arXiv subjects

Victor Aldaya

Publications and source records attributed to Victor Aldaya.

10 recordsLinked to original sources

arXiv2024

Non-relativistic tachyons: a new representation of the Galilei group

An algebraic characterization of the contractions of the Poincaré group permits a proper construction of a non-relativistic limit of its tachyonic representation. We arrive at a consistent, nonstandard representation of the Galilei group which was disregarded long ago by supposedly unphysical properties. The corresponding quantum (and classical) theory shares with the relativistic one their fundamentals, and serves as a toy model to better comprehend the unusual behavior of the tachyonic representation. For instance, we see that evolution takes place in a spatial coordinate rather than time, as for relativistic tachyons, but the modulus of the three-momentum is the same for all Galilean observers, leading to a new dispersion relation for a Galilean system. Furthermore, the tachyonic objects described by the new representation cannot be regarded as localizable in the standard sense.

hep-th↗

arXiv2023

Tachyons in "momentum-space'' representation

The momentum space associated with "tachyonic particles" proves to be rather intricate, departing very much from the ordinary dual to Minkowski space directly parametrized by space-time translations of the Poincaré group. In fact, although described by the constants of motion (Noether invariants) associated with space-time translations, they depend non-trivially on the parameters of the rotation subgroup. However, once the momentum space is parametrized by the Noether invariants, it behaves exactly as that of ordinary particles. On the other hand, the evolution parameter is no longer the one associated with time translation, whose Noether invariant, $P_o$, is now a basic one. Evolution takes place in a spatial direction. These facts not only make difficult the computation of the corresponding representation, but also force us to a sound revision of several traditional ingredients related to Cauchy hypersurface, scalar product and, of course, causality. After that, the theory becomes consistent and could shed new light on some special physical situations like inflation or traveling inside a black hole.

gr-qc↗

arXiv2020

A proper scalar product for tachyon representations in configuration space

We propose a new inner product for scalar fields that are solutions of the Klein-Gordon equation with $m^2<0$. This inner product is non-local, bearing an integral kernel including Bessel functions of the second kind, and the associated norm proves to be positive definite in the subspace of oscillatory solutions, as opposed to the conventional one. Poincaré transformations are unitarily implemented on this subspace, which is the support of a unitary and irreducible representation of the proper orthochronous Poincaré group. We also provide a new Fourier Transform between configuration and momentum spaces which is unitary, and recover the projection onto the representation space. This new scenario suggests a revision of the corresponding quantum field theory.

hep-th↗

arXiv2019

$SU(2)$-particle sigma model: Momentum-space quantization of a particle on the sphere $S^3$

We perform the momentum-space quantization of a spin-less particle moving on the $SU(2)$ group manifold, that is, the three-dimensional sphere $S^{3}$, by using a non-canonical method entirely based on symmetry grounds. To achieve this task, non-standard (contact) symmetries are required as already shown in a previous article where the configuration-space quantization was given. The Hilbert space in the momentum space representation turns out to be made of a subset of (oscillatory) solutions of the Helmholtz equation in four dimensions. The most relevant result is the fact that both the scalar product and the generalized Fourier transform between configuration and momentum spaces deviate notably from the naively expected expressions, the former exhibiting now a non-trivial kernel, under a double integral, traced back to the non-trivial topology of the phase space, even though the momentum space as such is flat. In addition, momentum space itself appears directly as the carrier space of an irreducible representation of the symmetry group, and the Fourier transform as the unitary equivalence between two unitary irreducible representations.

math-ph↗

arXiv2016

$SU(2)$ particle sigma model: the role of non-point symmetries in global quantization

In this paper we achieve the quantization of a particle moving on the $SU(2)$ group manifold, that is, the three-dimensional sphere $S^{3}$, by using group-theoretical methods. For this purpose, a fundamental role is played by contact, non-point symmetries, i.e., symmetries that leave the Poincaré-Cartan form semi-invariant at the classical level, although not necessarily the Lagrangian. Special attention is paid to the role played by the basic quantum commutators, which depart from the canonical, Heisenberg-Weyl ones, as well as the relationship between the integration measure in the Hilbert space of the system and the non-trivial topology of the configuration space. Also, the quantization on momentum space is briefly outlined.

math-ph↗

arXiv2015

Contact Symmetries in Non-Linear Mechanics: a preliminary step to (Non-Canonical) Quantization

In this paper we exploit the use of symmetries of a physical system so as to characterize algebraically the corresponding solution manifold by means of Noether invariants. This constitutes a necessary preliminary step towards the correct quantization in non-linear cases, where the success of Canonical Quantization is not guaranteed in general. To achieve this task "point symmetries" of the Lagrangian are generally not enough, and the notion of contact transformations is in order: the solution manifold can not be in general parametrized by means of Noether invariants associated with basic point symmetries. The use of the contact structure given by the Poincaré-Cartan form permits the definition of the symplectic form on the solution manifold, through some sort of Hamilton-Jacobi transformation. It also provides the required basic symmetries, realized as Hamiltonian vector fields associated with global functions on the solution manifold (thus constituting an inverse of the Noether Theorem), lifted back to the evolution space through the inverse of this Hamilton-Jacobi mapping. In this framework, solutions and symmetries, as a whole, are somehow identified and this correspondence is also kept at a perturbative level. We present non-trivial examples of this interplay between symmetries and solutions pointing out the usefulness of this mechanism in approaching the corresponding quantization. In particular, we achieve the proper generalization of the Heisenberg-Weyl algebra for the non-linear particle sigma model in $S^3$ within this framework, and notice that a subset of the classical symmetries corresponding with this quantizing algebra (those generalizing boosts) are necessarily of non-point character.

math-ph↗

arXiv2014

Interrelations between different canonical descriptions of dissipative systems

There are many approaches for the description of dissipative systems coupled to some kind of environment. This environment can be described in different ways; only effective models will be considered here. In the Bateman model, the environment is represented by one additional degree of freedom and the corresponding momentum. In two other canonical approaches, no environmental degree of freedom appears explicitly but the canonical variables are connected with the physical ones via non-canonical transformations. The link between the Bateman approach and those without additional variables is achieved via comparison with a canonical approach using expanding coordinates since, in this case, both Hamiltonians are constants of motion. This leads to constraints that allow for the elimination of the additional degree of freedom in the Bateman approach. These constraints are not unique. Several choices are studied explicitly and the consequences for the physical interpretation of the additional variable in the Bateman model are discussed.

physics.class-ph↗

arXiv2011

Chained Quantum Arnold Transformations

We put forward the concatenation of Quantum Arnold Transformations as a tool to obtain the wave function of a particle subjected to a harmonic potential which is switched on and off successively. This simulates the capture and release process of an ion in a trap and provides a mathematical picture of this physical process.

quant-ph↗

arXiv2011

Harmonic states for the free particle

Different families of states, which are solutions of the time-dependent free Schrödinger equation, are imported from the harmonic oscillator using the Quantum Arnold Transformation introduced in a previous paper. Among them, infinite series of states are given that are normalizable, expand the whole space of solutions, are spatially multi-localized and are eigenstates of a suitably defined number operator. Associated with these states new sets of coherent and squeezed states for the free particle are defined representing traveling, squeezed, multi-localized wave packets. These states are also constructed in higher dimensions, leading to the quantum mechanical version of the Hermite-Gauss and Laguerre-Gauss states of paraxial wave optics. Some applications of these new families of states and procedures to experimentally realize and manipulate them are outlined.

quant-ph↗

arXiv2010

The quantum Arnold transformation

By a quantum version of the Arnold transformation of classical mechanics, all quantum dynamical systems whose classical equations of motion are non-homogeneous linear second-order ordinary differential equations, including systems with friction linear in velocity, can be related to the quantum free-particle dynamical system. This transformation provides a basic (Heisenberg-Weyl) algebra of quantum operators, along with well-defined Hermitian operators which can be chosen as evolution-like observables and complete the entire Schrödinger algebra. It also proves to be very helpful in performing certain computations quickly, to obtain, for example, wave functions and closed analytic expressions for time-evolution operators.

quant-ph↗