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.

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 Hamiltonian theory of Landau-Lifschitz equation and the gauge transformations;

Effects of gauge transformations on symmetries of Birkhoffian system

With the gauge transformation between AKNS spectral problem and Geng spectral problem,the associated extensive function derivative of the Geng spectral problem is obtained.

By making use of standard transformation,we try to solve the time equation.

Starting from equation of motion of the active RLC circuit, we adopt the technique of the unitary transformation, transform the charge and current into the unitary variable and then have them quantization by making use of standard transformation, we try to solve the having time Schrdinger equation.

By introducing the standard transformation, physical defects in the mathematical expression of engineering mechanical vectors are removed, and the standardized mechanical vectors that are equivalent to the mechanical tensors in physics are obtained, which lay the foundation of invariance of new type elastic mechanics, and provide a theoretical guarantee for using it in various coordinates.

Through the canonical transformation,we first established a harmonic oscillator model of black holes.

Though the canonical transformation,we first establish a harmonic oscillator model of black holes.

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