By Cvitanovic P., Artuso R., Dahlqvist P.

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**Extra resources for Classical and quantum chaos**

**Example text**

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.