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Background independence in gauge theories. (English) Zbl 1436.83032
The authors develop an algebraic approach to background independence in gauge theories. In their approach one considers the bundle of observable algebras over the manifold of background configurations and constructs a flat connection on it. Those sections which are flat, i.e., covariantly constant, w.r.t. this connection are considered as background independent observables.
The authors start with the example of scalar field theory expanded around a classical solution \(\bar\phi\), i.e., the basic scalar field \(\Phi\) is split into \(\bar\phi\), which is kept classical at the quantum level, i.e., it commutes with all quantum fields, and a dynamical field \(\phi\) which is viewed as fluctuations around \(\bar\phi\) and is quantized in perturbation theory. Since any local functional \(F[\Phi]\) induces local functionals \(F[\bar\phi,\phi] := F[ \bar\phi + \phi]\) for different backgrounds \(\bar\phi\), their background independence can be stated schematically via functional derivatives \({\mathcal D}F=0\) where \[ {\mathcal D}:=\frac{\delta}{\delta\bar{\phi}}-\frac{\delta}{\delta\phi}. \] The problem is that the derivative w.r.t. the background field \(\bar\phi\) has no obvious meaning on the corresponding local algebra of observables \({\mathbf W}_{\bar{\phi}}\), as one is comparing elements of different algebras. The authors show that if ‘perturbative agreement’ holds for changes in the mass of the scalar field, which is the case for the \(\Phi^4\) theory, one can replace this derivative with the retarded variation which is the infinitesimal version of the Møller operator relating the algebras on different backgrounds. The resulting derivative \({\mathfrak D}\) is flat similarly to \({\mathcal D}\). Hence, the authors define the background independent observables as sections which are covariantly constant w.r.t. \({\mathfrak D}\).
To analyze the issue of background independence for gauge theories the authors consider the pure Yang-Mills theory which is the theory of a \(G\)-connection \({\mathcal A}\) on a principal bundle, subject to the Yang-Mills field equations. Similarly to the scalar case, the classical Yang-Mills action is ignorant of the way one might split \({\mathcal A}\) into a background connection \(\bar{\mathcal A}\) and a dynamical \(g\)-valued 1-form \(A\) which will be quantized in perturbation theory. However, for the purpose of perturbative quantization, one has to fix the gauge, which necessarily breaks this split independence. The gauge-fixed action exhibits a residual BV-BRST symmetry which acts by a nilpotent operator \(s\), and the physical, i.e. gauge invariant, observables are obtained as the cohomology of \(s\). To quantize, one considers the subalgebra of physical (gauge invariant) observables which is given by the cohomology of the renormalized interacting BRST charge \(Q_{\bar{\mathcal A}}\). Therefore, for background independence to hold, the desired connection \({\mathfrak D}\) has to be ‘well defined’ on the BRST cohomology. If this is the case, background independent observables can be defined as those sections of the observable algebra bundle which are flat w.r.t. \({\mathfrak D}\) modulo \(\mathrm{Im}\,Q_{\bar{\mathcal A}}\). The authors show that there are potential obstructions (anomalies) for the construction of such a connection. However, for pure Yang-Mills theory in \(D = 4\) spacetime dimensions, these turn out to be trivial.
Finally the authors sketch the application of their framework to perturbative quantum gravity and show that as a consequence of power counting non-renormalizability there are infinitely many nontrivial potential obstructions to background independence contrary to the pure Yang-Mills case.
The paper is well written and gives a rivetting account of an algebraic approach to a conceptual problem in quantum field theory.

MSC:
83C47 Methods of quantum field theory in general relativity and gravitational theory
81T13 Yang-Mills and other gauge theories in quantum field theory
83C45 Quantization of the gravitational field
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
70S15 Yang-Mills and other gauge theories in mechanics of particles and systems
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