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What Quantum Engineering Brings to Quantum Measurement Theory: the Equivalence Between Von Neumann-type Projective Operators and Fokker-Planck Differential Operators on Density Matrices

John A. Sidles

Published May 1, 2001• 0 citations
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We again note that Langevin equations written for observables (– operators in quantum mechanics) conform to the Heisenberg view. On the other hand, an entity that embodies the wave function of the Schrödinger picture is the von Neumann density operator, usually denoted by t( )ρ , whose classical counterpart is the probability distribution function (pdf) discussed earlier. It is helpful to introduce a formalism for the pdf completely at par with the Langevin description. The resultant Fokker-Planck (FP) equations serve as the subjects of this chapter. The simplest FP equation can be derived from the master equation (2.25) for a stationary Markov process, as shown below (Agarwal 1983, Risken 1996).

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Abstract This chapter provides an overview of the phase space method for treating the physics of systems of bosons and fermions in regimes where quantum degeneracy occurs. Key concepts such as modes, creation and annihilation operators, distribution functions in phase space depending on c-number (bosons) or Grassmann (fermions) phase space variables, quantum correlation functions, and phase space integrals are introduced, together with how the dynamics of the system density operator as described via Liouville–von Neumann or master equations is translated into Fokker–Planck equations for the distribution function, and then how phase space integrals for quantum correlation functions can be determined via Langevin stochastic equations that are equivalent to the Fokker–Planck equations, and which involve stochastic phase space variables. The extension of the phase space approach to systems with large mode numbers, involving field annihilation and creation operators, distribution functionals, and functional Fokker–Planck and Langevin stochastic field equations is also reviewed.

•Nov 13, 2014•0 citations
We again note that Langevin equations written for observables (– operators in quantum mechanics) conform to the Heisenberg view. On the other hand, an entity that embodies the wave function of the Schrödinger picture is the von Neumann density operator, usually denoted by t( )ρ , whose classical counterpart is the probability distribution function (pdf) discussed earlier. It is helpful to introduce a formalism for the pdf completely at par with the Langevin description. The resultant Fokker-Planck (FP) equations serve as the subjects of this chapter. The simplest FP equation can be derived from the master equation (2.25) for a stationary Markov process, as shown below (Agarwal 1983, Risken 1996).
In an algebraic understanding of holographic gravity, a major question is: how can a boundary theory with Type-III von Neumann algebras of local observables (where density operators and entropy are ill-defined) be equivalent to a bulk theory with Type-I von Neumann algebras of local observables (where density operators and entropy are well-defined)? Replica wormholes and problems like the factorization problem suggest that instead of full equivalence, there is an approximation. When approximate equality breaks down, corrections to bulk theory predictions are required, just as in replica wormholes. Quantum measurements are then described as operations that can only be described as identity operations in the bulk theory but are non-trivial in the boundary theory. The breakdown of approximate equality between the boundary theory and the bulk theory generates illusions of wavefunction collapse, when there actually is no wavefunction collapse in the boundary theory with Type-III von Neumann algebras.
Abstract This chapter provides an overview of the phase space method for treating the physics of systems of bosons and fermions in regimes where quantum degeneracy occurs. Key concepts such as modes, creation and annihilation operators, distribution functions in phase space depending on c-number (bosons) or Grassmann (fermions) phase space variables, quantum correlation functions, and phase space integrals are introduced, together with how the dynamics of the system density operator as described via Liouville–von Neumann or master equations is translated into Fokker–Planck equations for the distribution function, and then how phase space integrals for quantum correlation functions can be determined via Langevin stochastic equations that are equivalent to the Fokker–Planck equations, and which involve stochastic phase space variables. The extension of the phase space approach to systems with large mode numbers, involving field annihilation and creation operators, distribution functionals, and functional Fokker–Planck and Langevin stochastic field equations is also reviewed.