Cosmological Acceleration as a Consequence of Quantum de Sitter Symmetry
aa r X i v : . [ phy s i c s . g e n - ph ] A p r Cosmological Acceleration as a Consequence of Quantum deSitter Symmetry
Felix M. Lev
Artwork Conversion Software Inc., 509 N. Sepulveda Blvd, Manhattan Beach, CA90266, USA (Email: [email protected])
Abstract
Physicists usually understand that physics cannot (and should not) derivethat c ≈ · m/s and ¯ h ≈ . · − kg · m /s . At the same time they usuallybelieve that physics should derive the value of the cosmological constant Λ andthat the solution of the dark energy problem depends on this value. However,background space in General Relativity (GR) is only a classical notion whileon quantum level symmetry is defined by a Lie algebra of basic operators. Weprove that the theory based on Poincare Lie algebra is a special degeneratecase of the theories based on de Sitter (dS) or anti-de Sitter (AdS) Lie algebrasin the formal limit R → ∞ where R is the parameter of contraction from thelatter algebras to the former one, and R has nothing to do with the radius ofbackground space. As a consequence, R is necessarily finite, is fundamentalto the same extent as c and ¯ h , and a question why R is as is does not arise.Following our previous publications, we consider a system of two free bodiesin dS quantum mechanics and show that in semiclassical approximation thecosmological dS acceleration is necessarily nonzero and is the same as in GR ifthe radius of dS space equals R and Λ = 3 /R . This result follows from basicprinciples of quantum theory. It has nothing to do with existence or nonex-istence of dark energy and therefore for explaining cosmological accelerationdark energy is not needed. The result is obtained without using the notion ofdS background space (in particular, its metric and connection) but simply asa consequence of quantum mechanics based on the dS Lie algebra. Therefore,Λ has a physical meaning only on classical level and the cosmological constantproblem and the dark energy problem do not arise. In the case of dS and AdSsymmetries all physical quantities are dimensionless and no system of units isneeded. In particular, the quantities ( c, ¯ h, s ), which are the basic quantities inthe modern system of units, are not so fundamental as in relativistic quantumtheory. ”Continuous time” is a part of classical notion of space-time contin-uum and makes no sense beyond this notion. In particular, description of theinflationary stage of the Universe by times (10 − s, − s ) has no physicalmeaning. Keywords: dark energy, quantum theory, de Sitter symmetry1
Brief overview of the cosmological constant prob-lem and dark energy
The history of General Relativity (GR) is described in a vast literature. The La-grangian of GR is linear in Riemannian curvature R c , but from the point of viewof symmetry requirements there exist infinitely many Lagrangians satisfying such re-quirements. For example, f ( R c ) theories of gravity are widely discussed, where therecan be many possibilities for choosing the function f . Then the effective gravitationalconstant G eff can considerably differ from standard gravitational constant G . It isalso argued that GR is a low energy approximation of more general theories involv-ing higher order derivatives. The nature of gravity on quantum level is a problem,and standard canonical quantum gravity is not renormalizable. For those reasons thequantity G can be treated only as a phenomenological parameter but not fundamentalone. Let us restrict ourselves with the consideration of standard GR. Here theEinstein equations depend on two arbitrary parameters G and Λ where Λ is thecosmological constant (CC). In the formal limit when matter disappears, space-timebecomes Minkowski space when Λ = 0, de Sitter (dS) space when Λ >
0, and anti-deSitter (AdS) space when Λ < = 0.However, according to Gamow, after Friedman’s results and Hubble’s discovery of theUniverse expansion, Einstein changed his mind and said that inclusion of Λ was thegreatest blunder of his life.The usual philosophy of GR is that curvature is created by matter andtherefore Λ should be equal to zero. This philosophy has been advocated even instandard textbooks written before 1998. For example, the authors of Ref. [1] say that”...there are no convincing reasons, observational and theoretical, for introducing anonzero value of Λ” and that ”... introducing to the density of the Lagrange functiona constant term which does not depend on the field state would mean attributing tospace-time a principally ineradicable curvature which is related neither to matter norto gravitational waves”.However, the data of Ref. [2] on supernovae have shown that Λ > G is treated as fundamental andthe value of Λ should be extracted from the vacuum expectation value of the energy-momentum tensor. The theory contains strong divergencies and a reasonable cutoffgives for Λ a value exceeding the experimental one by 120 orders of magnitude. Thisresult is expected because in units c = ¯ h = 1 the dimension of G is m , the dimensionof Λ is m − and therefore one might think than Λ is of the order of 1 /G what exceedsthe experimental value by 120 orders of magnitude.Several authors argue that the CC problem does not exists. For exam-ple, the authors of Ref. [4] titled ”Why all These Prejudices Against a Constant?”note that since the solution of the Einstein equations depends on two arbitrary phe-nomenological constants G and Λ it is not clear why we should choose only a specialcase Λ = 0. If Λ is as small as given in Ref. [2] then it has no effect on the data inSolar System and the contribution of Λ is important only at cosmological distances.Also theorists supporting Loop Quantum Gravity say that the preferable choice ofMinkowski background contradicts the background independence principle. Neverthe-less, the majority of physicists working in this field believe that the CC problem doesexist and the solution should be sought in the framework of dark energy, quintessenceand other approaches. Theories dealing with foundation of physics are called fundamental. In this sectionwe discuss some of those theories and their comparisons. One of the known examplesis the comparison of nonrelativistic theory (NT) with relativistic one (RT). One ofthe reasons why RT can be treated as more fundamental is that it contains a finiteparameter c and NT can be treated as a special degenerate case of RT in the formallimit c → ∞ . Therefore, by choosing a large value of c , RT can reproduce any resultof NT with a high accuracy. On the contrary, when the limit is already taken onecannot return back from NT to RT and NT cannot reproduce all results of RT. Itcan reproduce only results obtained when v ≪ c . Other known examples are thatclassical theory is a special degenerate case of quantum one in the formal limit ¯ h → R → ∞ where R is the parameter of contraction from the dS or AdS algebrasto the Poincare algebra (see below). A question arises whether it is possible to give ageneral definition when theory A is more fundamental than theory B. In view of theabove examples, we propose the following Definition:
Let theory A contain a finite parameter and theory B be ob-tained from theory A in the formal limit when the parameter goes to zero or infinity.Suppose that with any desired accuracy theory A can reproduce any result of theory Bby choosing a value of the parameter. On the contrary, when the limit is already takenthen one cannot return back to theory A and theory B cannot reproduce all results oftheory A. Then theory A is more fundamental than theory B and theory B is a specialdegenerate case of theory A .A problem arises how to justify this
Definition not only from physicalbut also from mathematical considerations.In relativistic quantum theory the usual approach to symmetry on quan-tum level follows. Since the Poincare group is the group of motions of Minkowskispace, quantum states should be described by representations of this group. Thisimplies that the representation generators commute according to the commutationrelations of the Poincare group Lie algebra:[ P µ , P ν ] = 0 , [ P µ , M νρ ] = − i ( η µρ P ν − η µν P ρ ) , [ M µν , M ρσ ] = − i ( η µρ M νσ + η νσ M µρ − η µσ M νρ − η νρ M µσ ) (1)where µ, ν = 0 , , , P µ are the operators of the four-momentum and M µν arethe operators of Lorentz angular momenta. This approach is in the spirit of Klein’sErlangen program in mathematics.However, background space is only a classical notion which does not existin quantum theory. For example, although in QED, QCD and electroweak theorythe Lagrangian density depends on the four-vector x which is associated with a pointin Minkowski space, this is only the integration parameter which is used in the in-termediate stage. The goal of the theory is to construct the S -matrix and whenthe theory is already constructed one can forget about Minkowski space because nophysical quantity depends on x . This is in the spirit of the Heisenberg S -matrixprogram according to which in relativistic quantum theory it is possible to describeonly transitions of states from the infinite past when t → −∞ to the distant futurewhen t → + ∞ . For those reasons, as argued in Ref. [5], the approach should be theopposite. Each system is described by a set of linearly independent operators. Bydefinition, the rules how they commute with each other define the symmetry algebra.In particular, by definition , Poincare symmetry on quantum level means that the op-erators commute according to Eq. (1). This definition does not involve Minkowskispace at all.Such a definition of symmetry on quantum level has been proposed in Ref.[6] and in subsequent publications of those authors. I am very grateful to LeonidAvksent’evich Kondratyuk for explaining me this definition during our collabora-tion. I believe that this replacement of the standard paradigm is fundamental for4nderstanding quantum theory, and I did not succeed in finding a similar idea in theliterature. Our goal is to compare four theories: classical (i.e. non-quantum) the-ory, nonrelativistic quantum theory, relativistic quantum theory and dS or AdSquantum theory. All those theories are described by representations of the sym-metry algebra containing ten linearly independent operators A α ( α = 1 , , ... A α where α = 1 , , , A α where α = 5 , , A α where α = 8 , ,
10 referto Galilei or Lorentz boost operators. Let [ A α , A β ] = ic αβγ A γ where summation overrepeated indices is assumed. In the theory of Lie algebras the quantities c αβγ arecalled the structure constants.Let S be a set of ( α, β ) pairs such that c αβγ = 0 for all values of γ and S be a set of ( α, β ) pairs such that c αβγ = 0 at least for some values of γ . Since c αβγ = − c βαγ it suffices to consider only such ( α, β ) pairs where α < β . If ( α, β ) ∈ S then the operators A α and A β commute while if ( α, β ) ∈ S then they do not commute.Let ( S A , S A ) be the sets ( S , S ) for theory A and ( S B , S B ) be the sets( S , S ) for theory B. As noted above, we will consider only theories where α, β =1 , , ...
10. Then one can prove the following
Statement:
Let theory A contain a finite parameter and theory B beobtained from theory A in the formal limit when the parameter goes to zero or infinity.If the sets S A and S B are different and S A ⊂ S B (what equivalent to S B ⊂ S A ) thentheory A is more fundamental than theory B and theory B is a special degenerate caseof theory A. Proof: Let ˜ S be the set of ( α, β ) pairs such that ( α, β ) ∈ S A and ( α, β ) ∈ S B . Then, in theory B, c αβγ = 0 for any γ . One can choose the parameter suchthat in theory A all the quantities c αβγ are arbitrarily small. Therefore, by choosinga value of the parameter, theory A can reproduce any result of theory B with anydesired accuracy. When the limit is already taken then, in theory B, [ A α , A β ] = 0 forall ( α, β ) ∈ ˜ S . This means that the operators A α and A β become fully independentand therefore there is no way to return to the situation when they do not commute.Therefore for theories A and B the conditions of Definition are satisfied.It is sometimes stated that the expressions in Eq. (1) are not generalenough because they are written in the system of units c = ¯ h = 1. Let us considerthis problem in more details. The operators M µν in Eq. (1) are dimensionless. Inparticular, standard angular momentum operators ( J x , J y , J z ) = ( M , M , M ) aredimensionless and satisfy the commutation relations[ J x , J y ] = iJ z , [ J z , J x ] = iJ y , [ J y , J z ] = iJ x (2)If one requires that the operators M µν should have the dimension kg · m /s thenthey should be replaced by M µν / ¯ h , respectively. In that case the new commutationrelations will have the same form as in Eqs. (1) and (2) but the right-hand-sides willcontain the additional factor ¯ h . 5he result for the components of angular momentum depends on the sys-tem of units. As shown in quantum theory, in units ¯ h = 1 the result is given bya half-integer 0 , ± / , ± , ... . We can reverse the order of units and say that inunits where the angular momentum is a half-integer l , its value in kg · m /s is1 . · − · l · kg · m /s . Which of those two values has more physical sig-nificance? In units where the angular momentum components are half-integers, thecommutation relations (2) do not depend on any parameters. Then the meaning of l isclear: it shows how large the angular momentum is in comparison with the minimumnonzero value 1/2. At the same time, the measurement of the angular momentumin units kg · m /s reflects only a historic fact that at macroscopic conditions on theEarth between the 18th and 21st centuries people measured the angular momentumin such units.We conclude that for quantum theory itself the quantity ¯ h is not needed.However, it is needed for the transition from quantum theory to classical one: weintroduce ¯ h , then the operators M µν have the dimension kg · m /s , and since theright-hand-sides of Eqs. (1) and (2) in this case contain an additional factor ¯ h , allthe commutation relations disappear in the formal limit ¯ h →
0. Therefore in classicaltheory the set S is empty and all the ( α, β ) pairs belong to S . Since in quantumtheory there exist ( α, β ) pairs such that the operators A α and A β do not commutethen in quantum theory the set S is not empty and, as follows from Statement ,classical theory is the special degenerate case of quantum one in the formal limit¯ h →
0. Since in classical theory all operators commute with each other then in thistheory operators are not needed and one can work only with physical quantities. Aquestion why ¯ h is as is does not arise since the answer is: because people want tomeasure angular momenta in kg · m /s .Consider now the relation between RT and NT. If we introduce the Lorentzboost operators L j = M j ( j = 1 , ,
3) then Eqs. (1) can be written as[ P , P j ] = 0 , [ P j , P k ] = 0 , [ J j , P ] = 0 , [ J j , P k ] = iǫ jkl P l , [ J j , J k ] = iǫ jkl J l , [ J j , L k ] = iǫ jkl L l , [ L j , P ] = iP j (3)[ L j , P k ] = iδ jk P , [ L j , L k ] = − iǫ jkl J l (4)where j, k, l = 1 , , ǫ jkl is the fully asymmetric tensor such that ǫ = 1, δ jk isthe Kronecker symbol and a summation over repeated indices is assumed. If we nowdefine the energy and Galilei boost operators as E = P c and G j = L j /c ( j = 1 , , G j , P k ] = iδ jk E/c , [ G j , G k ] = − iǫ jkl J l /c (5)Note that for relativistic theory itself the quantity c is not needed. In thistheory the primary quantities describing particles are their momenta p and energies E while the velocity v of a particle is defined as v = p /E . This definition does notinvolve meters and seconds, and the velocities v are dimensionless quantities such6hat | v | ≤ c only for transitionfrom RT to NT: when we introduce c then the velocity of a particle becomes p c /E ,and its dimension becomes m/s . In this case, instead of the operators P and L j we work with the operators E and G j , respectively. If M is the Casimir operatorfor the Poincare algebra defined such that M c = E − P c then in the formallimit c → ∞ the first expression in Eq. (5) becomes [ G j , P k ] = iδ jk M while thecommutators in the second expression become zero. Therefore in NT the ( α, β ) pairswith α, β = 8 , ,
10 belong to S while in RT they belong to S . Therefore, as followsfrom Statement , NT is a special degenerate case of RT in the formal limit c → ∞ .The question why c = 3 · m/s and not, say c = 7 · m/s does not arise since theanswer is: because people want to measure c in m/s .From the mathematical point of view, c is the parameter of contractionfrom the Poincare algebra to the Galilei one. This parameter must be finite: theformal case c = ∞ corresponds to the situation when the Poincare algebra does notexist because it becomes the Galilei algebra.In his famous paper ”Missed Opportunities” [7] Dyson notes that RT ismore fundamental than NT, and dS and AdS theories are more fundamental thanRT not only from physical but also from pure mathematical considerations. Poincaregroup is more symmetric than Galilei one and the transition from the former to thelatter at c → ∞ is called contraction. Analogously dS and AdS groups are moresymmetric than Poincare one and the transition from the former to the latter at R → ∞ (described below) also is called contraction. At the same time, since dSand AdS groups are semisimple they have a maximum possible symmetry and cannotbe obtained from more symmetric groups by contraction. However, since we treatsymmetry not from the point of view of a group of motion for the correspondingbackground space but from the point of view of commutation relations in the sym-metry algebra, we will discuss the relations between the dS and AdS algebra on onehand and the Poincare algebra on the other.By analogy with the definition of Poincare symmetry on quantum level,the definition of dS symmetry on quantum level should not involve the fact thatthe dS group is the group of motions of dS space. Instead, the definition is thatthe operators M ab ( a, b = 0 , , , , M ab = − M ba ) describing the system underconsideration satisfy the commutation relations of the dS Lie algebra so(1,4), i.e. [ M ab , M cd ] = − i ( η ac M bd + η bd M ac − η ad M bc − η bc M ad ) (6)where η ab is the diagonal metric tensor such that η = − η = − η = − η = − η = 1. The definition of AdS symmetry on quantum level is given by the sameequations but η = 1.With such a definition of symmetry on quantum level, dS and AdS sym-metries are more natural than Poincare symmetry. In the dS and AdS cases all theten representation operators of the symmetry algebra are angular momenta while inthe Poincare case only six of them are angular momenta and the remaining four op-erators represent standard energy and momentum. If we define the operators P µ as7 µ = M µ /R where R is a parameter with the dimension length then in the formallimit when R → ∞ , M µ → ∞ but the quantities P µ are finite, Eqs. (6) become Eqs.(1). This procedure is called contraction and in the given case it is the same for the dSor AdS symmetry. As follows from Eqs. (1) and (6), if α, β = 1 , , , α, β )pairs belong to S in RT and to S in dS and AdS theories. Therefore, as follows from Statement , RT is indeed a special degenerate case of dS and AdS theories in theformal limit R → ∞ . By analogy with the abovementioned fact that c must be finite, R must be finite too: the formal case R = ∞ corresponds to the situation when thedS and AdS algebras do not exist because they become the Poincare algebra.One of the consequences is that the CC problem described in Sec. 1 doesnot exist because its formulation is based on the incorrect assumption that RT ismore fundamental than dS and AdS theories. Note that the operators in Eq. (6) donot depend on R at all. This quantity is needed only for transition from dS quantumtheory to Poincare quantum theory. In full analogy with the above discussion ofquantities ¯ h and c , a question why R is as is does not arise and the answer is: becausepeople want to measure distances in meters.On classical level, dS space is usually treated as the four-dimensional hy-persphere in the five-dimensional space such that x + x + x + x − x = R ′ (7)where R ′ is the radius of dS space and at this stage it is not clear whether or not R ′ coincide with R . Transformations from the dS group are usual and hyperbolicrotations of this space. They can be parametrized by usual and hyperbolic anglesand do not depend on R ′ . In particular, if instead of x a we introduce the quantities ξ a = x a /R ′ then the dS space can be represented as a set of points ξ + ξ + ξ + ξ − ξ = 1 (8)Therefore in classical dS theory itself the quantity R ′ is not needed at all. It is neededonly for transition from dS space to Minkowski one: we choose R ′ in meters, then thecurvature of this space is Λ = 3 /R ′ and a vicinity of the point x = R ′ or x = − R ′ becomes Minkowski space in the formal limit R ′ → ∞ . Analogous remarks are validfor the transition from AdS theory to Poincare one, and in this case Λ = − /R ′ .We have proved that all the three discussed comparisons satisfy the condi-tions formulated in Definition above. Namely, the more fundamental theory containsa finite parameter and the less fundamental theory can be treated as a special de-generate case of the former in the formal limit when the parameter goes to zero orinfinity. The more fundamental theory can reproduce all results of the less fundamen-tal one by choosing some value of the parameter. On the contrary, when the limit isalready taken one cannot return back from the less fundamental theory to the morefundamental one.In Ref. [8] we considered properties of dS quantum theory and argued thatdS symmetry is more natural than Poincare one. However, the above discussion provesthat dS and AdS symmetries are not only more natural than Poincare symmetry but8ore fundamental. In particular, R is fundamental to the same extent as ¯ h and c and, as noted above, R must be finite . Let us stress that the above proof that dS symmetry is more fundamental thanPoincare one has been performed on pure quantum level. In particular, the proofdoes not involve the notion of background space and the notion of Λ. Therefore aproblem arises whether this result can be used for explaining that experimental datacan be described in the framework of GR with Λ > χ ( v ) on the Lorenz 4-velocity hyperboloid with thepoints v = ( v , v ) , v = (1 + v ) / such that R | χ ( v ) | dρ ( v ) < ∞ and dρ ( v ) = d v /v is the Lorenz invariant volume element. For positive energy IRs the value of energy is E = mv where m is the particle mass defined as the positive square root ( E − P ) / .Therefore for massive IRs, m > χ ( v ) , χ ( v )) such that Z ( | χ ( v ) | + | χ ( v ) | ) dρ ( v ) < ∞ In Poincare limit one dS IR splits into two IRs of the Poincare algebra with positiveand negative energies and, as argued in Ref. [8], this implies that one IR of the dSalgebra describes a particle and its antiparticle simultaneously. Since in the presentpaper we do not deal with antiparticles and neglect spin effects, we give only expres-sions for the action of the operators on the upper hyperboloid in the case of zero spin[8]: J = l ( v ) , L = − iv ∂∂ v , B = m dS v + i [ ∂∂ v + v ( v ∂∂ v ) + 32 v ] E = m dS v + iv ( v ∂∂ v + 32 ) (9)9here B = { M , M , M } , l ( v ) = − i v × ∂/∂ v , E = M and m dS is a positivequantity. This implementation of the IR is convenient for the transition to Poincarelimit. Indeed, the operators of the Lorenz algebra in Eq. (9) are the same as inthe IR of the Poincare algebra. Suppose that the limit of m dS /R when R → ∞ is finite and denote this limit as m . Then in the limit R → ∞ we get standardexpressions for the operators of the IR of the Poincare algebra where m is standardmass, E = E /R = mv and P = B /R = m v . For this reason m dS has the meaningof the dS mass. Since Poincare symmetry is a special case of dS one, m dS is morefundamental than m . Since Poincare symmetry works with a high accuracy, the valueof R is supposed to be very large (but, as noted above, it cannot be infinite).Consider the non-relativistic approximation when | v | ≪
1. If we wish towork with units where the dimension of velocity is meter/sec , we should replace v by v /c . If p = m v then it is clear from the expression for B in Eq. (9) that p becomesthe real momentum P only in the limit R → ∞ . At this stage we do not have anycoordinate space yet. However, by analogy with standard quantum mechanics, wecan define the position operator r as i∂/∂ p .In classical approximation we can treat p and r as usual vectors. Then asfollows from Eq. (9) P = p + mc r /R, H = p / m + c pr /R, L = − m r (10)where H = E − mc is the classical nonrelativistic Hamiltonian. As follows from theseexpressions, H ( P , r ) = P m − mc r R (11)The last term in Eq. (11) is the dS correction to the non-relativisticHamiltonian. It is interesting to note that the non-relativistic Hamiltonian dependson c although it is usually believed that c can be present only in relativistic theory.This illustrates the fact mentioned in Sec. 2 that the transition to nonrelativistictheory understood as | v | ≪ c → ∞ . Thepresence of c in Eq. (11) is a consequence of the fact that this expression is written instandard units. In nonrelativistic theory c is usually treated as a very large quantity.Nevertheless, the last term in Eq. (11) is not large since we assume that R is verylarge. As follows from Eq. (11) and the Hamilton equations, in dS theory a freeparticle moves with the acceleration given by a = r c /R (12)where a and r are the acceleration and the radius vector of the particle, respectively.Since R is very large, the acceleration is not negligible only at cosmological distanceswhen | r | is of the order of R .Following our results in Ref. [8], we now consider whether the result (12) iscompatible with GR. As noted in Sec. 2, the dS space is a four-dimensional manifold10n the five-dimensional space defined by Eq. (7). In the formal limit R ′ → ∞ theaction of the dS group in a vicinity of the point (0 , , , , x = R ′ ) becomes the actionof the Poincare group on Minkowski space. With this parameterization the metrictensor on dS space is g µν = η µν − x µ x ν / ( R ′ + x ρ x ρ ) (13)where µ, ν, ρ = 0 , , , η µν is the Minkowski metric tensor, and a summation overrepeated indices is assumed. It is easy to calculate the Christoffel symbols in theapproximation where all the components of the vector x are much less than R ′ :Γ µ,νρ = − x µ η νρ /R ′ . Then a direct calculation shows that in the nonrelativisticapproximation the equation of motion for a single particle is the same as in Eq. (12)if R ′ = R . Another way to show that Eq. (12) is compatible with GR follows. Theknown result of GR is that if the metric is stationary and differs slightly from theMinkowskian one then in the nonrelativistic approximation the curved space-time canbe effectively described by a gravitational potential ϕ ( r ) = ( g ( r ) − / c . We nowexpress x in Eq. (7) in terms of a new variable t as x = t + t / R ′ − t x / R ′ .Then the expression for the interval becomes ds = dt (1 − r /R ′ ) − d r − ( r d r /R ′ ) (14)Therefore, the metric becomes stationary and ϕ ( r ) = − r / R ′ in agreement with Eq.(12) if R ′ = R . The fact that in classical approximation the parameter R definingcontraction from quantum dS symmetry to quantum Poincare symmetry becomesequal the radius of dS space in GR does not mean that R can be always identifiedwith this radius because on quantum level the notion of background space does nothave the physical meaning.Consider now a system of two free classical bodies in GR. Let ( r i , a i )( i = 1 ,
2) be their radius vectors and accelerations, respectively. Then Eq. (12) isvalid for each particle if ( r , a ) is replaced by ( r i , a i ), respectively. Now if we definethe relative radius vector r = r − r and the relative acceleration a = a − a thenthey will satisfy the same Eq. (12) which shows that the dS antigravity is repulsive.Let us now consider a system of two free bodies in the framework of therepresentation of the dS algebra. The particles are described by the variables P j and r j ( j = 1 , P = P + P , q = ( m P − m P ) / ( m + m ) R = ( m r + m r ) / ( m + m ) , r = r − r (15)Then, as follows from Eq. (10), in the nonrelativistic approximation the two-particlequantities P , E and L are given by P = P , E = M + P M − M c R R , L = − M R (16)where M = M ( q , r ) = m + m + H nr ( r , q ) , H nr ( r , q ) = q m − m c r R (17)11nd m is the reduced two-particle mass. Here the operator M acts in the space offunctions χ ( q ) such that R | χ ( q ) | d q < ∞ and r acts in this space as r = i∂/∂ q .It now follows from Eq. (9) that M has the meaning of the two-bodymass and therefore M ( q , r ) is the internal two-body Hamiltonian. Then, by analogywith the derivation of Eq. (12), it can be shown from the Hamilton equations that insemiclassical approximation the relative acceleration is given by the same expression(12) but now a is the relative acceleration and r is the relative radius vector. ( c, ¯ h, s ) funda-mental? In the literature the notion of the c ¯ hG cube of physical theories is sometimes used. Themeaning is that any relativistic theory should contain c , any quantum theory shouldcontain ¯ h and any gravitation theory should contain G . The more fundamental atheory is the greater number of those parameters it contains. In particular, relativisticquantum theory of gravity is treated as the most fundamental because it containsall the three parameters c , ¯ h and G while nonrelativistic classical theory withoutgravitation is the least fundamental because it contains none of those parameters.However, as noted in Sec. 1, since the nature of gravity is not clear yet,the quantity G is not fundamental. Also, as follows from the above discussion, theset of parameters ( c, ¯ h, R ) is more adequate than the set ( c, ¯ h, G ) and, in contrast tousual statements, the situation is the opposite: relativistic theory should not contain c , quantum theory should not contain ¯ h and dS or AdS theories should not contain R . Those three parameters are needed only for transitions from more general theoriesto less general ones. The most general dS and AdS quantum theories do not containdimensionful quantities at all while the least general nonrelativistic classical theorycontains three dimensionful quantities ( kg, m, s ).Indeed, as noted above, the angular momenta are dimensionless but forhistorical reasons people want to measure them in kg · m /s and that’s why thequantity ¯ h arises. Analogously, in particle theory, velocities are dimensionless butsince people want to measure them in m/s the quantity c comes into play. However,when a system under consideration is strongly quantum and Poincare symmetry doesnot work, neither the quantities ( kg, m, s ) nor the quantities ( c, ¯ h, R ) have a physicalmeaning and those quantities are not present in the theory at all.Nevertheless, physicists usually believe that the quantities ( c, ¯ h ) are fun-damental and do not change over time. This belief has been implemented in themodern system of units where the basic quantities are not ( kg, m, s ) but ( c, ¯ h, s ) andit is postulated that the quantities ( c, ¯ h ) do not change over time. By definition, it ispostulated that from now on c = 299792458 m/s and ¯ h = 1 . · − kg · m /s .As a consequence, now the quantities ( kg, m ) are not basic ones because they can beexpressed in terms of ( c, ¯ h, s ) while the second remains the basic quantity.The motivation for the modern system of units is based on several factsof quantum theory based on Poincare invariance. First of all, since it is postulated12hat the photon is massless, its speed c is always the same for any photons with anyenergies. Another postulate is that for any photon its energy is always proportional toits frequency and the coefficient of proportionality always equals ¯ h . Let us note thatthis terminology might be misleading for the following reasons. Since the photon isthe massless elementary particle, it is characterized only by energy, momentum, spinand helicity and is not characterized by frequency and wave length. The latter areonly classical notions characterizing a classical electromagnetic wave containing manyphotons. Quantum theory predicts the energy distribution of photons in blackbodyradiation but experimentally we cannot follow individual photons and can measureonly the frequency distribution in the radiation. Then the theory agrees with experi-ment if formally the photon with the energy E is attributed the frequency ω = E/ ¯ h .A typical theoretical justification is that the photon wave function con-tains exp ( − iEt/ ¯ h ). Note that the problem of time is one of the most fundamentalproblems of standard quantum theory. The usual point of view is that there is notime operator and time is simply a classical parameter such that the wave functioncontains the factor exp ( − iEt/ ¯ h ) (see e.g. the discussion in Ref. [10]). This agreeswith the facts that in classical approximation the Schr¨odinger equation becomes theHamilton-Jacobi equations and that with such a dependence of the wave functionon time one can describe trajectories of photons in classical approximation (see e.g.the discussion in Ref. [11]). At the same time, there is no experimental proof thatthis dependence takes place on quantum level and, as noted in Sec. 2, fundamentalquantum theories proceed from the Heisenberg S-matrix program that in quantumtheory one can describe only transitions of states from the infinite past when t → −∞ to the distant future when t → + ∞ .Consider now the description of the photon in AdS and dS quantum theo-ries but first let us make the following remarks. Dyson’s paper [7] explaining why deSitter symmetries are more fundamental than Poincare symmetry appeared in 1972.One might think that this paper should be a good stimulus for physicists to gen-eralize fundamental quantum theories from Poincare invariant theories to de Sitterinvariant ones. However, no big steps in this direction have been made. One of thearguments is that since R is much greater than sizes of elementary particles thende Sitter corrections will be negligible. However, as explained below, in de Sitterand Poincare invariant theories the structures of IRs describing elementary particlesare considerably different. The analogy is that relativistic theory cannot be treatedsimply as nonrelativistic one with the cutoff c for velocities: as a consequence of thefact that c is finite the theories considerably differ in several aspects.Consider first IRs of the AdS algebra. For the first time the constructionof such IRs has been given by Evans [12] (see also Ref. [13]). As noted above,the AdS analog of the energy operator is M . A common feature of the AdS andPoincare cases is that there are IRs containing either only positive or only negativeenergies and the latter can be associated with antiparticles. In the AdS case theminimum value of the energy in IRs with positive energies can be treated as the massby analogy with the Poincare case. However, the essential difference between theAdS and Poincare cases is that the IRs in the former belong to the discrete series13f IRs and the photon mass cannot be exactly zero. In the AdS analog of masslessPoincare IR, the AdS mass equals m AdS = 1 + s where s is the spin. From the pointof view of Poincare symmetry, this is an extremely small quantity since the Poincaremass m equals m AdS /R . However, since m AdS is not exactly zero, there is a nonzeroprobability that the photon can be even in the rest state, i.e. its speed will be zero. Ingeneral, the speed of the photon can be in the range [0 , c does not have the fundamental meaning asin Poincare theory.In addition, as a consequence of the fact that AdS analogs of massless IRscontain the rest state, particles described by such IRs necessarily have two valuesof helicity, not one as in Poincare case. Note that in Poincare theory the photon isnot described by an IR of the pure Poincare algebra because it is massless and hastwo helicities: it is described by an IR of the Poincare algebra with spatial reflectionadded. For example, if in Poincare theory neutrino is treated as massless then in AdStheory it will have two helicities. However, if its AdS energy is much greater thanits AdS mass then the probability to have the second helicity is very small (but notzero). Consider now IRs of the dS algebra. They have been constructed in Ref.[8] by using the results on IRs of the dS group in the excellent book by Mensky [9].Here the situation drastically differs from the Poincare case because there are no IRswith only positive and negative energies: one IR necessarily contains both positive andnegative energies. As argued in Ref. [8], this implies that a particle and its antiparticlebelong to the same IR. This means that the very notion of particles and antiparticlesis only approximate and the conservation of electric charge and baryon and leptonquantum numbers also is only approximate because transitions particle ↔ artiparticleare not strongly prohibited. One IR of the dS algebra splits into two IRs of thePoincare algebra in the formal limit R → ∞ . IRs of the dS algebra are characterizedby the dS mass m dS such that m dS cannot be zero and the relation between dS andPoincare masses is again m dS = Rm . So even the photon is necessarily massive. InPoincare theory there is a discussion what is the upper bound for the photon massand different authors give the values in the range (10 − ev, − ev ). These seemto be extremely tiny quantities but even if m = 10 − ev and R is of the order of10 meters as usually accepted than m dS is of the order of 10 , i.e. a very largequantity. We conclude that in the dS case the quantity c cannot have a fundamentalmeaning, as well as in the AdS case.Consider now whether the quantity ¯ h can be treated as fundamental inde Sitter invariant theories. For such theories it is not even clear how to defineenergy and time such that the wave function depends on time as exp ( − iEt/ ¯ h ) evenin classical approximation. For example, in the dS case the operator M is on thesame footing as the operators M j ( j = 1 , ,
3) and only in Poincare limit it becomesthe energy operator. In Sec. 3 this problem is solved in the approximation 1 /R butin the general case the problem remains open.While in the modern system of units, c and ¯ h are treated as exact quan-14ities, the second is treated only as an approximate quantity. Since there is no timeoperator, it is not even legitimate to say whether time should be discrete or contin-uous. The second is defined as the duration of 9192631770 periods of the radiationcorresponding to the transition between the two hyperfine levels of the ground state ofthe cesium 133 atom. The physical quantity describing the transition is the transitionenergy ∆ E , and the frequency of the radiation is defined as ∆ E/ ¯ h . The transition en-ergy cannot be the exact quantity because the width of the transition energies cannotbe zero. In addition, the transition energy depends on gravitational and electromag-netic fields and on other phenomena. In view of all those phenomena the accuracyof one second given in the literature is in the range (10 − s, − s ), and the betteraccuracy cannot be obtained in principle. In summary, ”continuous time” is a partof classical notion of space-time continuum and makes no sense beyond this notion.In modern inflationary models the inflation period of the Universe lasted inthe range (10 − s, − s ) after the Big Bang. In addition to the fact that such timescannot be measured in principle, at this stage of the Universe there were no nucleiand atoms and so it is unclear whether time can be defined at all. The philosophyof classical physics is that any physical quantity can be measured with any desiredaccuracy. However the state of the Universe at that time could not be classical,and in quantum theory the definition of any physical quantity is a description howthis quantity can be measured, at least in principle. In quantum theory it is notacceptable to say that ”in fact” some quantity exists but cannot be measured. So inour opinion, description of the inflationary period by times (10 − s, − s ) has nophysical meaning.In summary, since in dS and AdS theories all physical quantities are di-mensionless, here no system of units is needed. Dimensionful quantities ( c, ¯ h, s ) aremeaningful only at special conditions when Poincare symmetry works with a highaccuracy and measurements can be performed in a system which is classical (i.e.non-quantum) with a high accuracy. In Sec. 2 we have proved that dS and AdS quantum theories are more fundamentalthan Poincare quantum theory. The transition from the former to the latter is de-scribed by contraction R → ∞ . The parameter R has nothing to do with the radiusof background space and must be finite. As shown in Sec. 3, as a consequence ofthose results, in semiclassical approximation two free bodies have a relative accelera-tion defined by the same expression as in GR if the the radius of dS space equals R and Λ = 3 /R . This result has been obtained without using dS space, its metric, con-nection etc.: it is simply a consequence of dS quantum mechanics of two free bodiesand the calculation does not involve any geometry . In our opinion this result is moreimportant than the result of GR because any classical result should be a consequenceof quantum theory in semiclassical approximation.Therefore, as follows from basic principles of quantum theory, correct de-15cription of nature in GR implies that Λ must be nonzero, and the problem why Λ isas is does not arise. This has nothing to do with gravity, existence or nonexistenceof dark energy and with the problem whether or not empty space-time should benecessarily flat.As argued in Sec. 4, since all physical quantities in dS and AdS quantumtheories are dimensionless, here no system of units is needed. The quantities ( c, ¯ h, s )(which are basic ones in the modern system of units) have a physical meaning onlyat special conditions when Poincare symmetry works with a high accuracy and mea-surements are performed in a system which is classical (i.e. non-quantum) with ahigh accuracy. In particular, statements that the inflationary stage of the Universelasted in the range (10 − s, − s ) have no physical meaning. Acknowledgement:
I am grateful to Bernard Bakker and Vladimir Kar-manov for numerous important discussions.
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