SQUARES OF WHITE NOISE, SL(2, ℂ) AND KUBO–MARTIN–SCHWINGER STATES

Abstract

We investigate the structure of Kubo -Martin -Schwinger (KMS) states on some extension of the universal enveloping algebra of SL(2, C). We find that there exists a one-to-one correspondence between the set of all covariant KMS states on this algebra and the set of all probability measures dµ on the real half-line [0, +∞), which decrease faster than any inverse polynomial. This problem is connected to the problem of KMS states on square of white noise algebra.

C*-Non-Linear Second Quantization

“…This result motivated a large number of papers extending it in different directions and exhibiting connections with almost all fields of mathematics, see for example [22] for the case of free white noise, [1] for the connection with infinite divisibility and for the identification of the vacuum distributions of the generalized fields with the three nonstandard Meixner classes, [3] and [21] for finite temperature representations, [9] for the construction of the Fock functor, the survey [5] and the paper [6] for the connections with conformal field theory and with the Virasoro-Zamolodchikov hierarchy, [7] for the connections between renormalization and central extensions.…”

“…Proof. By definition, the Lie brackets of two generators defined by (21), (22) are a multiple of the generators. To verify that the Jacobi identity is satisfied notice that, for any i, j, k ∈ {0, 1, .…”

Quantum Probability, Renormalization and Infinite-Dimensional *-Lie Algebras

“…The problem of constructing the most general KMS states (for the free quadratic evolution) on the RSWN algebra was attacked with algebraic techniques in the paper [26] but its solution was obtained later, with a purely analytical approach by Prohorenko [57].…”