We investigate the Casimir effect of a rough membrane within the framework of the Hořava-Lifshitz... more We investigate the Casimir effect of a rough membrane within the framework of the Hořava-Lifshitz theory in 2+1 dimensions. Quantum fluctuations are induced by an anisotropic scalar field subject to Dirichlet boundary conditions. We implement a coordinate transformation to render the membrane completely flat, treating the remaining terms associated with roughness as a potential. The spectrum is obtained through perturbation theory and regularized using the ζ-function method. We present an explicit example of a membrane with periodic border. Additionally, we consider the effect of temperature. Our findings reveal that the Casimir energy and force depend on roughness, the anisotropic scaling factor and temperature.
We perform the Batalin-Fradkin-Vilkovisky quantization of the anisotropic conformal Hořava theory... more We perform the Batalin-Fradkin-Vilkovisky quantization of the anisotropic conformal Hořava theory in d spatial dimensions. We introduce a model with a conformal potential suitable for any dimension. We define an anisotropic and local gauge-fixing condition that accounts for the spatial diffeomorphisms and the anisotropic Weyl transformations. We show that the BRST transformations can be expressed mainly in terms of a spatial diffeomorphism along a ghost field plus a conformal transformation with another ghost field as argument. We study the quantum Lagrangian in the d ¼ 2 case, obtaining that all propagators are regular, except for the fields associated with the measure of the second-class constraints. This behavior is qualitatively equal to the nonconformal case.
We study the Casimir effect of a membrane embedded in 2 + 1 dimensions flat cone generated by a m... more We study the Casimir effect of a membrane embedded in 2 + 1 dimensions flat cone generated by a massive particle located at the origin of the coordinate system. The flat cone is an exact solution of the nonprojectable Hořava theory, similar to general relativity. We consider a scalar field satisfying Dirichlet boundary conditions, and regularize the spectrum using the ζ-function technique. In addition, we include the effects of temperature in our analysis. Our results show that the Casimir force depends on three factors: the anisotropic scaling z, the mass of the point particle, and the temperature.
We present the interesting case of the 2 + 1 nonprojectable Hořava theory formulated at the criti... more We present the interesting case of the 2 + 1 nonprojectable Hořava theory formulated at the critical point, where it does not posses local degrees of freedom. The critical point is defined by the value of a coupling constant of the theory. We discuss how, in spite of the absence of degrees of freedom, the theory admits solutions with nonflat or nonconstant curvature. We consider the theory without cosmological constant and without terms of higher order derivatives, hence this is an effect that can be seen at the same order of 2 + 1 general relativity. We present an exact nonflat solution that is not asymptotically flat. The presence of solutions with nontrivial curvature seems to be related to the relaxing of the asymptotically flat condition. We discuss that there is no analogue of Newtonian potential in this theory, and a broad class of asymptotically flat geometries leads to the restriction that the only solutions that can be found among them are the flat ones.
11TH INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS 2013: ICNAAM 2013, 2013
This work deals with thermo-statistical approaches for describing phenomenon of gravitational clu... more This work deals with thermo-statistical approaches for describing phenomenon of gravitational clustering in cosmology. We have introduced a family of N-body Hamiltonian toy models that describe self-gravitating gas embedded on a n-sphere Sn within a n+1-dimensional real space Rn+1. The closed character of these Riemannian manifolds avoids the incidence of evaporation and other bordering effects, while consideration of microcanonical description enables the access to states with negative heat capacities associated with gravitational clustering at low energies. Preliminary description of the particular case of two-dimensional sphere S2 evidences the existence of a microcanonical discontinuous phase transition from a homogeneous phase with positive heat capacities at high energies towards a clustered phase with negative heat capacities at low energies.
We show that the solution corresponding to the gravitational field of a point particle at rest in... more We show that the solution corresponding to the gravitational field of a point particle at rest in 2 + 1 nonprojectable Hořava is exactly the same as 2 + 1 General Relativity with the same source. In General Relativity this solution is well known, it is a flat cone whose deficit angle is proportional to the mass of the particle. To establish the system we couple the Hořava theory to a point particle with relativistic action. Motivated by this solution, we postulate the condition of asymptotic flatness exactly in the same way of 2 + 1 General Relativity. A remarkable feature of this condition is that the dominant mode is not fixed, but affected by the mass of the configuration. In this scheme, another important coincidence with 2 + 1 General Relativity under asymptotic flatness is that the energy is the same (except for some coupling constants involved), the z = 1 term with the derivative of the lapse function does not contribute.
We show that the Batalin-Fradkin-Vilkovisky (BFV) quantization scheme can be implemented in the n... more We show that the Batalin-Fradkin-Vilkovisky (BFV) quantization scheme can be implemented in the nonprojectable (2 þ 1)-Hořava theory. This opens the possibility of imposing more general gauge conditions in the quantization of this theory. The BFV quantization is based on the canonical formalism, which is suitable to incorporate the measure associated to the second-class constraints that the theory has. Special features of the Hamiltonian density and the matrix of second-class constraints allow that the system be involutive in terms of Dirac brackets, which is a nontrivial requisite for implementing the BFV formalism. We present the Becchi-Rouet-Stora-Tyutin symmetry transformations in the canonical variables. The theory is of rank one in the classification introduced by Fradkin and Fradkina. The originally called relativistic gauge-fixing conditions of the BFV formalism can be implemented in the nonprojectable Hořava theory, extended to nonrelativistic forms. We show that the nonlocal gauge condition introduced in the projectable theory can be included among these gauges.
We contrast the dynamics of the Hořava theory with anisotropic Weyl symmetry with the case when t... more We contrast the dynamics of the Hořava theory with anisotropic Weyl symmetry with the case when this symmetry is explicitly broken, which is called the kinetic-conformal Hořava theory. To do so we perform the canonical formulation of the anisotropic conformal theory at the classical level with a general conformal potential. Both theories have the generator of the anisotropic Weyl transformations as a constraint but it changes from first to second-class when the Weyl symmetry is broken. The FDiff-covariant vector a i = ∂ i ln N plays the role of gauge connection for the anisotropic Weyl transformations. A Lagrange multiplier plays also the role of gauge connection, being the time component. The anisotropic conformal theory propagates the same number of degrees of freedom of the kinetic-conformal theory, which in turn are the same of General Relativity. This is due to exchange of a second-class constraint in the kinetic-conformal case by a gauge symmetry in the anisotropic conformal case. An exception occurs if the conformal potential does not depend on the lapse function N , as is the case of the so called (Cotton) 2 potential, in which case one of the physical modes becomes odd. We develop in detail two explicit anisotropic conformal models. One of them depends on N whereas the other one is the (Cotton) 2 model. We also study conformally flat solutions in the anisotropic conformal and the kineticconformal theories, defining as conformally flat the spatial metric, but leaving for N a form different to the one dictated by the anisotropic Weyl transformations. We find that in both theories these configurations have vanishing canonical momentum and they are critical points of the potential. In the kinetic-conformal theory we find explicitly an exact, nontrivial, conformally flat solution.
The Hořava theory in 2 + 1 dimensions can be formulated at a critical point in the space of coupl... more The Hořava theory in 2 + 1 dimensions can be formulated at a critical point in the space of coupling constants where it has no local degrees of freedom. This suggests that this critical case could share many features with 2 + 1 general relativity, in particular its large-distance effective action that is of second order in derivatives. To deepen on this relationship, we study the asymptotically flat solutions of the effective action. We take the general definition of asymptotic flatness from 2 + 1 general relativity, where an asymptotically flat region with a nonfixed conical angle is approached. We show that a class of regular asymptotically flat solutions are totally flat. The class is characterized by having nonnegative energy (when the coupling constant of the Ricci scalar is positive). We present a detailed canonical analysis on the effective action showing that the dynamics of the theory forbids local degrees of freedom. Another similarity with 2 + 1 general relativity is the absence of a Newtonian force. In contrast to these results, we find evidence against the similarity with 2 + 1 general relativity: we find an exact nonflat solution of the same effective theory. This solution is out of the set of asymptotically flat solutions.
We perform an analysis of the ultraviolet divergences of the quantum nonprojectable Hořava gravit... more We perform an analysis of the ultraviolet divergences of the quantum nonprojectable Hořava gravity. We work the quantum-field theory directly in the Hamiltonian formalism provided by the Batalin-Fradkin-Vilkovisky quantization. In this way the second-class constraints can be incorporated into the quantization. A known local gauge-fixing condition leads to a local canonical Lagrangian. Although the canonical fields acquire regular propagators, irregular propagators persist dangerous subdivergences. We show that all these loops cancel exactly between them due to a perfect matching between the propagators and vertices of the fields and ghosts forming the loops. The rest of the divergences behave similarly to the projectable theory, they can be removed by local counterterms. This result points to the renormalization of the nonprojectable Hořava theory.
We perform the Batalin-Fradkin-Vilkovisky quantization of the anisotropic conformal Hořava theory... more We perform the Batalin-Fradkin-Vilkovisky quantization of the anisotropic conformal Hořava theory in d spatial dimensions. We introduce a model with a conformal potential suitable for any dimension. We define an anisotropic and local gauge-fixing condition that accounts for the spatial diffeomorphisms and the anisotropic Weyl transformations. We show that the BRST transformations can be expressed mainly in terms of a spatial diffeomorphism along a ghost field plus a conformal transformation with another ghost field as argument. We study the quantum Lagrangian in the d = 2 case, obtaining that all propagators are regular, except for the fields associated with the measure of the second-class constraints. This behavior is qualitatively equal to the nonconformal case.
We present an analysis on the BRST symmetry transformations of the Hořava theory under the BFV qu... more We present an analysis on the BRST symmetry transformations of the Hořava theory under the BFV quantization, both in the nonprojectable and projectable cases. We obtain that the BRST transformations are intimately related to a particular spatial diffeomorphism along one of the ghost vector fields. We show explicitly the invariance of the quantum action and the nilpotence of the BRST transformations, using this diffeomorphism largely. The BRST symmetry is verified in the whole phase space, inside and outside the constrained surface. When restricted to the constrained surface, the BRST transformations are completely local. The consistency of the BRST symmetry is a fundamental feature of a quantum field theory, specially for renormalization. The BFV quantization is independent of the chosen gauge-fixing condition. This allows us to study the unitarity of the quantum Hořava theory, covering the gauge required for renormalization. We prove that, if the solution for the lapse function exists, then the unitarity of the theory is established.
We present an analysis on the Becchi-Rouet-Stora-Tyutin (BRST) symmetry transformations of the Ho... more We present an analysis on the Becchi-Rouet-Stora-Tyutin (BRST) symmetry transformations of the Hořava theory under the Batalin-Fradkin-Vilkovisky quantization, both in the nonprojectable and projectable cases. We obtain that the BRST transformations are intimately related to a particular spatial diffeomorphism along one of the ghost vector fields. We show explicitly the invariance of the quantum action and the nilpotence of the BRST transformations, using this diffeomorphism largely. The BRST symmetry is verified in the whole phase space, inside and outside the constrained surface. When restricted to the constrained surface, the BRST transformations are completely local. The consistency of the BRST symmetry is a fundamental feature of a quantum field theory, specially for renormalization. The BFV quantization is independent of the chosen gauge-fixing condition. This allows us to study the unitarity of the quantum Hořava theory, covering the gauge required for renormalization. We prove that, if the solution for the lapse function exists, then the unitarity of the theory is established.
We investigate the Casimir effect of a rough membrane within the framework of the Hořava-Lifshitz... more We investigate the Casimir effect of a rough membrane within the framework of the Hořava-Lifshitz theory in 2+1 dimensions. Quantum fluctuations are induced by an anisotropic scalar field subject to Dirichlet boundary conditions. We implement a coordinate transformation to render the membrane completely flat, treating the remaining terms associated with roughness as a potential. The spectrum is obtained through perturbation theory and regularized using the ζ-function method. We present an explicit example of a membrane with periodic border. Additionally, we consider the effect of temperature. Our findings reveal that the Casimir energy and force depend on roughness, the anisotropic scaling factor and temperature.
We perform the Batalin-Fradkin-Vilkovisky quantization of the anisotropic conformal Hořava theory... more We perform the Batalin-Fradkin-Vilkovisky quantization of the anisotropic conformal Hořava theory in d spatial dimensions. We introduce a model with a conformal potential suitable for any dimension. We define an anisotropic and local gauge-fixing condition that accounts for the spatial diffeomorphisms and the anisotropic Weyl transformations. We show that the BRST transformations can be expressed mainly in terms of a spatial diffeomorphism along a ghost field plus a conformal transformation with another ghost field as argument. We study the quantum Lagrangian in the d ¼ 2 case, obtaining that all propagators are regular, except for the fields associated with the measure of the second-class constraints. This behavior is qualitatively equal to the nonconformal case.
We study the Casimir effect of a membrane embedded in 2 + 1 dimensions flat cone generated by a m... more We study the Casimir effect of a membrane embedded in 2 + 1 dimensions flat cone generated by a massive particle located at the origin of the coordinate system. The flat cone is an exact solution of the nonprojectable Hořava theory, similar to general relativity. We consider a scalar field satisfying Dirichlet boundary conditions, and regularize the spectrum using the ζ-function technique. In addition, we include the effects of temperature in our analysis. Our results show that the Casimir force depends on three factors: the anisotropic scaling z, the mass of the point particle, and the temperature.
We present the interesting case of the 2 + 1 nonprojectable Hořava theory formulated at the criti... more We present the interesting case of the 2 + 1 nonprojectable Hořava theory formulated at the critical point, where it does not posses local degrees of freedom. The critical point is defined by the value of a coupling constant of the theory. We discuss how, in spite of the absence of degrees of freedom, the theory admits solutions with nonflat or nonconstant curvature. We consider the theory without cosmological constant and without terms of higher order derivatives, hence this is an effect that can be seen at the same order of 2 + 1 general relativity. We present an exact nonflat solution that is not asymptotically flat. The presence of solutions with nontrivial curvature seems to be related to the relaxing of the asymptotically flat condition. We discuss that there is no analogue of Newtonian potential in this theory, and a broad class of asymptotically flat geometries leads to the restriction that the only solutions that can be found among them are the flat ones.
11TH INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS 2013: ICNAAM 2013, 2013
This work deals with thermo-statistical approaches for describing phenomenon of gravitational clu... more This work deals with thermo-statistical approaches for describing phenomenon of gravitational clustering in cosmology. We have introduced a family of N-body Hamiltonian toy models that describe self-gravitating gas embedded on a n-sphere Sn within a n+1-dimensional real space Rn+1. The closed character of these Riemannian manifolds avoids the incidence of evaporation and other bordering effects, while consideration of microcanonical description enables the access to states with negative heat capacities associated with gravitational clustering at low energies. Preliminary description of the particular case of two-dimensional sphere S2 evidences the existence of a microcanonical discontinuous phase transition from a homogeneous phase with positive heat capacities at high energies towards a clustered phase with negative heat capacities at low energies.
We show that the solution corresponding to the gravitational field of a point particle at rest in... more We show that the solution corresponding to the gravitational field of a point particle at rest in 2 + 1 nonprojectable Hořava is exactly the same as 2 + 1 General Relativity with the same source. In General Relativity this solution is well known, it is a flat cone whose deficit angle is proportional to the mass of the particle. To establish the system we couple the Hořava theory to a point particle with relativistic action. Motivated by this solution, we postulate the condition of asymptotic flatness exactly in the same way of 2 + 1 General Relativity. A remarkable feature of this condition is that the dominant mode is not fixed, but affected by the mass of the configuration. In this scheme, another important coincidence with 2 + 1 General Relativity under asymptotic flatness is that the energy is the same (except for some coupling constants involved), the z = 1 term with the derivative of the lapse function does not contribute.
We show that the Batalin-Fradkin-Vilkovisky (BFV) quantization scheme can be implemented in the n... more We show that the Batalin-Fradkin-Vilkovisky (BFV) quantization scheme can be implemented in the nonprojectable (2 þ 1)-Hořava theory. This opens the possibility of imposing more general gauge conditions in the quantization of this theory. The BFV quantization is based on the canonical formalism, which is suitable to incorporate the measure associated to the second-class constraints that the theory has. Special features of the Hamiltonian density and the matrix of second-class constraints allow that the system be involutive in terms of Dirac brackets, which is a nontrivial requisite for implementing the BFV formalism. We present the Becchi-Rouet-Stora-Tyutin symmetry transformations in the canonical variables. The theory is of rank one in the classification introduced by Fradkin and Fradkina. The originally called relativistic gauge-fixing conditions of the BFV formalism can be implemented in the nonprojectable Hořava theory, extended to nonrelativistic forms. We show that the nonlocal gauge condition introduced in the projectable theory can be included among these gauges.
We contrast the dynamics of the Hořava theory with anisotropic Weyl symmetry with the case when t... more We contrast the dynamics of the Hořava theory with anisotropic Weyl symmetry with the case when this symmetry is explicitly broken, which is called the kinetic-conformal Hořava theory. To do so we perform the canonical formulation of the anisotropic conformal theory at the classical level with a general conformal potential. Both theories have the generator of the anisotropic Weyl transformations as a constraint but it changes from first to second-class when the Weyl symmetry is broken. The FDiff-covariant vector a i = ∂ i ln N plays the role of gauge connection for the anisotropic Weyl transformations. A Lagrange multiplier plays also the role of gauge connection, being the time component. The anisotropic conformal theory propagates the same number of degrees of freedom of the kinetic-conformal theory, which in turn are the same of General Relativity. This is due to exchange of a second-class constraint in the kinetic-conformal case by a gauge symmetry in the anisotropic conformal case. An exception occurs if the conformal potential does not depend on the lapse function N , as is the case of the so called (Cotton) 2 potential, in which case one of the physical modes becomes odd. We develop in detail two explicit anisotropic conformal models. One of them depends on N whereas the other one is the (Cotton) 2 model. We also study conformally flat solutions in the anisotropic conformal and the kineticconformal theories, defining as conformally flat the spatial metric, but leaving for N a form different to the one dictated by the anisotropic Weyl transformations. We find that in both theories these configurations have vanishing canonical momentum and they are critical points of the potential. In the kinetic-conformal theory we find explicitly an exact, nontrivial, conformally flat solution.
The Hořava theory in 2 + 1 dimensions can be formulated at a critical point in the space of coupl... more The Hořava theory in 2 + 1 dimensions can be formulated at a critical point in the space of coupling constants where it has no local degrees of freedom. This suggests that this critical case could share many features with 2 + 1 general relativity, in particular its large-distance effective action that is of second order in derivatives. To deepen on this relationship, we study the asymptotically flat solutions of the effective action. We take the general definition of asymptotic flatness from 2 + 1 general relativity, where an asymptotically flat region with a nonfixed conical angle is approached. We show that a class of regular asymptotically flat solutions are totally flat. The class is characterized by having nonnegative energy (when the coupling constant of the Ricci scalar is positive). We present a detailed canonical analysis on the effective action showing that the dynamics of the theory forbids local degrees of freedom. Another similarity with 2 + 1 general relativity is the absence of a Newtonian force. In contrast to these results, we find evidence against the similarity with 2 + 1 general relativity: we find an exact nonflat solution of the same effective theory. This solution is out of the set of asymptotically flat solutions.
We perform an analysis of the ultraviolet divergences of the quantum nonprojectable Hořava gravit... more We perform an analysis of the ultraviolet divergences of the quantum nonprojectable Hořava gravity. We work the quantum-field theory directly in the Hamiltonian formalism provided by the Batalin-Fradkin-Vilkovisky quantization. In this way the second-class constraints can be incorporated into the quantization. A known local gauge-fixing condition leads to a local canonical Lagrangian. Although the canonical fields acquire regular propagators, irregular propagators persist dangerous subdivergences. We show that all these loops cancel exactly between them due to a perfect matching between the propagators and vertices of the fields and ghosts forming the loops. The rest of the divergences behave similarly to the projectable theory, they can be removed by local counterterms. This result points to the renormalization of the nonprojectable Hořava theory.
We perform the Batalin-Fradkin-Vilkovisky quantization of the anisotropic conformal Hořava theory... more We perform the Batalin-Fradkin-Vilkovisky quantization of the anisotropic conformal Hořava theory in d spatial dimensions. We introduce a model with a conformal potential suitable for any dimension. We define an anisotropic and local gauge-fixing condition that accounts for the spatial diffeomorphisms and the anisotropic Weyl transformations. We show that the BRST transformations can be expressed mainly in terms of a spatial diffeomorphism along a ghost field plus a conformal transformation with another ghost field as argument. We study the quantum Lagrangian in the d = 2 case, obtaining that all propagators are regular, except for the fields associated with the measure of the second-class constraints. This behavior is qualitatively equal to the nonconformal case.
We present an analysis on the BRST symmetry transformations of the Hořava theory under the BFV qu... more We present an analysis on the BRST symmetry transformations of the Hořava theory under the BFV quantization, both in the nonprojectable and projectable cases. We obtain that the BRST transformations are intimately related to a particular spatial diffeomorphism along one of the ghost vector fields. We show explicitly the invariance of the quantum action and the nilpotence of the BRST transformations, using this diffeomorphism largely. The BRST symmetry is verified in the whole phase space, inside and outside the constrained surface. When restricted to the constrained surface, the BRST transformations are completely local. The consistency of the BRST symmetry is a fundamental feature of a quantum field theory, specially for renormalization. The BFV quantization is independent of the chosen gauge-fixing condition. This allows us to study the unitarity of the quantum Hořava theory, covering the gauge required for renormalization. We prove that, if the solution for the lapse function exists, then the unitarity of the theory is established.
We present an analysis on the Becchi-Rouet-Stora-Tyutin (BRST) symmetry transformations of the Ho... more We present an analysis on the Becchi-Rouet-Stora-Tyutin (BRST) symmetry transformations of the Hořava theory under the Batalin-Fradkin-Vilkovisky quantization, both in the nonprojectable and projectable cases. We obtain that the BRST transformations are intimately related to a particular spatial diffeomorphism along one of the ghost vector fields. We show explicitly the invariance of the quantum action and the nilpotence of the BRST transformations, using this diffeomorphism largely. The BRST symmetry is verified in the whole phase space, inside and outside the constrained surface. When restricted to the constrained surface, the BRST transformations are completely local. The consistency of the BRST symmetry is a fundamental feature of a quantum field theory, specially for renormalization. The BFV quantization is independent of the chosen gauge-fixing condition. This allows us to study the unitarity of the quantum Hořava theory, covering the gauge required for renormalization. We prove that, if the solution for the lapse function exists, then the unitarity of the theory is established.
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Papers by Byron Droguett