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

Video Lecture:

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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