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J. D. Vergara

Publications and source records attributed to J. D. Vergara.

14 recordsLinked to original sources
arXiv2016

Gauge Invariance and the Quantum Metric Tensor

The quantum metric tensor was introduced for defining the distance in the parameter space of a system. However, it is also useful for other purposes, like predicting quantum phase transitions. Due to the physical information this tensor provides, its gauge independence sounds reasonable. More over, its original construction was made by looking for this gauge independence. The aim of this paper, however, is to prove that the quantum metric tensor does depend on the gauge. In addition, a real gauge invariant quantum metric tensor is introduced. In this paper, the gauge dependence is explicitly shown by computing the quantum metric tensor of the Landau problem in different gauges. Then, a real gauge independent metric tensor is proposed and computed for the same Landau problem. Since the gauge dependence has not been observed before, the results of this paper might lead to a new study of topics that were believed to be completely understood.

quant-ph↗
arXiv2009

An Alternative Canonical Approach to the Ghost Problem in a Complexified Extension of the Pais-Uhlenbeck Oscillator

Our purpose in this paper is to analyze the Pais-Uhlenbeck (PU) oscillator using complex canonical transformations. We show that starting from a Lagrangian approach we obtain a transformation that makes the extended PU oscillator, with unequal frequencies, to be equivalent to two standard second order oscillators which have the original number of degrees of freedom. Such extension is provided by adding a total time derivative to the PU Lagrangian together with a complexification of the original variables further subjected to reality conditions in order to maintain the required number of degrees of freedom. The analysis is accomplished at both the classical and quantum levels. Remarkably, at the quantum level the negative norm states are eliminated, as well as the problems of unbounded below energy and non-unitary time evolution. We illustrate the idea of our approach by eliminating the negative norm states in a complex oscillator. Next, we extend the procedure to the Pais-Uhlenbeck oscillator. The corresponding quantum propagators are calculated using Schwinger's quantum action principle. We also discuss the equal frequency case at the classical level.

math-ph↗
arXiv2009

The Photon Sector in the Quantum Myers-Pospelov Model: An improved description

The quantization of the electromagnetic sector of the Myers-Pospelov model coupled to standard fermions is studied. Our main objective is to construct an effective quantum theory that results in a genuine perturbation of QED, such that setting zero the Lorentz invariance violation (LIV) parameters will reproduce it. This is achieved by introducing an additional low energy scale $M$, together with a physically motivated prescription to take the QED limit. The prescription is successfully tested in the calculation of the electron self-energy in the one loop approximation. The LIV radiative corrections turn out to be properly scaled by very small factors for any reasonable values of the parameters, no fine-tuning problems are found at this stage and the choice for $M$ to be of the order of the electroweak symmetry breaking scale is consistent with the stringent bounds for the LIV parameters, in particular with those arising from induced dimension three operators.

hep-ph↗
arXiv2008

Quantization of the Myers-Pospelov model: the photon sector interacting with standard fermions as a perturbation of QED

We study the quantization of the electromagnetic sector of the Myers-Pospelov model coupled to standard fermions. Our main objective, based upon experimental and observational evidence, is to construct an effective theory which is a genuine perturbation of QED, such that setting zero the Lorentz invariance violation parameters will reproduce it. To this end we provide a physically motivated prescription, based on the effective character of the model, regarding the way in which the model should be constructed and how the QED limit should be approached. This amounts to the introduction of an additional coarse-graining physical energy scale $M$, under which we can trust the effective field theory formulation. The prescription is successfully tested in the calculation of the Lorentz invariance violating contributions arising from the electron self-energy. Such radiative corrections turn out to be properly scaled by very small factors for any reasonable values of the parameters and no fine-tuning problems are found. Microcausality violations are highly suppressed and occur only in a space-like region extremely close to the light-cone. The stability of the model is guaranteed by restricting to concordant frames satisfying $1-|\mathbf{v}_{max}|> 6.5\times10^{-11}$.

hep-ph↗
arXiv2008

The scalar sector in the Myers-Pospelov model

We construct a perturbative expansion of the scalar sector in the Myers-Pospelov model, up to second order in the Lorentz violating parameter and taking into account its higher-order time derivative character. This expansion allows us to construct an hermitian positive-definite Hamiltonian which provides a correct basis for quantization. Demanding that the modified normal frequencies remain real requires the introduction of an upper bound in the magnitude |k| of the momentum, which is a manifestation of the effective character of the model. The free scalar propagator, including the corresponding modified dispersion relations, is also calculated to the given order, thus providing the starting point to consider radiative corrections when interactions are introduced.

hep-ph↗
arXiv2007

Quantization of the Myers-Pospelov model: a progress report

The Myers-Pospelov (MP) model is an effective field theory, including dimension five operators, which describes the phenomenology of active Lorentz invariance violation produced by a preferred reference frame. We concentrate here in the case of the modified electrodynamics. The point of view taken in this work is that the Lorentz violating part of the action in the MP model, which includes higher order time derivative (HOTD) operators, is to be considered as a perturbation over the dynamics described by standard Electrodynamics, particularly in the quantum case. In order to cope with the challenges posed by HOTD theories it will be necessary to deal with a modified perturbation scheme which is well described in the literature. We apply such methods to this specific model providing a quantization of the free sector of the theory. The calculation of interacting processes, together with radiative corrections, is beyond the scope of the present article.

hep-ph↗
arXiv2007

Noncommutative spaces, the quantum of time and the Lorentz symmetry

We introduce three space-times that are discrete in time and compatible with the Lorentz symmetry. We show that these spaces are no commutative, with commutation relations similar to the relations of the Snyder and Yang spaces. Furthermore, using a reparametrized relativistic particle we obtain a realization of the Snyder type spaces and we construct an action for them.

hep-th↗
arXiv2001

Normal Coordinates and Primitive Elements in the Hopf Algebra of Renormalization

We introduce normal coordinates on the infinite dimensional group $G$ introduced by Connes and Kreimer in their analysis of the Hopf algebra of rooted trees. We study the primitive elements of the algebra and show that they are generated by a simple application of the inverse Poincaré lemma, given a closed left invariant 1-form on $G$. For the special case of the ladder primitives, we find a second description that relates them to the Hopf algebra of functionals on power series with the usual product. Either approach shows that the ladder primitives are given by the Schur polynomials. The relevance of the lower central series of the dual Lie algebra in the process of renormalization is also discussed, leading to a natural concept of $k$-primitiveness, which is shown to be equivalent to the one already in the literature.

hep-th↗
arXiv1999

Hamiltonian LGT in the complete Fourier analysis basis

The main problem in the Hamiltonian formulation of Lattice Gauge Theories is the determination of an appropriate basis avoiding the over-completeness arising from Mandelstam relations. We short-cut this problem using Harmonic analysis on Lie-Groups and intertwining operators formalism to explicitly construct a basis of the Hilbert space. Our analysis is based only on properties of the tensor category of Lie-Group representations. The Hamiltonian of such theories is calculated yielding a sparse matrix whose spectrum and eigenstates could be exactly derived as functions of the coupling $g^2$

hep-lat↗
arXiv1999

The basis of the physical Hilbert space of lattice gauge theories

Non-linear Fourier analysis on compact groups is used to construct an orthonormal basis of the physical (gauge invariant) Hilbert space of Hamiltonian lattice gauge theories. In particular, the matrix elements of the Hamiltonian operator involved are explicitly computed. Finally, some applications and possible developments of the formalism are discussed.

hep-lat↗
arXiv1999

Real sector of the nonminimally coupled scalar field to self-dual gravity

A scalar field nonminimally coupled to gravity is studied in the canonical framework, using self-dual variables. The corresponding constraints are first class and polynomial. To identify the real sector of the theory, reality conditions are implemented as second class constraints, leading to three real configurational degrees of freedom per space point. Nevertheless, this realization makes non-polynomial some of the constraints. The original complex symplectic structure reduces to the expected real one, by using the appropriate Dirac brackets. For the sake of preserving the simplicity of the constraints, an alternative method preventing the use of Dirac brackets, is discussed. It consists of converting all second class constraints into first class by adding extra variables. This strategy is implemented for the pure gravity case.

gr-qc↗
arXiv1997

Loop Variables for compact two-dimensional quantum electrodynamics

Variables parametrized by closed and open curves are defined to reformulate compact U(1) Quantum Electrodynamics in the circle with a massless fermion field. It is found that the gauge invariant nature of these variables accommodates into a regularization scheme for the Hamiltonian and current operators that is specially well suited for the study of the compact case. The zero mode energy spectrum, the value of the axial anomaly and the anomalous commutators this model presents are hence determined in a manifestly gauge invariant manner. Contrary to the non compact case, the zero mode spectrum is not equally spaced and consequently the theory does not lead to the spectrum of a free scalar boson. All the states are invariant under large gauge transformations. In particular, that is the case for the vacuum, and consequently the $θ$-dependence does not appear.

hep-th↗