Geometry of the electron local observables

Essential Fierz identities for a fermionic field

“…The 21 relations in Ref. [16] are not exactly the same as the 21 relations deduced directly from the algebraic structure of a regular spin 2-form S in this work. Here, both sets of 21 identities are compared, establishing that are linearly related between them and deduced from the sixteen essential relations given by the algebraic structure of S. Refs.…”

“…A similar type of relations involving two different spinors, in particular a field and its derivative [19], appears when the energy tensor of the field is analyzed [20], or in the formulation the Maxwell-Dirac equations based on the bilinear covariants [21]. For a compendium of historical references on this topic, see the introduction in [16] and [14], where a continuous media model governed by the Dirac equation is studied.…”

“…A fermionic field is usually described by a four-component spinor Ψ, Ψ(x) ∈ C 4 at each space-time event x. Given a basis of the space-time exterior algebra, say {Γ A } 16 A=1 , a set of sixteen bilinear forms, Ψ † Γ A Ψ, can be constructed from Ψ, where Ψ † is the hermitian conjugate spinor of Ψ. These bilinear forms reveal physical properties of the field and behave in a specific tensorial manner under Lorentz transformations.…”

“…Moreover, the products of any two bilinear forms, when expressed in terms of linear combinations of all of them, satisfy certain algebraic relations, which are usually referred as Fierz identities. They are derived from the completeness relations that give the canonical basis of the exterior algebra in terms of the basis under consideration {Γ A } 16 A=1 (see, for instance, [15][16][17]). The Fierz identities are also known as Pauli-Fierz identities [18].…”

“…By 'essential' we means non redundant, mathematically manageable, and able to retain the whole physical information of the Fierz identities. In [16], Minogin obtained a reduced set of 21 Fierz type relations, providing six different geometric representations for the 16 electron observables, depending on the chosen laboratory frame and the spinor field parameterization. The 21 relations in Ref.…”