×

zbMATH — the first resource for mathematics

Anomalies in time-ordered products and applications to the BV-BRST formulation of quantum gauge theories. (English) Zbl 1431.83066
Summary: We show that every (graded) derivation on the algebra of free quantum fields and their Wick powers in curved spacetimes gives rise to a set of anomalous Ward identities for time-ordered products, with an explicit formula for their classical limit. We study these identities for the Koszul-Tate and the full BRST differential in the BV-BRST formulation of perturbatively interacting quantum gauge theories, and clarify the relation to previous results. In particular, we show that the quantum BRST differential, the quantum antibracket and the higher-order anomalies form an \(L_\infty\) algebra. The defining relations of this algebra ensure that the gauge structure is well-defined on cohomology classes of the quantum BRST operator, i.e., observables. Furthermore, we show that one can determine contact terms such that also the interacting time-ordered products of multiple interacting fields are well defined on cohomology classes. An important technical improvement over previous treatments is the fact that all our relations hold off-shell and are independent of the concrete form of the Lagrangian, including the case of open gauge algebras.

MSC:
83C47 Methods of quantum field theory in general relativity and gravitational theory
81T50 Anomalies in quantum field theory
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
Software:
pAQFT
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Hawking, SW, Particle creation by black holes, Commun. Math. Phys., 43, 199 (1975) · Zbl 1378.83040
[2] Unruh, WG, Notes on black-hole evaporation, Phys. Rev. D, 14, 870 (1976)
[3] Crispino, LCB; Higuchi, A.; Matsas, GEA, The Unruh effect and its applications, Rev. Mod. Phys., 80, 787 (2008) · Zbl 1205.83030
[4] Planck collaboration, Ade, P.A.R., et al.: Planck 2015 results. XIII. Cosmological parameters. Astron. Astrophys. 594, A13 (2016). arXiv:1502.01589
[5] Planck collaboration, Ade, P.A.R..: Planck 2015 results. XVII. Constraints on primordial non-Gaussianity. Astron. Astrophys. 594, A17 (2016). arXiv:1502.01592
[6] Planck collaboration, Ade. P.A.R., et al.: Planck 2015 results. XX. Constraints on inflation. Astron. Astrophys. 594, A20 (2016). arXiv:1502.02114
[7] 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
[8] Hollands, S.; Wald, RM, Local Wick polynomials and time ordered products of quantum fields in curved spacetime, Commun. Math. Phys., 223, 289 (2001) · Zbl 0989.81081
[9] Hollands, S.; Wald, RM, Existence of local covariant time ordered products of quantum fields in curved spacetime, Commun. Math. Phys., 231, 309 (2002) · Zbl 1015.81043
[10] Hollands, S., Renormalized quantum Yang-Mills fields in curved spacetime, Rev. Math. Phys., 20, 1033 (2008) · Zbl 1161.81022
[11] Hollands, S.: Renormalized quantum Yang-Mills fields in curved spacetime (updated version). arXiv:0705.3340v4 · Zbl 1161.81022
[12] Fredenhagen, K.; Rejzner, K., Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory, Commun. Math. Phys., 317, 697 (2013) · Zbl 1263.81245
[13] Rejzner, K., Remarks on local symmetry invariance in perturbative algebraic quantum field theory, Ann. H. Poincaré, 16, 205 (2015) · Zbl 1306.81137
[14] Dütsch, M.; Boas, FM, The Master Ward Identity, Rev. Math. Phys., 14, 977 (2002) · Zbl 1037.81074
[15] Dütsch, M., Proof of perturbative gauge invariance for tree diagrams to all orders, Ann. Phys. (Leipzig), 14, 438 (2005) · Zbl 1081.83011
[16] Brennecke, F.; Dütsch, M., Removal of violations of the Master Ward Identity in perturbative QFT, Rev. Math. Phys., 20, 119 (2008) · Zbl 1149.81017
[17] Medeiros, P.; Hollands, S., Superconformal quantum field theory in curved spacetime, Class. Quantum Gravity, 30, 175015 (2013) · Zbl 1276.83046
[18] Taslimi Tehrani, M., Self-consistency of conformally coupled ABJM theory at the quantum level, JHEP, 11, 153 (2017) · Zbl 1383.83193
[19] 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
[20] Brunetti, R.; Fredenhagen, K.; Hack, T-P; Pinamonti, N.; Rejzner, K., Cosmological perturbation theory and quantum gravity, JHEP, 08, 032 (2016) · Zbl 1390.83059
[21] Fröb, MB, Gauge-invariant quantum gravitational corrections to correlation functions, Class. Quantum Gravity, 35, 055006 (2018) · Zbl 1386.83062
[22] Fröb, MB; Lima, WCC, Propagators for gauge-invariant observables in cosmology, Class. Quantum Gravity, 35, 095010 (2018) · Zbl 1391.83140
[23] Fredenhagen, K., Rejzner, K.: Perturbative algebraic quantum field theory. In: Proceedings, Winter School in Mathematical Physics: Mathematical Aspects of Quantum Field Theory: Les Houches, France, January 29—February 3, 2012, p. 17. Springer (2015). arXiv:1208.1428. https://doi.org/10.1007/978-3-319-09949-1_2 · Zbl 1315.81070
[24] Hollands, S.; Wald, RM, Quantum fields in curved spacetime, Phys. Rep., 574, 1 (2015) · Zbl 1357.81144
[25] Fewster, C.J., Verch, R.: Algebraic quantum field theory in curved spacetimes. In: Brunetti, R., Dappiaggi, C., Fredenhagen, K., Yngvason, J. (eds.), Advances in Algebraic Quantum Field Theory, p. 125. Springer International Publishing, Cham, (2015). arXiv:1504.00586. https://doi.org/10.1007/978-3-319-21353-8_4
[26] Fredenhagen, K.; Rejzner, K., Quantum field theory on curved spacetimes: axiomatic framework and examples, J. Math. Phys., 57, 031101 (2016) · Zbl 1338.81299
[27] Hack, T.-P.: Cosmological Applications of Algebraic Quantum Field Theory in Curved Spacetimes. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-21894-6 · Zbl 1337.81003
[28] Bär, C., Ginoux, N., Pfäffle, F.: Wave Equations on Lorentzian Manifolds and Quantization. European Mathematical Society Publishing House, Zürich, Switzerland (2007) · Zbl 1118.58016
[29] Fröb, MB; Taslimi Tehrani, M., Green’s functions and Hadamard parametrices for vector and tensor fields in general linear covariant gauges, Phys. Rev. D, 97, 025022 (2018)
[30] Hörmander, L.: The analysis of linear partial differential operators I, 2nd edn. Springer, Berlin (2003). https://doi.org/10.1007/978-3-642-61497-2 · Zbl 1028.35001
[31] Brunetti, R.; Dütsch, M.; Fredenhagen, K., Perturbative algebraic quantum field theory and the renormalization groups, Adv. Theor. Math. Phys., 13, 1541 (2009) · Zbl 1201.81090
[32] Fredenhagen, K.; Rejzner, K., Batalin-Vilkovisky formalism in the functional approach to classical field theory, Commun. Math. Phys., 314, 93 (2012) · Zbl 1418.70034
[33] Radzikowski, MJ, Micro-local approach to the Hadamard condition in quantum field theory on curved space–time, Commun. Math. Phys., 179, 529 (1996) · Zbl 0858.53055
[34] 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) · Zbl 0923.58052
[35] Brunetti, R.; Fredenhagen, K., Microlocal analysis and interacting quantum field theories: renormalization on physical backgrounds, Commun. Math. Phys., 208, 623 (2000) · Zbl 1040.81067
[36] Hollands, S.; Wald, RM, Conservation of the stress tensor in interacting quantum field theory in curved spacetimes, Rev. Math. Phys., 17, 227 (2005) · Zbl 1078.81062
[37] Zahn, J., Locally covariant charged fields and background independence, Rev. Math. Phys., 27, 1550017 (2015) · Zbl 1327.81273
[38] Hollands, S.; Wald, RM, On the renormalization group in curved spacetime, Commun. Math. Phys., 237, 123 (2003) · Zbl 1059.81138
[39] Dütsch, M.; Fredenhagen, K., Causal perturbation theory in terms of retarded products, and a proof of the action Ward identity, Rev. Math. Phys., 16, 1291 (2004) · Zbl 1084.81054
[40] Khavkine, I.; Melati, A.; Moretti, V., On Wick polynomials of boson fields in locally covariant algebraic QFT, Ann. H. Poincaré, 20, 929 (2019) · Zbl 1414.83022
[41] Dütsch, M.; Fredenhagen, K., Algebraic quantum field theory, perturbation theory, and the loop expansion, Commun. Math. Phys., 219, 5 (2001) · Zbl 1019.81041
[42] Achilles, R.; Bonfiglioli, A., The early proofs of the theorem of Campbell, Baker, Hausdorff, and Dynkin, Arch. Hist. Exact Sci., 66, 295 (2012) · Zbl 1245.01002
[43] Dütsch, M.; Fredenhagen, K., The Master Ward Identity and generalized Schwinger-Dyson equation in classical field theory, Commun. Math. Phys., 243, 275 (2003) · Zbl 1049.70017
[44] Epstein, H.; Glaser, V., The role of locality in perturbation theory, Ann. H. Poincaré, A19, 211 (1973) · Zbl 1216.81075
[45] Bernal, AN; Sánchez, M., Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions, Lett. Math. Phys., 77, 183 (2006) · Zbl 1126.53043
[46] Ward, JC, An identity in quantum electrodynamics, Phys. Rev., 78, 182 (1950) · Zbl 0041.33012
[47] Rohrlich, F., Quantum electrodynamics of charged particles without spin, Phys. Rev., 80, 666 (1950) · Zbl 0040.28003
[48] Takahashi, Y., On the generalized Ward identity, Nuovo Cim., 6, 371 (1957) · Zbl 0078.20202
[49] Buchholz, D.; D’Antoni, C.; Fredenhagen, K., The universal structure of local algebras, Commun. Math. Phys., 111, 123 (1987) · Zbl 0645.46048
[50] Buchholz, D.; Verch, R., Scaling algebras and renormalization group in algebraic quantum field theory, Rev. Math. Phys., 7, 1195 (1995) · Zbl 0842.46052
[51] Yngvason, J., The role of type III factors in quantum field theory, Rept. Math. Phys., 55, 135 (2005) · Zbl 1140.81427
[52] Sakai, S., Derivations of \(W^*\)-algebras, Ann. Math., 83, 273 (1966) · Zbl 0139.30601
[53] Kadison, RV, Derivations of operator algebras, Ann. Math., 83, 280 (1966) · Zbl 0139.30503
[54] Becchi, C.; Rouet, A.; Stora, R., Renormalization of Gauge theories, Ann. Phys., 98, 287 (1976)
[55] Batalin, I.; Vilkovisky, G., Gauge algebra and quantization, Phys. Lett. B, 102, 27 (1981)
[56] Batalin, I.; Vilkovisky, G., Quantization of gauge theories with linearly dependent generators, Phys. Rev. D, 28, 2567 (1983)
[57] Batalin, I.; Vilkovisky, G., Erratum: Quantization of gauge theories with linearly dependent generators, Phys. Rev. D, 30, 508 (1984)
[58] Henneaux, M., Lectures on the antifield-BRST formalism for gauge theories, Nucl. Phys. B Proc. Suppl., 18A, 47 (1990) · Zbl 0957.81711
[59] Gomis, J.; Paris, J.; Samuel, S., Antibracket, antifields and gauge-theory quantization, Phys. Rep., 259, 1 (1995)
[60] Barnich, G.; Brandt, F.; Henneaux, M., Local BRST cohomology in gauge theories, Phys. Rep., 338, 439 (2000) · Zbl 1097.81571
[61] Nakanishi, N., Covariant quantization of the electromagnetic field in the Landau gauge, Prog. Theor. Phys., 35, 1111 (1966)
[62] Lautrup, B., Canonical quantum electrodynamics in covariant gauges, Mat. Fys. Medd. Dan. Vid. Selsk., 35, 11 (1967)
[63] Brandt, F.; Henneaux, M.; Wilch, A., Global symmetries in the antifield formalism, Phys. Lett. B, 387, 320 (1996)
[64] Brandt, F.; Henneaux, M.; Wilch, A., Extended antifield formalism, Nucl. Phys. B, 510, 640 (1998) · Zbl 0953.81099
[65] Townsend, PK, Covariant quantization of antisymmetric tensor gauge fields, Phys. Lett. B, 88, 97 (1979)
[66] Namazie, MA; Storey, D., On secondary and higher-generation ghosts, J. Phys. A, 13, l161 (1980)
[67] Thierry-Mieg, J., BRS structure of the antisymmetric tensor gauge theories, Nucl. Phys. B, 335, 334 (1990)
[68] Siegel, W., Hidden ghosts, Phys. Lett. B, 93, 170 (1980)
[69] Kimura, T., Counting of ghosts in quantized antisymmetric tensor gauge field of third rank, J. Phys. A, 13, l353 (1980)
[70] Kimura, T., Quantum theory of antisymmetric higher rank tensor gauge field in higher dimensional space–time, Prog. Theor. Phys., 65, 338 (1981)
[71] Piguet, O., Sorella, S.P.: Algebraic Renormalization: Perturbative Renormalization, Symmetries and Anomalies. Springer, Berlin (1995) · Zbl 0845.58069
[72] Batalin, IA; Vilkovisky, GA, Closure of the gauge algebra, generalized Lie equations and Feynman rules, Nucl. Phys. B, 234, 106 (1984)
[73] Batalin, IA; Vilkovisky, GA, Existence theorem for gauge algebra, J. Math. Phys., 26, 172 (1985)
[74] Tyutin, I.V.: Gauge Invariance in Field Theory and Statistical Physics in Operator Formalism. arXiv:0812.0580
[75] Taslimi Tehrani, M., Quantum BRST charge in gauge theories in curved space-time, J. Math. Phys., 60, 012304 (2019) · Zbl 1406.81067
[76] Yang, C-N; Mills, RL, Conservation of Isotopic Spin and Isotopic Gauge Invariance, Phys. Rev., 96, 191 (1954) · Zbl 1378.81075
[77] Ferrara, S.; Zumino, B., Supergauge invariant Yang-Mills theories, Nucl. Phys. B, 79, 413 (1974)
[78] Salam, A.; Strathdee, J., Super-symmetry and non-Abelian gauges, Phys. Lett. B, 51, 353 (1974)
[79] Wit, B.; Freedman, DZ, Combined supersymmetric and gauge-invariant field theories, Phys. Rev. D, 12, 2286 (1975)
[80] Fierz, M., Zur Fermischen Theorie des \(\beta \)-Zerfalls, Z. Physik, 104, 553 (1937) · Zbl 0016.09403
[81] Freedman, D.Z., Van Proeyen, A.: Supergravity. Cambridge University Press, Cambridge (2012) · Zbl 1245.83001
[82] Wess, J.; Zumino, B., Consequences of anomalous Ward identities, Phys. Lett. B, 37, 95 (1971)
[83] Drago, N.; Hack, T-P; Pinamonti, N., The generalised principle of perturbative agreement and the thermal mass, Ann. H. Poincaré, 18, 807 (2017) · Zbl 1362.81064
[84] Adler, SL, Axial-Vector vertex in spinor electrodynamics, Phys. Rev., 177, 2426 (1969)
[85] Bell, JS; Jackiw, R., A PCAC puzzle: \( \pi ^0\rightarrow \gamma \gamma\) in the \(\sigma \)-model, Nuovo Cim. A, 60, 47 (1969)
[86] Fujikawa, K., Path-Integral measure for gauge-invariant fermion theories, Phys. Rev. Lett., 42, 1195 (1979)
[87] Geng, CQ; Marshak, RE, Uniqueness of quark and lepton representations in the standard model from the anomalies viewpoint, Phys. Rev. D, 39, 693 (1989)
[88] Minahan, JA; Ramond, P.; Warner, RC, Comment on anomaly cancellation in the standard model, Phys. Rev. D, 41, 715 (1990)
[89] 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
[90] Lada, T.; Stasheff, J., Introduction to SH Lie algebras for physicists, Int. J. Theor. Phys., 32, 1087 (1993) · Zbl 0824.17024
[91] Hohm, O.; Zwiebach, \(B., L_{\infty }\) algebras and field theory, Fortsch. Phys., 65, 1700014 (2017) · Zbl 1371.81261
[92] Henneaux, M., Teitelboim, C.: Quantization of Gauge systems. Princeton University Press, Princeton (1992) · Zbl 0838.53053
[93] Zahn, J.: Private communication (2018)
[94] Piguet, O.; Sibold, K., The anomaly in the Slavnov identity for \(N=1\) supersymmetric Yang-Mills theories, Nucl. Phys. B, 247, 484 (1984)
[95] Brandt, F., Extended BRST cohomology, consistent deformations and anomalies of four-dimensional supersymmetric gauge theories, JHEP, 04, 035 (2003)
[96] Junker, W.; Schrohe, E., Adiabatic vacuum states on general spacetime manifolds: definition, construction, and physical properties, Ann. H. Poincaré, 3, 1113 (2002) · Zbl 1038.81052
[97] Fewster, CJ; Pfenning, MJ, A quantum weak energy inequality for spin-one fields in curved space-time, J. Math. Phys., 44, 4480 (2003) · Zbl 1062.81115
[98] Duch, P., Weak adiabatic limit in quantum field theories with massless particles, Ann. H. Poincaré, 19, 875 (2018) · Zbl 1394.81123
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.