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Teubner-Texte zur Physik

Many-Body Systems

H. Eschrig

Published Jan 1, 1996• 0 citations
Publisher

Abstract

The dynamics of a quantum system is governed by the Hamiltonian \( \hat H.If|\Psi \rangle\) is a quantum state of the system in an abstract Hilbert space representation, its time-evolution is given by $$ - \frac{\hbar }{i}\frac{\partial }{{\partial t}}|\Psi \rangle = \hat H|\Psi \rangle$$ (1.1)

DOI
10.1007/978-3-322-97620-8_2

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The dynamics of a quantum system is governed by the Hamiltonian ( \hat H.If|\Psi \rangle) is a quantum state of the system in an abstract Hilbert space representation, its time-evolution is given by −ℏi∂∂t∣Ψ⟩=H^∣Ψ⟩ - \frac{\hbar }{i}\frac{\partial }{{\partial t}}|\Psi \rangle = \hat H|\Psi \rangle−iℏ​∂t∂​∣Ψ⟩=H^∣Ψ⟩ (1.1)
Abstract We investigate the time evolution for quantum systems with a dynamical Hilbert space within the framework of the pseudo-Hermitian representation of quantum mechanics. Each representation of quantum mechanics is characterized by a Hilbert space, a Hamiltonian, and a set of observables. The representation-transformation law of the time-evolution operator is derived from its formal solution in the dynamical Hilbert space. It ensures the unitarity of the dynamics and the representation independence of the transition amplitudes. In addition, we show that the equation of time evolution in the position space is independent of the representation, regardless of whether the Hilbert space is stationary or dynamical. Furthermore, we demonstrate the representation independence of the position wave function itself. As a concrete example of the representation independence of the quantum mechanics, we derive the reciprocity theorem in the dynamical Hilbert space. The material in the present paper makes a topic which can be covered in a graduate course on quantum mechanics.
In quantum mechanics it is usual to represent physical reality as a vector in Hilbert space at a particular time, with evolution being governed by Schrodinger's equation which involves an externally imposed time parameter. This leads to difficulties if one wishes to regard the universe as a quantum mechanical system, because there should no time external to such a system. The approach in this paper is to represent the totality of reality as a vector in Hilbert space. The author shows how time evolution follows, where time now is defined in terms of the states of a quantum mechanical clock which is part of the system. Rather than the correlation between the clock states and the states of the rest of the system arising because both are governed by an imposed law involving an external time parameter, it is seen that this correlation is of the separation-independent Einstein-Podolsky-Rosen type. The total reality vector, which incorporates the whole history of the system, is shown to be a zero-energy eigenstate of the system Hamiltonian. He discusses systems of finite and infinite lifetime, and is able to answer the question: what was the state before the initial state? He concludes that the quantum mechanical system of this paper is a reasonable representation of the observed universe.
Abstract To admit a canonically conjugate time operator, the Hamiltonian has to be a generator of translations (like the momentum operator generates translations in space), so its spectrum must be unbounded. But the Hamiltonian governing our world is thought to be bounded from below. Also, judging by the number of fields and parameters of the Standard Model, the Hamiltonian seems much more complicated. In this article I give examples of worlds governed by Hamiltonians generating translations. They can be expressed as a partial derivative operator just like the momentum operator, but when expressed in function of other observables they can exhibit any level of complexity. The examples include any quantum world realizing a standard ideal measurement, any quantum world containing a clock or a free massless fermion, the quantum representation of any deterministic time-reversible dynamical system without time loops, and any quantum world that cannot return to a past state. Such worlds are as sophisticated as our world, but they admit a time operator. I show that, despite having unbounded Hamiltonian, they do not decay to infinite negative energy any more than any quantum or classical world. Since two such quantum systems of the same Hilbert space dimension are unitarily equivalent even if the physical content of their observables is very different, they are concrete counterexamples to Hilbert Space Fundamentalism (HSF). Taking the observables into account removes the ambiguity of HSF and the clock ambiguity problem attributed to the Page-Wootters formalism, also caused by assuming HSF. These results provide additional motivations to restore the spacetime symmetry in the formulation of Quantum Mechanics and for the Page-Wootters formalism.