nLab
quantum anomaly

Idea

There are at least two things that are called quantum anomalies in the context of quantum field theory

• anomalous action functional: the action functional (in path integral quantization) is not a globally well defined function, but instead a section of a line bundle on configuration space;

• anomalous symmetry (gauge anomaly): a symmetry of the action functional does not extend to a symmetry of the exponentiated action times the path integral measure; or equivalently the action of a group on classical phase space is not preserved by deformation quantization.

Definition

Anomalous action functional

There are two major kinds of action functionals that may be anomalous in that they are not actually functions/functionals on the configuration space of fields, but just sections of some line bundle:

• theories with fermions (see e.g. spinors in Yang-Mills theory)

• gauge theories with higher degree gauge fields (differential cocycles of higher degree.)

Fermionic anomalies

The path integral for a quantum field theory with fermions can be decomposed into a fermionic path integral (see there for more details) over the fermionic fields followed by that over the bosonic fields. The former, a Berezin integral, is typically well defined for a fixed configuration of the bosonic fields, but does not produce a well defined function on the space of all bosonic fields: but a twisted function , a section of some line bundle called a determinant line bundle or, in $8k+2$ dimensions, its square root, the Pfaffian line bundle.

So to even start making sense of the remaining path integral over the bosonic degree of freedom, this determinant line bundle or the corresponding Pfaffian line bundle has to be trivializale. Its non-trivializability is the fermionic anomaly .

Higher gauge-theoreric anomalies

For the moment see Green-Schwarz mechanism for more.

Anomalous symmetry

{AnomalousSymmetry}

under construction

Let

$$S:C\to ℝ$$S : C \to \mathbb{R}

be a (well defined) action functional. Write $P$ for its resolved covariant phase space in dg-geometry and

$$S^{\mathrm{BV}}:P\to ℝ$$S^{BV} : P \to \mathbb{R}

for the BV-action functional, both as given by BRST-BV formalism.

If the action functional is local (comes from a Lagrangian on jet space) the covariant phase space $P$ a priori only carries a presymplectic structure. But by BV-theory there exists an equivalent (homotopical) derived action functional $S_{\Psi }^{\mathrm{BV}}:P\to ℝ$ such that $S_{\Psi }^{\mathrm{BV}}$ does induce a genuine symplectic structure on the derived space $P$.

For ordinary Poisson manifolds and hence symplectic manifolds Maxim Kontsevich’s theorem says that their deformation quantization always exist. But if $S$ is the action functional of a gauge theory then $P$ is in general a nontrivial derived infinity-Lie algebroid (its function algebra has “ghosts” and “ghosts of ghost”: the Chevalley-Eilenberg algebra generators) and the theorem does not apply. Instead, the quantization of the derived symplectic space $P$ exists only if the first and second infinity-Lie algebroid cohomology of $P$ vanishes:

These two cohomology groups

$${\mathrm{Anom}}_{\mathrm{gauge}}=H^1(\mathrm{CE}(P))\oplus H^2(\mathrm{CE}(P))$$Anom_{gauge} = H^1(CE(P)) \oplus H^2(CE(P))

are called the gauge anomaly of the system. Only if they vanish does the quantization of the gauge theory encoded by $S$ exist.

More concretely, the function algebra on $P$ is a graded-commutative dg-algebra equipped with a graded Poisson bracket $\{-,-{\}}_{\mathrm{BV}}$ and an element $Q\in C^{\mathrm{\infty }}(P)$ (the BV-BRST charge) whose Hamiltonian vector field is the derivation that is the differential of the dg-algebra $C^{\mathrm{\infty }}(X)$. If the gauge anomaly does not vanish, then, while the deformation quantization of the graded algebra $C^{\mathrm{\infty }}(P)$ to a non commutative graded algebra with commutator $[-,-]$ will exist, it may happen that the image $S$ of $Q$ under the quantization no longer satisfies the quantum master equation $[S,S]=\hslash \Delta S$.

Therefore the derivation $[S,-]$ will not define a quantized differential and therefore the quantization of the graded-commutative dg-algebra $C^{\mathrm{\infty }}(P)$ will only be a noncommutative algebra, not a non-commutative dg-algebra, hence will not be functions on a non-commutative space in derived geometry.

Examples

Anomalous action functional

Spinning particles and super-branes

The sigma-model for a supersymmetric fundamental brane on a target space $X$ has an anomaly coming from the nontriviality of Pfaffian line bundles associated with the fermionic fields on the worldvolume. These anomalies disappear (i.e. these bundles are trivializable) when the structure group of the tangent bundle of $X$ has a sufficiently high lift through the Whitehead tower of $O(n)$.

• Spin structure the worldline anomaly for the spinning particle/superparticle vanishes when $X$ has Spin structure

  This is a classical result. A concrete derivation is in

  • Edward Witten, Global anomalies in String theory in Symposium on anomalies, geometry, topology , World Scientific Publishing, Singapore (1985)

• String structure the worldsheet anomaly for the spinning string/superstring in heterotic string theory vanishes (essentially) when $X$ has String structure

  This is originally due to Killingback and Witten. A commented list of literature is here. Recently Ulrich Bunke gave the rigorous proof

  • Ulrich Bunke, String structures and trivialisations of a Pfaffian line bundle (arXiv)

in terms of differential cohomology in general and differential string structures in particular.

• Fivebrane structure the worldvolume anomaly for the super-5-brane in dual heterotic string theory vanishes (essentially) when $X$ has Fivebrane structure. See there.

Anomalous symmetry

Conformal anomaly

For the moment see Liouville cocycle.

References

• References on anomalous action functionals

• References on gauge anomalies

Anomalous action functional

• L. Faddeev and S. Shatashvili, “Algebraic and Hamiltonian Methods in the theory of Nonabelian Anomalies,” Theor. Math. Fiz., 60 (1984) 206; english transl. Theor. Math. Phys. 60 (1984) 770.

• B. Zumino, “Chiral anomalies and differential geometry,” in Relativity, Groups and Topology II, proceedings of the Les Houches summer school, B.S. DeWitt and R. Stora, eds. North-Holland, 1984.

• Luis Alvarez-Gaumé and Edward Witten, Gravitational Anomalies Nucl. Phys. B234 (1984) 269.

• Luis Alvarez-Gaumé and Paul Ginsparg, The structure of gauge and gravitational anomalies , Ann. Phys. 161 (1985) 423.

• Edward Witten, Global gravitational anomalies , Commun. Math. Phys. 100 (1985) 197. (EUCLID)

A survey of these results is in the slides

• Paolo Di Vecchia, Green-Schwarz anomaly cancellation (2010) (pdf)

The mathematical formulation of this in terms of index theory is due to

• Michael Atiyah, Isadore Singer, Dirac operators coupled to vector potentials, Proc. Nat. Acad. Sci. USA 81, 2597-2600 (1984)

• J.-M. Bismut and Daniel Freed, The analysis of elliptic families. I. Metrics and connections on determinant bundles , Comm. Math. Phys. 106 (1986), no. 1, 159–176.

• J.-M. Bismut and Daniel Freed, The analysis of elliptic families. II. Dirac operators, eta invariants, and the holonomy theorem , Comm. Math. Phys. 107 (1986), no. 1, 103–163.

A physicists’ monograph is

• Reinhold A. Bertlmann, Anomalies in quantum field theory, Oxford Science Publ., 1996, 2000

A clear description of the quantum anomalies for higher gauge theories is in

• Dan Freed, Dirac charge quantization and generalized differential cohomology (arXiv)

As an application of this, a detailed discussion of the cancellation of the anomaly of the supergravity C-field in 11-dimensional supergravity is in

• Dan Freed, Greg Moore, Setting the quantum integrand of M-theory (arXiv:hep-th/0409135)

The role of spin structures as the anomaly cancellation condition for the spinning particle is discussed in

• Edward Witten, Global anomalies in String theory in Symposium on anomalies, geometry, topology , World Scientific Publishing, Singapore (1985)

The anomaly line bundles for self-dual higher gauge theory is discussed in

• Samuel Monnier, The anomaly line bundle of the self-dual field theory (arXiv:1109.2904)

Gauge anomaly

In BV-BRST formulation

General discussion in the context of BRST-BV formalism (breaking of the quantum master equation by quantum corrections) is discussed in

• W. Troost, P. van Nieuwenhuizen, A. van Proeyen, Anomalies and the Batalin-Vilkovisky lagrangian formalism (web)

• P.S. Howe, U. Lindström and P. White, Anomalies And Renormalization In The BRST-BV Framework , Phys. Lett. B246 (1990) 430.

• J. Paris, W. Troost, Higher loop anomalies and their consistency conditions in nonlocal regularization , Nucl. Phys. B482 (1996) 373 (arXiv:hep-th/9607215)

• Glenn Barnich, Classical and quantum aspects of the extended antifield formalism (arXiv:hep-th/0011120)

The fact that the anomly sits in degree-1 BRST cohomology corresponds to the consistency condition discussed in

• Julius Wess and B. Zumino, Consequences Of Anomalous Ward Identities , Phys. Lett. B37 (1971) 95.

Discussion of special applications in

• F. De Jonghe, J. Paris and W. Troost, The BPHZ renormalised BV master equation and Two-loop Anomalies in Chiral Gravities , Nucl. Phys. B476 (1996) 559 (arXiv:hep-th/9603012)

• J. Paris, Nonlocally regularized antibracket - antifield formalism and anomalies in chiral $W(3)$ gravity , Nucl. Phys. B450 (1995) 357 (arXiv:hep-th/9502140)

• R. Amorim, N.R.F.Braga, R. Thibes, Axial and gauge anomalies in the field antifield quantization of the generalized Schwinger model (arXiv:hep-th/9712014)

Discussion in the context of AQFT with functional analysis taken into account is in section 5.3.3 of

• Katarzyna Rejzner, Batalin-Vilkovisky formalism in locally covariant field theory PhD thesis, Hamburg (2011) (arXiv:1111.5130)

and

• Klaus Fredenhagen, Katarzyna Rejzner, Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory (arXiv:1110.5232)

Other

An interpretation of gauge anomalies as failures of Hamiltonians to have self-adjoint extensions is in

• J. G. Esteve, Origin of the anomalies: the modified Heisenberg equation (arXiv:hep-th/0207164)

In finite-dimensional quantum mechanics

• A. P. Balachandran, Amilcar R. de Queiroz, Mixed states from anomalies, arxiv/1108.3898
• Carlos Alcalde, Daniel Sternheimer, Analytic vectors, anomalies and star representations, Lett. Math. Phys. 17 (1989), no. 2, 117–127. MR90h:22012, doi (the last section has also the field theory case)