From Special Relativity to Feynman Diagrams.pdf

X x d 4 d 4 x l ϕ α x μ ϕ α x x d 4 d σ μ δ x

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x ), x ) D 4 d 4 x L α ( x ), ∂ μ ϕ α ( x ), x ) + D 4 d σ μ δ x μ L α ( x ), ∂ μ ϕ α ( x ), x ), (8.157) where, in the last integral, we have replaced L α ( x ), ∂ μ ϕ α ( x ), x ) with L α ( x ), μ ϕ α ( x ), x ) , since their difference, being multiplied by δ x μ would have been an infinitesimal of higher order (see the analogous equation( 8.54 )). On the other hand the difference between the first two integrals can be written as follows: ( 8.52 ): D 4 d 4 x L α ( x ), ∂ μ ϕ α ( x ), x ) L α ( x ), ∂ μ ϕ α ( x ), x ) = D 4 d 4 x L ∂ϕ α δϕ α + L ∂(∂ μ ϕ α ) δ∂ μ ϕ α = D 4 d 4 x L ∂ϕ α μ L ∂(∂ μ ϕ α ) δϕ α ( x ) + D 4 d 4 x μ L ∂(∂ μ ϕ α ) δϕ α . (8.158) where, as usual, we have applied the property δ(∂ μ ϕ α ) = μ δϕ α . Finally we substitute ( 8.157 ) and ( 8.158 ) into ( 8.152 ) obtaining, for the variation of the action (see ( 8.53 )): c δ S = D 4 d 4 x L ∂ϕ α μ L ∂(∂ μ ϕ α ) δϕ α ( x ) + D 4 d 4 x μ L ∂(∂ μ ϕ α ) δϕ α + D 4 d σ μ δ x μ L α ( x ), ∂ μ ϕ α ( x ), x ). (8.159) If the Euler–Lagrange equations( 8.121 ) are satisfied, the first integral in ( 8.159 ) vanishes; moreover the last integral can be written as an integral on D 4 by use of the four-dimensional Gauss (or divergence theorem) theorem in reverse: D 4 d σ μ δ x μ L = D 4 d 4 x μ x μ L ). (8.160) We have thus obtained: δ S = 1 c D 4 d 4 x μ L ∂(∂ μ ϕ α ) δϕ α + δ x μ L . (8.161)
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8.7 Symmetry and the Noether Theorem 249 The above equation gives the desired result: it states that when δ S = 0, the integral in ( 8.161 ) is zero. Taking into account that the integration domain is arbitrary, we must have: μ J μ = 0 , (8.162) where J μ = L ∂(∂ μ ϕ α ) δϕ α + δ x μ L . (8.163) In terms of the infinitesimal, global parameters δθ r , r = 1 , . . . , g of the contin- uous transformation group G, the infinitesimal variations δϕ α and δ x μ can be written as δϕ α = δθ r α r ; δ x μ = δθ r X μ r . (8.164) where α r and X μ r are, in general, functions of the fields ϕ α and coordinates x μ . Thus we may write: J μ = δθ r J μ r , where J μ r = L ∂(∂ μ ϕ α ) α r + X μ r L . (8.165) Taking into account that the δθ r are independent, constant parameters, we can state that we have a set of g conserved currents μ J μ r = 0 . To each conserved current J μ r there corresponds a conserved charge Q r : Q r = R 3 d 3 x J 0 r , (8.166) where we take as V the entire three-dimensional space R 3 . Indeed: dQ r dt = c R 3 d 3 x x 0 J 0 r = − R 3 d 3 x x i J i r = − S d 2 σ 3 i = 1 J i r n i = 0 . where the last surface integral is zero being evaluated at infinity where the currents are supposed to vanish.
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250 8 Lagrangian and Hamiltonian Formalism 8.8 Space–Time Symmetries As already stressed in the first Chapters of this book, in order to satisfy the princi- ple of relativity a physical theory must fulfil the requirement of invariance under the Poincaré group . The latter was discussed in detail in Chap.4 and contains, as subgroups, the Lorentz group and the four-dimensional translation group. Invariance
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