Local exponents and infinitesimal generators of canonical transformations on Boson Fock spaces

Details Local exponents and infinitesimal generators of canonical transformations on Boson Fock spaces Local exponents and infinitesimal generators of canonical transformations on Boson Fock spaces

2003-09-18

arxiv.org/abs/math-ph/0309044v1

Physics

Mathematical Physics

Scientific paper

A one-parameter symplectic group $\{e^{t\dA}\}_{t\in\RR}$ derives proper canonical transformations on a Boson Fock space. It has been known that the unitary operator $U_t$ implementing such a proper canonical transformation gives a projective unitary representation of $\{e^{t\dA}\}_{t\in\RR}$ and that $U_t$ can be expressed as a normal-ordered form. We rigorously derive the self-adjoint operator $\D(\dA)$ and a phase factor $e^{i\int_0^t\TA(s)ds}$ with a real-valued function $\TA$ such that $U_t=e^{i\int_0^t\TA(s)ds}e^{it\D(\dA)}$. Key words: Canonical transformations(Bogoliubov transformations), symplectic groups, projective unitary representations, one-parameter unitary groups, infinitesimal self-adjoint generators, local factors, local exponents, normal-ordered quadratic expressions.

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