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arXiv:quant-ph/0104078

Time Interval Operators

M. Ruzzi, D. Galetti

Published Apr 16, 2001• 0 citations
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Abstract

In this work we will advance farther along a line previously developed concerning our proposal of a time interval operator, on finite dimensional spaces. The time interval operator is Hermitian, and its eigenvalues are time values with a precise and interesting role on the dynamics. With the help of the Discrete Phase Space Formalism (DPSF) previously developed, we show that the time interval operator is the complementary pair of the Hamiltonian. From that, a simple system is proposed as a quantum clock. The only restriction is that our results do not apply to all possible Hamiltonians.

DOI
10.48550/arxiv.quant-ph/0104078

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In this work we will advance farther along a line previously developed concerning our proposal of a time interval operator, on finite dimensional spaces. The time interval operator is Hermitian, and its eigenvalues are time values with a precise and interesting role on the dynamics. With the help of the Discrete Phase Space Formalism (DPSF) previously developed, we show that the time interval operator is the complementary pair of the Hamiltonian. From that, a simple system is proposed as a quantum clock. The only restriction is that our results do not apply to all possible Hamiltonians.
In this chapter, we finally introduce time evolution. The development is carried out for finite, enumerable systems (i.e., systems whose states live in a finite dimensional, inner product space) which, of course, allows a straight forward extension to the continuum and infinite dimensional state spaces. The necessary mathematical background of unitary operators, and their properties relevant to quantum mechanics are first introduced. In particular, the fact that preservation of norm under the action of an operator implies unitarity, and that every unitary operator can be expressed as the exponential of i timesa Hermitian operator are introduced. Time evolution is then introduced in its integral form through the time evolution operator, and the Hamiltonian operator is introduced in this context. This approach, however, necessitates the restriction to conservative systems. Next, the differential form of the law of time evolution, i.e., Schrodinger equation, is derived from the time evolution operator, and this is postulated to be the general law of time evolution. We then demonstrate how, for conservative systems, the solution of the Schrodinger equation essentially reduces to solving the eigenvalue problem of the Hamiltonian. Constants of motion and Heisenberg picture have been relegated to the problem section.
The main aspects of a discrete phase space formalism are presented and the discrete dynamical bracket, suitable for the description of time evolution in finite-dimensional spaces, is discussed. A set of operator bases is defined in such a way that the Weyl-Wigner formalism is shown to be obtained as a limiting case. In the same form, the Moyal bracket is shown to be the limiting case of the discrete dynamical bracket. The dynamics in quantum discrete phase spaces is shown not to be attained from discretization of the continuous case.
In the search of a quantum momentum operator with discrete spectrum, we obtain some properties of the discrete momentum operator for nonequally spaced spectrum. We find the inverse operator. We use the matrix representation of these operators, and we find that there is one more eigenvalue and eigenfunction than the dimension of the matrix. We apply the results to obtain the discrete adjoint of the momentum operator. We conclude that we can have discrete operators which can be self-adjoint and that it is possible to define a self-adjoint extension of the corresponding Hilbert space. These results help us understand the quantum time operator.
In this paper we wish to explore the possibility of the existence of a quantity that can be regarded as the complement of the Hamiltonian for a quantum system with discrete energy levels, recognizing that if this quantity is represented by an operator the appropriate commutator must be a generalized form of Eq. 2. Although this quantity will have dimensions of time, it will not be appropriate to refer to such an operator as a time operator. As an operator it would represent an observable of the quantum system and not time in the abstract or coordinate sense or as a reading on an external clock. Its eigenstates would represent a state of the quantum system and some measurement on the system, at least in principle, should tell us about the quantity involved. Although this quantity is not time, we would hope that for a particular ideal system its expectation value may change linearly with time, for example, so that a measurement of the quantity would also give a measure of time. In this case, we would be using the system as a clock. Before seeking an operator for the complement of the Hamiltonian we first represent this quantity by a nonorthogonal probability-operator measure, which is a more general concept 6.
It is shown that a Hermitian phase operator exists for quantum spins. Its spectrum is not continuous but has the values (2 pi /(2S+1))n, 0<or=n<or=2S for spin S. The precession of a high-S spin in a magnetic field is shown to consist essentially of a set of jumps from one value of phase to the next one, at equal time intervals. At intermediate times the value of the phase is uncertain. By going to a larger Hilbert space than the spin space, a Hermitian phase operator whose eigenvalue spectrum covers the entire real line is defined, and its relation to the formerly defined operator is clarified. Possible applications to spin and other systems are outlined.
We develop a path integral representation for the dynamics of quantum systems with a finite-dimensional Hilbert space, formulated entirely within a discrete phase space. Starting from the discrete Wigner function on Zd×Zd{\mathbb{Z}_d} \times {\mathbb{Z}_d}Zd​×Zd​ (ddd an odd prime) and the associated Weyl transform built from generalized displacement operators, we derive an exact kernel that propagates the discrete Wigner function in time and, by iterating its composition law through a short-time approximation, obtain a sum-over-paths expression weighted by a discrete phase-space action -- the natural finite-dimensional counterpart of Marinov's functional. For Hamiltonians linear in the phase-space coordinates and at times strictly commensurate with the lattice, the fluctuation sum collapses to a deterministic shift, realizing the discrete analog of classical Hamiltonian flow. Applying the formalism to one and to two interacting qutrits (d=3d=3d=3), we show that the full entanglement dynamics -- captured by a closed-form linear entropy valid for all times -- requires the coherent contribution of all fluctuation sectors; the boundary-term (mean-field) sector alone fails to reproduce it. For a non-stabilizer Hamiltonian, where the short-time kernel is only approximate, the time-sliced path integral converges to the exact dynamics, including the dynamical generation of Wigner negativity. We discuss implications for the semiclassical simulation of many-body spin systems and for the characterization of non-classicality through Wigner negativity.