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Birkhäuser Boston eBooks

Quantum Stochastic Processes

Luigi Accardi

Published Jan 1, 1985• 226 citations
Publisher

Abstract

Let ℬ be *-algebra with identity (usually it wil be a C*- or a W*-algebra). A quantum stochastic process over ℬ indexed by ℝ is defined by a triple { A , (j t ) t∈ℝ , φ } where A is a *-algebra with identity. j t : ℬ ↪ A is an embedding (t∈ℝ). φ is a State on A . These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-1-4899-6653-7_16

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Let ℬ be -algebra with identity (usually it wil be a C- or a W*-algebra). A quantum stochastic process over ℬ indexed by ℝ is defined by a triple { A , (j t ) t∈ℝ , φ } where A is a *-algebra with identity. j t : ℬ ↪ A is an embedding (t∈ℝ). φ is a State on A . These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
In Section 21 we have already seen how the classical stochastic processes with independent increments can be realised as suitable linear combinations of the creation, conservation and annihilation operators in the boson Fock space Γ s (ℋ) over a Hilbert space ℋ. This includes, in particular, the Brownian motion and Poisson process. Since a well-developed theory of stochastic integration with respect to these classical processes exists, it is natural to wonder whether it would be possible to introduce quantum probabilistic analogues of stochastic integrals with respect to processes defined by the creation, conservation and annihilation operators. To this end one needs the notions of filtration and adapted processes. This, in turn, depends on the notion of time. We adopt the view that time is an ℝ+-valued observable in a Hilbert space ℋ which, as a spectral measure, has no jumps. The absence of jumps is assumed more as a matter of mathematical convenience than a philosophical or conceptual necessity. We shall now examine how such a notion of time leads to a filtration and the definition of adapted processes. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
A discrete quantum process is defined as a sequence of local states ρtρ_tρt​, t=0,1,2,...t=0,1,2,...t=0,1,2,..., satisfying certain conditions on an L2L_2L2​ Hilbert space HHH. If ρ=lim⁡ρtρ=\limρ_tρ=limρt​ exists, then ρρρ is called a global state for the system. In important cases, the global state does not exist and we must then work with the local states. In a natural way, the local states generate a sequence of quantum measures which in turn define a single quantum measure μμμ on the algebra of cylinder sets $\cscript$. We consider the problem of extending μμμ to other physically relevant sets in a systematic way. To this end we show that μμμ can be properly extended to a quantum measure $\mutilde$ on a "quadratic algebra" containing $\cscript$. We also show that a random variable fff can be "quantized" to form a self-adjoint operator $\fhat$ on HHH. We then employ $\fhat$ to define a quantum integral $\int fd\mutilde$. Various examples are given
A covariant version of the dilation theory for quantum dynamical semigrops is established, including both Bhat-Parthasarathy type and Evans-Hudson type dilation. It is shown in particular that every uniformly continuous quantum dynamical semigroup on a separable C*-algebra or a von Neumann algebra, which is covariant under a suitable group-action, admits a covariant Evans-Hudson dilation in some appropriate Fock space. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Quantum mechanics contains certain novel mathematical concepts. Among these are complex numbers, Hilbert spaces with their unitary and self-adjoint operators, states represented by complex vectors, superpositions of states, collapse of wave functions, Born's rule for probabilities and others. If we accept that quantum mechanics is probabilistic, then these concepts can be derived and they become secondary. In this work, we begin with what we call a conditional stochastic process, which is based on real numbers and probabilities. As we shall see, such processes are defined by three simple axioms. We then use conditional stochastic processes to derive quantum mechanics by employing a correspondence called a dictionary. We also show that the converse holds. That is, beginning with a quantum system, we employ the dictionary to derive a conditional stochastic process.