Nonlinear Transformation Group of CAR Fermion Algebra

Based on our previous work on the recursive fermion system in the Cuntz algebra, it is shown that a nonlinear transformation group of the CAR fermion algebra is induced from a U(2) action on the Cuntz algebra O2p with an arbitrary positive integer p. In general, these nonlinear transformations are expressed in terms of finite polynomials in generators. Some Bogoliubov transformations are involved as special cases. E-mail: [email protected] E-mail: [email protected] – 1 – In our previous paper, we have introduced the recursive fermion system (RFSp) which gives embeddings of CAR into the Cuntz algebra O2p with an arbitrary positive integer p. As for the special case, which we call the standard RFSp, CAR is embedded onto the U(1)-invariant ∗-subalgebra O U(1) 2p of O2p. Here, the U(1) action on Od is defined by γz : si 7→ z si, z ∈ C, |z| = 1 (1) with si (i = 1, . . . , d) being the generators of Od. Since an automorphism of O2p , which is described by a U(2) action, also keeps O U(1) 2p invariant, it induces a transformation of CAR. We find this type of transformations is nonlinear with respect to generators {am, a ∗ n |m, n = 1, 2, . . . } in general, and includes some Bogoliubov transformations as special cases. First, let us recall that the Cuntz algebra Od is a C -algebra generated by si, i = 1, 2, . . . , d, which satisfy the following relations: s∗i sj = δi,jI, (2) d ∑ i=1 si s ∗ i = I. (3) We often use the brief description such as si1i2···im; jn ··· j2 j1 ≡ si1si2 · · · sims ∗ jn · · · s∗j2s ∗ j1 , m+n ≧ 1. The U(1)-invariant subalgebra O U(1) d of Od is given by a linear space spanned by si1···im; jm···j1, m = 1, 2, . . . . We consider an automorphism αu of Od obtained from a natural U(d) action α as follows: α : U(d) y Od, αu(si) ≡ d ∑ k=1 skuki, u ∈ U(d), i = 1, 2, . . . , d. (4) Indeed, from the equality αu1 ◦ αu2 = αu1u2 , u1, u2 ∈ U(d), (5) α becomes a U(d) action on Od. Since αu ◦ γz = γz ◦ αu, restriction of αu to O U(1) d gives an automorphism of O U(1) d : αu|OU(1) d ∈ AutO U(1) d , αu(si1···im; jm···j1) = d ∑ k1,... ,lm=1 uk1i1 · · ·ukmimu ∗ j1l1 · · ·u∗jmlmsk1···km; lm···l1 ∈ O U(1) d . (6) Using the homogeneous embedding Ψ of Odr (r = 2, 3, . . . ) into Od defined by 3) Ψ : Odr →֒ Od, Ψ (Si) ≡ sj1 · · · sjr , i = 1, 2, . . . , d ; j1, . . . , jr = 1, 2, . . . , d, i− 1 = r ∑