Broken Charges Associated with Classical Spacetime Symmetries under Canonical Transformation in Real Scalar Field Theory
Author(s)
Susobhan Mandal
Affiliation(s)
Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur, India.
Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur, India.
ABSTRACT
We know from Noether’s theorem that there is a conserved charge for every continuous symmetry. In General Relativity, Killing vectors describe the spacetime symmetries and to each such Killing vector field, we can associate conserved charge through stress-energy tensor of matter which is mentioned in the article. In this article, I show that under simple set of canonical transformation of most general class of Bogoliubov transformation between creation, annihilation operators, those charges associated with spacetime symmetries are broken. To do that, I look at stress-energy tensor of real scalar field theory (as an example) in curved spacetime and show how it changes under simple canonical transformation which is enough to justify our claim. Since doing Bogoliubov transformation is equivalent to coordinate transformation which according to Einstein’s equivalence principle is equivalent to turn on effect of gravity, therefore, we can say that under the effect of gravity those charges are broken.
We know from Noether’s theorem that there is a conserved charge for every continuous symmetry. In General Relativity, Killing vectors describe the spacetime symmetries and to each such Killing vector field, we can associate conserved charge through stress-energy tensor of matter which is mentioned in the article. In this article, I show that under simple set of canonical transformation of most general class of Bogoliubov transformation between creation, annihilation operators, those charges associated with spacetime symmetries are broken. To do that, I look at stress-energy tensor of real scalar field theory (as an example) in curved spacetime and show how it changes under simple canonical transformation which is enough to justify our claim. Since doing Bogoliubov transformation is equivalent to coordinate transformation which according to Einstein’s equivalence principle is equivalent to turn on effect of gravity, therefore, we can say that under the effect of gravity those charges are broken.
Cite this paper
Mandal, S. (2019) Broken Charges Associated with Classical Spacetime Symmetries under Canonical Transformation in Real Scalar Field Theory. Journal of High Energy Physics, Gravitation and Cosmology, 5, 167-180. doi: 10.4236/jhepgc.2019.51009.
Mandal, S. (2019) Broken Charges Associated with Classical Spacetime Symmetries under Canonical Transformation in Real Scalar Field Theory. Journal of High Energy Physics, Gravitation and Cosmology, 5, 167-180. doi: 10.4236/jhepgc.2019.51009.
References
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https://doi.org/10.1142/S0217732394003518 [5] Das, A. and De Benedictis, A. (2012) The General Theory of Relativity: A Mathematical Exposition. Springer-Verlag, New York. [6] Steven, W. (1972) Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. Wiley, Hoboken. [7] Sigbjorn, H.Y.G. (2007) Einstein’s General Theory of Relativity. Springer, Berlin. [8] Goldstone, J., Salam, A. and Weinberg, S. (1962) Broken Symmetries. Physical Review, 127, 965-970.
https://doi.org/10.1103/PhysRev.127.965 [9] Schakel, A.M.J. (1998) Boulevard of Broken Symmetries. [10] Birrell, N.D. and Davies, P.C.W. (1984) Quantum Fields in Curved Space of Cambridge Monographs on Mathematical Physics. CUP Edition, Cambridge University Press, Cambridge. [11] Ford, L.H. (1997) Quantum Field Theory in Curved Space-Time. Particles and Fields. Proceedings, 9th Jorge Andre Swieca Summer School, Campos do Jordao, 16-28 February 1997, 345-388. [12] Parker, L. and Toms, D. (2009) Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity of Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge. [13] Ambrus, V.E. (2014) Dirac Fermions on Rotating Space-Times. PhD Thesis, University of Sheffield. [14] Arovas, D. (2013) Lecture Notes on Thermodynamics and Statistical Mechanics (A Work in Progress). [15] Vilenkin, A. (1980) Quantum Field Theory at Finite Temperature in a Rotating System. Physical Review D, 21, 2260-2269.
https://doi.org/10.1103/PhysRevD.21.2260 [16] Das, A. (1997) Finite Temperature Field Theory. [17] Kapusta, J.I. (1989) Finite-Temperature Field Theory of Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge. [18] Laine, M. and Vuorinen, A. (2016) Basics of Thermal Field Theory. Lecture Notes in Physics, Vol. 925, Springer, Berlin.
https://doi.org/10.1007/978-3-319-31933-9 [19] Yang, Y. (2011) An Introduction to Thermal Field Theory. Imperial College, London. [20] Ahmadzadegan, A. (2017) Probing the Unruh and Hawking Effects Using Unruh-DeWitt Detectors. PhD Thesis, U. Waterloo (Main). [21] Birrell, N.D. and Davies, P.C.W. (1980) Massive Particle Production in Anisotropic Space-Times. Journal of Physics A: Mathematical and General, 13, 2109.
https://doi.org/10.1088/0305-4470/13/6/032 [22] Biswas, S., Guha, J. and Sarkar, N.G. (1995) Particle Production in de Sitter Space. Classical and Quantum Gravity, 12, 1591.
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https://doi.org/10.1103/RevModPhys.80.787 [25] Davies, P.C.W. (1975) Scalar Production in Schwarzschild and Rindler Metrics. Journal of Physics A: Mathematical and General, 8, 609.
https://doi.org/10.1088/0305-4470/8/4/022 [26] Degner, A. and Verch, R. (2010) Cosmological Particle Creation in States of Low Energy. Journal of Mathematical Physics, 51, 022302.
https://doi.org/10.1063/1.3271106 [27] Duru, I.H. and ünal, N. (1986) Particle Production in Expanding Universes with Path Integrals. Physical Review D, 34, 959.
https://doi.org/10.1103/PhysRevD.34.959 [28] Frieman, J.A. (1989) Particle Creation in Inhomogeneous Spacetimes. Physical Review D, 39, 389-398.
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https://doi.org/10.1103/PhysRevD.72.104011 [30] Buchholz, D. and Verch, R. (2015) Macroscopic Aspects of the Unruh Effect. Classical and Quantum Gravity, 32, 245004.
https://doi.org/10.1088/0264-9381/32/24/245004 [31] Blasone, M., Lambiase, G. and Luciano, G.G. (2017) Nonthermal Signature of the Unruh Effect in Field Mixing. Physical Review, D96, 025023.
https://doi.org/10.1103/PhysRevD.96.025023 [32] Ford, G.W. and O’Connell, R.F. (2006) Is There Unruh Radiation? Physics Letters, A350, 17-26.
https://doi.org/10.1016/j.physleta.2005.09.068 [33] Nikolic, H. (2001) Inappropriateness of the Rindler Quantization. Modern Physics Letters A, 16, 579-581.
https://doi.org/10.1142/S0217732301003681 [34] Rosu, H.C. (2001) Hawking Like Effects and Unruh Like Effects: Toward Experiments? Gravitation and Cosmology, 7, 1-17.
[1] Mandal, S.U. (2018) Symmetry Breaking under Canonical Transformation in Real Scalar Field Theory. [2] Perelomov, A.M. (1986) Generalized Coherent States and Their Applications of Modern Methods of Plant Analysis. Springer-Verlag, Berlin. [3] Umezawa, H., Matsumoto, H. and Tachiki, M. (1982) Thermo Field Dynamics and Condensed States. [4] Sato, H.-T. and Suzuki, H. (1994) On Bogoliubov Transformation of Scalar Wave Functions in de Sitter Space. Modern Physics Letters A, 9, 3673-3684.
https://doi.org/10.1142/S0217732394003518 [5] Das, A. and De Benedictis, A. (2012) The General Theory of Relativity: A Mathematical Exposition. Springer-Verlag, New York. [6] Steven, W. (1972) Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. Wiley, Hoboken. [7] Sigbjorn, H.Y.G. (2007) Einstein’s General Theory of Relativity. Springer, Berlin. [8] Goldstone, J., Salam, A. and Weinberg, S. (1962) Broken Symmetries. Physical Review, 127, 965-970.
https://doi.org/10.1103/PhysRev.127.965 [9] Schakel, A.M.J. (1998) Boulevard of Broken Symmetries. [10] Birrell, N.D. and Davies, P.C.W. (1984) Quantum Fields in Curved Space of Cambridge Monographs on Mathematical Physics. CUP Edition, Cambridge University Press, Cambridge. [11] Ford, L.H. (1997) Quantum Field Theory in Curved Space-Time. Particles and Fields. Proceedings, 9th Jorge Andre Swieca Summer School, Campos do Jordao, 16-28 February 1997, 345-388. [12] Parker, L. and Toms, D. (2009) Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity of Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge. [13] Ambrus, V.E. (2014) Dirac Fermions on Rotating Space-Times. PhD Thesis, University of Sheffield. [14] Arovas, D. (2013) Lecture Notes on Thermodynamics and Statistical Mechanics (A Work in Progress). [15] Vilenkin, A. (1980) Quantum Field Theory at Finite Temperature in a Rotating System. Physical Review D, 21, 2260-2269.
https://doi.org/10.1103/PhysRevD.21.2260 [16] Das, A. (1997) Finite Temperature Field Theory. [17] Kapusta, J.I. (1989) Finite-Temperature Field Theory of Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge. [18] Laine, M. and Vuorinen, A. (2016) Basics of Thermal Field Theory. Lecture Notes in Physics, Vol. 925, Springer, Berlin.
https://doi.org/10.1007/978-3-319-31933-9 [19] Yang, Y. (2011) An Introduction to Thermal Field Theory. Imperial College, London. [20] Ahmadzadegan, A. (2017) Probing the Unruh and Hawking Effects Using Unruh-DeWitt Detectors. PhD Thesis, U. Waterloo (Main). [21] Birrell, N.D. and Davies, P.C.W. (1980) Massive Particle Production in Anisotropic Space-Times. Journal of Physics A: Mathematical and General, 13, 2109.
https://doi.org/10.1088/0305-4470/13/6/032 [22] Biswas, S., Guha, J. and Sarkar, N.G. (1995) Particle Production in de Sitter Space. Classical and Quantum Gravity, 12, 1591.
https://doi.org/10.1088/0264-9381/12/7/005 [23] Mo Chitre, D. and Hartle, J.B. (1977) Path-Integral Quantization and Cosmological Particle Production: An Example. Physical Review D, 16, 251.
https://doi.org/10.1103/PhysRevD.16.251 [24] Crispino, L.C.B., Higuchi, A. and Matsas, G.E.A. (2008) The Unruh Effect and Its Applications. Reviews of Modern Physics, 80, 787-838.
https://doi.org/10.1103/RevModPhys.80.787 [25] Davies, P.C.W. (1975) Scalar Production in Schwarzschild and Rindler Metrics. Journal of Physics A: Mathematical and General, 8, 609.
https://doi.org/10.1088/0305-4470/8/4/022 [26] Degner, A. and Verch, R. (2010) Cosmological Particle Creation in States of Low Energy. Journal of Mathematical Physics, 51, 022302.
https://doi.org/10.1063/1.3271106 [27] Duru, I.H. and ünal, N. (1986) Particle Production in Expanding Universes with Path Integrals. Physical Review D, 34, 959.
https://doi.org/10.1103/PhysRevD.34.959 [28] Frieman, J.A. (1989) Particle Creation in Inhomogeneous Spacetimes. Physical Review D, 39, 389-398.
https://doi.org/10.1103/PhysRevD.39.389 [29] Winitzki, S. (2005) Cosmological Particle Production and the Precision of the WKB Approximation. Physical Review D, 72, Article ID: 104011.
https://doi.org/10.1103/PhysRevD.72.104011 [30] Buchholz, D. and Verch, R. (2015) Macroscopic Aspects of the Unruh Effect. Classical and Quantum Gravity, 32, 245004.
https://doi.org/10.1088/0264-9381/32/24/245004 [31] Blasone, M., Lambiase, G. and Luciano, G.G. (2017) Nonthermal Signature of the Unruh Effect in Field Mixing. Physical Review, D96, 025023.
https://doi.org/10.1103/PhysRevD.96.025023 [32] Ford, G.W. and O’Connell, R.F. (2006) Is There Unruh Radiation? Physics Letters, A350, 17-26.
https://doi.org/10.1016/j.physleta.2005.09.068 [33] Nikolic, H. (2001) Inappropriateness of the Rindler Quantization. Modern Physics Letters A, 16, 579-581.
https://doi.org/10.1142/S0217732301003681 [34] Rosu, H.C. (2001) Hawking Like Effects and Unruh Like Effects: Toward Experiments? Gravitation and Cosmology, 7, 1-17.