Groups in physics: Collective model of the nucleus canonical transformations in quantum mechanics

M. Moshińsky

J. Deenen, C. Quesne

The present series of papers deals with various realizations of the dynamical group 𝒮𝓅c(2d,R) of microscopic collective states for an A nucleon system in d dimensions, defined as those A particle states invariant under the orthogonal group O(n) associated with the n=A−1 Jacobi vectors. In the present paper, we derive two boson representations of 𝒮𝓅c(2d,R), namely the Dyson representation and the Holstein–Primakoff (HP) one. Our starting point is a representation of microscopic collective states, as introduced in the first paper of the present series, in a Barut Hilbert space ℱc of analytic functions in ν =(1/2)d(d+1) complex variables. Basis functions in ℱc, classified according to the chain 𝒮𝓅c(2d,R)⊇𝒰c(d), can be put into one-to-one correspondence with basis functions, classified according to the chain 𝔘(ν)⊇𝔘(d), in a Bargmann Hilbert space ℬ of analytic functions in ν complex variables representing ν-dimensional boson states. By equating the complex variables of ℱc and their conjugate momenta with those of ℬ, we get the non-Hermitian Dyson representation of 𝒮𝓅c(d,R). We then go from the latter to the Hermitian HP representation by means of a canonical transformation that restores the Hermiticity properties of the variables and conjugate momenta. The inverse of the HP representation gives the unitary representation in quantum mechanics of the classical canonical transformation relating the oscillator Hamiltonians of the microscopic collective model and the boson macroscopic one. From the ν boson creation and annihilation operators, it is possible to build the generators of a 𝔘(ν) group, which in the physical three-dimensional case reduces to 𝔘(6). The latter is finally compared with the U(6) group appearing in the interacting boson model.

James Lindesay

Alastair Brodlie

p-mechanics is a consistent physical theory which describes both classical and quantum mechanics simultaneously through the representation theory of the Heisenberg group. In this paper we describe how nonlinear canonical transformations affect p-mechanical observables and states. Using this we show how canonical transformations change a quantum mechanical system. We seek an operator on the set of p-mechanical observables which corresponds to the classical canonical transformation. In order to do this we derive a set of integral equations which when solved will give us the coherent state expansion of this operator. The motivation for these integral equations comes from the work of Moshinsky and a variety of collaborators. We consider a number of examples and discuss the use of these equations for non-bijective transformations.

Philippe Manjakasoa Randriantsoa, Ravo Tokiniaina Ranaivoson, Raoelina Andriambololona +3 more

Recent developments in the unification of quantum mechanics and relativity have emphasized the necessity of generalizing classical phase space into a relativistic quantum phase space which is a framework that inherently incorporates the uncertainty principle and relativistic covariance. In this context, the present work considers the derivation of linear and quadratic Casimir operators corresponding to representations of the Linear Canonical Transformations (LCT) group associated with a five-dimensional spacetime of signature (1,4). This LCT group, which emerges naturally as the symmetry group of the relativistic quantum phase space, is isomorphic to the symplectic group Sp(2,8). The latter notably contains the de Sitter group SO(1,4) as a subgroup. This geometric setting provides a unified framework for extending the Standard Model of particle physics while incorporating cosmological features. Previous studies have shown that the LCT group admits both fermionic-like and bosonic-like representations. Within this framework, a novel classification of quarks and leptons, including sterile neutrinos, has also been proposed. In this work, we present a systematic derivation of the linear and quadratic Casimir operators associated with these representations, motivated by their fundamental role in the characterization of symmetry groups in physics. The construction is based on the relations between the LCT group and the pseudo-unitary group U(1,4). Three linears and three quadratics Casimir operators are identified: two corresponding to the fermionic-like representation, two to the bosonic-like representation, and two hybrid operators linking the two representations. The complete eigenvalue spectra and corresponding eigenstates for each operator are subsequently computed and identified