# Symmetries and Metamorphoses

## Abstract

**:**

## 1. Introduction

## 2. Two Levels of Description in Quantum Field Theory

## 3. Spontaneous Breakdown of Symmetry

## 4. Dynamical Rearrangement of Symmetry

## 5. Homogeneous and Non-Homogeneous Boson Condensation

## 6. The Origin of Metamorphoses

## 7. Boundary Effects in Spontaneous Breakdown of Symmetry

## 8. Fractals and Coherence

## 9. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Spontaneous Breakdown of Symmetry and Nambu-Goldstone Bosons

## Appendix B. Symmetry Rearrangement

## Appendix C. Infrared Contributions

## References

- Darwin, C. On the Origin of Species; John Murray: London, UK, 1860; p. 490. [Google Scholar]
- Schweber, S.S. An Introduction to Relativistic Quantum Field Theory; Harper and Row Publ. Inc.: New York, NY, USA, 1961. [Google Scholar]
- Bogoliubov, N.N.; Logunov, A.A.; Todorov, I.T. Axiomatic Quantum Field Theory; Benjamin: New York, NY, USA, 1975. [Google Scholar]
- Bogoliubov, N.N. Lectures on qUantum Statistics, Quasi-Averages; Macdonald & Co. Ltd.: London, UK, 1971; Volume 2, ISBN 0356026981, ISBN 978-0356026985. [Google Scholar]
- Umezawa, H.; Matsumoto, H.; Tachiki, M. Thermo Field Dynamics and Condensed States; North-Holland: Amsterdam, The Netherlands, 1982. [Google Scholar]
- Umezawa, H. Advanced Field Theory: Micro, Macro and Thermal Concepts; American Institute of Physics: New York, NY, USA, 1993. [Google Scholar]
- Vitiello, G. Dynamical rearrangement of symmetry. Diss. Ab. Intern.
**1975**, 36/02, 769-B. [Google Scholar] - Blasone, M.; Jizba, J.; Vitiello, G. Quantum Field Theory and Its Macroscopic Manifestations; Imperial College Press: London, UK, 2011. [Google Scholar]
- Umezawa, H. Developments in concepts in quantum field theory in half century. Math. Jpn.
**1995**, 41, 109–124. [Google Scholar] - von Neumann, J. Mathematical Foundations of Quantum Mechanics; Princeton University Press: Princeton, NY, USA, 1955. [Google Scholar]
- Del Giudice, E.; Manka, R.; Milani, M.; Vitiello, G. Non-constant order parameter and vacuum evolution. Phys. Lett. B
**1988**, 206, 661–664. [Google Scholar] [CrossRef] - Bratteli, O.; Robinson, D.W. Operator Algebras and Quantum Statistical Mechanics; Springer: Berlin, Germany, 1979. [Google Scholar]
- Vitiello, G. The world opacity and knowledge. In The Systemic Turn in Human and Natural Sciences. The Rock in the Pond; Urbani Ulivi, L., Ed.; Springer Nature Switzerland AG: Cham, Switzerland, 2019; pp. 41–51. (In Italian) [Google Scholar]
- Umezawa, H. Dynamical rearrangement of symmetries. Nuovo Cim. A
**1965**, 40, 450–475. [Google Scholar] [CrossRef] - Leplae, L.; Sen, R.N.; Umezawa, H. Asymmetric ground states in invariant many-body theories. Nuovo Cim. B
**1967**, 49, 1–31. [Google Scholar] [CrossRef] - Matsumoto, H.; Umezawa, H.; Vitiello, G.; Wyly, J.K. Spontaneous breakdown of a non-Abelian symmetry. Phys. Rev. D
**1974**, 9, 2806–2813. [Google Scholar] [CrossRef] - Shah, M.N.; Umezawa, H.; Vitiello, G. Relation among spin operators and magnons. Phys. Rev. B
**1974**, 10, 4724–4726. [Google Scholar] [CrossRef] - Shah, M.N.; Vitiello, G. Self-consistent formulation of itinerant electron ferromagnet. Nuovo Cim. B
**1975**, 30, 21–42. [Google Scholar] [CrossRef] - De Concini, C.; Vitiello, G. Spontaneous breakdown of symmetry and group contraction. Nucl. Phys. B
**1976**, 116, 141–156. [Google Scholar] [CrossRef] - Celeghini, E.; Graziano, E.; Vitiello, G. Classical limit and spontaneous breakdown of symmetry as an environment effect in quantum field theory. Phys. Lett. A
**1990**, 145, 1–6. [Google Scholar] [CrossRef] - Goldstone, J.; Salam, A.; Weinberg, S. Broken Symmetries. Phys. Rev.
**1962**, 127, 965–970. [Google Scholar] [CrossRef] - Inönü, E.; Wigner, E.P. On the contraction of groups and their representations. Proc. Nat. Acad. Sci. USA
**1953**, 39, 510–524. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Segal, I.E. A class of operator algebras which are determined by groups. Duke Math. J.
**1951**, 18, 221–265. [Google Scholar] [CrossRef] - Saletan, E.J. Contraction of Lie groups. J. Math. Phys.
**1961**, 2, 1–21. [Google Scholar] [CrossRef] - Vitiello, G. Dynamical rearrangement of SU(3) symmetry. Phys. Lett. A
**1976**, 58, 293–294. [Google Scholar] [CrossRef] - Weimar, E. What does the centre of the universal enveloping algebra tell us about the deformations of representations? Nuovo Cim.
**1973**, 15, 245–256. [Google Scholar] [CrossRef] - Klauder, J.R.; Sudarshan, E.C.G. Fundamentals of Quantum Optics; Benjamin: New York, NY, USA, 1968. [Google Scholar]
- Perelomov, A. Generalized Coherent States and Their Applications; Springer: Berlin, Germany, 1986. [Google Scholar]
- Lakhno, V.D. Translation-invariant bipolarons and superconductivity. Condens. Matter
**2020**, 5, 30. [Google Scholar] [CrossRef] - Higgs, P.W. Spontaneous Symmetry Breakdown without Massless Bosons. Phys. Rev.
**1966**, 145, 1156–1163. [Google Scholar] [CrossRef] [Green Version] - Kibble, T.W.B. Symmetry breaking in nonAbelian gauge theories. Phys. Rev.
**1967**, 155, 1554–1561. [Google Scholar] [CrossRef] - Matsumoto, H.; Papastamatiou, N.J.; Umezawa, H.; Vitiello, G. Dynamical rearrangement in Anderson-Higgs-Kibble mechanism. Nucl. Phys. B
**1975**, 97, 61–89. [Google Scholar] [CrossRef] - Vitiello, G. Topological defects, fractals and the structure of quantum field theory. In Vision of Oneness; Licata, I., Sakaji, A.J., Eds.; Aracne Edizioni: Rome, Italy, 2011; pp. 155–180. [Google Scholar]
- Celeghini, E.; De Martino, S.; De Siena, S.; Rasetti, M.; Vitiello, G. Quantum groups, squeezing, Bloch and theta functions. Mod. Phys. Lett. B
**1993**, 7, 1321–1329. [Google Scholar] [CrossRef] - Celeghini, E.; De Martino, S.; De Siena, S.; Rasetti, M.; Vitiello, G. Quantum groups, coherent states, squeezing, and lattice quantum mechanics. Ann. Phys.
**1995**, 241, 50–67. [Google Scholar] [CrossRef] [Green Version] - Holstein, T.; Primakoff, H. Field Dependence of the Intrinsic Domain Magnetization of a Ferromagnet. Phys. Rev.
**1940**, 58, 1098–1113. [Google Scholar] [CrossRef] - Umezawa, H.; Vitiello, G. Quantum Mechanics; Bibliopolis: Napoli, Italy, 1985. [Google Scholar]
- Alfinito, E.; Vitiello, G. Formation and life-time of memory domains in the dissipative quantum model of brain. Int. J. Mod. Phys. B
**2000**, 14, 853–868. [Google Scholar] [CrossRef] [Green Version] - Dyson, F.J. General Theory of Spin-Wave Interactions. Phys. Rev.
**1956**, 102, 1217–1230. [Google Scholar] [CrossRef] - Adler, S.L. Consistency Conditions on the Strong Interactions Implied by a Partially Conserved Axial-Vector Current. II. Phys. Rev. B
**1965**, 139, 1638–1643. [Google Scholar] [CrossRef] - Vitiello, G. Coherent states, fractals and brain waves. New Math. Nat. Comput.
**2009**, 5, 245–264. [Google Scholar] [CrossRef] [Green Version] - Vitiello, G. Fractals, coherent states and self-similarity induced noncommutative geometry. Phys. Lett. A
**2012**, 376, 2527–2532. [Google Scholar] [CrossRef] [Green Version] - Vitiello, G. On the isomorphism between dissipative systems, fractal self-similarity and electrodynamics. Toward an integrated vision of nature. Systems
**2014**, 2, 203–216. [Google Scholar] [CrossRef] - Peitgen, H.O.; Jürgens, H.; Saupe, D. Chaos and Fractals. New frontiers of Science; Springer: Berlin, Germany, 1986. [Google Scholar]
- Bunde, A.; Havlin, S. (Eds.) Fractals in Science; Springer: Berlin, Germany, 1995. [Google Scholar]
- Vitiello, G. …And Kronos ate his sons. In Beyond Peaceful Coexistence. The Emergence of Space, Time and Quantum; Licata, I., Ed.; Imperial College Press: London, UK, 2016; pp. 465–486. [Google Scholar]
- Celeghini, E.; Rasetti, M.; Vitiello, G. Quantum Dissipation. Ann. Phys.
**1992**, 215, 156–170. [Google Scholar] [CrossRef] - Vitiello, G. Links. Relating different physical systems through the common QFT algebraic structure. In Quantum Analogues: From Phase Transitions to Black Holes and Cosmology; Lectures Notes in Physics 718; Unruh, W.G., Schuetzhold, R., Eds.; Springer: Berlin, Germany, 2007; pp. 165–205. [Google Scholar]
- Celeghini, E.; Rasetti, M.; Vitiello, G. Squeezing and Quantum Groups. Phys. Rev. Lett.
**1991**, 66, 2056–2059. [Google Scholar] [CrossRef] [PubMed] - Celeghini, E.; De Martino, S.; De Siena, S.; Iorio, A.; Rasetti, M.; Vitiello, G. Thermo field dynamics and quantum algebras. Phys. Lett. A
**1998**, 244, 455–461. [Google Scholar] [CrossRef] [Green Version] - Sivasubramanian, S.; Srivastava, Y.N.; Vitiello, G.; Widom, A. Quantum dissipation induced noncommutative geometry. Phys. Lett. A
**2003**, 311, 97–105. [Google Scholar] [CrossRef] [Green Version] - Vitiello, G. Classical chaotic trajectories in quantum field theory. Int. J. Mod. Phys. B
**2004**, 18, 785–792. [Google Scholar] [CrossRef] [Green Version] - Sabbadini, S.A.; Vitiello, G. Entanglement and phase-mediated correlations in quantum field theory. Application to brain-mind states. Appl. Sci.
**2019**, 9, 3203. [Google Scholar] [CrossRef] [Green Version] - Blasone, M.; Jizba, P.; Vitiello, G. Dissipation and quantization. Phys. Lett. A
**2001**, 287, 205–210. [Google Scholar] [CrossRef] [Green Version] - Gerry, C.C.; Knight, P.L. Introductory Quantum Optics; Cambridge University Press: Cambridge, UK, 2005. [Google Scholar]
- Hooft, G.T. Quantum gravity as a dissipative deterministic system. Class. Quant. Grav.
**1999**, 16, 3263–3279. [Google Scholar] [CrossRef] [Green Version] - Hooft, G.T. A mathematical theory for deterministic quantum mechanics. J. Phys. Conf. Ser.
**2007**, 67, 012015. [Google Scholar] [CrossRef] - Martellini, M.; Sodano, P.; Vitiello, G. Vacuum Structure for a Quantum Field Theory in Curved Space-Time. Nuovo Cim. A
**1978**, 48, 341–358. [Google Scholar] [CrossRef] - Alfinito, E.; Vitiello, G. Canonical quantization and expanding metrics. Phys. Lett. A
**1999**, 252, 5–10. [Google Scholar] [CrossRef] [Green Version] - Alfinito, E.; Vitiello, G. Vacuum structure for expanding geometry. Class. Quant. Grav.
**2000**, 17, 93–111. [Google Scholar] [CrossRef] - Alfinito, E.; Vitiello, G. Double universe and the arrow of time. J. Phys. Conf. Ser.
**2007**, 67, 012010. [Google Scholar] [CrossRef] - Alfinito, E.; Vitiello, G. Time reversal violation as loop-antiloop symmetry breaking: The Bessel equation, group contraction and dissipation. Mod. Phys. Lett. B
**2003**, 17, 1–12. [Google Scholar] [CrossRef] - Hooft, G.T. Foreword. In Beyond Peaceful Coexistence. The Emergence of Space, Time and Quantum; Licata, I., Ed.; Imperial College Press: London, UK, 2016; pp. ix–x. [Google Scholar]
- Licata, I. From peaceful coexistence to co-emergence. In Beyond Peaceful Coexistence. The Emergence of Space, Time and Quantum; Licata, I., Ed.; Imperial College Press: London, UK, 2016; pp. xi–xx. [Google Scholar]
- Capolupo, A.; Lambiase, G.; Vitiello, G. Thermal Condensate Structure and Cosmological Energy Density of the Universe. Adv. High Energy Phys.
**2016**, 2016, 3127597. [Google Scholar] [CrossRef] [Green Version] - Hilborn, R. Chaos and Nonlinear Dynamics; Oxford University Press: Oxford, UK, 1994. [Google Scholar]
- De Concini, C.; Vitiello, G. Relation between projective geometry and group contraction in spontaneously broken symmetry theories. Phys. Lett. B
**1977**, 70, 355–357. [Google Scholar] [CrossRef] - Celeghini, E.; Magnollay, P.; Tarlini, M.; Vitiello, G. Non linear realizations and contraction of group representations. Phys. Lett. B
**1985**, 162, 133–136. [Google Scholar] [CrossRef] - Vitiello, G. Simmetrie e metamorfosi. Atque
**2019**, 24, 139–160. [Google Scholar] - Del Giudice, E.; Doglia, S.; Milani, M.; Vitiello, G. A quantum field theoretical approach to the collective behavior of biological systems. Nucl. Phys. B
**1985**, 251, 375–400. [Google Scholar] [CrossRef] - Del Giudice, E.; Doglia, S.; Milani, M.; Vitiello, G. Electromagnetic field and spontaneous symmetry breakdown in biological matter. Nucl. Phys. B
**1986**, 275, 185–199. [Google Scholar] [CrossRef] - Vitiello, G. Dissipation and memory capacity in the quantum brain model. Int. J. Mod. Phys. B
**1995**, 9, 973–989. [Google Scholar] [CrossRef] - Loppini, A.; Capolupo, A.; Cherubini, C.; Gizzi, A.; Bertolaso, M.; Filippi, S.; Vitiello, G. On the coherent behavior of pancreatic beta cell clusters. Phys. Lett. A
**2014**, 378, 3210–3217. [Google Scholar] [CrossRef] [Green Version] - Vitiello, G. My Double Unveiled; John Benjamins: Amsterdam, The Netherlands, 2001. [Google Scholar]
- Freeman, W.J.; Vitiello, G. Nonlinear brain dynamics as macroscopic manifestation of underlying many-body dynamics. Phys. Life Rev.
**2006**, 3, 93–117. [Google Scholar] [CrossRef] [Green Version] - Freeman, W.J.; Vitiello, G. Matter and Mind are entangled in two streams of images guiding behavior and informing the subject through awareness. Mind Matter
**2016**, 14, 7–24. [Google Scholar] - Kurian, P.; Capolupo, A.; Craddock, T.J.A.; Vitiello, G. Water-mediated correlations in DNA-enzyme interactios. Phys. Lett. A
**2016**, 382, 33–43. [Google Scholar] [CrossRef] [PubMed] - Montagnier, L.; Aïssa, J.; Capolupo, A.; Craddock, T.J.A.; Kurian, P.; Lavallee, C.; Polcari, A.; Romano, P.; Tedeschi, A.; Vitiello, G. Water bridging dynamics of polymerase chain reaction in the gauge theory paradigm of quantum fields. Water
**2017**, 9, 339, Addendum 9, 436. [Google Scholar] [CrossRef] - Piattelli-Palmarini, M.; Vitiello, G. Linguistics and Some Aspects of Its Underlying Dynamics. Biolinguistics
**2015**, 9, 96–115. [Google Scholar] - Longo, G. Confusing biological rhythms and physical clocks. Today’s ecological relevance of Bergson-Einstein debate on time. In Proceedings of the Conference “What is time? Einstein and Bergson 100 years later”, L’Aquila, Italy, 4–6 April 2019. in press. [Google Scholar]
- Damasco, A.; Giuliani, A. A resonance based model of biological evolution. Physics A
**2017**, 471, 750–756. [Google Scholar] [CrossRef] - Camponeschi, I.; Damasco, A.; Uversky, V.N.; Giuliani, A.; Bianchi, M.M. Phenotypic suppression caused by resonance with light-dark cycles indicates the presence of a 24-h oscillator in yeast and suggests a new role of intrinsically disordered protein regions as internal mediators. J. Biomol. Struct. Dyn.
**2020**, accepted. [Google Scholar] [CrossRef] - Fadini, U.; Pieri, P.F. (Eds.) Prefazione. In Metamorfosi Del Vivente. Atque; Moretti & Vitali: Bergamo, Italy, 2019; Volume 24, pp. 9–17. [Google Scholar]

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Vitiello, G.
Symmetries and Metamorphoses. *Symmetry* **2020**, *12*, 907.
https://doi.org/10.3390/sym12060907

**AMA Style**

Vitiello G.
Symmetries and Metamorphoses. *Symmetry*. 2020; 12(6):907.
https://doi.org/10.3390/sym12060907

**Chicago/Turabian Style**

Vitiello, Giuseppe.
2020. "Symmetries and Metamorphoses" *Symmetry* 12, no. 6: 907.
https://doi.org/10.3390/sym12060907