From Special Relativity to Feynman Diagrams.pdf

In the rest of this section we shall treat

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In the rest of this section we shall treat exclusively the case of spin 0 fields, that is fields that are scalar under Lorentz transformations. We shall consider a complex scalar field , φ , or equivalently two real scalar fields (see Chap.7 , Sect.7.4 ). In this case the equation of motion ( 10.9 ) can be derived from the Hamilton principle of stationary action, starting from the following Lagrangian density ( 8.198 ): L = c 2 μ φ μ φ m 2 c 2 2 φ φ . (10.11) 1 Extension of the invariance to the full Poincaré group is obvious.
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10.2 The Klein–Gordon Equation 309 Indeed in this case the Euler–Lagrange equations L ∂φ( x ) μ L ∂∂ μ φ( x ) = 0 ; L ∂φ( x ) μ L ∂∂ μ φ( x ) = 0 , give: + m 2 c 2 2 φ( x ) = 0 . (10.12) together with its complex conjugate. As a complete set of solutions we can take the plane waves ( 9.113 ) p ( x ) e i p μ x μ , (10.13) with wave number k = p / and angular frequency ω = E / . These are the eigen- functions of the operator ˆ P μ which describe the wave functions of particles with definite value of energy E and momentum p , see Chap.9 . Substituting the exponen- tials ( 10.13 ) in ( 10.12 ) we find E 2 c 2 − | p | 2 = m 2 c 2 , (10.14) or E = ± E p = ± | p | 2 c 2 + m 2 c 4 . (10.15) We see that solutions exist for both positive and negative values of the energy corresponding to the exponentials: e i ( E p t p · x ) ; e i ( E p t + p · x ) . (10.16) Strictly speaking this is not a problem as long as we consider only free fields. Indeed the conservation of energy would forbid transition between positive and negative energy solutions and a positive energy state will remain so. Therefore we could regard as physical only those solutions corresponding to positive energy E > 0 . However the very notion of free particle is far from reality since real particles interact with each other, usually in scattering processes. During an interaction transitions between quantum states are induced, according to perturbation theory. Therefore we cannot neglect the existence of negative energy states. For example, a particle with energy E = + E p could decay into a particle of energy E = − E p , through the emission of a photon of energy 2 E p . Moreover the existence of negative energies is in some sense contradictory since, as shown in the following, from a field theoretical point of view, the Hamiltonian of the theory is positive definite . 2 Thus, the existence of negative energy solutions is a true problem when trying to achieve a relativistic generalization of the Schroedinger equation. 2 Furthermore, erasing the negative energy solutions would spoil the completeness of the eigenstates of ˆ P μ and the expansion in plane waves would be no longer correct.
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310 10 Relativistic Wave Equations A second problem arises when trying to give a probabilistic interpretation to the wave function ψ( x , t ). As we have anticipated in the introduction with each solution to the Schroedinger equation we can associate a a positive probability ρ = | ψ( x , t ) | 2 , and a current density j = i 2 m ψ ψ ψ) satisfying the continuity equation ( 10.2 ), which assures that the total probability is conserved.
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