Symmetry Breaking in Classical Particle Mechanics

The U(1) gauge field is usually induced from the gauge principle, that is, the extension of global U(1) phase transformation for matter field. However the phase itself is realized only for quantum theory. This makes us feel that the gauge fields are induced only in quantum theory by gauge principle. To make clear this point, we start from the non-relativistic classical particle, and we transform its action to the field theoretical form classically. We then have a multiplier field which insures the current conservation, and we get a global symmetry which translate this multiplier field. Using the gauge-principle to extend this global translation to the local one, we can introduce the gauge field, and obtain the minimal gauge coupling with matter. This shows that the gauge principle is alive for the symmetry of translation in classical theory without any phase variable to obtain the gauge field. Using this field theory we show the spontaneous symmetry breaking and Higgs mechanism which is similar to the super-conductivity. The relation of these two phenomenon is discussed. ∗Email: [email protected]