Canonical anticommutation relations

Summary

The canonical anticommutation relations (CAR) are the algebraic rules satisfied by fermionic creation and annihilation operators. For each vector $f$ of a one-particle Hilbert space there is an operator $a(f)$, and these operators satisfy $\{a(f), a(g)\} = 0$ and $\{a(f), a^*(g)\} = \ip{f}{g}$, where $\{A, B\} = AB + BA$. They express the Pauli exclusion principle algebraically, since they imply $a^*(f)^2 = 0$. The C*-algebra generated by such operators, the CAR algebra, is unique up to isomorphism and is a basic object of quantum statistical mechanics.