THE LOOP PHASE-FACTOR APPROACH TO GAUGE FIELDS

Abstract：

The gauge field theory is formulated via loop phase factors with a fixed point Oas their initial and final point. Let Gbe the gauge group. When the base space is the Minkowski space E ₄, we introduce a set of standard paths Ox（for example, the set of line segments Ox）, where x is arbitrary. The phase factor for the infinitesimal loop Oxx+ dxOcorresponds to an element in the Lie algebra gand can be expressed as a g-valued differential form k（ x, dx） which satisfies the following conditions of consistency （a） k（ O, dx）=0, （b） k（ x, v）=0, where v is the tangential vector of Oxat x. It is shown that an equivalent class of gauge fields is determined by k（ x, dx） or （ ad a） k（ x, dx） where a is a fixed element of G. Hence if we adopt k（ x, dx） for the fundamental physical quantity of a gauge field then a great part of gauge indefiniteness is eliminated. Moreover if the phase factors Φ_xofor standard paths Oxare given then the phase factors for differential arcs x x+ dxare easily calculated, and hence a gauge field in the equivalent class is extracted. We call the set of phase factors for standard paths a gauge and k（ x, dx） may be interpretated as the gauge potential under a special gauge under which Φ_xo=the unit element of G.The method is useful in considering the equivalence problem and the spacetime symmetry of gauge fields. For example, it is quite easy to determine all spherically symmetric gauge fields if they are free of singularities. By using the method it can also be proved that if two gauge fields have the same gauge and the same field strength then their gauge potentials are equal to each other. Consequently, under a given gauge in the above sense the field strength determines the gauge potential completely.For a general base manifold M_n, every equivalent class of gauge fields over M_ncan be defined by loop phase factors also. In this case, M_nis expressed as the sum of a set of neighborhoods each of which is homeomorphic to the Euclidean space. The standard paths are constructed according a certain rule, the phase factors for standard differential loops are also introduced. The transition functions and gauge potentials of a gauge field in the given equivalent class are derived as the phase factors for some finite loops and standard differential loops respectively. Further it is remarkable that a global gauge field is determined completely by the field strength and some discrete loop factors, if the phase factors for the standard paths are gwen.In mathematical terminology principal G-bundle structure as well as a connection in it is determined by the holonomic mapping which maps the set of loops through a fixed point into the group G, provided the mapping is differentible in a certain.The author is very grateful to Prof. Yang Chen Ning for many helpful discussions.