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THE CANONICAL ANTICOMMUTATION RELATIONS Lecture notes for Mathematics 208
 

Summary: THE CANONICAL ANTICOMMUTATION RELATIONS
Lecture notes for Mathematics 208
William Arveson
24 November 1998
In these notes we discuss the canonical anticommutation relations, the C
-
algebra associated with them (the CAR algebra), second quantization, and the
construction of KMS states for so-called free Fermi gasses. We only scratch the
surface. For more, I refer you to Gert Pedersen's book C
-algebras and their au-
tomorphism groups [3] and volume 2 of Operator algebras and quantum statistical
mechanics, by Ola Bratteli and Derek Robinson [2].
Two operators X, Y are said to anticommute if XY + Y X = 0. Suppose we are
given two sets of self-adjoint operators p1, . . . , pn, q1, . . . , qn acting on some Hilbert
space (or more generally, belonging to some C
-algebra) which satisfy the following
pkpj + pjpk = qkqj + qjqk = 2jk1
pkqj + qjpk = 0
for all k, j. These are the canonical anticommutation relations in their self-adjoint
form for a Fermionic quantum system having n degrees of freedom. Taking j = k

  

Source: Arveson, William - Department of Mathematics, University of California at Berkeley

 

Collections: Mathematics