# zbMATH — the first resource for mathematics

Mode-sum construction of the two-point functions for the Stueckelberg vector fields in the Poincaré patch of de Sitter space. (English) Zbl 1294.81135
Summary: We perform canonical quantization of the Stueckelberg Lagrangian for massive vector fields in the conformally flat patch of de Sitter space in the Bunch-Davies vacuum and find their Wightman two-point functions by the mode-sum method. We discuss the zero-mass limit of these two-point functions and their limits where the Stueckelberg parameter {$$\xi$$} tends to zero or infinity. It is shown that our results reproduce the standard flat-space propagator in the appropriate limit. We also point out that the classic work of B. Allen and T. Jacobson [Commun. Math. Phys. 103, 669–692 (1986; Zbl 0632.53060)] for the two-point function of the Proca field and a recent work by N. C. Tsamis and R. P. Woodard [J. Math. Phys. 48, No. 5, 052306, 14 p. (2007; Zbl 1144.81417)] for that of the transverse vector field are two limits of our two-point function, one for $$\xi \to \infty$$ and the other for $$\xi \to 0$$. Thus, these two works are consistent with each other, contrary to the claim by the latter authors.{
©2014 American Institute of Physics}

##### MSC:
 81T20 Quantum field theory on curved space or space-time backgrounds 81T70 Quantization in field theory; cohomological methods 70S05 Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems 81V22 Unified quantum theories 81R40 Symmetry breaking in quantum theory 83F05 Relativistic cosmology
##### Keywords:
Wightman two-point functions
Full Text:
##### References:
 [1] Allen, B.; Jacobson, T., Vector two-point functions in maximally symmetric spaces, Commun. Math. Phys., 103, 669, (1986) · Zbl 0632.53060 [2] Proca, A., Sur les équations fondamentales des particules élémentaires, C. R. Acad. Sci., 202, 1490, (1936) · JFM 62.1001.02 [3] Tsamis, N. C.; Woodard, R. P., Maximally symmetric vector propagator, J. Math. Phys., 48, 052306, (2007) · Zbl 1144.81417 [4] Stueckelberg, E. C. G., Die Wechselwirkungskräfte in der Elektrodynamik und in der Feldtheorie der Kernkräfte. Teil II und III, Helv. Phys. Acta, 11, 299, (1938) · JFM 64.1492.03 [5] Schomblond, C.; Spindel, P., Propagateurs des champs spinoriels et vectoriels dans l’univers de de Sitter, Bull. Cl. Sci. Acad. Roy. Belg., LXII, 124-134, (1976) · Zbl 0329.53012 [6] Higuchi, A., Quantization of scalar and vector fields inside the cosmological event horizon and its application to the Hawking effect, Class. Quantum Grav., 4, 721, (1987) [7] Gazeau, J.-P.; Takook, M. V., Massive’ vector field in de Sitter space, J. Math. Phys., 41, 5920, (2000) · Zbl 0978.81062 [8] Cotăescu, I. I., Polarized vector bosons on the de Sitter expanding universe, Gen. Rel. Grav., 42, 861, (2010) · Zbl 1188.83043 [9] Itzykson, C.; Zuber, J. B., Quantum Field Theory, (1980), McGraw-Hill: McGraw-Hill, New York · Zbl 0453.05035 [10] See for NIST Digital Library of Mathematical Functions. [11] Candelas, P.; Raine, D. J., General-relativistic quantum field theory: An exactly soluble model, Phys. Rev. D, 12, 965, (1975) [12] Youssef, A., Infrared behavior and gauge artifacts in de Sitter spacetime: The photon field, Phys. Rev. Lett., 107, 021101, (2011) [13] Smirnov, V. A., Evaluating Feynman Integrals, (2005), Springer: Springer, Berlin/Heidelberg [14] Schwinger, J. S., Field theory commutators, Phys. Rev. Lett., 3, 296, (1959) [15] Boulware, D. G.; Deser, S., Stress-Tensor commutators and Schwinger terms, J. Math. Phys., 8, 1468, (1967) [16] Jacob, G.; Stech, B., Commutators and Lorentz covariance, Z. Phys., 239, 379, (1970) [17] Dashen, R. F.; Lee, S. Y., Existence of the covariant time-ordered product of currents, Phys. Rev., 187, 2017, (1969) [18] Nutbrown, D. A., Simple derivation of Seagull terms for propagator functions, Phys. Rev. D, 3, 2981, (1971) [19] Gross, D. J.; Jackiw, R., Construction of covariant and gauge invariant T* products, Nucl. Phys. B, 14, 269, (1969) [20] Brown, S. G., Covariance and the cancellation of Schwinger and Seagull terms in applications of current algebras, Phys. Rev., 158, 1444, (1967) [21] Kahya, E. O.; Woodard, R. P., Charged scalar self-mass during inflation, Phys. Rev. D, 72, 104001, (2005) [22] Higuchi, A.; Lee, Y. C., How to use retarded Green’s functions in de Sitter spacetime, Phys. Rev. D, 78, 084031, (2008) [23] Higuchi, A.; Lee, Y. C.; Nicholas, J. R., More on the covariant retarded Green’s function for the electromagnetic field in de Sitter spacetime, Phys. Rev. D, 80, 107502, (2009) [24] Kahya, E. O.; Woodard, R. P., One loop corrected mode functions for SQED during inflation, Phys. Rev. D, 74, 084012, (2006) [25] Prokopec, T.; Tsamis, N. C.; Woodard, R. P., Two loop scalar bilinears for inflationary SQED, Class. Quantum Grav., 24, 201-230, (2007) · Zbl 1133.83341 [26] Prokopec, T.; Tsamis, N. C.; Woodard, R. P., Two loop stress-energy tensor for inflationary scalar electrodynamics, Phys. Rev. D, 78, 043523, (2008) [27] Coleman, S. R.; Weinberg, E. J., Radiative corrections as the origin of spontaneous symmetry breaking, Phys. Rev. D, 7, 1888-1910, (1973) [28] Allen, B., Phase transitions in de Sitter space, Nucl. Phys. B, 226, 228, (1983) [29] Prokopec, T.; Tsamis, N.; Woodard, R., Stochastic inflationary scalar electrodynamics, Ann. Phys., 323, 1324-1360, (2008) · Zbl 1151.81030 [30] Nielsen, N., On the gauge dependence of spontaneous symmetry breaking in gauge theories, Nucl. Phys. B, 101, 173, (1975) [31] Aitchison, I.; Fraser, C., Gauge invariance and the effective potential, Ann. Phys., 156, 1, (1984) · Zbl 1216.81109 [32] Johnston, D., Nielsen identities in the ’t Hooft gauge, Nucl. Phys. B, 253, 687, (1985) [33] Tsamis, N. C.; Woodard, R. P., The structure of perturbative quantum gravity on a de Sitter background, Commun. Math. Phys., 162, 217-248, (1994) · Zbl 0809.53084 [34] Bičák, J.; Krtouš, P., Accelerated sources in de Sitter space-time and the insufficiency of retarded fields, Phys. Rev. D, 64, 124020, (2001) [35] Woodard, R. P.; Liu, J. T.; Duff, M. J.; Stelle, K. S.; Woodard, R. P., De Sitter breaking in field theory, Proceedings of Deserfest: A Celebration of the Life and Works of Stanley Deser, Ann Arbor, USA, 3-5 April 2004, 339, (2006), World Scientific Publishing: World Scientific Publishing, Singapore · Zbl 1126.81047 [36] Faci, S.; Huguet, E.; Renaud, J., Conformal use of retarded Green’s functions for the Maxwell field in de Sitter space, Phys. Rev. D, 84, 124050, (2011) [37] Fröb, M. B.; Roura, A.; Verdaguer, E., One-loop gravitational wave spectrum in de Sitter spacetime, JCAP, 1208, 009, (2012) [38] Tanaka, T.; Urakawa, Y., Loops in inflationary correlation functions, Class. Quantum Grav., 30, 233001, (2013) · Zbl 1284.83013 [39] Marolf, D.; Morrison, I. A., The IR stability of de Sitter: Loop corrections to scalar propagators, Phys. Rev. D, 82, 105032, (2010) [40] Higuchi, A.; Marolf, D.; Morrison, I. A., On the equivalence between Euclidean and in-in formalisms in de Sitter QFT, Phys. Rev. D, 83, 084029, (2011) [41] Korai, Y.; Tanaka, T., QFT in the flat chart of de Sitter space, Phys. Rev. D, 87, 024013, (2013) [42] Bailey, W. N., Some infinite integrals involving Bessel functions, Proc. London Math. Soc., s2-s40, 37, (1936) · JFM 61.0398.01
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.