Massive gauge field theory without Higgs mechanism
It has been shown that the massive non-Abelian gauge field theory in which all the gauge fields have the same mass may really be set up on the gauge-invariance principle without the help of the Higgs mechanism. The essential points of achieving this conclusion are : (1) The massive gauge field must be viewed as a constrained system in the whole space of vector potential. Therefore, the Lorentz condition, as a necessary constraint, must be introduced from the onset and imposed on the massive Yang-Mills Lagrangian so as to restrict the unphysical degrees of freedom involved in the Lagrangian. That is to say, the massive Yang-Mills Lagrangian itself is not complete to describe the dynamics of the massive gauge field; (2) The gauge-invariance of gauge field dynamics should be more generally required to the action of the field other than the Lagrangian because the action is of more fundamental dynamical meaning. In particular, the gauge-invariance for the constrained system should be required to the action written in the physical subspace defined by the Lorentz condition in which the fields exist and move only; (3) In the physical subspace , only the infinitesimal gauge transformations are possible to exist and necessary to be considered in inspection of whether the theory is gauge-invariant or not; (4) To construct a correct gauge field theory, the residual gauge degrees of freedom existing in the physical subspace must be eliminated by the constraint condition on the gauge group. This constraint condition may be determined by requiring the action to be gauge-invariant. Thus, the theory is set up from beginning to end on the gauge-invariance principle. These points which are not realized clearly in the past are important to build up a correct quantum massive non-Abelian gauge field theory. The quantization of the theory can well be performed in the Hamiltonian path-integral formalism or in the Lagrangian path-integral formalism. To achieve a Lorentz-covariant quantization, it is necessary to incorporate the Lorentz condition into the massive Yang-Mills Lagrangian so that each component of a vector potential acquires a canonically conjugate counterpart. This makes the Lorentz-covariant formulation of the Hamiltonian path-integral quantization become possible. The result of this quantization is confirmed by the quantization carried out in the Lagrangian path-integral formalism by means of the Lagrange undetermined multiplier method. The latter method of quantization is proposed first and proved to be equivalent to the Faddeev-Popov approach. The renormalizability and unitarity of the theory has also been exactly proved to be no problems.