From Special Relativity to Feynman Diagrams.pdf

Sits in a tensor representation of the lorentz group

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sits in a tensor representation of the Lorentz group. For example, while the scalar field is a Lorentz singlet, the electromagnetic field A μ ( x ) transforms in the defining representation of the Lorentz group. A fermi- onic field, like the spin 1/2 Dirac field, transforms instead in the spinor representation (or for half-integer spins in higher spinor representations). In Chap.8 we have given the fundamental Poisson brackets between the classical field α ( x ) and its conjugate momentum density π α ( x ). We have also pointed out that, in a quantum theory, the dynamic variables α ( x ) and their conjugate momenta π α ( x ) are promoted to linear operators ˆ α ( x ), ˆ π α ( x ) acting on the Hilbert space of the physical states. As mentioned in the introduction their commutation properties depend on their being bosonic or fermionic. For every boson field φ α ( x ) the quantiza- tion procedure is effected through the canonical Heisenberg equal time commutation rules through the prescription ( 11.1 ). A bosonic quantum field theory will thus be characterized by the following commutators between the field operators : ˆ φ α ( x , t ), ˆ π β ( y , t ) = i δ α β δ 3 ( x y ), ˆ φ α ( x , t ), ˆ φ β ( y , t ) = ˆ π α ( x , t ), ˆ π β ( y , t ) = 0 . (11.2)
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11.2 Quantization of the Klein–Gordon Field 361 Taking into account that the complex conjugation of a classical dynamic variable must be replaced by the hermitian conjugate of the corresponding quantum operator, we can also write the hermitian conjugate counterparts of ( 11.2 ): ˆ φ α ( x , t ), ˆ π β ( y , t ) = i δ α β δ 3 ( x y ), ˆ φ α ( x , t ), ˆ φ β ( y , t ) = ˆ π α ( x , t ), ˆ π β ( y , t ) = 0 . (11.3) Note that the classical relation ( 8.208 ) now becomes ˆ π α ( x , t ) = t ˆ φ α ( x , t ) ; π α ( x , t ) = t ˆ φ α ( x , t ). (11.4) The same replacement ( 11.1 ) implies that the classical Hamilton equations, given in terms of the Poisson brackets in ( 8.221 ) and ( 8.222 ), at the quantum level become i t ˆ φ α ( x , t ) = ˆ φ α ( x , t ), ˆ H , i t ˆ π α ( x , t ) = ˆ π α ( x , t ), ˆ H , (11.5) where the Hamiltonian operator is obtained from the classical expression ( 8.209 ) and ( 8.210 ) by promoting the field variables to quantum operators. We note that this replacement implies time evolution in the quantum system to be described in the Heisenberg picture since the classical dynamic variables are time dependent. Thus the quantum state of the system is time independent. In this section we restrict our discussion to the dynamics of a free complex scalar field , which, as discussed in Sects.8.8.1 and 8.8.9 is equivalent to two real scalar fields. Since by definition a scalar field sits in the trivial representation of the Lorentz group, it corresponds to a spin-0 field, carrying no representation indices. Its classical description is given in terms of the Lagrangian ( 10.11 ) from which the classical Klein–Gordon equation ( 10.12 ) is derived. In that case, following ( 11.1 ), the Poisson brackets ( 8.226 ) become the equal-time commutators ( 11.2 ) with no indices α, β : ˆ φ( x , t ), ˆ π( y , t ) = i δ 3 ( x y ), ˆ φ ( x , t ), ˆ π ( y , t ) = i δ 3 ( x y ), (11.6) all the other possible commutators being zero.
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