Construction of currents in a Gauge-invariant spinor theory

Abstract

A 2-component scale-invariant Weyl-spinor theory involving noncanonical spinor fields of dimension 1/2 with quartic self-interaction in which the coupling constant is adjusted in such a way that—according to an earlier prescription—the theory is invariant under a phase-gauge transformation (gauge invariance of the second kind), is investigated. In this theory the formal «current» :ψ*σμψ(x) plays the role of the gauge-variant vector potential. The current operators are constructed as the finite local generators of the unitary transformation in the quantum-mechanical state space corresponding to constant phase transformations of the field operators. The gauge-invariant currents turn out of be of the form\( \sim :\psi *\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftrightarrow}$}} {\partial } _\aleph \sigma _\mu \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftrightarrow}$}} {\partial } ^\aleph \psi :(x)\). They are not conserved because to an Adler-type term which occurs in a definite form. This current is essentially identical to the gauge-invariant current\(:\psi _c^* \bar \sigma _\mu \psi _c :(x)\) if the canonical field is identified as ψc∼iσμ∂μψ. Expressions for the energy-momentum operatorTμν and the relativistic angular momentum operatorsMλμν are given and their conservation indicated. As a second step a 4-component Dirac theory with (v-a) coupling is considered which is adjusted to be invariant under chiral phase-gauge transformations (U1⊗U1 of the second kind). This theory is intimately related to the 2-component Weyl theory. Consequently one finds that both the vector and the axial vector currents invariant under both gauge transformations are not conserved. The conservation of the vector current, however, can be established by relaxing the invariance of the currents under γ5-gauge transformations whereas the conservation of the axial vector can only be arranged if the invariance under both gauge transformations is given up.