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

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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)

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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)

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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)

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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)

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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)

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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)

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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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