Sugawara Construction

In the late sixties, attempts to understand the strong interaction lead to constructions of field theories in terms of observable currents only, rather than using a Lagrangian. In these constructions, equal time commutators of currents (“current algebra”) serve as a substitute for the canonical commutation relations of “canonical fields” that appear in the Lagrangian. Sugawara [1] considered four-dimensional models with vector currents V ^a _μ (x) and axial currents A ^a _μ (x) transforming in the adjoint representation of SU(3), in which the stress-energy tensor θ_{μ ν} can be expressed in terms of the currents. He constructed a simple four-dimensional model in which the stress-energy tensor is quadratic in the currents and does not contain derivatives of currents:

where κ is the inverse of the Killing form of SU(3) and where Cis the eigenvalue of the central element of the current algebra. In this model the Heisenberg equations for the currents, conservation of energy and momentum,...