The need for a gauge theory

In this chapter we examine some of the problems inherent in the current-current form of the weak interaction Lagrangian, and are led to the idea of the weak force being mediated by the exchange of vector mesons. This picture too runs into diflSiculties which, however, can be alleviated in gauge-invariant theories. The latter are introduced and discussed at some length. 

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Theory provides calculated values of absolute shieldings, that is, the shielding relative to a bare nucleus with no electrons. As we see in Section 4.3, experimental measurements provide information on shieldings relative to some selected standard. For comparison between theory and experiment, additional data are needed. For example, it can be shown that trP calculated with the gauge origin at a particular nucleus in a small molecule is proportional to the molecular spin—rotation constant of that nucleus, which can be independently measured by microwave spectroscopy, because crD can be calculated precisely, this combination permits the establishment of an absolute experimental shielding scale for various nuclei. For hydrogen, simultaneous measurements of NMR and the electronic... 

Correspondingly, the same law and formulae, as in affine theory, also serve for the projective differentiation of a completely general tensor. There are admittedly still further laws, not available in affine theory because they depend on the speciaf nature of a gauge transformation. We shall develop such laws only when we need them. 

In order to derive a quantum mechanical expression for the frequency-dependent polarizability we can make use of time-dependent response theory as described in Section 3.11. We need therefore to evaluate the time-dependent expectation value of the electric dipole operator (4 o(i (f)) Pa o( (t))) in the presence of a time-dependent electric field, Eq. (7.11). Employing the length gauge, Eqs. (2.122) - (2.124), which implies that the time-dependent electric field enters the Hamiltonian via the scalar potential in Eq. (2.105), the perturbation Hamilton operator for the periodic and spatially uniform electric field of the electromagnetic wave is given as... 

The study of matter at a molecular and sub-molecular level requires a quantum mechanical framework. In this chapter, we provide an overview of the essential concepts and notation of quantum theory needed to understand the development of relativistic many-electron quantum mechanics. This introductory overview starts from basic axioms with a focus on the nonrelativistic theory. Gauge and Lorentz invariance properties as introduced in the classical theory are then discussed for the basic quantum mechanical equations in the following two chapters. 

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