Symmetry and Noethers theorem

Noether s theorem will be proved here for a classical relativistic theory defined by a generic field j , which may have spinor or tensor indices. The Lagrangian density ( / , 9/x / ) is assumed to be Lorentz invariant and to depend only on scalar forms defined by spinor or tensor fields. It is assumed that coordinate displacements are described by Jacobi s theorem S(d4x) = d4x 9/x ix/l. The most general variation of the action integral, evaluated over a closed space-time region 2, is 

Any variation offtakes the form S p = S0 p + 9/( / 5x/x, where 5o omits coordinate variations. The full variation 8C is 

The last term here vanishes when / satisfies the field equations. Using CB Sx 1 + (B,C)8x11 = 3/x( 5x/l), the total variation of the action about a field solution takes the form 

Because the assumed hypervolume can be reduced to an infinitesimal, stationary or invariant action implies the local form of Noether s theorem, 3/27/2 = 0, an equation of continuity in space-time for the generalized current density determined by the field / . 

If the variations of both coordinates and field are determined by parameters denoted by aq, then 

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