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arXiv:quant-ph/9608013 · Journal of Physics A Mathematical and General

Time-of-arrival formalism for the relativistic particle

J León

Published Jul 7, 1997• 57 citations
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Abstract

A suitable operator for the time-of-arrival at a detector is defined for the free relativistic particle in (3 + 1) dimensions. For each detector position there exists a subspace of detected states in the Hilbert space of solutions to the Klein - Gordon equation. Orthogonality and completeness of the eigenfunctions of the time-of-arrival operator apply inside this subspace, opening up a standard probabilistic interpretation.

DOI
10.1088/0305-4470/30/13/027

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A suitable operator for the time-of-arrival at a detector is defined for the free relativistic particle in (3 + 1) dimensions. For each detector position there exists a subspace of detected states in the Hilbert space of solutions to the Klein - Gordon equation. Orthogonality and completeness of the eigenfunctions of the time-of-arrival operator apply inside this subspace, opening up a standard probabilistic interpretation.
A formalism is presented giving the development of $\ensuremath{\psi}(\overline{x})$ in time with the interpretation that $\ensuremath{\psi}(\overline{x})$ is the probability amplitude for observing an event at a space-time point x‾\overline{x}x. No properties other than the four space-time coordinates are associated with an event. A Hilbert space is defined in which $\ensuremath{\psi}(\overline{x})$ is the result of a scalar product. The space-time position operators defined in this Hilbert space have no association with particle properties, such as mass. These operators cannot be defined in the Hilbert space spanned by solutions of a Schr\"odinger equation, since the operators lead out of the Hilbert subspace belonging to a given mass. It is shown that state vectors in Hilbert space that are eigenvectors of ${P}_{\ensuremath{\mu}}{P}^{\ensuremath{\mu}}$ produce position amplitudes satisfying the Klein-Gordon equation. The relation between this Hilbert space and the one introduced by Dirac is discussed. Spin is not considered.
We develop a general framework for the construction of probabilities for the time of arrival in quantum systems. The time of arrival is identified with the time instant when a transition in the detector's degrees of freedom takes place. Thus, its definition is embedded within the larger issue of defining probabilities with respect to time for general quantum transitions. The key point in our analysis is that we manage to reduce the problem of defining a quantum time observable to a mathematical model where time is associated to a transition from a subspace of the Hilbert space of the total system to its complementary subspace. This property makes it possible to derive a general expression for the probability for the time of transition, valid for any quantum system, with the only requirement that the time of transition is correlated with a definite macroscopic record. The framework developed here allows for the consideration of any experimental configuration for the measurement of the time of arrival, and it also applies to relativistic systems with interactions described by quantum field theory. We use the method in order to describe time-of-arrival measurements in high-energy particle reactions and for a rigorous derivation of the time-integrated probabilities in particle oscillations.
We study the problem of computing the probability for the time of arrival of a quantum particle at a given spatial position. We consider a solution to this problem based on the spectral decomposition of the particle's (Heisenberg) state into the eigenstates of a suitable operator, which we denote as the ``time-of-arrival'' operator. We discuss the general properties of this operator. We construct the operator explicitly in the simple case of a free nonrelativistic particle and compare the probabilities it yields with the ones estimated indirectly in terms of the flux of the Schr\"odinger current. We derive a well-defined uncertainty relation between time of arrival and energy; this result shows that the well-known arguments against the existence of such a relation can be circumvented. Finally, we define a ``time representation'' of the quantum mechanics of a free particle, in which the time of arrival is diagonal. Our results suggest that, contrary to what is commonly assumed, quantum mechanics exhibits a hidden equivalence between independent (time) and dependent (position) variables, analogous to the one revealed by the parametrized formalism in classical mechanics. \textcopyright{} 1996 The American Physical Society.
We study the problem of computing the probability for the time-of-arrival of a quantum particle at a given spatial position. We consider a solution to this problem based on the spectral decomposition of the particle's (Heisenberg) state into the eigenstates of a suitable operator, which we denote as the ``time-of-arrival'' operator. We discuss the general properties of this operator. We construct the operator explicitly in the simple case of a free nonrelativistic particle, and compare the probabilities it yields with the ones estimated indirectly in terms of the flux of the Schr\\"odinger current. We derive a well defined uncertainty relation between time-of-arrival and energy; this result shows that the well known arguments against the existence of such a relation can be circumvented. Finally, we define a ``time-representation'' of the quantum mechanics of a free particle, in which the time-of-arrival is diagonal. Our results suggest that, contrary to what is commonly assumed, quantum mechanics exhibits a hidden equivalence between independent (time) and dependent (position) variables, analogous to the one revealed by the parametrized formalism in classical mechanics.
The time of arrival at an arbitrary position in configuration space can be given as a function of the phase space variables for the Liouville integrable systems of classical mechanics, but only for them. We review the Jacobi-Lie transformation that explicitly implements this function of positions and momenta. We then discuss the recently developed quantum formalism for the time of arrival. We first analyze the case of free particles in one and three space dimensions. Then, we apply the quantum version of the Jacobi-Lie transformation to work out the time of arrival operator in the presence of interactions. We discuss the formalism and its interpretation. We finish by disclosing the presence (absence) of "instantaneous" tunneling for thin (thick) barriers.