By means of the action S=∫ t t 1 L d t′=S(x μ) as a function of the coordinates at the upper limit of integration and its uncertainty, the local gauge invariance — ψ is equivalent to ψ′=e -iqα(x μ) ψ is derived.

The hidden local gauge invariance in the one-dimensional Heisenberg XXZ model associated with general boundary K matrices is discussed.

The derivative expansion method and the gauge invariance;

The relationship between gauge invariance and the law of charge conservtion;

However,the kenetic equations are gauge invariance,so the derivative expansion method must not violate the aguge invariance.

From the point of symmetry of the lag rangian of clasiscal meChanics,and by making use of the gauge invariance,the Maxwell s equations have been derived.

A theoretical study and demonstration for Slavnon-Taylor identity in quantum chromodynamics were developed by the gauge invariance and Becchi-Rouet-Stora transformation,and it provided a good tool for the research and proof of the renormalization of non-Abelian gauge theory.

Based upon Weyl\'s electromagnetic theory,the paper gives a brief review of the construction of non-integrable phase factor and the theory of local gauge transformation in gauge field theory.