Background independence in gauge theories.

*(English)*Zbl 1436.83032The 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.

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.

Reviewer: Farhang Loran (Isfahan)

##### 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 |

##### Software:

pAQFT
PDF
BibTeX
XML
Cite

\textit{M. T. Tehrani} and \textit{J. Zahn}, Ann. Henri Poincaré 21, No. 4, 1135--1190 (2020; Zbl 1436.83032)

Full Text:
DOI

##### References:

[1] | Abbott, LF, Introduction to the background field method, Acta Phys. Pol. B, 13, 33 (1982) |

[2] | Brunetti, R.; Fredenhagen, K.; Rejzner, K., Quantum gravity from the point of view of locally covariant quantum field theory, Commun. Math. Phys., 345, 741 (2016) · Zbl 1346.83001 |

[3] | Becker, D.; Reuter, M., En route to background independence: broken split-symmetry, and how to restore it with bi-metric average actions, Ann. Phys., 350, 225 (2014) · Zbl 1344.83024 |

[4] | Hollands, S.; Wald, RM, Local Wick polynomials and time ordered products of quantum fields in curved space-time, Commun. Math. Phys., 223, 289 (2001) · Zbl 0989.81081 |

[5] | Hollands, S.: Background independence in quantum field theory. Unpublished notes (2011) |

[6] | Hollands, S.: Constructing quantum field theories with fedosov quantization, 2012, Talk given at workshop Mathematical Aspects of Quantum Field Theory and Quantum Statistical Mechanics, Hamburg (2012). https://www.lqp2.org/node/1492. Accessed 3 Jan 2018 |

[7] | Fedosov, BV, A simple geometrical construction of deformation quantization, J. Differ. Geom., 40, 213 (1994) · Zbl 0812.53034 |

[8] | Reuter, M., Quantum mechanics as a gauge theory of metaplectic spinor fields, Int. J. Mod. Phys. A, 13, 3835 (1998) · Zbl 0934.81025 |

[9] | Witten, E.: Conference on highlights of particle and condensed matter physics (SALAMFEST) Trieste, Italy, 8-12 March 1993, pp. 257-275 (1993). arXiv:hep-th/9306122 |

[10] | Sen, A.; Zwiebach, B., Quantum background independence of closed string field theory, Nucl. Phys. B, 423, 580 (1994) · Zbl 0925.53041 |

[11] | Brunetti, R.; Fredenhagen, K.; Verch, R., The generally covariant locality principle: a new paradigm for local quantum field theory, Commun. Math. Phys., 237, 31 (2003) · Zbl 1047.81052 |

[12] | Zahn, J., The renormalized locally covariant Dirac field, Rev. Math. Phys., 26, 1330012 (2014) · Zbl 1287.81086 |

[13] | Hollands, S.; Wald, RM, Conservation of the stress tensor in perturbative interacting quantum field theory in curved spacetimes, Rev. Math. Phys., 17, 227 (2005) · Zbl 1078.81062 |

[14] | Brennecke, F.; Dütsch, M., Removal of violations of the master ward identity in perturbative QFT, Rev. Math. Phys., 20, 119 (2008) · Zbl 1149.81017 |

[15] | Collini, G.: Fedosov quantization and perturbative quantum field theory. Ph.D. dissertation, Universität Leipzig (2016). arXiv:1503.03754 |

[16] | Drago, N.; Hack, TP; Pinamonti, N., The generalised principle of perturbative agreement and the thermal mass, Ann. Henri Poincaré, 18, 807 (2017) · Zbl 1362.81064 |

[17] | Khavkine, I., Local and gauge invariant observables in gravity, Class. Quantum Grav., 32, 185019 (2015) · Zbl 1327.83125 |

[18] | Bergmann, PG; Komar, AB, Poisson brackets between locally defined observables in general relativity, Phys. Rev. Lett., 4, 432 (1960) |

[19] | Benini, M.; Schenkel, A.; Schreiber, U., The stack of Yang-Mills fields on Lorentzian manifolds, Commun. Math. Phys., 359, 765 (2018) · Zbl 1423.14007 |

[20] | Arms, JM, The structure of the solution set for the Yang-Mills equations, Math. Proc. Camb. Philos. Soc., 90, 361 (1981) · Zbl 0462.53036 |

[21] | Kluberg-Stern, H.; Zuber, JB, Renormalization of nonabelian gauge theories in a background field gauge. 1. Green functions, Phys. Rev. D, 12, 482 (1975) |

[22] | Wald, RM, General Relativity (1984), Chicago: University of Chicago Press, Chicago |

[23] | Chilian, B.; Fredenhagen, K., The time slice axiom in perturbative quantum field theory on globally hyperbolic spacetimes, Commun. Math. Phys., 287, 513 (2009) · Zbl 1171.81015 |

[24] | Brunetti, R.; Fredenhagen, K., Microlocal analysis and interacting quantum field theories: renormalization on physical backgrounds, Commun. Math. Phys., 208, 623 (2000) · Zbl 1040.81067 |

[25] | Hollands, S., Renormalized quantum Yang-Mills fields in curved spacetime, Rev. Math. Phys., 20, 1033 (2008) · Zbl 1161.81022 |

[26] | Hollands, S.; Wald, RM, Quantum fields in curved spacetime, Phys. Rep., 574, 1 (2015) · Zbl 1357.81144 |

[27] | Rejzner, K., Perturbative Algebraic Quantum Field Theory (2016), Berlin: Springer, Berlin · Zbl 1347.81011 |

[28] | Hollands, S.; Wald, RM, Existence of local covariant time ordered products of quantum fields in curved space-time, Commun. Math. Phys., 231, 309 (2002) · Zbl 1015.81043 |

[29] | DeWitt, BS; Brehme, RW, Radiation damping in a gravitational field, Ann. Phys., 9, 220 (1960) · Zbl 0092.45003 |

[30] | Epstein, H.; Glaser, V., The role of locality in perturbation theory, Annales de l’IHP Physique théorique A, 19, 211 (1973) · Zbl 1216.81075 |

[31] | Zahn, J., Locally covariant charged fields and background independence, Rev. Math. Phys., 27, 1550017 (2015) · Zbl 1327.81273 |

[32] | Taslimi Tehrani, M., Quantum BRST charge in gauge theories in curved space-time, J. Math. Phys., 60, 012304 (2019) · Zbl 1406.81067 |

[33] | Chrusciel, PT; Shatah, J., Global existence of solutions of the Yang-Mills equations on globally hyperbolic four dimensional Lorentzian manifolds, Asian J. Math., 1, 530 (1997) · Zbl 0912.58039 |

[34] | Becchi, C.; Rouet, A.; Stora, R., Renormalization of gauge theories, Ann. Phys., 98, 287 (1976) |

[35] | Batalin, I.; Vilkovisky, G., Gauge algebra and quantization, Phys. Lett. B, 102, 27 (1981) |

[36] | Fredenhagen, K.; Rejzner, K., Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory, Commun. Math. Phys., 317, 697 (2013) · Zbl 1263.81245 |

[37] | Rejzner, K., Fermionic fields in the functional approach to classical field theory, Rev. Math. Phys., 23, 1009 (2011) · Zbl 1242.81112 |

[38] | DeWitt, BS, Quantum theory of gravity. 2. The manifestly covariant theory, Phys. Rev., 162, 1195 (1967) · Zbl 0161.46501 |

[39] | ’t Hooft, G., An algorithm for the poles at dimension four in the dimensional regularization procedure, Nucl. Phys. B, 62, 444 (1973) |

[40] | Honerkamp, J., The question of invariant renormalizability of the massless Yang-Mills theory in a manifest covariant approach, Nucl. Phys. B, 48, 269 (1972) |

[41] | Boulware, DG, Gauge dependence of the effective action, Phys. Rev. D, 23, 389 (1981) |

[42] | Iyer, V.; Wald, RM, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D, 50, 846 (1994) |

[43] | Barnich, G.; Brandt, F.; Henneaux, M., Local BRST cohomology in gauge theories, Phys. Rep., 338, 439 (2000) · Zbl 1097.81571 |

[44] | Mañes, J.; Stora, R.; Zumino, B., Algebraic study of chiral anomalies, Commun. Math. Phys., 102, 157 (1985) · Zbl 0573.53054 |

[45] | Peierls, RE, The commutation laws of relativistic field theory, Proc. R. Soc. Lond., A214, 143 (1952) · Zbl 0048.44606 |

[46] | DeWitt, B.; DeWitt-Morette, C., From the Peierls bracket to the Feynman functional integral, Ann. Phys., 314, 448 (2004) · Zbl 1078.81026 |

[47] | Grassi, PA, Stability and renormalization of Yang-Mills theory with background field method: a regularization independent proof, Nucl. Phys. B, 462, 524 (1996) |

[48] | Ferrari, R.; Picariello, M.; Quadri, A., Algebraic aspects of the background field method, Ann. Phys., 294, 165 (2001) · Zbl 1016.81043 |

[49] | Anselmi, D., Background field method, Batalin-Vilkovisky formalism and parametric completeness of renormalization, Phys. Rev. D, 89, 045004 (2014) |

[50] | Becchi, C.; Collina, R., Further comments on the background field method and gauge invariant effective actions, Nucl. Phys. B, 562, 412 (1999) |

[51] | Gérard, C.; Wrochna, M., Hadamard states for the linearized Yang-Mills equation on curved spacetime, Commun. Math. Phys., 337, 253 (2015) · Zbl 1314.83025 |

[52] | Wrochna, M.; Zahn, J., Classical phase space and Hadamard states in the BRST formalism for gauge field theories on curved spacetime, Rev. Math. Phys., 29, 1750014 (2017) · Zbl 1447.81165 |

[53] | Zahn, J., Locally covariant chiral fermions and anomalies, Nucl. Phys. B, 890, 1 (2014) · Zbl 1326.81135 |

[54] | Kugo, T.; Ojima, I., Local covariant operator formalism of nonabelian gauge theories and quark confinement problem, Prog. Theor. Phys. Suppl., 66, 1 (1979) |

[55] | Dütsch, M.; Fredenhagen, K., A local (perturbative) construction of observables in gauge theories: the example of QED, Commun. Math. Phys., 203, 71 (1999) · Zbl 0938.81028 |

[56] | Fröb, MB, Anomalies in time-ordered products and applications to the BV-BRST formulation of quantum gauge theories, Commun. Math. Phys., 372, 281 (2019) · Zbl 1431.83066 |

[57] | Schenkel, A.; Zahn, J., Global anomalies on Lorentzian space-times, Ann. Henri Poincaré, 18, 2693 (2017) · Zbl 1373.81343 |

[58] | Hörmander, L., The Analysis of Linear Partial Differential Operators. I (2003), Berlin: Springer, Berlin |

[59] | Ichinose, S., BRS symmetry on background field, Kallosh theorem and renormalization, Nucl. Phys. B, 395, 433 (1993) |

[60] | Brunetti, R., Cosmological perturbation theory and quantum gravity, JHEP, 08, 032 (2016) · Zbl 1390.83059 |

[61] | Fröb, MB; Hack, TP; Higuchi, A., Compactly supported linearised observables in single-field inflation, JCAP, 1707, 043 (2017) |

[62] | Barnich, G.; Brandt, F.; Henneaux, M., General solution of the Wess-Zumino consistency condition for Einstein gravity, Phys. Rev. D, 51, 1435 (1995) |

[63] | Brunetti, R.; Fredenhagen, K.; Fauser, B.; Tolksdorf, J.; Zeidler, E., Quantum gravity, Towards a Background Independent Formulation of Perturbative Quantum Gravity, 151-159 (2006), Basel: Springer, Basel |

[64] | Rejzner, K., Remarks on local symmetry invariance in perturbative algebraic quantum field theory, Ann. Henri Poincaré, 16, 205 (2015) · Zbl 1306.81137 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.