M. Gadella

Moyal quantization of 2+1‐dimensional Galilean systems

Stratonovich–Weyl kernels are constructed for some of the coadjoint orbits of the two‐dimensional extended Galilean group G(2+1). As an intermediate step, the unitary irreducible representations associated with a given group orbit are obtained by using the Kirillov–Mackey theory. Star products are defined, in the sense of Moyal, for functions on each of these orbits. The central extension of G(2+1) with parameter k is also analyzed, which results from the commutator between the generators of boosts, to conclude that it originates a sort of nonrelativistic remainder of the Thomas precession.

A rigged Hilbert space of Hardy‐class functions: Applications to resonances

The explicit construction of a dense subspace Φ of square integrable functions on the positive half of the real line is given. This space Φ has the properties that: (1) it is endowed with a metrizable nuclear topology, (2) it is stable under multiplication by x, and (3) the functions in Φ have suitable analytical continuation to a half plane. The space Φ* of functions which are conjugate to elements of Φ is also considered. Then the triplets Φ⊆ L2 (0,∞)⊆Φ′ and Φ*⊆ L2 (0,∞)⊆Φ*′ are used to give a description of resonances.

Gamow vectors and decaying states

Gamow vectors are generalized eigenvectors of the Hamiltonian with complex eigenvalues that describe exponentially decaying (or growing) states. The energy wavefunctions corresponding to Gamow vectors have a pole immediately below (or above) the real axis in the complex energy plane. Although complex energy values were introduced more than half a century ago for the theory of alpha decay, they have become disreputable and have been banished from quantum mechanics textbooks because of mathematical problems and incorrect physical interpretation. Developments in modern mathematics have now provided a mathematical foundation that has led to the correct physical interpretation. In this article, energy eigenvectors with complex eigenvalues are first introduced by explicitly considering a specific simple decaying system. Then, an elementary account of the properties of Gamow vectors is given. The results from the decaying system provide a motivation for the introduction of Gamow vectors and some of the backgroun...

Quantum mechanical irrebersibility

Microphysical irreversibility is distinguished from the extrinsic irreversibility of open systems. The rigged Hilbert space (RHS) formulation of quantum mechanics is justified based on the foundations of quantum mechanics. Unlike the Hilbert space formulation of quantum mechanics, the rigged Hilbert space formulation of quantum mechanics allows for the description of decay and other irreversible processes because it allows for a preferred direction of time for time evolution generated by a semi-bounded, essentially self-adjoint Hamiltonian. This quantum mechanical arrow of time is obtained and applied to a resonance scattering experiment. Within the cintext of a resonance scattering experiment, it is shown how the dichotomy of state and observable leads to a pair of RHSs, one for states and one for observables. Using resonance scattering, it is shown how the Gamow vectors describing decaying states with complex energy eigenvalues (ER − iγ/2) emerge from the first-order resonance poles of the S-matrix. Then, these considerations are extended to S-matrix poles order N and it shown that this leads to Gamow vectors of higher order k = 0, 1, …, N − 1 which are also Jordan vectors of degree k + 1 = 1, 2,…,N. The matrix elements of the self-adjoint Hamiltonian between these vectors from a Jordan block of degree N. The two semigroups of time evolution generated by the Hamiltonian are obtained for Gamow vectors of any order. It is shown how the irreversible time evolution of Gamow vectors enables the derivation of an exact Golden Rule for the calculation of decay probabilities, from which the standard (approximate) Golden Rule is obtained as the Born approximation in the limit γR ⪡ ER.

A pedestrian introduction to Gamow vectors

The Gamow vector description of resonances is compared with the S matrix and the Green function descriptions using the example of the square barrier potential. By imposing different boundary conditions on the time independent Schrodinger equation, we obtain either eigenvectors corresponding to real eigenvalues and the physical spectrum or eigenvectors corresponding to complex eigenvalues (Gamow vectors) and the resonance spectrum. We show that the poles of the S matrix are the same as the poles of the Green function and are the complex eigenvalues of the Schrodinger equation subject to a purely outgoing boundary condition. The intrinsic time asymmetry of the purely outgoing boundary condition is discussed. Finally, we show that the probability of detecting the decay within a shell around the origin of the decaying state follows an exponential law if the Gamow vector (resonance) contribution to this probability is the only contribution that is taken into account.

Relativistic Gamow Vectors

The Friedrichs model has often been used in order to obtain explicit formulas for eigenvectors associated to complex eigenvalues corresponding to lifetimes. Such eigenvectors are called Gamow vectors and they acquire meaning in extensions of the conventional Hilbert space of quantum theory to the so-called rigged Hilbert space. In this paper, Gamow vectors are constructed for a solvable model of an unstable relativistic field. As a result, we obtain a time asymmetric relativistic extension of the Fock space. This extension leads to two distinct Poincare semigroups. The time reversal transformation maps one semigroup to the other. As a result, the usual PCT invariance should be extended. We show that irreversibility as expressed by dynamical semigroups is compatible with the requirements of relativity.

The Stratonovich–Weyl correspondence for one‐dimensional kinematical groups

The Stratonovich–Weyl correspondence is a restatement of the Moyal quantization where the phase space is a manifold and where a group of transformations acts on it transitively. The first and most important step is to define a mapping from the manifold into the set of self‐adjoint operators on a Hilbert space, under suitable conditions. This mapping is called a Stratonovich–Weyl kernel. The construction of this mapping is discussed on coadjoint orbits of the one‐dimensional Galilei, Poincare, and Newton–Hooke groups as well as the two‐dimensional Euclidean group.

Gamow vectors for degenerate scattering resonances

In this paper we construct Gamow vectors for resonances given by poles of the analytic continuation of the S matrix of any finite order. We study their modes of decay (or growth). We obtain Jordan block structures for the extended Hamiltonians and evolution operators on the subspaces spanned by these Gamow vectors. We perform this study within the context of the rigged Hilbert space extension of quantum theory. We construct an explicit Friedrichs model with a double pole resonance to illustrate the general formulation.

Feynman Formulas for Particles with Position-Dependent Mass

In this paper, we obtain Feynman formulas for solutions to equations describing the diffusion of particles with mass depending on the particle position and to Schrodinger-type equations describing the evolution of quantum particles with similar properties. Such particles (to be more precise, quasi-particles) arise in, e.g., models of semiconductors. Tens of papers studying such models have been published (see [1] and the references therein), but representations of solutions to the arising Schrodinger- and heat-type equations, which go back to Feynman, have not been considered so far. One of the possible reasons is that the traditional application of Feynman’s approach involves integrals with respect to diffusion processes whose transition probabilities have no explicit representation (in terms of elementary functions) in the situation under consideration. In this paper, instead of these transition probabilities, we use their approximations, which can be expressed in terms of elementary functions. Apparently, similar approximations were first applied in [4, 5] to study the diffusion and the quantum evolution of particles of constant mass on Riemannian manifolds. It turns out that the central idea of the approach developed in [4, 5] can also be applied (after appropriate modifications) to the situation considered in this paper. In what follows, we assume that solutions to the Cauchy problems for the equations under examination exist and are unique; thus, we can and shall consider not only solutions to equations but also the corresponding semigroups. Somewhat changing the terminology of [2, 6], we define a real (complex) Schrodinger semigroup as e – tH (respectively, e ith ), where H is a self-adjoint positive operator on a Hilbert space or the generator of a diffusion process.

Rigged Hilbert Space Treatment of Continuous Spectrum

The ability of the Rigged Hilbert Space formalism to deal with continuous spectrum is demonstrated within the example of the square barrier poten- tial. The non-square integrable solutions of the time-independent Schrodinger equation are used to define Dirac kets, which are (generalized) eigenvectors of the Hamiltonian. These Dirac kets are antilinear functionals over the space of physical wave functions. They are also basis vectors that expand any physical wave function in a Dirac basis vector expansion. It is shown that an accept- able physical wave function must fulfill stronger conditions than just square integrability—the space of physical wave functions is not the whole Hilbert space but rather a dense subspace of the Hilbert space. We construct the position and energy representations of the Rigged Hilbert Space generated by the square barrier potential Hamiltonian. We shall also construct the unitary operator that transforms from the position into the energy representation. We shall see that in the energy representation the Dirac kets act as the anti- linear Schwartz delta functional. In constructing the Rigged Hilbert Space of the square barrier potential, we will find a systematic procedure to construct the Rigged Hilbert Space of a large class of spherically symmetric potentials. The example of the square barrier potential will also make apparent that the natural framework for the solutions of a Schrodinger operator with continuous spectrum is the Rigged Hilbert Space rather than just the Hilbert space.