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Dirac fields in flat FLRW cosmology: uniqueness of the Fock quantization. (English) Zbl 1364.81194
Summary: We address the issue of the infinite ambiguity that affects the construction of a Fock quantization of a Dirac field propagating in a cosmological spacetime with flat compact sections. In particular, we discuss a physical criterion that restricts to a unique possibility (up to unitary equivalence) the infinite set of available vacua. We prove that this desired uniqueness is guaranteed, for any possible choice of spin structure on the spatial sections, if we impose two conditions. The first one is that the symmetries of the classical system must be implemented quantum mechanically, so that the vacuum is invariant under the symmetry transformations. The second and more important condition is that the constructed theory must have a quantum dynamics that is implementable as a (non-trivial) unitary operator in Fock space. Actually, this unitarity of the quantum dynamics leads us to identify as explicitly time dependent some very specific contributions of the Dirac field. In doing that, we essentially characterize the part of the dynamics governed by the Dirac equation that is unitarily implementable. The uniqueness of the Fock vacuum is attained then once a physically motivated convention for the concepts of particles and antiparticles is fixed.

MSC:
81T20 Quantum field theory on curved space or space-time backgrounds
81T70 Quantization in field theory; cohomological methods
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[1] Penzias, A. A.; Wilson, R. W., Astrophys. J., 142, 419, (1965)
[2] Dicke, R. H.; Peebles, P. J.E.; Roll, P. G.; Wilkinson, D. T., Astrophys. J., 142, 414, (1965)
[3] Hingshaw, G., ApJS, 208, 19, (2013)
[4] P.A.R. Ade, et al. (Planck Collaboration), arXiv:1502.01589.
[5] Liddle, A. R.; Lyth, D. H., Cosmological inflation and large-scale structure, (2000), Cambridge University Press Cambridge, U.K
[6] Mukhanov, V., Physical foundations of cosmology, (2005), Cambridge University Press Cambridge, U.K · Zbl 1095.83002
[7] Langlois, D., Lecture Notes in Phys., 800, 1, (2010)
[8] Wald, R. M., Quantum field theory in curved spacetime and black hole thermodynamics, (1994), Chicago University Press Chicago · Zbl 0842.53052
[9] Parker, L., Phys. Rev. D, 3, 346, (1971)
[10] Dolgov, A. D.; Kirilova, D. P., Sov. J. Nucl. Phys., 51, 172, (1990)
[11] Kuzmin, V.; Tkachev, I., Phys. Rev. D, 59, (1999)
[12] Giudice, G. F.; Peloso, M.; Riotto, A.; Tkachev, I., J. High Energy Phys., 08, 014, (1999)
[13] Greene, P. B.; Kofman, L., Phys. Lett. B, 448, 6, (1999)
[14] Chung, D. J.H.; Everett, L. L.; Yoo, H.; Zhou, P., Phys. Lett. B, 712, 147, (2012)
[15] Enqvist, K.; Figueroa, D. G.; Meriniemi, T., Phys. Rev. D, 86, (2012)
[16] Chung, D. J.H.; Yoo, H.; Zhou, P., Phys. Rev. D, 91, (2015)
[17] Parker, L., Phys. Rev., 183, 1057, (1969)
[18] Kay, B., Comm. Math. Phys., 62, 55, (1978)
[19] Baez, J. C.; Segal, I. V.; Zhou, Z., Introduction to algebraic and constructive quantum field theory, (1992), Princeton University Press Princeton
[20] Cortez, J.; Mena Marugán, G. A.; Velhinho, J. M., Ann. Phys. (New York), 363, 36, (2015)
[21] Cortez, J.; Mena Marugán, G. A., Phys. Rev. D, 72, (2005)
[22] Corichi, A.; Cortez, J.; Mena Marugán, G. A., Phys. Rev. D, 73, (2006)
[23] Corichi, A.; Cortez, J.; Mena Marugán, G. A., Phys. Rev. D, 73, (2006)
[24] Corichi, A.; Cortez, J.; Mena Marugán, G. A.; Velhinho, J. M., Classical Quantum Gravity, 23, 6301, (2006) · Zbl 1117.83025
[25] Corichi, A.; Cortez, J.; Mena Marugán, G. A.; Velhinho, J. M., Phys. Rev. D, 76, (2007)
[26] Cortez, J.; Mena Marugán, G. A.; Velhinho, J. M., Phys. Rev. D, 75, (2007)
[27] Cortez, J.; Mena Marugán, G. A.; Velhinho, J. M., Classical Quantum Gravity, 25, (2008)
[28] Cortez, J.; Mena Marugán, G. A.; Velhinho, J. M., Phys. Rev. D, 81, (2010)
[29] Cortez, J.; Mena Marugán, G. A.; Olmedo, J.; Velhinho, J. M., Classical Quantum Gravity, 28, (2011)
[30] Cortez, J.; Mena Marugán, G. A.; Olmedo, J.; Velhinho, J. M., Phys. Rev. D, 86, (2012)
[31] Castelló Gomar, L.; Cortez, J.; Martín-de Blas, D.; Mena Marugán, G. A.; Velhinho, J. M., J. Cosmol. Astropart. Phys., 11, 001, (2012)
[32] Castelló Gomar, L.; Cortez, J.; Martín-de Blas, D.; Mena Marugán, G. A.; Velhinho, J. M., EJTP, 11, 43, (2014)
[33] Cortez, J.; Mena Marugán, G. A.; Olmedo, J.; Velhinho, J. M., Phys. Rev. D, 83, (2011)
[34] Cortez, J.; Elizaga Navascués, B.; Martín-Benito, M.; Mena Marugán, G. A.; Velhinho, J. M., Phys. Rev. D, 92, (2015)
[35] Cortez, J.; Elizaga Navascués, B.; Martín-Benito, M.; Mena Marugán, G. A.; Velhinho, J. M., Phys. Rev. D, 93, (2016)
[36] D’Eath, P. D.; Halliwell, J. J., Phys. Rev. D, 35, 1100, (1987)
[37] Cortez, J.; Elizaga Navascués, B.; Martín-Benito, M.; Mena Marugán, G. A.; Velhinho, J. M.
[38] Geroch, R., J. Math. Phys., 9, 1739, (1968)
[39] Lawson, H. B.; Michelson, M. L., Spin geometry, (1989), Princeton University Press Princeton
[40] Dimock, J., Trans. Amer. Math. Soc., 269, 133, (1982)
[41] Berezin, F. A., The method of second quantization, (1966), Academic New York · Zbl 0151.44001
[42] Nelson, J. E.; Teitelboim, C., Ann. Phys. (New York), 116, 86, (1978)
[43] Isham, C. J., Modern differential geometry for physicists, (1999), World Scientific Singapore · Zbl 0931.53002
[44] Dirac, P. A.M., Lectures on quantum mechanics, (1964), Belfer Graduate School of Science, Yeshiva University New York · Zbl 0141.44603
[45] Casalbuoni, R., Neural Comput., 33A, 115, (1976)
[46] Friedrich, Th., Colloq. Math., 48, 57, (1984)
[47] Roe, J., Elliptic operators, topology and asymptotic methods, (1999), Chapman & Hall/CRC Boca Raton
[48] Kirillov, A. A., Elements of theory of representations, (1976), Springer-Verlag New York · Zbl 0342.22001
[49] Shale, D., Trans. Amer. Math. Soc., 103, 149, (1962)
[50] Dereziński, J., Lecture Notes in Phys., 695, 63, (2006)
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