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Mathematics of Quantization and Quantum Fields by Christian Gérard, Jan Derezinski

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10

Symplectic invariance of CCR in finite-dimensions

This is the first chapter devoted to the symplectic invariance of the CCR. In this chapter we restrict ourselves to regular CCR representations over finite-dimensional symplectic spaces.

In an infinite-dimensional symplectic space there is no distinguished topology. This problem is absent in a finite-dimensional space. This motivates a separate discussion of the finite-dimensional case.

The chapter is naturally divided in two parts. In the first three sections we consider symplectic invariance without invoking any conjugation on the symplectic space. We consider an arbitrary irreducible regular CCR representation over a finite-dimensional symplectic space and do not explicitly use the Schrödinger ...

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