From Special Relativity to Feynman Diagrams.pdf

For the lagrangian formalism consider a theory

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for the Lagrangian formalism Consider a theory describing a field ϕ( x ) (let us suppress the internal index α for the time being). Just as we did in Sect.8.5.1 , we divide the three-dimensional domain V in which we study the system, into tiny cells of volume δ V i , defining the Lagrangian coordinates ϕ i ( t ) as the mean value of ϕ( x , t ) within the i th cell. We thus have a discrete dynamic system and define the momenta p i conjugate to ϕ i as p i = L ( t ) ˙ ϕ i ( t ) . (8.204) The Hamiltonian of the system is given by H = i p i ˙ ϕ i L , (8.205) with equations of motion: ˙ ϕ i = H p i ; ˙ p i = − H ∂ϕ i . (8.206) Recall now from the discussion in Sect.8.5.1 that, in the continuum limit ( δ V i infinitesimal)
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258 8 Lagrangian and Hamiltonian Formalism p i ( t ) = L ˙ ϕ i ( t ) = δ V i δ L δ ˙ ϕ( x , t ) = δ V i L ( x , t ) ˙ ϕ( x , t ) = δ V i π( x , t ), (8.207) where x δ V i and we have defined the field π( x , t ) as π( x , t ) L ( x , t ) ˙ ϕ( x , t ) , (8.208) so that p i ( t ) represents the mean value of π( x , t ) within the i th cell δ V i . The field π( x ) is the momentum conjugate to the ϕ( x ) . Expressing p i ( t ) in terms of π( x ) through ( 8.207 ), upon identifying in the continuum limit δ V i = d 3 x , we may write the Hamiltonian ( 8.205 ) as H = V [ π( x ) ˙ ϕ( x ) L ( x ) ] d 3 x , (8.209) where we have used the definition ( 8.110 ) of Lagrangian density. The integrand in the above equation: H = π( x ) ˙ ϕ( x ) L ( x ), (8.210) defines the Hamiltonian density . Using the notion of functional derivative, the Hamilton equations of motion can be derived in a way analogous to ( 8.106 ): δ H ( t ) δϕ( x , t ) = lim δ V i 0 1 δ V i H ( t ) ∂ϕ i ( t ) , δ H ( t ) δπ( x , t ) = lim δ V i 0 1 δ V i H ( t ) ∂π i ( t ) , (8.211) and combining ( 8.106 ), ( 8.211 ) with ( 8.206 ) and ( 8.207 ) we obtain: ˙ π( x ) = − δ H ( t ) δϕ( x ) = − H ( x ) ∂ϕ( x ) , (8.212) ˙ ϕ( x ) = δ H ( t ) δπ( x , t ) = H ( x ) ∂π( x ) . (8.213) Using the same limiting procedure one can see that the Poisson brackets of two functionals F [ ϕ, π ] , G [ ϕ, π ] is defined as { F , G } = V δ F δϕ( x ) δ G δπ( x ) δ F δπ( x ) δ G δϕ( x ) d 3 x , (8.214) so that, the time derivative of F gives:
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8.9 Hamiltonian Formalism in Field Theory 259 ˙ F ( t ) = F t + V δ F δϕ( x ) ˙ ϕ( x ) + δ F δπ( x ) ˙ π( x ) d 3 x = F t + V δ F δϕ( x ) δ H δπ( x ) δ F δπ( x ) δ H δϕ( x ) d 3 x = F t + { F , H } , (8.215) where we have used ( 8.212 )–( 8.213 ). In particular, if F does not have an explicit dependence on time: ˙ F ( t ) = { F , H } . In this case the dynamic variable F is conserved if and only if its Poisson bracket with the Hamiltonian vanishes. Writing: ϕ( x , t ) = V δ 3 ( x x )ϕ( x , t ) d 3 x , π( x , t ) = V δ 3 ( x x )π( x , t ) d 3 x , from the definition of functional derivative we have: δϕ( x , t ) δϕ( x , t ) = δπ( x , t ) δπ( x , t ) = δ 3 ( x x ). (8.216) Applying this relations we find: { ϕ( x , t ), H } = δ H δπ( x , t ) , (8.217) { π( x , t ), H } = − δ H δϕ( x , t ) , (8.218) and using the Hamilton ( 8.212 )–( 8.213 ): ˙ ϕ( x , t ) = { ϕ( x , t ), H } , (8.219) ˙ π( x , t ) = { π( x , t ), H } . (8.220) From ( 8.216 ) we also derive the fundamental relations: { ϕ( x , t ), π( x , t ) } = δ 3 ( x x ), (8.221) { ϕ( x , t ), ϕ( x , t ) } = { π( x , t ), π( x , t ) } = 0 . (8.222)
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260 8 Lagrangian and Hamiltonian Formalism In order to simplify notation, we have developed the Hamilton formalism using just one field. If we have several fields in some non trivial representation of a group
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