We present a formalism of Galilean quantum mechanics in non-inertial reference frames and discuss its implications for the equivalence principle. This extension of quantum mechanics rests on the Galilean line group, the semidirect product of the real line and the group of analytic functions from the real line to the Euclidean group in three dimensions. This group provides transformations between all inertial and non-inertial reference frames and contains the Galilei group as a subgroup. We construct a certain class of unitary representations of the Galilean line group and show that these representations determine the structure of quantum mechanics in non-inertial reference frames. Our representations of the Galilean line group contain the usual unitary projective representations of the Galilei group, but have a more intricate cocycle structure. The transformation formula for the Hamiltonian under the Galilean line group shows that in a non-inertial reference frame it acquires a fictitious potential energy term that is proportional to the inertial mass, suggesting the equivalence of inertial mass and gravitational mass in quantum mechanics.

Particle systems admit a variety of tensor product structures (TPSs) depending on the algebra of observables chosen for analysis. Global symmetry transformations and dynamical transformations may be resolved into local unitary operators with respect to certain TPSs and not with respect to others. Symmetry-invariant and dynamical-invariant TPSs are defined and various notions of entanglement are considered for scattering states.PACS numbers: 03.67. Mn, 03.65.Nk, The interaction of particle systems via scattering is a fundamental theoretical and experimental paradigm. The quantum information theory of particle scattering is, however, still in its infancy. Results, theoretical and computational, exist for the entanglement between the momenta [1] or the angular momenta [2] of two particles generated in scattering, but many problems remain open. The challenges are partly technical due to the greater complexity of dealing with entanglement in continuous variable systems [3] and partly conceptual as in defining a measure of entanglement that has meaningful properties under space-time symmetry transformations. See, for example, the literature on spin-entanglement of relativistic particles [4,5] where different types of entanglement (between two particles, between two particles' spins, and between a single particle's spin and momentum) have been discussed and occasionally confused.In this letter, we examine how some of these difficulties may be resolved by combining two approaches: (1) the generalized tensor product structures (TPSs) and observable-dependent entanglement developed by Zanardi and others [6], and (2) the representation theory of space-time symmetry groups, which has a long and fruitful history in quantum mechanics. Using these methods, TPSs for single particle and multi-particle systems are explored. These methods allow one to distinguish between TPSs that are symmetry invariant and/or dynamically invariant and TPSs that are not, and, in the latter case, to obtain quantitative expressions for the change of entanglement. The reason why certain TPSs have entanglement measures which are symmetry or dynamically invariant is that the space-time symmetries or the time evolution operator, respectively, act as a product of local unitaries with respect to these TPSs.As an application of these general concepts and methods, we will study non-relativistic elastic scattering of two particles. In this context, several interesting results emerge. First, there are single particle TPSs that are invariant under transformations between inertial reference frames, and these TPSs allow one to define intraparticle entanglement between momentum and spin degrees of freedom in a Galilean invariant manner. Second, there are multiple, inequivalent two particle TPSs that are symmetry invariant. In particular, these TPSs can be used to define Galilean invariant entanglement between the internal and external degrees of freedom of the two particle system. Finally, this internal-external entanglement is also dynamically invariant, i.e., i...

Particle systems admit a variety of tensor product structures (TPSs) depending on the complete system of commuting observables chosen for the analysis. Different notions of entanglement are associated with these different TPSs. Global symmetry transformations and dynamical transformations factor into products of local unitary operators with respect to certain TPSs and not with respect to others. Symmetry-invariant and dynamical-invariant TPSs and corresponding measures of entanglement are defined for particle scattering systems.

Rigged Hilbert spaces of Hardy functions lead to a consistent theory of resonance scattering and decay. Contrary to the claims of a recent article [8], the theory holds for a wide range of potentials and rigorously describes the asymmetric time evolution of resonances.

We propose a theory of resonances by combining the S-matrix approach with the Bakamjian–Thomas (BT) construction. Characterization of resonances by the poles of the S-matrix has many advantages. Foremost among them is perhaps the gauge invariance of the definitions of resonance mass and width, a problem with which some definitions based on field theoretical approaches suffer. The BT construction provides a general framework for constructing Poincaré generators for an interacting quantum system. While much of what we develop here can be cast in the language of quantum field theory, in the spirit of BT construction, which does not assume the existence of local field mediating interactions, we will work at the fundamental level of an interacting Poincaré algebra. Our construction shows that a subset of this Poincaré algebra integrates to a representation of the semigroup of causal transformations of relativistic space-time. These representations are characterized by the spin and S-matrix complex pole position of the resonance. The state vectors that transform under these representations also show an exact exponential decay, the signature of a decaying state. In this sense, the semigroup representations developed here tie together resonances and decaying states into a single theoretical description.