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Linear Canonical Transformations
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aB(f adjoint annihilation operators antisymmetric arbitrary bilinear form Bose bosons bounded operator calculation called canonical trans commutation relations continual integral converges strongly degrees of freedom denote dense set derivatives differential difficult to verify domain of definition elements equal Evidently exists expression factor Fermi fermions finite vectors finite-dimensional follows formal power series Fredholm determinant functional A(a functional corresponding Grassmann algebra Hence Hermitian Hilbert space Hilbert-Schmidt operator inequality infinite number inner product invariant with respect inverse kernel Lemma limit linear canonical transformation matrix form norm number of degrees operator ft orthogonal orthonormal basis particles polynomial projection operator proof proper canonical transformation proved quadratic operator representation right member satisfying the condition scattering operator self-adjoint operator sequence space with involution square-integrable functions subsection symmetric trace-class operator unitary operator vacuum vector variables x a*(xl