Quantum Vacuum, Dark Matter, Dark Energy, and Spontaneous...

Review ArticleQuantum Vacuum Dark Matter Dark Energy and SpontaneousSupersymmetry Breaking

Antonio Capolupo

Dipartimento di Fisica ER Caianiello and INFN Gruppo Collegato di Salerno Universita di Salerno 84084 Fisciano Italy

Correspondence should be addressed to Antonio Capolupo capoluposainfnit

Received 11 August 2017 Revised 19 October 2017 Accepted 5 February 2018 Published 10 April 2018

Academic Editor Hernando Quevedo

Copyright copy 2018 Antonio Capolupo This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

We study the behavior of the vacuum condensates characterizing many physical phenomena We show that condensates due tothermal states to fields in curved space and to neutrino mixing may represent new components of the dark matter whereas thecondensate due to axion-photonmixing can contribute to the dark energy Moreover by considering a supersymmetric frameworkwe show that the nonzero energy of vacuum condensates may induce a spontaneous supersymmetry breaking

1 Introduction

In recent years many efforts have been made to understandthe nature and the origin of the dark energy and of the darkmatter [1ndash29] which represent almost the 68 and the 27respectively of the matter and energy content of the universe[30ndash34]

Another important field of study is the supersymmetry(SUSY) [35ndash40] Such a symmetry associates any boson toa fermion (called superpartner) with the same mass andinternal quantum numbers and vice versa However up tonow no superpartner has been detected Therefore SUSYmust be a broken symmetry which permits the existenceof superpartners heavier than the corresponding particlesor it must be ruled out as a fundamental symmetry Inparticular intensive study has been devoted to the analysisof the possibility of SUSY breaking

Here we report on recent results [41ndash47] obtained bystudying the condensate structure of the vacuum of manyphenomena such asHawking effect and fields in curved space[48ndash61] In the framework of the standard model of particleswe show that the thermal vacuum of the hot plasma presentat the center of a galaxy cluster the vacuum fluctuations offields in curved space [62] and the flavor neutrino vacuumgive nontrivial contributions to the energy of the universeand have a pressure equal to zero Thus such condensates

in the presence of ordinary matter can aggregate structuresand to represent a component of the dark matter A behaviorcompletely different is the one of the vacuum condensateinduced by the axion-photon mixing Indeed in this case thecondensate has a state equation which coincides with one ofthe cosmological constants and can contribute to the darkenergy of the universe [41 42] Then the condensates canprovide both the effects of the dark part of the universe thatis the dynamical and the gravitational ones

It is worth stressing that the condensate contributionsare different from the usual zero-point energy contributionof fields Indeed they are not originated from radiativecorrections but they derive from the property of QFT ofbeing characterized by infinitely many representations of thecanonical (anti)commutation relations in the infinite volumelimit Values compatible with the estimated ones for darkmatter and dark energy are obtained by using reasonablevalues of the cut-offs on the momenta This result is dueto the drastic decrease in the degree of divergency for highmomenta of integrals describing the energy-momentumtensor of the condensates in comparison with the case ofthe zero-point energy of a free field Indeed the usual zero-point energy contribution goes like 1198704 for high momentaSuch divergence and 1198702 one are removed in the vacuumcondensates [63ndash67]

HindawiAdvances in High Energy PhysicsVolume 2018 Article ID 9840351 7 pageshttpsdoiorg10115520189840351

2 Advances in High Energy Physics

We also consider the framework of the supersymmetryand by using the freeWessndashZuminomodel we show that thepresence of nonvanishing vacuum energy at the Lagrangianlevel implies that SUSY is spontaneously broken by thecondensates [43ndash47]

The structure of the paper is the following in Section 2weintroduce the Bogoliubov transformations in quantum fieldtheory (QFT) and we study the energy-momentum tensordensity for vacuum condensates of boson and fermion fieldsIn Sections 3 and 4 we present the contribution given to theenergy of the universe by thermal states with reference to theHawking and Unruh effects by fields in curved space andby particle mixing phenomena SUSY breaking induced byvacuum condensate is presented in Section 5 and Section 6 isdevoted to the conclusions

2 Bogoliubov Transformation and Energy-Momentum Tensor of Vacuum Condensate

Many phenomena ranging from the BCS theory of super-conductivity [51] to the Casimir and the Schwinger effects[50] are represented in the context of QFT by a Bogoliubovtransformation [55] Such a transformation for bosons isexpressed as 119886k(120585 119905) = 119880퐵k 119886k(119905) minus 119881퐵minusk119886daggerminusk(119905) with 120585 parameterdepending on the phenomenon analyzed 119886k annihilatorsof the vacuum |0⟩ and 119880퐵k 119881퐵k coefficients satisfying theconditions 119880퐵k = 119880퐵minusk 119881퐵k = 119881퐵minusk and |119880퐵k |2 minus |119881퐵k |2 = 1(similar discussion holds for fermions)

Introducing the generator 119869(120585 119905) of the Bogoliubov trans-formation one can write 119886k(120585 119905) = 119869minus1(120585 119905)119886k(119905)119869(120585 119905) Here119869(120585 119905) is a unitary operator 119869minus1(120585) = 119869(minus120585) which allowsrelating the original vacua |0⟩ to the vacua |0(120585 119905)⟩ annihi-lated by 119886k(120585 119905) |0(120585 119905)⟩ = 119869minus1(120585 119905)|0⟩ Such a relation is aunitary operation in quantummechanics where k assumes adiscrete range of values but it is not a unitary transformationin QFT where k assumes a continuous infinity of values Inthis case |0(120585 119905)⟩ and |0⟩ are unitarily inequivalent and theproper physical vacua are |0(120585 119905)⟩ [54 55] Such vacua have acondensate structure which is responsible for an expectationvalue of the number operator119873k = 119886daggerk119886k on |0(120585 119905)⟩ differentfrom zero ⟨0(120585 119905)|119886daggerk119886k|0(120585 119905)⟩ = |119881k|2 This fact impliesa nontrivial result of the expectation value of the energy-momentum tensor density 119879휇](119909) for free fields on |0(120585 119905)⟩Ξ휆휇](119909) =휆 ⟨0(120585 119905)| 119879휆휇](119909) |0(120585 119905)⟩휆 Here sdot sdot sdot denotesthe normal ordering with respect to the original vacuum |0⟩휆Notice that the off-diagonal components of Ξ휆휇](119909) are zeroΞ휆푖푗(119909) = 0 for 119894 = 119895 then the condensates act as a perfectfluid and one can define their energy density and pressureby means of the (0 0) and the (119895 119895) components of Ξ휆휇](119909) as120588휆 = Ξ휆00(119909) and 119901휆 = Ξ휆푗푗(119909) [41 42]

For bosons in the particular case of the isotropy of themomenta 1198961 = 1198962 = 1198963 the energy density the pressure andthe state equation 119908퐵 = 119901퐵120588퐵 are given by [41 42]

120588퐵 = 121205872 intinfin

01198891198961198962120596푘 10038161003816100381610038161003816119881퐵푘 100381610038161003816100381610038162 (1)

119901퐵 = 161205872 intinfin

01198891198961198962 [ 1198962120596푘

10038161003816100381610038161003816119881퐵푘 100381610038161003816100381610038162

minus ( 1198962120596푘 +311989822120596푘 )

10038161003816100381610038161003816119880퐵푘 10038161003816100381610038161003816 10038161003816100381610038161003816119881퐵푘 10038161003816100381610038161003816 cos (120596푘119905)] (2)

119908퐵 = 13int 1198893k (1198962120596푘) 10038161003816100381610038161003816119881퐵푘 100381610038161003816100381610038162int1198893k120596푘 1003816100381610038161003816119881퐵푘 10038161003816100381610038162 minus 13

sdot int 1198893k (1198962120596푘 + 311989822120596푘)119880퐵푘119881퐵푘 cos (120596푘119905)int 1198893k120596푘 1003816100381610038161003816119881퐵푘 10038161003816100381610038162

(3)

respectively For Majorana fermions one has [41 42]

120588퐹 = 11205872 intinfin

01198891198961198962120596푘 10038161003816100381610038161003816119881퐹푘 100381610038161003816100381610038162 (4)

119901퐹 = 131205872 intinfin

0119889119896 1198964120596푘

10038161003816100381610038161003816119881퐹푘 100381610038161003816100381610038162 (5)

119908퐹 = 13int 1198893k (1198962120596푘) 10038161003816100381610038161003816119881퐹푘 100381610038161003816100381610038162int1198893k120596푘 1003816100381610038161003816119881퐹푘 10038161003816100381610038162 (6)

The equations presented above hold for all the systemsdescribed by Bogoliubov transformations in QFT By usingsuch equations and the explicit form of the Bogoliubovcoefficients we analyze the cases of thermal states and offields in curved spaces Then we study the particle mixingphenomenon which at the level of the annihilators isdescribed by a Bogoliubov transformation and a rotationThis fact implies that for particle mixing (1)ndash(6) must bemodified as shown in Section 4

3 Dark Matter Components from ThermalStates Hawking and Unruh Effects andFields in Curved Background

We show that thermal vacuum states and vacuum condensateof fields in curved space can be interpreted as components ofthe dark matter

We start by considering the thermal vacuum state|0(120585(120573))⟩휆 introduced in the framework of ThermofieldDynamics (TFD) (where 120573 equiv 1(119896퐵119879) and 119896퐵 is the Boltz-mann constant and120582 = 119861 119865) [53ndash55] In such a context everydegree of freedom is doubled and the thermal Bogoliubovtransformation allows defining at nonzero temperature thestate |0(120585(120573))⟩휆 such that the thermal statistical average isgiven by 휆⟨0(120585(120573))|119873휒k |0(120585(120573))⟩휆 where 119873휒k = 120594daggerk120594k (120594 = 119886for bosons and 120572 for fermions) is the number operator [54]The Bogoliubov coefficients are 119880푇k = radic119890훽휔k(119890훽휔k plusmn 1) and119881푇k = radic1(119890훽휔k plusmn 1) with minus for bosons and + for fermionsand 120596k = radic1198962 + 1198982 The energy density and pressure of|0(120585(120573))⟩휆 obtained by (1)ndash(6) with 119880k = 119880푇k and 119881k = 119881푇k permit studying many thermal systems

We obtain that for temperatures of order of the cosmicmicrowave radiation (ie 119879 = 272K) only photons and

Advances in High Energy Physics 3

very light particles contribute to the energy radiation with120588 sim 10minus51 GeV4 and state equations 119908 = 13 whereasnonrelativistic particles give negligible contributions [68]On the other hand the thermal vacuum of the hot plasmafilling the center of galaxy clusters with temperatures (10 divide100) times 106 K can contribute to the dark matter Indeed sucha vacuum has an energy density of (10minus48ndash10minus47)GeV4 and apressure 119901 sim 0 Moreover since the temperatures related toUnruh and Hawking effects are very low their contributionsto the energy of the universe are negligible [68]

Let us now consider fields in curved background In sucha case the Bogoliubov coefficients depend on the metricanalyzed [58] Here we study boson fields in spatially flatFriedmann Robertson-Walker metric 1198891199042 = 1198891199052 minus1198862(119905)119889x2 =1198862(120578)(1198891205782 minus 119889x2) where 119905 is the comoving time 119886 is the scalefactor and 120578(119905) = int푡

푡0(119889119905119886(119905)) is the conformal time with 1199050

arbitrary constantThe energy density and pressure are [69 70]

120588curv = 21205871198862 int퐾0

1198891198961198962 (10038161003816100381610038161003816120601耠푘100381610038161003816100381610038162 + 1198962 1003816100381610038161003816120601푘10038161003816100381610038162 + 1198982 1003816100381610038161003816120601푘10038161003816100381610038162) 119901curv = 21205871198862 int퐾

01198891198961198962 (10038161003816100381610038161003816120601耠푘100381610038161003816100381610038162 minus 11989623 1003816100381610038161003816120601푘10038161003816100381610038162 minus 1198982 1003816100381610038161003816120601푘10038161003816100381610038162)

(7)

where 119870 is the cut-off on the momenta 120601푘 are modefunctions and 120601耠푘 are the derivative of 120601푘 with respect to theconformal time 120578 The numerical value of 120588curv of 119901curv andof the state equation 119908curv are depending on the choice of 119870However it has been shown in [62] that in infrared regimeby using a cut-off on the momenta much smaller than thecomoving mass of the field 119870 ≪ 119898119886 and setting 119898 ≫ 119867one obtains the state equation of the dark matter 119908curv ≃ 0The energy density in this case is120588curv = 11989811987031212058721198863 whichis much smaller than 1198984121205872 Therefore the contributionsof light particles are compatible with the value estimated forthe dark matter

4 Dark Matter Component from NeutrinoMixing and Dark Energy Contribution fromMixed Bosons

We now compute the contributions to the energy given bythe vacuum of mixed fields The particle mixing concernsfermions as neutrinos and quarks and bosons as axionskaons 1198610 1198630 and 120578-120578耠 systems The mixing of two fields isexpressed as

1205931 (120579 119909) = 1205931 (119909) cos (120579) + 1205932 (119909) sin (120579) 1205932 (120579 119909) = minus1205931 (119909) sin (120579) + 1205932 (119909) cos (120579) (8)

where 120579 is the mixing angle 120593푖(119909) are free fields and 120593푖(120579 119909)are mixed fields with 119894 = 1 2

The generator of the mixing 119869(120579 119905) allows writing (8) as120593푖(120579 119909) equiv 119869minus1(120579 119905)120593푖(119909)119869(120579 119905) and the mixed annihilationoperators as 120594푟k푖(120579 119905) equiv 119869minus1(120579 119905)120594푟k푖(119905)119869(120579 119905) Here wedenote with 120594푟k푖 the annihilators of bosons 119886k푖 and the

ones of fermions 120572푟k푖 [59 60] One can see that the mixingtransformation at the level of annihilators is a rotation plusa Bogoliubov transformation [59 60] Then the physicalvacuum |0(120579 119905)⟩ equiv 119869minus1(120579 119905)|0⟩12 (where |0⟩12 is the vacuumannihilated by 120594푟k푖) generates a condensation density givenby

⟨0 (120579 119905)| 120594푟daggerk푖120594푟k푖 |0 (120579 119905)⟩ = sin2 120579 10038161003816100381610038161003816Υ휆k 100381610038161003816100381610038162 (9)

where 120582 = 119861 119865 The Bogoliubov coefficient Υ휆k appearing inthe condensates assumes the following form for bosons andfermions

10038161003816100381610038161003816Υ퐵k 10038161003816100381610038161003816 = 12 (radicΩ푘1Ω푘2 minus radicΩ푘2Ω푘1) (10)

10038161003816100381610038161003816Υ퐹k 10038161003816100381610038161003816 = (Ω푘1 + 1198981) minus (Ω푘2 + 1198982)2radicΩ푘1Ω푘2 (Ω푘1 + 1198981) (Ω푘2 + 1198982) |k| (11)

respectively where Ω푘푖 are the energies of the free fields 119894 =1 2 The other coefficients of the transformations are Σ퐵k =radic1 + |Υ퐵k |2 and Σ퐹k = radic1 minus |Υ퐹k |2

In the following we use the relation 119869minus1(120579 119905) = 119869dagger(120579 119905) =119869(minus120579 119905) to write휆⟨0 (120579 119905)| 119879휆휇] (119909) |0 (120579 119905)⟩휆 =

=휆 ⟨0| 119869minus1휆 (minus120579 119905) 119879휆휇] (119909) 119869휆 (minus120579 119905) |0⟩휆 (12)

Then we denote with Θ(minus120579 119909) = 119869minus1(minus120579 119905)Θ(119909)119869(minus120579 119905)the operators transformed by 119869(minus120579 119905) Let us now analyzethe contributions induced by the mixing of bosons and offermions

(i) Boson Mixing The energy density and pressure of thevacuum condensate induced by mixed bosons are

120588퐵mix = 12 ⟨0| sum푖

[1205872푖 (minus120579 119909) + [997888rarrnabla120601푖 (minus120579 119909)]2

+ 1198982푖 1206012푖 (minus120579 119909)] |0⟩ (13)

119901퐵mix = ⟨0| sum푖

([120597푗120601푖 (minus120579 119909)]2 + 12 [1205872푖 (minus120579 119909)minus [997888rarrnabla120601푖 (minus120579 119909)]2 minus 1198982푖 1206012푖 (minus120579 119909)]) |0⟩

(14)

respectively One can see that the kinetic and gradient termsof the mixed vacuum are equal to zero [41 42]

⟨0| sum푖

1205872푖 (minus120579 119909) |0⟩ = ⟨0| sum푖

[997888rarrnabla120601푖 (minus120579 119909)]2 |0⟩= ⟨0| sum

푖

[120597푗120601푖 (minus120579 119909)]2 |0⟩ = 0(15)


Therefore (13) and (14) reduce to

120588퐵mix = minus119901퐵mix = ⟨0| sum푖

1198982푖 1206012푖 (minus120579 119909) |0⟩ (16)

and the state equation is the ones of the cosmologicalconstant 119908퐵mix = minus1 Denoting by 119870 the cut-off on themomenta and by setting Δ1198982 = |11989822 minus 11989821| one has

120588퐵mix = Δ1198982 sin2 12057981205872 int퐾0

1198891198961198962 ( 1120596푘1 minus1120596푘2) (17)

Such an integral solved analytically gives the followingresults

(i) For axion-photon mixing considering magnetic fieldstrength 119861 isin [106ndash1017]G axion mass 119898푎 ≃ 2 times 10minus2 eVsin2푎 120579 sim 10minus2 and a Planck scale cut-off 119870 sim 1019 GeV oneobtains a value of the energy density120588axion

mix = 23times10minus47 GeV4which is of the same order of the estimated upper bound onthe dark energy

(ii) For the mixing of neutrino superpartners assuming1198981 = 10minus3 eV 1198982 = 9 times 10minus3 eV and sin2120579 = 03 one has120588퐵mix = 7 times 10minus47 GeV4 for 119870 = 10 eV For smaller values ofsin2120579 one obtains 120588퐵mix sim 10minus47 GeV4 also in the case inwhich119870 = 1019 GeV [41 42]

(ii) Fermion Mixing The energy density and pressure of thecondensate induced by mixed fermions are

120588퐹mix = minus ⟨0| sum푖

[120595dagger푖 (minus120579 119909) 1205740120574푗120597푗120595푖 (minus120579 119909)+ 119898120595dagger푖 (minus120579 119909) 1205740120595푖 (minus120579 119909)] |0⟩

119901퐹mix = 119894 ⟨0| sum푖

[120595dagger푖 (minus120579 119909) 1205740120574푗120597푗120595푖 (minus120579 119909)] |0⟩ (18)

where 120595푖(minus120579 119909) denote the mixed fields With the kineticterms equal to zero that is

⟨0| sum푖

120595푖 (minus120579 119909) 120574푗120597푗120595푖 (minus120579 119909) |0⟩ =⟨0| sum

푖

[120595dagger푖 (minus120579 119909) 1205740120574푗120597푗120595푖 (minus120579 119909)] |0⟩ = 0 (19)

then (18) reduces to


[119898푖120595dagger푖 (minus120579 119909) 1205740120595푖 (minus120579 119909)] |0⟩ 119901퐹mix = 0

(20)

Then the vacuum condensate of fermion mixing behaves asa dark matter component being 119908퐹mix = 0 Explicitly one has

120588퐹mix = Δ119898 sin2 12057921205872 int퐾0

1198891198961198962 ( 1198982120596푘2 minus1198981120596푘1) (21)

By considering 119898푖 sim 10minus3 eV and 119870 = 1198981 + 1198982 one has120588퐹mix = 4times10minus47 GeV4 For Plank scale cut-off one has 120588퐹mix simtimes10minus46 GeV4

We also note that the condensates of quarks kaons andother mesons should not contribute to the dark matter and tothe dark energy since the quarks confinement inhibits otherinteractions

5 SUSY Breaking and Vacuum Condensate

Up to now we have analyzed condensed systems in thecontext of the standard model Let us now extend ourformalism to a supersymmetric frameworkWe show that thevacuum condensate provides a new mechanism of sponta-neous SUSY breaking To do that we consider a system inwhich SUSY is preserved at the Lagrangian level Then inorder not to break SUSY explicitly we consider a Bogoliubovtransformation which acts simultaneously and with the sameparameters on boson and on fermion operators Such atransformation leads to a vacuum condensate with nonzeroenergy In general a nonzero vacuum energy implies thespontaneous SUSY breaking in any field theory which hasmanifest supersymmetry at the Lagrangian level [37] Hencethe nontrivial energy of vacuum condensate breaks SUSYspontaneously

We consider the WessndashZumino Lagrangian which isinvariant under supersymmetry transformations [71]

L = 1198942120595120574휇120597휇120595 + 12120597휇119878120597휇119878 + 12120597휇119875120597휇119875 minus 1198982 120595120595minus 11989822 (1198782 + 1198752)

(22)

Here 120595 is a Majorana spinor field 119878 is a scalar field and 119875 is apseudoscalar field Denoting by 120572푟k 119887k and 119888k the annihilatorsfor 120595 119878 and 119875 respectively the vacuum annihilated by suchoperators is defined as |0⟩ = |0⟩휓 otimes |0⟩푆 otimes |0⟩푃 Let usnow carry out simultaneous Bogoliubov transformations onfermion and boson annihilators

120572푟k (120585 119905) = 119880휓k 120572푟k (119905) + 119881휓minusk120572푟daggerminusk (119905) (23)

119887k (120578 119905) = 119880푆k119887k (119905) minus 119881푆minusk119887daggerminusk (119905) (24)

119888k (120578 119905) = 119880푃k 119888k (119905) minus 119881푃minusk119888daggerminusk (119905) (25)

where the coefficients for scalar and pseudoscalar fieldsare equal to each other 119880푆k = 119880푃k and 119881푆k = 119881푃k Wedenote such quantities as 119880퐵k and 119881퐵k respectively Theannihilators in (23)ndash(25) can be expressed as 120594푟k(120585 119905) =119869minus1(120585 120578 119905)120594푟k(119905)119869(120585 120578 119905) with 120594k = 120572k 119887k 119888k and 119869(120585 120578 119905) =119869휓(120585 119905)119869푆(120578 119905)119869푃(120578 119905) where 119869휓 119869푆 and 119869푃 are the generatorsof transformations (23) (24) and (25) respectively

The supersymmetric transformed vacuum which is thephysical one is given by |0(120585 120578 119905)⟩ = |0(120585 119905)⟩휓 otimes |0(120578 119905)⟩푆 otimes|0(120578 119905)⟩푃 Here |0(120585 119905)⟩휓 = 119869minus1휓 (120585 119905)|0⟩휓 |0(120578 119905)⟩푆 =119869minus1푆 (120578 119905)|0⟩푆 and |0(120578 119905)⟩푃 = 119869minus1푃 (120578 119905)|0⟩푃 are the trans-formed vacua of fermion and boson fields Then one has|0(120585 120578 119905)⟩ = 119869minus1(120585 120578 119905)|0⟩ As shown above the vacua|0(120585 119905)⟩휓 |0(120578 119905)⟩푆 and |0(120578 119905)⟩푃 have nontrivial value oftheir energy density This fact implies a nonzero energy


densities for |0(120585 120578 119905)⟩ Indeed denoting by 119867 = 119867휓 + 119867퐵(where 119867퐵 = 119867푆 + 119867푃) the free Hamiltonian correspondingto the Lagrangian (22) one has

⟨0 (120585 120578 119905)1003816100381610038161003816 119867휓 10038161003816100381610038160 (120585 120578 119905)⟩= minusint1198893k120596k (1 minus 2 10038161003816100381610038161003816119881휓k 100381610038161003816100381610038162)

⟨0 (120585 120578 119905)1003816100381610038161003816 119867퐵 10038161003816100381610038160 (120585 120578 119905)⟩ = int1198893k120596k (1 + 2 10038161003816100381610038161003816119881퐵k 100381610038161003816100381610038162) (26)

Then the total energy density is

⟨0 (120585 120578 119905)1003816100381610038161003816 119867 10038161003816100381610038160 (120585 120578 119905)⟩= 2int1198893k120596k (10038161003816100381610038161003816119881휓k 100381610038161003816100381610038162 + 10038161003816100381610038161003816119881퐵k 100381610038161003816100381610038162) (27)

which is different from zero and positive Such a result holdsfor all the phenomena characterized by vacuum condensateRecent experiments on cold atoms-molecules trapped in two-dimensional optical lattices [72] could permit testing theSUSY breaking mechanism here presented [44 45]

6 Conclusions

We have shown that in the framework of the standardmodel of the particles the vacuum condensates of manysystems can contribute to the energy of the universe Inparticular dark matter components can derive by thermalvacuum of intercluster medium by the vacuum of fields incurved space and by the neutrino flavor vacuum On theother hand the condensate induced by axion-photon mixingcan contribute to the dark energy We have also studied asupersymmetric field theory In this framework consideringthe WessndashZumino model we have shown that vacuumcondensates may lead to spontaneous SUSY breaking

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

Partial financial support from MIUR and INFN is acknowl-edged

References

[1] L Amendola and S Tsujikawa Dark Energy Theory andObservations Cambridge University Press 2010

[2] S Nojiri S D Odintsov and V K Oikonomou ldquoModifiedgravity theories on a nutshell inflation bounce and late-timeevolutionrdquo Physics Reports vol 692 pp 1ndash104 2017

[3] C F Martins and P Salucci ldquoAnalysis of rotation curves inthe framework of Rn gravityrdquo Monthly Notices of the RoyalAstronomical Society vol 381 no 3 pp 1103ndash1108 2007

[4] V C Rubin ldquoThe rotation of spiral galaxiesrdquo Science vol 220no 4604 pp 1339ndash1344 1983

[5] S Nojiri and S D Odintsov ldquoUnified cosmic history inmodified gravity from F(R) theory to Lorentz non-invariantmodelsrdquo Physics Reports vol 505 no 2 p 59 2011

[6] A Silvestri and M Trodden ldquoApproaches to understandingcosmic accelerationrdquo Reports on Progress in Physics vol 72 no9 2009

[7] J A Frieman M S Turner and D Huterer ldquoDark energy andthe accelerating universerdquo Annual Review of Astronomy andAstrophysics vol 46 pp 385ndash432 2008

[8] R Durrer and R Maartens ldquoDark energy and dark gravitytheory overviewrdquoGeneral Relativity andGravitation vol 40 no2 pp 301ndash328 2008

[9] M Sami ldquoModels of dark energyrdquo in The Invisible UniverseDark Matter and Dark Energy vol 720 of Lecture Notes inPhysics pp 219ndash256 Springer Berlin Germany 2007

[10] E J Copeland M Sami and S Tsujikawa ldquoDynamics of darkenergyrdquo International Journal of Modern Physics D GravitationAstrophysics Cosmology vol 15 no 11 pp 1753ndash1935 2006

[11] G Lambiase S Mohanty and A R Prasanna ldquoNeutrinocoupling to cosmological background a review on gravitationalbaryoleptogenesisrdquo International Journal of Modern Physics Dvol 22 no 12 Article ID 1330030 2013

[12] T P Sotiriou and V Faraoni ldquof(R) theories of gravityrdquo Reviewsof Modern Physics vol 82 no 1 pp 451ndash497 2010

[13] A De Felice and S J Tsujikawa ldquof(R) theoriesrdquo Living Reviewsin Relativity vol 13 p 3 2010

[14] R Schutzhold ldquoSmall Cosmological Constant from the QCDTrace Anomalyrdquo Physical Review Letters vol 89 no 8 2002

[15] E C Thomas F R Urban and A R Zhitnitsky ldquoThe cosmo-logical constant as a manifestation of the conformal anomalyrdquoJournal of High Energy Physics vol 2009 no 8 article 043 2009

[16] F R Klinkhamer andG EVolovik ldquoGluonic vacuum q-theoryand the cosmological constantrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 79 no 6 Article ID063527 2009

[17] F R Urban and A R Zhitnitsky ldquoCosmological constant fromthe ghost a toy modelrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 80 no 6 Article ID 0630012009

[18] F R Klinkhamer and G E Volovik ldquoVacuum energy densitykicked by the electroweak crossoverrdquo Physical Review D Parti-cles Fields Gravitation and Cosmology vol 80 no 8 Article ID083001 2009

[19] S Alexander T Biswas and G Calcagni ldquoErratum Cosmo-logical Bardeen-Cooper-Schrieffer condensate as dark energyrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 81 no 6 Article ID 069902 2010

[20] N J Poplawski ldquoCosmological constant from quarks andtorsionrdquo Annalen der Physik vol 523 pp 291ndash295 2011

[21] G Bertone D Hooper and J Silk ldquoParticle dark matterevidence candidates and constraintsrdquo Physics Reports vol 405no 5-6 pp 279ndash390 2005

[22] I De Martino ldquof(R)-gravity model of the Sunyaev-Zeldovichprofile of the Coma cluster compatible with Planck datardquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 93 Article ID 124043 2016

[23] I De Martino M De Laurentis F Atrio-Barandela and SCapozziello ldquoConstraining f(R) gravity with planck data ongalaxy cluster profilesrdquoMonthly Notices of the Royal Astronom-ical Society vol 442 no 2 pp 921ndash928 2014


[24] GDrsquoAmicoMKamionkowski andK Sigurdson ldquoDarkMatterAstrophysicsrdquo httpsarxivorgabs09071912

[25] G Bertone Particle Dark Matter Observations Models andSearches (Paris Inst Astrophys) Cambridge UK Univ PrCambridge UK 2010

[26] A Bottino and N Fornengo Dark matter and its particlecandidates Trieste 1998 Non-accelerator particle physics 1988

[27] S Derraro F Schmidt and W Hu ldquoCluster abundance in f(R)gravity modelsrdquo Physical Review D Particles Fields Gravitationand Cosmology vol 83 Article ID 063503 2011

[28] L Lombriser F Schmidt T Baldauf R Mandelbaum U Seljakand R E Smith ldquoCluster density profiles as a test of modifiedgravityrdquo Physical Review D Particles Fields Gravitation andCosmology vol 85 no 10 Article ID 102001 2012

[29] F Schmidt A Vikhlinin and W Hu ldquoCluster constraints onf(R) gravityrdquoPhysical ReviewD Particles Fields Gravitation andCosmology vol 80 no 8 Article ID 083505 2009

[30] P de Bernardis P A R Ade and J J Bock ldquoA flat Universe fromhigh-resolution maps of the cosmic microwave backgroundradiationrdquo Nature vol 404 pp 955ndash959 2000

[31] D N Spergel L Verde and H V Peiris ldquoFirst-Year Wilkin-son Microwave Anisotropy Probe (WMAP) ObservationsDetermination of Cosmological ParametersrdquoThe AstrophysicalJournal Supplement Series vol 148 no 1 pp 175ndash194 2003

[32] S Dodelson V K Narayanan and M Tegmark ldquoThe three-dimensional power spectrum from angular clustering of galax-ies in early sloan digital sky survey datardquo The AstrophysicalJournal vol 572 no 1 p 140 2002

[33] A S Szalay B Jain and T Matsubara ldquoKarhunen-Loeveestimation of the power spectrum parameters from the angulardistribution of galaxies in early sloan digital sky survey datardquoThe Astrophysical Journal vol 591 pp 1ndash11 2003

[34] AG Riess L-G Strolger and J Tonry ldquoType Ia SupernovaDis-coveries at z gt 1 from the Hubble Space Telescope Evidence forPast Deceleration and Constraints on Dark Energy EvolutionrdquoThe Astrophysical Journal vol 607 no 2 pp 665ndash687 2004

[35] P Fayet and J Iliopoulos ldquoSpontaneously broken supergaugesymmetries and goldstone spinorsrdquo Physics Letters B vol 51 no5 pp 461ndash464 1974

[36] L OrsquoRaifeartaigh ldquoSpontaneous symmetry breaking for chiralsscalar superfieldsrdquo Nuclear Physics B vol 96 no 2 pp 331ndash3521975

[37] E Witten ldquoDynamical breaking of supersymmetryrdquo NuclearPhysics B vol 188 no 3 pp 513ndash554 1981

[38] Y Shadmi and Y Shirman ldquoDynamical supersymmetry break-ingrdquo Reviews of Modern Physics vol 72 no 1 article 25 2000

[39] A Das Finite Temperature Field Theory World ScientificSingapore 1997

[40] D Buchholz and I Ojima ldquoSpontaneous collapse of supersym-metryrdquo Nuclear Physics B Theoretical Phenomenological andExperimental High Energy Physics Quantum Field Theory andStatistical Systems vol 498 no 1-2 pp 228ndash242 1997

[41] A Capolupo ldquoDark Matter and Dark Energy Induced byCondensatesrdquo Advances in High Energy Physics vol 2016Article ID 8089142 p 10 2016

[42] A Capolupo ldquoCondensates as components of dark matter anddark energyrdquo Journal of Physics Conference Series vol 880 p012059 2017

[43] A Capolupo M Di Mauro and A Iorio ldquoMixing-inducedspontaneous supersymmetry breakingrdquo Physics Letters A vol375 no 39 pp 3415ndash3418 2011

[44] A Capolupo and M Di Mauro ldquoSpontaneous supersymmetrybreaking induced by vacuum condensatesrdquo Physics Letters Avol 376 no 45 pp 2830ndash2833 2012

[45] A Capolupo and M Di Mauro ldquoVacuum condensates flavormixing and spontaneous supersymmetry breakingrdquo JagellonianUniversity Institute of Physics Acta Physica Polonica B vol 44no 1 pp 81ndash89 2013

[46] A Capolupo and G Vitiello ldquoSpontaneous supersymmetrybreaking probed by geometric invariantsrdquo Advances in HighEnergy Physics vol 2013 Article ID 850395 5 pages 2013

[47] A Capolupo and M Di Mauro ldquoVacuum Condensatesas a Mechanism of Spontaneous Supersymmetry BreakingrdquoAdvances in High Energy Physics vol 2015 Article ID 9293626 pages 2015

[48] S W Hawking ldquoParticle creation by black holes Erratum 462 (1976)rdquo Communications in Mathematical Physics vol 43 no3 pp 199ndash220 1975

[49] W G Unruh ldquoNotes on black-hole evaporationrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 14no 4 pp 870ndash892 1976

[50] J Schwinger ldquoOn gauge invariance and vacuum polarizationrdquoPhysical Review A Atomic Molecular and Optical Physics vol82 pp 664ndash679 1951

[51] J Bardeen L N Cooper and J R Schrieffer ldquoTheory ofsuperconductivityrdquo Physical Review Letters vol 108 pp 1175ndash1204 1957

[52] A Iorio ldquoWeyl-gauge symmetry of graphenerdquoAnnals of Physicsvol 326 no 5 pp 1334ndash1353 2011

[53] Y Takahashi and H Umezawa ldquoThermo field dynamicsrdquoCollective Phenomena vol 2 pp 55ndash80 1975

[54] H Umezawa H Matsumoto and M Tachiki Thermo fielddynamics and condensed states North-Holland PublishingCompany 1982

[55] H Umezawa Advanced Field Theory Micro Macro and Ther-mal Physics AIP New York NY USA 1993

[56] H B G Casimir andD Polder ldquoThe influence of retardation onthe London-van der Waals forcesrdquo Physical Review A AtomicMolecular and Optical Physics vol 73 no 4 pp 360ndash372 1948

[57] H B G Casimir ldquoOn the attraction between two perfectlyconducting platesrdquo Proceedings of the Koninklijke NederlandseAkademie van Wetenschappen vol 51 pp 793ndash795 1948

[58] N D Birrell and P C W Davies Quantum Fields in CurvedSpace CambridgeMonographs onMathematical Physics Cam-bridge University Press Cambridge UK 1984

[59] M Blasone A Capolupo and G Vitiello ldquoQuantum fieldtheory of three flavor neutrino mixing and oscillations with CPviolationrdquo Physical Review D Particles Fields Gravitation andCosmology vol 66 no 2 Article ID 025033 2002

[60] M Blasone A Capolupo O Romei and G Vitiello ldquoQuantumfield theory of boson mixingrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 63 no 12 Article ID125015 2001

[61] A Capolupo C-R Ji Y Mishchenko and G Vitiello ldquoPhe-nomenology of flavor oscillations with non-perturbative effectsfrom quantum field theoryrdquo Physics Letters B vol 594 no 1-2pp 135ndash140 2004

[62] F D Albareti J A R Cembranos and A L Maroto ldquoVacuumenergy as dark matterrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 90 no 12 Article ID 1235092014


[63] A Capolupo S Capozziello and G Vitiello ldquoNeutrino mixingas a source of dark energyrdquo Physics Letters A vol 363 no 1-2pp 53ndash56 2007

[64] A Capolupo S Capozziello and G Vitiello ldquoDark energycosmological constant and neutrino mixingrdquo InternationalJournal of Modern Physics A vol 23 no 31 2008

[65] M Blasone A Capolupo S Capozziello and G Vitiello ldquoNeu-trinomixing flavor states and dark energyrdquoNuclear Instrumentsand Methods in Physics Research Section A Accelerators Spec-trometers Nuclear Instruments and Methods in Physics ResearchSection A Accelerators Spectrometers Detectors and AssociatedEquipment vol 588 no 1-2 pp 272ndash275 2008

[66] M Blasone A Capolupo and G Vitiello ldquoParticle mixingflavor condensate and dark energyrdquo Progress in Particle andNuclear Physics vol 64 no 2 pp 451ndash453 2010

[67] M Blasone A Capolupo S Capozziello S Carloni and GVitiello ldquoNeutrino mixing contribution to the cosmologicalconstantrdquo Physics Letters A vol 323 no 3-4 pp 182ndash189 2004

[68] A Capolupo G Lambiase and G Vitiello ldquoThermal conden-sate structure and cosmological energy density of the universerdquoAdvances in High Energy Physics vol 2016 Article ID 31275976 pages 2016

[69] L Parker and S A Fulling ldquoAdiabatic regularization of theenergy-momentum tensor of a quantized field in homogeneousspacesrdquo Physical Review D Particles Fields Gravitation andCosmology vol 9 no 2 pp 341ndash354 1974

[70] S A Fulling and L Parker ldquoRenormalization in the theory ofa quantized scalar field interacting with a Robertson-Walkerspace-timerdquo Annalen der Physik D vol 87 pp 176ndash204 1974

[71] J Wess and B Zumino ldquoA lagrangian model invariant undersupergauge transformationsrdquo Physics Letters B vol 49 no 1 pp52ndash54 1974

[72] Y Yu and K Yang ldquoSimulating theWess-Zumino supersymme-try model in optical latticesrdquo Physical Review Letters vol 105no 15 Article ID 150605 4 pages 2010

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components


Shock and Vibration


High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Acoustics and VibrationAdvances in



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

Antennas andPropagation

International Journal of




Geophysics

Advances inOpticalTechnologies

Hindawiwwwhindawicom

Volume 2018

Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018

Advances inOptoElectronics


Volume 2018


Mathematical PhysicsAdvances in


ChemistryAdvances in


Journal of

Chemistry


Advances inPhysical Chemistry


RotatingMachinery



Journal ofEngineeringVolume 2018

Submit your manuscripts atwwwhindawicom


We also consider the framework of the supersymmetryand by using the freeWessndashZuminomodel we show that thepresence of nonvanishing vacuum energy at the Lagrangianlevel implies that SUSY is spontaneously broken by thecondensates [43ndash47]

The structure of the paper is the following in Section 2weintroduce the Bogoliubov transformations in quantum fieldtheory (QFT) and we study the energy-momentum tensordensity for vacuum condensates of boson and fermion fieldsIn Sections 3 and 4 we present the contribution given to theenergy of the universe by thermal states with reference to theHawking and Unruh effects by fields in curved space andby particle mixing phenomena SUSY breaking induced byvacuum condensate is presented in Section 5 and Section 6 isdevoted to the conclusions

2 Bogoliubov Transformation and Energy-Momentum Tensor of Vacuum Condensate

Many phenomena ranging from the BCS theory of super-conductivity [51] to the Casimir and the Schwinger effects[50] are represented in the context of QFT by a Bogoliubovtransformation [55] Such a transformation for bosons isexpressed as 119886k(120585 119905) = 119880퐵k 119886k(119905) minus 119881퐵minusk119886daggerminusk(119905) with 120585 parameterdepending on the phenomenon analyzed 119886k annihilatorsof the vacuum |0⟩ and 119880퐵k 119881퐵k coefficients satisfying theconditions 119880퐵k = 119880퐵minusk 119881퐵k = 119881퐵minusk and |119880퐵k |2 minus |119881퐵k |2 = 1(similar discussion holds for fermions)

Introducing the generator 119869(120585 119905) of the Bogoliubov trans-formation one can write 119886k(120585 119905) = 119869minus1(120585 119905)119886k(119905)119869(120585 119905) Here119869(120585 119905) is a unitary operator 119869minus1(120585) = 119869(minus120585) which allowsrelating the original vacua |0⟩ to the vacua |0(120585 119905)⟩ annihi-lated by 119886k(120585 119905) |0(120585 119905)⟩ = 119869minus1(120585 119905)|0⟩ Such a relation is aunitary operation in quantummechanics where k assumes adiscrete range of values but it is not a unitary transformationin QFT where k assumes a continuous infinity of values Inthis case |0(120585 119905)⟩ and |0⟩ are unitarily inequivalent and theproper physical vacua are |0(120585 119905)⟩ [54 55] Such vacua have acondensate structure which is responsible for an expectationvalue of the number operator119873k = 119886daggerk119886k on |0(120585 119905)⟩ differentfrom zero ⟨0(120585 119905)|119886daggerk119886k|0(120585 119905)⟩ = |119881k|2 This fact impliesa nontrivial result of the expectation value of the energy-momentum tensor density 119879휇](119909) for free fields on |0(120585 119905)⟩Ξ휆휇](119909) =휆 ⟨0(120585 119905)| 119879휆휇](119909) |0(120585 119905)⟩휆 Here sdot sdot sdot denotesthe normal ordering with respect to the original vacuum |0⟩휆Notice that the off-diagonal components of Ξ휆휇](119909) are zeroΞ휆푖푗(119909) = 0 for 119894 = 119895 then the condensates act as a perfectfluid and one can define their energy density and pressureby means of the (0 0) and the (119895 119895) components of Ξ휆휇](119909) as120588휆 = Ξ휆00(119909) and 119901휆 = Ξ휆푗푗(119909) [41 42]

For bosons in the particular case of the isotropy of themomenta 1198961 = 1198962 = 1198963 the energy density the pressure andthe state equation 119908퐵 = 119901퐵120588퐵 are given by [41 42]

120588퐵 = 121205872 intinfin

01198891198961198962120596푘 10038161003816100381610038161003816119881퐵푘 100381610038161003816100381610038162 (1)

119901퐵 = 161205872 intinfin

01198891198961198962 [ 1198962120596푘

10038161003816100381610038161003816119881퐵푘 100381610038161003816100381610038162

minus ( 1198962120596푘 +311989822120596푘 )

10038161003816100381610038161003816119880퐵푘 10038161003816100381610038161003816 10038161003816100381610038161003816119881퐵푘 10038161003816100381610038161003816 cos (120596푘119905)] (2)

119908퐵 = 13int 1198893k (1198962120596푘) 10038161003816100381610038161003816119881퐵푘 100381610038161003816100381610038162int1198893k120596푘 1003816100381610038161003816119881퐵푘 10038161003816100381610038162 minus 13

sdot int 1198893k (1198962120596푘 + 311989822120596푘)119880퐵푘119881퐵푘 cos (120596푘119905)int 1198893k120596푘 1003816100381610038161003816119881퐵푘 10038161003816100381610038162

(3)

respectively For Majorana fermions one has [41 42]

120588퐹 = 11205872 intinfin

01198891198961198962120596푘 10038161003816100381610038161003816119881퐹푘 100381610038161003816100381610038162 (4)

119901퐹 = 131205872 intinfin

0119889119896 1198964120596푘

10038161003816100381610038161003816119881퐹푘 100381610038161003816100381610038162 (5)

119908퐹 = 13int 1198893k (1198962120596푘) 10038161003816100381610038161003816119881퐹푘 100381610038161003816100381610038162int1198893k120596푘 1003816100381610038161003816119881퐹푘 10038161003816100381610038162 (6)

The equations presented above hold for all the systemsdescribed by Bogoliubov transformations in QFT By usingsuch equations and the explicit form of the Bogoliubovcoefficients we analyze the cases of thermal states and offields in curved spaces Then we study the particle mixingphenomenon which at the level of the annihilators isdescribed by a Bogoliubov transformation and a rotationThis fact implies that for particle mixing (1)ndash(6) must bemodified as shown in Section 4

3 Dark Matter Components from ThermalStates Hawking and Unruh Effects andFields in Curved Background

We show that thermal vacuum states and vacuum condensateof fields in curved space can be interpreted as components ofthe dark matter

We start by considering the thermal vacuum state|0(120585(120573))⟩휆 introduced in the framework of ThermofieldDynamics (TFD) (where 120573 equiv 1(119896퐵119879) and 119896퐵 is the Boltz-mann constant and120582 = 119861 119865) [53ndash55] In such a context everydegree of freedom is doubled and the thermal Bogoliubovtransformation allows defining at nonzero temperature thestate |0(120585(120573))⟩휆 such that the thermal statistical average isgiven by 휆⟨0(120585(120573))|119873휒k |0(120585(120573))⟩휆 where 119873휒k = 120594daggerk120594k (120594 = 119886for bosons and 120572 for fermions) is the number operator [54]The Bogoliubov coefficients are 119880푇k = radic119890훽휔k(119890훽휔k plusmn 1) and119881푇k = radic1(119890훽휔k plusmn 1) with minus for bosons and + for fermionsand 120596k = radic1198962 + 1198982 The energy density and pressure of|0(120585(120573))⟩휆 obtained by (1)ndash(6) with 119880k = 119880푇k and 119881k = 119881푇k permit studying many thermal systems

We obtain that for temperatures of order of the cosmicmicrowave radiation (ie 119879 = 272K) only photons and






120588curv = 21205871198862 int퐾0

1198891198961198962 (10038161003816100381610038161003816120601耠푘100381610038161003816100381610038162 + 1198962 1003816100381610038161003816120601푘10038161003816100381610038162 + 1198982 1003816100381610038161003816120601푘10038161003816100381610038162) 119901curv = 21205871198862 int퐾

01198891198961198962 (10038161003816100381610038161003816120601耠푘100381610038161003816100381610038162 minus 11989623 1003816100381610038161003816120601푘10038161003816100381610038162 minus 1198982 1003816100381610038161003816120601푘10038161003816100381610038162)

(7)

















120588퐵mix = 12 ⟨0| sum푖


+ 1198982푖 1206012푖 (minus120579 119909)] |0⟩ (13)

119901퐵mix = ⟨0| sum푖


(14)


⟨0| sum푖

1205872푖 (minus120579 119909) |0⟩ = ⟨0| sum푖


푖

[120597푗120601푖 (minus120579 119909)]2 |0⟩ = 0(15)




1198982푖 1206012푖 (minus120579 119909) |0⟩ (16)


120588퐵mix = Δ1198982 sin2 12057981205872 int퐾0

1198891198961198962 ( 1120596푘1 minus1120596푘2) (17)








119901퐹mix = 119894 ⟨0| sum푖



⟨0| sum푖


푖





(20)


120588퐹mix = Δ119898 sin2 12057921205872 int퐾0

1198891198961198962 ( 1198982120596푘2 minus1198981120596푘1) (21)







(22)










⟨0 (120585 120578 119905)1003816100381610038161003816 119867퐵 10038161003816100381610038160 (120585 120578 119905)⟩ = int1198893k120596k (1 + 2 10038161003816100381610038161003816119881퐵k 100381610038161003816100381610038162) (26)


⟨0 (120585 120578 119905)1003816100381610038161003816 119867 10038161003816100381610038160 (120585 120578 119905)⟩= 2int1198893k120596k (10038161003816100381610038161003816119881휓k 100381610038161003816100381610038162 + 10038161003816100381610038161003816119881퐵k 100381610038161003816100381610038162) (27)


6 Conclusions




Acknowledgments


References














































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery










120588curv = 21205871198862 int퐾0

1198891198961198962 (10038161003816100381610038161003816120601耠푘100381610038161003816100381610038162 + 1198962 1003816100381610038161003816120601푘10038161003816100381610038162 + 1198982 1003816100381610038161003816120601푘10038161003816100381610038162) 119901curv = 21205871198862 int퐾

01198891198961198962 (10038161003816100381610038161003816120601耠푘100381610038161003816100381610038162 minus 11989623 1003816100381610038161003816120601푘10038161003816100381610038162 minus 1198982 1003816100381610038161003816120601푘10038161003816100381610038162)

(7)

















120588퐵mix = 12 ⟨0| sum푖


+ 1198982푖 1206012푖 (minus120579 119909)] |0⟩ (13)

119901퐵mix = ⟨0| sum푖


(14)


⟨0| sum푖

1205872푖 (minus120579 119909) |0⟩ = ⟨0| sum푖


푖

[120597푗120601푖 (minus120579 119909)]2 |0⟩ = 0(15)




1198982푖 1206012푖 (minus120579 119909) |0⟩ (16)


120588퐵mix = Δ1198982 sin2 12057981205872 int퐾0

1198891198961198962 ( 1120596푘1 minus1120596푘2) (17)








119901퐹mix = 119894 ⟨0| sum푖



⟨0| sum푖


푖





(20)


120588퐹mix = Δ119898 sin2 12057921205872 int퐾0

1198891198961198962 ( 1198982120596푘2 minus1198981120596푘1) (21)







(22)










⟨0 (120585 120578 119905)1003816100381610038161003816 119867퐵 10038161003816100381610038160 (120585 120578 119905)⟩ = int1198893k120596k (1 + 2 10038161003816100381610038161003816119881퐵k 100381610038161003816100381610038162) (26)


⟨0 (120585 120578 119905)1003816100381610038161003816 119867 10038161003816100381610038160 (120585 120578 119905)⟩= 2int1198893k120596k (10038161003816100381610038161003816119881휓k 100381610038161003816100381610038162 + 10038161003816100381610038161003816119881퐵k 100381610038161003816100381610038162) (27)


6 Conclusions




Acknowledgments


References














































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery








1198982푖 1206012푖 (minus120579 119909) |0⟩ (16)


120588퐵mix = Δ1198982 sin2 12057981205872 int퐾0

1198891198961198962 ( 1120596푘1 minus1120596푘2) (17)








119901퐹mix = 119894 ⟨0| sum푖



⟨0| sum푖


푖





(20)


120588퐹mix = Δ119898 sin2 12057921205872 int퐾0

1198891198961198962 ( 1198982120596푘2 minus1198981120596푘1) (21)







(22)










⟨0 (120585 120578 119905)1003816100381610038161003816 119867퐵 10038161003816100381610038160 (120585 120578 119905)⟩ = int1198893k120596k (1 + 2 10038161003816100381610038161003816119881퐵k 100381610038161003816100381610038162) (26)


⟨0 (120585 120578 119905)1003816100381610038161003816 119867 10038161003816100381610038160 (120585 120578 119905)⟩= 2int1198893k120596k (10038161003816100381610038161003816119881휓k 100381610038161003816100381610038162 + 10038161003816100381610038161003816119881퐵k 100381610038161003816100381610038162) (27)


6 Conclusions




Acknowledgments


References














































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery








⟨0 (120585 120578 119905)1003816100381610038161003816 119867퐵 10038161003816100381610038160 (120585 120578 119905)⟩ = int1198893k120596k (1 + 2 10038161003816100381610038161003816119881퐵k 100381610038161003816100381610038162) (26)


⟨0 (120585 120578 119905)1003816100381610038161003816 119867 10038161003816100381610038160 (120585 120578 119905)⟩= 2int1198893k120596k (10038161003816100381610038161003816119881휓k 100381610038161003816100381610038162 + 10038161003816100381610038161003816119881퐵k 100381610038161003816100381610038162) (27)


6 Conclusions




Acknowledgments


References














































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery



























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery





Quantum Vacuum, Dark Matter, Dark Energy, and Spontaneous...

Documents

Transcript of Quantum Vacuum, Dark Matter, Dark Energy, and Spontaneous...