From Special Relativity to Feynman Diagrams.pdf

16 somewhat improperly by the word representation

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16 Somewhat improperly, by the word representation people often refer to the carrier space V p of a representation. We shall also do this to simplify the exposition and thus talk about a basis of a representation when referring to a basis of the corresponding carrier space.
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8.5 Lagrangian and Hamiltonian Formalism in Field Theories 233 on the values of the fields ϕ α ( x , t ) and t ϕ α ( x , t ) at every point in the domain V of the three-dimensional space. We say in this case that the Lagrangian is a functional of ϕ α ( x , t ) and t ϕ α ( x , t ) , viewed as functions of x . It will be convenient in the following to denote by ϕ α ( t ) the function ϕ α ( x , t ) of the point x in space at a given time t , and by ˙ ϕ α ( t ) its time derivative ˙ ϕ α ( x , t ) t ϕ α ( x , t ) . We shall presently explore some property of functionals. Let us consider a functional F [ ϕ ] , and perform an independent variation of ϕ( x ) , at each space point x . The corresponding variation of F [ ϕ ] will be: δ F [ ϕ ] ≡ F [ ϕ + δϕ ] − F [ ϕ ] = δ F [ ϕ ] δϕ( x ) δϕ( x ) d 3 x , (8.103) where by definition , δ F [ ϕ ] δϕ( x ) is the functional derivative of F [ ϕ ] with respect to ϕ at the point x . Here we have suppressed the possible dependence on time of ϕ and of the functional F either explicitly or through ϕ : ϕ = ϕ( x , t ), F = F [ ϕ( t ), t ] . From its definition it is easy to verify that the functional derivation enjoys the same properties as the ordinary one, namely it is a linear operator, vanishes on constants and satisfies the Leibnitz rule. When the functional depends on more than a single function, its definition can be extended correspondingly, as for ordinary derivatives. Of particular relevance for us is the additional dependence of F on the time derivative t ϕ( x , t ) of ϕ( x , t ). Moreover we may consider a set of fields ϕ α labeled by the index α pertaining to a given representation of a group G. This is the case of the Lagrangian F = L α ( t ), ˙ ϕ α ( t ), t ) , where we recall once again that, in writing ϕ( t ), ˙ ϕ( t ) among the arguments of the Lagrangian, we mean that L depends on the values ϕ( x , t ), ˙ ϕ( x , t ) of these fields in every point x in space at a given time t . Applying the definition ( 8.103 ) to the two functions ϕ α ( t ) and ˙ ϕ α ( t ) we have: δ L α ( t ), ˙ ϕ α ( t ), t ) = d 3 x δ L δϕ α ( x , t ) δϕ α ( x , t ) + δ L δ ˙ ϕ α ( x , t ) δ ˙ ϕ α ( x , t ) . (8.104) Note that the Lagrangian depends on t either through ϕ α and ˙ ϕ α or explicitly. The Lagrangian, as a functional with respect to the space-dependence of the fields, can be thought of as the continuous limit of a function of infinitely many discrete variables: L i ( t ), ˙ ϕ i ( t ), t ) i x −→ L (ϕ( t ), ˙ ϕ( t ), t ). Here and in the following we shall often omit the index α if not essential to our considerations. Correspondingly, we can show that the functional derivative defined above can be thought of as a suitable continuous limit of the ordinary derivative with respect to a discrete set of degrees of freedom q i , described by a Lagrangian L ( q i , ˙ q i ).
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