CAR algebra

The CAR algebra is an algebra considered in the mathematical field of functional analysis. It is a C * algebra , which closely with the quantum mechanics studied canonical anticommutation (engl. C anonical a nticommutation r elation, hence the name CAR) is connected and therefore Fermionenalgebra is called.

construction

If the C * -algebra denotes the complex matrices, one can use the isometric * homomorphism ${\displaystyle M_{n}}$ ${\displaystyle n\times n}$ ${\displaystyle M_{2^{n}}}$

${\displaystyle M_{2^{n}}\rightarrow M_{2^{n+1}},\quad X\mapsto {\begin{pmatrix}X&0\\0&X\end{pmatrix}}}$

as a sub- algebra of. On the union of all matrix algebras lying in this way one then has a norm that continues each of the C * norms and therefore has all the properties of a C * norm except for completeness . The completion is then a C * -algebra, which is called the CAR-algebra. ${\displaystyle M_{2^{n+1}}}$ ${\displaystyle M_{2^{n}}}$

Canonical anti-commutation relations

Let a separable Hilbert space and a linear mapping into the C * -algebra of the continuous, linear operators on with the following properties be assumed: ${\displaystyle H}$ ${\displaystyle \alpha :H\rightarrow L(H)}$ ${\displaystyle L(H)}$ ${\displaystyle H}$

${\displaystyle \alpha (x)\alpha (y)+\alpha (y)\alpha (x)\,=\,0}$

${\displaystyle \alpha (x)\alpha (y)^{*}+\alpha (y)^{*}\alpha (x)\,=\,\langle x,y\rangle \mathrm {id} _{H}}$

for all vectors . ${\displaystyle x,y\in H}$

It is said that the canonical anti-commutation relations are fulfilled; these are fulfilled by the creation and annihilation operators for fermions considered in quantum mechanics . Such images can be implemented on the jib room , for example . The isomorphism class of the C * -algebra generated by the operators turns out to be independent of the special selection of the mapping , because: ${\displaystyle \alpha }$ ${\displaystyle \alpha }$ ${\displaystyle \alpha (x)}$ ${\displaystyle \alpha }$

• The C * -algebra generated by all operators is isomorphic to the CAR-algebra. ${\displaystyle \alpha (x),\,x\in H}$

If is an orthonormal basis of , the embedding can be identified with the above embedding ( here stands for the C * algebra generated by the operators listed in brackets). ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle H}$ ${\displaystyle C^{*}(\alpha (x_{1}),\ldots ,\alpha (x_{n}))\subset C^{*}(\alpha (x_{1}),\ldots ,\alpha (x_{n+1}))}$ ${\displaystyle M_{2^{n}}\rightarrow M_{2^{n+1}}}$ ${\displaystyle C^{*}(\ldots )}$

As UHF algebra and AF algebra

According to its construction, the CAR algebra is a UHF algebra, namely the one for the supernatural number (see also the article UHF algebra ). As a UHF algebra, it is also an AF-C * algebra and therefore distinguished among all AF-C * algebras by its ordered scaled group . This is on the scale given by [0.1]. stands for the set of all rational numbers whose denominator is a power of two. ${\displaystyle 2^{\infty }}$ ${\displaystyle K_{0}}$ ${\displaystyle \mathbb {Z} \left[{\frac {1}{2}}\right]}$ ${\displaystyle \mathbb {Z} \left[{\frac {1}{2}}\right]}$

Product conditions and type III factors

For each one can recursively define states , where ${\displaystyle \lambda \in [0,1]}$ ${\displaystyle \varphi _{\lambda }^{(n)}:M_{2^{n}}\rightarrow \mathbb {C} }$

• ${\displaystyle \varphi _{\lambda }^{(0)}:M_{2^{0}}\cong \mathbb {C} \rightarrow \mathbb {C} }$ let the identical mapping be and
• ${\displaystyle \varphi _{\lambda }^{(n)}(x)=\lambda \varphi _{\lambda }^{(n-1)}(x_{1,1})+(1-\lambda )\varphi _{\lambda }^{(n-1)}(x_{2,2})}$for each , where as -Matrix with elements from is written. ${\displaystyle n>0}$ ${\displaystyle x=(x_{i,j})}$ ${\displaystyle 2\times 2}$ ${\displaystyle M_{2^{n-1}}}$

Then the restriction of to is equal , because according to the embedding of to is considered here ${\displaystyle \varphi _{\lambda }^{(n)}}$ ${\displaystyle M_{2^{n-1}}}$ ${\displaystyle \varphi _{\lambda }^{(n-1)}}$ ${\displaystyle M_{2^{n-1}}}$ ${\displaystyle M_{2^{n}}}$

${\displaystyle (\varphi _{\lambda }^{(n)}|_{M_{2^{n-1}}})(x)=\varphi _{\lambda }^{(n)}\left({\begin{pmatrix}x&0\\0&x\end{pmatrix}}\right)=\lambda \varphi _{\lambda }^{(n-1)}(x)+(1-\lambda )\varphi _{\lambda }^{(n-1)}(x)=\varphi _{\lambda }^{(n-1)}(x)}$.

Hence there is a unique state on CAR algebra that is consistent with on all . This is called the associated product condition. The term product state comes from the fact that it can also be obtained using tensor product constructions, which is not explained here. According to J. Glimm , factors of type III can be constructed using these states as follows . ${\displaystyle \varphi _{\lambda }}$ ${\displaystyle M_{2^{n}}}$ ${\displaystyle \varphi _{\lambda }^{(n)}}$ ${\displaystyle \lambda }$

A Hilbert space representation on a Hilbert space belongs to the state by means of a GNS construction . For the image is a C * -algebra, the closure of which in the weak operator topology is a factor of type III. Any two such factors for different numbers from the open interval are not isomorphic. ${\displaystyle \varphi _{\lambda }}$ ${\displaystyle \pi _{\lambda }:A\rightarrow L(H_{\lambda })}$ ${\displaystyle H_{\lambda }}$ ${\displaystyle 0<\lambda <{\frac {1}{2}}}$ ${\displaystyle \pi _{\lambda }(A)\subset L(H_{\lambda })}$ ${\displaystyle \left(0,{\frac {1}{2}}\right)}$

GICAR algebra

Let be a mapping that satisfies the canonical anti-commutation relations defined above. If with , then the canonical anti-commutation relations are also fulfilled, as one can easily calculate. Since the C * -algebra generated by the (s) , with the Hilbert space running through, is the CAR-algebra in both cases , it can be shown that an automorphism is obtained which is called a gauge automorphism . ${\displaystyle \alpha :H\rightarrow L(H)}$ ${\displaystyle \mu \in \mathbb {C} }$ ${\displaystyle |\mu |=1}$ ${\displaystyle \beta :H\rightarrow L(H),\,x\mapsto \alpha (\mu x)}$ ${\displaystyle \alpha (x)}$ ${\displaystyle \beta (x)}$ ${\displaystyle x}$ ${\displaystyle A}$ ${\displaystyle \sigma _{\mu }:A\rightarrow A}$

The C * subalgebra of those elements of which are invariant under all gauge automorphisms is called GICAR algebra. GI stands for gauge-invariant (German: eich-invariant). One can show that the GICAR algebra is an AF-C * algebra. While the CAR algebra is simple, that is, it has no non-trivial two-sided ideals , the GICAR algebra has a rich ideal structure that can be read from its Bratteli diagram . This has the shape of Pascal's triangle : ${\displaystyle A}$ ${\displaystyle \sigma _{\mu },|\mu |=1}$

${\displaystyle {\begin{matrix}&&&&&&1&\\&&&&&\nearrow &&\\&&&&1&&&\ldots \\&&&\nearrow &&\searrow &&\\&&1&&&&3&\\&\nearrow &&\searrow &&\nearrow &&\\1&&&&2&&&\ldots \\&\searrow &&\nearrow &&\searrow &&\\&&1&&&&3&\\&&&\searrow &&\nearrow &&\\&&&&1&&&\ldots \\&&&&&\searrow &&\\&&&&&&1&\end{matrix}}}$

Individual evidence

1. ^ KR Davidson: C * -Algebras by Example. American Mathematical Society, 1996, ISBN 0-8218-0599-1 , Example III.5.4.
2. ^ KR Davidson: C * -Algebras by Example. American Mathematical Society, 1996, ISBN 0-8218-0599-1 , Example IV.3.4.
3. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups. Academic Press Inc., 1979, ISBN 0-12-549450-5 , Theorem 6.5.15.
4. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups. Academic Press Inc., 1979, ISBN 0-12-549450-5 , Theorem 8.15.13.
5. ^ KR Davidson: C * -Algebras by Example. American Mathematical Society, 1996, ISBN 0-8218-0599-1 : Example III.5.5.