Construction of a fermionic vacuum and the fermionic operators of creation and annihilation in the theory of algebraic s
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nstruction of a Fermionic Vacuum and the Fermionic Operators of Creation and Annihilation in the Theory of Algebraic Spinors V. V. Monakhov St. Petersburg State University, St. Petersburg, 198504 Russia e-mail: [email protected] Abstract⎯In complex modules over real Clifford algebras of even dimension, fermionic variables, which are an analogue of the Witt basis, are introduced. Based on them, primitive idempotents are built which represent the equivalent Clifford vacua. It is shown that modules of algebras are decomposed into a direct sum of minimal left ideals, generated by these idempotents, and that fermionic variables can be considered as more fundamental mathematical objects than spinors. DOI: 10.1134/S1063779617050318
Let us consider a complex module over a real Clifford algebra of the even dimension, n = 2m, of the signature ( p, q), n = p + q. We split the basis eα of the algebra generators into m pairs: e1 with e2 , e3 with e4 , …, and e2m −1 with e2m. We introduce the quantities sα = 1 with
( eα ) 2 = 1 and sα = i with ( eα ) 2 = –1. Then,
(s α )2 = (eα ) 2. In this case, {eα , eβ } = (s α )2 δβα . Here and after, except for Eq. (4), no summation over recurrent indices is carried out. We introduce the variables E α, such that eα = s α E α, E α = (s α ) −1eα, and (E α ) 2 = 1 for all α values, and the variables θ α and θα, which will be called fermionic variables: E − iE 2α E + iE 2α θ = 2α−1 , θα = 2α−1 . 2 2 α
(1)
From (1) it follows that (θ α )2 = (θα )2 = 0 and {θ α, θβ} = δ βα. In complex Clifford algebras Cl(2m), quantities like θ α and θα are referred to as the Witt basis [1, 2]. However, the Witt basis transforms by orthogonal rotations, while transformation of fermionic variables is performed using pseudo-orthogonal rotations which preserve (eα ) 2 for all α. The matrix representations θ α and θ α of fermionic variables θ α and θα are built from gamma matrices ⎛0 0⎞ ⎛0 1⎞ based on (1). For a pair, these are ⎜ and ⎜ ⎟ ⎟. ⎝1 0⎠ ⎝0 0⎠ For higher-dimensional Clifford algebras, we use the algorithm [2] for constructing gamma matrices in higher-dimensional spaces on the basis of a direct
product of 2 × 2 matrices. We will call this form of matrices of fermionic variables the canonical form. Algebraic spinors are the elements of the left ideal which is obtained from multiplying elements of the Clifford space by a primitive idempotent. As such an idempotent, we use
I V = θ1θ1 θ2θ 2...θmθ m.
(2)
Since (I V ) 2 = (θ1θ1) 2 (θ2θ 2 ) 2...(θmθ m ) 2 = I V , then I V is really an idempotent. The multipliers of the kind of θαθ α in (2) commute and are the Hermitian conjugates, therefore, = (θ1θ1 θ2θ 2...θmθ m ) +
(θmθ m )...(θ2θ 2 )(θ1θ1) = θ1θ1 θ2θ 2...θmθ m, i.e., it is the Hermitian conjugate, I V + = I V . Let us prove that I V is primitive. Let it be not the case, and I V consists of a sum of orthogonal idempotents: I V = I 1 + I 2, where (I 1) 2 = I 1, (I 2 ) 2 = I 2, and I 1I 2 = I 2I 1 = 0. Then, I 1I V = I 1(I 1 + I 2 ) = I 1 and I V I 1 = (I 1 + I 2 )I 1 = I 1. In this case, θk I 1 = θk I V
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