Review Article | Open Access
Antonio Capolupo, "Quantum Vacuum, Dark Matter, Dark Energy, and Spontaneous Supersymmetry Breaking", Advances in High Energy Physics, vol. 2018, Article ID 9840351, 7 pages, 2018. https://doi.org/10.1155/2018/9840351
Quantum Vacuum, Dark Matter, Dark Energy, and Spontaneous Supersymmetry Breaking
We study the behavior of the vacuum condensates characterizing many physical phenomena. We show that condensates due to thermal states, to fields in curved space, and to neutrino mixing, may represent new components of the dark matter, whereas the condensate due to axion-photon mixing can contribute to the dark energy. Moreover, by considering a supersymmetric framework, we show that the nonzero energy of vacuum condensates may induce a spontaneous supersymmetry breaking.
In recent years, many efforts have been made to understand the nature and the origin of the dark energy and of the dark matter [1–29], which represent almost the 68% and the 27%, respectively, of the matter and energy content of the universe [30–34].
Another important field of study is the supersymmetry (SUSY) [35–40]. Such a symmetry associates any boson to a fermion (called superpartner) with the same mass and internal quantum numbers and vice versa. However, up to now, no superpartner has been detected. Therefore, SUSY must be a broken symmetry, which permits the existence of superpartners heavier than the corresponding particles, or it must be ruled out as a fundamental symmetry. In particular, intensive study has been devoted to the analysis of the possibility of SUSY breaking.
Here we report on recent results [41–47] obtained by studying the condensate structure of the vacuum of many phenomena such as Hawking effect and fields in curved space [48–61]. In the framework of the standard model of particles, we show that the thermal vacuum of the hot plasma present at the center of a galaxy cluster, the vacuum fluctuations of fields in curved space , and the flavor neutrino vacuum give nontrivial contributions to the energy of the universe and have a pressure equal to zero. Thus, such condensates, in the presence of ordinary matter, can aggregate structures and to represent a component of the dark matter. A behavior completely different is the one of the vacuum condensate induced by the axion-photon mixing. Indeed, in this case the condensate has a state equation which coincides with one of the cosmological constants and can contribute to the dark energy of the universe [41, 42]. Then, the condensates can provide both the effects of the dark part of the universe, that is, the dynamical and the gravitational ones.
It is worth stressing that the condensate contributions are different from the usual zero-point energy contribution of fields. Indeed they are not originated from radiative corrections, but they derive from the property of QFT of being characterized by infinitely many representations of the canonical (anti)commutation relations in the infinite volume limit. Values compatible with the estimated ones for dark matter and dark energy are obtained by using reasonable values of the cut-offs on the momenta. This result is due to the drastic decrease in the degree of divergency, for high momenta, of integrals describing the energy-momentum tensor of the condensates, in comparison with the case of the zero-point energy of a free field. Indeed, the usual zero-point energy contribution goes like for high momenta. Such divergence and one are removed in the vacuum condensates [63–67].
We also consider the framework of the supersymmetry and, by using the free Wess–Zumino model, we show that the presence of nonvanishing vacuum energy at the Lagrangian level implies that SUSY is spontaneously broken by the condensates [43–47].
The structure of the paper is the following: in Section 2, we introduce the Bogoliubov transformations in quantum field theory (QFT) and we study the energy-momentum tensor density for vacuum condensates of boson and fermion fields. In Sections 3 and 4, we present the contribution given to the energy of the universe by thermal states, with reference to the Hawking and Unruh effects, by fields in curved space and by particle mixing phenomena. SUSY breaking induced by vacuum condensate is presented in Section 5 and Section 6 is devoted to the conclusions.
2. Bogoliubov Transformation and Energy-Momentum Tensor of Vacuum Condensate
Many phenomena, ranging from the BCS theory of superconductivity  to the Casimir and the Schwinger effects , are represented, in the context of QFT, by a Bogoliubov transformation . Such a transformation for bosons is expressed as , with parameter depending on the phenomenon analyzed, annihilators of the vacuum , and , , coefficients satisfying the conditions , , and (similar discussion holds for fermions).
Introducing the generator of the Bogoliubov transformation, one can write Here is a unitary operator, , which allows relating the original vacua to the vacua annihilated by ; . Such a relation is a unitary operation in quantum mechanics, where assumes a discrete range of values, but it is not a unitary transformation in QFT, where assumes a continuous infinity of values. In this case, and are unitarily inequivalent and the proper physical vacua are [54, 55]. Such vacua have a condensate structure which is responsible for an expectation value of the number operator on different from zero; This fact implies a nontrivial result of the expectation value of the energy-momentum tensor density for free fields on ; Here, denotes the normal ordering with respect to the original vacuum . Notice that the off-diagonal components of are zero, , for , then the condensates act as a perfect fluid, and one can define their energy density and pressure, by means of the and the components of , as and [41, 42].
For bosons, in the particular case of the isotropy of the momenta, , the energy density, the pressure, and the state equation, , are given by [41, 42]respectively. For Majorana fermions, one has [41, 42]The equations presented above hold for all the systems described by Bogoliubov transformations in QFT. By using such equations and the explicit form of the Bogoliubov coefficients, we analyze the cases of thermal states and of fields in curved spaces. Then, we study the particle mixing phenomenon which, at the level of the annihilators, is described by a Bogoliubov transformation and a rotation. This fact implies that, for particle mixing, (1)–(6) must be modified, as shown in Section 4.
3. Dark Matter Components from Thermal States, Hawking and Unruh Effects, and Fields in Curved Background
We show that thermal vacuum states and vacuum condensate of fields in curved space can be interpreted as components of the dark matter.
We start by considering the thermal vacuum state introduced in the framework of Thermofield Dynamics (TFD) (where , and is the Boltzmann constant and ) [53–55]. In such a context, every degree of freedom is doubled and the thermal Bogoliubov transformation allows defining, at nonzero temperature, the state such that the thermal statistical average is given by , where , ( for bosons and for fermions) is the number operator . The Bogoliubov coefficients are and , with − for bosons and + for fermions, and . The energy density and pressure of , obtained by (1)–(6) with and , permit studying many thermal systems.
We obtain that, for temperatures of order of the cosmic microwave radiation (i.e., ), only photons and very light particles contribute to the energy radiation with and state equations, , whereas nonrelativistic particles give negligible contributions . On the other hand, the thermal vacuum of the hot plasma filling the center of galaxy clusters with temperatures can contribute to the dark matter. Indeed, such a vacuum has an energy density of and a pressure . Moreover, since the temperatures related to Unruh and Hawking effects are very low, their contributions to the energy of the universe are negligible .
Let us now consider fields in curved background. In such a case, the Bogoliubov coefficients depend on the metric analyzed . Here, we study boson fields in spatially flat Friedmann Robertson-Walker metric, , where is the comoving time, is the scale factor, and is the conformal time, with arbitrary constant.
The energy density and pressure are [69, 70]where is the cut-off on the momenta, are mode functions, and are the derivative of with respect to the conformal time . The numerical value of , of , and of the state equation are depending on the choice of . However, it has been shown in  that, in infrared regime, by using a cut-off on the momenta much smaller than the comoving mass of the field, , and setting , one obtains the state equation of the dark matter, . The energy density, in this case, is , which is much smaller than . Therefore, the contributions of light particles are compatible with the value estimated for the dark matter.
4. Dark Matter Component from Neutrino Mixing and Dark Energy Contribution from Mixed Bosons
We now compute the contributions to the energy given by the vacuum of mixed fields. The particle mixing concerns fermions as neutrinos and quarks and bosons as axions, kaons, , , and systems. The mixing of two fields is expressed aswhere is the mixing angle, are free fields, and are mixed fields, with .
The generator of the mixing, , allows writing (8) as and the mixed annihilation operators as . Here we denote with the annihilators of bosons and the ones of fermions [59, 60]. One can see that the mixing transformation, at the level of annihilators, is a rotation plus a Bogoliubov transformation [59, 60]. Then the physical vacuum, (where is the vacuum annihilated by ), generates a condensation density given bywhere . The Bogoliubov coefficient, , appearing in the condensates, assumes the following form for bosons and fermions:respectively, where are the energies of the free fields, . The other coefficients of the transformations are and .
In the following, we use the relation to writeThen, we denote with the operators transformed by . Let us now analyze the contributions induced by the mixing of bosons and of fermions.
(i) Boson Mixing. The energy density and pressure of the vacuum condensate induced by mixed bosons arerespectively. One can see that the kinetic and gradient terms of the mixed vacuum are equal to zero [41, 42]:Therefore, (13) and (14) reduce toand the state equation is the ones of the cosmological constant, . Denoting by the cut-off on the momenta and by setting , one hasSuch an integral, solved analytically, gives the following results.
(i) For axion-photon mixing, considering magnetic field strength , axion mass , , and a Planck scale cut-off, , one obtains a value of the energy density , which is of the same order of the estimated upper bound on the dark energy.
(ii) Fermion Mixing. The energy density and pressure of the condensate induced by mixed fermions are where denote the mixed fields. With the kinetic terms equal to zero, that is,then (18) reduces toThen, the vacuum condensate of fermion mixing behaves as a dark matter component, being . Explicitly, one hasBy considering and , one has . For Plank scale cut-off, one has .
We also note that the condensates of quarks, kaons, and other mesons should not contribute to the dark matter and to the dark energy, since the quarks confinement inhibits other interactions.
5. SUSY Breaking and Vacuum Condensate
Up to now we have analyzed condensed systems in the context of the standard model. Let us now extend our formalism to a supersymmetric framework. We show that the vacuum condensate provides a new mechanism of spontaneous SUSY breaking. To do that, we consider a system in which SUSY is preserved at the Lagrangian level. Then, in order not to break SUSY explicitly, we consider a Bogoliubov transformation which acts simultaneously and with the same parameters on boson and on fermion operators. Such a transformation leads to a vacuum condensate with nonzero energy. In general, a nonzero vacuum energy implies the spontaneous SUSY breaking in any field theory which has manifest supersymmetry at the Lagrangian level . Hence, the nontrivial energy of vacuum condensate breaks SUSY spontaneously.
We consider the Wess–Zumino Lagrangian which is invariant under supersymmetry transformations :Here is a Majorana spinor field, is a scalar field, and is a pseudoscalar field. Denoting by , , and the annihilators for , , and , respectively, the vacuum annihilated by such operators is defined as . Let us now carry out simultaneous Bogoliubov transformations on fermion and boson annihilators:where the coefficients for scalar and pseudoscalar fields are equal to each other, and . We denote such quantities as and , respectively. The annihilators in (23)–(25) can be expressed as , with and , where , and are the generators of transformations (23), (24), and (25), respectively.
The supersymmetric transformed vacuum, which is the physical one, is given by . Here , , and are the transformed vacua of fermion and boson fields. Then one has As shown above, the vacua , , and have nontrivial value of their energy density. This fact implies a nonzero energy densities for . Indeed, denoting by (where ) the free Hamiltonian corresponding to the Lagrangian (22), one hasThen the total energy density iswhich is different from zero and positive. Such a result holds for all the phenomena characterized by vacuum condensate. Recent experiments on cold atoms-molecules trapped in two-dimensional optical lattices  could permit testing the SUSY breaking mechanism here presented [44, 45].
We have shown that, in the framework of the standard model of the particles, the vacuum condensates of many systems can contribute to the energy of the universe. In particular, dark matter components can derive by thermal vacuum of intercluster medium, by the vacuum of fields in curved space, and by the neutrino flavor vacuum. On the other hand, the condensate induced by axion-photon mixing can contribute to the dark energy. We have also studied a supersymmetric field theory. In this framework, considering the Wess–Zumino model, we have shown that vacuum condensates may lead to spontaneous SUSY breaking.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Partial financial support from MIUR and INFN is acknowledged.
- L. Amendola and S. Tsujikawa, Dark Energy: Theory and Observations, Cambridge University Press, 2010.
- S. Nojiri, S. D. Odintsov, and V. K. Oikonomou, “Modified gravity theories on a nutshell: inflation, bounce and late-time evolution,” Physics Reports, vol. 692, pp. 1–104, 2017.
- C. F. Martins and P. Salucci, “Analysis of rotation curves in the framework of Rn gravity,” Monthly Notices of the Royal Astronomical Society, vol. 381, no. 3, pp. 1103–1108, 2007.
- V. C. Rubin, “The rotation of spiral galaxies,” Science, vol. 220, no. 4604, pp. 1339–1344, 1983.
- S. Nojiri and S. D. Odintsov, “Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models,” Physics Reports, vol. 505, no. 2, p. 59, 2011.
- A. Silvestri and M. Trodden, “Approaches to understanding cosmic acceleration,” Reports on Progress in Physics, vol. 72, no. 9, 2009.
- J. A. Frieman, M. S. Turner, and D. Huterer, “Dark energy and the accelerating universe,” Annual Review of Astronomy and Astrophysics, vol. 46, pp. 385–432, 2008.
- R. Durrer and R. Maartens, “Dark energy and dark gravity: theory overview,” General Relativity and Gravitation, vol. 40, no. 2, pp. 301–328, 2008.
- M. Sami, “Models of dark energy,” in The Invisible Universe: Dark Matter and Dark Energy, vol. 720 of Lecture Notes in Physics, pp. 219–256, Springer, Berlin, Germany, 2007.
- E. J. Copeland, M. Sami, and S. Tsujikawa, “Dynamics of dark energy,” International Journal of Modern Physics D: Gravitation, Astrophysics, Cosmology, vol. 15, no. 11, pp. 1753–1935, 2006.
- G. Lambiase, S. Mohanty, and A. R. Prasanna, “Neutrino coupling to cosmological background: a review on gravitational baryo/leptogenesis,” International Journal of Modern Physics D, vol. 22, no. 12, Article ID 1330030, 2013.
- T. P. Sotiriou and V. Faraoni, “f(R) theories of gravity,” Reviews of Modern Physics, vol. 82, no. 1, pp. 451–497, 2010.
- A. De Felice and S. J. Tsujikawa, “f(R) theories,” Living Reviews in Relativity, vol. 13, p. 3, 2010.
- R. Schützhold, “Small Cosmological Constant from the QCD Trace Anomaly?” Physical Review Letters, vol. 89, no. 8, 2002.
- E. C. Thomas, F. R. Urban, and A. R. Zhitnitsky, “The cosmological constant as a manifestation of the conformal anomaly?” Journal of High Energy Physics, vol. 2009, no. 8, article 043, 2009.
- F. R. Klinkhamer and G. E. Volovik, “Gluonic vacuum, q-theory, and the cosmological constant,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 79, no. 6, Article ID 063527, 2009.
- F. R. Urban and A. R. Zhitnitsky, “Cosmological constant from the ghost: a toy model,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 80, no. 6, Article ID 063001, 2009.
- F. R. Klinkhamer and G. E. Volovik, “Vacuum energy density kicked by the electroweak crossover,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 80, no. 8, Article ID 083001, 2009.
- S. Alexander, T. Biswas, and G. Calcagni, “Erratum: Cosmological Bardeen-Cooper-Schrieffer condensate as dark energy,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 81, no. 6, Article ID 069902, 2010.
- N. J. Poplawski, “Cosmological constant from quarks and torsion,” Annalen der Physik, vol. 523, pp. 291–295, 2011.
- G. Bertone, D. Hooper, and J. Silk, “Particle dark matter: evidence, candidates and constraints,” Physics Reports, vol. 405, no. 5-6, pp. 279–390, 2005.
- I. De Martino, “f(R)-gravity model of the Sunyaev-Zeldovich profile of the Coma cluster compatible with Planck data,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 93, Article ID 124043, 2016.
- I. De Martino, M. De Laurentis, F. Atrio-Barandela, and S. Capozziello, “Constraining f(R) gravity with planck data on galaxy cluster profiles,” Monthly Notices of the Royal Astronomical Society, vol. 442, no. 2, pp. 921–928, 2014.
- G. D'Amico, M. Kamionkowski, and K. Sigurdson, “Dark Matter Astrophysics,” https://arxiv.org/abs/0907.1912.
- G. Bertone, Particle Dark Matter: Observations, Models and Searches, (Paris, Inst. Astrophys.), Cambridge, UK: Univ. Pr., Cambridge, UK, 2010.
- A. Bottino and N. Fornengo, Dark matter and its particle candidates, Trieste 1998, Non-accelerator particle physics, 1988.
- S. Derraro, F. Schmidt, and W. Hu, “Cluster abundance in f(R) gravity models,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 83, Article ID 063503, 2011.
- L. Lombriser, F. Schmidt, T. Baldauf, R. Mandelbaum, U. Seljak, and R. E. Smith, “Cluster density profiles as a test of modified gravity,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 85, no. 10, Article ID 102001, 2012.
- F. Schmidt, A. Vikhlinin, and W. Hu, “Cluster constraints on f(R) gravity,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 80, no. 8, Article ID 083505, 2009.
- P. de Bernardis, P. A. R. Ade, and J. J. Bock, “A flat Universe from high-resolution maps of the cosmic microwave background radiation,” Nature, vol. 404, pp. 955–959, 2000.
- D. N. Spergel, L. Verde, and H. V. Peiris, “First-Year Wilkinson Microwave Anisotropy Probe (WMAP)* Observations: Determination of Cosmological Parameters,” The Astrophysical Journal Supplement Series, vol. 148, no. 1, pp. 175–194, 2003.
- S. Dodelson, V. K. Narayanan, and M. Tegmark, “The three-dimensional power spectrum from angular clustering of galaxies in early sloan digital sky survey data,” The Astrophysical Journal, vol. 572, no. 1, p. 140, 2002.
- A. S. Szalay, B. Jain, and T. Matsubara, “Karhunen-Loève estimation of the power spectrum parameters from the angular distribution of galaxies in early sloan digital sky survey data,” The Astrophysical Journal, vol. 591, pp. 1–11, 2003.
- A. G. Riess, L.-G. Strolger, and J. Tonry, “Type Ia Supernova Discoveries at z > 1 from the Hubble Space Telescope: Evidence for Past Deceleration and Constraints on Dark Energy Evolution*,” The Astrophysical Journal, vol. 607, no. 2, pp. 665–687, 2004.
- P. Fayet and J. Iliopoulos, “Spontaneously broken supergauge symmetries and goldstone spinors,” Physics Letters B, vol. 51, no. 5, pp. 461–464, 1974.
- L. O'Raifeartaigh, “Spontaneous symmetry breaking for chirals scalar superfields,” Nuclear Physics B, vol. 96, no. 2, pp. 331–352, 1975.
- E. Witten, “Dynamical breaking of supersymmetry,” Nuclear Physics B, vol. 188, no. 3, pp. 513–554, 1981.
- Y. Shadmi and Y. Shirman, “Dynamical supersymmetry breaking,” Reviews of Modern Physics, vol. 72, no. 1, article 25, 2000.
- A. Das, Finite Temperature Field Theory, World Scientific, Singapore, 1997.
- D. Buchholz and I. Ojima, “Spontaneous collapse of supersymmetry,” Nuclear Physics. B. Theoretical, Phenomenological, and Experimental High Energy Physics. Quantum Field Theory and Statistical Systems, vol. 498, no. 1-2, pp. 228–242, 1997.
- A. Capolupo, “Dark Matter and Dark Energy Induced by Condensates,” Advances in High Energy Physics, vol. 2016, Article ID 8089142, p. 10, 2016.
- A. Capolupo, “Condensates as components of dark matter and dark energy,” Journal of Physics: Conference Series, vol. 880, p. 012059, 2017.
- A. Capolupo, M. Di Mauro, and A. Iorio, “Mixing-induced spontaneous supersymmetry breaking,” Physics Letters A, vol. 375, no. 39, pp. 3415–3418, 2011.
- A. Capolupo and M. Di Mauro, “Spontaneous supersymmetry breaking induced by vacuum condensates,” Physics Letters A, vol. 376, no. 45, pp. 2830–2833, 2012.
- A. Capolupo and M. Di Mauro, “Vacuum condensates, flavor mixing and spontaneous supersymmetry breaking,” Jagellonian University. Institute of Physics. Acta Physica Polonica B, vol. 44, no. 1, pp. 81–89, 2013.
- A. Capolupo and G. Vitiello, “Spontaneous supersymmetry breaking probed by geometric invariants,” Advances in High Energy Physics, vol. 2013, Article ID 850395, 5 pages, 2013.
- A. Capolupo and M. Di Mauro, “Vacuum Condensates as a Mechanism of Spontaneous Supersymmetry Breaking,” Advances in High Energy Physics, vol. 2015, Article ID 929362, 6 pages, 2015.
- S. W. Hawking, “Particle creation by black holes, Erratum: 46, 2, (1976),” Communications in Mathematical Physics, vol. 43, no. 3, pp. 199–220, 1975.
- W. G. Unruh, “Notes on black-hole evaporation,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 14, no. 4, pp. 870–892, 1976.
- J. Schwinger, “On gauge invariance and vacuum polarization,” Physical Review A: Atomic, Molecular and Optical Physics, vol. 82, pp. 664–679, 1951.
- J. Bardeen, L. N. Cooper, and J. R. Schrieffer, “Theory of superconductivity,” Physical Review Letters, vol. 108, pp. 1175–1204, 1957.
- A. Iorio, “Weyl-gauge symmetry of graphene,” Annals of Physics, vol. 326, no. 5, pp. 1334–1353, 2011.
- Y. Takahashi and H. Umezawa, “Thermo field dynamics,” Collective Phenomena, vol. 2, pp. 55–80, 1975.
- H. Umezawa, H. Matsumoto, and M. Tachiki, Thermo field dynamics and condensed states, North-Holland Publishing Company, 1982.
- H. Umezawa, Advanced Field Theory: Micro, Macro, and Thermal Physics, AIP, New York, NY, USA, 1993.
- H. B. G. Casimir and D. Polder, “The influence of retardation on the London-van der Waals forces,” Physical Review A: Atomic, Molecular and Optical Physics, vol. 73, no. 4, pp. 360–372, 1948.
- H. B. G. Casimir, “On the attraction between two perfectly conducting plates,” Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, vol. 51, pp. 793–795, 1948.
- N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, UK, 1984.
- M. Blasone, A. Capolupo, and G. Vitiello, “Quantum field theory of three flavor neutrino mixing and oscillations with CP violation,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 66, no. 2, Article ID 025033, 2002.
- M. Blasone, A. Capolupo, O. Romei, and G. Vitiello, “Quantum field theory of boson mixing,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 63, no. 12, Article ID 125015, 2001.
- A. Capolupo, C.-R. Ji, Y. Mishchenko, and G. Vitiello, “Phenomenology of flavor oscillations with non-perturbative effects from quantum field theory,” Physics Letters B, vol. 594, no. 1-2, pp. 135–140, 2004.
- F. D. Albareti, J. A. R. Cembranos, and A. L. Maroto, “Vacuum energy as dark matter,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 90, no. 12, Article ID 123509, 2014.
- A. Capolupo, S. Capozziello, and G. Vitiello, “Neutrino mixing as a source of dark energy,” Physics Letters A, vol. 363, no. 1-2, pp. 53–56, 2007.
- A. Capolupo, S. Capozziello, and G. Vitiello, “Dark energy, cosmological constant and neutrino mixing,” International Journal of Modern Physics A, vol. 23, no. 31, 2008.
- M. Blasone, A. Capolupo, S. Capozziello, and G. Vitiello, “Neutrino mixing, flavor states and dark energy,” Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, vol. 588, no. 1-2, pp. 272–275, 2008.
- M. Blasone, A. Capolupo, and G. Vitiello, “Particle mixing, flavor condensate and dark energy,” Progress in Particle and Nuclear Physics, vol. 64, no. 2, pp. 451–453, 2010.
- M. Blasone, A. Capolupo, S. Capozziello, S. Carloni, and G. Vitiello, “Neutrino mixing contribution to the cosmological constant,” Physics Letters A, vol. 323, no. 3-4, pp. 182–189, 2004.
- A. Capolupo, G. Lambiase, and G. Vitiello, “Thermal condensate structure and cosmological energy density of the universe,” Advances in High Energy Physics, vol. 2016, Article ID 3127597, 6 pages, 2016.
- L. Parker and S. A. Fulling, “Adiabatic regularization of the energy-momentum tensor of a quantized field in homogeneous spaces,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 9, no. 2, pp. 341–354, 1974.
- S. A. Fulling and L. Parker, “Renormalization in the theory of a quantized scalar field interacting with a Robertson-Walker space-time,” Annalen der Physik D, vol. 87, pp. 176–204, 1974.
- J. Wess and B. Zumino, “A lagrangian model invariant under supergauge transformations,” Physics Letters B, vol. 49, no. 1, pp. 52–54, 1974.
- Y. Yu and K. Yang, “Simulating the Wess-Zumino supersymmetry model in optical lattices,” Physical Review Letters, vol. 105, no. 15, Article ID 150605, 4 pages, 2010.
Copyright © 2018 Antonio Capolupo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.