mechanics on (pseudo)-Riemannian manifolds. Within this approach, we derive stochastic differential equations for massive spin-0 test particles charged under scalar potentials, vector potentials and gravity. Furthermore, we derive the associated Schrödinger equation. The resulting equations show that massive scalar particles must be conformally coupled to gravity in a theory of quantum gravity. We conclude with a discussion of some prospects of the…
In solving the problem of finding a temperature distribution which, at zero temperature, corresponds to superfluidity, i.e., to nonzero energy, the author tried to quantize free energy. This was done on the basis of supersecondary quantization whose special case is the usual secondary quantization for bosons and with the help of which new representations of the Schrödinger equation were obtained. The supersecondary quantization allowed the author to construct a variational method whose zero approximation are the Hartree-Fock and Bogolyubov-BCSch variational principles. This method works especi…
A new idea for the quantization of dynamic systems, as well as space time itself, using a stochastic metric is proposed. The quantum mechanics of a mass point is constructed on a space time manifold using a stochastic metric. A stochastic metric space is, in brief, a metric space whose metric tensor is given stochastically according to some appropriate distribution function. A mathematically consistent model of a space time manifold equipping a stochastic metric is proposed in this report. The quantum theory in the local Minkowski space can be recognized as a classical theory on the stochastic…
It has very recently been suggested that asymmetric coupling of electromagnetic fields to thermal reservoirs under nonequilibrium conditions can produce unexpected oscillatory behavior in the local photon statistics in layered structures. Better understanding of the predicted phenomena could enable useful applications related to thermometry, noise filtering, and enhancing optical interactions. In this work we briefly review the field quantization and study the local steady state temperature distributions in optical cavities formed of lossless and lossy media to show that also local field tempe…
Stochastic mechanics (SM), as proposed by Edward Nelson and others in the 20th century, aims to reconstruct quantum mechanics (QM) from a more fundamental theory of classical point particles interacting with a classical-like ether, where said interaction causes the particles to undergo a diffusion process that conserves their average total energy. However, Timothy Wallstrom and others have emphasized that SM runs into the problem that it cannot recover the Schroedinger equation of QM unless an ad hoc quantization condition is assumed. In this thesis, I reformulate SM so that the quantization c…
One of the most important applications of quantum mechanics is the thermodynamic description of quantum gases. Despite the fundamental importance of this topic, a comprehensive description of the thermodynamic properties of non-Hermitian quantum gases is still lacking. Here, we investigate the properties of bosonic and fermionic non-Hermitian systems at finite temperatures. We show that non-Hermitian systems exihibit oscillations both in temperature and imaginary time. As such, they can be a possible platform to realize an imaginary time crystal (iTC) phase. The Hatano-Nelson model is identifi…
Quantization Mapping on Dirac Dynamics via Voltage-Driven Charge Density in Monolayer Graphene: A Klein Paradox and Entropy-Ruled Wavevector Mechanics Study
KPR Institute of Engineering and Technology（KPR工程技术学院）
We show that when the thermal wavelength is comparable to the spatial size of a system, thermodynamic observables like Pressure and Volume have quantum fluctuations that cannot be ignored. They are now represented by operators; conventional (classical) thermodynamics is no longer applicable. We continue the work in earlier papers where quantization rules for thermodynamics were developed by analogy with optics and mechanics, by working out explicitly the quantum theory of van der Waals gases. We find a wave equation satisfied by the thermodynamic wave function as well as solutions ({\em cohere…
Non-canonical quantization is based on certain reducible representations of canonical commutation relations. Relativistic formalism for electromagnetic non-canonical quantum fields is introduced. Unitary representations of the Poincaré group at the level of fields and states are explicitly given. Multi-photon and coherent states are introduced. Statistics of photons in a coherent state is Poissonian if an appropriately defined thermodynamic limit is performed. Radiation fields having a correct $S$ matrix are constructed. The $S$ matrix is given by a non-canonical coherent-state displacement op…
In this paper it is argued how the dynamics of the classical Newtonian N-body system can be described in terms of the Schr$\ddot{o}$dinger-Poisson equations in the large $N$ limit. This result is based on the stochastic quantization introduced by Nelson, and on the Calogero conjecture. According to the Calogero conjecture, the emerging effective Planck constant is computed in terms of the parameters of the N-body system as $\hbar \sim M^{5/3} G^{1/2} (N/<ρ>)^{1/6}$, where is $G$ the gravitational constant, $N$ and $M$ are the number and the mass of the bodies, and $<ρ>$ is their average densit…
We develop a general stochastic thermodynamics of RLC electrical networks built on top of a graph-theoretical representation of the dynamics commonly used by engineers. The network is: open, as it contains resistors and current and voltage sources, nonisothermal as resistors may be at different temperatures, and driven, as circuit elements may be subjected to external parametric driving. The proper description of the heat dissipated in each resistor requires care within the white noise idealization as it depends on the network topology. Our theory provides the basis to design circuits-based th…
We present numerical evidence that in a system of interacting bosons there exists a correspondence between the spectral properties of the exact quantum Hamiltonian and the dynamical chaos of the associated mean field evolution. This correspondence, analogous to the usual quantum-classical correspondence, is related to the formal parallel between the second quantization of the mean field, which generates the exact dynamics of the quantum $N$-body system, and the first quantization of classical canonical coordinates. The limit of infinite density and the thermodynamic limit are then briefly disc…
