 
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 socalled 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 selfadjoint 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 selfadjoint
form for a Fermionic quantum system having n degrees of freedom. Taking j = k
