On deformations of the Galilei algebra

In this note we compute all deformations of the 4-dimensional classical Galilei algebra &. In particular, we find examples of quadratic, cubic and quartic Lie algebra deformations.

$$ \mathcal{N} $$ = 1, 2, 3 ℓ-conformal Galilei superalgebras

The issue of constructing $$ \mathcal{N} $$ N = 1, 2, 3 supersymmetric extensions of the ℓ-conformal Galilei algebra is reconsidered following the approach in [27]. Drawing a parallel between acceleration generators entering the superalgebra and irreducible supermultiplets of d = 1, $$ \mathcal{N} $$ N -extended superconformal group, a new $$ \mathcal{N} $$ N = 1 ℓ-conformal Galilei superalgebra, two new $$ \mathcal{N} $$ N = 2 variants, and two new $$ \mathcal{N} $$ N = 3 versions are built. Realisations in terms of differential operators in superspace are given.

The group structure of dynamical transformations between quantum reference frames

Recently, it was shown that when reference frames are associated to quantum systems, the transformation laws between such quantum reference frames need to be modified to take into account the quantum and dynamical features of the reference frames. This led to a relational description of the phase space variables of the quantum system of which the quantum reference frames are part of. While such transformations were shown to be symmetries of the system's Hamiltonian, the question remained unanswered as to whether they enjoy a group structure, similar to that of the Galilei group relating classical reference frames in quantum mechanics. In this work, we identify the canonical transformations on the phase space of the quantum systems comprising the quantum reference frames, and show that these transformations close a group structure defined by a Lie algebra, which is different from the usual Galilei algebra of quantum mechanics. We further find that the elements of this new algebra are in fact the building blocks of the quantum reference frames transformations previously identified, which we recover. Finally, we show how the transformations between classical reference frames described by the standard Galilei group symmetries can be obtained from the group of transformations between quantum reference frames by taking the zero limit of the parameter that governs the additional noncommutativity introduced by the quantum nature of inertial transformations.

Nonrelativistic k-contractions of the coadjoint Poincaré algebra

We study a class of extensions of the [Formula: see text]-contracted Poincaré algebra under the hypothesis of generalizing the Bargmann algebra and its central charge. As we will see this type of contractions will lead in a natural way to consider the codajoint Poincaré algebra and some of their contractions. Among them there is one such that considering the quotient of it by a suitable ideal, the (stringy) [Formula: see text]-brane Galilei algebra is recovered.

Polynomial Extensions of the Weyl C*-Algebra

We introduce higher order (polynomial) extensions of the unique (up to isomorphisms) nontrivial central extension of the Heisenberg algebra, which can be concretely realized as sub-Lie algebras of the polynomial algebra generated by the creation and annihilation operators in the Schrödinger representation. The simplest nontrivial of these extensions (the quadratic one) is isomorphic to the Galilei algebra, widely studied in quantum physics. By exponentiation of this representation we construct the corresponding polynomial analogue of the Weyl [Formula: see text]-algebra and compute the polynomial Weyl relations. From this we deduce the explicit form of the composition law of the associated nonlinear extensions of the 1-dimensional Heisenberg group. The above results are used to calculate a simple explicit form of the vacuum characteristic functions of the nonlinear field operators of the Galilei algebra, as well as of their moments. The corresponding measures turn out to be an interpolation family between Gaussian and Meixner, in particular Gamma.