From Special Relativity to Feynman Diagrams.pdf

15 reference for further reading see refs 2 5 14 15

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15 Reference For further reading see Refs. [2, 5, 14] 15 We shall use the convention of denoting by a hatted symbol O the quantum mechanical operator acting on wave functions, associated with the observable O . Occasionally, for the sake of notational simplicity, the hat will be omitted, provided the operator nature of the quantity be manifest from the context.
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Chapter 8 Lagrangian and Hamiltonian Formalism InthischapterwegiveashortaccountoftheLagrangianandHamiltonianformulation of classical non-relativistic and relativistic theories. For pedagogical reasons we first address the case of systems of particles, described by a finite number of degrees of freedom. Afterwards, starting from Sect.8.5 , we extend the formalism to fields , that is to dinamical quantities described by functions of the points in space. Their consideration implies the study of dynamic systems carrying a continuous infinity of canonical coordinates, labeled by the three spatial coordinates. 8.1 Dynamical System with a Finite Number of Degrees of Freedom 8.1.1 The Action Principle Let us consider a mechanical system consisting of an arbitrary number of point- like particles. We recall that the number of coordinates necessary to determine the configuration of the system at a given instant, defines the number of its degrees of freedom. These coordinates are not necessarily the Cartesian ones, but are parameters chosen in such a way as to characterize in the simplest way the properties of the system. They are referred to as generalized coordinates or Lagrangian coordinates, usuallydenotedby q i ( t ), i = 1 , . . . , n , where n isthenumberofdegreesoffreedom. The space parameterized by the Lagrangian coordinates is the configuration space . Each point P in this space, of coordinates P ( t ) ( q i ( t )), ( i = 1 , . . . , n ) , defines the configuration of the system, that is the position of all the particles at a given instant. During the time evolution of the dinamical system the point P will therefore describe a trajectory in the configuration space. The mechanical properties of the system are encoded in a Lagrangian , that is a function of the Lagrangian coordinates q i ( t ) , their time derivatives ˙ q i ( t ) and the time t : R. D’Auria and M. Trigiante, From Special Relativity to Feynman Diagrams , 207 UNITEXT, DOI: 10.1007/978-88-470-1504-3_8, © Springer-Verlag Italia 2012
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208 8 Lagrangian and Hamiltonian Formalism L = L ( q i ( t ), ˙ q i ( t )), t ), q = ( q 1 , q 2 , . . . , q n ). Given the Lagrangian, the time evolution of the system is then derived from Hamilton’s principle of stationary action . Let us define the action S of the system as the integral of the Lagrangian along some curve γ in configuration space between two points corresponding to the con- figurations of the system at the instants t 1 , t 2 1 : S [ q ; t 1 , t 2 ] = t 2 (γ ) t 1 L ( q ( t ), ˙ q ( t ), t ) dt . (8.1) Notice that while L depends on the values of q i and ˙ q i at a given time t , S depends on the functions q i , namely on all the values q i ( t ), with t 1 t t 2 , defining a path γ in the configuration space. Thus, for fixed t 1 , t 2 , S is said to be a functional of q i .
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