Cuntz代数中的递归费米子系统。II --表示的自同态、自同构和分支--
Recursive Fermion System in Cuntz Algebra. II -- Endomorphism, Automorphism and Branching of Representation --
摘要：
基于CAR代数在Cuntz代数${\mathcal O}_{2^p}$中的嵌入公式，通过对Cuntz代数的性质进行限制，研究了CAR代数的性质.明确地构造了Cuntz代数的各种$\ast$ -自同态，并将其转化为CAR代数的自同态.特别地，构造了CAR代数到其偶子代数的一组$\ast$ -自同态.根据合成表示和$\ast$ -自同态得到的分支公式，证明了CAR代数的KMS态是由Fock表示通过上述偶CAR自同态得到的.对${\mathcal O}_{2^p}$的$U(2^p)$作用诱导CAR代数的$\ast$ -自同构，这些自同构由用生成元中的多项式表示的非线性变换给出。结果表明，在CAR代数的这些$\ast$ -自同构中，存在一族描述费米子时间演化的单参数$\ast$ -自同构群，其中系统的粒子数随时间变化，而Fock真空保持不变.
英文摘要：
Based on an embedding formula of the CAR algebra into the Cuntz algebra ${\mathcal O}_{2^p}$, properties of the CAR algebra are studied in detail by restricting those of the Cuntz algebra. Various $\ast$-endomorphisms of the Cuntz algebra are explicitly constructed, and transcribed into those of the CAR algebra. In particular, a set of $\ast$-endomorphisms of the CAR algebra into its even subalgebra are constructed. According to branching formulae, which are obtained by composing representations and $\ast$-endomorphisms, it is shown that a KMS state of the CAR algebra is obtained through the above even-CAR endomorphisms from the Fock representation. A $U(2^p)$ action on ${\mathcal O}_{2^p}$ induces $\ast$-automorphisms of the CAR algebra, which are given by nonlinear transformations expressed in terms of polynomials in generators. It is shown that, among such $\ast$-automorphisms of the CAR algebra, there exists a family of one-parameter groups of $\ast$-automorphisms describing time evolutions of fermions, in which the particle number of the system changes by time while the Fock vacuum is kept invariant.
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