×

zbMATH — the first resource for mathematics

On Wick polynomials of boson fields in locally covariant algebraic QFT. (English) Zbl 1414.83022
Summary: This work presents some results about Wick polynomials of a vector field renormalization in locally covariant algebraic quantum field theory in curved spacetime. General vector fields are pictured as sections of natural vector bundles over globally hyperbolic spacetimes and quantized through the known functorial machinery in terms of local \(*\)-algebras. These quantized fields may be defined on spacetimes with given classical background fields, also sections of natural vector bundles, in addition to the Lorentzian metric. The mass and the coupling constants are in particular viewed as background fields. Wick powers of the quantized vector field are axiomatically defined imposing in particular local covariance, scaling properties, and smooth dependence on smooth perturbation of the background fields. A general classification theorem is established for finite renormalization terms (or counterterms) arising when comparing different solutions satisfying the defining axioms of Wick powers. The result is specialized to the case of general tensor fields. In particular, the case of a vector Klein-Gordon field and the case of a scalar field renormalized together with its derivatives are discussed as examples. In each case, a more precise statement about the structure of the counterterms is proved. The finite renormalization terms turn out to be finite-order polynomials tensorially and locally constructed with the backgrounds fields and their covariant derivatives whose coefficients are locally smooth functions of polynomial scalar invariants constructed from the so-called marginal subset of the background fields. The notion of local smooth dependence on polynomial scalar invariants is made precise in the text. Our main technical tools are based on the Peetre-Slovák theorem characterizing differential operators and on the classification of smooth invariants on representations of reductive Lie groups.

MSC:
83C47 Methods of quantum field theory in general relativity and gravitational theory
81T20 Quantum field theory on curved space or space-time backgrounds
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
53Z05 Applications of differential geometry to physics
81T16 Nonperturbative methods of renormalization applied to problems in quantum field theory
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Allen, B.; Folacci, A., Massless minimally coupled scalar field in de Sitter space, Phys. Rev. D, 35, 3771-3778, (1987)
[2] Anderson, I.M., Torre, C.G.: Two component spinors and natural coordinates for the prolonged Einstein equation manifolds. Tech. Rep., Utah State University (1994). Unpublished
[3] Anderson, IM; Torre, CG, Classification of local generalized symmetries for the vacuum Einstein equations, Commun. Math. Phys., 176, 479-539, (1996) · Zbl 0857.53058
[4] Atiyah, M.; Bott, R.; Patodi, VK, On the heat equation and the index theorem, Invent. Math., 19, 279-330, (1973) · Zbl 0257.58008
[5] Bär, C., Fredenhagen, K. (eds.): Quantum field theory on curved spacetimes: concepts and mathematical foundations. Lecture Notes in Physics, vol. 786. Springer (2009)
[6] Benini, M.; Dappiaggi, C.; Brunetti, R. (ed.); Dappiaggi, C. (ed.); Fredenhagen, K. (ed.); Yngvason, J. (ed.), Models of free quantum field theories on curved background, (2015), Berlin
[7] Brouder, C.; Dang, N V; Laurent-Gengoux, C.; Rejzner, K., Properties of field functionals and characterization of local functionals, J. Math. Phys., 59, 023508, (2017) · Zbl 1456.81296
[8] 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
[9] 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
[10] Christoffel, E.B.: Über die Transformation der homogenen Differentialausdrücke zweiten Grades. J. Reine Angew. Math. 70, 46-70 (1869). http://eudml.org/doc/148073
[11] Dappiaggi, C.; Drago, N., Constructing Hadamard states via an extended Møller operator, Lett. Math. Phys., 106, 1587-1615, (2016) · Zbl 1362.81073
[12] Drago, N.; Gérard, C., On the adiabatic limit of Hadamard states, Lett. Math. Phys., 107, 1409-1438, (2017) · Zbl 1374.81068
[13] Drago, N.; Hack, T-P; Pinamonti, N., The generalised principle of perturbative agreement and the thermal mass, Ann. Henri Poincaré, 18, 807-868, (2017) · Zbl 1362.81064
[14] Fredenhagen, K.; Rejzner, K., Batalin-Vilkovisky formalism in the functional approach to classical field theory, Commun. Math. Phys., 314, 93-127, (2012) · Zbl 1418.70034
[15] Fulton, W.: Young Tableaux: With Applications to Representation Theory and Geometry. London Mathematical Society Student Texts, vol. 35. Cambridge University Press, Cambridge (1996)
[16] Gilkey, PB, Curvature and the eigenvalues of the Laplacian for elliptic complexes, Adv. Math., 10, 344-382, (1973) · Zbl 0259.58010
[17] Goodman, R., Wallach, N.R.: Symmetry, Representations, and Invariants. Graduate Texts in Mathematics, vol. 255. Springer, New York (2009) · Zbl 1173.22001
[18] Hack, T-P; Pinamonti, N.; Brunetti, R. (ed.); Dappiaggi, C. (ed.); Fredenhagen, K. (ed.); Yngvason, J. (ed.), Cosmologial application of algebraic quantum field theory, (2015), Berlin
[19] Hollands, S., Renormalized quantum Yang-Mills fields in curved spacetime, Rev. Math. Phys., 20, 1033-1172, (2008) · Zbl 1161.81022
[20] Hollands, S.; Wald, RM, Local Wick polynomials and time ordered products of quantum fields in curved spacetime, Commun. Math. Phys., 223, 289-326, (2001) · Zbl 0989.81081
[21] Hollands, S.; Wald, RM, Existence of local covariant time ordered products of quantum fields in curved spacetime, Commun. Math. Phys., 231, 309-345, (2002) · Zbl 1015.81043
[22] Hollands, S.; Wald, RM, Conservation of the stress tensor in perturbative interacting quantum field theory in curved spacetimes, Rev. Math. Phys., 17, 227-311, (2005) · Zbl 1078.81062
[23] Jentsch, T.: The jet isomorphism theorem of pseudo-Riemannian geometry. arXiv:1509.08269
[24] Khavkine, I.; Moretti, V.; Brunetti, R. (ed.); Dappiaggi, C. (ed.); Fredenhagen, K. (ed.); Yngvason, J. (ed.), Algebraic QFT in curved spacetime and quasifree Hadamard states: an introduction, (2015), Berlin
[25] Khavkine, I.; Moretti, V., Analytic dependence is an unnecessary requirement in renormalization of locally covariant QFT, Commun. Math. Phys., 344, 581-620, (2016) · Zbl 1351.81081
[26] Kolař, I., Michor, P.W., Slovák, J.: Natural Operations in Differential Geometry. Springer, Berlin (1993) · Zbl 0782.53013
[27] Luna, D., Fonctions différentiables invariantes sous l’opération d’un groupe réductif, Ann. Inst. Fourier, 26, 33-49, (1976) · Zbl 0315.20039
[28] Michor, P.W.: Topics in Differential Geometry. American Mathematical Society, Providence, RI (2008) · Zbl 1175.53002
[29] Parlett, BN, The (matrix) discriminant as a determinant, Linear Algebra Appl., 355, 85-101, (2002) · Zbl 1018.15006
[30] Penrose, R., A spinor approach to general relativity, Ann. Phys., 10, 171-201, (1960) · Zbl 0091.21404
[31] Procesi, C.: Lie Groups: An Approach Through Invariants and Representations. Universitext. Springer, New York (2007) · Zbl 1154.22001
[32] Richardson, RW, Principal orbit types for real-analytic transformation groups, Am. J. Math., 95, 193-203, (1973) · Zbl 0277.57013
[33] Richardson, RW; Slodowy, PJ, Minimum vectors for real reductive algebraic groups, J. Lond. Math. Soc., 42, 409-429, (1990) · Zbl 0675.14020
[34] Rumberger, M., Finitely differentiable invariants, Math. Z., 229, 675-694, (1998) · Zbl 0930.58004
[35] Sahlmann, H.; Verch, R., Microlocal spectrum condition and Hadamard form for vector-valued quantum fields in curved spacetime, Rev. Math. Phys., 13, 1203-1246, (2001) · Zbl 1029.81053
[36] Schambach, M., Sanders, K.: The Proca field in curved spacetimes and its zero mass limit. arXiv:1709.01911 [math-ph]
[37] Schouten, J.A.: Ricci-calculus: An Introduction to Tensor Analysis and Its Geometrical Applications. Grundlehren der mathematischen Wissenschaften, vol. 10, 2nd edn. Springer, Berlin (1954) · Zbl 0057.37803
[38] Slovák, J., Peetre theorem for nonlinear operators, Ann. Global Anal. Geom., 6, 273-283, (1988) · Zbl 0636.58042
[39] Slovák, J.: On invariant operations on pseudo-Riemannian manifolds. Comment. Math. Univ. Carol. 33, 269-276 (1992). http://eudml.org/doc/247392
[40] Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C., Herlt, E.: Exact Solutions of Einstein’s Field Equations. Cambridge University Press, Cambridge (2003) · Zbl 1057.83004
[41] Stoetzel, H.: Quotients of real reductive group actions related to orbit type strata. PhD thesis, Ruhr-Universitat Bochum (2008). http://nbn-resolving.de/urn/resolver.pl?urn=urn:nbn:de:hbz:294-23168
[42] Thomas, T.Y.: Differential Invariants of Generalized Spaces. CUP, Cambridge (1934) · Zbl 0009.08503
[43] Wald, R.M.: General Relativity. The University of Chicago Press, Chicago (1984) · Zbl 0549.53001
[44] Zahn, J., The renormalized locally covariant Dirac field, Rev. Math. Phys., 26, 1330012, (2014) · Zbl 1287.81086
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.