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Stochastic particle annihilation: a relativistic model of quantum state reduction

Daniel Bedingham

Published Mar 1, 2007• 0 citations
OpenAlex

Abstract

A relativistic model of quantum state reduction involving a nonlinear stochastic extension of Schr\\"odinger's equation is outlined. The eigenstates of the annihilation operator are chosen as the preferred basis onto which reduction occurs. These are the coherent states which saturate the bound of the Heisenberg uncertainty relation, exhibiting classical-like behavior. The quantum harmonic oscillator is studied in detail before generalizing to relativistic scalar quantum field theory. The infinite rates of increase in energy density which have plagued recent relativistic proposals of dynamical state reduction are absent in this model. This is because the state evolution equation does not drive particle creation from the vacuum. It is demonstrated how state reduction to a charge density basis can be induced in fermionic matter via an appropriate coupling to a bosonic field undergoing this mechanism.

Related papers

Stochastic particle annihilation: a model of state reduction in relativistic quantum field theory

Daniel Bedingham

A model of state reduction in relativistic quantum field theory involving a nonlinear stochastic extension of Schrödinger's equation is outlined. The eigenstates of the annihilation operator are chosen as the preferred basis onto which reduction occurs. These are the coherent states which saturate the bound of the Heisenberg uncertainty relation, exhibiting classical-like behaviour. The quantum harmonic oscillator is studied in detail before generalizing to relativistic scalar quantum field theory. The infinite rates of increase in energy density which have plagued recent relativistic proposals of dynamical state reduction are absent in this model. This is because the state evolution equation does not drive particle creation from the vacuum. The model requires the specification of a preferred sequence of space-like hyper-surfaces supporting the time-like state evolution. However, it is shown that the choice of preferred surfaces has no effect on perturbative results to second order in the coupling parameter. It is demonstrated how state reduction to a charge density basis can be induced in fermionic matter via an appropriate coupling to a bosonic field undergoing this mechanism.

arXiv:quant-ph/0703011•Journal of Physics A Mathematical and Theoretical•Jul 3, 2007•3 citations

Defining coherent states: why must they be eigenstates of the annihilation operator?

Juan Pablo Paz, Augusto J. Roncaglia

This is a pedagogical paper where we present a physically motivated approach to introduce the coherent states of a harmonic oscillator from which it is simple to rigorously derive their mathematical definition. We do this in two different ways that turn out to be equivalent but emphasize two related but different aspects of classicality. First, we analyze which are the quantum states that are the closest one can get to a point in phase space and demonstrate the validity of the following theorem: (i) The product of the uncertainty in position and that of momentum saturates the bound imposed by Heisenberg uncertainty relations for all times if and only if the state is an eigenstate of the annihilation operator. Second, we analyze the way in which the difference between the expectation value of the energy and the energy associated with the expectation values of position and momentum depends on the state, and show the validity of the following theorem (ii) the difference between the expectation value of the energy and the energy associated with the expectation values is minimal if and only if the state is an eigenstate of the annihilation operator. We also show that the reason why coherent states are chosen as the most classical ones by the decoherence process induced by coupling the particle to an environment in the standard Quantum Brownian motion model, is precisely due to the validity of the two above theorems.

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Generalized stochastic Schrödinger equations for state vector collapse

Stephen L Adler, Todd A Brun

A number of authors have proposed stochastic versions of the Schrödinger equation, either as effective evolution equations for open quantum systems or as alternative theories with an intrinsic collapse mechanism. Here we discuss two directions for the generalization of these equations. First, we study a general class of norm preserving stochastic evolution equations, and show that even after making several specializations there is an infinity of possible stochastic Schrödinger equations for which state vector collapse is provable. Second, we explore the problem of formulating a relativistic stochastic Schrödinger equation, using a manifestly covariant equation for a quantum field system based on the interaction picture of Tomonaga and Schwinger. The stochastic noise term in this equation can couple to any local scalar density that commutes with the interaction energy density, and leads to collapse onto spatially localized eigenstates. However, as found in a similar model by Pearle, the equation predicts an infinite rate of energy nonconservation proportional to δ 3 ( ), arising from the local double commutator in the drift term.

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Relativistic state reduction model

Daniel Bedingham

In order to address the measurement problem of quantum theory we make the assumption that quantum state reduction should be regarded as a genuine physical process deserving of a dynamical description. Generalizing the nonrelativistic spontaneous localization models of Ghirardi, Rimini, Weber, and Pearle, a relativistic state reduction mechanism is proposed. The mechanism involves nonlinear stochastic modifications to the standard description of unitary state evolution and the introduction of a mediating field to facilitate smearing of quantum field interactions.

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Relativistic state reduction model

Daniel Bedingham

In order to address the measurement problem of quantum theory we make the assumption that quantum state reduction should be regarded as a genuine physical process deserving of a dynamical description. Generalizing the nonrelativistic spontaneous localization models of Ghirardi, Rimini, Weber, and Pearle, a relativistic state reduction mechanism is proposed. The mechanism involves nonlinear stochastic modifications to the standard description of unitary state evolution and the introduction of a mediating field to facilitate smearing of quantum field interactions.

•Jan 1, 2011•20 citations

Lévy models for collapse of the wave function

Dorje C. Brody, Lane P. Hughston

Abstract Recently there has been much progress in the development of stochastic models for state reduction in quantum mechanics. In such models, the collapse of the wave function is a physical process, governed by a nonlinear stochastic differential equation that generalizes the Schrödinger equation. The present paper considers energy-based stochastic extensions of the Schrödinger equation. Most of the work carried out hitherto in this area has been concerned with models where the process driving the stochastic dynamics of the quantum state is Brownian motion. Here, the Brownian framework is broadened to a wider class of models where the noise process is of the Lévy type, admitting stationary and independent increments. The properties of such models are different from those of Brownian reduction models. In particular, for Lévy models the decoherence rate depends on the overall scale of the energy. Thus, in Lévy reduction models, a macroscopic quantum system will spontaneously collapse to an eigenstate even if the energy level gaps are small.

arXiv:2207.12241•Journal of Physics A Mathematical and Theoretical•Feb 23, 2023•7 citations
A relativistic model of quantum state reduction involving a nonlinear stochastic extension of Schr\"odinger's equation is outlined. The eigenstates of the annihilation operator are chosen as the preferred basis onto which reduction occurs. These are the coherent states which saturate the bound of the Heisenberg uncertainty relation, exhibiting classical-like behavior. The quantum harmonic oscillator is studied in detail before generalizing to relativistic scalar quantum field theory. The infinite rates of increase in energy density which have plagued recent relativistic proposals of dynamical state reduction are absent in this model. This is because the state evolution equation does not drive particle creation from the vacuum. It is demonstrated how state reduction to a charge density basis can be induced in fermionic matter via an appropriate coupling to a bosonic field undergoing this mechanism.
A model of state reduction in relativistic quantum field theory involving a nonlinear stochastic extension of Schrödinger's equation is outlined. The eigenstates of the annihilation operator are chosen as the preferred basis onto which reduction occurs. These are the coherent states which saturate the bound of the Heisenberg uncertainty relation, exhibiting classical-like behaviour. The quantum harmonic oscillator is studied in detail before generalizing to relativistic scalar quantum field theory. The infinite rates of increase in energy density which have plagued recent relativistic proposals of dynamical state reduction are absent in this model. This is because the state evolution equation does not drive particle creation from the vacuum. The model requires the specification of a preferred sequence of space-like hyper-surfaces supporting the time-like state evolution. However, it is shown that the choice of preferred surfaces has no effect on perturbative results to second order in the coupling parameter. It is demonstrated how state reduction to a charge density basis can be induced in fermionic matter via an appropriate coupling to a bosonic field undergoing this mechanism.
This is a pedagogical paper where we present a physically motivated approach to introduce the coherent states of a harmonic oscillator from which it is simple to rigorously derive their mathematical definition. We do this in two different ways that turn out to be equivalent but emphasize two related but different aspects of classicality. First, we analyze which are the quantum states that are the closest one can get to a point in phase space and demonstrate the validity of the following theorem: (i) The product of the uncertainty in position and that of momentum saturates the bound imposed by Heisenberg uncertainty relations for all times if and only if the state is an eigenstate of the annihilation operator. Second, we analyze the way in which the difference between the expectation value of the energy and the energy associated with the expectation values of position and momentum depends on the state, and show the validity of the following theorem (ii) the difference between the expectation value of the energy and the energy associated with the expectation values is minimal if and only if the state is an eigenstate of the annihilation operator. We also show that the reason why coherent states are chosen as the most classical ones by the decoherence process induced by coupling the particle to an environment in the standard Quantum Brownian motion model, is precisely due to the validity of the two above theorems.
A number of authors have proposed stochastic versions of the Schrödinger equation, either as effective evolution equations for open quantum systems or as alternative theories with an intrinsic collapse mechanism. Here we discuss two directions for the generalization of these equations. First, we study a general class of norm preserving stochastic evolution equations, and show that even after making several specializations there is an infinity of possible stochastic Schrödinger equations for which state vector collapse is provable. Second, we explore the problem of formulating a relativistic stochastic Schrödinger equation, using a manifestly covariant equation for a quantum field system based on the interaction picture of Tomonaga and Schwinger. The stochastic noise term in this equation can couple to any local scalar density that commutes with the interaction energy density, and leads to collapse onto spatially localized eigenstates. However, as found in a similar model by Pearle, the equation predicts an infinite rate of energy nonconservation proportional to δ 3 ( ), arising from the local double commutator in the drift term.
In order to address the measurement problem of quantum theory we make the assumption that quantum state reduction should be regarded as a genuine physical process deserving of a dynamical description. Generalizing the nonrelativistic spontaneous localization models of Ghirardi, Rimini, Weber, and Pearle, a relativistic state reduction mechanism is proposed. The mechanism involves nonlinear stochastic modifications to the standard description of unitary state evolution and the introduction of a mediating field to facilitate smearing of quantum field interactions.
In order to address the measurement problem of quantum theory we make the assumption that quantum state reduction should be regarded as a genuine physical process deserving of a dynamical description. Generalizing the nonrelativistic spontaneous localization models of Ghirardi, Rimini, Weber, and Pearle, a relativistic state reduction mechanism is proposed. The mechanism involves nonlinear stochastic modifications to the standard description of unitary state evolution and the introduction of a mediating field to facilitate smearing of quantum field interactions.
Abstract Recently there has been much progress in the development of stochastic models for state reduction in quantum mechanics. In such models, the collapse of the wave function is a physical process, governed by a nonlinear stochastic differential equation that generalizes the Schrödinger equation. The present paper considers energy-based stochastic extensions of the Schrödinger equation. Most of the work carried out hitherto in this area has been concerned with models where the process driving the stochastic dynamics of the quantum state is Brownian motion. Here, the Brownian framework is broadened to a wider class of models where the noise process is of the Lévy type, admitting stationary and independent increments. The properties of such models are different from those of Brownian reduction models. In particular, for Lévy models the decoherence rate depends on the overall scale of the energy. Thus, in Lévy reduction models, a macroscopic quantum system will spontaneously collapse to an eigenstate even if the energy level gaps are small.