Boson Field Description of Fermions in Two Dimensions

Abstract

It is shown that any interacting spinor system in two dimensions can be equivalently described by the scalar system. The equivalency is complete in the sense that both theories have the common Hamiltonian. In the solvable case like the massless Thirring model, the scalar field version of the Hamiltonian is bilinear on the field variables. It becomes of the standard form of the free Hamiltonian by the suitable choice of the field variables. The spinor field version of the Hamiltonian also becomes of the standard form of the free Hamiltonian by the corresponding choice of the field variables. The canonical Hamiltonian in the spinor version does not necessarily satisfy the integrability condition. Then the use of the integrable Hamiltonian is the essential point in this paper. The vector-meson system is also describable by the scalar system. § I. Introduction Recently, quantum field theories of extended objects have been discussed in connection with the dynamics underlying duality. Among various attempts in this direction the so-called soliton solutions in two-dimensional nonlinear field the­ ories have interested us as instructive ones. In particular Colemann has shown that the quantized sine-Gordon system is equivalent to the charge-zero sector of the massive Thirring model and has conjectured that the quantum soliton of the sine-Gordon equation is the fermion of the massive Thirring model. After a while, the above conjecture has been justified by Mandelstam.> Then, the fermion system in the massive Thirring model can be equivalently described by the scalar field. It is one of the most important characters of the theories that two theories, being apparently different from each other, are equivalent. Therefore how to rec­ ognize the equivalence character is a matter of consequence. At this point, we want to note the parallelism between