By Cvitanovic P., Artuso R., Dahlqvist P.
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Extra resources for Classical and quantum chaos
It states the absence of magnetic charges and Faraday’s law. For electrodynamics one has the freedom of gauge transformations. 6) the electric and magnetic fields are unchanged, Fµν −→ Fµν + (∂µ ∂ν χ − ∂ν ∂µ χ) = Fµν . 7) The equations of motion for the fields Aµ become ✷Aµ − ∂µ (∂ν Aν ) = jµ . 8) This equation is not affected by a gauge transformation. 10) of which the solutions give the Li´enard-Wiechert potentials. The equation in vacuum, ✷A µ = 0, moreover, shows that the electromagnetic fields correspond to massless particles.
The integrand of the first term must vanish, leading to the Euler-Lagrange equations, δL δL ∂µ = . 21) leading to (✷ + M 2 )φ(x) = 0. 24) which can be considered as the sum of the lagrangian densities for two scalar fields φ 1 and φ2 with φ √ = (φ1 + iφ2 )/ 2. One easily obtains (✷ + M 2 )φ(x) = 0, 2 ∗ (✷ + M )φ (x) = 0. 28) where the second line is not symmetric but in the action only differs from the symmetric version by a surface term (partial integration). 29) and similarly from the variation with respect to ψ ψ ← i∂ / +M = 0.
G. µνρσ will change sign, the aij elements of a tensor will not change sign. Examples are r −→ −r (vector), t −→ t (scalar), p −→ −p (vector), H J λ(p) K −→ H −→ J (scalar), (axial vector), −→ −λ(p) −→ −K (pseudoscalar), (vector). The behavior is the same for classical quantities, generators, etc. From the definition of the representations (0, 21 ) and ( 12 , 0) (via operators J and K) one sees that under parity 1 1 (0, ) −→ ( , 0). 1) In nature parity turns (often) out to be a good quantum number for elementary particle states.