CCR and CAR algebras

In and  CCR algebras (after s) and CAR algebras (after canonical anticommutation relations) arise from the  study of s and s respectively. They play a prominent role in and.

CCR and CAR as *-algebras
Let $$V$$ be a  equipped with a  real   $$(\cdot,\cdot)$$ (i.e. a ). The  generated by elements of $$V$$ subject to the relations


 * $$fg-gf=i(f,g) \, $$
 * $$ f^*=f, \, $$

for any $$f,~g$$ in $$V$$ is called the canonical commutation relations (CCR) algebra. The uniqueness of the representations of this algebra when $$V$$ is is discussed in the.

If $$V$$ is equipped with a real  $$(\cdot,\cdot)$$ instead, the unital *-algebra generated by the elements of $$V$$ subject to the relations


 * $$fg+gf=(f,g), \,$$
 * $$ f^*=f, \,$$

for any $$f,~g$$ in $$V$$ 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 $$H$$ be a real symplectic vector space with nonsingular symplectic form $$(\cdot,\cdot)$$. In the theory of s, the CCR algebra over $$H$$ is the unital generated by elements $$ \{W(f):~f\in H\}$$ subject to
 * $$ W(f)W(g)=e^{-i(f,g)}W(f+g), \,$$
 * $$ W(f)^*=W(-f). \,$$

These are called the Weyl form of the canonical commutation relations and, in particular, they imply that each $$W(f)$$ is and $$W(0)=1$$. It is well known that the CCR algebra is a simple non-separable algebra and is unique up to isomorphism.

When $$H$$ is a and $$(\cdot,\cdot)$$ is given by the imaginary part of the inner-product, the CCR algebra is  on the  over $$H$$ by setting
 * $$ W(f)\left(1,g,\frac{g^{\otimes 2}}{2!},\frac{g^{\otimes 3}}{3!},\ldots\right)= e^{-\frac{1}{2}||f||^2-\langle f,g\rangle }\left(1,f+g,\frac{(f+g)^{\otimes 2}}{2!},\frac{(f+g)^{\otimes 3}}{3!},\ldots\right), \, $$

for any $$f,g\in H$$. The field operators $$B(f)$$ are defined for each $$ f\in H$$ as the of the one-parameter unitary group $$(W(tf))_{t\in\mathbb{R}}$$ on the symmetric Fock space. These are s, however they formally satisfy
 * $$ B(f)B(g)-B(g)B(f)=2i\mathrm{Im}\langle f,g\rangle. \,$$

As the assignment $$f\mapsto B(f)$$ is real-linear, so the operators $$B(f)$$ define a CCR algebra over $$(H,2\mathrm{Im}\langle\cdot,\cdot\rangle)$$ in the sense of.

The C*-algebra of CAR
Let $$H$$ be a Hilbert space. In the theory of operator algebras the CAR algebra is the unique of the complex unital *-algebra generated by elements $$\{b(f),b^*(f):~f\in H\}$$ subject to the relations
 * $$b(f)b^*(g)+b^*(g)b(f)=\langle f,g\rangle, \,$$
 * $$b(f)b(g)+b(g)b(f)=0, \,$$
 * $$\lambda b^*(f)=b^*(\lambda f), \,$$
 * $$b(f)^*=b^*(f), \, $$

for any $$f,g\in H$$, $$\lambda\in\mathbb{C}$$. When $$H$$ is separable the CAR algebra is an and in the special case $$H$$ is infinite dimensional it is often written as $${M_{2^\infty}(\mathbb{C})}$$.

Let $$F_a(H)$$ be the over $$H$$ and let $$P_a$$ be the  orthogonal projection onto antisymmetric vectors:
 * $$P_a: \bigoplus_{n=0}^\infty H^{\otimes n} \to F_a(H). \, $$

The CAR algebra is faithfully represented on $$F_a(H)$$ by setting
 * $$ b^*(f)P_a(g_1\otimes g_2\otimes\cdots\otimes g_n)=P_a(f\otimes g_1\otimes g_2\otimes\cdots\otimes g_n) \, $$

for all $$ f,g_1,\ldots,g_n\in H$$ and $$n\in\mathbb{N}$$. 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 s. Moreover, the field operators $$B(f):=b^*(f)+b(f)$$ satisfy
 * $$ B(f)B(g)+B(g)B(f)=2\mathrm{Re}\langle f,g\rangle, \, $$

giving the relationship with.

Superalgebra generalization
Let $$V$$ be a real $$\mathbb{Z}_2$$- equipped with a nonsingular antisymmetric bilinear superform $$(\cdot,\cdot)$$ (i.e. $$(g,f)=-(-1)^{|f||g|}(f,g) $$ ) such that $$(f,g)$$ is real if either $$f$$ or $$g$$ is an even element and if both of them are odd. The unital *-algebra generated by the elements of $$V$$ subject to the relations


 * $$fg-(-1)^{|f||g|}gf=i(f,g) \,$$
 * $$f^*=f,~g^*=g\,$$

for any two pure elements $$f,~g$$ in $$V$$ is the obvious 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.

In mathematics, the abstract structure of the CCR and CAR algebras, over any field, not just the complex numbers, is studied by the name of and s, where many significant results have accrued. One of these is that the generalizations of  and  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 and.