From Special Relativity to Feynman Diagrams.pdf

To derive the quantum equations of motion we first

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all the other possible commutators being zero. To derive the quantum equations of motion we first need to compute the Hamiltonian operator. Recall that the classical Hamiltonian density is given by ( 10.46 ), that here we rewrite here for convenience: H = ππ + c 2 φ φ + m 2 c 4 2 φ φ. (11.7) Caution is however required when trading the classical fields in the above expression by their quantum counterparts ˆ ϕ( x ), ˆ ϕ x ), ˆ π( x ), ˆ π ( x ), since operators appearing
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362 11 Quantization of Boson and Fermion Fields in products in ( 11.6 ), being computed at the same space–time point x μ = y μ , do not in general commute (see Sect.11.4 below). This implies that a certain order must be chosen. The convention we shall use will be shown in the sequel to lead to consistent results in the development of the theory. It consists in the following substitutions: ( classical fields ) ( field operators ), π ( x )π( x ) ˆ π( x ) ˆ π ( x ), φ ( x )φ( x ) ˆ φ ( x ) ˆ φ( x ), φ ( x ) · φ( x ) ˆ φ ( x ) · ˆ φ( x ). (11.8) The resulting Hamiltonian operator reads: ˆ H = d 3 x ˆ π( x ) ˆ π ( x ) + c 2 ˆ φ ( x ) · ˆ φ( x ) + m 2 c 4 2 ˆ φ ( x ) ˆ φ( x ) . (11.9) Let us now use this Hamiltonian in the quantum Hamilton’s equations ( 11.5 ) i t ˆ φ( x , t ) = ˆ φ( x , t ), ˆ H ; i t ˆ π( x , t ) = ˆ π( x , t ), ˆ H . (11.10) and show that it reproduces the quantum version of the classical Klein–Gordon equation. Applying ( 11.6 ) to the the first of ( 11.10 ) we find i t ˆ φ( y , t ) = ˆ φ( x , t ), ˆ H ( t ) = d 3 y ˆ φ( x , t ), ˆ π( y , t ) ˆ π ( y , t ) = i ˆ π ( x , t ), (11.11) which is the expression ( 11.4 ) of the conjugate momentum operators. The same computations applied to the last of ( 11.10 ) (or better to its hermitian conjugate), yields i t ˆ π ( x , t ) = c 2 d 3 y ˆ π ( x , t ), ˆ φ y i ˆ φ y i ( y , t ) + m 2 c 2 2 d 3 y ˆ π ( x , t ), ˆ φ ( y , t ) ˆ φ( y , t ) = − c 2 d 3 y ˆ π ( x , t ), ˆ φ ( y , t ) 2 ˆ φ( y , t ) i m 2 c 2 ˆ φ( x , t ) = i c 2 2 ˆ φ m 2 c 2 2 ˆ φ ( x , t ). (11.12) Substituting in the left hand side the value of π given by ( 11.11 ) we obtain
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11.2 Quantization of the Klein–Gordon Field 363 1 c 2 2 ˆ φ t 2 − ∇ 2 ˆ φ m 2 c 2 2 ˆ φ = 0 , (11.13) so that the quantum field operator obeys the same Klein–Gordon equation as its classical counterpart. In an analogous way we reproduce the quantum version of equation( 10.52 ) ˆ P ( t ) = − d 3 y ˆ π( y ) ˆ φ( y ) + ˆ φ( y ) ˆ π ( y ) , (11.14) where y = ( ct , y ), which yields the right transformation property of the field operator under infinitesimal space-translations (see ( 9.39 )) δ ˆ φ( x , t ) = i ˆ φ( x , t ), · ˆ P ( t ) = − i d 3 y ˆ φ( x , t ), ˆ π( y , t ) · ˆ φ( y , t ) = · ˆ φ( x , t ). (11.15) We can thus also write: i ˆ φ( x , t ) = ˆ φ( x , t ), ˆ P ( t ) . (11.16) Recalling that ˆ P is the three-dimensional counterpart of the four-momentum ˆ P μ = ( ˆ H / c , ˆ P ) of the field, ( 11.13 ) and ( 11.16 ) can be written in a Lorentz covariant form as ˆ φ( x , t ), ˆ P μ ( t ) = i μ ˆ φ( x , t ) = i η μν ν ˆ φ( x , t ). (11.17) Solving the quantum Klein–Gordon theory means to explicitly construct the Hilbert space of states V ( c ) and the dynamic variables ˆ φ α , ˆ
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