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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Stochastic representation of arbitrary fermion sectors in supersymmetric quantum mechanics on Riemannian manifolds

Robert Graham, D. Roekaerts

Published Oct 15, 1986• 20 citations
Publisher

Abstract

In the zero-fermion and n-fermion sectors of Euclidean supersymmetric quantum theories on an n-dimensional manifold a transformation is known which reduces the functional integrals to Gaussian functional integrals. The transformation, often called a Nicolai map, can be interpreted as a Markovian stochastic process. We investigate two different possibilities to generalize this result to an arbitrary p-fermion sector (0\ensuremath{\le}p\ensuremath{\le}n), and find that only one of them is viable, in general. We give a general representation of the solution of the Schr\"odinger equation in the p-fermion sector in terms of a Markovian stochastic process which extends the stochastic process of the zero- and n-fermion sectors.

DOI
10.1103/physrevd.34.2312

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In the zero-fermion and n-fermion sectors of Euclidean supersymmetric quantum theories on an n-dimensional manifold a transformation is known which reduces the functional integrals to Gaussian functional integrals. The transformation, often called a Nicolai map, can be interpreted as a Markovian stochastic process. We investigate two different possibilities to generalize this result to an arbitrary p-fermion sector (0\ensuremath{\le}p\ensuremath{\le}n), and find that only one of them is viable, in general. We give a general representation of the solution of the Schr"odinger equation in the p-fermion sector in terms of a Markovian stochastic process which extends the stochastic process of the zero- and n-fermion sectors.
We use the stochastic quantization method to construct a supersymmetric version of the quantum spherical model. This is based on the equivalence between the Brownian motion described by a Langevin equation and the supersymmetric quantum mechanics, which is connected with the existence of the Nicolai map. We investigate the critical behavior of the supersymmetric model at finite as well as at zero temperatures, showing that it exhibits both classical and quantum phase transitions, and determine the critical dimensions.
This paper considers the extension of the non-Markovian stochastic approach for quantum open systems strongly coupled to a fermionic bath to the models in which the system operators commute with the fermionic bath. This technique can also be a useful tool for studying open quantum systems coupled to a spin-chain environment, which can be further transformed into an effective fermionic bath. We derive an exact stochastic Schr\"odinger equation (SSE), called the fermionic quantum state diffusion (QSD) equation, from the first principle by using the fermionic coherent state representation. The reduced density operator for the open system can be recovered from the stochastic average of the solutions to the QSD equation over the Grassmann-type noise. By employing the exact fermionic QSD equation, we can derive the corresponding exact master equation. The power of our approach is illustrated by the applications of our stochastic approach to several models of interest including the one-qubit dissipative model, the coupled two-qubit dissipative model, the quantum Brownian motion model, and the NNN-fermion model coupled to a fermionic bath. Different effects caused by the fermionic and bosonic baths on the dynamics of open systems are also discussed.
We establish a novel correspondence between the spacetime correlators of one-dimensional (1D) N-particle classical stochastic models described by a Langevin equation with that of the ground-state dynamics of a class of integrable 1D interacting many-body quantum models of the supersymmetric elliptic type. We show that these seemingly different concepts of stochastic systems, supersymmetry and quantum integrability can be viewed in a unified framework. Starting with an Lax matrix, we show that row (column) sums driven by a Gaussian noise term may be interpreted as a set of forward (backward) Langevin equations. Then, following functional path integral methods of stochastic quantization, we straightforwardly find an associated supersymmetric 1D quantum Hamiltonian. If, further, the classical stochastic system consists of two-body interactions only and we also want the quantum interactions to be of two-body type, we find that the only class of interactions permissible for the quantum models corresponds to the elliptic models. The algebraic structure that emerges very naturally reproduces the proof of integrability and allows the identification of the ground-state wavefunction of these quantum models.
Within a stochastic picture quantum systems can be described in terms of kinematic and dynamic equations where the particles follow a conservative diffusion process. Pavon has shown that these equations and the Schrödinger equation can be derived from a quantum Hamilton principle, generalizing the Hamilton principle from classical mechanics. As shown recently by Köppe et al for the one-dimensional case, the reformulation of the quantum Hamilton principle as a stochastic optimal control problem leads to quantum Hamilton equations of motion which can be seen as a non relativistic generalization of Hamilton's equations of motion to the quantum world. In this report we will extend this to higher dimensional problems and derive, by incorporating the concept of supersymmetric quantum mechanics, a general scheme that allows the determination of all bound states within the stochastic framework. Physical principles from classical mechanics like the decoupling of the center-of-mass motion in multi-particle systems or the solution of the Kepler problem, as a special case of the two-body problem, using space-time symmetries, are translated to this formulation of quantum mechanics. We will present numerical results for the ground state and lower excited states of the hydrogen atom in Cartesian coordinates, and, additionally, we will show that for spherically symmetric potentials the solution of the two-body problem can be reduced to a system of equations concerning only the radial part.