In 1983, L. Alvarez-Gaumé and E. Witten (AGW) wrote a fundamental article in which they calculated the one-loop gravitational anomalies (anomalies in the local Lorentz symmetry of (4k + 2)-dimensional Minkowskian quantum field theories coupled to external gravity) of complex chiral spin-½ and spin- fields and real self-dual antisymmetric tensor fields. They used two methods: a straightforward Feynman graph calculation in 4k + 2 dimensions with Pauli-Villars regularization, and a quantum mechanical (QM) path integral method in which corresponding nonlinear sigma models appeared. The former has been discussed in detail in an earlier book. The latter method is the subject of this book. AGW applied their formulas to N =2B supergravity in 10 dimensions, which contains precisely one field of each kind, and found that the sum of the gravitational anomalies cancels. Soon afterwards, M. B. Green and J. H. Schwarz calculated the gravitational anomalies in one-loop string amplitudes, and concluded that these anomalies cancel in string theory, and therefore should also cancel in N = 1 supergravity in 10 dimensions with suitable gauge groups for the N = 1 matter couplings.

Path integrals play an important role in modern quantum field theory. One usually first encounters them as useful formal devices to derive Feynman rules. For gauge theories they yield straightforwardly the Ward identities. Namely, if BRST symmetry (the “quantum gauge invariance” discovered by Becchi, Rouet, Stora and Tyutin) holds at the quantum level, certain relations between Green functions can be derived from path integrals, but details of the path integral (for example, the precise form of the measure) are not needed for this purpose. Once the BRST Ward identities for gauge theories have been derived, unitarity and renormalizability can be proven, and at this point one may forget about path integrals if one is only interested in perturbative aspects of quantum field theories. One can compute higher-loop Feynman graphs without ever using path integrals.

However, for nonperturbative aspects, path integrals are essential. The first place where one encounters path integrals in nonperturbative quantum field theory is in the study of instantons and solitons. Here advanced methods based on path integrals have been developed. For example, in the case of instantons the correct measure for integration over their collective coordinates (corresponding to the zero modes) is needed. In particular, for supersymmetric nonabelian gauge theories, there are only contributions from these zero modes, while the contributions from the nonzero modes cancel between bosons and fermions.

We now start the second part of this book, namely the computation of anomalies in higher-dimensional quantum field theories using quantum mechanical (QM) path integrals. Anomalies arise when the symmetries of a classical system cannot all be preserved by the quantization procedure. Those symmetries which turn out to be violated by the quantum corrections are called anomalous. The anomalous behavior is encoded in the quantum effective action which fails to be invariant: its nonvanishing variation is called the anomaly. As we shall see, the ordinary Dirac action for a chiral fermion in n dimensions has anomalies which can be computed by using an N = 1 supersymmetric (susy) nonlinear sigma model in one (timelike) dimension. Although this relation between a nonsusy quantum field theory (QFT) and a susy QM system may seem surprising at first sight, it becomes plausible if one notices that the Dirac operator γµDµ contains hermitian Dirac matrices γm (where γµ = γmemµ, with emµ being the inverse vielbein field) satisfying the same Clifford algebra {γl, γm} = 2δlm (with l, m = 1, …, n flat indices) as the equal-time anti-commutation rules of a real (Majorana) fermionic quantum mechanical point particle ψa(t) with a = 1, …, n, namely {ψa(t), ψb(t)} = ħδab.

In this chapter we discuss quantum mechanical path integrals defined by time slicing. Our starting point is an arbitrary but fixed Hamiltonian operator Ĥ. We obtain the Feynman rules for nonlinear sigma models, first for bosonic point particles xi(t) with curved indices i = 1, …, n and then for fermionic point particles ψa(t) with flat indices a=1, …, n. In the bosonic case we first discuss in detail configuration-space path integrals, and then return to the corresponding phase-space integrals. In the fermionic case we use coherent states to define bras and kets, and we discuss the proper treatment of Majorana fermions, both in the operatorial and in the path integral approach. Finally, we compute directly the transition element 〈z|e−(β/ħ)Ĥ|y〉 to order β (two-loop order) using operator methods, and compare the answer with the results of a similar calculation based on the perturbative evaluation of the path integral with time slicing regularization. Complete agreement is found. These results were obtained in. Additional discussions are found in.

The quantum action, i.e. the action to be used in the path integral, is obtained from the quantum Hamiltonian by mathematical identities, and the quantum Hamiltonian is fixed by the quantum field theory, the anomalies of which we study in Part II of the book. Hence, there is no ambiguity in the quantum action.

In this chapter we discuss path integrals defined by dimensional regularization (DR). In contrast to the previous time slicing (TS) and mode regularization (MR) schemes, this type of regularization seems to have no meaning outside perturbation theory. However, it leads to the simplest set up for perturbative calculations. In fact, the associated counterterm VDR turns out to be covariant, and the additional vertices obtained by expanding VDR, needed at higher loops, can be obtained with relative ease (using, for example, Riemann normal coordinates).

Dimensional regularization is based on the analytic continuation in the number of dimensions of the momentum integrals corresponding to Feynman graphs (1 → D + 1 with arbitrary complex D, in our case). At complex D we assume that the regularization of ultraviolet (UV) divergences is achieved by the analytic continuation as usual. The limit D → 0 is taken at the end. Again one does not expect divergences to arise in quantum mechanics when the regulator is removed (D → 0), and thus no infinite counterterms are necessary to renormalize the theory: potential divergences are canceled by the ghosts.

To derive the dimensional regularization scheme, one can employ a set up quite similar to the one described in the previous chapter for mode regularization. The only difference will be the prescriptions of how to regulate ambiguous diagrams.

In the main text we showed that gravitational and gauge anomalies cancel in 10 dimensions for N = 1 supergravity coupled to Yang–Mills theory if the gauge group G is either SO(32) or E8 × E8. None of the other “classical groups” (SO(n), SU(n) and Sp(n)) was allowed. We now complete this analysis by discussing the exceptional groups, namely G2, F4, E6, E7 and E8. As we have shown in the main text, cancellation of gravitational anomalies allows only Lie groups with 496 generators. There are clearly many products of simple Lie algebras with this number of generators. In particular, there are semisimple Lie algebras with one or more exceptional groups as simple factors. However, we can at once rule out these exceptional groups if we study one-loop hexagon graphs with six gauge fields all belonging to the same exceptional Lie group, and if factorization of the kind discussed in the main text does not occur. In four dimensions gauge anomalies are proportional to the symmetrized trace of three generators, dabc(R) = tr(Ta{Tb, Tc}), where Ta are the generators of the gauge group in a representation R, and thus real or pseudoreal representations do not carry anomalies. A representation R can only carry an anomaly if dabc(R) is nonvanishing, and this is only possible if there exists a cubic Casimir operator for the group.

We now turn to a second class of anomalies, namely the trace anomalies. These are anomalies in the local scale invariance of actions for scalar fields, spin-½ fields and certain vector and antisymmetric tensor fields (vectors in n = 4, antisymmetric tensors with two indices in n = 6, etc.). One needs gauge-fixing terms and ghosts, but the trace anomalies of these vector and antisymmetric tensor fields are independent of the gauge chosen. From a technical point of view, these anomalies are very interesting, because one needs higher loop graphs on the worldline to compute them. In fact, due to the β dependence of the measure of the quantum mechanical path integrals, A = (2πħβ)−n/2, one needs (½n + 1)-loop calculations in quantum mechanics for the one-loop trace anomalies of n-dimensional quantum field theories. Already in two dimensions one needs two-loop graphs and in four dimensions three-loop graphs. Another interesting technical point regards the fermions. In the path integral they now have antiperiodic boundary conditions. Originally we devised a path integral approach in which fermions were still treated by an operator formalism and in which actions were operator-valued. Here we shall instead present a complete path integral approach, with ordinary actions, in which the fermions are described in the path integral by Grassmann fields. Contrary to the case of chiral anomalies, we shall not use a background field formalism for the fermions because the background fermions are constant and thus cannot accomodate anti-periodic boundary conditions. Instead we shall directly use fermionic quantum fields with antiperiodic boundary conditions. The results we find agree with the results in the literature for trace anomalies obtained by different methods (see, for example).