PHYS
From Special Relativity to Feynman Diagrams.pdf

Set of vectors n n n a n b 1154 each of the above

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set of vectors: |{ N }; { N } ≡ |{ N } ( a ) ⊗ |{ N } ( b ) , (11.54) Each of the above states are constructed by repeatedly applying a and b operators to the “vacuum” state: | 0 ≡ | 0 , 0 , . . . , 0 ( a ) ⊗ | 0 , 0 , . . . , 0 ( b ) , (11.55) For instance a ( p ) | 0 = | 0 , . . . , 0 , 1 , 0 , . . . , 0 ( a ) ⊗ | 0 , . . . , 0 ( b ) , b ( p ) | 0 = | 0 , . . . , 0 ( a ) ⊗ | 0 , . . . , 0 , 1 , 0 , . . . , 0 ( b ) , where the position of the entry 1 corresponds to type- ( a ), respectively ( b ), oscillator state labeled by the momentum p . The states |{ N ( a ) }; { N ( b ) } are, by construction, eigenstates of all the number operators ˆ N ( a ) p , ˆ N ( b ) p and the Hilbert space they generate is called Fock space. Recall now the expression of the momentum operator ˆ P of the field in the contin- uous as well as in the discrete (i.e. finite volume) notations ˆ P = d 3 p ( 2 π ) 3 V p ˆ N ( a ) p + ˆ N ( b ) p p p ˆ N ( a ) p + ˆ N ( b ) p , (11.56) which completes, with the Hamiltonian operator ˆ H in ( 11.49 ), the four momentum operator: ˆ P μ = d 3 p ( 2 π ) 3 V p μ ˆ N ( a ) p + ˆ N ( a ) p p p μ ˆ N ( a ) p + ˆ N ( a ) p . (11.57) Just as we did for the quantized electromagnetic field, the quantum field states are interpreted as describing a multiparticle system: Each type- ( a ) and type- ( b ) oscil- lator defines a single particle state with definite momentum p and the occupation number N p is interpreted as the number of particles in that state. This time how- ever the quantized excitations of the field are described in terms of two kinds of particles , according to the type of oscillator. Conventionally those describing exci- tations of types- ( a ) and ( b ) oscillators are referred to as particles and antiparticles , respectively. For instance the state |{ N }; { N } describes N p 1 particles and N p 1 antiparticles with momentum p 1 ; N p 2 particles and N p 2 antiparticles with momentum p 2 , and so on.
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374 11 Quantization of Boson and Fermion Fields With this interpretation the quantum Hamiltonian and momentum operators are simply understood as the sum of the energies and momenta of the particles and antiparticles in the system, each carrying a quantum of energy E p and of momentum p . Every single-particle (antiparticle) state contributes to the energy and momentum of the total field state |{ N }; { N } an amount N p E p and N p p ( N p E p and N p p ), respectively, proportional to the corresponding occupation number. Therefore when this number varies by a unit, the total energy and momentum of the state vary by E p and p , respectively. It is important to note that even if antiparticles are associated with negative energy solutions to the classical Klein–Gordon equation, they contribute a positive energy E p to the Hamiltonian, that is antiparticles are positive energy particles. Let us observe in this respect that the photon, associated with the excitations of the electromagnetic field, coincides with its own antiparticle, since in that case, as often pointed out, the field ˆ A μ ( x ) is hermitian, thus implying a = b .
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