Abelian field theory

It is then apparent that the Hamiltonian for this non-Abelian field theory is going to contain quartic terms in addition to the quadratic terms seen in abelian field theory, such as U(l) electromagnetism. 

The procedure for obtaining the QED potential cannot be directly taken over into QCD to calculate the properties of quarkonium. The reason is that in contrast to QED, QCD is a non-Abelian field theory. Because of this, the running coupling of QCD is weak at very small distances between quarks (asymptotic freedom), but becomes strong at large interquark separations. Therefore, we can use lowest-order QCD perturbation theory to describe the quark-antiquark potential only at very small distances. 

Classical vacuum in non-abelian gauge theory is infinitely degenerate and numbered by Chern-Simons number Ncs of vacuum gauge fields Af ... 

In order to understand interferometry at a fundamental level in gauge field theory, the starting point must be the non-Abelian Stokes theorem [4]. The theorem is generated by a round trip or closed loop in Minkowski spacetime using covariant derivatives, and in its most general form is given [17] by... 

The example candidate for the topological field theory defining the l.h.s. of the non-Abelian Stokes theorem could be given by the (classical) action... 

To consider magnetic flux density components of IAIV, Q must have the units of weber and R, the scalar curvature, must have units of inverse square meters. In the flat spacetime limit, R 0, so it is clear that the non-Abelian part of the field tensor, Eq. (6), vanishes in special relativity. The complete field tensor F vanishes [1] in flat spacetime because the curvature tensor vanishes. These considerations refute the Maxwell-Heaviside theory, which is developed in flat spacetime, and show that 0(3) electrodynamics is a theory of conformally curved spacetime. Most generally, the Sachs theory is a closed field theory that, in principle, unifies all four fields gravitational, electromagnetic, weak, and strong. 

Equation (B.26) has the structure of a quaternion-valued non-Abelian gauge field theory. If we denote... 

Now for non-Abelian electromagnetic field theory, we have the 3-Lie index component of the field, and for the magnetic field B1"3 1, it equals... 

In this chapter we discuss the close relationship between the Born-Oppenheimer treatment of molecular systems and field theory as applied to elementary particles. The theory is based on the Born-Oppenheimer non-adiabatic coupling terms which are known to behave as vector potentials in electromagnetic dynamics. Treating the time-dependent Schrodinger equation for the electrons and the nuclei we show that enforcing diabatization produces for non-Abelian time-dependent systems the four-component Curl equation as obtained by Yang and Mills (Phys. Rev. 95, 631 (1954)). 

The diabatization within the time-dependent framework produced the expected potential matrix W presented in equation (56) but enforced the four vector curl equation which is given in equation (54). This set of equations contains not only derivatives with respect to the spatial coordinates but also with respect to time. In fact this non-Abelian curl equation is completely identical to YM curl equation which has its origin in field theory. 

In an Abelian theory [for which I (r, R) in Eq. (90) is a scalar rather than a vector function, Al=l], the introduction of a gauge field g(R) means premultiplication of the wave function x(R) by exp(igR), where g(R) is a scalar. This allows the definition of a gauge -vector potential, in natural units... 

In a non-Abelian theory (where the Hamiltonian contains noncommuting matrices and the solutions are vector or spinor functions, with N in Eq. (90) >1) we also start with a vector potential Af, [In the manner of Eq. (94), this can be decomposed into components A, in which the superscript labels the matrices in the theory). Next, we define the field intensity tensor through a covaiiant curl by... 

---