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25) In a bit more systematic way we obtain the dispersion: e−i(n− n )ωT = e|γ| 2 (e−iωT −1+iωT ) . 26) Expanding this to second order in ωT we get (n − n )2 = n 2 − n or 2 ≡ ( n)2 = |γ|2 = n , 1 n =√ . 28) For large quantum numbers, which corresponds to the classical limit, the fluctuations become relatively small. 56 5 Time-Cycle or Schwinger-Keldysh Formulation Now consider a more general variational statement than in Eq. 29) we see that since we can change the source functions at will, and make very localized changes, it makes sense to define the variational derivatives δ K± = |y+ (t)| K ± , ∗ (t) | δK + δ † | K ± = |y− (t)| K ± .
We shall supply it, in the framework of a particular example. The property of gauge invariance will be both a valuable guide, and an aid to simplifying the calculations. The time displacement of a complete physical system identifies its total energy. This suggests that time displacement of a part of the system provides energetic information about that portion. The ultimate limit of this spatial subdivision, a local description, should appear in response to an (infinitesimal) time displacement that varies arbitrarily in space as well as in time, δt (r, t).
Trivially, the field part of L remains unchanged. 53) p → p + ∇λ. 54) where w= a ea λ(ra , t). 5 Gauge Invariance and the Conservation of Charge 25 What is the physical consequence of adding a total time derivative to a Lagrangian? It does not change the equations of motion, so the system is unaltered. 56) the whole effect is a redefinition of the generators, G, G = G + δw. 57) This alteration reflects the fact that the Lagrangian itself is ambiguous up to a total time derivative term. 58) and apply this result to an infinitesimal gauge transformation, λ → δλ.