D. Vassilevich - Academia.edu (original) (raw)
Papers by D. Vassilevich
Letters in Mathematical Physics, Aug 19, 2002
We compute the first 5 terms in the short-time heat trace asymptotics expansion for an operator o... more We compute the first 5 terms in the short-time heat trace asymptotics expansion for an operator of Laplace type with transfer boundary conditions using the functorial properties of these invariants.
Theoretical and Mathematical Physics, 2011
ABSTRACT This Chapter introduces main notions of quantum field theories (QFT). The commutation re... more ABSTRACT This Chapter introduces main notions of quantum field theories (QFT). The commutation relations are imposed on operator functionals defined in terms of a relativistic inner product on a Cauchy surface. The product is uniquely defined for any two solutions to a field equation and is used for normalization of the solutions. The chosen way of quantization is more convenient in ‘mode by mode’ calculations of spectral functions in free QFT’s on non-trivial classical backgrounds, which is a central subject of this book. A distinctive feature of the exposition in this Chapter is that fairy general constructions are explained in details for fields which are important for physical applications (spins zero, one-half and one). The material also includes mode decompositions, Bose and Fermi statistics, creation and annihilation operators, Bogoliubov transformations, description of gauge theories. Much space is devoted to QFT’s on stationary classical backgrounds, relation to canonical quantization, the single-particle modes and eigenvalue problems which define the spectra of single-particle energies. Among the traditional issues are Green’s functions and calculations of averages. The Chapter ends with a brief introduction to a theory of quasinormal modes. These modes, although physically relevant, serve as an example of excitations which are not quantized.
Theoretical and Mathematical Physics, 2011
ABSTRACT This Chapter introduces basic notions from differential geometry and classical field the... more ABSTRACT This Chapter introduces basic notions from differential geometry and classical field theories, including some less standard material, such as, e.g. boundaries and singularities. The Chapter begins with definitions from the Riemannian geometry with a focus on mathematical foundations of the general relativity theory. The material includes gravity actions, examples of dynamical equations and their physically important solutions. Description of isometries contains definitions of Killing vectors and Killing spinors. Characteristics of hypersurfaces are explained in detail to set a stage for theories with boundaries. Defects of geometry located on codimension one and codimension two hypersurfaces are considered as examples of singularities when the curvature behaves as a distribution. These two cases are among the base manifolds where a heat kernel operator is studied in next chapters. Discussion of field models starts with a brief introduction to fiber bundles and the structure groups including basic features of the spin group. Properties of the modern elementary particle physics are exposed by using models of dynamical scalar, spinor, massive vector, and gauge fields propagating in classical backgrounds.
Theoretical and Mathematical Physics, 2011
ABSTRACT
Theoretical and Mathematical Physics, 2011
ABSTRACT
Theoretical and Mathematical Physics, 2011
ABSTRACT
Theoretical and Mathematical Physics, 2011
Theoretical and Mathematical Physics, 2011
Theoretical and Mathematical Physics, 2011
This Chapter describes violation of classical symmetries by quantum effects. The aim here is to d... more This Chapter describes violation of classical symmetries by quantum effects. The aim here is to demonstrate how different types of anomalies are calculated by using spectral functions. The Chapter starts with formulation of the Noether theorems and derivation of Noether currents in a number of important examples with gauge and space-time symmetries. The following anomalies are derived by using the zeta-function techniques: axial anomaly, Einstein and Lorentz anomalies, conformal anomaly, and parity anomaly. For pedagogical purposes all calculations are done straightforwardly for two- or three-dimensional models and take into account all relevant terms which are sometime missing in original scientific papers. The results of this Chapter are also used to get with the help of anomalous transformations non-local effective actions in gravitational and gauge backgrounds, including important cases of the Polyakov-Liouville action and the Chern-Simons actions.
We consider quantum p-form fields interacting with a background dilaton. We calculate the variati... more We consider quantum p-form fields interacting with a background dilaton. We calculate the variation with respect to the dilaton of a difference of the effective actions in the models related by a duality transformation. We show that this variation is defined essentially by the supertrace of the twisted de Rham complex. The supertrace is then evaluated on a manifold of an arbitrary dimension, with or without boundary.
We study several classes of non-associative algebras as possible candidates for deformation quant... more We study several classes of non-associative algebras as possible candidates for deformation quantization in the direction of a Poisson bracket that does not satisfy Jacobi identities. We show that in fact alternative deformation quantization algebras require the Jacobi identities on the Poisson bracket and, under very general assumptions, are associative. At the same time, flexible deformation quantization algebras exist for any Poisson bracket.
The spectral action for a non-compact commutative spectral triple is computed covariantly in a ga... more The spectral action for a non-compact commutative spectral triple is computed covariantly in a gauge perturbation up to order 2 in full generality. In the ultraviolet regime, p→∞, the action decays as 1/p^4 in any even dimension.
We analyze the Casimir interaction of doped graphene. To this end we derive a simple expression f... more We analyze the Casimir interaction of doped graphene. To this end we derive a simple expression for the finite temperature polarization tensor with a chemical potential. It is found that doping leads to a strong enhancement of the Casimir force reaching almost 60% in quite realistic situations. This result should be important for planning and interpreting the Casimir measurements, especially taking into account that the Casimir interaction of undoped graphene is rather weak.
We analyse the ultraviolet divergencies in the ground state energy for a penetrable sphere and a ... more We analyse the ultraviolet divergencies in the ground state energy for a penetrable sphere and a dielectric ball. We argue that for massless fields subtraction of the "empty space" or the "unbounded medium" contribution is not enough to make the ground state energy finite whenever the heat kernel coefficient a_2 is not zero. It turns out that a_2 0 for a penetrable sphere, a general dielectric background and the dielectric ball. To our surprise, for more singular configurations, as in the presence of sharp boundaries, the heat kernel coefficients behave to some extend better than in the corresponding smooth cases, making, for instance, the dilute dielectric ball a well defined problem.
We consider different methods of calculating the (fractional) fermion number of solitons based on... more We consider different methods of calculating the (fractional) fermion number of solitons based on the heat kernel expansion. We derive a formula for the localized eta function a more systematic version of the derivative expansion for spectral assymmetry and that provides a more systematic version of the derivative expansion for spectral asymmetry and compute the fermion number in a multiflavour extension of the Goldstone-Wilczek model.We also propose an improved expansionof the heat kernelthat allows the tackling ofthe convergence issues and permits an automated computation of the coefficients
Abstract. We use invariance theory to compute the divergence term a d+δ m+2,m in the super trace ... more Abstract. We use invariance theory to compute the divergence term a d+δ m+2,m in the super trace for the twisted de Rham complex for a closed Riemannian manifold. 1.
We prove that no local diffeomorphism invariant two-dimensional theory of the metric and the dila... more We prove that no local diffeomorphism invariant two-dimensional theory of the metric and the dilaton without higher derivatives can describe the exact string black hole solution found a decade ago by Dijkgraaf, Verlinde and Verlinde. One of the key points in this proof is the concept of dilaton-shift invariance. We present and solve (classically) all dilaton-shift invariant theories of two-dimensional dilaton gravity. Two such models, resembling the exact string black hole and generalizing the CGHS model, are discussed explicitly.
We consider quantum p-form fields interacting with a background dilaton. We calculate the variati... more We consider quantum p-form fields interacting with a background dilaton. We calculate the variation with respect to the dilaton of a difference of the effective actions in the models related by a duality transformation. We show that this variation is defined essentially by the supertrace of the twisted de Rham complex. The supertrace is then evaluated on a manifold of an arbitrary dimension, with or without boundary.
Exploiting methods of Quantum Field Theory we compute the bulk polarization tensor and bulk diele... more Exploiting methods of Quantum Field Theory we compute the bulk polarization tensor and bulk dielectric functions for Dirac materials in the presence of a mass gap, chemical potential, and finite temperature. Using these results (and neglecting eventual boundary effects), we study the Casimir interaction of Dirac materials. We describe in detail the characteristic features of the dielectric functions and their influence on the Casimir pressure.
Letters in Mathematical Physics, Aug 19, 2002
We compute the first 5 terms in the short-time heat trace asymptotics expansion for an operator o... more We compute the first 5 terms in the short-time heat trace asymptotics expansion for an operator of Laplace type with transfer boundary conditions using the functorial properties of these invariants.
Theoretical and Mathematical Physics, 2011
ABSTRACT This Chapter introduces main notions of quantum field theories (QFT). The commutation re... more ABSTRACT This Chapter introduces main notions of quantum field theories (QFT). The commutation relations are imposed on operator functionals defined in terms of a relativistic inner product on a Cauchy surface. The product is uniquely defined for any two solutions to a field equation and is used for normalization of the solutions. The chosen way of quantization is more convenient in ‘mode by mode’ calculations of spectral functions in free QFT’s on non-trivial classical backgrounds, which is a central subject of this book. A distinctive feature of the exposition in this Chapter is that fairy general constructions are explained in details for fields which are important for physical applications (spins zero, one-half and one). The material also includes mode decompositions, Bose and Fermi statistics, creation and annihilation operators, Bogoliubov transformations, description of gauge theories. Much space is devoted to QFT’s on stationary classical backgrounds, relation to canonical quantization, the single-particle modes and eigenvalue problems which define the spectra of single-particle energies. Among the traditional issues are Green’s functions and calculations of averages. The Chapter ends with a brief introduction to a theory of quasinormal modes. These modes, although physically relevant, serve as an example of excitations which are not quantized.
Theoretical and Mathematical Physics, 2011
ABSTRACT This Chapter introduces basic notions from differential geometry and classical field the... more ABSTRACT This Chapter introduces basic notions from differential geometry and classical field theories, including some less standard material, such as, e.g. boundaries and singularities. The Chapter begins with definitions from the Riemannian geometry with a focus on mathematical foundations of the general relativity theory. The material includes gravity actions, examples of dynamical equations and their physically important solutions. Description of isometries contains definitions of Killing vectors and Killing spinors. Characteristics of hypersurfaces are explained in detail to set a stage for theories with boundaries. Defects of geometry located on codimension one and codimension two hypersurfaces are considered as examples of singularities when the curvature behaves as a distribution. These two cases are among the base manifolds where a heat kernel operator is studied in next chapters. Discussion of field models starts with a brief introduction to fiber bundles and the structure groups including basic features of the spin group. Properties of the modern elementary particle physics are exposed by using models of dynamical scalar, spinor, massive vector, and gauge fields propagating in classical backgrounds.
Theoretical and Mathematical Physics, 2011
ABSTRACT
Theoretical and Mathematical Physics, 2011
ABSTRACT
Theoretical and Mathematical Physics, 2011
ABSTRACT
Theoretical and Mathematical Physics, 2011
Theoretical and Mathematical Physics, 2011
Theoretical and Mathematical Physics, 2011
This Chapter describes violation of classical symmetries by quantum effects. The aim here is to d... more This Chapter describes violation of classical symmetries by quantum effects. The aim here is to demonstrate how different types of anomalies are calculated by using spectral functions. The Chapter starts with formulation of the Noether theorems and derivation of Noether currents in a number of important examples with gauge and space-time symmetries. The following anomalies are derived by using the zeta-function techniques: axial anomaly, Einstein and Lorentz anomalies, conformal anomaly, and parity anomaly. For pedagogical purposes all calculations are done straightforwardly for two- or three-dimensional models and take into account all relevant terms which are sometime missing in original scientific papers. The results of this Chapter are also used to get with the help of anomalous transformations non-local effective actions in gravitational and gauge backgrounds, including important cases of the Polyakov-Liouville action and the Chern-Simons actions.
We consider quantum p-form fields interacting with a background dilaton. We calculate the variati... more We consider quantum p-form fields interacting with a background dilaton. We calculate the variation with respect to the dilaton of a difference of the effective actions in the models related by a duality transformation. We show that this variation is defined essentially by the supertrace of the twisted de Rham complex. The supertrace is then evaluated on a manifold of an arbitrary dimension, with or without boundary.
We study several classes of non-associative algebras as possible candidates for deformation quant... more We study several classes of non-associative algebras as possible candidates for deformation quantization in the direction of a Poisson bracket that does not satisfy Jacobi identities. We show that in fact alternative deformation quantization algebras require the Jacobi identities on the Poisson bracket and, under very general assumptions, are associative. At the same time, flexible deformation quantization algebras exist for any Poisson bracket.
The spectral action for a non-compact commutative spectral triple is computed covariantly in a ga... more The spectral action for a non-compact commutative spectral triple is computed covariantly in a gauge perturbation up to order 2 in full generality. In the ultraviolet regime, p→∞, the action decays as 1/p^4 in any even dimension.
We analyze the Casimir interaction of doped graphene. To this end we derive a simple expression f... more We analyze the Casimir interaction of doped graphene. To this end we derive a simple expression for the finite temperature polarization tensor with a chemical potential. It is found that doping leads to a strong enhancement of the Casimir force reaching almost 60% in quite realistic situations. This result should be important for planning and interpreting the Casimir measurements, especially taking into account that the Casimir interaction of undoped graphene is rather weak.
We analyse the ultraviolet divergencies in the ground state energy for a penetrable sphere and a ... more We analyse the ultraviolet divergencies in the ground state energy for a penetrable sphere and a dielectric ball. We argue that for massless fields subtraction of the "empty space" or the "unbounded medium" contribution is not enough to make the ground state energy finite whenever the heat kernel coefficient a_2 is not zero. It turns out that a_2 0 for a penetrable sphere, a general dielectric background and the dielectric ball. To our surprise, for more singular configurations, as in the presence of sharp boundaries, the heat kernel coefficients behave to some extend better than in the corresponding smooth cases, making, for instance, the dilute dielectric ball a well defined problem.
We consider different methods of calculating the (fractional) fermion number of solitons based on... more We consider different methods of calculating the (fractional) fermion number of solitons based on the heat kernel expansion. We derive a formula for the localized eta function a more systematic version of the derivative expansion for spectral assymmetry and that provides a more systematic version of the derivative expansion for spectral asymmetry and compute the fermion number in a multiflavour extension of the Goldstone-Wilczek model.We also propose an improved expansionof the heat kernelthat allows the tackling ofthe convergence issues and permits an automated computation of the coefficients
Abstract. We use invariance theory to compute the divergence term a d+δ m+2,m in the super trace ... more Abstract. We use invariance theory to compute the divergence term a d+δ m+2,m in the super trace for the twisted de Rham complex for a closed Riemannian manifold. 1.
We prove that no local diffeomorphism invariant two-dimensional theory of the metric and the dila... more We prove that no local diffeomorphism invariant two-dimensional theory of the metric and the dilaton without higher derivatives can describe the exact string black hole solution found a decade ago by Dijkgraaf, Verlinde and Verlinde. One of the key points in this proof is the concept of dilaton-shift invariance. We present and solve (classically) all dilaton-shift invariant theories of two-dimensional dilaton gravity. Two such models, resembling the exact string black hole and generalizing the CGHS model, are discussed explicitly.
We consider quantum p-form fields interacting with a background dilaton. We calculate the variati... more We consider quantum p-form fields interacting with a background dilaton. We calculate the variation with respect to the dilaton of a difference of the effective actions in the models related by a duality transformation. We show that this variation is defined essentially by the supertrace of the twisted de Rham complex. The supertrace is then evaluated on a manifold of an arbitrary dimension, with or without boundary.
Exploiting methods of Quantum Field Theory we compute the bulk polarization tensor and bulk diele... more Exploiting methods of Quantum Field Theory we compute the bulk polarization tensor and bulk dielectric functions for Dirac materials in the presence of a mass gap, chemical potential, and finite temperature. Using these results (and neglecting eventual boundary effects), we study the Casimir interaction of Dirac materials. We describe in detail the characteristic features of the dielectric functions and their influence on the Casimir pressure.