Thermo Field Dynamics and quantum algebras.pdf

Thermo Field Dynamics and quantum algebras 8 † ∗ ∗ ∗∗ + 9 E.Celeghini , S.De Martino , S.De Siena , A.Iorio , M.Rasetti and 9 1 G.Vitiello∗ n a J 7 † Dipartimento di Fisica, Universit`a di Firenze, and INFN-Firenze, 1 v 1 I-50125 Firenze, Italy 3 0 ∗ Dipartimento di Fisica, Universit`a di Salerno, and INFN-Salerno, 1 0 8 I-84100 Salerno, Italy 9 / h ∗∗ School of Mathematics, Trinity College, Dublin, Ireland t - p + e Dipartimento di Fisica and Unit`a INFM, Politecnico di Torino, h : v I-10129 Torino, Italy i X r a Abstract The algebraic structure of Thermo Field Dynamics lies in the q-deformation of the algebra of creation and annihilation operators. Doubling of the degrees of freedom, tilde-conjugation rules, and Bogoliubov transformation for bosons and fermions are recognized as algebraic properties of h (1) and of h (1|1), respectively. q q PACS: 03.70.+k, 03.65.F, 11.10.-z Keywords: Thermal field theories, q-groups, Lie-Hopf algebras, Bogoliubov trans- formations, thermal variables, unitarily inequivalent representations. 2 1 Introduction One central ingredient of Hopf algebras [1] is the operator doubling implied by the coalgebra. The coproduct operation is indeed a map ∆ : A → A⊗A which duplicates the algebra. Lie-Hopf algebras are commonly used in the familiar addition of energy, momentum and angular momentum. On the other hand, the doubling of th