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)

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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)

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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)

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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)

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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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