In this view, the difficulties of quantum gravity are due to the dimensionality of the coupling constant, /C. Another important issue that arises in a calculation of such complexity is the correctness of the result. Past experience with computer-aided calculations has shown that the most likely cause of error is human intervention in the calculation steps.
The on-shell part of the out-of-shell computation was reconciled with the in-shell computation. In Section 2, we give a brief overview of regularization and renormalization, as well as some formalism of the background field method of quantum field theory. The presence of anomalies in these theories can be considered as a failure of the regulation scheme.
Therefore, the redefinitions of the local field are seen not to affect the S matrix of the theory. Therefore, the divergences of quantum theory can be absorbed by a redefinition of the dimensionless coupling constant. In order to be somewhat self-contained, we now briefly review some of the more obvious features of the background field method.
However, the quantities of physical relevance are the amplitudes on the scale, which are anyway independent of the choice of meter.
Method of Integration
With two loops, the problem reduces to the evaluation of integrals of the form where all masses and external moments were set to zero. 3.6) (and any other scalar logarithmic integral) can be reduced to integrals of the form (3.7) and. To perform subtraction integrals, we first combine two (or one) denominators containing subdivergent moments using Feynman parameters, and then the full part of the integral follows from the standard formula.
The method is adapted for minimal discounting and makes implicit use of the gauge invariance of the regulator. This way, we will always be sure that we are subtracting only the pole part of the subdivergent loop, since their M-loop subdivergence can produce a. The immense complexity of the Einstein action (see Appendix A for more details) makes it appropriate to choose a gauge that leads to the simplest possible propagator.
This is the De Donder gauge, the gravitational analogue of the famous Feynman gauge for quantum electrodynamics. The resulting Lagrangian used in the derivation of Feynman's rules is simply the sum of the terms in Eqs. For simplicity, we illustrate the details of the procedure in the simpler case of the Yang–Mills theory, and at the end of the section we give the corresponding results for Einsteinian background gravity.
However, the algebra of matrices is well defined and the compact description in terms of a single field is clearly more convenient than the usual one obtained by considering separately the Faddeev-Popov phantoms in the complex representation. On this note, ghost fields are reminiscent of scalar modes that arise in an a la Kaluza-Klein reduction of the vector potential. Finally, the anticommuting nature of standard phantoms has the only real effect of introducing a minus sign for closed loops.
The other effect of the standard anti-commuting ghosts is to introduce a sign change in each vertex and propagator when the orientation of the loop changes. The resulting quantity is then symmetrized with respect to the exchange of the quantum fields to achieve a vertex. However, if we perform all loop integrals in momentum space and then transform back to x-space, we see that there is no need for explicit symmetry of the outer lines.
One associates with each graph a combinatorial factor that is one over the product of the number of symmetries of internal legs and the number of symmetries of external legs. For example, the graph in figure 3 has a factor of ~ for the internal symmetries of the interchange of the upper and lower lines and a factor of ~ x ~ for the symmetries of the two pairs of external legs.
One-Loop Gravity
The answer is that higher orders in a perturbative expansion of the integrand of Eq. Therefore, the lowest order expansion of the topological relation in 2n dimensions is always a total manifest divergence. However, if one were to compute a one-loop vertex correction and not apply these Fierz-like antisymmetric identities, one can determine the renormalization of the topological density.
We now describe the computation of one-loop divergences in the De Donder gauge, including the renormalization of the topological invariant. Thus, with a renormalization of the topological term, all one-loop divergences cancel out in the Einstein gravity S matrix. The calculation of the divergent part of the propagation diagram in Fig. 5 reproduces the result of ref.
We wish to emphasize that, both here and in the subsequent discussion of two-loop divergences in the next section, a fully shell vertex diagram is defined by satisfying moments. This is analogous to what happens for two-loop peak correction which will be discussed in the next section, and can be recognized simply by looking at the shape of the peak. The coefficient of the topological term in Eq. 5.12) agrees with a previous result [24], obtained by calculating the functional determinant with one loop of the background field method in (- function adjustment.
It is clear that the conceptual simplicity of the approach. discussed in this section is achieved at the price of some algebraic complications. Lagrangian in Eq. 5.13) has a measure SO (N) in the variance associated with the rotations of the fields ¢i·. This is not a consequence of the symmetry of the Riemann tensor, nor of the Bianchi identities, but can be seen by examining the topological invariant in 6 dimensions.
The discussion of the one-loop case presented suggests how to proceed in the two-loop case. At the higher levels, one would need a Fierz-like identity to relate the expansions of the two invariants. We now have to look to see what the effect of the one-loop field redefinition is on the two-loop level.
So, with a little extra effort, we have essentially repeated the computation of the vertex correction to the scale in the normal field theory approach. The results for the polar parts of the individual graphs on the scale are shown in Table 1.
Conclusions
We have shown that the procedure for working with subtracted integrals is very convenient, especially when working in the background field method. Moreover, it is very simple to extract pole parts from dimensionally regularized Feynman integrals, since the overlapping divergences are removed term by term. With this approach, there is very little difference between calculation in the background field method and in normal field theory.
The only remaining ghost is the need to consider, in addition to the extended metric tensor, a pair of projection operators that satisfy a very simple commutative algebra. Although these techniques are clearly very efficient, they do not by themselves make the problem of the divergences in quantum gravity tractable. The large number of indices and momenta present in a perturbative expansion of gravity makes it impractical to apply the methods described so far without extensive use of computers.
In this calculation, almost every step was performed by the computer using a number of relatively small programs (typically no more than 1000 lines of code) written by us in the C language. The C language is useful for manipulating characters, and the relatively small size of the programs makes them relative. By limiting the applicability of the programs, one can easily gain a factor of about 1,000 in speed over the general purpose programs.
For example, one of the most difficult steps in a calculation is making graphs, especially in the case of out-of-shell. A large number of expressions are generated at the intermediate stages, so it is essential that we are very efficient in allocating memory. This corresponds to finding the minimal set of bits that describe a typical structure, thereby obtaining a non-trivial factor in the amount of memory required.
Resorting to computers for specific algebraic problems may become quite common practice in Theoretical Physics in the coming years. UCB-PTH-85/42, September 1985, to appear in "Proceedings of the 4th Marcel Grossmann Meeting on Recent Developments in General Relativity". 30] The subject moves very quickly, and it is impossible to provide a complete list of references.
For completeness, we give below the extension of the Lagrangian of gravity to fourth order in quantum fields, including the calibration term in eq. The quantum field is denoted by hf.Lv• and it is implicitly assumed that all other quantities are constructed from the background metric, gf.LV". Indices are incremented and decremented using the background metric.
