Background field method

Abstract: Contrary to the situation which holds for the canonical theory described in the first paper of this series, there exists at present no tractable pure operator language on which to base a manifestly covariant quantum theory of gravity. One must construct the theory by analogy with conventional $S$-matrix theory, using the $c$-number language of Feynman amplitudes when nothing else is available. The present paper undertakes this construction. It begins at an elementary level with a treatment of the propagation of small disturbances on a classical background. The classical background plays a fundamental role throughout, both as a technical instrument for probing the vacuum (i.e., analyzing virtual processes) and as an arbitrary fiducial point for the quantum fluctuations. The problem of the quantized light cone is discussed in a preliminary way, and the formal structure of the invariance group is displayed. A condensed notation is adopted which permits the Yang-Mills field to be studied simultaneously with the gravitational field. Generally covariant Green's functions are introduced through the imposition of covariant supplementary conditions on small disturbances. The transition from the classical to the quantum theory is made via the Poisson bracket of Peierls. Commutation relations for the asymptotic fields are obtained and used to define the incoming and outgoing states. Because of the non-Abelian character of the coordinate transformation group, the separation of propagated disturbances into physical and nonphysical components requires much greater care than in electrodynamics. With the aid of a canonical form for the commutator function, two distinct Feynman propagators relative to an arbitrary background are defined. One of these is manifestly covariant, but propagates nonphysical as well as physical quanta; the other propagates physical quanta only, but lacks manifest covariance. The latter is used to define external-line wave functions and non-radiatively-corrected amplitudes for scattering, pair production, and pair annihilation by the background field. The group invariance of these amplitudes is proved. A fully covariant generalization of the complete $S$ matrix is next proposed, and Feynman's tree theorem on the group invariance of non-radiatively-corrected $n$-particle amplitudes is derived. The big problem of radiative corrections is then confronted. The resolution of this problem is carried out in steps. The single-loop contribution to the vacuum-to-vacuum amplitude is first computed with the aid of the formal theory of continuous determinants. This contribution is then functionally differentiated to obtain the lowest-order radiative corrections to the $n$-quantum amplitudes. These amplitudes split automatically into Feynman baskets, i.e., sums over tree amplitudes (bare scattering amplitudes) in which all external lines are on the mass shell. This guarantees their group invariance. The invariance can be made partially manifest by converting from the noncovariant Feynman propagator to the covariant one, and this leads to the formal appearance of fictitious quanta which compensate the nonphysical modes carried by the covariant propagator. Although avoidable in principle, these quanta necessarily appear whenever manifestly covariant expressions are employed, e.g., in renormalization theory. The fictitious quanta, however, appear only in closed loops and are coupled to real quanta through vertices which vanish when the invariance group is Abelian. The vertices are nonsymmetric and always occur with a uniform orientation around any fictitious quantum loop. The problem of splitting radiative corrections into Feynman baskets becomes more difficult in higher orders, when overlapping loops occur. This problem is approached with the aid of the Feynman functional integral. It is shown that the "measure" or "volume element" for the functional integration plays a fundamental role in the decomposition into Feynman baskets and in guaranteeing the invariance of radiative corrections under arbitrary changes in the choice of basic field variables. The "measure" has two effects. Firstly, it removes from all closed loops the non-causal chains of cyclically connected advanced (or retarded) Green's functions, thereby breaking them open and ensuring that at least one segment of every loop is on the mass shell. Secondly it adds certain nonlocal corrections to the operator field equations, which vanish in the classical limit $\ensuremath{\hbar}\ensuremath{\rightarrow}0$. The question arises why these removals and corrections are always neglected in conventional field theory without apparent harm. It is argued that the usual procedures of renormalization theory automatically take care of them. In practice the criteria of locality and unitarity are replaced by analyticity statements and Cutkosky rules. It is virtually certain that the "measure" may be similarly ignored (set equal to unity) in gravity theory, and that attention may therefore be confined to primary diagrams, i.e., diagrams which contain Feynman propagators only, with no noncausal chains removed. A general algorithm is given for obtaining the primary diagrams of arbitrarily high order, including all fictitious quantum loops, and the group invariance of the amplitudes thereby defined is proved. Essential to all these derivations is the use of a background field satisfying the classical "free" field equations. It is never necessary to employ external sources, and hence the well-known difficulties arising with sources in a non-Abelian context are avoided.