# Background independence in gauge theory

In Quantum Field Theory (QFT), one frequently considers the quantum fluctuations around classical field configurations. Examples are:

• Spontaneous symmetry breaking in the Standard Model, where one considers quantum fluctuations around a non-trivial classical configuration of the Higgs field;
• The background field method, which is an efficient tool, for example, for the computation of the renormalization group flow;
• Perturbative Quantum Gravity, where one has to use a non-trivial background metric, providing the necessary structure for the formulation of a QFT.

Hence, the issue of background independence is of high conceptual importance. There are basically two approaches to deal with it in the literature. One is the Riemannian path integral framework, which faces the problem that, in the presence of non-trivial background fields, the relation between correlation functions on Riemannian spaces, and the QFT on Lorentzian space-time in which one is ultimately interested, is unclear. In particular, in the absence of an Osterwalder-Schrader theorem, it is not clear whether such correlation functions define a QFT in the sense of observables represented by operators on some Hilbert space. The other approach is to treat the background field as an infinitesimal perturbation around a fixed flat reference background. However, for a full proof of background independence, one should treat the background field non-perturbatively. Then one faces the problem that on generic backgrounds there is no unique vacuum state and that the usual renormalization techniques based on momentum space are not available. A further common shortcoming of these approaches is that they are not “operational” in the sense that they do not address the following question:

Given a background configuration and an observable defined w.r.t. this background, what is the same observable on a different background?

In view of the difficulties mentioned above, it seems advantageous to follow the algebraic approach, i.e., to directly (perturbatively) construct the algebras of observables for the different background configurations, using locally covariant renormalization techniques developed in the context of QFT on curved space-times. Background independence for us then means that we can unambiguously identify observables on different backgrounds (at least for infinitesimally close backgrounds). As suggested by Hollands, this can be formulated in the spirit of Fedosov quantization: One considers the bundle of observable algebras over the manifold of background configurations and constructs a flat connection on it. The sections that are flat, i.e., covariantly constant, w.r.t. this connection provide a consistent assignment of an observable to each background.

As a warm-up, consider a scalar field $\Phi$, split into a classical (background) part $\bar \phi$ (a solution to the classical field equation) and a perturbation $\phi$, which is going to be quantized:
$\Phi = \bar \phi + \phi.$
Considering classical observables $F(\bar \phi, \phi)$ depending on the background and the perturbation, we can characterise the background independent ones as fulfilling

\label{eq:classicalBGind}
\mathcal{D}_{\bar \varphi} F := ( \bar \delta_{\bar \varphi} – \delta_{\bar \varphi} ) F = 0, \qquad \text{ where } \bar \delta_{\bar \varphi} F = \langle \tfrac{\delta}{\delta \bar \phi} F, \quad \bar \varphi \rangle, \delta_{\bar \varphi} F = \langle \tfrac{\delta}{\delta \phi} F, \bar \varphi \rangle

for all infinitesimal variations $\bar \varphi$ of the background. This completely characterises the background independent classical functionals. Furthermore, as the connection $\mathcal{D}_{\bar \varphi}$ is flat, $[ \mathcal{D}_{\bar \varphi}, \mathcal{D}_{\bar \varphi’} ] = \mathcal{D}_{[\bar \varphi, \bar \varphi’]}$ with $[\bar \varphi, \bar \varphi’]$ the Lie brackets of vector fields on the manifold of classical solutions, one can uniquely extend extend an observable $F_{\bar \phi}(\phi)$ defined on a background $\bar \phi$ to all backgrounds, thus answering the question posed above.

When extending this to the interacting quantum theory, and further to gauge theories, several difficulties occur:

1. The interacting observables defined on different backgrounds are elements of different algebras. It is thus not clear how to define the derivative $\bar \delta_{\bar \varphi}$ w.r.t. the background field, which was part of the definition of $\mathcal{D}$. To overcome this difficulty, one uses that, for an interaction $\lambda \Phi^4$ with an infra-red cut-off, i.e., with $\lambda$ a function of compact support, one can canonically identify the algebras for different backgrounds, by identifying them in the past of the support of $\lambda$. The canonical isomorphism thus defined is called the Moller operators. Its infinitesimal version is called the retarded variation $\delta^{\mathrm r}_{\bar \varphi}$. The principle of perturbative agreement, first formulated by Hollands and Wald (2005), asserts that this retarded variation should act naturally on the generators of interacting observables, which implies

\label{eq:quantumBGind}
\mathfrak{D}_{\bar \varphi} T^{\mathrm{int}}(e^{i F[\bar \phi, -]}) := ( \delta^{\mathrm r}_{\bar \varphi} – \delta_{\bar \varphi} ) T^{\mathrm{int}}(e^{i F[\bar \phi, -]}) = i T^{\mathrm{int}}(\mathcal{D}_{\bar \varphi} F \otimes e^{i F[\bar \phi, -]}),

where $T^\mathrm{int}$ are the interacting time-ordered products and $F$ is a classical local functional. Hence, if perturbative agreement holds, then the classically background independent local functionals, i.e., those fulfilling \eqref{eq:classicalBGind}, give rise to interacting background independent observables. Perturbative agreement is to be seen as a renormalization condition, which in fact encompasses the well-known Wess-Zumino consistency conditions in gauge theory. For variations in the background scalar field, it can indeed be fulfilled. Up to this point, background independence (and in particular its relation to Fedosov quantization) had been worked out by Hollands and his student Collini. A crucial ingredient in this treatment is the fact that the action itself is background independent.

2. When one considers gauge theories, more difficulties occur. As an example, consider Yang-Mills theory, where one splits the connection $\mathcal{A} = \bar{\mathcal{A}} + A$ into a classical background connection $\bar{\mathcal{A}}$ and a dynamical vector potential $A$. On a technical level, implementing the infra-red cutoff is more involved if one aims to keep covariance w.r.t. changes in the background connection $\bar{\mathcal{A}}$. At a more fundamental level, one has the problem that it is necessary to fix the gauge for the perturbation $A$. Doing this in a way which respects covariance w.r.t. changes in the background connection $\bar{\mathcal{A}}$ implies that the gauge fixed action is no longer background independent. It follows that the identity \eqref{eq:quantumBGind} will not hold. However, in a quantized gauge theory, not all functionals are physical observables. The physical ones are given by the cohomology of $[Q^{\mathrm{int}}, -]$ with $Q^{\mathrm{int}}$ the interacting BRST charge. This means that physical observables need to commute with $Q^{\mathrm{int}}$ and that observables which can be written as an anti-commutator with $Q^{\mathrm{int}}$ are considered trivial. A connection $\mathfrak{D}_{\bar a}$ which is well-defined on the observable algebra thus has to commute with $[Q^{\mathrm{int}}, -]$. Furthermore, in order to uniquely identify observables on different backgrounds, the connection has to be flat on this cohomology. It turns out that it is possible to construct a connection fulfilling the desired properties. For $F$ a generator of the algebra of interacting observables, one than has, modulo trivial elements,

\label{eq:YMBGind}
\mathfrak{D}_{\bar a} T^{\mathrm{int}}(e^{i F}) = i T^{\mathrm{int}}( \{ \hat{\mathcal{D}}_{\bar a} F + \hat A_{\bar a}(e^F) \} \otimes e^{i F}).

Here $\hat{\mathcal{D}}_{\bar a}$ is the appropriate connection on the classical observables (to be well-defined on the observables, a correction to the definition in \eqref{eq:classicalBGind} has to be introduced), and the anomaly term $\hat A_{\bar a}(e^F)$ incorporates quantum mechanical anomalies. In the absence of these quantum corrections, the classically background independent observables again give rise to interacting background independent observables. \eqref{eq:YMBGind} and the well-definedness and flatness of $\mathfrak{D}_{\bar a}$ on the observables hold, provided that perturbative agreement holds and a certain anomaly (basically the anomaly of the variation of the gauge condition) vanishes. It can be shown that there is indeed a renormalization scheme in which this is the case. Power counting renormalizability is a crucial ingredient in the proof.

For (renormalizable) Yang-Mills theories there thus seems to be a satisfactory definition and proof of background independence. One can consider the analogous problem for perturbative quantum gravity. There, however, due to power counting non-renormalizability, one can not exclude the occurrence of the anomaly obstructing background independence by arguments similar to those used in the Yang-Mills case. This seems to be an interesting topic for further investigations.