Canonical Gauge- and Lorentz-Invariant Quantization of the Yang-Mills Field

Abstract

The Yang-Mills field, interacting with itself but not with other fields, is quantized using canonical equal-time commutation relations that do not involve the time component of the Yang-Mills potentials or the conjugate operator. Instead of decomposing the field into physical components and gauge variables, all three spatial components of the Yang-Mills potentials are retained. This keeps the equal-time commutators simple and provides, in the Schr\"odinger representation, a simple solution of the problem of constraints: "Good states, i.e., states satisfying all constraints, are represented by gauge-invariant functionals of the spatial Yang-Mills potentials. A finite scalar product for good states is given as a functional integral over a "tube" in configuration space, i.e., the space of the spatial Yang-Mills potentials over all of three-dimensional space. This tube is constructed in a certain manner around a manifold $\ensuremath{\Xi}$ of representatives for the gauge-invariant manifolds. It is shown that this scalar product does not depend on the choice of $\ensuremath{\Xi}$. For the purpose of proving Lorentz and gauge invariance of the theory, a modified Hamiltonian is considered which does not give rise to any constraints by itself, and which results in field equations which reduce to the Yang-Mills equations when applied to good states. The conventional primary and secondary constraints show up as conditions selecting the subspace of good states. Gauge invariance is proven for the complete theory, applied to good states. Lorentz invariance is proven for the equal-time commutation relations, using the method of Heisenberg and Pauli, for the conditions selecting the subspace of good states, and for the equations of motion applied to good states. Lorentz invariance of the scalar product of good states is shown by proving self-adjointness of the energy-momentum density operators, relative to the scalar product. The self-adjointness of these operators is assured by the particular construction of the tube in configuration space, over which the functional integral is taken. One may take the limit of the scalar product, letting the tube thickness go to zero, and arrive at a functional integral over $\ensuremath{\Xi}$.