From Special Relativity to Feynman Diagrams.pdf

Let us indeed regard the values of ϕ x t at each

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Let us indeed regard the values of ϕ( x , t ) at each point x as independent canonical coordinates. To deal with a continuous infinity of canonical coordinates, we divide the 3-space into tiny cells of volume δ V i . Let ϕ i ( t ) be the mean value of ϕ( x , t ) inside the ith cell and L ( t ) = L i ( t ), ˙ ϕ i ( t ), t ) be the Lagrangian, depending on
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234 8 Lagrangian and Hamiltonian Formalism the values ϕ i ( t ), ˙ ϕ i ( t ) of the field and its dime derivative in every cell. The variation δ L i , ˙ ϕ i ) can be written as δ L i ( t ), ˙ ϕ i ( t ), t ) = i L ∂ϕ i δϕ i + L ˙ ϕ i δ ˙ ϕ i = i 1 δ V i L ∂ϕ i δϕ i + L ˙ ϕ i δ ˙ ϕ δ V i , (8.105) If we compare this expression with ( 8.104 ), in the continuum limit one can make the following identification: δ L δϕ( x , t ) lim δ V i 0 1 δ V i L ∂ϕ i , δ L δ ˙ ϕ( x , t ) lim δ V i 0 1 δ V i L ∂( ˙ ϕ i ) , (8.106) where x is in the i th cell. In the limit δ V i 0 we can set δ V i d 3 x . Thus the functional derivative δ L ( t )/δϕ( x , t ) is essentially proportional to the derivative of L with respect to the value of ϕ at the point x . Since in the discretized notation the action principle leads to the equations of motion: L ( t ) ∂ϕ i t L ( t ) ˙ ϕ i ( t ) = 0 (8.107) in the continuum limit the Euler–Lagrange equations become: δ L δϕ α ( x , t ) t δ L δ ˙ ϕ α ( x , t ) = 0 . (8.108) where we have reintroduced the index α of the general case. In the discretized notation we shall assume the Lagrangian L , which depends on the values of the fields and their time derivatives in every cell, to be the sum of quantities L i defined in each cell: L i depends on the values of the field ϕ α i ( t ) , its gradient ϕ α i and its time derivative ˙ ϕ α i ( t ) in the i th cell only: L α i ( t ), ˙ ϕ α i ( t ), t ) = i L i α i ( t ), ϕ α i ( t ), ˙ ϕ α i ( t ), t ). (8.109) Multiplying and dividing the right hand side by δ V i and taking the continuum limit δ V i d 3 x , the above equality becomes L α ( t ), ˙ ϕ α ( t ), t ) = V d 3 x L α ( x ), ϕ α ( x ), ˙ ϕ α ( x ) ; x , t ), (8.110) where x ( x μ ) = ( ct , x ) and we have defined the Lagrangian density L as
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8.5 Lagrangian and Hamiltonian Formalism in Field Theories 235 L α ( x ), ϕ α ( x ), ˙ ϕ α ( x ) ; x , t ) lim δ V i 0 1 δ V i L i α i ( t ), ϕ α i ( t ), ˙ ϕ α i ( t ), t ). Just as L i depends, at a time t , on the dynamic variables referred to the i -th cell only, L is a local quantity in Minkowski space in that it depends on both x and t . We note the appearance in L ( x ) of the space derivatives ϕ α ( x , t ). This follows from the fact that in order to have an action which is a scalar under Lorentz transformations, L itself must be a Lorentz scalar. Since Lorentz transformations will in general shuffle time and space derivatives, L should then depend on all of them. The action, in terms of the Lagrangian density, will read: S [ ϕ α ; t 1 , t 2 ] = t 2 t 1 L ( t ) dt = dtd 3 x L ( x ) = 1 c D 4 d 4 x L ( x ), (8.111) where D 4 is a space–time domain: An event x ( x μ ) in D 4 occurs at a time t between t 1 and t 2 and at a point x in the volume V .
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