By Scharlau W.

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12 QUANTUM SYMMETRIES The discussion of symmetries and conservation laws in quantum mechanics differs considerably from the classical theory. In quantum mechanics, the states of a system are not described by points in a phase space, but are instead described in terms of Hilbert spaces. A Hilbert space is a vector space (usually infinite-dimensional) over the complex number field equipped with an inner product and having the property that every Cauchy sequence of vectors converges. 12 Quantum Symmetries 27 denoted by (HP, O).

The velocity of the particle at any point on the path is a tangent vector at that point. If v is a tangent vector at the identity, we may choose a path h(t) passing through the identity with velocity v at time t = 0. The action of an element g of the group carries this path into a path g <> h(t) passing through the point g at time t = 0 with a certain velocity, which is a tangent vector at g. This tangent vector at g actually depends only on the vector v, and not on the particular choice for the path h(t).

It is easy to exhibit the unitary symmetry of a system of harmonic oscillators directly. We adopt the quantum mechanical point of view because it is simpler, but the same type of discussion can be given in classical mechanics. The Hamiltonian for a system of n identical noninteracting harmonic oscillators is In discussing harmonic oscillators, it is natural and convenient to choose units in which m = k = h = 1. 14 Lie Subgroups and Homomorphisms 31 The creation and annihilation operators satisfy the commutation relations with each other, and with the Hamiltonian.