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Introduction to Quantum Mechanics
Dirac Spaces and Generalized
Groups and Their Represen
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algebra structure ators automorphism classical mechanics commutator compact complex numbers complex vector space condition connected convergence curve defined DEFINITION denote DGCV differential equation Dirac direct sum eigenvalues eigenvectors elements of G EXERCISE finite dimensional formula Fourier transform function f g e G hence Hermitian operators Hilbert space homomorphism inner product integral invariant irreducible representations isomorphic lemma Let G Lie algebra Lie group linear transformations Lorentz mathematical measure metric space n x n notation observables one-parameter group one-parameter subgroup operators on H orthogonal orthonormal basis particle physicists physics Poisson bracket Proof prove quantum mechanics quotient rapidly decreasing real numbers real vector space relation representation of G right hand side satisfies Schrodinger Section skew-Hermitian operators solution space H subalgebra subgroup of G subset subspace Suppose G symmetric tation THEOREM tion topological group trace unitary operators zero