Lecture 08 - Spectra and perturbation theory (Schuller's Lectures on Quantum Theory)

Video Lecture:

Summary of contents: resolvent map and resolvent set of an operator; spectrum of an operator; eigenvalues and eigenvectors; difference between spectrum and eigenvalues; pure point spectrum, point embedded in continuum spectrum, purely continuous spectrum; point and continuous spectrum; proof that the eigenvalues of a self-adjoint operator are real; proof that the point spectrum coincides with the eigenvalues; proof by contrapositive; eigenspace and degenerate eigenvalues; proof that the eigenvectors of a self-adjoint operator corresponding to distinct eigenvalues are orthogonal; perturbation theory for the point spectrum; formal power series ansatz; Big O notation; fixing phase and normalisation of perturbed eigenvectors; order-by-order decomposition of the perturbed eigenvalue problem: explicit calculation of first and second-order corrections.

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Lecture 07 - Self-adjoint and essentially self-adjoint operators (Schuller's Lectures on Quantum Theory)

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Summary of contents: densely defined and adjoint operators, proof of well-definedness; adjoint of sum; kernel and range (or image) of an operator; injective, surjective, and invertible operators; ker(A^*) = ran(A)^perp; extension of an operator; relation between extensions and adjoints; symmetric operators; remark on Hermitian operators; self-adjoint operators and self-adjoint extensions; closable operators; closure of an operator; closed operators; a symmetric operator is necessarily closable; essentially self-adjoint operators; unique self-adjoint extension; defect indices; criteria for self-adjointness and essential self-adjointness without calculating the adjoint; proofs and examples.

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Lecture 25 - Covariant Derivatives (Schuller's Geometric Anatomy of Theoretical Physics)

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Summary of contents: proof of the equivalence of local sections and G-equivariant functions; linear actions on associated vector fibre bundles; matrix Lie group; construction of the covariant derivative for local sections on the base manifold.

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Lecture 06 - Integration of measurable functions (Schuller's Lectures on Quantum Theory)

Video Lecture:

Summary of contents: characteristic and simple functions; integration of non-negative, measurable, simple functions; example: integration with respect to the counting measure; basic properties of integrals of simple functions; proof that the integral against a fixed simple function defines a measure; sigma-algebra on the extended real line; real measurable functions; integration of non-negative, measurable functions; notation for integrals; Markov inequality; Monotone converge theorem and applications; proof that the integral of a non-negative measurable function is zero if and only if the function is almost everywhere zero; integral of the Dirichlet function; Lebesgue integrable functions;  f^+ and f^-; integration of complex integrable functions; proof that L^1 is a vector space; Dominated convergence theorem; L^p spaces, with p in [0,infinity]; essential supremum; proof that L^2 is a complex vector space; equivalence relations, equivalence classes and quotient sets; proof that "almost everywhere equal" is an equivalence relation; Höolder's inequality; inner product on L^2.

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Lecture 05 - Measure Theory (Schuller's Lectures on Quantum Theory)

Video Lecture:

Summary of contents: sigma-algebras; De Morgan's Laws; measurable spaces and measurable subsets; extended real line; measures and measure spaces; example: N and the counting measure; proof of basic properties of measures; proof that measures are continuous from above and from below; proof that measures are countably sub-additive; finite measures; sigma-algebras generated by a collection of subsets; Borel sigma-algebra on a topological space; example: the Borel sigma-algebra of R; null sets and almost everywhere; complete and translation-invariant measures; Lebesgue measure on R^d; proof that the Lebesgue measure is finite; measurable maps; composition of measurable maps; pointwise converging sequences of measurable maps; push-forward of a measure.

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Lecture 24 - Curvature and Torsion on Principal Bundles (Schuller's Geometric Anatomy of Theoretical Physics)

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Summary of contents: exterior covariant derivative; curvature two-form; characterisation of the curvature two-form with proof; Yang-Mills field strength; First Bianchi identity; solder (ing) form; torsion two-form; Second Bianchi identity.

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Lecture 23 - Parallel Transport (Schuller's Geometric Anatomy of Theoretical Physics)

Video Lecture:

Summary of contents: horizontal lifts of a curve to the principal bundle; ODE characterising horizontal lifts; explicit solution in the case of a matrix Lie group; path-ordered exponential; parallel transport map; loops and holonomy groups; horizontal lifts to the associated bundle; parallel transport map on the associated bundle; covariant derivative of a section.

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Lecture 04 - Projectors, bras and kets (Schuller's Lectures on Quantum Theory)

Video Lecture:

Summary of contents: projection and orthogonal complement; infinite-dimensional Pythagoras' theorem; open and closed sets; proof that a closed subset of a complete metric space is complete; a closed linear subspace of a Hilbert space is a sub-Hilbert space; proof that the orthogonal complement of a linear subspace is a closed linear subspace; orthogonal projector; proof of properties of orthogonal projectors; Riesz representation theorem with proof; Riesz map; critical discussion of Dirac's bra-ket notation.

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Lecture 03 - Separable Hilbert spaces (Schuller's Lectures on Quantum Theory)

Video Lecture:

Summary of contents: Hilbert spaces; norm induced by an inner product; proof of the Cauchy-Schwarz inequality; proof that the induced norm is a norm; detailed proof, step by step, of the Jordan - von Neumann theorem: a norm is induced by an inner product if and only if it satisfies the parallelogram identity, and the inner product is determined by the polarisation identity, adapting hints from Friedberg, Insel, Spence's Linear algebra; proof that the dual of a Hilbert space is a Hilbert space; proof that inner products are sequentially continuous; Hamel basis and dimension of a vector space; Schauder basis and properties; orthonormal Schauder basis and separable Hilbert spaces; coefficients with respect to a Schauder basis; Pythagoras' theorem; structure-preserving maps and classification of structures; unitary map (or unitary operator); proof that a surjective map which preserves the inner product is a unitary map; square-summable complex sequences; classification of separable Hilbert spaces: proof that every separable Hilbert space is unitarily equivalent to l^2(N).

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Lecture 02 - Banach Spaces (Schuller's Lectures on Quantum Theory)

Video Lecture:

Summary of contents: metric spaces; converges of sequences; Cauchy sequences; completeness; norms and normed spaces; metric induced by a norm; detailed example: C^0[0,1], the continuous complex-valued functions on [0,1] form a Banach space; bounded operators; equivalent conditions for boundedness; operator norm; example: C^1[0,1], first derivative operator is unbounded; proof of continuity of addition, complex scalar multiplication and norm; proof that the bounded linear operators form a Banach space; dual space and functionals; weak convergence; proof that strong convergence implies weak convergence; sequential characterisation of dense subsets; extensions of a linear operator; proof of the BLT theorem  (bounded linear transformation).

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Lecture 01 - Axioms of Quantum Mechanics (Schuller's Lectures on Quantum Theory)

Video Lecture:

Summary of contents: brief comparison of classical and quantum mechanics; discrete and continuous spectrum; impact of measurement on the observation; Stern-Gerlach experiment; probabilistic nature of quantum mechanical predictions; axioms of quantum mechanics; quantum systems and states; pure and mixed states; complex Hilbert space; sesqui-linear inner product; completeness; operators: domain, densely defined, positive, trace-class; observables; adjoint operators and self-adjoint operators; measurement; projection-valued measure; unitary dynamics; evolution operator; projective dynamics.

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Lecture 22 - Local Representations of a Connection on the Base Manifold: Yang-Mills Fields (Schuller's Geometric Anatomy of Theoretical Physics)

Video Lecture:

Summary of contents: Yang-Mills field as pull-back of a connection one form along a local section; local trivialisations of a principal bundle; local representation of a connection one-form; Maurer-Cartan form; example: the Yang-Mills fields on the frame bundle, Christoffel symbol; example: calculation of the Maurer-Cartan form of the general linear group GL(n,R); patching Yang-Mills fields on different domains; the gauge map; example: the gauge map on the frame bundle.

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Lecture 21 - Connections and Connection 1-Forms (Schuller's Geometric Anatomy of Theoretical Physics)

Video Lecture:

Summary of contents: vertical and horizontal subspaces at a point; decomposition in vertical and horizontal parts; connection on a principal bundle; connection one-form; properties of connection one-forms with proof.

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Lecture 20 - Associated Fibre Bundles (Schuller's Geometric Anatomy of Theoretical Physics)

Video Lecture:

Summary of contents: associated fibre bundle to a principal bundle; detailed example: the frame bundle; scalar and tensor densities on a manifold; associated bundle maps and isomorphisms; trivial associated bundles; restrictions and extensions of a principal bundle; examples.

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Lecture 19 - Principal Fibre Bundles (Schuller's Geometric Anatomy of Theoretical Physics)

Video Lecture:

Summary of contents: left and right Lie group actions; example: actions from representations; proof: right actions from left actions; equivariance of smooth maps; orbits, orbit space and stabilisers; free and transitive actions; examples; smooth and principal bundles; detailed example: the frame bundle; principal bundle morphisms and isomorphisms (or diffeomorphisms); trivial bundles; proof that a bundle is trivial if and only if it admits a global section.

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Lecture 18 - Reconstruction of a Lie Group from its Algebra (Schuller's Geometric Anatomy of Theoretical Physics)

Video Lecture:

Summary of contents: integral curves to a vector field; maximal integral curves; complete vector fields; every vector field on a compact manifold is complete; exponential map; the image of exp is the connected component of the Lie group containing the identity; examples: orthogonal group, special orthogonal group; (restricted) Lorentz group: proper/improper orthochronous/non-orthochronous transformations; Lorentz algebra; one-parameter subgroups; flow of a vector field; the exponential map commutes with smooth maps.

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Lecture 17 - Representation Theory of Lie Groups and Lie Algebras (Schuller's Geometric Anatomy of Theoretical Physics)

Video Lecture:

Summary of contents: representations of a Lie algebras; representation spaces and dimension of a representation; examples of representations; homomorphism and isomorphism of representations; trivial and adjoint representations; faithful representations; direct sum and tensor product representations; invariant subspaces, reducible and irreducible representations; highest weights; Killing form associated to a representation; Casimir operator; proof that the Casimir operator commutes with the representation; Schur's lemma; worked examples; automorphism group; representation of Lie groups; Adjoint representation.

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Lecture 16 - Dynkin Diagrams from Lie Algebras, and Vice Versa (Schuller's Geometric Anatomy of Theoretical Physics)

Video Lecture:

Summary of contents: proof that sl(2,C) is simple; Cartan subalgebra of sl(2,C); roots and fundamental roots of sl(2,C); the Dynkin diagram of sl(2,C); the A2 Dynkin diagram; detailed reconstruction of A2 from its Dynkin diagram.

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Lecture 15 - The Lie Group SL(2,C) and its Lie Algebra sl(2,C) (Schuller's Geometric Anatomy of Theoretical Physics)

Video Lecture:

Summary of contents: the complex special linear group SL(2,C): as a set, as a group, as a topological space, as a topological manifold, as a complex differentiable manifold, as a Lie group; the Lie algebra sl(2,C) of the Lie group SL(2,C); detailed calculation of the structure constants of sl(2,C); determination of the Lie bracket between left-invariant vector fields on SL(2,C).

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Lecture 14 - Classification of Lie Algebras and Dynkin Diagrams (Schuller's Geometric Anatomy of Theoretical Physics)

Video Lecture:

Summary of contents: complex Lie algebras; abelian Lie algebras; the trivial Lie algebra; ideal of a Lie algebra; trivial ideals; simple and semi-simple Lie algebras; derived subalgebra; solvability; direct and semi-direct sum of Lie algebras; Levi's theorem on the decomposition of finite-dimensional complex Lie algebras; adjoint map and ad; proof that ad is a Lie algebra homomorphism; Killing form; proof of the invariance (or associativity, or anti-symmetry) of the Killing form; a Lie algebra is semi-simple if and only if the Killing is non-degenerate; structure constants; components of adjoint maps and the Killing form in terms of the structure constants; Cartan subalgebra, rank of a Lie algebra and Cartan-Weyl basis; roots and fundamental roots; proof that the restriction of the Killing form on a Cartan subalgebra is a pseudo inner product; real inner product; length and angle between roots; Weyl transformations and Weyl group; Cartan matrix; bond number; Dynkin diagrams and classification of finite-dimensional semi-simple complex Lie algebras.

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Lecture 13 - Lie groups and their Lie algebras (Schuller's Geometric Anatomy of Theoretical Physics)

Video Lecture:

Summary of contents: Lie groups; dimension of a Lie group; examples of Lie groups: n-dimensional translation group, unitary group U(1), general linear GL(n,R), orthogonal group O(p,q); pseudo-inner products on a vector space; Lie group homomorphism and isomorphism; proof that the left translation map is a diffeomorphism; push-forward of the left translation map; left-invariant vector fields; proof that the space of left-invariant vector fields is isomorphic to the tangent space at the identity; proof that the left-invariant vector fields form a Lie algebra, the Lie algebra of the Lie group. Lie algebra homomorphisms and isomorphic Lie algebras.

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Lecture 12 - Grassmann algebra and de Rham cohomology (Schuller's Geometric Anatomy of Theoretical Physics)

Video Lecture:

Summary of contents: differential n-forms; orientable manifolds; degree of a differential form; pull-back of a differential form; wedge (or exterior) product of differential forms; local expression of a differential form; proof that the pull-back distributes over the wedge product; Grassmann algebra; Grassmann numbers; proof that the wedge product is graded commutative; exterior derivative; Lie bracket (or commutator) of vector fields; example: exterior derivative of a differential one-form; proof that the exterior derivative is graded additive; commutation of the exterior derivative with the pull-back; Maxwell's electrodynamics and Maxwell's equations expressed using differential forms; symplectic forms and classical mechanics; closed and exact forms; proof that d^2=0; symmetrisation and anti-symmetrisation of indices with examples; every exact form is closed; kernel and image of a linear map; Z^n and B^n; Poincaré lemma; cohomology groups.

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Lecture 11 - Tensor space theory II: over a ring (Schuller's Geometric Anatomy of Theoretical Physics)

Video Lecture:

Summary of contents: vector fields as smooth sections of the tangent bundle; vector fields as linear maps on the space of smooth maps; push-forward of a smooth map as a map between tangent bundles; push-forward of a vector field; structure of the set of vector fields; rings: commutative, unital and division (or skew) rings; examples; modules of a unital ring; examples of modules admitting and not admitting a basis; Zorn's lemma; partial orders and partially ordered sets (posets); total order and totally ordered sets; upper bounds; proof that every module over a division ring (and hence every vector space) admits a Hamel basis; direct sum of modules; finitely generated, free and projective modules; homomorphism of modules (or linear maps); Serre-Swan-et al.'s theorem; pull-back of forms; tensor fields as multilinear maps; tensor product of tensor fields.

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Lecture 10 - Construction of the Tangent Bundle (Schuller's Geometric Anatomy of Theoretical Physics)

Video Lecture:

Summary of contents: cotangent space and tensor space at a point of a manifold; differential of a smooth map; gradient of a real function on a manifold; dual coordinate-induced basis and gradients of coordinate functions; push-forward and pull-back of smooth maps at a point; push-forward of tangent vectors and pull-back of covectors; immersions and immersed submanifolds; embedding and embedded submanifolds; Whitney's theorem; definition of tangent bundle; proof that the tangent bundle is a smooth manifold.

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Lecture 09 - Differential structures: the pivotal concept of tangent vector spaces (Schuller's Geometric Anatomy of Theoretical Physics)

Video Lecture:

Summary of contents: the space of smooth maps on a manifold; smooth curves on a manifold; directional derivative operator; tangent vectors at a point and tangent space at a point; proof that the sum of tangent vectors is a tangent vector; alternative definitions of tangent space (via equivalence classes of smooth curves, derivations at a point on germs of functions, and physical tangent vectors); algebras over an algebraic field; associative, unital and commutative algebras; Lie algebras, Lie bracket and Jacobi identity; commutator; derivations on an algebra; detailed examples; proof that derivations on a algebra constitute a Lie algebra; proof of equality of manifold dimension and tangent space dimension: dim M = dim TpM; coordinate-induced basis of tangent spaces; change of coordinates under a change of coordinate-induced bases.

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Lecture 08 - Tensor space theory I: over a field (Schuller's Geometric Anatomy of Theoretical Physics)

Video Lecture:

Summary of contents: algebraic fields; vector spaces over an arbitrary field; vector (or linear) subspaces; linear maps; linear isomorphisms and isomorphic vector spaces; Hom-spaces; endomorphisms and automorphisms; dual vector space and linear functionals (covectors/one-forms); bilinear and multilinear maps; tensors and tensor product; examples; equivalence of endomorphisms and (1,1)-tensors; Hamel bases; linear independence and spanning set; dimension; double dual; dual bases and isomorphism of a vector space and its dual in finite dimensions; components of vectors and tensors; change of basis formulas; Einstein's summation convention and examples; column and row vectors and matrices; change of components under a change of basis; bilinear forms; permutations, symmetric group, transpositions, and signature of a transposition; totally anti-symmetric tensor; n-forms; volume-form and volume; determinant of an endomorphism.

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Lecture 07 - Differentiable structures: definition and classification (Schuller's Geometric Anatomy of Theoretical Physics)

Video Lecture:

Summary of contents: refining a maximal atlas; C^k and smooth compatibility of charts; Cauchy-Riemann equations; differentiable atlas; compatibility of differentiable atlases; examples; proof of well-definedness of the definition of differentiability of maps; smooth maps and diffeomorphisms; diffeomorphic manifolds; classification of smooth structure on manifolds; Betti numbers.

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