The contribution of the quantum vacuum to the cosmological constant is zero: proof that vacuum energy does not gravitate
aa r X i v : . [ phy s i c s . g e n - ph ] M a y The contribution of the quantum vacuum to thecosmological constant is zero: proof that vacuum energydoes not gravitate
G. B. Mainland a, ∗ , Bernard Mulligan b a Department of Physics, The Ohio State University at Newark, 1179 University Dr.,Newark, OH 43055, USA b Department of Physics, The Ohio State University, Columbus, OH 43210, USA
Abstract
The consensus among many theoretical physicists is that the calculated con-tribution of the quantum vacuum to the total energy density of the universeis approximately 10 times the observed energy density. This is thought tobe one of the worst theoretical predictions of all time. However, as shownhere, this immense vacuum energy cannot in and of itself exert forces onnormal matter. As a result the huge vacuum energy density predicted byquantum field theory does not contribute to the ordinary energy density ofthe universe, is not a source for gravitational fields, and, as a result, does notcontribute to the value of the cosmological constant.
Keywords: cosmological constant; vacuum energy density; quantumvacuum; bound particle-antiparticle vacuum fluctuation; photon-antiphotonvacuum fluctuation;
1. Introduction: the quantum vacuum
The present consensus is that quantum field theory predicts a vacuumenergy density that is much too large. The Planck CMB anisotropy measu-rements [1] determined that the energy density in the universe is, to within ∗ Corresponding author
Email addresses: [email protected] (G. B. Mainland), [email protected] (Bernard Mulligan)
Preprint submitted to Physics of the Dark Universe 15 mai 2020 .4%, equal to the critical energy density. However, in a standard calcula-tion, if no cutoff is imposed, the vacuum energy density resulting just fromphotons is infinite; if the maximum energy of a photon is cut off at thePlanck energy, the contribution of the vacuum energy density of photons isapproximately 10 times the observed energy density of the universe. (In-cluding the contribution from gravitons and gluons increases the value of thetheoretical calculation by about an order of magnitude. The contributionfrom fundamental massive particles is less.) The huge disparity between theobserved energy density of the universe and the calculation of the vacuumenergy density is known as the “vacuum catastrophe” [2, 3].Since it is certain experimentally that the gigantic vacuum energy is notpresent as normal energy, the theoretical response has generally been to ig-nore it: the contribution of the vacuum energy to the total energy density ofthe universe has been simply assumed to be small or zero although the the-oretical mechanism is unknown. Blau and Guth[4], for example, write,“thistheoretical estimate [of the vacuum energy density] is regarded by many tobe one of the deepest mysteries of physics.”The “vacuum catastrophe” is a long-standing problem. In their 2002review article, Rugh and Zinkernagel[5] summarized the situation with regardto the cosmological constant Λ by focusing on the relation between quantumfield theory and general relativity. Such a focus requires a resolution ofwhy the vacuum energy density appears to give the cosmological constanta value vastly larger than the observed value. Explaining why the effectivecosmological constant is not large is known as “the old cosmological constantproblem”[6, 7], which is the topic primarily discussed in this article. Bycarefully examining the predictions of quantum field theory and employingwell-known physics principles, it is demonstrated here that the huge vacuumenergy density predicted by quantum field theory cannot exert a gravitationalforce; the vacuum energy density makes no contribution to the cosmologicalconstant.As will be discussed, the manifestation of vacuum energy is through theappearance of vacuum fluctuations. Accordingly, the plan of this article is asfollows: The structure and properties of vacuum fluctuations are presentedin Sec. 2. In Sec. 3 quantum field theory is used to prove that vacuumfluctuations with their structure and accompanying properties must exist,and expressions are derived for the vacuum expectation value of the energyassociated with these fluctuations. In Sec. 4 expressions are derived for theenergy density of bound, massive particle-antiparticle vacuum fluctuations2nd photon-antiphoton vacuum fluctuations, the latter of which, when acutoff is imposed at the Planck energy, is shown to be approximately 10 times the observed energy density of the universe.In Sec. 5 Einstein’s field equation is used to show that the vacuum expec-tation value of the energy-momentum tensor has the form of a cosmologicalconstant. The vacuum energy density of the universe is shown to be constant,implying that the vacuum energy density is conserved independently of theconservation of energy for normal matter. As will be discussed in Sec. 6, aconsequence of conservation of the vacuum energy density is that vacuumenergy cannot exert forces on normal matter, either directly or indirectlythrough the presence of vacuum fluctuations. Therefore the vacuum energydensity cannot be a source for gravitational fields . Because vacuum energycannot exert a force on normal matter, it cannot contribute to the value ofthe cosmological constant. Thus the contribution to the cosmological con-stant from vacuum energy is zero. In Sec. 7 the results of the article aresummarized and the consequences of those results are discussed.
2. Structure and properties of vacuum fluctuations
As has been emphasized by Dimopoulos, Raby, and Wilczek[9] and furtherdeveloped by Wilczek[10] in terms of his concept of the Grid, the quantumvacuum is filled with fluctuations that may interact with normal (meaningreal, ordinary, and observable) test particles placed in the vacuum. Physi-cists use the term “vacuum fluctuation” to describe two very different entitiesthat have been designated “type 1,” which have observable consequences, and“type 2,” which have no observable consequences[8]. Type 2 vacuum fluc-tuations are sometimes called vacuum diagrams[11] or vacuum bubbles[12],a class of Feynman diagrams for which virtual particles appear from andthen vanish back into the vacuum. These diagrams, and as a consequencetype 2 vacuum fluctuations, do not contribute to physical processes. Type 2vacuum fluctuations consist of virtual particles that are off shell, appear asinternal particles in Feynman diagrams, and exist for a time ∆ t permittedby the Heisenberg uncertainty principle. The restricted ways in which vacuum fluctuations can interact with normal matter arediscussed in [8] and make it possible to calculate the permittivity and the speed of lightin the vacuum. t permitted by the Heisenberg uncertainty principle.[13, 14, 12].Normal particles can, under very restricted circumstances, be used astest particles to observe characteristics of the Grid. The calculation by theauthors[15, 8] of the permittivity ǫ of the vacuum, the speed c of light inthe vacuum, and the fine structure constant α validates the statement byDimopoulos, Raby, and Wilczek[9] that “the vacuum is a dielectric,” andestablishes that the major contribution to the dielectric constant of the va-cuum results from VFs of charged lepton-antilepton pairs that appear in thevacuum as bound states. A formula for the permittivity ǫ of the vacuumwas calculated by examining the interaction of photons with VFs that arebound, charged lepton-antilepton pairs. Then a formula for c was immedia-tely obtained from the equation c = 1 / √ ǫ µ , where µ is the permeabilityof the vacuum. Using the formulas for ǫ and c , a value for the fine-structureconstant was calculated[15] that, to lowest order in α , agrees with the expe-rimental value to within a few percent.The calculation of c must satisfy three conditions, one from special rela-tivity and two from electrodynamics. In Einstein’s 1905 paper[16] “On the4lectrodynamics of moving bodies” in which he introduced special relativity,one of the two postulates on which his work is based is that the speed of lightin the vacuum is the same in every inertial reference frame. Furthermore,Einstein was compelled to introduce the assumption that there is no prefer-red inertial frame of reference, as demonstrated by Michelson and Morley in1887[17] and by experiments that continued to search for such a frame[18].As Leonhardt et al.[19] state,“In free space the vacuum is Lorentz invariant,so a uniformly moving observer would not see any effect due to motion . . . .”Since the vacuum is at rest with respect to every inertial reference frame andthe speed of light in the vacuum is determined by the interaction of photonswith VFs[15, 8], it follows that the speed of light is the same in every iner-tial frame, providing a theoretical explanation for why Einstein’s postulateis true and obviating the need for the postulate. To make the equations ofphysics the same in all inertial frames of reference required modification ofthe equations describing the motion of particles, but not the field equationsof electricity and magnetism . .For the derivations of ǫ and c to be consistent with Maxwell’s equations,the derivations must satisfy the following two general relations when res-tricted to the vacuum: The electric displacement D in a dielectric satisfies D = ǫ E + P , where E and P are, respectively, the electric field and the po-larization density. In the vacuum this relation becomes D = ǫ E , a relationthat was used in the derivation of the formula for ǫ . The second electro-dynamics relation that must be satisfied follows from the general equation B = µ ( H + M ), where B , H and M are, respectively, the magnetic field, themagnetic field strength, and the magnetization. In the vacuum the relationbecomes B = µ H , which the derivations satisfy because bound, chargedlepton-antilepton vacuum fluctuations have zero total angular momentumand, therefore, have zero magnetism M [20].
3. The creation and zero-point energy of vacuum fluctuations
Vacuum fluctuations result from fluctuations of free fields so they are onshell. As discussed in the previous section, they always consist of particle- In this article, as in references [8, 15], SI units are used throughout. The presentarticle deals with fundamental issues that need to be understood by nonspecialists. Thespecialist who finds the explicit use of S.I. units a nuisance does not need the presence of¯ h and c for clarification. h M c ≡ L Z (1)that gives the VF a size. In the above equation M is the mass of the bound,massive particle-antiparticle VF. The volume V Z ≡ L resulting from zitter-bewegung is the average volume in which one VF exists. The amplitude L Z associated with the zitterbewegung depends only on the mass of the particle,not its spin[23]. A VF or PVF must have spin-0 .Because the quanta of a free field behave as free particles, to understandthe creation of a VF or a PVF it is necessary to represent a VF or PVF by aquantum field. Field theory provides a proof that VFs exist and a formula forthe source of the energy available for the creation of VFs. The structure of aVF is not important when discussing the creation of the VF. What is mostimportant is that the VF’s total angular momentum is zero. Thus a VF in itsground state with zero angular momentum can be approximately representedby the Klein-Gordon field φ ( x ) for a free, neutral, spin-0 particle[24, 25, 14].To show that VFs must exist, first note that the free field φ ( x ) contains twoterms, one proportional to a creation operator a † k and the other proportio-nal to an annihilation operator a k . The vacuum expectation value of eachoperator is zero, so the average value of a free field in the vacuum is zero,(0 | φ ( x ) |
0) = 0 . (2)The expectation value of the product of a free field at two different locations x and x ′ is written in terms of the angular wave number k = p / ¯ h and the In the case of a particle with spin = 0, the spin of the particle is expressed by arotation of the particle around an axis of orientation of the particle. Whether the particleis spinning or not, the amplitude of the zitterbewegung is that stated in (1)[23]. ω k = E/ ¯ h where p and E are, respectively, the relativisticmomentum and energy[14, 12],(0 | φ ( x ) φ ( x ′ ) |
0) = ¯ h Z k = ω k /c d k (2 π ) ω k e − ik µ ( x µ − x ′ µ ) . (3)The expression in (3). is nonzero because the product φ ( x ) φ ( x ′ ) contains aterm proportional to a k a † k ′ that has a nonzero vacuum expectation value[26].Eq. (3) has the feature that (0 | φ ( x ) |
0) is infinite. However, as a result ofzitterbewegung, any VF has a finite size V Z over which it must be averaged,thus removing the infinity in (0 | φ ( x ) | φ ( x ) φ ( x ′ ) is nonzero, the free field φ ( x ) in the vacuumcannot be zero everywhere although its average value given in (2) is zero.The Hamiltonian H of the Klein-Gordon field[25, 12] is H = X k ¯ h ω k (cid:18) a † k a k + 12 (cid:19) , ω k = + r c k + M c ¯ h . (4)The above Hamiltonian has the same form as that of a harmonic oscillator.Since the annihilation operator acting on the vacuum is zero, a k |
0) = 0, theexpectation value of the Hamiltonian in the vacuum is(0 | H |
0) = X k (0 | ¯ h ω k (cid:18) a † k a k + 12 (cid:19) |
0) = 12 X k ¯ h ω k . (5)The energy in the vacuum, the zero-point energy, is the term on the right-hand side of (5). Because there are an infinite number of cells in k -space, theenergy in the vacuum is infinite. However, in any inertial frame a VF mustappear in the vacuum at rest so its center-of-mass momentum p = ¯ h k = 0 . In (5) the only energy available to create a VF is the finite energy ¯ hω k = / | H | V F = 12 ¯ h ω k = = 12 M c . (6)Since VFs are represented by noninteracting fields, the quanta associatedwith the fields are on shell: from (3) it follows that the integral over three-dimensional k -space satisfies the condition k = ω k /c . Using the expressionfor ω k in (4), it is easy to verify that VFs satisfy the on-shell condition E − ( p c ) = (¯ hω k ) − (¯ h k c ) = (¯ hck ) − (¯ hc k ) = (¯ hc ) k µ k µ = ( M c ) .7ince the mass of a photon is zero, polarization and helicity are measuresof the same quantity: a single photon can be represented either by positiveor negative helicity. Angular momentum is conserved when a PVF appearsin the vacuum with the two photons moving in opposite directions, eachwith the same helicity. The zero-point energy of a PVF is then given by theright-hand side of (5)[27] after the mass M is set to zero and the two helicitystates are summed over. Since there are two photons in a PVF,(0 | H | PVF = (2 photons)(2 helicities) 12 X k ¯ h ω k (cid:12)(cid:12) M =0 = 2 X k ¯ hc | k | . (7)The extension of the 1911 Planck result for photons[28] to include all quan-tum fields goes at least as far back as Marshall[29].
4. Energy density of VFs and PVFs and their interaction withnormal matter
In an inertial frame of reference, the number density of VFs, which appearas bound states, is 1 /L Z [8]. The energy density is then the energy of a VFmultiplied by the number density, ρ energy V F = M c L z = 4 ( M c ) (¯ hc ) . (8)To obtain the total energy density resulting from all massive particle-antiparticleVFs, (8) must be summed over all fundamental, massive particles that formspin-0, particle-antiparticle bound states. For each different particle the mass M in (8) is the mass corresponding to the energy of the state with spin-0and minimum bound-state energy.The charged fermion-antifermion VF that makes the smallest contribu-tion to this sum is the electron-positron VF that appears in the vacuum asparapositronium. Then M = 2 m e − binding energy, where m e is the mass ofthe electron. Neglecting the binding energy, which is very small in compari-son with 2 m e , ρ energyparapositroniumVF ∼ = 9 . × J / m . (9)The experimental value of the energy density of the universe[30] is ρ energyuniverse = 7 . × − J / m . (10)8hus the energy density resulting from parapositronium VFs alone is 10 times the observed energy density of the universe.The energy density of photon-antiphoton vacuum fluctuations is calcu-lated as follows[2, 3]: In (7) the entire k -space is taken to be a cube withvolume L , and that space is divided into cells labeled by integers. Periodicboundary conditions[31, 3, 32] are imposed: along the x-axis the boundarycondition is e ik x L = 1 or k x L = 2 πn x , (11)where n x is a positive or negative integer. Corresponding relations are ob-tained for the y- and z-axes:( n x , n y , n z ) = L π ( k x , k y , k z ) . (12)From (7) the energy density of PVFs is ρ energyPVF = 1 L X k hc | k | . (13)The sum over k in discrete k -space becomes a sum over integers that, inturn, can be expressed as an integral: ρ energyPVF = 2¯ hcL X n x X n y X n z | k | → hcL Z d n | k | . (14)From (12), d n = (cid:18) L π (cid:19) d k . (15)Using (15), (14) becomes ρ energyPVF = 2¯ hc (2 π ) Z d k | k | . (16)Transforming from Cartesian to spherical coordinates and noting that theintegrand is spherically symmetric, the angular integration can be performedimmediately, yielding a factor of 4 π , ρ energyPVF = 4¯ hc (2 π ) Z ∞ d | k | | k | . (17)9mposing a cutoff | k | max for the divergent integral, ρ energyPVF = ¯ hc (2 π ) | k | . (18)To obtain the total energy density resulting from all massless particle-antiparticle VFs, (18) must be summed over all fundamental, massless par-ticles. In addition to photons, the vacuum energy of gravitons and the eightgluons must be included. This will increase (18) by about an order of mag-nitude, which will not change any conclusions, so the effects of gluons andgravitons will not be included.Choosing the cutoff so that the maximum energy of a photon is thePlanck energy E P = p ¯ hc /G , where G is the gravitational coupling con-stant, | k | max = E P / (¯ hc ) = 6 . × m − . Eq. (18) then becomes ρ energyPVF = 1 . × J/m . (19)The energy density resulting from PVFs is 10 times the observed energydensity of the universe, the result stated in the Abstract.
5. The effective cosmological constant
Einstein’s field equation[2, 5, 33, 6] with a bare cosmological constantΛ bare is R µν − R g µν − Λ bare g µν = − πGc T µν . (20)In (20) R µν is the Ricci tensor, R ≡ R σσ is the Ricci scalar, g µν is the metrictensor, G is the gravitational force constant, and T µν is the energy-momentumtensor. As will be shown, the vacuum expectation value of T µν , namely(0 | T µν | T µν that has the mathematical form of a cosmologicalconstant in (20)[34]. To isolate the contribution from this term, the energy-momentum tensor resulting entirely from the presence of normal matter is,by definition, T matter µν ≡ T µν − (0 | T µν | , (21)because it has an expectation value of zero in the vacuum. In terms of T matter µν ,(20) becomes R µν − R g µν − Λ bare g µν = − πGc [ T matter µν + (0 | T µν | . (22)10o show that the vacuum expectation value of T µν has the form of a cos-mological constant[34], (0 | T µν |
0) will first be considered in Minkowski spacewhere (0 | T µν |
0) is a 4 × | T µν |
0) must be the same in any inertial reference frame.The only 4 × η µν , chosen here with diagonal ele-ments (-1,+1,+1,+1); therefore, (0 | T µν |
0) must be proportional to η µν , andthe proportionality constant must be a scalar, denoted by S ,(0 | T µν |
0) = S η µν . (23)The vacuum cannot conduct heat and has no shear stresses or viscosity,which are precisely the properties of a perfect fluid; therefore, the energy-momentum tensor of the vacuum is that of a perfect fluid[35]. T µν = (cid:18) ρ mass + Pc (cid:19) U µ U ν + P η µν , (24)where ρ mass is the mass density, P is the isotropic pressure, and U µ is a 4-velocity vector. Since the vacuum is at rest with respect to every inertialframe, in an inertial frame the three-velocity v of the perfect fluid is zero.Using U µ ( v = 0) = ( c, , ,
0) and the fact that the energy density ρ energy = ρ mass c , a perfect fluid at rest has the diagonal energy-momentum tensor T µ,ν where T = ρ energy , T = T = T = P . (25)Defining ρ energyvac ≡ (0 | ρ energy |
0) and P vac ≡ (0 | P |
0) and then substituting (25)into (23) establishes that (23) is satisfied provided S = − ρ energyvac , (26a) P vac = − ρ energyvac . (26b)Using (26a), (23) becomes(0 | T µν |
0) = − ρ energyvac η µν . (27)The generalization of (27) to curved space-time is(0 | T µν |
0) = − ρ energyvac g µν . (28)11sing (28), (22) becomes R µν − R g µν − (Λ bare + 8 πGc ρ energyvac ) g µν = − πGc T matter µν , (29)from which it follows that the effective cosmological constant Λ eff isΛ eff = Λ bare + 8 πGc ρ energyvac . (30)As a result of (26b), which is the equation of state for the vacuum, thevacuum energy density remains constant as the universe expands (or con-tracts) adiabatically[34, 36, 10]. As a result the vacuum energy density isneither a function of time nor position: ρ energyvac is a constant, independent oftime and uniform throughout all space. Since vacuum energy density is con-served, vacuum energy is conserved independently of ordinary energy in thephysical world; energy conservation in the physical world is well establishedwithout regard to vacuum energy.Eq. (30) is the source of “the old cosmological constant problem.” Foran order-of-magnitude estimate of the final term in (30), if ρ energyvac is appro-ximated (but underestimated) by the vacuum energy density of photons asgiven in (19), 8 πGc ρ energyvac ∼ πGc ρ energyPVF ∼ m − . (31)The experimental value of the cosmological constant follows from Ω Λ =0 . ρ critical = 8 . × − kg/m [30] of theuniverse: Λ eff = 1 . × − m − , which is approximately 10 − times sma-ller than the final term on the right-hand-side of (31). This is “the oldcosmological problem”.
6. Why the contribution of the quantum vacuum to the cosmolo-gical constant is zero
Because the vacuum energy density is conserved, VFs and PVFs caninteract with normal matter only in specific, limited ways. As an example,a normal photon traveling through the vacuum can interact with a VF tocreate a photon-excited VF. The progress of the photon through the vacuumis therefore slowed. When the VF annihilates, the energy borrowed fromthe vacuum for its creation must be returned; therefore, a photon identical12o the original photon is emitted and continues through the vacuum. Thisinteraction determines the speed of light in the vacuum[15, 8].There is a second way that VFs can interact with normal matter: consi-der a VF that occurs for a specific type of particle-antiparticle pair such asan electron-positron pair. A normal electron could annihilate with the posi-tron that was part of the VF, returning to the vacuum the energy originallyborrowed to create the VF. The electron that was part of the VF would thenbecome a normal electron at a location different from the original, normalelectron, giving rise to zitterbewegung[14].A solution to “the old cosmological constant problem” thus follows im-mediately from an understanding of vacuum fluctuations. Eq. (30) appearsto show that the bare cosmological constant Λ bare is increased by an amount8 πGρ energyvac /c as a result of the presence of the vacuum energy density ρ energyvac .Appearances are deceiving.So why doesn’t the vacuum energy density contribute to the cosmologicalconstant? The presence of the term proportional to ρ energyvac in (30) depends onthe assumption that the vacuum energy density can exert a classical gravita-tional force on a normal particle. However, in an inertial frame the vacuumenergy density cannot . If vacuum energy could, it would do work on nor-mal particles, permanently transferring energy between the vacuum and thephysical world and reducing the vacuum energy density.Since vacuum fluctuations are the quantum manifestation of vacuumenergy, if vacuum energy could exert a gravitational force, that force wo-uld result from the interaction of normal matter with a vacuum fluctuation.When the vacuum fluctuation vanished back into the vacuum, the energyassociated with any work done by the vacuum fluctuation would be remainbehind as ordinary energy, permanently decreasing the vacuum energy den-sity and violating the separate conservation of energy in the physical worldand energy density in the vacuum. From a quantum perspective, if a VFhas not interacted with normal matter and gained energy, it cannot spon-taneously emit quanta unless it reabsorbs identical quanta: all it can do isvanish back into the vacuum, returning to the vacuum the energy borrowedfor its creation. If a VF has formed a quasi-stationary state with a normalboson such as a photon or graviton, when the VF vanished back into the va-cuum, to conserve energy, momentum, and angular momentum, all the VFcan do is emit a boson that is identical to the original. As was shown by theauthors[15, 8], this is precisely the mechanism by which photons (and almostcertainly gravitons[37]) travel through the vacuum. Since a VF cannot “per-13anently” exchange a boson associated with a force with normal matter, itcannot exert a force on normal matter.From the conservation of the energy density, an observer in an inertialreference frame will conclude that vacuum energy cannot exert a force onnormal matter so the vacuum energy of the vacuum does not contribute tothe energy density of normal energy in the universe, explaining why thereis no “vacuum catastrophe”. Similarly, in an inertial frame vacuum energycannot exert a gravitational force on normal matter so vacuum energy cannotbe a source for a gravitational field. Thus in (30) ρ energyvac contributes nothingto the cosmological constant and Λ eff = Λ bare . That is, quantum vacuumenergy does not gravitate.
7. Results and discussion
VFs are on shell, massive particle-antiparticle pairs. In an inertial framethey appear at rest in the least energetic bound state that has zero angularmomentum. PVFs are photon-antiphoton pairs. In an inertial frame the twophotons appear with zero center-of-mass momentum and the same helicityso that each pair has zero total angular momentum. Field theory predictsthe existence of both VFs and PVFs as well as their energy in the vacuum.The vacuum energy density of VFs and PVFs is both time-independentand homogeneous so in an inertial frame of reference, vacuum energy is con-served independently of normal energy. A consequence is that in an inertialframe, vacuum energy cannot exert a force on normal matter. The calcula-tions presented in this paper directly address the issue raised by Wang, Zhu,and Unruh[6] with respect to the combination of quantum field theory andgeneral relativity: “the equivalence principle of general relativity requiresthat every form of energy gravitates in the same way.” The conclusion of thepresent article is that in an inertial frame of reference vacuum energy cannotexert a gravitational force because exerting such a force would violate conser-vation of vacuum energy density. Accordingly, in an inertial frame, vacuumenergy does not gravitate. Therefore, in an inertial frame vacuum energydensity does not contribute to the normal energy density in the universe,explaining why there is no “vacuum catastrophe”.Physicists have struggled to explain why the cosmological constant isso small. Before the cosmological constant was found to be nonzero[38, 7],Barrow and Shaw[39] wrote,“many particle physicists suspected that somefundamental principle must force [the value of the cosmological constant] to14e precisely zero. Perlmutter et al.[7] wrote,“ . . . presumably some symmetryof the particle physics model is causing cancellations of this vacuum energydensity.” Since vacuum energy cannot exert a gravitational force in a iner-tial frame, vacuum energy is not a source for a gravitational field and doesnot contribute to the value of the cosmological constant, solving “the oldcosmological constant problem.”Determining why the cosmological constant has the observed value is verymuch an open question[40, 41]. As Guth[42] wrote in 1981,“The reason Λ isso small is of course one of the deep mysteries of physics. The value of Λ isnot determined by the particle theory alone, but must be fixed by whatevertheory couples particles to quantum gravity.”
Bibliografie [1] Y. Akrami, et al. , Planck 2018 results. X. Constraints on inflation, As-tron. and Astrophys. J. arXiv:1807.06211v1(astro-ph).[2] S. Weinberg, The cosmological constant problem, Rev. Mod. Phys. 61(1989) 1–82.[3] R. J. Adler, B. Casey, O. C. Jacob, Vacuum catastrophe: An elementaryexposition of the cosmological constant problem, Am. J. Phys. 63 (1995)620–626.[4] S. K. Blau, A. H. Guth, Inflationary cosmology, in: S. W. Hawking,W. Israel (Eds.), Three Hundred Years of Gravitationt, Cambridge Univ.Press, Cambridge, 1987, pp. 524–603, p. 540.[5] S. E. Rugh, H. Zinkernagel, The quantum vacuum and the cosmologicalconstant problem, Stud. Hist. Philos. Mod. Phys. 33 (2002) 663–705,p. 664.[6] Q. Wang, Z. Zhu, G. U. Unruh, How the huge energy of quantum vacuumgravitates to drive the slow accelerating expansion of the universe, Phys.Rev. D. 95 (2017) 013504–1–34.[7] S. Perlmutter, et al. , Measurements of Ω and Λ from 42 high-redshiftsupernovae, Astrophys. J. 517 (1999) 565–586, p. 584.158] G. B. Mainland, B. Mulligan, Polarization of vacuum fluctuations: so-urce of the vacuum permittivity and speed of light, Found of Phy-sics doi:10.1007/s10701-020-00339-3 .[9] S. Dimopoulos, S. A. Raby, F. Wilczek, Unification of couplings, PhysicsToday 23 (1991) 25–33, p. 27.[10] F. Wilczek, The Lightness of Being, Basic Books, New York, 2008,pp. 88-111; p. 108.[11] J. M. Jauch, F. Rohrlich, The Theory of Photons and Electrons, 2nded., Springer, New York, 1976, pp. 176-178, p. 285.[12] J. D. Bjorken, S. D. Drell, Relativistic Quantum Fields, McGraw-Hill,New York, 1965, pp. 29-30; pp. 35-37; pp. 187-189.[13] W. Heisenberg, The physical content of quantum kinematics and mecha-nics, in: J. A. Wheeler, H. Zurek (Eds.), Quantum Theory and Measu-rement, Princeton University Press, Princeton, 1983, pp. 62–84, Englishtranslation of W. Heisenberg, ¨Uber den anschaulichen Inhalt der qu-antentheoretischen kinematik und mechanik, Zeitschrift f¨ur Physik (1927) 172-198.[14] W. E. Thirring, Principles of Quantum Electrodynamics, AcademicPress, New York, 1958, p. 77; p. 81 and p. 6.[15] G. B. Mainland, B. Mulligan, How vacuum fluctuations determine theproperties of the vacuum, J. Phys.: Conf. Ser. 1239 (2019) 012016. doi:10.1088/1742-6596/1239/1/012016 .[16] A. Einstein, On the electrodynamics of moving bodies (Zur Elektrody-namik bewegter K¨orper), Annalen der Physik 17 (1905) 891–921.[17] A. A. Michelson, E. M. Morley, On the relative motion of the earth andthe luminiferous ether, American Journal of Science 34 (1887) 333–345.[18] H. C. Ohanian, Einstein’s Mistakes, Norton, New York, 2008, pp. 22-33.[19] U. Leonhardt, I. Griniasty, S. Wildeman, E. Fort, M. Fink, Classicalanalog of the Unruh effect, Phys. Rev. A 98 (2018) 022118–1–022118–11, p. 022118-1. 1620] W. C. Sauder, R. D. Desolates, Zeeman effect in positronium annihila-tion at low temperatures, Journal of Research of the National Bureauof Standards, 71A (1967) 347–353, p. 349.[21] L. de Broglie, The Theory of Particles of Spin-1/2, Gauthiers-Villars,Paris, 1952, see pp. 97-103, English translation of La th´eorie des particlesde spin-1/2 ´Electrons de Dirac, Gauthier-Villars, Paris, 1951.[22] A. O. Barut, A. J. Bracken, Zitterbewegung and the internal geometryof the electron, Phys. Rev. D 23 (1981) 2454–2463, p. 2456.[23] E. Corinaldesi, Relativistic Wave Mechanics, North-Holland, Amster-dam, 1963, (Dover pp. 64-66 and p. 152).[24] W. Pauli, V. Weisskopf, The quantization of the scalar relativistic waveequation, in: A. I. Miller (Ed.), Early Quantum Electrodynamics: ASource Book, Cambridge University Press, Cambridge, 1994, pp. 188–205, English translation of W. Pauli and V. Weisskopf, Helvetica PhysicaActa , 709 (1934).[25] G. Wentzel, Quantum Theory of Fields, Dover, New York, 2003, seep. 48–53, English translation of G. Wentzel, Einf ¨ uhrung in die Quan-tentheorie der Wellenfelder Vienna: Franz Deuticke, 1942. See Sec. 8.[26] A. Bohm, P. Kielanowski, G. B. Mainland, Quantum Physics: States,Observables and Their Time Evolution, Springer, New York, 2019, p. 11.[27] E. M. Lifshitz, V. B. Berestetski, L. P. Pitaevskii, Quantum Electro-dynamics (Course of Theoretical Physics, Vol. 4), Pergamon, Oxford,1982, pp. 24-29.[28] M. Planck, A new radiation hypothesis (Eine neue Strahlungshypo-these), Verh. Dtsch. Phys. Ges 13 (1911) 138–148.[29] T. W. Marshall, Random electrodynamics, Proc. R. Soc. A 276 (1963)475–491.[30] Tanabashi, et al. (Particle Data Group), Review of particle physics,Phys. Rev. D 98 (2018) 030001.[31] D. Saxon, Elementary Quantum Mechanics, McGraw-Hill, New York,1968, pp. 215-216 and 267. 1732] S. Gasiorowicz, Quantum Physics, 3rd ed., Wiley, New Jersey, 2003, p.241.[33] J. Martin, Everything you always wanted to know about the cosmologi-cal constant problem (but were afraid to ask), C. R. Physique 13 (2012)566–665.[34] S. M. Carroll, W. H. Press, The cosmological constant, Annu. Rev.Astron. Astrophys. 30 (1992) 499–542, pp. 501-502.[35] C. Misner, K. S. Thorne, J. A. Wheeler, Gravitation, Freeman, NewYork, 1973, pp. 562-563.[36] S. M. Carroll, The cosmological constant, Living Rev. Relativity 3 (2001)1–61, p. 11.[37] L. H. Ford, L. Parker, Quantized gravitational wave perturbations inrobertson-walker universes, Phys. Rev. D 16 (1977) 1601–1608.[38] A. G. Riess, et al. , Observational evidence from supernovae for an ac-celerating universe and a cosmological constant, Astron. J. 116 (1998)1009–1038.[39] J. D. Barrow, D. J. Shaw, The value of the cosmologicalconstant, Gen Relativ Gravit 43 (2011) 2555–2560, p. 2556. doi:10.1007/s10714-011-1199-1 .[40] E. J. Copeland, M. Sami, S. Tsujikawa, Dynamics of darkenergy, Int. J. of Mod. Phys. D 15 (2006) 1753–1935. doi:10.1142/S021827180600942X .[41] P. Brax, What makes the universe accelerate? A review on whatdark energy could be and how to test it, Rep. Prog. Phys. 81 (2018)016902(52pp), p. 3. doi:10.1088/1361-6633/aa8e64doi:10.1088/1361-6633/aa8e64