From Special Relativity to Feynman Diagrams.pdf

We conclude that the operators a p and b p create a

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We conclude that the operators a ( p ) and b ( p ) create a particle and an antiparticle with momentum p , respectively, while a ( p ) and b ( p ) destroy them. In an analogous way, denoting by ˆ φ + ( x ) and ˆ φ ( x ) the positive and negative energy components of the field operator ˆ φ( x ) in ( 11.49 ): ˆ φ( x ) = ˆ φ + ( x ) + ˆ φ ( x ), ˆ φ + ( x ) = d 3 p ( 2 π ) 3 V 2 E p a ( p ) e i p · x = p 2 E p V a ( p ) e i p · x , ˆ φ ( x ) = d 3 p ( 2 π ) 3 V 2 E p b ( p ) e i p · x = p 2 E p V b ( p ) e i p · x , (11.58) the former destroys a particle at the space–time point x ( x μ ) (since it contains a ( p ) ) while the latter creates an antiparticle at x (since it contains b ( p ) ). The reverse is true for ˆ φ ( x ), ˆ φ + ( x ), defined as the negative and positive energy components of ˆ φ ( x ), respectively: ˆ φ ( x ) = d 3 p ( 2 π ) 3 V 2 E p a ( p ) e i p · x , ˆ φ + ( x ) = d 3 p ( 2 π ) 3 V 2 E p b ( p ) e i p · x , (11.59) It is implicit from the above discussion that we are working in the Heisenberg picture in which operators, like ˆ φ( x , t ) depend on time while states are constant. This is necessary in order to have a relativistically covariant framework, see Sect.6.2.1 of Chap.6 . The Fock space formalism is particularly suited for providing a multiparticle description of a quantum relativistic free field theory. It is however interesting to write down the familiar non-relativistic wave function of a system of particles, using
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11.2 Quantization of the Klein–Gordon Field 375 the x -representation instead of the Fock representation. We define a state describing n particles located at the points x 1 , . . . , x n at a time t as | x 1 , . . . , x n ; t ˆ φ ( x 1 , t ) . . . ˆ φ ( x n , t ) | 0 , (11.60) where the effect of ˆ φ ( x i , t ) is that of creating a particle in x i at the time t . On the other hand a generic n particle state in the Fock-representation is defined as | N 1 , N 2 , . . . ( a ) = ( a 1 ) N 1 ( a 2 ) N 2 . . . ( N 1 ! N 2 ! . . . ) 1 2 | 0 ( a ) , (11.61) where N 1 + N 2 + · · · = n and we have used the short-hand notation N 1 N p 1 , N 2 N p 2 , and so on. The wave function φ ( n ) N 1 , N 2 ,... ( x 1 , . . . , x n , t ) realizing the coordinate representation of the state ( 11.61 ) therefore reads φ ( n ) N 1 , N 2 ,... ( x 1 , . . . , x n , t ) = x 1 , . . . , x n ; t | N 1 , N 2 , . . . ( a ) . (11.62) By the same token we construct the coordinate representation of a multi-antiparticle state | N 1 , N 2 , . . . ( b ) or of a generic particle-antiparticle state |{ N }; { N } . We con- clude from this that the multi-particle wave function (describing both particles and antiparticles) is completely symmetric with respect to the exchange of the particles (antiparticles) since the operators ˆ φ ( x i , t ) and ˆ φ ( x j , t ) ( ˆ φ ( x i , t ) and ˆ φ ( x j , t ) ) commute. In other words: Spin-zero particles obey the Bose–Einstein statistics . This result, which obviously holds for the photon field, can be shown to be valid for all particles of integer spin.
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