From Special Relativity to Feynman Diagrams.pdf

If we have several fields in some non trivial

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one field. If we have several fields in some non trivial representation of a group G , we need an additional index α. The extension of the previous formalism to several fields is, however, straightforward. For example the momenta conjugate to the fields are defined as π α ( x ) L ( x ) ∂ϕ α ( x ) . (8.223) Similarly, in defining the Poisson brackets, we need, besides the integration on the x variable, also a sum over the index α : { F , G } = α V δ F δϕ α ( x ) δ G δπ α ( x ) δ F δπ α ( x ) δ G δϕ α ( x ) d 3 x . (8.224) Furthermore, the relations ( 8.221 )–(8.699) generalize as follows: { ϕ α ( x , t ), π β ( x , t ) } = δ α β δ 3 ( x x ), (8.225) { ϕ α ( x , t ), ϕ β ( x , t ) } = { π α ( x , t ), π β ( x , t ) } = 0 . (8.226) Animportantcaseisthatoftworealscalarfields ϕ 1 , ϕ 2 which,asshownin Sect.8.8.1 , is equivalent to a single complex scalar field and its complex conjugate. In this case, using the real notation we have indices α, β = 1 , 2 . If however, as we shall mostly do in the next Chapters, we use the complex scalar fields ϕ( x , t ), ϕ ( x , t ) , then the Poisson brackets ( 8.225 ) become { ϕ( x , t ), π( y , t ) } = δ 3 ( x y ), (8.227) { ϕ ( x , t ), π ( y , t ) } = δ 3 ( x y ), (8.228) all the other Poisson brackets being zero. 8.9.1 Symmetry Generators in Field Theories We have seen in Sect.8.4.1 that the infinitesimal generators of continuous canonical transformations δθ r G r ( t ) generate transformations δ q i , δ p i leaving the Hamilton equations in the standard form ( 8.67 )–( 8.212 ). Moreover, when the Hamiltonian is left invariant , H = H , for each parameter θ r associated with the continuous symmetry group G , the infinitesimal generator G r ( t ) provides a constant of motion which coincides with the “charge” given by Noether theorem. The same of course applies for continuous theories described by fields, namely, the generators of canonical symmetry transformations of a field theory are precisely
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8.9 Hamiltonian Formalism in Field Theory 261 the conserved Noether charges. Therefore, in analogy with ( 8.92 ) and ( 8.93 ), we may write: δϕ α ( x , t ) = −{ ϕ α ( x , t ), G ( t ) } (8.229) δπ α ( x , t ) = −{ π α ( x , t ), G ( t ) } . (8.230) where G ( t ) δθ r G r ( t ). When the Hamiltonian is left invariant it coincides with the charge Q ( t ) δθ r Q r ( t ) of the Noether theorem. In the case of Poincaré transformations given by space–time translations and Lorentz transformations, let us show that the infinitesimal generator has the following form: G ( t ) = − μ P μ ( t ) + 1 2 δθ μν J μν ( t ), (8.231) where the explicit expression of the generators is obtained from ( 8.169 ) and ( 8.190 ) identifying L ˙ ϕ α π α : P ρ = π α ( x , t )∂ ρ ϕ α ( x , t ) η 0 ρ L ( x ) δ 3 x , (8.232) J ρσ = − π α ( x , t )( L ρσ ) α β ϕ β ( x , t ) + ( x ρ P σ x σ P ρ d 3 x . (8.233) Let us first consider the case of space–time translations, that is we take G ( t ) = μ P μ . Taking into account the fundamental Poisson brackets ( 8.225 ) and the gen- eral formulae ( 8.229 ), ( 8.230 ), we obtain: δϕ α = −{ ϕ α ( x , t ), [− ρ P ρ ( t ) ]} = ρ δ P ρ ( t ) δπ α ( x , t ) = ρ ρ ϕ α ( x , t ), (8.234) so that, for time or space translations we find, respectively: δϕ α = δ t { ϕ α ( x , t ), H ( t ) } = δ t t ϕ α , (8.235) δϕ α = i { ϕ α ( x , t ), P i ( t ) } = i i ϕ α = · ϕ α , (8.236) where ( i ).
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