Duality and self-duality of the spin-1 model in the covariant operator formalism

We perform the covariant operator quantization of the spin-$1$ model in $2+1$ spacetime dimensions to rigorously establish its dualities. For this purpose, the Kugo–Ojima–Nakanishi formalism, based on an indefinite metric Hilbert space in the Heisenberg picture, is used. We show that it is possible to extract a massive physical excitation constructed from a linear combination of the vector field $A_{\mu}$ and the $B$-field. In turn, we also show that this excitation generates the Maxwell–Chern–Simons theory. This is achieved by exploring the two-point function of the vector field.

WIGHTMAN FUNCTIONS IN THE COVARIANT OPERATOR FORMALISM AND THE OSTENDORF RULES

The Ostendorf rules, which generalize Feynman rules for τ functions to Wightman functions, are discussed in the light of the consistency with the operator solution and the energy-positivity condition (or the spectral condition). For the exactly solvable model called the one-loop model, it is shown that the Ostendorf rules are strictly justified.