Examples of conservation laws

Consider an infinitesimal displacement Sa1 in space-time. The action is invariant if for x = x + Sa, p (x ) = / (x) for a field / , where the displaced field function 

The conserved current density determined by Noether s theorem is 

These formulas are valid for the nonrelativistic one-electron Schrodinger equation. The Lagrangian density is 

Under a Lorentz transformation A, a covariant field pa(x) transforms according to 

For an infinitesimal transformation, the irreducible representation matrix is Dab = Sab + SDab, and the coordinate transformation returning to the original coordinate value is (A 1) x,v = x 1 - )Jdx v = x 1. Defining the transformation with this reverse step makes hx1 = 0, while the functional form of the field changes according to 

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