This book provides advanced physics and mathematics students with an accessible yet detailed unde... more This book provides advanced physics and mathematics students with an accessible yet detailed understanding of the fundamentals of differential geometry and symmetries in classical physics. Most of the topics covered in this book have previously only been scattered across many different sources of literature, therefore this is the first book to coherently present this treatment of topics in one comprehensive volume.
- - Key features:
-> Contains a modern, streamlined presentation of classical topics, which are normally taught separately.
-> Includes several advanced topics, such as the Belinfante energy-momentum tensor, the Weyl-Schouten theorem, the derivation of Noether currents for diffeomorphisms, and the definition of conserved integrals in general relativity.
-> Focuses on the clear presentation of the mathematical notions and calculational technique.
- - - Table of Contents:
- Chapter 1. Manifolds and Tensors.
- Chapter 2. Geometry and Integration on Manifolds.
- Chapter 3. Symmetries of Manifolds.
- Chapter 4. Newtonian Mechanics.
- Chapter 5. Lagrangian Methods and Symmetry.
- Chapter 6. Relativistic Mechanics.
- Chapter 7. Lie Groups.
- Chapter 8. Lie Algebras.
- Chapter 9. Representations.
- Chapter 10. Rotations and Euclidean Symmetry.
- Chapter 11. Boosts and Galilei Symmetry.
- Chapter 12. Lorentz Symmetry.
- Chapter 13. Poincare Symmetry.
- Chapter 14. Conformal Symmetry.
- Chapter 15. Lagrangians and Noether's Theorem.
- Chapter 16. Spacetime Symmetries of Fields.
- Chapter 17. Gauge Symmetry.
- Chapter 18. Connection and Geodesics.
- Chapter 19. Riemannian Curvature.
- Chapter 20. Symmetries of Riemannian Manifolds.
- Chapter 21. Einstein's Gravitation.
- Chapter 22. Lagrangian Formulation.
- Chapter 23. Conservation Laws and Further Symmetries.
- - Appendices
- A) Notation and Conventions.
- Physical Units and Dimensions.
- Mathematical Conventions.
- Abbreviations.
- B) Mathematical Tools.
- Tensor Algebra.
- Matrix Exponential.
- Pauli and Dirac Matrices.
- Dirac Delta Distribution.
- Poisson and Wave Equation.
- Variational Calculus.
- Volume Element and Hyperspheres.
- Hypersurface Elements.
- C) Weyl Rescaling Formulae.
- D) Spaces and Symmetry Groups.
The framework of field theory allows the definition of a class of transformations other than spac... more The framework of field theory allows the definition of a class of transformations other than spacetime transformations. These are the internal transformations which, due to Noether’s theorem, also lead to conserved currents and charges. We first discuss global phase transformations and subsequently the more general local phase transformations of fields, also called gauge transformations. Gauge symmetry provides a powerful guiding principle in order to define the interaction between different types of fields. The gauge symmetry leads to a coupling prescription that fully determines the interaction between fields. The Maxwell electromagnetic field interacting with a matter field represents the prime example. In this chapter, we discuss the resulting Lagrangian densities, the equations of motion, and the associated conserved currents and energy momentum tensors.
Galilei symmetry is the symmetry of nonrelativistic spacetime. It corresponds to the maximally ad... more Galilei symmetry is the symmetry of nonrelativistic spacetime. It corresponds to the maximally admissible set of transformations between Galileian inertial reference frames. Beyond this global definition, it is enlightening to construct the Galilei group from the ground up. To this end, one starts with the introduction of a time parameter and the group of velocity boosts. Then one moves on to the construction of the homogeneous Galilei group as the semidirect product of velocity boosts and rotations. In a final step, one reaches to the full Galilei group by constructing the semidirect product of translations with the homogeneous Galilei group. In the last section of this chaoter the Galilei algebra is derived.
In this chapter we study the consequences of Noether’s theorem in field theory for the case of pu... more In this chapter we study the consequences of Noether’s theorem in field theory for the case of pure spacetime transformations. The symmetry groups considered are the Poincaré group and the conformal group. In this context, the energy-momentum tensor turns out to be the central quantity to focus on, since it allows the definition of all conserved quantities under the conformal group. After a detailed description of the Belinfante prescription for the construction of a symmetric energy-momentum tensor, numerous concrete examples from relevant field theories are given. In the last section, we discuss the meaning and the necessary conditions for achieving conformal symmetry in a field theory.
In this chapter we construct locally and globally conserved quantities based on spacetime symmetr... more In this chapter we construct locally and globally conserved quantities based on spacetime symmetries of general relativity. The existence of Killing vector fields ensures that one can define conserved quantities from geodesics and from the energy-momentum tensor of matter fields. We discuss the notion of energy of the gravitational field and the challenges this conception poses. By employing the Noether currents of diffeomorphisms generated by suitable Killing vectors, we are able to define the so-called Komar integrals, which can measure the total energy and angular momentum of a spacetime. In the final part, we examine how Weyl rescalings affect diverse physical theories and under which conditions Weyl rescaling invariance can be achieved.
Geometry, Symmetries, and Classical Physics: A Mosaic, Dec 29, 2021
This book provides advanced physics and mathematics students with an accessible yet detailed unde... more This book provides advanced physics and mathematics students with an accessible yet detailed understanding of the fundamentals of differential geometry and symmetries in classical physics. Most of the topics covered in this book have previously only been scattered across many different sources of literature, therefore this is the first book to coherently present this treatment of topics in one comprehensive volume.
- - Key features:
-> Contains a modern, streamlined presentation of classical topics, which are normally taught separately.
-> Includes several advanced topics, such as the Belinfante energy-momentum tensor, the Weyl-Schouten theorem, the derivation of Noether currents for diffeomorphisms, and the definition of conserved integrals in general relativity.
-> Focuses on the clear presentation of the mathematical notions and calculational technique.
- - - Table of Contents:
- Chapter 1. Manifolds and Tensors.
- Chapter 2. Geometry and Integration on Manifolds.
- Chapter 3. Symmetries of Manifolds.
- Chapter 4. Newtonian Mechanics.
- Chapter 5. Lagrangian Methods and Symmetry.
- Chapter 6. Relativistic Mechanics.
- Chapter 7. Lie Groups.
- Chapter 8. Lie Algebras.
- Chapter 9. Representations.
- Chapter 10. Rotations and Euclidean Symmetry.
- Chapter 11. Boosts and Galilei Symmetry.
- Chapter 12. Lorentz Symmetry.
- Chapter 13. Poincare Symmetry.
- Chapter 14. Conformal Symmetry.
- Chapter 15. Lagrangians and Noether's Theorem.
- Chapter 16. Spacetime Symmetries of Fields.
- Chapter 17. Gauge Symmetry.
- Chapter 18. Connection and Geodesics.
- Chapter 19. Riemannian Curvature.
- Chapter 20. Symmetries of Riemannian Manifolds.
- Chapter 21. Einstein's Gravitation.
- Chapter 22. Lagrangian Formulation.
- Chapter 23. Conservation Laws and Further Symmetries.
- - Appendices
- A) Notation and Conventions.
- Physical Units and Dimensions.
- Mathematical Conventions.
- Abbreviations.
- B) Mathematical Tools.
- Tensor Algebra.
- Matrix Exponential.
- Pauli and Dirac Matrices.
- Dirac Delta Distribution.
- Poisson and Wave Equation.
- Variational Calculus.
- Volume Element and Hyperspheres.
- Hypersurface Elements.
- C) Weyl Rescaling Formulae.
- D) Spaces and Symmetry Groups.
In this chapter we proceed to the next step in the generalization of spacetime symmetries and dis... more In this chapter we proceed to the next step in the generalization of spacetime symmetries and discuss conformal symmetry. We consider conformal transformations that either act on Minkowski space or act as structure-preserving transformations on conformal space. We systematically derive the conformal group and the single transformations of which it is comprised. Two new types of transformations appear, the dilatations and the special conformal transformations. Then we move on to the derivation of the Lie algebra for tensor fields in complete generality. We introduce the notion of primary fields and their relation to tensor densities. A special feature of the special conformal transformations is their nonlinearity. The conformal group can act linearly if we employ a representation on a six-dimensional pseudo-Euclidean space or a representation on a four-complex-dimensional space.
Galilei symmetry is the symmetry of nonrelativistic spacetime. It corresponds to the maximally ad... more Galilei symmetry is the symmetry of nonrelativistic spacetime. It corresponds to the maximally admissible set of transformations between Galileian inertial reference frames. Beyond this global definition, it is enlightening to construct the Galilei group from the ground up. To this end, one starts with the introduction of a time parameter and the group of velocity boosts. Then one moves on to the construction of the homogeneous Galilei group as the semidirect product of velocity boosts and rotations. In a final step, one reaches to the full Galilei group by constructing the semidirect product of translations with the homogeneous Galilei group. In the last section of this chapter the Galilei algebra is derived.
This chapter introduces the most basic facts about the Lagrangian formalism. It formulates classi... more This chapter introduces the most basic facts about the Lagrangian formalism. It formulates classical mechanics by starting from Hamilton's global variational principle, which employs the action of the system. This approach leads to the Euler-Lagrange equations of motion and their specific invariances. The Lagrange formulation of mechanics is particularly well-suited for treating questions concerning symmetries and conservation. The concepts of the Lagrangian formalism and the Noether theorem can be nicely extended to relativistic mechanics and to the framework of classical field theory. Noether's theorem provides the framework in which one can relate conserved quantities to corresponding symmetries of the theory.
In this chapter we introduce and develop field theory as a powerful framework for building relati... more In this chapter we introduce and develop field theory as a powerful framework for building relativistically invariant theories. The focus is entirely on Lagrangian methods. We start with Hamilton’s principle of stationary action and derive the Euler-Lagrange equations as the dynamical field equations. We illustrate the methods by providing basic examples of scalar, spinor, and vector fields, with the Maxwell field representing the most important classical case. Then we turn to the formulation of symmetry in a relativistic field theory. This is accomplished by the famous Noether theorem, which we derive and discuss.
This chapter introduces the metric as an additional structure, whose central role is to define th... more This chapter introduces the metric as an additional structure, whose central role is to define the geometry of a manifold. It considers the conditions of isometry and conformality between different geometric manifolds. The chapter provides several examples of classical geometries that are relevant for later purposes. It develops the rudiments of integration on manifolds. The chapter introduces differential forms and the exterior derivative so that it can define integration measures for curves, hypersurfaces, and volumes in a unified way. It discusses the integral theorem of Stokes, the Gauss divergence theorem, and a special version of Stokes' theorem for antisymmetric tensor fields.
Given a Lie group or algebra, a representation of it yields objects that satisfy that symmetry fr... more Given a Lie group or algebra, a representation of it yields objects that satisfy that symmetry from the outset. In this sense, a representation makes a symmetry more concrete. The adjoint representation has a dimension equal to the dimension of the Lie group and Lie algebra. For our applications in physics, people are interested in tensor fields that display a specific transformation behavior under basic symmetry transformations of interest. Technically, the tensor fields are viewed as functions of discrete tensorial indices and continuous spacetime indices. In the last section isometry transformations and conformal transformations are considered, and their Lie algebras of Killing vector fields are derived.
To grasp the notion of symmetry and explore its possible realizations, we begin with abstract gro... more To grasp the notion of symmetry and explore its possible realizations, we begin with abstract group theory. We consider the basic definitions of groups, subgroups, and the con- struction of larger groups from existing ones. The next central notion is that of a group representation, which provides a concrete realization of the group structure at hand. Subsequently, we specialize to Lie groups and identify symmetry transformations as the elements of these Lie groups. Initially, we introduce Lie groups in a general manner, but in the follo- wing we restrict ourselves to matrix Lie groups. The reason for this focus is twofold: first, matrix Lie groups provide the primary cases for our applications, and second, the central results of Lie theory for this class of groups can be obtained in the simplest way.
In this chapter we discuss the notion of symmetric space, as a manifold with geometric symmetries... more In this chapter we discuss the notion of symmetric space, as a manifold with geometric symmetries expressed through the existence of Killing vector fields. We derive the maximum possible number of distinct Killing vector fields that can exist in a geometric manifold and provide examples. Then we move to a different question and consider Weyl rescalings of the metric and the corresponding Weyl group acting on a manifold. We investigate under which conditions a geometric manifold is conformally flat. The answer is given by the classical Weyl-Schouten theorem, whose complete proof is given. Finally, we discuss the algebraic structure of the set of general coordinate transformations of a given manifold forming the infinite-dimensional group of diffeomorphisms.
In this chapter we turn back to differential geometry and introduce the notions of connection and... more In this chapter we turn back to differential geometry and introduce the notions of connection and covariant derivative on a general manifold. These additional structures allow us to formulate tensor field equations on a manifold in an invariant way. There is a natural connection on every geometric manifold, the Levi-Civita connection, which we develop in detail. In addition, the integral formulae of Gauss and Stokes are revisited for the case of geometric manifolds. In the last section we consider the notion of parallel transport of a vector along a curve. It is particularly fruitful to consider the parallel transport of the tangent vector of a curve along that curve. This leads to geodesics and a generalized concept of minimal distance between points in a geometric manifold.
A Lie algebra is derived naturally from a Lie group as the tangent space of the group manifold at... more A Lie algebra is derived naturally from a Lie group as the tangent space of the group manifold at the group identity. A Lie algebra describes group elements near the identity and encodes the essential properties of the corresponding Lie group. The great advantage of Lie algebras is that they are much easier to handle than Lie groups. The exponential acts as the mediator between a Lie algebra and the associated Lie group. The Baker-Campbell-Hausdorff formula is derived and discussed. Lie algebras can be introduced abstractly as vector spaces with a commutator product, and the connection between Lie algebras and Lie groups can be established separately.
Minkowski space is an affine-linear space, which means in particular that the origin of inertial ... more Minkowski space is an affine-linear space, which means in particular that the origin of inertial reference frames can be shifted. Therefore, in addition to the Lorentz transformations considered so far, we must now consider spacetime translations. These two transformations are combined to the Poincaré transformations, which are the maximally allowed symmetry transformations of Minkowski space. The importance of Poincaré transformations arises from the fact that we require classical physical laws to have the same form in all inertial reference frames. Equivalently, we require that the physical laws be invariant under Poincaré transformations. In this chapter we study the foundations of the Poincaré group and its Lie algebra.
Lorentz transformations contain the usual space rotations and Lorentz boosts and intertwine them ... more Lorentz transformations contain the usual space rotations and Lorentz boosts and intertwine them into a unified relativistic formalism. In this chapter, we derive the Lorentz transformations group-theoretically and study the corresponding Lie group. Our focus is on the restricted Lorentz group which excludes space and time inversions. For both the Lorentz group and the Lorentz algebra, we derive the existence of spinors. We then derive the finite-dimensional tensorial representations for scalars, vectors, and tensors, as well as the representations for Weyl and Dirac spinors. Finally, we derive the infinite-dimensional orbital representations where the Lorentz group acts on functions of spacetime coordinates.
This chapter addresses the question of how one can meaningfully define a notion of symmetry for g... more This chapter addresses the question of how one can meaningfully define a notion of symmetry for geometric manifolds. It considers general diffeomorphisms defined by vector fields and study how arbitrary tensor fields transform. This leads and to the central notion of the Lie derivative. The symmetry transformations of a geometric manifold are identified with the isometries of the manifold. In many cases the symmetry transformations are as characteristic as the defining properties of the space. One can reverse the procedure then and actually define the space based on its invariance under certain transformations. It is important to compare conformal transformations with Weyl rescalings. The Killing vector fields of a given geometric manifold M constitute a real Lie algebra, with respect to the commutator of vector fields.
In this chapter we construct locally and globally conserved quantities based on spacetime symmetr... more In this chapter we construct locally and globally conserved quantities based on spacetime symmetries of general relativity. The existence of Killing vector fields ensures that one can define conserved quantities from geodesics and from the energy-momentum tensor of matter fields. We discuss the notion of energy of the gravitational field and the challenges this conception poses. By employing the Noether currents of diffeomorphisms generated by suitable Killing vectors, we are able to define the so-called Komar integrals, which can measure the total energy and angular momentum of a spacetime. In the final part, we examine how Weyl rescalings affect diverse physical theories and under which conditions Weyl rescaling invariance can be achieved.
This book provides advanced physics and mathematics students with an accessible yet detailed unde... more This book provides advanced physics and mathematics students with an accessible yet detailed understanding of the fundamentals of differential geometry and symmetries in classical physics. Most of the topics covered in this book have previously only been scattered across many different sources of literature, therefore this is the first book to coherently present this treatment of topics in one comprehensive volume.
- - Key features:
-> Contains a modern, streamlined presentation of classical topics, which are normally taught separately.
-> Includes several advanced topics, such as the Belinfante energy-momentum tensor, the Weyl-Schouten theorem, the derivation of Noether currents for diffeomorphisms, and the definition of conserved integrals in general relativity.
-> Focuses on the clear presentation of the mathematical notions and calculational technique.
- - - Table of Contents:
- Chapter 1. Manifolds and Tensors.
- Chapter 2. Geometry and Integration on Manifolds.
- Chapter 3. Symmetries of Manifolds.
- Chapter 4. Newtonian Mechanics.
- Chapter 5. Lagrangian Methods and Symmetry.
- Chapter 6. Relativistic Mechanics.
- Chapter 7. Lie Groups.
- Chapter 8. Lie Algebras.
- Chapter 9. Representations.
- Chapter 10. Rotations and Euclidean Symmetry.
- Chapter 11. Boosts and Galilei Symmetry.
- Chapter 12. Lorentz Symmetry.
- Chapter 13. Poincare Symmetry.
- Chapter 14. Conformal Symmetry.
- Chapter 15. Lagrangians and Noether's Theorem.
- Chapter 16. Spacetime Symmetries of Fields.
- Chapter 17. Gauge Symmetry.
- Chapter 18. Connection and Geodesics.
- Chapter 19. Riemannian Curvature.
- Chapter 20. Symmetries of Riemannian Manifolds.
- Chapter 21. Einstein's Gravitation.
- Chapter 22. Lagrangian Formulation.
- Chapter 23. Conservation Laws and Further Symmetries.
- - Appendices
- A) Notation and Conventions.
- Physical Units and Dimensions.
- Mathematical Conventions.
- Abbreviations.
- B) Mathematical Tools.
- Tensor Algebra.
- Matrix Exponential.
- Pauli and Dirac Matrices.
- Dirac Delta Distribution.
- Poisson and Wave Equation.
- Variational Calculus.
- Volume Element and Hyperspheres.
- Hypersurface Elements.
- C) Weyl Rescaling Formulae.
- D) Spaces and Symmetry Groups.
The framework of field theory allows the definition of a class of transformations other than spac... more The framework of field theory allows the definition of a class of transformations other than spacetime transformations. These are the internal transformations which, due to Noether’s theorem, also lead to conserved currents and charges. We first discuss global phase transformations and subsequently the more general local phase transformations of fields, also called gauge transformations. Gauge symmetry provides a powerful guiding principle in order to define the interaction between different types of fields. The gauge symmetry leads to a coupling prescription that fully determines the interaction between fields. The Maxwell electromagnetic field interacting with a matter field represents the prime example. In this chapter, we discuss the resulting Lagrangian densities, the equations of motion, and the associated conserved currents and energy momentum tensors.
Galilei symmetry is the symmetry of nonrelativistic spacetime. It corresponds to the maximally ad... more Galilei symmetry is the symmetry of nonrelativistic spacetime. It corresponds to the maximally admissible set of transformations between Galileian inertial reference frames. Beyond this global definition, it is enlightening to construct the Galilei group from the ground up. To this end, one starts with the introduction of a time parameter and the group of velocity boosts. Then one moves on to the construction of the homogeneous Galilei group as the semidirect product of velocity boosts and rotations. In a final step, one reaches to the full Galilei group by constructing the semidirect product of translations with the homogeneous Galilei group. In the last section of this chaoter the Galilei algebra is derived.
In this chapter we study the consequences of Noether’s theorem in field theory for the case of pu... more In this chapter we study the consequences of Noether’s theorem in field theory for the case of pure spacetime transformations. The symmetry groups considered are the Poincaré group and the conformal group. In this context, the energy-momentum tensor turns out to be the central quantity to focus on, since it allows the definition of all conserved quantities under the conformal group. After a detailed description of the Belinfante prescription for the construction of a symmetric energy-momentum tensor, numerous concrete examples from relevant field theories are given. In the last section, we discuss the meaning and the necessary conditions for achieving conformal symmetry in a field theory.
In this chapter we construct locally and globally conserved quantities based on spacetime symmetr... more In this chapter we construct locally and globally conserved quantities based on spacetime symmetries of general relativity. The existence of Killing vector fields ensures that one can define conserved quantities from geodesics and from the energy-momentum tensor of matter fields. We discuss the notion of energy of the gravitational field and the challenges this conception poses. By employing the Noether currents of diffeomorphisms generated by suitable Killing vectors, we are able to define the so-called Komar integrals, which can measure the total energy and angular momentum of a spacetime. In the final part, we examine how Weyl rescalings affect diverse physical theories and under which conditions Weyl rescaling invariance can be achieved.
Geometry, Symmetries, and Classical Physics: A Mosaic, Dec 29, 2021
This book provides advanced physics and mathematics students with an accessible yet detailed unde... more This book provides advanced physics and mathematics students with an accessible yet detailed understanding of the fundamentals of differential geometry and symmetries in classical physics. Most of the topics covered in this book have previously only been scattered across many different sources of literature, therefore this is the first book to coherently present this treatment of topics in one comprehensive volume.
- - Key features:
-> Contains a modern, streamlined presentation of classical topics, which are normally taught separately.
-> Includes several advanced topics, such as the Belinfante energy-momentum tensor, the Weyl-Schouten theorem, the derivation of Noether currents for diffeomorphisms, and the definition of conserved integrals in general relativity.
-> Focuses on the clear presentation of the mathematical notions and calculational technique.
- - - Table of Contents:
- Chapter 1. Manifolds and Tensors.
- Chapter 2. Geometry and Integration on Manifolds.
- Chapter 3. Symmetries of Manifolds.
- Chapter 4. Newtonian Mechanics.
- Chapter 5. Lagrangian Methods and Symmetry.
- Chapter 6. Relativistic Mechanics.
- Chapter 7. Lie Groups.
- Chapter 8. Lie Algebras.
- Chapter 9. Representations.
- Chapter 10. Rotations and Euclidean Symmetry.
- Chapter 11. Boosts and Galilei Symmetry.
- Chapter 12. Lorentz Symmetry.
- Chapter 13. Poincare Symmetry.
- Chapter 14. Conformal Symmetry.
- Chapter 15. Lagrangians and Noether's Theorem.
- Chapter 16. Spacetime Symmetries of Fields.
- Chapter 17. Gauge Symmetry.
- Chapter 18. Connection and Geodesics.
- Chapter 19. Riemannian Curvature.
- Chapter 20. Symmetries of Riemannian Manifolds.
- Chapter 21. Einstein's Gravitation.
- Chapter 22. Lagrangian Formulation.
- Chapter 23. Conservation Laws and Further Symmetries.
- - Appendices
- A) Notation and Conventions.
- Physical Units and Dimensions.
- Mathematical Conventions.
- Abbreviations.
- B) Mathematical Tools.
- Tensor Algebra.
- Matrix Exponential.
- Pauli and Dirac Matrices.
- Dirac Delta Distribution.
- Poisson and Wave Equation.
- Variational Calculus.
- Volume Element and Hyperspheres.
- Hypersurface Elements.
- C) Weyl Rescaling Formulae.
- D) Spaces and Symmetry Groups.
In this chapter we proceed to the next step in the generalization of spacetime symmetries and dis... more In this chapter we proceed to the next step in the generalization of spacetime symmetries and discuss conformal symmetry. We consider conformal transformations that either act on Minkowski space or act as structure-preserving transformations on conformal space. We systematically derive the conformal group and the single transformations of which it is comprised. Two new types of transformations appear, the dilatations and the special conformal transformations. Then we move on to the derivation of the Lie algebra for tensor fields in complete generality. We introduce the notion of primary fields and their relation to tensor densities. A special feature of the special conformal transformations is their nonlinearity. The conformal group can act linearly if we employ a representation on a six-dimensional pseudo-Euclidean space or a representation on a four-complex-dimensional space.
Galilei symmetry is the symmetry of nonrelativistic spacetime. It corresponds to the maximally ad... more Galilei symmetry is the symmetry of nonrelativistic spacetime. It corresponds to the maximally admissible set of transformations between Galileian inertial reference frames. Beyond this global definition, it is enlightening to construct the Galilei group from the ground up. To this end, one starts with the introduction of a time parameter and the group of velocity boosts. Then one moves on to the construction of the homogeneous Galilei group as the semidirect product of velocity boosts and rotations. In a final step, one reaches to the full Galilei group by constructing the semidirect product of translations with the homogeneous Galilei group. In the last section of this chapter the Galilei algebra is derived.
This chapter introduces the most basic facts about the Lagrangian formalism. It formulates classi... more This chapter introduces the most basic facts about the Lagrangian formalism. It formulates classical mechanics by starting from Hamilton's global variational principle, which employs the action of the system. This approach leads to the Euler-Lagrange equations of motion and their specific invariances. The Lagrange formulation of mechanics is particularly well-suited for treating questions concerning symmetries and conservation. The concepts of the Lagrangian formalism and the Noether theorem can be nicely extended to relativistic mechanics and to the framework of classical field theory. Noether's theorem provides the framework in which one can relate conserved quantities to corresponding symmetries of the theory.
In this chapter we introduce and develop field theory as a powerful framework for building relati... more In this chapter we introduce and develop field theory as a powerful framework for building relativistically invariant theories. The focus is entirely on Lagrangian methods. We start with Hamilton’s principle of stationary action and derive the Euler-Lagrange equations as the dynamical field equations. We illustrate the methods by providing basic examples of scalar, spinor, and vector fields, with the Maxwell field representing the most important classical case. Then we turn to the formulation of symmetry in a relativistic field theory. This is accomplished by the famous Noether theorem, which we derive and discuss.
This chapter introduces the metric as an additional structure, whose central role is to define th... more This chapter introduces the metric as an additional structure, whose central role is to define the geometry of a manifold. It considers the conditions of isometry and conformality between different geometric manifolds. The chapter provides several examples of classical geometries that are relevant for later purposes. It develops the rudiments of integration on manifolds. The chapter introduces differential forms and the exterior derivative so that it can define integration measures for curves, hypersurfaces, and volumes in a unified way. It discusses the integral theorem of Stokes, the Gauss divergence theorem, and a special version of Stokes' theorem for antisymmetric tensor fields.
Given a Lie group or algebra, a representation of it yields objects that satisfy that symmetry fr... more Given a Lie group or algebra, a representation of it yields objects that satisfy that symmetry from the outset. In this sense, a representation makes a symmetry more concrete. The adjoint representation has a dimension equal to the dimension of the Lie group and Lie algebra. For our applications in physics, people are interested in tensor fields that display a specific transformation behavior under basic symmetry transformations of interest. Technically, the tensor fields are viewed as functions of discrete tensorial indices and continuous spacetime indices. In the last section isometry transformations and conformal transformations are considered, and their Lie algebras of Killing vector fields are derived.
To grasp the notion of symmetry and explore its possible realizations, we begin with abstract gro... more To grasp the notion of symmetry and explore its possible realizations, we begin with abstract group theory. We consider the basic definitions of groups, subgroups, and the con- struction of larger groups from existing ones. The next central notion is that of a group representation, which provides a concrete realization of the group structure at hand. Subsequently, we specialize to Lie groups and identify symmetry transformations as the elements of these Lie groups. Initially, we introduce Lie groups in a general manner, but in the follo- wing we restrict ourselves to matrix Lie groups. The reason for this focus is twofold: first, matrix Lie groups provide the primary cases for our applications, and second, the central results of Lie theory for this class of groups can be obtained in the simplest way.
In this chapter we discuss the notion of symmetric space, as a manifold with geometric symmetries... more In this chapter we discuss the notion of symmetric space, as a manifold with geometric symmetries expressed through the existence of Killing vector fields. We derive the maximum possible number of distinct Killing vector fields that can exist in a geometric manifold and provide examples. Then we move to a different question and consider Weyl rescalings of the metric and the corresponding Weyl group acting on a manifold. We investigate under which conditions a geometric manifold is conformally flat. The answer is given by the classical Weyl-Schouten theorem, whose complete proof is given. Finally, we discuss the algebraic structure of the set of general coordinate transformations of a given manifold forming the infinite-dimensional group of diffeomorphisms.
In this chapter we turn back to differential geometry and introduce the notions of connection and... more In this chapter we turn back to differential geometry and introduce the notions of connection and covariant derivative on a general manifold. These additional structures allow us to formulate tensor field equations on a manifold in an invariant way. There is a natural connection on every geometric manifold, the Levi-Civita connection, which we develop in detail. In addition, the integral formulae of Gauss and Stokes are revisited for the case of geometric manifolds. In the last section we consider the notion of parallel transport of a vector along a curve. It is particularly fruitful to consider the parallel transport of the tangent vector of a curve along that curve. This leads to geodesics and a generalized concept of minimal distance between points in a geometric manifold.
A Lie algebra is derived naturally from a Lie group as the tangent space of the group manifold at... more A Lie algebra is derived naturally from a Lie group as the tangent space of the group manifold at the group identity. A Lie algebra describes group elements near the identity and encodes the essential properties of the corresponding Lie group. The great advantage of Lie algebras is that they are much easier to handle than Lie groups. The exponential acts as the mediator between a Lie algebra and the associated Lie group. The Baker-Campbell-Hausdorff formula is derived and discussed. Lie algebras can be introduced abstractly as vector spaces with a commutator product, and the connection between Lie algebras and Lie groups can be established separately.
Minkowski space is an affine-linear space, which means in particular that the origin of inertial ... more Minkowski space is an affine-linear space, which means in particular that the origin of inertial reference frames can be shifted. Therefore, in addition to the Lorentz transformations considered so far, we must now consider spacetime translations. These two transformations are combined to the Poincaré transformations, which are the maximally allowed symmetry transformations of Minkowski space. The importance of Poincaré transformations arises from the fact that we require classical physical laws to have the same form in all inertial reference frames. Equivalently, we require that the physical laws be invariant under Poincaré transformations. In this chapter we study the foundations of the Poincaré group and its Lie algebra.
Lorentz transformations contain the usual space rotations and Lorentz boosts and intertwine them ... more Lorentz transformations contain the usual space rotations and Lorentz boosts and intertwine them into a unified relativistic formalism. In this chapter, we derive the Lorentz transformations group-theoretically and study the corresponding Lie group. Our focus is on the restricted Lorentz group which excludes space and time inversions. For both the Lorentz group and the Lorentz algebra, we derive the existence of spinors. We then derive the finite-dimensional tensorial representations for scalars, vectors, and tensors, as well as the representations for Weyl and Dirac spinors. Finally, we derive the infinite-dimensional orbital representations where the Lorentz group acts on functions of spacetime coordinates.
This chapter addresses the question of how one can meaningfully define a notion of symmetry for g... more This chapter addresses the question of how one can meaningfully define a notion of symmetry for geometric manifolds. It considers general diffeomorphisms defined by vector fields and study how arbitrary tensor fields transform. This leads and to the central notion of the Lie derivative. The symmetry transformations of a geometric manifold are identified with the isometries of the manifold. In many cases the symmetry transformations are as characteristic as the defining properties of the space. One can reverse the procedure then and actually define the space based on its invariance under certain transformations. It is important to compare conformal transformations with Weyl rescalings. The Killing vector fields of a given geometric manifold M constitute a real Lie algebra, with respect to the commutator of vector fields.
In this chapter we construct locally and globally conserved quantities based on spacetime symmetr... more In this chapter we construct locally and globally conserved quantities based on spacetime symmetries of general relativity. The existence of Killing vector fields ensures that one can define conserved quantities from geodesics and from the energy-momentum tensor of matter fields. We discuss the notion of energy of the gravitational field and the challenges this conception poses. By employing the Noether currents of diffeomorphisms generated by suitable Killing vectors, we are able to define the so-called Komar integrals, which can measure the total energy and angular momentum of a spacetime. In the final part, we examine how Weyl rescalings affect diverse physical theories and under which conditions Weyl rescaling invariance can be achieved.
Uploads
Papers by Manousos Markoutsakis
- - Key features:
-> Contains a modern, streamlined presentation of classical topics, which are normally taught separately.
-> Includes several advanced topics, such as the Belinfante energy-momentum tensor, the Weyl-Schouten theorem, the derivation of Noether currents for diffeomorphisms, and the definition of conserved integrals in general relativity.
-> Focuses on the clear presentation of the mathematical notions and calculational technique.
- - - Table of Contents:
- Chapter 1. Manifolds and Tensors.
- Chapter 2. Geometry and Integration on Manifolds.
- Chapter 3. Symmetries of Manifolds.
- Chapter 4. Newtonian Mechanics.
- Chapter 5. Lagrangian Methods and Symmetry.
- Chapter 6. Relativistic Mechanics.
- Chapter 7. Lie Groups.
- Chapter 8. Lie Algebras.
- Chapter 9. Representations.
- Chapter 10. Rotations and Euclidean Symmetry.
- Chapter 11. Boosts and Galilei Symmetry.
- Chapter 12. Lorentz Symmetry.
- Chapter 13. Poincare Symmetry.
- Chapter 14. Conformal Symmetry.
- Chapter 15. Lagrangians and Noether's Theorem.
- Chapter 16. Spacetime Symmetries of Fields.
- Chapter 17. Gauge Symmetry.
- Chapter 18. Connection and Geodesics.
- Chapter 19. Riemannian Curvature.
- Chapter 20. Symmetries of Riemannian Manifolds.
- Chapter 21. Einstein's Gravitation.
- Chapter 22. Lagrangian Formulation.
- Chapter 23. Conservation Laws and Further Symmetries.
- - Appendices
- A) Notation and Conventions.
- Physical Units and Dimensions.
- Mathematical Conventions.
- Abbreviations.
- B) Mathematical Tools.
- Tensor Algebra.
- Matrix Exponential.
- Pauli and Dirac Matrices.
- Dirac Delta Distribution.
- Poisson and Wave Equation.
- Variational Calculus.
- Volume Element and Hyperspheres.
- Hypersurface Elements.
- C) Weyl Rescaling Formulae.
- D) Spaces and Symmetry Groups.
- - Bibliography.
- - Index.
- - Key features:
-> Contains a modern, streamlined presentation of classical topics, which are normally taught separately.
-> Includes several advanced topics, such as the Belinfante energy-momentum tensor, the Weyl-Schouten theorem, the derivation of Noether currents for diffeomorphisms, and the definition of conserved integrals in general relativity.
-> Focuses on the clear presentation of the mathematical notions and calculational technique.
- - - Table of Contents:
- Chapter 1. Manifolds and Tensors.
- Chapter 2. Geometry and Integration on Manifolds.
- Chapter 3. Symmetries of Manifolds.
- Chapter 4. Newtonian Mechanics.
- Chapter 5. Lagrangian Methods and Symmetry.
- Chapter 6. Relativistic Mechanics.
- Chapter 7. Lie Groups.
- Chapter 8. Lie Algebras.
- Chapter 9. Representations.
- Chapter 10. Rotations and Euclidean Symmetry.
- Chapter 11. Boosts and Galilei Symmetry.
- Chapter 12. Lorentz Symmetry.
- Chapter 13. Poincare Symmetry.
- Chapter 14. Conformal Symmetry.
- Chapter 15. Lagrangians and Noether's Theorem.
- Chapter 16. Spacetime Symmetries of Fields.
- Chapter 17. Gauge Symmetry.
- Chapter 18. Connection and Geodesics.
- Chapter 19. Riemannian Curvature.
- Chapter 20. Symmetries of Riemannian Manifolds.
- Chapter 21. Einstein's Gravitation.
- Chapter 22. Lagrangian Formulation.
- Chapter 23. Conservation Laws and Further Symmetries.
- - Appendices
- A) Notation and Conventions.
- Physical Units and Dimensions.
- Mathematical Conventions.
- Abbreviations.
- B) Mathematical Tools.
- Tensor Algebra.
- Matrix Exponential.
- Pauli and Dirac Matrices.
- Dirac Delta Distribution.
- Poisson and Wave Equation.
- Variational Calculus.
- Volume Element and Hyperspheres.
- Hypersurface Elements.
- C) Weyl Rescaling Formulae.
- D) Spaces and Symmetry Groups.
- - Bibliography.
- - Index.
- - Key features:
-> Contains a modern, streamlined presentation of classical topics, which are normally taught separately.
-> Includes several advanced topics, such as the Belinfante energy-momentum tensor, the Weyl-Schouten theorem, the derivation of Noether currents for diffeomorphisms, and the definition of conserved integrals in general relativity.
-> Focuses on the clear presentation of the mathematical notions and calculational technique.
- - - Table of Contents:
- Chapter 1. Manifolds and Tensors.
- Chapter 2. Geometry and Integration on Manifolds.
- Chapter 3. Symmetries of Manifolds.
- Chapter 4. Newtonian Mechanics.
- Chapter 5. Lagrangian Methods and Symmetry.
- Chapter 6. Relativistic Mechanics.
- Chapter 7. Lie Groups.
- Chapter 8. Lie Algebras.
- Chapter 9. Representations.
- Chapter 10. Rotations and Euclidean Symmetry.
- Chapter 11. Boosts and Galilei Symmetry.
- Chapter 12. Lorentz Symmetry.
- Chapter 13. Poincare Symmetry.
- Chapter 14. Conformal Symmetry.
- Chapter 15. Lagrangians and Noether's Theorem.
- Chapter 16. Spacetime Symmetries of Fields.
- Chapter 17. Gauge Symmetry.
- Chapter 18. Connection and Geodesics.
- Chapter 19. Riemannian Curvature.
- Chapter 20. Symmetries of Riemannian Manifolds.
- Chapter 21. Einstein's Gravitation.
- Chapter 22. Lagrangian Formulation.
- Chapter 23. Conservation Laws and Further Symmetries.
- - Appendices
- A) Notation and Conventions.
- Physical Units and Dimensions.
- Mathematical Conventions.
- Abbreviations.
- B) Mathematical Tools.
- Tensor Algebra.
- Matrix Exponential.
- Pauli and Dirac Matrices.
- Dirac Delta Distribution.
- Poisson and Wave Equation.
- Variational Calculus.
- Volume Element and Hyperspheres.
- Hypersurface Elements.
- C) Weyl Rescaling Formulae.
- D) Spaces and Symmetry Groups.
- - Bibliography.
- - Index.
- - Key features:
-> Contains a modern, streamlined presentation of classical topics, which are normally taught separately.
-> Includes several advanced topics, such as the Belinfante energy-momentum tensor, the Weyl-Schouten theorem, the derivation of Noether currents for diffeomorphisms, and the definition of conserved integrals in general relativity.
-> Focuses on the clear presentation of the mathematical notions and calculational technique.
- - - Table of Contents:
- Chapter 1. Manifolds and Tensors.
- Chapter 2. Geometry and Integration on Manifolds.
- Chapter 3. Symmetries of Manifolds.
- Chapter 4. Newtonian Mechanics.
- Chapter 5. Lagrangian Methods and Symmetry.
- Chapter 6. Relativistic Mechanics.
- Chapter 7. Lie Groups.
- Chapter 8. Lie Algebras.
- Chapter 9. Representations.
- Chapter 10. Rotations and Euclidean Symmetry.
- Chapter 11. Boosts and Galilei Symmetry.
- Chapter 12. Lorentz Symmetry.
- Chapter 13. Poincare Symmetry.
- Chapter 14. Conformal Symmetry.
- Chapter 15. Lagrangians and Noether's Theorem.
- Chapter 16. Spacetime Symmetries of Fields.
- Chapter 17. Gauge Symmetry.
- Chapter 18. Connection and Geodesics.
- Chapter 19. Riemannian Curvature.
- Chapter 20. Symmetries of Riemannian Manifolds.
- Chapter 21. Einstein's Gravitation.
- Chapter 22. Lagrangian Formulation.
- Chapter 23. Conservation Laws and Further Symmetries.
- - Appendices
- A) Notation and Conventions.
- Physical Units and Dimensions.
- Mathematical Conventions.
- Abbreviations.
- B) Mathematical Tools.
- Tensor Algebra.
- Matrix Exponential.
- Pauli and Dirac Matrices.
- Dirac Delta Distribution.
- Poisson and Wave Equation.
- Variational Calculus.
- Volume Element and Hyperspheres.
- Hypersurface Elements.
- C) Weyl Rescaling Formulae.
- D) Spaces and Symmetry Groups.
- - Bibliography.
- - Index.