States on Clifford algebras

Abstract

We study states on Clifford algebras from the point of view of C*algebras. A criterium is given under which the odd-point functions vanish. A particular set of states, called quasi-free states is extensively studied and explicit representations are given; as an application we give an approximate calculation of the ground state of a Fermion system.

Fermion mixing in quasifree states

“…Instead of the vacuum states described above we shall use a more general quasi-free state ω for the CAR algebra. As noted above, this is determined by the two-point correlation functions which define a complex linear operator R and a conjugate linear operator S by [3,4,5]. We note that if K = i(2R − 2S − 1) defines a complex structure on W , that is K 2 = −1, then the state ω is a Fock state for some choice of complex structure.…”

On optimal currents of indistinguishable particles

“…Therefore, when we rescale these parameters as g  A A and g  D D, it directly follows, from expression (50), that g  J J max max . The results of figure 2 imply that for any such value of γ, the current J(51) can be optimised, for l g  , by appropriately designing the system according to (58) and (57). However, because the value of the bound increases with γ, it is conceivable that a large value of γ (and thus large rates of particle exchange between system and reservoirs) can lead to large currents, even for slow coherent time scales, i.e., l g < .…”

“…Different states may, however, lead to inequivalent representations, which typically happens in the thermodynamic limit of many particle-systems. As an example one may consider Bardeen-Cooper-Schrieffer theory [51][52][53][54][55][56][57][58], where states with a finite particle density in the thermodynamic limit must be represented in a different Hilbert space than the Fock space which is constructed by exciting the physical vacuum (see section 2.1). The GNS construction is a key result in algebraic quantum physics, which stresses that the properties of the system's state are essential prerequisites to study physical models.…”

On the best constants in noncommutative Khintchine-type inequalities

“…It turns out (see [1] and [2]) that a gauge-invariant quasi-free state ω on A is completely determined by one truncated function ω T . More precisely, a functional ω T (·, ·) over the monomials in a * (f ) and a(g), ∀f, g ∈ H , which is linear in the first argument and conjugate-linear in the second determines a gauge-invariant quasi-free state ω on A if and only if…”