Gauge-invariant formulation of the dynamics of the quantized Yang-Mills field

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Abstract
The quantum theory of the Yang-Mills field is formulated in terms of gauge-invariant, path-independent potentials and conjugate momenta. These nonlocal variables are a generalization to the non-Abelian case of the gauge invariants used by Dirac in his gauge-invariant formulation of quantum electrodynamics, and they are a path-independent, symmetrically ordered modification of the Mandelstam-displaced operators. The commutation relations, constraints, and equations of motion satisfied by the gauge invariants are derived from a canonical foundation and are seen to form Schwinger's consistent system of symmetrically factor-ordered gauge-field equations. The gauge invariance of the equations is manifest since only gauge-invariant quantities are involved. All equations are satisfied strongly, and if gauge-invariant operators are used to raise states from a gauge-invariant vacuum, nonphysical states will not be introduced into the theory. A simple relation holds between the local, canonical variables and the gauge invariants which allows the energy-momentum tensor density to be expressed either in terms of the canonical variables or the gauge invariants. Elimination of the local canonical variables in favor of the gauge invariants shows, from a different point of view, the origin of the nonclassical terms in Schwinger's Hamiltonian and equations of motion. It is shown that these terms are necessary in order to satisfy integrability conditions on the field equations. Working with the canonical variables permits a straightforward evaluation of the energy-momentum density commutators which are needed to verify the Lie-algebra relations of the inhomogeneous Lorentz group and the local conservation of the energy-momentum density operator. The inhomogeneous Lorentz-group boost-transformation equations are derived in a manner natural to the development given here. Schwinger's transformations are found and his assertion of Lorentz invariance is confirmed.