From Special Relativity to Feynman Diagrams.pdf

To make contact with our previous non relativistic

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To make contact with our previous non-relativistic discussion, we have introduced the basis of vectors | x , α and used it to define the relativistic wave-function. There are however problems in defining such states within a relativistic framework. Let us mention some of them:
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292 9 Quantum Mechanics Formalism • The state | x , α would describe a particle located in x at a time t . Due to the possibility, in relativistic processes, of creating new particles, provided the energy involved is large enough, and in the light of Heisenberg’s uncertainty principle, there is a physical obstruction in determining the position of a particle at a given time with indefinite precision: The smaller the distances we wish to probe, in order to locate a particle with a sufficiently high precision, the larger the momentum and thus the energy we need to transfer to the particle and, if the energy transferred is large enough to produce one or more particles identical to the original one, we may end up with a system of virtually undistinguishable particles, thus making our initial position measurement meaningless. This is also related to the problem with interpreting the relativistic field α ( x ) as the wave function associated with a given single-particle state, like we did in the non-relativistic theory. We shall comment on this in some more detail in the introduction to next chapter. • The normalization ( 9.95 ), which guarantees that all the states of the basis have a positivenorm,isnotLorentzinvariant.Indeed,while δ 4 ( x x ) isLorentzinvariant, δ α,β would be invariant only if the representation D were unitary . There is however a property in group theory which states that unitary representations of the Lorentz group can only be infinite dimensional (like the one acting on the infinitely many independent quantum states of a particle). Being D finite-dimensional, it cannot be unitary, namely D D = 1 . As an example, suppose the representation D is the fundamental (defining) representation of the Lorentz group, that is D ( ) = = ( μ ν ) . If these matrices were unitary, being real, they would be orthogonal. We have learned, however, that are pseudo-orthogonal matrices, namely they leave the Minkowski metric η μν , rather than the Euclidean one δ μν , invariant. In other words T = 1 . 13 These are some of the reasons why the coordinate basis {| x , α } is ill defined in a relativistic context. We find it however pedagogically useful to use such states in order to introduce the main objects and notations of relativistic field theory starting from non-relativistic quantum mechanics. From now on the main object associated with a given particle will be the relativistic field α ( x ) . 13 A problem related to the non-unitarity of D is the fact that if we defined α ( x ) = x , α | a , as we did in the non-relativistic theory, it would no longer have the correct transformation property ( 9.99 ) under Poincaré transformations. For spin 1 / 2 and 1 particles, we can however define a real symmetric matrix γ = αβ ) squaring to the identity γ 2 = 1 , such that γ D ( ) γ = D
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