PROGRAMMA SVOLTO DEL CORSO DI MATHEMATICAL PHYSICS
(MORETTI 2016)

dalle dispense: Mathematical Foundations of Quantum Mechanics: An Advanced Short Course  (unitamente al libro Spectral Theory and Quantum Mechanics per alcuni richiami e dettagli)

1 Introduction: Summary of elementary facts of QM
1.1 Physical facts about Quantum Mechanics
1.1.1 When a physical system is quantum
1.1.2 General properties of quantum systems
1.2 Elementary formalism for the nite dimensional case
1.3 Time evolution
1.4 Composite systems
1.5 A first look to the infinite dimensional case, CCR and quantization procedures

2 Observables in infinite dimensional Hilbert spaces: Spectral Theory
2.1 Classes of (especially unbounded) operators in Hilbert spaces
2.2 Spectrum of an operator
2.3 Spectral measures
2.4 Spectral Decomposition and Representation Theorems
2.5 Mesurable functional calculus
2.6 Elementary formalism for the in nite dimensional case
2.7 Technical Interemezzo: Three Operator Topologies

3 More Fundamental Quantum Structures
3.1 The Boolean logic of CM
3.2 The non-Boolean Logic of QM, the reason why observables are self-adjoint operators
3.3 Recovering the Hilbert space structure
3.4 States as measures on L(H): Gleason's Theorem
3.4.1 Trace class operators
3.4.2 The notion of quantum state and the crucial theorem by Gleason

3.6 Quantum Symmetries: unitary projective representations
3.6.1 Wigner and Kadison theorems, groups of symmetries
3.6.2 Groups of quantum symmetries
3.6.3 One-parameter strongly continuous unitary groups: von Neumann and Stone theorems
3.6.4 Time evolution, Heisenberg picture and quantum Noether theorem


dalle dispense: Multi-Linear Algebra, Tensors and Spinors in Mathematical Physics
1 Introduction and some useful notions and results

2 Multi-linear Mappings and Tensors
2.1 Dual space and conjugate space, pairing, adjoint operator
2.2 Multi linearity: tensor product, tensors, universality theorem
2.2.1Tensors as multi linear maps.
2.2.2 Universality theorem and its applications
2.2.3 Abstract de nition of the tensor product of vector spaces
2.2.4 Hilbertian tensor product of Hilbert spaces (solo accenno del risultato)

3 Tensor algebra, abstract index notation and some applications
3.1 Tensor algebra generated by a vector space
3.2 The abstract index notation and rules to handle tensors
3.2.1 Canonical bases and abstract index notation
3.2.2 Rules to compose, decompose, produce tensors form tensors
3.2.3 Linear transformations of tensors are tensors too
3.3 Physical invariance of the form of laws and tensors
3.4 Tensors on Affine and Euclidean spaces.
3.4.1 Tensor spaces, Cartesian tensors
3.4.2 Applied tensors
3.4.3 The natural isomorphism between V and V* for Cartesian vectors in Euclidean spaces

4 Some application to group theory
4.1 Some notions about Groups
4.1.1 Basic notions on groups and matrix groups
4.1.2 Direct product and semi-direct product of groups
4.2 Tensor products of group representations
4.2.1 Linear representation of groups and tensor representations
4.2.2 An example from Continuum Mechanics
4.2.3 An example from Quantum Mechanics.
4.3 Permutation group and symmetry of tensors
4.4 Grassmann algebra, also known as Exterior algebra (and something about the space of symmetric tensors)
4.4.1 The exterior product and Grassmann algebra  (solo definizione)

5 Scalar Products and Metric Tools
5.1 Scalar products
5.2 Natural isomorphism between V and V* and metric tensor
5.2.1 Signature of pseudo-metric tensor, pseudo-orthonormal bases and pseudo-orthonormal groups
5.2.2 Raising and lowering indices of tensors

8 Minkowski spacetime and Poincaré-Lorentz group
8.1 Minkowski spacetime
8.1.1 General Minkowski spacetime structure.
8.1.2 Time orientation
8.1.3 Curves as stories
8.1.4 Minkowskian reference frames, time dilatation and contraction of length
8.2 Lorentz and Poincaré groups
8.2.1 Lorentz group
8.2.2 Poincaré group

dalle dispense: Tensor Analysis on Manifolds in Mathematical Physics with Applications to Relativistic Theories

1 Basics on differential geometry: topological and di erentiable manifolds
1.1 Basics of general topology
1.1.1 The topology of R^n
1.1.2 Topological Manifolds
1.2 Differentiable Manifolds
1.2.1 Local charts and atlas
1.2.2 Differentiable structures
1.2.3 Differentiable functions and diffeomorphisms

2 Tensor Fields in Manifolds and Associated Geometric Structures
2.1 Tangent and cotangent space in a point
2.1.1 Vectors as derivations
2.1.2 Cotangent space
2.2 Tensor fields
2.2.1 Lie brackets
2.3 Tangent and cotangent space manifolds (solo accenno)

3 Differential mapping and Submanifolds
3.1 Push forward
3.2 Rank of a di erentiable map: immersions and submersions
3.3 Submanifolds
3.3.1 Theorem of regular values (solo eneunciato)

4 Riemannian and pseudo Riemannian manifolds
4.1 Local and global flatness
4.3 Induced metric and isometries

5 Covariant Derivative. Levi-Civita's Connection
5.1 Affine connections and covariant derivatives
5.1.1 Affine connections
5.1.2 Connection coefficients
5.1.3 Transformation rule of the connection coefficients
5.1.4 Torsion tensor
5.1.5 Assignment of a connection
5.2 Covariant derivative of tensor field
5.3 Levi-Civita's connection
5.4 Geodesics: parallel transport approach
5.4.1 Parallel transport and geodesics
5.4.2 Back on the meaning of the covariant derivative
5.5 Geodesics: variational approach (solo accenno)

6 Some advanced geometric tools
6.1 Exponential map and its applications in General Relativity
6.1.1The exponential map and normal coordinates about a point
6.1.2 Riemannian normal coordinates adapted to a di erentiable curve (solo enunciato)


7 Curvature
7.1 Curvature tensor and Riemann's curvature tensor
7.1.1 Flatness and curvature tensor
7.2 Properties of curvature tensor. Bianchi's identity (solo enunciati e idea della dim dell'identità di Bianchi)
7.3 Ricci's tensor and Einstein's tensor
7.4 Flatness and Riemann's curvature tensor: the whole story (accenno)
7.4.1 Frobenius' theorem (accenno)
7.4.2 The crucial theorem (accenno)

dalle dispense:  Teoria della Relativita' Speciale: formulazione matematica (Con un'introduzione alla formulazione della Relatività generale)

5 Dinamica in Relatività Speciale: covarianza delle leggi siche ed equazioni
della dinamica
5.1 Nozione di massa, quadriforza e quadrimpulso per punti materiali
5.1.1 Teorema delle forze vive" relativistico
5.2 Conservazione del quadri impulso e principio di equivalenza massa-energia
5.2.1 Legge di conservazione del quadri impulso
5.2.2 Il principio di equivalenza massa energia
5.3 Il tensore energia-impulso
5.3.1 Teorema della divergenza in forma covariante
5.3.2 Il tensore energia impulso per il uido di materia non interagente
5.3.3 Il tensore energia impulso
5.3.4 Il tensore energia impulso del uido perfetto

8 Le idee fisico-matematiche alla base della teoria Generale della Relatività
8.1 Fisica: il Principio di Equivalenza di Einstein
8.2 Matematica: l'exponential map
8.2.1 L'exponential map e le coordinate normali attorno ad un punto (già visto nelle dispense precedenti, riferirisi a quelle)
8.2.2 Coordinate normali adattate ad una curva assegnata (già visto nelle dispense precedenti, riferirisi a quelle)
8.3 La versione geometrica di RG3 e nozione relativistica di gravità
8.3.1 L'interpretazione di RG3: sistemi di coordinate localmente inerziali
8.3.2Il principio di equivalenza in forma forte e l'equazione di conservazione del tensore energia impulso
8.3.3 La deviazione geodetica e la gravità come curvatura dello spaziotempo
8.4 Le equazioni del campo gravitazionale di Einstein
8.4.1 Il limite classico dell'equazione della geodetica (solo ipotesi e risultato)
8.4.2 Le equazioni di Einstein del campo gravitazionale