Gauge Fields and Flavor Oscillations
In this chapter we discuss the general formulation of gauge fields in the quantum theory, both abelian and nonabelian. A generalization of the elementary Stueckelberg diagram (Fig. 2.1 ), demonstrating a “classical” picture of pair annihilation and creat
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Gauge Fields and Flavor Oscillations
In this chapter we discuss the general formulation of gauge fields in the quantum theory, both abelian and nonabelian. A generalization of the elementary Stueckelberg diagram (Fig. 2.1), demonstrating a “classical” picture of pair annihilation and creation, provides a similar picture of a process involving two or more vertices (diagrams of this type appear in Feynman’s paper in 1949 (Feynman 1949) with sharp instantaneous vertices). A single vertex, as in Stueckelberg’s original diagram, in the presence of a nonabelian gauge field, can induce a flavor change on the particle line, resulting in a transition to an antiparticle with different identity. An even number of such transitions can result in flavor oscillations, such as in the simple case of neutrino oscillations. On the quark constituent level, such transitions can be associated with K , B or D meson oscillations as well. The construction of the Lorentz force acting on particles with abelian or nonabelian gauge will also be discussed, with results consistent with the assumptions for the semiclassical model. In view of our discussion of the previous chapter, it will also be shown that this picture could provide a fundamental mechanism for CP violation.
4.1
Abelian Gauge Fields
In his original paper Stueckelberg (1941) introduced the electromagnetic vector gauge fields, as we shall explain below, as compensation fields for the derivatives on the wave functions representing the four-momenta. For a Hamiltonian of the form (2.4), i.e., p μ pμ , 2M for which the Stueckelberg-Schrödinger equation is K =
i
∂ ψτ (x) = K ψτ (x), ∂τ
© Springer Science+Business Media Dordrecht 2015 L.P. Horwitz, Relativistic Quantum Mechanics, Fundamental Theories of Physics 180, DOI 10.1007/978-94-017-7261-7_4
(4.1)
(4.2)
51
52
4 Gauge Fields and Flavor Oscillations
one must introduce so-called compensation fields to retain the form of the equation when the wave function is modified by a (differentiable) phase function at every point. Thus, for (4.3) ψ(x) = eie(x) ψ(x), the relation is satisfied if
( p μ − e Aμ (x) )ψ(x) = eie ( p μ − e Aμ (x))ψ(x),
(4.4)
Aμ (x) = Aμ (x) + ∂ μ .
(4.5)
One sees that the gauge transformation induced on the compensation field is of the same form as the gauge transformations of the Maxwell potentials, and therefore this procedure may be thought of as an underlying theory for electromagnetism (Wu 1975). Stueckelberg (1941) noted that he was unable to explain the diagram of Fig. 2.1 with this form of the electromagnetic interaction. The reason is that the canonical velocity is p μ − e Aμ , (4.6) x˙ μ = M so that ( p μ − e Aμ )( pμ − e Aμ ) ds . (4.7) x˙ μ x˙μ = −( )2 = dτ M2 This expression is proportional to the conserved Hamiltonian (for a closed system), so that the proper time cannot go through zero. To avoid this difficulty, he added an extra force term in the equations of motion. However, this construction did not take into account the compensation field required for the τ derivative in the
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