Giampiero Esposito – författare

This book is written for theoretical and mathematical physicists and mat- maticians interested in recent developments in complex general relativity and their application to classical and quantum gravity. Calculations are presented by paying attention to those details normally omitted in research papers, for pedagogical r- sons. Familiarity with fibre-bundle theory is certainly helpful, but in many cases I only rely on two-spinor calculus and conformally invariant concepts in gravitational physics. The key concepts the book is devoted to are complex manifolds, spinor techniques, conformal gravity, ?-planes, ?-surfaces, Penrose transform, complex 3 1 – – space-time models with non-vanishing torsion, spin- fields and spin- potentials. 2 2 Problems have been inserted at the end, to help the reader to check his und- standing of these topics. Thus, I can find at least four reasons for writing yet another book on spinor and twistor methods in general relativity: (i) to write a textbook useful to - ginning graduate students and research workers, where two-component spinor c- culus is the unifying mathematical language.

This 2004 textbook provides a pedagogical introduction to the formalism, foundations and applications of quantum mechanics. Part I covers the basic material which is necessary to understand the transition from classical to wave mechanics. Topics include classical dynamics, with emphasis on canonical transformations and the Hamilton-Jacobi equation, the Cauchy problem for the wave equation, Helmholtz equation and eikonal approximation, introduction to spin, perturbation theory and scattering theory. The Weyl quantization is presented in Part II, along with the postulates of quantum mechanics. Part III is devoted to topics such as statistical mechanics and black-body radiation, Lagrangian and phase-space formulations of quantum mechanics, and the Dirac equation. This book is intended for use as a textbook for beginning graduate and advanced undergraduate courses. It is self-contained and includes problems to aid the reader's understanding.

The Dirac operator has many useful applications in theoretical physics and mathematics. This book provides a clear, concise and self-contained introduction to the global theory of the Dirac operator and to the analysis of spectral asymptotics with local or non-local boundary conditions. The theory is introduced at a level suitable for graduate students. Numerous examples are then given to illustrate the peculiar properties of the Dirac operator, and the role of boundary conditions in heat-kernel asymptotics and quantum field theory. Topics covered include the introduction of spin-structures in Riemannian and Lorentzian manifolds; applications of index theory; heat-kernel asymptotics for operators of Laplace type; quark boundary conditions; one-loop quantum cosmology; conformally covariant operators; and the role of the Dirac operator in some recent investigations of four-manifolds. This volume provides graduate students with a rigorous introduction and researchers with a valuable reference to the Dirac operator and its applications in theoretical physics.

This 2004 textbook provides a pedagogical introduction to the formalism, foundations and applications of quantum mechanics. Part I covers the basic material which is necessary to understand the transition from classical to wave mechanics. Topics include classical dynamics, with emphasis on canonical transformations and the Hamilton-Jacobi equation, the Cauchy problem for the wave equation, Helmholtz equation and eikonal approximation, introduction to spin, perturbation theory and scattering theory. The Weyl quantization is presented in Part II, along with the postulates of quantum mechanics. Part III is devoted to topics such as statistical mechanics and black-body radiation, Lagrangian and phase-space formulations of quantum mechanics, and the Dirac equation. This book is intended for use as a textbook for beginning graduate and advanced undergraduate courses. It is self-contained and includes problems to aid the reader's understanding.

This book is written for theoretical and mathematical physicists and mat- maticians interested in recent developments in complex general relativity and their application to classical and quantum gravity. Calculations are presented by paying attention to those details normally omitted in research papers, for pedagogical r- sons. Familiarity with fibre-bundle theory is certainly helpful, but in many cases I only rely on two-spinor calculus and conformally invariant concepts in gravitational physics. The key concepts the book is devoted to are complex manifolds, spinor techniques, conformal gravity, ?-planes, ?-surfaces, Penrose transform, complex 3 1 – – space-time models with non-vanishing torsion, spin- fields and spin- potentials. 2 2 Problems have been inserted at the end, to help the reader to check his und- standing of these topics. Thus, I can find at least four reasons for writing yet another book on spinor and twistor methods in general relativity: (i) to write a textbook useful to - ginning graduate students and research workers, where two-component spinor c- culus is the unifying mathematical language.

This book reflects our own struggle to understand the semiclassical behaviour of quantized fields in the presence of boundaries. Along many years, motivated by the problems of quantum cosmology and quantum field theory, we have studied in detail the one-loop properties of massless spin-l/2 fields, Euclidean Maxwell the­ ory, gravitino potentials and Euclidean quantum gravity. Hence our book begins with a review of the physical and mathematical motivations for studying physical theories in the presence of boundaries, with emphasis on electrostatics, vacuum v Maxwell theory and quantum cosmology. We then study the Feynman propagator in Minkowski space-time and in curved space-time. In the latter case, the corre­ sponding Schwinger-DeWitt asymptotic expansion is given. The following chapters are devoted to the standard theory of the effective action and the geometric im­ provement due to Vilkovisky, the manifestly covariant quantization of gauge fields, zeta-function regularization in mathematics and in quantum field theory, and the problem of boundary conditions in one-loop quantum theory. For this purpose, we study in detail Dirichlet, Neumann and Robin boundary conditions for scalar fields, local and non-local boundary conditions for massless spin-l/2 fields, mixed boundary conditions for gauge fields and gravitation. This is the content of Part I. Part II presents our investigations of Euclidean Maxwell theory, simple super­ gravity and Euclidean quantum gravity.

Introducing a geometric view of fundamental physics, starting from quantum mechanics and its experimental foundations, this book is ideal for advanced undergraduate and graduate students in quantum mechanics and mathematical physics. Focusing on structural issues and geometric ideas, this book guides readers from the concepts of classical mechanics to those of quantum mechanics. The book features an original presentation of classical mechanics, with the choice of topics motivated by the subsequent development of quantum mechanics, especially wave equations, Poisson brackets and harmonic oscillators. It also presents new treatments of waves and particles and the symmetries in quantum mechanics, as well as extensive coverage of the experimental foundations.

Introducing a geometric view of fundamental physics, starting from quantum mechanics and its experimental foundations, this book is ideal for advanced undergraduate and graduate students in quantum mechanics and mathematical physics. Focusing on structural issues and geometric ideas, this book guides readers from the concepts of classical mechanics to those of quantum mechanics. The book features an original presentation of classical mechanics, with the choice of topics motivated by the subsequent development of quantum mechanics, especially wave equations, Poisson brackets and harmonic oscillators. It also presents new treatments of waves and particles and the symmetries in quantum mechanics, as well as extensive coverage of the experimental foundations.

Introducing a geometric view of fundamental physics, starting from quantum mechanics and its experimental foundations, this book is ideal for advanced undergraduate and graduate students in quantum mechanics and mathematical physics. Focusing on structural issues and geometric ideas, this book guides readers from the concepts of classical mechanics to those of quantum mechanics. The book features an original presentation of classical mechanics, with the choice of topics motivated by the subsequent development of quantum mechanics, especially wave equations, Poisson brackets and harmonic oscillators. It also presents new treatments of waves and particles and the symmetries in quantum mechanics, as well as extensive coverage of the experimental foundations.

Several scientists learn only a first course in complex analysis, and hence they are not familiar with several important properties: every polygenic function defines a congruence of clocks; the basic properties of algebraic functions and abelian integrals; how mankind arrived at a rigorous definition of Riemann surfaces; the concepts of dianalytic structures and Klein surfaces; the Weierstrass elliptic functions; the automorphic functions discovered by Poincare’ and their links with the theory of Fuchsian groups; the geometric structure of fractional linear transformations; Kleinian groups; the Heisenberg group and geometry of the complex ball; complex powers of elliptic operators and the theory of spectral zeta-functions; an assessment of the Poincare’ and Dieudonne’ definitions of the concept of asymptotic expansion. The book is unique both for the selection of topics and for the readable access that it offers to the otherwise too large landscape of modern complex analysis.

Questo libro è scritto per studenti della laurea magistrale in Fisica o in Matematica che desiderano apprendere i fondamenti della teoria classica dei campi, con particolare riguardo al linguaggio matematico ed alle applicazioni fisiche. Ciò che lo rende unico è la visione d'insieme che lo studente acquisisce, come pure lo stimolo a sviluppare il pensiero astratto in varie forme. Nel primo capitolo mi raccordo ai corsi di elettrodinamica classica. Nel secondo capitolo sono introdotti alcuni concetti di algebra astratta che non vengono insegnati in corsi obbligatori per gli studenti di Fisica. Nel terzo capitolo si studiano le varietà topologiche e le varietà differenziabili, mentre il quarto capitolo studia le varietà regolari di R^{n}, che possono essere applicate per comprendere le varietà vincolari dei capitoli 25 e 26. I capitoli dal 5 al 7 svolgono il calcolo su varietà senza mai far uso di metriche. A volte gli studenti sono bloccati dal potere di sintesi della notazione moderna, dunque ho cercato di svolgere in dettaglio vari calcoli ed esempi e controesempi. I capitoli da 8 a 12 sono dedicati ai gruppi topologici, i gruppi di Lie e le algebre di Lie, e alle loro applicazioni fisiche. La geometria riemanniana viene studiata nei capitoli da 13 a 17, mentre la teoria dei fibrati principali e vettoriali viene introdotta nei capitoli da 18 a 20. Nel capitolo 21 si studia il nastro di Möbius, mentre il capitolo 22 approfondisce la fibrazione di Hopf in modo pedagogico. Principi variazionali e simmetrie sono studiati nei capitoli 23 e 24. Nel capitolo 25 vengono studiati i sistemi hamiltoniani soggetti a vincoli, poiché in essi rientrano anche tutte le teorie di gauge delle interazioni fondamentali. Infatti il capitolo 26 applica tale teoria all'elettrodinamica nello spaziotempo di Minkowski. Nel capitolo 27 si studia poi l'elettrodinamica classica in spaziotempo curvo, e vengono confrontati in modo pedagogico i linguaggi vettoriale, tensoriale e delle forme differenziali, con applicazione ai potenziali di Hertz. La relatività generale viene delineata nel capitolo 28: il principio di equivalenza, il principio di covarianza generale, il funzionale d'azione. La teoria dei gruppi di Lie a dimensione infinita viene introdotta nel capitolo 29, dimostrando alfine il teorema di Freifeld: esistono diffeomorfismi arbitrariamente prossimi all'identità che non giacciono su sottogruppi ad un parametro. Le trasformazioni lineari frazionarie usate nel capitolo 11 per studiare il gruppo di Lorentz vengono classificate in ellittiche, paraboliche, iperboliche e lossodromiche nel capitolo 30. Nel capitolo 31 si dimostra il teorema di Hodge sullo spazio delle forme lisce su una varietà di Riemann. Nel capitolo 32 il lettore apprende che tutte le teorie di gauge note ricadono nelle famiglie di tipo I, II e III, in base alla forma delle parentesi di Lie dei campi vettoriali che lasciano invariato il funzionale d'azione. Inoltre, esiste una forma di connessione nonlocale per tutte le teorie di gauge. Questo capitolo prepara gli studenti che poi, studiando la fisica quantistica, vorranno apprendere l'approccio globale alla teoria quantistica dei campi in corsi successivi.

Questo libro è scritto per studenti della laurea magistrale in Fisica o in Matematica che desiderano apprendere i fondamenti della teoria classica dei campi, con particolare riguardo al linguaggio matematico ed alle applicazioni fisiche. Ciò che lo rende unico è la visione d'insieme che lo studente acquisisce, come pure lo stimolo a sviluppare il pensiero astratto in varie forme. Nel primo capitolo mi raccordo ai corsi di elettrodinamica classica. Nel secondo capitolo sono introdotti alcuni concetti di algebra astratta che non vengono insegnati in corsi obbligatori per gli studenti di Fisica. Nel terzo capitolo si studiano le varietà topologiche e le varietà differenziabili, mentre il quarto capitolo studia le varietà regolari di R^{n}, che possono essere applicate per comprendere le varietà vincolari dei capitoli 25 e 26. I capitoli dal 5 al 7 svolgono il calcolo su varietà senza mai far uso di metriche. A volte gli studenti sono bloccati dal potere di sintesi della notazione moderna, dunque ho cercato di svolgere in dettaglio vari calcoli ed esempi e controesempi. I capitoli da 8 a 12 sono dedicati ai gruppi topologici, i gruppi di Lie e le algebre di Lie, e alle loro applicazioni fisiche. La geometria riemanniana viene studiata nei capitoli da 13 a 17, mentre la teoria dei fibrati principali e vettoriali viene introdotta nei capitoli da 18 a 20. Nel capitolo 21 si studia il nastro di Möbius, mentre il capitolo 22 approfondisce la fibrazione di Hopf in modo pedagogico. Principi variazionali e simmetrie sono studiati nei capitoli 23 e 24. Nel capitolo 25 vengono studiati i sistemi hamiltoniani soggetti a vincoli, poiché in essi rientrano anche tutte le teorie di gauge delle interazioni fondamentali. Infatti il capitolo 26 applica tale teoria all'elettrodinamica nello spaziotempo di Minkowski. Nel capitolo 27 si studia poi l'elettrodinamica classica in spaziotempo curvo, e vengono confrontati in modo pedagogico i linguaggi vettoriale, tensoriale e delle forme differenziali, con applicazione ai potenziali di Hertz. La relatività generale viene delineata nel capitolo 28: il principio di equivalenza, il principio di covarianza generale, il funzionale d'azione. La teoria dei gruppi di Lie a dimensione infinita viene introdotta nel capitolo 29, dimostrando alfine il teorema di Freifeld: esistono diffeomorfismi arbitrariamente prossimi all'identità che non giacciono su sottogruppi ad un parametro. Le trasformazioni lineari frazionarie usate nel capitolo 11 per studiare il gruppo di Lorentz vengono classificate in ellittiche, paraboliche, iperboliche e lossodromiche nel capitolo 30. Nel capitolo 31 si dimostra il teorema di Hodge sullo spazio delle forme lisce su una varietà di Riemann. Nel capitolo 32 il lettore apprende che tutte le teorie di gauge note ricadono nelle famiglie di tipo I, II e III, in base alla forma delle parentesi di Lie dei campi vettoriali che lasciano invariato il funzionale d'azione. Inoltre, esiste una forma di connessione nonlocale per tutte le teorie di gauge. Questo capitolo prepara gli studenti che poi, studiando la fisica quantistica, vorranno apprendere l'approccio globale alla teoria quantistica dei campi in corsi successivi.

This book is addressed to mathematics and physics students who want to develop an interdisciplinary view of mathematics, from the age of Riemann, Poincaré and Darboux to basic tools of modern mathematics. It enables them to acquire the sensibility necessary for the formulation and solution of difficult problems, with an emphasis on concepts, rigour and creativity. It consists of eight self-contained parts: ordinary differential equations; linear elliptic equations; calculus of variations; linear and non-linear hyperbolic equations; parabolic equations; Fuchsian functions and non-linear equations; the functional equations of number theory; pseudo-differential operators and pseudo-differential equations. The author leads readers through the original papers and introduces new concepts, with a selection of topics and examples that are of high pedagogical value.

This book is aimed at theoretical and mathematical physicists and mathematicians interested in modern gravitational physics. I have thus tried to use language familiar to readers working on classical and quantum gravity, paying attention both to difficult calculations and to existence theorems, and discussing in detail the current literature. The first aim of the book is to describe recent work on the problem of boundary conditions in one-loop quantum cosmology. The motivation of this research was to under­ stand whether supersymmetric theories are one-loop finite in the presence of boundaries, with application to the boundary-value problemsoccurring in quantum cosmology. Indeed, higher-loop calculations in the absence of boundaries are already available in the litera­ ture, showing that supergravity is not finite. I believe, however, that one-loop calculations in the presence of boundaries are more fundamental, in that they provide a more direct check of the inconsistency of supersymmetric quantum cosmology from the perturbative point of view. It therefore appears that higher-order calculations are not strictly needed, if the one-loop test already yields negative results. Even though the question is not yet settled, this research has led to many interesting, new applications of areas of theoretical and mathematical physics such as twistor theory in flat space, self-adjointness theory, the generalized Riemann zeta-function, and the theory of boundary counterterms in super­ gravity. I have also compared in detail my work with results by other authors, explaining, whenever possible, the origin of different results, the limits of my work and the unsolved problems.

Aside from the obvious statement that it should be a theory capable of unifying general relativity and quantum field theory, not much is known about the true nature of quantum gravity.New ideas - and there are many of them for this is an exciting field of research - often diverge to a degree where it seems impossible to decide in which of the many possible direction(s) the ongoing developments should be further sustained.The division of the book in two (overlapping) parts reflects the duality between the physical vision and the mathematical construction. The former is represented by tutorial reviews on non-commutative geometry, on space-time discretization and renormalization and on gauge field path integrals.  The latter one by lectures on cohomology, on stochastic geometry and on mathematical tools for the effective action in quantum gravity.The book will benefit everyone working or entering the field of quantum gravity research.

This book is aimed at theoretical and mathematical physicists and mathematicians interested in modern gravitational physics. I have thus tried to use language familiar to readers working on classical and quantum gravity, paying attention both to difficult calculations and to existence theorems, and discussing in detail the current literature. The first aim of the book is to describe recent work on the problern of boundary conditions in one-loop quantum cosmology. The motivation of this research was to under­ stand whether supersymmetric theories are one-loop finite in the presence of boundaries, with application to the boundary-value problems occurring in quantum cosmology. Indeed, higher-loop calculations in the absence of boundaries are already available in the litera­ ture, showing that supergravity is not finite. I believe, however, that one-loop calculations in the presence of boundaries are more fundamental, in that they provide a more direct check of the inconsistency of Supersymmetrie quantum cosmology from the perturbative point of view. It therefore appears that higher-order calculations are not strictly needed, if the one-loop test already yields negative results. Even though the question is not yet settled, this research has led to many interesting, new applications of areas of theoretical and mathematical physics such as twistor theory in flat space, self-adjointness theory, the generalized lliemann zeta-function, and the theory of boundary counterterms in super­ gravity.

This book is aimed at theoretical and mathematical physicists and mathematicians interested in modern gravitational physics. I have thus tried to use language familiar to readers working on classical and quantum gravity, paying attention both to difficult calculations and to existence theorems, and discussing in detail the current literature. The first aim of the book is to describe recent work on the problem of boundary conditions in one-loop quantum cosmology. The motivation of this research was to under­ stand whether supersymmetric theories are one-loop finite in the presence of boundaries, with application to the boundary-value problemsoccurring in quantum cosmology. Indeed, higher-loop calculations in the absence of boundaries are already available in the litera­ ture, showing that supergravity is not finite. I believe, however, that one-loop calculations in the presence of boundaries are more fundamental, in that they provide a more direct check of the inconsistency of supersymmetric quantum cosmology from the perturbative point of view. It therefore appears that higher-order calculations are not strictly needed, if the one-loop test already yields negative results. Even though the question is not yet settled, this research has led to many interesting, new applications of areas of theoretical and mathematical physics such as twistor theory in flat space, self-adjointness theory, the generalized Riemann zeta-function, and the theory of boundary counterterms in super­ gravity. I have also compared in detail my work with results by other authors, explaining, whenever possible, the origin of different results, the limits of my work and the unsolved problems.

This book is written for theoretical and mathematical physicists and mat- maticians interested in recent developments in complex general relativity and their application to classical and quantum gravity. Calculations are presented by paying attention to those details normally omitted in research papers, for pedagogical r- sons. Familiarity with fibre-bundle theory is certainly helpful, but in many cases I only rely on two-spinor calculus and conformally invariant concepts in gravitational physics. The key concepts the book is devoted to are complex manifolds, spinor techniques, conformal gravity, ?-planes, ?-surfaces, Penrose transform, complex 3 1 – – space-time models with non-vanishing torsion, spin- fields and spin- potentials. 2 2 Problems have been inserted at the end, to help the reader to check his und- standing of these topics. Thus, I can find at least four reasons for writing yet another book on spinor and twistor methods in general relativity: (i) to write a textbook useful to - ginning graduate students and research workers, where two-component spinor c- culus is the unifying mathematical language.

This book reflects our own struggle to understand the semiclassical behaviour of quantized fields in the presence of boundaries. Along many years, motivated by the problems of quantum cosmology and quantum field theory, we have studied in detail the one-loop properties of massless spin-l/2 fields, Euclidean Maxwell the­ ory, gravitino potentials and Euclidean quantum gravity. Hence our book begins with a review of the physical and mathematical motivations for studying physical theories in the presence of boundaries, with emphasis on electrostatics, vacuum v Maxwell theory and quantum cosmology. We then study the Feynman propagator in Minkowski space-time and in curved space-time. In the latter case, the corre­ sponding Schwinger-DeWitt asymptotic expansion is given. The following chapters are devoted to the standard theory of the effective action and the geometric im­ provement due to Vilkovisky, the manifestly covariant quantization of gauge fields, zeta-function regularization in mathematics and in quantum field theory, and the problem of boundary conditions in one-loop quantum theory. For this purpose, we study in detail Dirichlet, Neumann and Robin boundary conditions for scalar fields, local and non-local boundary conditions for massless spin-l/2 fields, mixed boundary conditions for gauge fields and gravitation. This is the content of Part I. Part II presents our investigations of Euclidean Maxwell theory, simple super­ gravity and Euclidean quantum gravity.