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We define a time-dependent extension of the quantum geometric tensor to describe the geometry of the time-parameter space for a quantum state, by considering small variations in both time and wave function parameters. Compared to the standard quantum geometric tensor, this tensor introduces new temporal components, enabling the analysis of systems with non-time-separable or explicitly time-dependent quantum states and encoding new information about these systems. In particular, the time-time component of this tensor is related to the energy dispersion of the system. We applied this framework to a harmonic/inverted oscillator, a time-dependent harmonic oscillator, and a chain of generalized harmonic/inverted oscillators. We show some results on the scalar curvature associated with the time-dependent quantum geometric tensor and the generalized Berry curvature behavior on the transition from harmonic oscillators to inverted ones. Furthermore, we analyze the entanglement for the chain through purity analysis, obtaining that the purity for any excited state is zero in the mentioned transitions., Comment: 37 pages, 21 figures

This work introduces a geometrical object that generalizes the quantum geometric tensor; we call it $N$-bein. Analogous to the vielbein (orthonormal frame) used in the Cartan formalism, the $N$-bein behaves like a ``square root'' of the quantum geometric tensor. Using it, we present a quantum geometric tensor of two states that measures the possibility of moving from one state to another after two consecutive parameter variations. This new tensor determines the commutativity of such variations through its anti-symmetric part. In addition, we define a connection different from the Berry connection, and combining it with the $N$-bein allows us to introduce a notion of torsion and curvature \`{a} la Cartan that satisfies the Bianchi identities. Moreover, the torsion coincides with the anti-symmetric part of the two-state quantum geometric tensor previously mentioned, and thus, it is related to the commutativity of the parameter variations. We also describe our formalism using differential forms and discuss the possible physical interpretations of the new geometrical objects. Furthermore, we define different gauge invariants constructed from the geometrical quantities introduced in this work, resulting in new physical observables. Finally, we present two examples to illustrate these concepts: a harmonic oscillator and a generalized oscillator, both immersed in an electric field. We found that the new tensors quantify correlations between quantum states that were unavailable by other methods., Comment: 21 pages, 3 figures. Accepted version (Journal of Physics A: Mathematical and Theoretical)

In this paper, we extend the quantum geometric tensor for parameter-dependent curved spaces to higher dimensions, and introduce an equivalent definition that generalizes the Zanardi, et al, formulation of the tensor. The parameter-dependent metric modifies the behavior of both the quantum metric tensor and Berry curvature in a purely geometric way. Our focus is on understanding the distinctions in higher dimensions that emerge when using the generalized tensor compared to the conventional one. Through a comparative analysis, illustrated with examples in two dimensions, we highlight unique quantum geometric properties for both the quantum metric tensor and the Berry curvature. Additionally, we explore differences between analytical and perturbative approaches in solving the problems., Comment: 21 pages

We compute both analytically and numerically the geometry of the parameter space of the anharmonic oscillator employing the quantum metric tensor and its scalar curvature. A novel semiclassical treatment based on a Fourier decomposition allows to construct classical analogues of the quantum metric tensor and of the expectation values of the transition matrix elements. A detailed comparison is presented between exact quantum numerical results, a perturbative quantum description and the semiclassical analysis. They are shown to coincide for both positive and negative quadratic potentials, where the potential displays a double well. Although the perturbative method is unable to describe the region where the quartic potential vanishes, it is remarkable that both the perturbative and semiclassical formalisms recognize the negative oscillator parameter at which the ground state starts to be delocalized in two wells.

The quantum geometric tensor, composed of the quantum metric tensor and Berry curvature, fully encodes the parameter space geometry of a physical system. We first provide a formulation of the quantum geometrical tensor in the path integral formalism that can handle both the ground and excited states, making it useful to characterize excited state quantum phase transitions (ESQPT). In this setting, we also generalize the quantum geometric tensor to incorporate variations of the system parameters and the phase-space coordinates. This gives rise to an alternative approach to the quantum covariance matrix, from which we can get information about the quantum entanglement of Gaussian states through tools such as purity and von Neumann entropy. Second, we demonstrate the equivalence between the formulation of the quantum geometric tensor in the path integral formalism and other existing methods. Furthermore, we explore the geometric properties of the generalized quantum metric tensor in depth by calculating the Ricci tensor and scalar curvature for several quantum systems, providing insight into this geometric information., Comment: In this version we corrected typos and expanded Section 2. Results unchanged

It has recently been proposed classical analogs of the purity, linear quantum entropy, and von Neumann entropy for classical integrable systems, when the corresponding quantum system is in a Gaussian state. We generalized these results by providing classical analogs of the generalized purities, Bastiaans-Tsallis entropies, R\'enyi entropies, and logarithmic negativity for classical integrable systems. These classical analogs are entirely characterized by the classical covariance matrix. We compute these classical analogs exactly in the cases of linearly coupled harmonic oscillators, a generalized harmonic oscillator chain, and a one-dimensional circular lattice of oscillators. In all of these systems, the classical analogs reproduce the results of their quantum counterparts whenever the system is in a Gaussian state. In this context, our results show that quantum information of Gaussian states can be reproduced by classical information., Comment: 17 pages. 9 figures

We introduce a quantum geometric tensor in a curved space with a parameter-dependent metric, which contains the quantum metric tensor as the symmetric part and the Berry curvature corresponding to the antisymmetric part. This parameter-dependent metric modifies the usual inner product, which induces modifications in the quantum metric tensor and Berry curvature by adding terms proportional to the derivatives with respect to the parameters of the determinant of the metric. The quantum metric tensor is obtained in two ways: By using the definition of the infinitesimal distance between two states in the parameter-dependent curved space and via the fidelity susceptibility approach. The usual Berry connection acquires an additional term with which the curved inner product converts the Berry connection into an object that transforms as a connection and density of weight one. Finally, we provide three examples in one dimension with a nontrivial metric: an anharmonic oscillator, a Morse-like potential, and a generalized anharmonic oscillator; and one in two dimensions: the coupled anharmonic oscillator in a curved space., Comment: 22 pages, 6 figures

We obtain a classical analog of the quantum covariance matrix by performing its classical approximation for any continuous quantum state, and we illustrate this approach with the anharmonic oscillator. Using this classical covariance matrix, we propose classical analogs of the purity, linear quantum entropy, and von Neumann entropy for classical integrable systems, when the quantum counterpart of the system under consideration is in a Gaussian state. As is well known, this matrix completely characterizes the purity, linear quantum entropy, and von Neumann entropy for Gaussian states. These classical analogs can be interpreted as quantities that reveal how much information from the complete system remains in the considered subsystem. To illustrate our approach, we calculate these classical analogs for three coupled harmonic oscillators and two linearly coupled oscillators. We find that they exactly reproduce the results of their quantum counterparts. In this sense, it is remarkable that we can calculate these quantities from the classical viewpoint., Comment: 16 pages, 3 figures. To be published in Physical Review A

We study the classical analog of the quantum metric tensor and its scalar curvature for two well-known quantum physics models. First, we analyze the geometry of the parameter space for the Dicke model with the aid of the classical and quantum metrics and find that, in the thermodynamic limit, they have the same divergent behavior near the quantum phase transition, as opposed to their corresponding scalar curvatures which are not divergent there. On the contrary, under resonance conditions, both scalar curvatures exhibit a divergence at the critical point. Second, we present the classical and quantum metrics for the Lipkin-Meshkov-Glick model in the thermodynamic limit and find a perfect agreement between them. We also show that the scalar curvature is only defined on one of the system's phases and that it approaches a negative constant value. Finally, we carry out a numerical analysis for the system's finite sizes, which clearly shows the precursors of the quantum phase transition in the metric and its scalar curvature and allows their characterization as functions of the parameters and of the system's size., Comment: 16 pages, 12 figures

We study the quantum metric tensor and its scalar curvature for a particular version of the Lipkin-Meshkov-Glick model. We build the classical Hamiltonian using Bloch coherent states and find its stationary points. They exhibit the presence of a ground state quantum phase transition, where a bifurcation occurs, showing a change of stability associated with an excited state quantum phase transition. Symmetrically, for a sign change in one Hamiltonian parameter, the same phenomenon is observed in the highest energy state. Employing the Holstein-Primakoff approximation, we derive analytic expressions for the quantum metric tensor and compute the scalar and Berry curvatures. We contrast the analytic results with their finite-size counterparts obtained through exact numerical diagonalization and find an excellent agreement between them for large sizes of the system in a wide region of the parameter space, except in points near the phase transition where the Holstein-Primakoff approximation ceases to be valid., Comment: 14 pages

The geometry of the parameter space is encoded by the quantum geometric tensor, which captures fundamental information about quantum states and contains both the quantum metric tensor and the curvature of the Berry connection. We present a formulation of the Berry connection and the quantum geometric tensor in the framework of the phase space or Wigner function formalism. This formulation is obtained through the direct application of the Weyl correspondence to the geometric structure under consideration. In particular, we show that the quantum metric tensor can be computed using only the Wigner functions, which opens an alternative way to experimentally measure the components of this tensor. We also address the non-Abelian generalization and obtain the phase space formulation of the Wilczek-Zee connection and the non-Abelian quantum geometric tensor. In this case, the non-Abelian quantum metric tensor involves only the non-diagonal Wigner functions. Then, we verify our approach with examples and apply it to a system of $N$ coupled harmonic oscillators, showing that the associated Berry connection vanishes and obtaining the analytic expression for the quantum metric tensor. Our results indicate that the developed approach is well adapted to study the parameter space associated with quantum many-body systems., Comment: 22 pages

The local geometry of the parameter space of a quantum system is described by the quantum metric tensor and the Berry curvature, which are two fundamental objects that play a crucial role in understanding geometrical aspects of condensed matter physics. We consider classical integrable systems and report a new approach to obtain the classical analogs of the quantum metric tensor and the Berry curvature. An advantage of this approach is that it can be applied to a wide variety of classical systems corresponding to quantum systems with bosonic and fermionic degrees of freedom. Our approach arises from the semiclassical approximation of the Berry curvature and the quantum metric tensor in the Lagrangian formalism. We also exploit this semiclassical approximation to establish, for the first time, the relation between the quantum metric tensor and its classical counterpart. We illustrate and validate our approach by applying it to five systems: the generalized harmonic oscillator, the symmetric and linearly coupled harmonic oscillators, the singular Euclidean oscillator, and a spin-half particle in a magnetic field. Finally, we mention some potential applications of this approach and possible generalizations that can be of interest in the field of condensed matter physics, Comment: 16 pages, accepted for publication in Annalen der Physik

We introduce a new method to compute the Quantum Geometric Tensor, this procedure allows us to compute the Quantum Information Metric and the Berry curvature perturbatively for a theory with an arbitrary interaction Hamiltonian. The calculation uses the generating function method, and it is illustrated with the harmonic oscillator with a linear and a quartic perturbation., Comment: 19 pages

We present a classical analog of the quantum metric tensor, which is defined for classical integrable systems that undergo an adiabatic evolution governed by slowly varying parameters. This classical metric measures the distance, on the parameter space, between two infinitesimally different points in phase space, whereas the quantum metric tensor measures the distance between two infinitesimally different quantum states. We discuss the properties of this metric and calculate its components, exactly in the cases of the generalized harmonic oscillator, the generalized harmonic oscillator with a linear term, and perturbatively for the quartic anharmonic oscillator. Finally, we propose alternative expressions for the quantum metric tensor and Berry's connection in terms of quantum operators., Comment: 13 pages

Starting from ${\cal N}=4$ SYM and using an appropriate Higgs mechanism we reconsider the construction of a scalar field theory non-minimally coupled to a Coulomb potential with a relativistic SO(4) symmetry and check for scalar field consistency conditions. This scalar field theory can also be obtained from a relativistic particle Lagrangian with a proper implementation of the non-minimal coupling. We provide the generalization of the non-relativistic construction of the Runge-Lenz vector to the relativistic case and show explicitly that this new vector generates the SO(4) algebra. Using the power of the SO(4) symmetry, we calculate the relativistic hydrogen atom spectrum. We provide a generalization of the Kustaanheimo-Stiefel transformation to the relativistic case and relate our results with the corresponding relativistic oscillator. Finally, in the light of these results, we reconsider the calculation of the hydrogen atom spectrum from the cusp anomalous dimension given in [2]., Comment: 17 pages. Enhaced version matching the published JHEP version. Typos corrected. The argument of concistence at the end of section 2 was corrected

We investigate a quantum geometric space in the context of what could be considered an emerging effective theory from Quantum Gravity. Specifically we consider a two-parameter class of twisted Poincar\'e algebras, from which Lie-algebraic noncommutativities of the translations are derived as well as associative star-products, deformed Riemannian geometries, Lie-algebraic twisted Minkowski spaces and quantum effects that arise as noncommutativities. Starting from a universal differential algebra of forms based on the above mentioned Lie-algebraic noncommutativities of the translations, we construct the noncommutative differential forms and Inner and Outer derivations, which are the noncommutative equivalents of the vector fields in the case of commutative differential geometry. Having established the essentials of this formalism we construct a bimodule, required to be central under the action of the Inner derivations in order to have well defined contractions and from where the algebraic dependence of its coefficients is derived. This again then defines the noncommutative equivalent of the geometrical line-element in commutative differential geometry. We stress, however, that even though the components of the twisted metric are by construction symmetric in their algebra valuation, this is not so for their inverse and thus to construct it we made use of Gel'fand's theory of quasi-determinants, which is conceptually straightforward but computationally becoming quite complicate beyond an algebra of 3 generators. The consequences of the noncommutativity of the Lie-algebra twisted geometry are further discussed., Comment: 43 pages, 1 figure

The aim of this paper is to introduce and analyze a new gauge symmetry that appears in complex holomorphic systems. This symmetry allow us to project the system, using different gauge conditions, to several real systems which are connect by gauge transformations in the complex space. We prove that the space of solutions of one system is related to the other by the gauge transformation. The gauge transformations are in some cases canonical transformations. However, in other cases are more general transformations that change the symplectic structure, but there is still a map between the systems. In this way our construction extend the group of canonical transformations in classical mechanics. Also, we show how to extend the analysis to the quantum case using path integrals by means of the Batalin-Fradkin-Vilkovisky theorem and within the canonical formalism, where we show explicitly that solutions of the Schr\"odinger equation are gauge related., Comment: 18 pages

We take as a starting point an expression for the quantum geometric tensor recently derived in the context of the gauge/gravity duality. We proceed to generalize this formalism in such way it is possible to compute the geometrical phases of quantum systems. Our scheme provides a conceptually complete description and introduce a different point of view of earlier works. Using our formalism, we show how this expression can be applied to well-known quantum mechanical systems., Comment: 33 pages, references added and minor comments

The Dirac's formalism for constrained systems is applied to the analysis of time-dependent Hamiltonians in the extended phase space. We show that the Lewis invariant is a reparametrization invariant and we calculate the Feynman propagator using the extended phase description. We show that the quantum phase of the Feynman propagator is given by the boundary term of the canonical transformation of the extended phase space., Comment: 16 pages

The Polymer Quantization of the Fourier modes of the real scalar field is studied within algebraic scheme. We replace the positive linear functional of the standard Poincar\'e invariant quantization by a singular one. This singular positive linear functional is constructed as mimicking the singular limit of the complex structure of the Poincar\'e invariant Fock quantization. The resulting symmetry group of such Polymer Quantization is the subgroup $\mbox{SDiff}(\mathbb{R}^4)$ which is a subgroup of $\mbox{Diff}(\mathbb{R}^4)$ formed by spatial volume preserving diffeomorphisms. In consequence, this yields an entirely different irreducible representation of the Canonical Commutation Relations, non-unitary equivalent to the standard Fock representation. We also compared the Poincar\'e invariant Fock vacuum with the Polymer Fourier vacuum., Comment: 29 pages

It is computed, using instanton methods, the first allowed energy band for the polymer harmonic oscillator. The result is consistent with the band structure of the standard quantum pendulum but with pure point spectrum. An effective infinite degeneracy emerges in the formal limit $\mu/l_0 \to 0$ where $l_0$ is the characteristic length of the vacuum eigenfunction of a quantum harmonic oscillator. As an additional result, it is shown along the article the role played by the lattice reference point $\lambda$ in the full quantization of the polymer harmonic oscillator., Comment: 32 pages, 8 figures

We study the effect of a hidden gauge symmetry on complex holomorphic systems. For this purpose, we show that intrinsically any holomorphic system has this gauge symmetry. We establish that this symmetry is related to the Cauchy-Riemann equations, in the sense that the associated constraint is a first class constraint only in the case that the potential be holomorphic. As a consequence of this gauge symmetry on the complex space, we can fix the gauge condition in several ways and project from the complex phase-space to real phase space. Different projections are gauge related on the complex phase-space but are not directly related on the real physical phase-space., Comment: 20 pages, to be published in Phys. Lett. A

In this work, we propose three different modified relativistic particles. In the first case, we propose a particle with metrics depending on the momenta and we show that the quantum version of these systems includes different field theories, as Lifshitz field theories. As a second case we propose a particle that implies a modified symplectic structure and we show that the quantum version of this system gives different noncommutative space-times, for example the Snyder space-time. In the third case, we combine both structures before mentioned, namely noncommutative space-times and momentum dependent metrics. In this last case, we show that anisotropic field theories can be seen as a limit of noncommutative field theory., Comment: 26 pages. Accepted for publication in Mod.Phys.Lett. A

This work addresses the quantization of a self-interacting higher order time derivative theory using path integrals. To quantize this system and avoid the problems of energy not bounded from below and states of negative norm, we observe the following steps: 1) We extend the theory to the complex plane and in this sense we double the degrees of freedom. 2) We add a total derivative to fix the convenient boundary conditions. 3) We check that the complex structure is consistent. 4) To map from the complex space to the real space we introduce reality conditions as second class constraints and we check that the interactions do not generate more constraints. 5) We built the measure of the complex path integral and we show that including currents the theory is projected to a self-interacting real theory that is renormalizable., Comment: 26 pages

A twisted ${\mathcal C}^\star $- algebra of the extended (noncommutative) Heisenberg-Weyl group has been constructed which takes into account the Uncertainty Principle for coordinates in the Planck length regime. This general construction is then used to generate an appropriate Hilbert space and observables for the noncommutative theory which, when applied to the Bianchi I Cosmology, leads to a new set of equations that describe the quantum evolution of the universe. We find that this formulation matches theories based on a reticular Heisenberg-Weyl algebra in the bouncing and expanding regions of a collapsing Bianchi universe. There is, however, an additional effect introduced by the dynamics generated by the noncommutativity. This is an oscillation in the spectrum of the volume operator of the universe, within the bouncing region of the commutative theories. We show that this effect is generic and produced by the noncommutative momentum exchange between the degrees of freedom in the cosmology. We give asymptotic and numerical solutions which show the above mentioned effects of the noncommutativity., Comment: 31 pages, 10 figures, to appear in PRD

The quantization of higher order time derivative theories including interactions is unclear. In this paper in order to solve this problem, we propose to consider a complex version of the higher order derivative theory and map this theory to a real first order theory. To achieve this relationship, the higher order derivative formulation must be complex since there is not a real canonical transformation from this theory to a real first order theory with stable interactions. In this manner, we work with a non-Hermitian higher order time derivative theory. To quantize this complex theory, we introduce reality conditions that allow us to map the complex higher order theory to a real one, and we show that the resulting theory is regularizable and renormalizable for a class of interactions., Comment: 21 pages, 10 figures

In this paper we implement scale anisotropic transformations in the space-time in classical mechanics. The resulting system is consistent with the dispersion relation of gravity at a Lifshitz point recently considered in arXiv:0901.3775. Also, we show that our model is a generalization of the conformal mechanics of Alfaro, Fubini and Furlan. For arbitrary $z$ we construct the dynamical symmetries that correspond to the Schroedinger algebra. Furthermore, we obtain the Boltzman distribution for a gas of free particles compatible with anisotropic scaling transformations and compare our result with the corresponding thermodynamics of the recent anisotropic black branes proposed in arXiv:0907.4755., Comment: 6 pages

In previous work [Rosenbaum M. et al., J. Phys. A: Math. Theor. 40 (2007), 10367-10382, hep-th/0611160] we have shown how for canonical parametrized field theories, where space-time is placed on the same footing as the other fields in the theory, the representation of space-time diffeomorphisms provides a very convenient scheme for analyzing the induced twisted deformation of these diffeomorphisms, as a result of the space-time noncommutativity. However, for gauge field theories (and of course also for canonical geometrodynamics) where the Poisson brackets of the constraints explicitely depend on the embedding variables, this Poisson algebra cannot be connected directly with a representation of the complete Lie algebra of space-time diffeomorphisms, because not all the field variables turn out to have a dynamical character [Isham C.J., Kuchar K.V., Ann. Physics 164 (1985), 288-315, 316-333]. Nonetheless, such an homomorphic mapping can be recuperated by first modifying the original action and then adding additional constraints in the formalism in order to retrieve the original theory, as shown by Kuchar and Stone for the case of the parametrized Maxwell field in [Kuchar K.V., Stone S.L., Classical Quantum Gravity 4 (1987), 319-328]. Making use of a combination of all of these ideas, we are therefore able to apply our canonical reparametrization approach in order to derive the deformed Lie algebra of the noncommutative space-time diffeomorphisms as well as to consider how gauge transformations act on the twisted algebras of gauge and particle fields. Thus, hopefully, adding clarification on some outstanding issues in the literature concerning the symmetries for gauge theories in noncommutative space-times., Comment: This is a contribution to the Special Issue on Deformation Quantization, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

A {\em complex} canonical transformation is found that takes the fourth order derivative Pais-Uhlenbeck oscillator into two independent harmonic oscillators thus showing that this model has energy bounded from below, unitary time-evolution and no negative norm states, or ghosts. Such transformation yields a positive definite inner product consistent with reality conditions in the Hilbert space. The method is illustrated by eliminating the negative norm states in a complex oscillator. Extensions to other higher order mechanical models and field theory are discussed., Comment: 5 pages

We investigate the incorporation of space noncommutativity into field theory by extending to the spectral continuum the minisuperspace action of the quantum mechanical harmonic oscillator propagator with an enlarged Heisenberg algebra. In addition to the usual $\star$-product deformation of the algebra of field functions, we show that the parameter of noncommutativity can occur in noncommutative field theory even in the case of free fields without self-interacting potentials., Comment: 13 pages

Within the spirit of Dirac's canonical quantization, noncommutative spacetime field theories are introduced by making use of the reparametrization invariance of the action and of an arbitrary non-canonical symplectic structure. This construction implies that the constraints need to be deformed, resulting in an automatic Drinfeld twisting of the generators of the symmetries associated with the reparametrized theory. We illustrate our procedure for the case of a scalar field in 1+1- spacetime dimensions, but it can be readily generalized to arbitrary dimensions and arbitrary types of fields., Comment: 16 pages

Using arbitrary symplectic structures and parametrization invariant actions, we develop a formalism, based on Dirac's quantization procedure, that allows us to consider theories with both space-space as well as space-time noncommutativity. Because the formalism has as a starting point an action, the procedure admits quantizing the theory either by obtaining the quantum evolution equations or by using the path integral techniques. For both approaches we only need to select a complete basis of commutative observables. We show that for certain choices of the potentials that generate a given symplectic structure, the phase of the quantum transition function between the admissible bases corresponds to a linear canonical transformation, by means of which the actions associated to each of these bases may be related and hence lead to equivalent quantizations. There are however other potentials that result in actions which can not be related to the previous ones by canonical transformations, and for which the fixed end-points, in terms of the admissible bases, can only be realized by means of a Darboux map. In such cases the original arbitrary symplectic structure is reduced to its canonical form and therefore each of these actions results in a different quantum theory. One interesting feature of the formalism here discussed is that it can be introduced both at the levels of particle systems as well as of field theory., Comment: 19 pages, submitted to Contemporary Mathematics

We show that introducing an extended Heisenberg algebra in the context of the Weyl-Wigner-Groenewold-Moyal formalism leads to a deformed product of the classical dynamical variables that is inherited to the level of quantum field theory, and that allows us to relate the operator space noncommutativity in quantum mechanics to the quantum group inspired algebra deformation noncommutativity in field theory., Comment: 17 pages, to be published in Phys. Lett. A

Using the parametrized relativistic particle we obtain the noncommutative Snyder space-time. In addition, we study the consistency conditions between the boundary conditions and the canonical gauges that give origin to noncommutative theories. Using these results we construct a first order action, in the reduced phase-space, for the Snyder particle with momenta fixed on the boundary., Comment: 10 pages

We consider the quantum mechanical equivalence of the Seiberg-Witten map in the context of the Weyl-Wigner-Groenewold-Moyal phase-space formalism in order to construct a quantum mechanics over noncommutative Heisenberg algebras. The formalism is then applied to the exactly soluble Landau and harmonic oscillator problems in the 2-dimensional noncommutative phase-space plane, in order to derive their correct energy spectra and corresponding Wigner distributions. We compare our results with others that have previously appeared in the literature., Comment: 19 pages

In this note we show that in a two-dimensional non-commutative space the area operator is quantized, this outcome is compared with the result obtained by Loop Quantum Gravity methods., Comment: 6 pages, references added, minor corrections

We investigate the Kepler problem using a symplectic structure consistent with the commutation rules of the noncommutative quantum mechanics. We show that a noncommutative parameter of the order of $10^{-58} \text m^2$ gives observable corrections to the movement of the solar system. In this way, modifications in the physics of smaller scales implies modifications at large scales, something similar to the UV/IR mixing., Comment: 10 pages

The analysis of the combinatorics resulting from the perturbative expansion of the transition amplitude in quantum field theories, and the relation of this expansion to the Hausdorff series leads naturally to consider an infinite dimensional Lie subalgebra and the corresponding enveloping Hopf algebra, to which the elements of this series are associated. We show that in the context of these structures the power sum symmetric functionals of the perturbative expansion are Hopf primitives and that they are given by linear combinations of Hall polynomials, or diagrammatically by Hall trees. We show that each Hall tree corresponds to sums of Feynman diagrams each with the same number of vertices, external legs and loops. In addition, since the Lie subalgebra admits a derivation endomorphism, we also show that with respect to it these primitives are cyclic vectors generated by the free propagator, and thus provide a recursion relation by means of which the (n+1)-vertex connected Green functions can be derived systematically from the n-vertex ones., Comment: 21 pages, accepted for publication in J.Geom.and Phys

We examine the extent to which the action for the membrane of M-theory (the eleven-dimensional construct which underlies and unifies all of the known string theories) simplifies in the so-called Open Membrane (OM) limit, a limit which lies at the root of the various manifestations of noncommutativity in the string context. In order for the discussion to be relatively self-contained, we start out by reviewing why the strings of ten-dimensional string theory are in fact membranes (M2-branes) living in eleven dimensions. After that, we recall the definition of OM theory, as well as the arguments showing that it is part of a larger, eleven-dimensional structure known as Galilean or Wrapped M2-brane (WM2) theory. WM2 theory is a rich theoretical construct which is interesting for several reasons, in particular because it is essentially a toy model of M-theory. We then proceed to deduce a membrane action for OM/WM2 theory, and spell out its implications for the four different types of M2-branes one can consider in this setting. For two of these types, the action in question can be simplified by gauge-fixing to a form which implies a discrete membrane spectrum. The boundary conditions for the remaining two cases do not allow this same gauge choice, and so their dynamics remain to be unraveled., Comment: LaTeX 2e, 8 pages; aimed at phenomenologists. Invited talk given by A. Guijosa at the X Mexican School of Particles and Fields, Playa del Carmen, Mexico, November 2002

A new method to compute the symplectic structure of a quantum field theory with non trivial boundary conditions is proposed. Following the suggestion in \cite{ho:gnus, ardalan}, we regard that the boundary conditions are second class constraints in the sense of the Dirac's method. However, we show that this proposal is more useful if we consider an inverse of the Holographic map between a theory defined in the boundary to another with constraints but without boundary., Comment: 18 pages, references added

In this work we show that corrections to the Newton's second law appears if we assume that the phase space has a symplectic structure consistent with the rules of commutation of noncommutative quantum mechanis. In the central field case we find that the correction term breaks the rotational symmetry. In particular, for the Kepler problem, this term takes the form of a Coriolis force produced by the weak gravitational field far from a rotating massive object., Comment: 7 pages, References added

Through direct examination of the effect of the OM limit on the M2-brane worldvolume action, we derive a membrane action for OM theory, and more generally, for the eleven-dimensional M-theoretic construct known as Galilean or Wrapped M2-brane (WM2) theory, which contains OM theory as a special class of states. In the static gauge, the action in question implies a discrete spectrum for the closed membrane of WM2 theory, which under double dimensional reduction is shown to reproduce the known NCOS/Wound closed string spectrum. We examine as well open membranes ending on each of the three types of M5-branes in WM2 theory (OM theory arising from the 'longitudinal' type), and show that the 'fully transverse' fivebrane is tensionless. As a prelude to the membrane, we also study the case of the string, where we likewise obtain a reparametrization-invariant action, and make contact with previous work., Comment: LaTeX 2e, 28 pages; v2: reference added

The exact solution of the Schwinger model with compact gauge group U(1) is presented. The compactification is imposed by demanding that the only surviving true electromagnetic degree of freedom has angular character. Not surprinsingly, this topological condition defines a version of the Schwinger model which is different from the standard one, where $c$ takes values on the line. The main consequences are: the spectra of the zero modes is not degenerated and does not correspond to the equally spaced harmonic oscillator, both the electric charge and a modified gauge invariant chiral charge are conserved (nevertheless, the axial-current anomaly is still present) and, finally, there is no need to introduce a $\theta$-vacuum. A comparison with the results of the standard Schwinger model is pointed out along the text., Comment: 14 pages, 2 figures. Accepted for publication in Mod. Phys. Letts. A

Conformal invariant new forms of $p$-brane and D$p$-brane actions are proposed. The field content of these actions are: an induced metric, gauge fields, an auxiliary metric and an auxiliary scalar field that implements the Weyl invariance. This scalar field transforms with a conformal weight that depends on the brane dimension. The proposed actions are Weyl invariant in any dimension and the elimination of the auxiliary metric and the scalar field reproduces the Nambu-Goto action for $p$-branes and the Born-Infeld action for D$p$-branes. As a consequence of the fact that in the $p$-brane case, the action is quadratic in $\partial_0 X$, we develop a complete construction for the associated canonical formalism, solving the problem in previous formulations of conformal $p$-brane actions where the Hamiltonian can not be constructed. We obtain the algebra of constraints and identify the generator of the Weyl symmetry. In the construction of the corresponding supersymmetric generalization of this conformal $p$-brane action we show that the associated $\kappa$-symmetry is consistent with the conformal invariance. In the D$p$-brane case, the actions are quadratic in the gauge fields. These actions can be used to construct new conformal couplings in any dimension $p$ to the auxiliary scalar field now promoted as a dynamical variable., Comment: 12 pages, one additional section

The complete exact solution of the Schwinger model with compact gauge group U(1), in the Hamiltonian approach, is presented . The compactification is imposed by demanding that the only surviving true electromagnetic degree of freedom has angular character. Not surprinsingly, this topological condition defines a version of the Schwinger model which is different from the standard one, where $c$ takes values on the line . The main consequences are: the spectra of the zero modes is not degenerated and does not correspond to the equally spaced harmonic oscillator, both the electric charge and a modified gauge invariant chiral charge are conserved (nevertheless, the axial-current anomaly is still present) and, finally, there is no need to introduce a $\theta$-vacuum. A comparison with the results of the standard Schwinger model is pointed out along the text., Comment: 34 pages, 3 figures

Starting from an associated reparametrization-invariant action, the generalization of the BRST-BFV method for the case of nonstationary systems is constructed. The extension of the Batalin-Tyutin conversional approach is also considered in the nonstationary case. In order to illustrate these ideas, the propagator for the time-dependent two-dimensional rotor is calculated by reformulating the problem as a system with only first class constraints and subsequently using the BRST-BFV prescription previously obtained., Comment: Latex, RevTeX, 13 pages

We show that the reality conditions to be imposed on Ashtekar variables to recover real gravity can be implemented as second class constraints a la Dirac. Thus, counting gravitational degrees of freedom follows accordingly. Some constraints of the real theory turn out to be non-polynomial, regardless of the form, polynomial or non-polynomial, taken for the reality conditions. We comment upon the compatibility of our approach with the recently proposed Wick transform point of view, as well as on some alternatives for dealing with such second class constraints., Comment: 16 pages, plain LaTeX, submitted to Class. Quant. Grav. E-mail: hugo@xanum.uam.mx

We introduce an operator version of the BRST-BFV effective action for arbitrary systems with first-class constraints. Using the Schwinger action principle we calculate the propagators corresponding to: (i) the parametrized non-relativistic free particle, (ii) the relativistic free particle and (iii) the spining relativistic free particle. Our calculation correctly imposes the BRST-invariance at the end-points. The precise use of the additional boundary terms required in the description of fermionic variables is also incorporated., Comment: 28 pages, LaTeX