From Special Relativity to Feynman Diagrams.pdf

In formulas we will write d 4 t 1 t 2 v m 4 since s

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In formulas we will write D 4 ≡ [ t 1 , t 2 ] × V M 4 . Since S does not depend only on the time interval [ t 1 , t 2 ] but also on the volume V in which the values of the fields and their derivatives are considered, we will write S S [ ϕ α ; D 4 ] . The boundary of D 4 , to be denoted by D 4 , consists of all the events occurring either at t = t 1 or at t = t 2 , and of events occurring at a generic t ∈ [ t 1 , t 2 ] in a point x belonging to the surface S V which encloses the volume V : x S V V . The measure of integration d 4 x dx 0 dx 1 dx 2 dx 3 = cdtd 3 x is invariant under Lorentz transformations = ( μ ν ) , since the absolute value | det ( ) | of the determinant of the corresponding Jacobian matrix , is equal to one: x μ −→ x μ = μ ν x ν d 4 x −→ d 4 x = | det ( ) | d 4 x = d 4 x . (8.112) It follows that in order to have a scalar Lagrangian density L must have the same dependence on ϕ α ( x , t ) as on ˙ ϕ α ( x , t ) , that is it must actually depend on the four-vector μ ϕ α ( x , t ). Moreover, being a scalar, it must depend on the fields and their derivatives μ ϕ α ( x , t ) only through invariants constructed out of them. For the same reason it cannot depend on t only, but, in general, on all the space–time coordinates x μ . Let us now consider arbitrary infinitesimal variations of the field ϕ α ( x ) which vanish at the boundary D 4 of D 4 : δϕ α ( x ) 0 if x D 4 . The corresponding variation of L can be computed by using ( 8.110 ): δ L = d 3 x L ( x , t ) ∂ϕ α ( x , t ) δϕ α ( x , t ) + L ( x , t ) ∂∂ i ϕ α ( x , t ) δ∂ i ϕ α ( x , t ) + L ( x , t ) ∂( ˙ ϕ α ( x , t )) δ ˙ ϕ α ( x , t ) = d 3 x L ( x , t ) ∂ϕ α ( x , t ) i L ( x , t ) ∂∂ i ϕ α ( x , t ) δϕ α ( x , t ) L ( x , t ) ˙ ϕ α ( x , t ) δ ˙ ϕ α ( x , t ) , (8.113)
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236 8 Lagrangian and Hamiltonian Formalism where we have written (∂ i ) i = 1 , 2 , 3 , used the property that δ∂ i ϕ α = i δϕ α and integrated the second term within the integral by parts, dropping the surface term, being δϕ α ( x ) = 0 for x S V V . Taking into account that the quantity inside the curly brackets defines the func- tional derivative of L , by comparison with ( 8.108 ) we find: δ L δϕ α ( x ) = L ( x ) ∂ϕ α ( x ) i L ( x ) ∂∂ i ϕ α ( x ) , δ L δ ˙ ϕ α ( x ) = L ( x ) ˙ ϕ α ( x ) . (8.114) Itisimportanttonotethat,usingthe Lagrangiandensity insteadoftheLagrangian,the derivatives of L ( x , t ) with respect to the fields in ( x , t ) are now ordinary derivatives , since they are computed at a particular point x . Using the equalities ( 8.114 ) the Euler–Lagrange equations ( 8.108 ) take the following form: t L ∂(∂ t ϕ α ) = L ∂ϕ α i L ∂(∂ i ϕ α ) , (8.115) or, using a Lorentz covariant notation: L ∂ϕ α μ L ∂(∂ μ ϕ α ) = 0 . (8.116) 8.5.2 The Hamilton Principle of Stationary Action In the previous paragraph the equations of motion for fields have been derived using the definition of functional derivative and performing the continuous limit of the Euler–Lagrange equations for a discrete system.
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