From Special Relativity to Feynman Diagrams.pdf

For infinitesimal canonical transformations generated

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For infinitesimal canonical transformations generated by J μν we find δϕ α = − δθ ρσ 2 { ϕ α ( x , t ), J ρσ ( t ) } = − δθ ρσ 2 δ J ρσ δπ α ( x , t ) = δθ μν 2 ( L μν ) α β ϕ β + ( x μ ν x ν μ α . (8.237) Let us now compute the infinitesimal change of the Hamiltonian:
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262 8 Lagrangian and Hamiltonian Formalism δ H = H (ϕ , π ) H (ϕ, π) = α δ H δϕ α δϕ α + δ H δπ α δπ α d 3 x . = − α δ H δϕ α δ G δπ α δ H δπ α δ G δϕ α d 3 x = −{ H , G ( t ) } . (8.238) If the transformations are a symmetry of the Hamiltonian, δ H = G t , see (8.99) and (8.100), and we recover the result that G ( t ) is a conserved quantity : dG dt = { G ( t ), H } + G t = 0 , (8.239) As an example let us consider Lorentz boosts for which δ H = 0 since it transforms as the 0-component of the four vector P μ ; infinitesimally we have: δ P 0 1 c δ H = − θ 0 i { H , J 0 i } . (8.240) On the other hand δ P 0 = δθ 0 μ P μ = δθ 0 i P i , so that combining the two expressions of δ P 0 we find: { H , J 0 i } = − cP i . Now if we considerthecomponent0 i of( 8.233 )weseethatwhentheLorentzindex ρ = 0,itcar- ries an explicit time dependence in the second term, namely d 3 x ( x 0 P i x i P 0 ) . It follows: d J 0 i dt = −{ H , J 0 i } + J 0 i t = cP i c d dt d 3 x t P i = cP i cP i = 0 , (8.241) and therefore J 0 i is also conserved, in agreement with the Noether theorem. Reference For further reading see Refs. [1], [2] (Vol. 1)
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Chapter 9 Quantum Mechanics Formalism 9.1 Introduction In this chapter we give a concise review of the quantum mechanics formalism from a perspective which generalizes the ordinary Schroedinger formulation. In this way we may reconsider the Schroedinger approach to quantum mechanics from a more geometrical and group-theoretical point of view and show the close relationship between the classical Hamiltonian theory and quantum mechanics. Moreover the formalism developed in this Chapter will be useful for an appropriate exposition of the relativistic wave equations in Chap.10 and the field quantization approach in Chap.11 . 9.2 Wave Functions, Quantum States and Linear Operators In elementary courses in quantum mechanics the state of a system is described by a wave function ψ α (ξ, t ) where the variables ξ denote the set of the coordinates on which the wave function depends and the suffix α refers to a set of (discrete) phys- ical quantities, or quantum numbers, which, together with ξ, define the state of the system. In the Schroedinger approach the variables ξ comprise the space coordinates x = ( x , y , z ) while, if spin is present, the variable α labels the corresponding polarization state. In this case ψ α ( x ; t ) is referred to as the wave function in the coordinate representation . In this section we wish to adopt the Dirac formalism which allows a quantum description of a system that is independent of its explicit coordinate representation . Since in all the considerations of this section we refer to states at a particular instant t , the time coordinate will not be indicated explicitly.
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