We exhibit a class of quantum field theories where particles interact with pair potentials and for which the time evolution exists in the Heisenberg representation. The essential condition for existence is stability in the thermodynamic sense and this is achieved by having the interaction fall off with the relative momenta of the particles. This can be done in a Galilei-invariant manner. We show that these systems have some mixing properties which one postulates in ergodic theory but which are difficult to prove for classical systems.
We exhibit a class of quantum field theories where particles interact with pair potentials and for which the time evolution exists in the Heisenberg representation. The essential condition for existence is stability in the thermodynamic sense and this is achieved by having the interaction fall off with the relative momenta of the particles. This can be done in a Galilei-invariant manner. We show that these systems have some mixing properties which one postulates in ergodic theory but which are difficult to prove for classical systems.
We construct a model for a chemical reaction in the Heisenberg representation. The time evolution exists in the thermodynamic limit, independent of the particle density and the initial state. In mathematical terms, we establish a C*-dynamical system. Thus we can benefit from general results in algebraic quantum statistical mechanics, showing, for example, that equilibrium states exist. Galilei invariance of our nonrelativistic model is demonstrated by defining it directly on the Galilean space-time manifold, without reference to any coordinate system. PACS 05.30.Fk, 03.70.+ k, ll.10.Cd, 11.15.Tk
This paper is devoted to the study of the Galilei group and its representations. The Galilei group presents a certain number of essential differences with respect to the Poincaré group. As Bargmann showed, its physical representations, here explicitly constructed, are not true representations but only up-to-a-factor ones. Consequently, in nonrelativistic quantum mechanics, the mass has a very special role, and in fact, gives rise to a superselection rule which prevents the existence of unstable particles. The internal energy of a nonrelativistic system is known to be an arbitrary parameter; this is shown to come also from Galilean invariance, because of a nontrivial concept of equivalence between physical representations. On the contrary, the behavior of an elementary system with respect to rotations, is very similar to the relativistic case. We show here, in particular, how the number of polarization states reduces to two for the zero-mass case (though in fact there are no physical zero-mass systems in nonrelativistic mechanics). Finally, we study the two-particle system, where the orbital angular momenta quite naturally introduce themselves through the decomposition of the tensor product of two physical representations.
Abstract Although it is usual to read that non-relativistic quantum mechanics is covariant, and even invariant, under the Galilei transformations, this topic has scarcely been addressed in the standard literature on the theory. This fact has, at least, two undesirable consequences. On the one hand, the meanings of the concepts of covariance and invariance are not precisely elucidated. On the other hand, one is left without a clear idea about in what sense the covariance and/or invariance of quantum mechanics can be asserted. This situation has its counterpart in the field of the interpretation of quantum mechanics: the relevance of the Galilei group is rarely discussed in the impressive amount of literature on the subject. The general premise underlying the present chapter is that the relationship between interpretation and Galilei transformations deserves to be seriously analyzed: the fact that the theory is covariant under the Galilei group does not guarantee the same property for the interpretation because, in general, interpretations add interpretative postulates to the formal structure of the theory. On this basis, and after clarifying the concepts of covariance and invariance, this chapter considers in what sense quantum mechanics is covariant and under what conditions it is invariant under the transformations of the Galilei group. This conceptual framework will provide the tools to study the result of the application of the Galilei transformations to the Modal-Hamiltonian interpretative postulates.
Multi-time wavefunctions are of particular interest in relativistic quantum mechanics. A multitime wavefunction has separate time-variables for each particle; this makes it a manifestly Lorentz-invariant object. The time-evolution equations are systems of Schroedinger equations; one for each particle's time variable and each with a certain Hamiltonian. We derive conditions under which these systems of equations have a common solution. Also, we derive three main results about concrete multi-time models. First we show that a model proposed by Duerr and Tumulka in 2001 is inconsistent. The second result is a consistent model for a constant number of particles with a cutoff pair potential. The third result is a consistent theory for a simple quantum field theoretic model with creation and annihilation of particles. Existence and uniqueness of solutions is proven for both models.
Despite its name, Quantum Field Theory (QFT) has been built to describe interactions between localizable particles. For this reason the actual formalism of QFT is partly based on a suitable generalization of the one already used for systems of point particles. This circumstance gives rise to a number of conceptual problems, stemming essentially from the fact that the existence within QFT of non-equivalent representations implies the existence of field theories allowing, within the same theory, different, inequivalent, descriptions of particles. This led some authors to claim that in QFT the concept itself of particle should be abandoned. In this paper we will shortly discuss the validity of this claim, as well as the possibilities, so far existing, of building alternative versions of QFT, not designed in advance to allow some kind of particle representation. We will also spend some words about the generalizations of the concept itself of particle which could grant for a better cohabitation of particles and fields within a wider formulation of QFT. The latter is indispensable if we want to extend the range of application of QFT from particle physics or simple condensed matter physics to other domains of scientific research.