Bogoliubov Hamiltonians and one parameter groups of Bogoliubov transformations

Details Bogoliubov Hamiltonians and one parameter groups of Bogoliubov transformations Bogoliubov Hamiltonians and one parameter groups of Bogoliubov transformations

2005-11-22

arxiv.org/abs/math-ph/0511069v1

Physics

Mathematical Physics

Scientific paper

On the bosonic Fock space, a family of Bogoliubov transformations corresponding to a strongly continuous one-parameter group of symplectic maps R(t) is considered. Under suitable assumptions on the generator A of this group, which guarantee that the induced representations of CCR are unitarily equivalent for all time t, it is known that the unitary operator U_{nat}(t) which implement this transformation gives a prjective unitary representation of R(t). Under rather general assumptions on the generator A, we prove that the corresponding Bogoliubov transformations can be implemented by a one-parameter group U(t) of unitary operators. The generator of U(t) will be called a Bogoliubov Hamiltonian. We will introduce two kinds of Bogoliubov Hamiltonians (type I and II) and give conditions so that they are well defined.

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