CCR and CAR algebras
In mathematics and physics CCR algebras (after canonical commutation relations) and CAR algebras (after canonical anticommutation relations) arise from the quantum mechanical study of bosons and fermions respectively. They play a prominent role in quantum statistical mechanics and quantum field theory.
CCR and CAR as *-algebras
for any in is called the canonical anticommutation relations (CAR) algebra.
The C*-algebra of CCR
There is a distinct, but closely related meaning of CCR algebra, called the CCR C*-algebra. Let be a real symplectic vector space with nonsingular symplectic form . In the theory of operator algebras the CCR algebra over is the unital C*-algebra generated by elements subject to
These are called the Weyl form of the canonical commutation relations and, in particular, they imply that each is unitary and . It is well known that the CCR algebra is a simple non-separable algebra and is unique up to isomorphism.
for any . The field operators are defined for each as the generator of the one-parameter unitary group on the symmetric Fock space. These are self-adjoint unbounded operators, however they formally satisfy
As the assignment is real-linear, so the operators define a CCR algebra over in the sense of Section 1.
The C*-algebra of CAR
Let be a Hilbert space. In the theory of operator algebras the CAR algebra is the unique C*-completion of the complex unital *-algebra generated by elements subject to the relations
Let be the antisymmetric Fock space over and let be the orthogonal projection onto antisymmetric vectors:
The CAR algebra is faithfully represented on by setting
for all and . The fact that these form a C*-algebra is due to the fact that creation and annihilation operators on antisymmetric Fock space are bona-fide bounded operators. Moreover the field operators satisfy
giving the relationship with Section 1.
Let be a real -graded vector space equipped with a nonsingular antisymmetric bilinear superform (i.e. ) such that is real if either or is an even element and imaginary if both of them are odd. The unital *-algebra generated by the elements of subject to the relations
for any two pure elements in is the obvious superalgebra generalization which unifies CCRs with CARs: if all pure elements are even, one obtains a CCR, while if all pure elements are odd, one obtains a CAR.
The graded generalizations of Weyl and Clifford algebras allow the basis-free formulation of the canonical commutation and anticommutation relations in terms of a symplectic and a symmetric non-degenerate bilinear form. In addition the binary elements in this graded Weyl algebra give a basis-free version of the commutation relations of the symplectic and pseudo-orthogonal[clarification needed] Lie algebras.
- Bose–Einstein statistics
- Fermi–Dirac statistics
- Glossary of string theory
- Heisenberg group
- Bogoliubov transformation
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- Evans, David E.; Kawahigashi, Yasuyuki (1998). Quantum Symmetries in Operator Algebras. Oxford University Press. ISBN 978-0-19-851175-5. .
- Roger Howe (1989). "Remarks on Classical Invariant Theory". Transactions of the American Mathematical Society 313: 539–570. doi:10.1090/S0002-9947-1989-0986027-X. JSTOR 2001418.