×

zbMATH — the first resource for mathematics

Remarks on local symmetry invariance in perturbative algebraic quantum field theory. (English) Zbl 1306.81137
Summary: We investigate various aspects of invariance under local symmetries in the framework of perturbative algebraic quantum field theory (pAQFT). Our main result is the proof that the quantum Batalin-Vilkovisky operator, on-shell, can be written as the commutator with the interacting BRST charge. Up to now, this was proven only for a certain class of fields in quantum electrodynamics and in Yang-Mills theory. Our result is more general and it holds in a wide class of theories with local symmetries, including general relativity and the bosonic string. We also comment on other issues related to local gauge invariance and, using the language of homological algebra, we compare different approaches to quantization of gauge theories in the pAQFT framework.

MSC:
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
46L60 Applications of selfadjoint operator algebras to physics
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
83C47 Methods of quantum field theory in general relativity and gravitational theory
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81R15 Operator algebra methods applied to problems in quantum theory
Software:
pAQFT
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Bahns, D., Rejzner, K., Zahn, J.: The effective theory of strings. arXiv.org:math-ph/1204.6263v2
[2] Barnich, G., Henneaux, M., Hurth, T., Skenderis, K.: Cohomological analysis of gauge-fixed gauge theories. Phys. Lett. B 492, 376 (2000). (arXiv:hep-th/9910201) · Zbl 0976.81119
[3] Barnich, G., Brandt, F., Henneaux, M.: Local BRST cohomology in gauge theories. Phys. Rept. 338, 439 (2000). (arXiv:hep-th/0002245) · Zbl 1097.81571
[4] Batalin, I.A.; Vilkovisky, G.A., Relativistic S matrix of dynamical systems with boson and fermion constraints, Phys. Lett. B, 69, 309, (1977)
[5] Batalin, I.A.; Vilkovisky, G.A., Gauge algebra and quantization, Phys. Lett. B, 102, 27, (1981)
[6] Batalin, I.A.; Vilkovisky, G.A., Feynman rules for reducible gauge theories, Phys. Lett. B, 120, 166, (1983)
[7] Batalin, I.A.; Vilkovisky, G.A., Quantization of gauge theories with linearly dependent generators, Phys. Rev. D, 28, 2567, (1983)
[8] Battle, C.; Gomis, J.; Paris, J.; Roca, J., Field-antifield formalism and Hamiltonian BRST approach, Nucl. Phys. B, 329, 139-154, (1990)
[9] Baulieu, L.; Thierry-Mieg, J., Algebraic structure of quantum gravity and the classification of the gravitational anomalies, Elsevier, 145, 53-60, (1984)
[10] Becchi, C.; Rouet, A.; Stora, R., Renormalization of the abelian Higgs-kibble model, Commun. Math. Phys., 42, 127, (1975)
[11] Becchi, C.; Rouet, A.; Stora, R., Renormalization of gauge theories, Ann. Phys., 98, 287, (1976)
[12] Boas, F.-M.: Gauge Theories in Local Causal Perturbation Theory. Ph.D. thesis, Hamburg (1999), Hamburg DESY-THESIS-1999-032, ISSN 1435-808
[13] Bogoliubov N.N., Shirkov D.V.: Introduction to the Theory of Quantized Fields. Interscience Publishers, Inc., New York (1959) · Zbl 0088.21701
[14] Brennecke, F.; Dütsch, M., Removal of violations of the master Ward identity in perturbative QFT, Rev. Math. Phys., 20, 119-172, (2008) · Zbl 1149.81017
[15] Brunetti, R.; Fredenhagen, K., Microlocal analysis and interacting quantum field theories: renormalization on physical backgrounds, Commun. Math. Phys., 208, 623-661, (2000) · Zbl 1040.81067
[16] Brunetti, R., Fredenhagen, K.: Towards a background independent formulation of perturbative quantum gravity. In: Fauser, B., et al. (eds.) Quantum gravity, pp. 151-159. Proceedings of Workshop on Mathematical and Physical Aspects of Quantum Gravity, Blaubeuren, Germany, 28 Jul-1 Aug 2005. (arXiv:gr-qc/0603079v3) · Zbl 1120.83016
[17] Brunetti, R., Fredenhagen, K., Köhler, M.: The microlocal spectrum condition and Wick polynomials of free fields on curved spacetimes. Commun. Math. Phys. 180, 633 (1996). (arXiv:gr-qc/9510056) · Zbl 0923.58052
[18] Brunetti, R., Fredenhagen, K., Rejzner, K.: Locally covariant quantum field theory as a way to quantum gravity. (arXiv:math-ph/1306.1058) · Zbl 1346.83001
[19] Brunetti, R.; Fredenhagen, K.; Verch, R., The generally covariant locality principle - A new paradigm for local quantum field theory, Commun. Math. Phys., 237, 31-68, (2003) · Zbl 1047.81052
[20] Brunetti, R., Dütsch, M., Fredenhagen, K.: Perturbative algebraic quantum field theory and the renormalization groups. Adv. Theor. Math. Phys. 13(5), 1541-1599 (2009). (arXiv:math-ph/0901.2038v2) · Zbl 1201.81090
[21] Chevalley, C.; Eilenberg, S., Cohomology theory of Lie groups and Lie algebras, Trans. Am. Math. Soc. (Providence: American Mathematical Society), 63, 85-124, (1948) · Zbl 0031.24803
[22] Dütsch, M.; Boas, F.-M., The master Ward identity, Rev. Math. Phys, 14, 977-1049, (2002) · Zbl 1037.81074
[23] Dütsch, M.; Fredenhagen, K., A local (perturbative) construction of observables in gauge theories: the example of QED, Commun. Math. Phys., 203, 71-105, (1999) · Zbl 0938.81028
[24] Dütsch, M., Fredenhagen, K.: Perturbative algebraic field theory, and deformation quantization. In: Proceedings of the Conference on Mathematical Physics in Mathematics and Physics, Siena June 20-25 2000. (arXiv:hep-th/0101079) · Zbl 1216.81075
[25] Dütsch, M., Fredenhagen, K.: Algebraic quantum field theory, perturbation theory, and the loop expansion. Commun. Math. Phys. 219, 5 (2001). (arXiv:hep-th/0001129) · Zbl 1019.81041
[26] Dütsch, M., Fredenhagen, K.: Causal perturbation theory in terms of retarded products, and a proof of the action Ward identity. Rev. Math. Phys. 16(10), 1291-1348 (2004). (arXiv:hep-th/0403213) · Zbl 1084.81054
[27] Dütsch, M., Fredenhagen, K.: The master Ward identity and generalized Schwinger-Dyson equation in classical field theory. Commun. Math. Phys. 243, 275 (2003). (arXiv:hep-th/0211242) · Zbl 1049.70017
[28] Epstein, H.; Glaser, V., The role of locality in perturbation theory, Ann. Inst. H. Poincaré A, 19, 211, (1973) · Zbl 1216.81075
[29] Fisch, J.M.L.; Henneaux, M., Antibracket-antifield formalism for constrained Hamiltonian systems, Phys. Lett. B, 226, 80-88, (1989)
[30] Fradkin, E.S.; Vasilev, M.A., Hamiltonian formalism, quantization and S matrix for supergravity, Phys. Lett. B, 72, 70, (1977) · Zbl 0967.81505
[31] Fradkin, E.S.; Vilkovisky, G.A., Quantization of relativistic systems with constraints, Phys. Lett. B, 55, 224, (1975) · Zbl 0967.81532
[32] Fradkin, E.S., Vilkovisky, G.A.: Quantization of relativistic systems with constraints: equivalence of canonical and covariant formalisms in quantum theory of gravitational field. CERN-TH-2332 · Zbl 0967.81532
[33] Fradkin, E.S.; Fradkina, T.E., Quantization of relativistic systems with boson and fermion first and second class constraints, Phys. Lett. B, 72, 343, (1978)
[34] Friedrich, H., Is general relativity “essentially understood”?, Ann. Phys. (Leipzig), 15, 84-108, (2006) · Zbl 1098.83005
[35] Fredenhagen, K.: Locally covariant quantum field theory. In: Proceedings of the XIVth International Congress on Mathematical Physics, Lisbon 2003, (hep-th/0403007) · Zbl 1221.81123
[36] Fredenhagen, K.: Algebraic structures in perturbative quantum field theory. A talk given at the CMTP Workshop “Two days in QFT” dedicated to the memory of Claudio D’Antoni, Rome, January 10-11, 2011 · Zbl 1040.81067
[37] Fredenhagen, K., Rejzner, K.: Batalin-Vilkovisky formalism in the functional approach to classical field theory. Commun. Math. Phys. 314, 93-127 (2012). (arXiv:math-ph/1101.5112) · Zbl 1418.70034
[38] Fredenhagen, K., Rejzner, K.: Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory. Commun. Math. Phys. 317, 697-725 (2013). (arXiv:math-ph/1110.5232) · Zbl 1263.81245
[39] Fulling, S.A.; Narcowich, F.J.; Wald, R.M., Singularity structure of the two-point function in quantum field theory in curved spacetime. II, Ann. Phys., 136, 243-272, (1981) · Zbl 0495.35054
[40] Haag, R.; Kastler, D., An algebraic approach to quantum field theory, J. Math. Phys., 5, 848, (1964) · Zbl 0139.46003
[41] Henneaux, M., Teitelboim, C.: Quantization of gauge systems, p 520. Princeton University Press, Princeton (1992) · Zbl 0838.53053
[42] Henneaux, M.: Lectures on the antifield—BRST formalism for gauge theories. Lectures given at 20th GIFT Int. Seminar on Theoretical Physics, Jaca, Spain, Jun 5-9, 1989, and at CECS, Santiago, Chile, June/July 1989, Nucl. Phys. B (Proc. Suppl.) A18, 47 (1990) · Zbl 1059.81138
[43] Hollands, S.: Renormalized quantum Yang-Mills fields in curved spacetime. Rev. Math. Phys. 20, 1033 (2008). (arXiv:gr-qc/0705.3340v3) · Zbl 1161.81022
[44] Hollands, S.; Wald, R.M., Local Wick polynomials and time ordered products of quantum fields in curved spacetime, Commun. Math. Phys., 223, 289, (2001) · Zbl 0989.81081
[45] Hollands, S.; Wald, R.M., Existence of local covariant time-ordered-products of quantum fields in curved spacetime, Commun. Math. Phys., 231, 309-345, (2002) · Zbl 1015.81043
[46] Hollands, S.; Wald, R.M., On the renormalization group in curved spacetime, Commun. Math. Phys., 237, 123-160, (2003) · Zbl 1059.81138
[47] Hollands, S., Wald, R.M.: Conservation of the stress tensor in interacting quantum field theory in curved spacetimes. Rev. Math. Phys. 17, 227 (2005). (arXiv:gr-qc/0404074) · Zbl 1078.81062
[48] Hörmander, L.: The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis. Springer, Berin (2003) · Zbl 1028.35001
[49] Keller, K.J.: Dimensional Regularization in Position Space and a Forest Formula for Regularized Epstein-Glaser Renormalization. Ph.D thesis, Hamburg (2010). (arXiv:math-ph/1006.2148v1)
[50] Kugo, T.; Ojima, I., Subsidiary conditions and physical \(S\)-matrix unitarity in indefinite metric quantum gravitational theory, Nucl. Phys., 144, 234, (1978)
[51] Kugo, T.; Ojima, I., Manifestly covariant canonical formulation of Yang-Mills theories physical state subsidiary conditions and physical S-matrix unitarity, Phys. Lett. B, 73, 459-462, (1978)
[52] Kugo, T., Ojima, I.: Local covariant operator formalism of non-abelian gauge theories and quark confinement problem. Suppl. Prog. Theor. Phys. 66, 1 (1979) (Prog. Theor. Phys. 71, 1121 (1984) (Erratum)) · Zbl 1098.81592
[53] Neeb, K.-H.: Monastir Lecture Notes on Infinite-Dimensional Lie Groups. http://www.math.uni-hamburg.de/home/wockel/data/monastir1
[54] Rejzner, K.: Fermionic fields in the functional approach to classical field theory. Rev. Math. Phys. 23, 1009-1033 (2011). (arXiv:math-ph/1101.5126v1) · Zbl 1242.81112
[55] Rejzner, K.: Batalin-Vilkovisky formalism in locally covariant field theory. Ph.D. thesis, DESY-THESIS-2011-041, Hamburg. (arXiv:math-ph/1110.5130) · Zbl 1263.81245
[56] Tyutin, I.V.: Gauge invariance in field theory and statistical physics in operator formalism. LEBEDEV-75-39 preprint (In Russian) p 62 (1975)
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.