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The Noether's theorem in field theory takes the form of the conservation of current: where the four-current J^{}_{} is defined as:, where is the index for the space-time, _{} indicates the component of the field, and ^{} represents a small variation in the ^{th} parameter denoted as . From this conserved current, we can also establish a conserved
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## Figure 01d Gauss Theorem |
charge by integrating the equation over a volume in the three dimensional space. Applying the Gauss theorem (see Figure 01d), the term with spatial derivative can be converted into a surface integral, which would vanish |

,

which implies the charge

The Noether's theorem can be illustrated in more details by using the two-component Klein-Gordon Equation as an example. The explicit form of the Lagrangian density in this case is:

,

where

or, in a more familiar form: | , |

where N and N' represent the sums of number operators (in momentum space). In this form, N can be interpreted as the number of particles carrying charge -e (with various momentum), while N' is the number of anti-particles with charge +e. The sum of these numbers is a constant within a volume containing no source or sink.

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