Quantum theory of redshift in de Sitter expanding universe
aa r X i v : . [ g r- q c ] F e b Quantum theory of redshift in de Sitterexpanding universe
Ion I. Cot˘aescu
West University of Timi¸soara,V. Pˆarvan Ave. 4, RO-1900 Timi¸soara, Romania
February 25, 2021
Abstract
The quantum theory of the Maxwell free field in Coulombgauge on the de Sitter expanding universe is completed with thetechnical elements needed for building a coherent quantum theoryof redshift. Paying a special attention to the conserved observ-ables and defining the projection operator selecting the detectedmomenta it is shown that the expectation values of the energies ofthe emitted and detected photons comply with the Lemaˆıtre ruleof Hubble’s law. Moreover, the quantum corrections to the disper-sions of the principal observables and new uncertainty relationsare derived.Pacs: 04.62.+v
Keywords: de Sitter spacetime; Maxwell field; Coulomb gauge; canon-ical quantization; one-particle operators; quantum redshift; dispersions;uncertainty relations. 1
Introduction
An important source of empirical data in the observational astrophysics isthe light emitted by different cosmic objects whose redshifts encapsulateinformation about the cosmic expansion and possible peculiar velocity ofthe observed objects [1]. For understanding these two contributions onecombined so far the Lemaˆıtre rule [2, 3] of Hubble’s law [4], describing thecosmological effect [5], with the usual theory of the Doppler effect of spe-cial relativity. Recently we proposed an improvement of this approachreplacing the special relativity with our de Sitter relativity [6, 7]. Weobtained thus a rdshift formula having a new term combining the cosmo-logical and kinetic contributions in a non-trivial manner [8]. Moreover,we related the black hole shadow and redshift for the Schwarzschild [9]and Reissner-Nordstrom [10] black holes moving freely in the de Sitterexpanding universe.The next step might be the quantum theory of redshift but this wasnever considered because of the real or presumed difficulties in construct-ing the quantum theory of light in curved backgrounds. In fact there isnothing much in it since we have already the classical and quantum theoryof the free Maxwell field on the de Sitter expanding universe [11] includ-ing the de Sitter QED in the first order of perturbations [12]. Therefore,we may build a quantum theory of redshift exploiting this frameworkand solving the specific difficulties of this problem. We devote this pa-per to this goal constructing step by step the redshift theory from theclassical level up to a new quantum approach able to reveal the quantumcorrections and the uncertainty relations related to this effect.The cornerstone here is the conformal covariance of the Maxwell equa-tions in Coulomb gauge allowing us to take over the all the results ofspecial relativity in the co-moving local charts (called here frames) withconformal coordinates of the de Sitter expanding universe [11]. In theseframes the quantisation of the Maxwell field can be done in canonicalmanner as in special relativity. The difference is that there is a richeralgebra of isometry generators giving rise to more conserved quantitiesof the classical theory that become conserved one-particle operators af-ter quantization [11]. Of a special interest is the energy operator whichdoes not commute with the components of the conserved momentumgenerating new uncertainty relations [13].On the other hand, the conformal coordinates are different from thephysical ones which are of the Painlev´e type [14] being related to theconformal ones through coordinate transformations depending on time.However, in the quantum theory these transformations change the time2volution picture as we have shown in Refs. [15–17]. Therefore, foravoiding this difficulty, we restrict ourselves to the conformal coordinatessetting the initial conditions at the time t when the scale factor a ( t ) = 1and the physical and conformal space coordinates coincide. Under suchcircumstances the physical effects may be studied by using exclusivelythe conserved one-particle operators.In addition, we pay attention to a pair of sensitive technical prob-lems which are crucial in our approach. The first one is related to themomentum dependent phase of the plane wave solutions of the Maxwellequations which determines the form of the energy operator. Here weset for the first time the phase which guarantees the correct flat limitof our theory. The second problem is related to the detector measuringthe redshift which has to select only the radiation emitted by a remotesource. For doing so we assume that the detector filters the momentain a desired domain of the momentum space whose associated projectionoperator helps us to derive the expectation values and dispersions of themeasured observables.We obtain thus a complete quantum theory of the redshift observedin the radiation emitted by a remote source without peculiar velocity.We show that the expectations values of the energies of the emittedand detected photons comply with the Lemaˆıtre rule of Hubble’s lawwhile the dispersions get new quantum corrections involved in a set ofnew uncertainty relations. However, it is less probable to identify suchcorrections in the astrophysical observations since these are very small inour actual expanding universe. Nevertheless, the methods developed hereare important as these can be adapted to any spatially flat Friedman-Lemaˆıtre-Robertson-Walker (FLRW) space-time including those studiedin the cosmology of early universe.We start in the second section with a brief review of the de Sittergeometry defining the conserved quantities and introducing the conformaland physical coordinates. In the next section we revisit the classicaltheory of redshift pointing out the role of the conserved quantities inderiving the Lemaˆıtre equation. The fourth section is devoted to theclassical theory of the Maxwell field showing how by fixing a convenientphase we assure the correct flat limit and deriving the principal conservedquantities which become one-particle operators after the quantizationperformed in the next section. In the last part of this section we showhow the wave packets can be measured by choosing a suitable projectionoperator for selecting the momenta of the modes which contribute tothe expectation values of the principal conserved observables. The nextsection is devoted to the quantum redshift for which we derive the new3uantum corrections and uncertainty relations. Finally we present someconcluding remarks.As here we develop a quantum approach, we introduce a special no-tation denoting by ω H = q Λ3 c the de Sitter Hubble constant (frequency)since H is reserved for the energy or Hamiltonian operator [13]. More-over, the Hubble time t H = ω H and the Hubble length l H = cω H will havethe same form in the natural Planck units with c = ~ = G = 1 we usehere. The manifold in which we would like to study the quantum theory ofredshift is the expanding portion M + of the de Sitter space-time M knownas the de Sitter expanding universe. The manifold M may be defined asthe hyperboloid of radius 1 /ω H in the five-dimensional flat spacetime( M , η ) of coordinates z A (labelled by the indices A, B, ... = 0 , , , , η = diag(1 , − , − , − , − { x } of coordinates x µ (of natural indices α, µ, ν, ... = 0 , , ,
3) canbe introduced on M or M + giving the set of functions z A ( x ) which solvethe hyperboloid equation, η AB z A ( x ) z B ( x ) = − ω H , (1)where ω H is the Hubble de Sitter constant (frequency) in our notations.In what follows we consider the co-moving frames with two sets oflocal coordinates, the conformal pseudo-Euclidean ones, { t c , x c } , and thephysical de Sitter-Painlev´e coordinates, { t, x } . The conformal time t c and the conformal Cartesian spaces coordinates x ic ( i, j, k, ... = 1 , , z ( x c ) = − ω H t c (cid:2) − ω H ( t c − x c ) (cid:3) ,z i ( x c ) = − ω H t x ic , (2) z ( x c ) = − ω H t c (cid:2) ω H ( t c − x c ) (cid:3) , written with the vector notation, x c = ( x c , x c , x c ) ∈ R ⊂ M . Theseframes cover the expanding portion M + for t c ∈ ( −∞ ,
0) and x c ∈ R M − is covered by similar charts with t c >
0. Inboth these cases we have the same conformal flat line element, ds = η AB dz A ( x c ) dz B ( x c )= g µν ( x c ) dx µc dx νc = 1 ω H t c (cid:0) dt c − d x c · d x c (cid:1) . (3)Here we restrict ourselves to the expanding portion M + which is a plau-sible model of our expanding universe.The de Sitter-Painlev´e coordinates { t, x } on the expanding portioncan be introduced directly by substituting t c = − ω H e − ω H t , x c = x e − ω H t , (4)where t ∈ ( −∞ , ∞ ) is the proper or cosmic time while x i are the ’physical’Cartesian space coordinates. Then the line element reads ds = g µν ( x ) dx µ dx ν = (1 − ω H x ) dt + 2 ω H x · d x dt − d x · d x . (5)Notice that this chart is useful in applications since in the flat limit (when ω H →
0) its coordinates become just the Cartesian ones of the Minkowskispacetime. In the charts with combined coordinates { t, x c } the metrictakes the FLRW form ds = dt − a ( t ) d x c · d x c , a ( t ) = e ω H t , (6)where a ( t ) is the scale factor of the expanding portion which can berewritten in the conformal chart, a ( t c ) ≡ a [ t ( t c )] = − ω H t c , (7)as a function defined for t c < SO (1 ,
4) of theembedding manifold M that leave invariant its metric and implicitly Eq.(1). Therefore, given a system of coordinates defined by the functions z = z ( x ), each transformation g ∈ SO (1 ,
4) defines an isometry, x → x ′ = φ g ( x ), derived from the system of equations z [ φ g ( x )] = g z ( x ) . (8)The frames related through such isometries play the role of the inertialframes similar of special relativity. Each isometry x → x ′ = φ g ( ξ ) ( x ),5epending on the group parameter ξ , gives rise to an associated Killingvector, k = ∂ ξ φ ξ | ξ =0 . In a canonical parametrization of the SO (1 , ξ AB = − ξ BA , any infinitesi-mal isometry, φ µ g ( ξ ) ( x ) = x µ + ξ AB k µ ( AB ) ( x ) + .... (9)depend on the components k ( AB ) µ = z A ∂ µ z B − z B ∂ µ z A , z A = η AB z B , (10)of the Killing vectors associated to the parameters ξ AB .The classical conserved quantities along geodesics have the generalform K ( AB ) ( x, P ) = ω H k ( AB ) µ p µ where the four-momentum components p µ = dx µ ( s ) dλ are the derivatives with respect to the afine parameter λ which satisfies ds = mdλ such that g µν p µ p ν = m . The conserved quan-tities with physical meaning [13] are, the energy E = ω H k (04) µ p µ , the an-gular momentum components, L i = ε ijk k ( jk ) µ p µ , and the components K i = k (0 i ) µ p µ and R i = k ( i µ p µ of two vectors related to the conservedmomentum P and its associated dual momentum Q as, [13] P = − ω H ( R + K ) , Q = ω H ( K − R ) . (11)satisfying the identity E − ω H L − P · Q = m , (12)corresponding to the first Casimir invariant of the so (1 ,
4) algebra [13].In the flat limit, ω H → − ω H t c →
1, we have Q → P such that thisidentity becomes just the usual null mass-shell condition E − P = m of special relativity. The conserved quantities K ( AB ) transform as a five-dimensional skew-symmetric tensor under the SO (1 ,
4) transformationsgenerating the de Sitter isometries.The simple isometries we meet here are the translations of parameters d = ( d , d , d ) that act as t c = t ′ c x c = x ′ c + d → t = t ′ x = x ′ + d e ω H t . (13)Hereby we can derive the Killing vector of components k j = 0 , k ij = ∂x ic ∂d j (cid:12)(cid:12)(cid:12)(cid:12) d =0 = δ ij = ω H (cid:0) k i (0 ,j ) + k i ( i, (cid:1) , (14)where k i ( AB ) = g ij ( x c ) k ( AB ) j result from Eq. (10). This give rise tothe conserved momentum of the classical approach and to the momen-tum operator of the quantum theory. We shall see in the next sectionthat these isometries transform the energy, angular momentum and dualmomentum but preserving the conserved momentum.6 Null geodesics and redshift
We consider now the null geodesics of the photons (with m = 0) denotingthe conserved quantities along these geodesics with capital letters. Inthe conformal chart the four-momentum components, denoted now by k µc = dx µc dλ , satisfy the identity g µν ( x c ) k µc k νc = a ( t c ) (cid:2) k c ( t c ) − k c ( t c ) (cid:3) = 0 , (15)resulted from the line element (3). In addition, we may consider thecomponents P i = − g jk ( x c ) k ji dx kc dλ = a ( t c ) dx jc dλ , (16)of the conserved momentum P = n P P ( P = | P | ). Then by using theprime integrals (15) and (16) we derive the energy and covariant momen-tum in this frame as k c ( t c ) = dt c dλ = Pa ( t c ) = ω H t c P , (17) k c ( t c ) = d x c dλ = P a ( t c ) = ω H t c P . (18)The null geodesic results simply as [18] x c ( t c ) = x c + n P ( t c − t c ) , (19)concluding that this geodesic is determined completely by the unit vector n P and the initial condition x c ( t c ) = x c .The corresponding physical quantities measured in the chart { t, x } ,may be obtained by substituting the physical coordinates according toEq. (4). Thus we find k ( t ) = dtdλ = P e − ω H t , (20) k ( t ) = d x dλ = P e − ω H t + ω H x ( t ) P e − ω H t , (21)which represent the measured energy and covariant momentum in thepoint [ t, ~x ( t )] of the null geodesic [18] x ( t ) = x e ω H ( t − t ) + n P e ω H ( t − t ) − ω H , (22)7hich is passing through the space point x ( t ) = x at the initial time t . The conserved quantities on this geodesic can calculated at any timeas E = e − ω H t [ P + ω H x ( t ) · P ] , (23) L = x ( t ) ∧ P e − ω H t , (24) Q = 2 ω H x ( t ) E + P e − ω H t [1 − ω H x ( t ) ] , (25)observing that these satisfy the identity (12) for m = 0.The momentum defined by Eq. (21) can be split as k ( t ) = ˆ k ( t ) + ¯ k ( t )where ˆ k ( t ) = P e − ω H t , ¯ k ( t ) = ω H x ( t ) k ( t ) , (26)are the peculiar and respectively recessional momenta we have definedrecently [19]. The prime integral derived from the line element (5) givesthe familiar identity k ( t ) − ˆ k ( t ) = 0 , (27)which is just the mass-shell condition of special relativity satisfied by theenergy and peculiar momentum along the null geodesics.Now we may analyse how two different observers measure a photonmoving on a null geodesic which is passing through the origins O and O ′ of their proper co-moving frames { t, x } O and { t, x ′ } O ′ . We assume thatthe photon is emitted in O ′ at the initial time t when the origin O ′ istranslated with respect to O as x ( t ) = x ′ ( t ) + d e ω H t , (28)where the translation parameter d = n d of the isometry (13) has thedirection OO ′ given by the unit vector n . If we know that the photon isemitted in x ′ ( t ) = 0 with the momentum k = − n k and energy k = k we ask which are the energy and momentum of this photon measured inthe origin O at the final time t f when the particle reach this point. Forsolving this problem we look first for the conserved momentum that isthe same in the points O ′ and O , P ′ = P = k e ω M t → P = k e ω M t , n P = − n , (29)since this is invariant under translations being associated to their gener-ators. Furthermore, we observe that the choice of the initial time t c = − ω H → t = 0 , (30)8hen a = 1 and, consequently, the conformal and physical space coor-dinates coincide . This simplifies the calculations allowing us to find thequantities measured by the observers O and O ′ at this moment derivedfrom Eqs. (23)-(25).The results are presented in the next table where we introduce intu-itive notations for the initial, E i , and final, E f , photon energies whichare the physical quantities involved in redshift.frame { t, x ′ } O ′ frame { t, x } O initial condition x ′ (0) = 0 x (0) = d conserved energy E ′ ≡ E i = k E ≡ E f = k (1 − ω H d )conserved momentum P ′ = k P = k angular momentum L ′ = 0 L = 0dual momentum Q ′ = k Q = k (1 − ω H d ) We observe that the translation (28) change the energy and the compo-nents of the adjoint momentum but preserving the invariant (12) with m = 0. Indeed, in the frame { t, x ′ } O ′ we have E i = k = P ′ · Q ′ . Inthe frame { t, x } O the photon arrives in x = 0 at the conformal time t cf = d − ω H corresponding to the cosmic time t f = − ω H ln( − ω H t cf ) = − ω H ln(1 − ω H d ) . (31)Hereby we can deduce that O measures the final peculiar momentum,ˆ k ( t f ) = k (1 − ω H d ) , (32)resulted from Eqs. (26) for t = t f . Obviously, this satisfies the mass-shell condition E f = ˆ k ( t f ) = P · Q . Hereby we conclude that P and Q are conserved quantities that cannot be measured directly but completeeach other for closing the identity (12). The only measurable quantitiesremain thus the energy, peculiar momentum and angular momentum.These results allow us to recover the Lemaˆıtre expression of Hubble’slaw giving the redshift z as11 + z = E f E i = 1 − ω H d = 1 − dl H , (33)and the physical distance observer-source at the time t f , d f = d − ω H t cf = d − ω H d , (34)9hich was increasing because of the space expansion during the photonpropagation. Thus we revisited the redshift in the particular case when O ′ does not have a peculiar velocity. The general result for arbitrarypeculiar velocity of O ′ was derived recently [8, 9].These results that hold at the level of the geometric optics in deSitter background are incomplete since at this level we neglect the wavebehaviour and polarization that may be studied considering the classicaland quantum theory of the Maxwell field. Let us consider first the classical approach denoting by A the electromag-netic potential of the Maxwell field minimally coupled to the de Sittergravity, whose action in an arbitrary chart { x } of ( M, g ) reads S [ A ] = Z d x √ g L = − Z d x √ g F µν F µν , (35)where g = | det( g µν ) | and F µν = ∂ µ A ν − ∂ ν A µ is the field strength. Fromthis action one derives the field equations ∂ ν ( √ g g να g µβ F αβ ) = 0 , (36)which are invariant under conformal transformations, g µν → g ′ µν = Ω g µν and A µ → A ′ µ = A µ A µ → A ′ µ = Ω − A µ . (37)The canonical variables A µ must obey, in addition, the Lorentz condition ∂ µ ( √ g g µν A ν ) = 0 , (38)which is no longer conformally invariant since ∂ µ ( p g ′ g ′ µν A ′ ν ) = ∂ µ ( √ g g µν A ν ) + √ gA µ ∂ µ Ω . (39)However, we may get over this inconvenience imposing the Coulombgauge, A = 0, in the conformal chart { t c , x c } since then the secondterm of Eq. (39) does not contribute. Under such circumstances we can write the solutions of the Maxwellequations in Coulomb gauge, ( ∂ t c − ∆ c ) A i = 0, and Lorentz condition,10 x ic A i = 0, taking over all the well-known results of special relativity.Thus we may write the plane wave solutions A i ( x c ) = A (+) i ( x c ) + A ( − ) i ( x c )= Z d k X λ h e i ( n k , λ ) ˆ f k ( x c )ˆ a ( k , λ ) + [ e i ( n k , λ ) ˆ f k ( x c )] ∗ ˆ a ∗ ( k , λ ) i , (40)in terms of wave functions in momentum representation, ˆ a ( k , λ ), polar-ization vectors, e i ( n k , λ ), and fundamental solutions of the d’Alambertequation, ˆ f k ( x c ) = 1(2 π ) / √ k e iδ ( k ) − ikt c + i k · x c , (41)where k = k n k is the momentum vector with k = | k | .The momentum-dependent phase δ ( k ) is introduced in order to assurethe correct flat limit of the plane wave solutions when ω H →
0. Accordingto Eqs. (4), we see that the entire phase of the function (41) behaves as iδ ( k ) − ik (cid:18) − ω H + t (cid:19) + i x c · k + O ( ω H ) , (42)having a pole in ω H = 0. For removing this singularity we must imposethe condition lim ω H → (cid:18) δ ( k ) + kω H (cid:19) = 0 , (43)giving the correct flat limit of special relativity,lim ω H → e iδ ( k ) − ikt c + i k · x c = e − ikt + i k · x . (44)For avoiding some difficulties related to this explicit phase, it is conve-nient to redefine f k ( x c ) = e − iδ ( k ) ˆ f k ( x c ) = 1(2 π ) / √ k e − ikt c + i k · x c , (45) a ( k , λ ) = e iδ ( k ) ˆ a ( k , λ ) , (46)substituting ˆ f k ( x c )ˆ a ( k , λ ) = f k ( x c ) a ( k , λ ) in Eq. (40). In Ref. [11]we neglected the problem of this phase which is solved here for the firsttime. This guarantees the correct flat limit as in the case of the rest framevacua we defined recently for the massive Dirac [20], Klein-Gordon [21]and Proca [22] showing that only these vacua assure correct flat limits[23]. 11he functions f k ( x ) are assumed to be of positive frequencies whilethose of negative frequencies are f k ( x ) ∗ . These solutions satisfy the or-thonormalization relations [11]( f k , f k ′ ) = − ( f ∗ k , f ∗ k ′ ) = δ ( k − k ′ ) , (47)( f k , f ∗ k ′ ) = 0 , (48)and the completeness condition i Z d k f ∗ k ( t c , x c ) ↔ ∂ t c f k ( t c , x ′ ) = δ ( x c − x ′ c ) , (49)with respect to the Hermitian form( f, g ) = i Z d x c f ∗ ( t c , x c ) ↔ ∂ t c g ( t c , x c ) . (50)where we denote f ↔ ∂ g = f ∂g − g∂f .The polarization vectors e ( n k , λ ) in Coulomb gauge must be orthog-onal to the momentum direction, k · e ( n k , λ ) = 0 , (51)for any polarization λ = ±
1. We remind the reader that the polariza-tion can be defined in different manners independent of the form of thescalar solutions f k . In general, the polarization vectors have c-numbercomponents which must satisfy [24] e ( n k , λ ) · e ( n k , λ ′ ) ∗ = δ λλ ′ , (52) X λ e i ( n k , λ ) e j ( n k , λ ) ∗ = δ ij − k i k j k . (53)Here we restrict ourselves to consider only the circular polarization forwhich the supplementary condition e ( n k , λ ) ∗ ∧ e ( n k , λ ) = iλ n k is re-quested.We obtained thus mode expansions in terms of transverse plane wavesof given momentum and helicity. The functions w i ( k ,λ ) = e i ( n k , λ ) f k , areof positive frequencies while those of negative frequencies are w ∗ i ( k ,λ ) . Wesay that these sets of fundamental solutions of the Maxwell equationdefine the momentum-helicity basis. Note that an energy-helicity basiscan also be defined [11]. 12 .2 Conserved quantities The Maxwell theory is invariant under SO (1 ,
4) isometries such thatunder a given isometry , x → x ′ = φ g ( ξ ) ( x ), depending on the groupparameter ξ , the vector field A transforms as A → A ′ = T ξ A , according tothe operator-valued representation ξ → T ξ of the isometry group definedby the well-known rule ∂φ νξ ( x ) ∂x µ ( T ξ A ) ν [ φ ( x )] = A µ ( x ) . (54)The corresponding generator, X K = i ∂ ξ T ξ | ξ =0 , has the action( X k A ) µ = − i ( k ν A µ ; ν + k ν ; µ A ν ) . (55)where k is the Killing vector associated to ξ . In a canonical parametriza-tion of the SO (1 ,
4) group we have the correspondence ξ AB → k ( AB ) → X k ( AB ) ≡ X ( AB ) , (56)which means that the generators X ( AB ) form a basis of the vector rep-resentation of the so (4 ,
1) algebra carried by the space of the vectorpotential, A . This algebra yields the principal observables, i. e. theenergy operator ˆ H = ωX (05) , the components of the momentum, ˆ P i = − ω H ( X ( i + X (0 i ) ), dual momentum, ˆ Q i = ω H ( X (0 i ) − X ( i ), and totalangular momentum, ˆ J i = ε ijk X ( jk ) ( i, j, ... = 1 , , A = 0) which are useful in applications. The energy andmomentum operators do not have spin parts, acting as [11]ˆ H A j = − iω H (cid:18) t c ∂∂t c + x ic ∂∂x ic + 1 (cid:19) A j , (57)ˆ P i A j = − i ∂∂x ic A j , (58)while the action of the total angular momentum readsˆ J i A j = ˆ L i A j − iε ijk A k , (59)where ˆ L = x c × P is the usual angular momentum operator. In addition,we define the Pauli-Lubanski (or helicity) operator ˆ W = ˆ P · ˆ J whoseaction depends only on the spin parts,ˆ W A i = ε ijk ∂∂x jc A k . (60)13his operator will define the polarization in the canonical basis of the so (3) algebra as in special relativity [11].However, the dual momentum has the most complicated action,ˆ Q i A j = − iω H (cid:20) x ic (cid:18) t c ∂∂t c + x kc ∂∂x kc (cid:19) + ( t c − x c ) ∂∂x ic (cid:21) A j − iω H (cid:0) δ ij x c · A + x ic A j − x jc A i (cid:1) , (61)we present here for the first time.We constructed thus the basis { ˆ H, ˆ P i , ˆ Q i , ˆ J i } of the vector represen-tation of the so (1 ,
4) algebra whose commutation rules are h ˆ H, ˆ J i i = 0 (62) h ˆ H, ˆ P i i = iω H ˆ P i , h ˆ H, ˆ Q i i = − iω H ˆ Q i , (63) h ˆ J i , ˆ P j i = iε ijk ˆ P k , h ˆ J i , ˆ Q j i = iε ijk ˆ Q k , (64) h ˆ Q i , ˆ Q j i = 0 , h ˆ P i , ˆ P j i = 0 , (65) h ˆ Q i , ˆ P j i = 2 iω H δ ij ˆ H + 2 iω H ε ijk ˆ J k , (66)from which we deduce h ˆ P i , ˆ W i = 0 , h ˆ H, ˆ W i = iω H ˆ W , (67)understanding that the maximal set of commuting operators we may useis { ˆ P i , ˆ W } . The first Casimir operator of this algebra has the form [13] C = ˆ H + 3 iω H ˆ H − ˆ Q · ˆ P − ω H ˆ J · ˆ J (68)giving the supplemental equation C A i = 0 which in the Coulomb gaugeis just the d’Alambert one.In what concerns the structure of the so (1 ,
4) algebra we observe thatthere are two Abelian sub-algebras generated by the momentum com-ponents, { ˆ P i } , and respectively by those of the dual momentum, { ˆ Q i } .Another specific feature is that the energy operators does not commutewith the momentum components as in special relativity. Regarding ournotations we must specify that the upper or lower positions of the spaceindices do not have here a meaning but we denoted the components of themomentum and dual momentum with upper indices since in the flat limitthese become ˆ Q i → ˆ P i → − i∂ i , i. e. the contravariant space componentsof the momentum operator with respect to the Minkowski metric.14he conserved quantities of our Lagrangian theory are related to theoperators of the so (1 ,
4) algebra via Noether’s theorem. From the action(35) we deduce that in Coulomb’s gauge the conserved quantities can bederived by using the Hermitian form (50) as [11] X → C [ X ] = 12 δ ij ( A i , XA j ) ∀ X ∈ so (1 , , (69)This integral can be expressed in terms of electric and magnetic com-ponents of the field strength [12] or as a mode integral in momentumrepresentation. Thus we can conclude that we outlined here a coherentde Sitter electrodynamics which has a correct flat limit. The next step is the quantization we may perform in canonical manneras in special relativity exploiting the global conformal invariance of thetheory in Coulomb gauge. There are many delicate problems that can beavoided if we restrict ourselves to the conserved operators in the Heisen-berg picture assuming that the quantum states are defined at the initialtime (30) when a (0) = 1 and the conserved and peculiar momenta coin-cide. Then we have to focus only on the conserved operators calculated atthis moment which are enough for analysing the quantum redshift. How-ever, this method is not suitable for studying other dynamic operatorsas, for example, the coordinate operator and that of peculiar momentum. We assume that the wave functions a of the field (40) becomes fieldoperators (with a ∗ → a † ) [24] such that the potentials (40) become fieldoperators denoted by A i . We assume that the field operators fulfill thestandard commutation relations in the momentum-helicity basis fromwhich the non-vanishing ones are[ a ( k , λ ) , a † ( k ′ , λ ′ )] = δ λλ ′ δ ( k − k ′ ) . (70)Then the Hermitian field A = A † is correctly quantized according to the canonical rule[ A i ( t c , x c ) , π j ( t c , x ′ c )] = [ A i ( t c , x ) , ∂ t c A j ( t c , x ′ c )] = i δ trij ( x c − x ′ c ) , (71)where π j = √ g δ L δ ( ∂ t c A j ) = ∂ t c A j (72)15s the momentum density in Coulomb gauge ( A = 0) and δ trij ( x c ) = 1(2 π ) Z d q (cid:18) δ ij − q i q j q (cid:19) e i q · x c (73)is the well-known transverse δ -function [24] arising from Eq. (53).As in special relativity we consider a unique vacuum state, | i , of theFock such that a ( k , λ ) | i = 0 , h | a † ( k , λ ) = 0 . (74)The sectors with a given number of particles may be constructed usingthe standard methods for obtaining the generalized momentum-helicitybasis of the the Fock space.The one-particle operators corresponding to the conserved quantities(69) can be calculated in Coulomb gauge as [11] X = 12 δ ij : ( A i , X A j ) (75)respecting the normal ordering of the operator products [24]. The obviousalgebraic properties[ X , A i ( x )] = − X A i ( x ) , (76)[ X , Y ] = 12 δ ij : ( A i , [ X, Y ] A j ) (77)are due to the canonical quantization adopted here. However, there aremany other conserved operators which do not have corresponding differ-ential operators but can be defined directly as mode expansions. Thesimplest example is the operator of the number of particles, N = Z d k X λ a † ( k , λ ) a ( k , λ ) , (78)The conserved one-particle operators whose mode expansions can bederived easily are the components of the conserved momentum operator, P l = 12 δ ij : (cid:16) A i , ˆ P l A j (cid:17) = Z d k k l X λ a † ( k , λ ) a ( k , λ ) , (79)and the Pauli-Lubanski operator, W = 12 δ ij : (cid:16) A i , ˆ W A j (cid:17) = Z d k k X λ λ a † ( k , λ ) a ( k , λ ) , (80)16hich commutes with the momentum componenta, [ P i , W ] = 0. Themomentum-helicity basis | i , | k , λ i = a † ( k , λ ) | i , | k , λ ; k ′ , λ ′ i = a † ( k , λ ) a † ( k ′ , λ ′ ) | i , ... (81)is formed by the eigenvectors of the set of commuting operators {W , P i } corresponding to the discrete polarizations, λ, λ + λ ′ , ... ∈ Z and mo-menta k , k + k ′ , ... of continuous spectrum R k .The problem of the energy operator is more delicate but can be solvedresorting to the identity( ˆ Hf k )( x ) = − iω H (cid:18) k i ∂ k i + 32 (cid:19) f k ( x ) , (82)satisfied by the functions (45). Then, after a few manipulation and ap-plying the Green theorem, we obtain the final result [11], H = 12 δ ij : (cid:16) A i , ˆ H A j (cid:17) := iω H Z d k k i X λ a † ( k , λ ) ↔ ∂ k i a ( k , λ ) . (83)Hereby we see that the form of the energy operator is strongly depen-dent on the phase of the operators a ( k , λ ) defined by Eq. (46). Thisis in accordance with the similar property of the energy operator thatholds for the Klein-Gordon, Dirac or Proca free fields on this background.This general behaviour is due to the space expansion giving the depen-dence of the energy operator on the translations that change the phase.Nevertheless, this behaviour does not change the commutation relations (cid:2) H , P i (cid:3) = i ω H P i , (84)[ H , W ] = i ω H W , (85)which are independent on the phase δ ( k ) as it results from Eqs. (63) and(77). A simple model which prevents us from complicated calculations is thatof the one-particle wave-packets. In our Heisenberg picture these aregiven by the time-independent one-particle states, | α i = Z d k X λ α λ ( k ) a † ( k , λ ) | i (86)17efined by the square integrable functions in momentum representation α λ ( k ) which must satisfy the normalization condition h α | α i = Z d k X λ | α λ ( k ) | = 1 . (87)The corresponding ’wave functions’ A [ α ] i ( x ) = h |A i ( x ) | α i = Z d k X λ e i ( n k , λ ) f k ( x ) α λ ( k ) , (88)are known as wave-packets. These are useful auxiliary functions relatedto those of the momentum representation through the inversion relations α λ ( k ) = δ ij e i ( n k , λ ) ∗ ( f k , A [ α ] j ) . (89)Moreover, the expectation values of the one-particle operators (75) in thestate | α i can be calculated simply as h α |X | α i = δ ij ( A [ α ] i , XA [ α ] j ) , (90)avoiding the tedious algebra of field operators.Once the wave-packet is prepared this evolves causally until an idealapparatus measures some parameters. More specific, this apparatus canmeasure all the eigenvalues of the operators W and P i which are diagonalin the momentum-helicity basis. In an experiment we can set this appa-ratus to select only the momenta included in a desired domain ∆ ⊂ R k by using a suitable projection operator Λ ∆ = Λ † ∆ (satisfying Λ = Λ ∆ )that can be represented asΛ ∆ = | ih | + Z ∆ d k X λ a † ( k , λ ) | ih | a ( k , λ ) + ... , (91)where the integral is restricted to the domain ∆. During the experimentthis operator filters only the momenta k ∈ ∆ transforming the stateof the system as | α i → Λ ∆ | α i . Then the expectation value hX i of aone-particle operator X can be calculated as [26] hX i = h α | Λ ∆ X | α ih α | Λ ∆ | α i (92)taking into account that our one-particle operators commute with Λ ∆ .The quantity h α | Λ ∆ | α i = Z ∆ d k X λ | α λ ( k ) | ≤ , (93)18ives the probability P ∆ = |h α | Λ ∆ | α i| of measuring any momentum k ∈ ∆. Obviously, when we can measure the whole continuous spectrum,∆ = R k , then Λ ∆ → , P ∆ = 1 and hX i = h α |X | α i .Furthermore, bearing in mind the role of the phase factor in Eq. (46)we assume that the functions α λ have the general form α λ ( k ) = e iδ ( k ) ˆ α λ ( k ) , (94)where ˆ α λ = ˆ α ∗ λ are real valued functions. Then we can derive the ex-pectation values of the operators which are diagonal in the basis (81) byusing the rule (90) as h ( P i ) n i = 1 h α | Λ ∆ | α i Z ∆ d k ( k i ) n X λ ˆ α λ ( k ) , (95) hW n i = 1 h α | Λ ∆ | α i Z ∆ d k X λ λ n ˆ α λ ( k ) , (96)For the energy operator which is not diagonal in this basis we may applythe same formula but using, in addition, the identity (82) and the Greentheorem which helps us to write hHi = 1 h α | Λ ∆ | α i ( iω H Z ∆ d k k i X λ α ∗ λ ( k ) ↔ ∂ k i α λ ( k ) ) = − h α | Λ ∆ | α i ( ω H Z ∆ d k (cid:2) k i ∂ k i δ ( k ) (cid:3) X λ ˆ α λ ( k ) ) , (97)since ˆ α ∗ λ ↔ ∂ ˆ α λ = 0 as these are real valued functions. Note that theoperator k i ∂ k i in momentum space is in fact a radial operator such thatthis does not affect the polarization vectors which depend only on theunit vector of the momentum direction, n k . Finally, by using again Eq.(90) we may write h α |H | α i = δ ij (cid:16) ˆ HA [ α ] i , ˆ HA [ α ] j (cid:17) , (98)which helps us to obtain the useful formula hH i = ω H h α | Λ ∆ | α i (Z ∆ d k (cid:2) k i ∂ k i δ ( k ) (cid:3) X λ ˆ α λ ( k ) + Z ∆ d k X λ (cid:20)(cid:18) k i ∂ k i + 32 (cid:19) ˆ α λ ( k ) (cid:21) ) , (99)we need in the next application. 19 Quantum redshift
Let us come back now to the problem of two translated observers, O ′ and O , preparing and respectively measuring a photon state | α i . We assumethat these observers use the same global ideal apparatus represented bythe operator algebra A ′ ∪ A formed by two sub-algebras including the fieldoperators and the conserved ones for which we use the self-explanatorynotations O ′ : A ′ ( x ′ c ) , ... H ′ , P ′ i , Q ′ i , L ′ i ... ∈ A ′ , (100) O : A ( x c ) , ... H , P i , Q i , L i .... ∈ A , (101)The state | α i is prepared at the initial time (30) when the observersare translated each other with the position vector d . The translationgenerators P ′ i = P i , which are the same in both the above sub-algebras,define the translation operator T ( d ) = exp (cid:0) id i P i (cid:1) , (102)which transforms these sub-algebras, T ( d ) : A ′ → A , such that any op-erator X ′ ∈ A ′ is transformed into X = T ( d ) X ′ T ( d ) † ∈ A . Particularly,the energy operator is translated as H = T ( d ) H ′ T ( d ) † = H ′ + ω H d i P i , (103)according to the commutation rule (84). In what follows, we simplify thegeometry be choosing an orthogonal frame { e , e , e } such that d = d e . The observer O ′ prepares the state | α i in his proper co-moving framewhere the principal parameters are the expectation values of energy, E ′ ,momentum components, P ′ i , and polarization, W ′ . These quantitiescan be calculated by using the simple rule (90) taking into account thatthe packet is defined by the functions (94) whose phase must be fixedaccording to the condition (43). The simplest choice is δ ( k ) = − kω H , (104)since then the expectation value of the energy reads E ′ ≡ h α |H ′ | α i = δ ij (cid:16) A [ α ] i , ˆ HA [ α ] j (cid:17) = Z d k k X λ ˆ α λ ( k ) (105)20s it results from Eq. (97) for ∆ = R k . The other expectation values canbe derived simpler by using Eq. (90) as P ′ i ≡ h α |P i | α i = δ ij (cid:16) A [ α ] i , ˆ P i A [ α ] j (cid:17) = Z d k k i X λ ˆ α λ ( k ) , (106) W ′ ≡ h α |W i | α i = δ ij (cid:16) A [ α ] i , ˆ W A [ α ] j (cid:17) = Z d k X λ λ ˆ α λ ( k ) , (107)since these do not depend on the phase (104). We observe that in ourframework with the phase (104) all these expectation values have thesame forms as in Minkowski space-time.However, these quantities are not accessible to the observer O whichfocuses on the observables, H ′ , H , P i and W selecting only the photonscoming from the source O ′ , whose momenta are parallel with e . Thismeans that the domain of momenta measured by O is∆ = (cid:26) k (cid:12)(cid:12)(cid:12)(cid:12) − ∆ k ≤ k ≤ ∆ k , − ∆ k ≤ k ≤ ∆ k , k < (cid:27) (108)where ∆ k is a small quantity. Then we may evaluate the integrals over∆ as Z ∆ d kF ( k ) = Z ∆ k − ∆ k dk Z ∆ k − ∆ k dk Z −∞ dk F ( k , k , k ) ≃ (∆ k ) Z ∞ dkF (0 , , − k ) , (109)according to the mean value theorem.Now we come back to our intuitive notations of Sec. 3 of the expecta-tion values of the initial, E i ≡ hH ′ i , and final, E f ≡ hHi , energies relatedto the conserved momentum of components P i ≡ hP i i that can be ob-served by O . These expectation values have to be calculated according toEq. (92) with the state | α i defined by the functions (94) with the phase(104) and the projection operator Λ ∆ corresponding to the domain (108).First we find that h α | Λ ∆ | α i = Z ∆ d k X λ ˆ α λ ( k ) = (∆ k ) κ , (110)where κ = Z ∞ dk X λ ˆ α λ (0 , , − k ) . (111)21urthermore, we calculate the expectation values defined by Eq. (95) for n = 1 P ≡ hP i = − κ Z ∞ dk k X λ ˆ α λ (0 , , − k ) , (112) hP i = hP i = 0 , (113)which do not depend on the phase (104). For the energy operators thesituation is different since their expectation values depend on this phaseas in Eq. (97) which allows us to write E i ≡ hH ′ i = 1 κ Z ∞ dk k X λ ˆ α λ (0 , , − k ) = − P , (114)deriving the expectation value of Eq. (103) as E f ≡ hHi = (1 − ω H d ) E i , (115)recovering thus the Lemaˆıtre form of Hubble’s law (33). Note that thisresult can be derived in a different manner observing that h α |H| α i = h α | T ( d ) H ′ T ( d ) † | α i = h ˜ α |H ′ | ˜ α i , (116)where now the translated state | ˜ α i is given by the functions (94) in whichwe must substitute δ ( k ) → ˜ δ ( k ) = − kω H − k · d . (117)With this new phase Eq. (97) gives just the result (115). We mustspecify that now the initial energy E i observed by O is different from E ′ measured by O ′ in contrast with the classical approach where these twoquantities coincide (as in the table of Sec. 3).In our experiment we select only the momenta oriented along e suchthat the polarizations vectors e ( e , ±
1) = √ ( e ∓ i e ) are in the plane { e , e } . The expectation value of the Pauli-Lubanski operator, W ≡ hWi = 1 κ Z ∞ dk (cid:2) ˆ α (0 , , − k ) − ˆ α − (0 , , − k ) (cid:3) (118)suggests us to introduce the polarization angle 0 ≤ θ ( k ) ≤ π such thatˆ α (0 , , k ) = cos θ ( k ) α ( k ) , ˆ α − (0 , , k ) = sin θ ( k ) α ( k ) (119)where the new function α ( k ) satisfies R ∞ dk α ( k ) = κ . In the particularcase when θ is a constant independent on k we have W = cos 2 θ .22 .2 Dispersions and uncertainty The next step is to study the dispersions of the observables measured by O applying the well-known ruledisp X = (∆ X ) = hX i − hX i . (120)We observe first that the operators H ′ and H commute alike with P i and W as in Eqs. (84) and (85) but do not commute with each other since[ H ′ , H ] = iω H d i P i = iω H d P . (121)Therefore, from the above equation and Eq. (84) we obtain the uncer-tainty relations disp H ′ disp P i ≥ ω H (cid:12)(cid:12) hP i i (cid:12)(cid:12) , (122)disp H disp P i ≥ ω H (cid:12)(cid:12) hP i i (cid:12)(cid:12) , (123)disp H ′ disp H ≥ ω H d (cid:12)(cid:12) hP i (cid:12)(cid:12) , (124)For i = 1 , P = disp P = 0 withoutviolating the uncertainty relations but along the third axis the relation(122) is non-trivial since P ≡ |hP i| 6 = 0. On the other hand, from Eq.(99) with the phase (104) we obtain hH ′ i = h ( P ) i + ω H χ . (125)The last terms of the above equations represents the quantum correctionswhich are proportional with the dimensionless quantity χ = 1 κ Z ∞ dk X λ (cid:20)(cid:18) k∂ k + 32 (cid:19) ˆ α λ (0 , , − k ) (cid:21) , (126)resulted from the last term of Eq. (99). This is generated by the de Sittergravity and depends exclusively on the form of the functions ˆ α λ . For de-riving the dispersion of the operator H we calculate first the expectationvalue (99) with the new phase (117) obtaining the identity hH i = (1 − ω H d ) h ( P ) i + ω H χ . (127)Finally, from Eqs. (125) and (127) combined with Eqs. (114) and re-spectively (115) we finddisp E i ≡ disp H ′ = disp P + ω H χ , (128)disp E f ≡ disp H = (1 − ω H d ) disp P + ω H χ , (129)23here we denote disp P ≡ disp P .Now we come back to the uncertainty relations (122) and (123) byusing Eqs. (128) and (129) for deriving the inequalitiesdisp E i,f (cid:0) disp E i,f − ω H χ (cid:1) ≥ ω H E i,f , (130)disp P (cid:0) disp P + ω H χ (cid:1) ≥ ω H P , (131)from which we deducedisp E i,f ≥ ω H (cid:16)q E i,f + ω H χ + ω H χ (cid:17) = ω H (cid:18) E i,f + ω H χ + 12 E i,f ω H χ (cid:19) + O ( ω H χ ) , (132)disp P ≥ ω H (cid:18)q P + ω H χ − ω H χ (cid:19) = ω H (cid:18) P − ω H χ + 12 P ω H χ (cid:19) + O ( ω H χ ) , (133)relating thus the dispersions to the corresponding expectation values.More interesting is Eq. (124) as depending explicitly on the distance d between O and O ′ . This can be rewritten in our new notations as,disp E i disp E f ≥ ω H d P , (134)and can be seen as the starting point for deriving new inequalities de-pending on d by using Eqs. (128) and (129). Our preliminary calculationsindicate that these are more complicated requiring a special analyticalan numerical study which will be performed elsewhere.Finally, let us analyse the dispersion of the Pauli-Lubanski operatorin the simple case when the polarization angle θ is independent on k .We have seen that then the expectation value has the form W ≡ hWi =cos 2 θ . Moreover, from Eqs. (96) we obtain hW i = 1 → disp W = sin θ , (135)while from Eq. (85) we derive the uncertainty relationdisp H disp W ≥ ω H |hWi| (136)giving the restriction tan 2 θ ≥ ω H ∆ E f , (137)preventing one from measuring total polarizations, i. e. θ = 0 for λ = 1or θ = π for λ = −
1. 24
Concluding remarks
We presented the complete classical and quantum theory of the Maxwellfield minimally coupled to the gravity of the de Sitter expanding universefocusing on the principal effect due to the space expansion, namely theredshift for which we derived the quantum corrections and the principaluncertainty relations.In the actual expanding universe the quantum corrections and thelimits of the uncertainty relations are extremely small since the actualvalue of ω H (or ~ ω H in SI units) is of the order 10 − eV such that itis less probable to be identified in astronomical observations. Moreover,the limitation predicted by the inequality (137) in an ideal universe istoo small to be separated from other polarization effects produced by thecosmic dust and plasma.However, these results are interesting as coming from the first com-plete and coherent classical and quantum theory of the Maxwell fieldcoupled to the gravity of an expanding universe. The methods devel-oped here can be applied to any spatially flat FLRW expanding universeincluding the actual models of early universe.On the other hand, the method of regularization of the momentumdependent phase plays the same role as the rest frame vacuum of themassive particles assuring the correct flat limit of the Maxwell field. Thuswe obtain a coherent quantum theory on the de Sitter expanding universein which we may apply the perturbation methods of the traditional theoryin Minkowski space-time. For this reason we hope that our approach willopen the door to a large field of applications not only in astrophysics andcosmology but even in particle physics. References [1] E. R. Harrison,
Cosmology: The Science of the Universe (New York:Cambridge Univ. Press, 1981).[2] G. E. Lemaˆıtre,
Ann. Soc. Sci. de Bruxelles (1927) 49.[3] G. E. Lemaˆıtre,
MNRAS (1931) 483.[4] E. Hubble, Proc. Nat. Acad. Sci. (1929) 168.[5] E. Harrison, Astrophys. J. (1993) 28.[6] I. I. Cot˘aescu,
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Phys. Rev. D (2013) 044016.[13] I. I. Cot˘aescu, GRG (2011) 1639.[14] P. Painleve, C. R. Acad. Sci. (Paris) (1921) 677.[15] I. I. Cot˘aescu,
Mod. Phys. Lett. A (2007) 2965.[16] I. I. Cot˘aescu, C. Crucean and A. Pop, Int. J. Mod. Phys. A (2008) 2563.[17] I. I. Cot˘aescu, Int. J. Mod. Phys. A (2020) 2030019.[18] I. I. Cot˘aescu, Mod. Phys. Lett. A (2017) 1750223.[19] I. I. Cot˘aescu, arXiv:2102.03211.[20] I. I. Cot˘aescu, Eur. Phys. J. C (2019) 696.[21] I. I. Cot˘aescu, Eur. Phys. J. C (2020) 621.[22] I. I. Cot˘aescu, Eur. Phys. J. C (2020) 535.[23] I. I. Cot˘aescu, Chin. Phys. C (2021) 1.[24] S. Weinberg, The Quantum Theory of Fields (Univ. Press, Cam-bridge 1995).[25] I. I. Cot˘aescu,