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Vacuum polarization of a scalar field on the nonunimodular lie groups

“…We have no way of presenting the method of K-orbits in ample detail and refer the reader to [2][3][4]. Here we recall only the main notions and definitions.…”

Polarization of a scalar field vacuum on a manifold conformally equivalent to the manifold R⊗G

“…(6) by the method of coadjoint representation orbits. We now designate by L the Lie algebra of the Lie group G. We need to define the λ-representation of the Lie algebra [4] arising in quantization of the Poisson bracket in the Darboux canonical coordinates in a nonsingular orbit of the coadjoint representation O λ on the group L * . The λ-representation is an irreducible representation of the Lie algebra in space of smooth functions on the Lagrange manifold to…”

“…The region Q of variables q is a complex cylinder with the axis directed along the imaginary axis: 1 1 , , q i S R = α + β α ∈ β ∈ . This is connected with the condition of real transfer functions to the Darboux canonical coordinates in the K-orbit [4]. We choose the measure dμ(q) in the region Q as follows: We note that operators of λ-representation (21) are antisymmetric relative to scalar product (7) and are generators of unitary (2j + 1)-dimensional irreducible representation of the group SO (3) in the space L 2 (Q, dμ(q)).…”

Method of orbits of coassociated representation in thermodynamics of the lie noncompact groups

“…After transition from the group space to the Lagrangian submanifold of the orbit O λ , the Bloch equation for the density matrix is reduced to the equation in the orbit with a smaller number of independent variables [10]:…”