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International Journal of Modern Physics B

A MODEL OF MEASUREMENT AS EVOLUTION DRIVEN BY BOTH PREVIOUS AND SUBSEQUENT CAUSES

Giuseppe Castagnoli, Mario Rasetti

Published Jan 30, 1996• 3 citations
Publisher

Abstract

The notion of the coexistence in the quantum framework of causality forward and backward in time, is used to develop a generalizazion of the conventional description of unitary evolutions, based on the assumption that evolutions are driven only by previous causes. Such a generalization is capable of describing basic aspects of state vector reduction.

DOI
10.1142/s021797929600012x

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The notion of the coexistence in the quantum framework of causality forward and backward in time, is used to develop a generalizazion of the conventional description of unitary evolutions, based on the assumption that evolutions are driven only by previous causes. Such a generalization is capable of describing basic aspects of state vector reduction.
Given a Hamiltonian HHH on a Hilbert space H\mathcal HH it is shown that, under the assumption that σ(H)=σac(H)=R+σ(H)=σ_{ac}(H)=R^+σ(H)=σac​(H)=R+, there exist unique positive operators TFT_FTF​ and TBT_BTB​ registering the Schrödinger time evolution generated by HHH in the forward (future) direction and backward (past) direction respectively. These operators may be considered as time observables for the quantum evolution. Moreover, it is shown that the same operators may serve as time observables in the construction of quantum stochastic differential equations and quantum stochastic processes in the framework of the Hudson-Parthasarathy quantum stochastic calculus. The basic mechanism enabling for the definition of the time observables originates from the recently developed semigroup decomposition formalism used in the description of the time evolution of resonances in quantum mechanical scattering problems.
Schroumloumloumldinger equation for joint bidirectional motion in time
The conventional time-dependent Schrödinger equation describes only unidirectional time evolution of the state of a physical system, i.e. forward or, less commonly, backward. This paper proposes a generalized quantum dynamics for the description of joint, and interactive, forward and backward time evolution within a physical system. The principal mathematical assumption for bidirectional evolution in general is that the space of states should be taken to be not merely a Hilbert space, but a more restricted entity known as a Kreĭn space, which is a complex Hilbert space with a Hermitian operator that has eigenvalues +1 and −1 only, and that therefore gives rise to an indefinite metric. The vector subspaces of states with positive or negative norm with respect to the indefinite metric will—for open channels—be construed to be states in forward or, respectively, backward evolution along the time axis. The quantum dynamics is generated by a pseudo-Hermitian Hamiltonian operator and conserves inner products with respect to the indefinite metric. Input and output states are defined in physically plausible ways such that the output comprises both reflected and transmitted states from a zone of interaction in time; a unitary transformation between input and output states is obtained from the pseudounitary transformation between the initial and final states. Three applications are studied: (1) a formal theory of collisions in terms of perturbation theory; (2) a relativistically invariant quantum field theory for a system that kinematically comprises the direct sum of two quantized real scalar fields, such that one subfield evolves forward and the other backward in time, and such that there is dynamical coupling between the subfields; (3) an argument that in the latter field theory, the dynamics predicts that in a range of values of the coupling constants, the expectation value of the vacuum energy of the universe is forced to be zero to high accuracy.
The measurement of nonlocal properties of relativistic quantum-mechanical systems, the compatibility (or lack of it) of two or more such measurements, and various other of their characteristics are considered; and with these notions in hand an old problem (which is to produce a covariant description of the state reduction associated with the measuring process) is attacked, and succumbs. The solution requires us to depart (not as we did in part I of the present work, but in an entirely different direction) from the usual picture of the time evolution of quantum states.