Hydrogen atom in Palatini theories of gravity
aa r X i v : . [ g r- q c ] J un The Hydrogen atom in Palatini theories of gravity
Gonzalo J. Olmo ∗ Perimeter Institute, 31 Caroline St. N, Waterloo, ON N2L 2Y5, Canada (Dated: 23rd February, 2008)We study the effects that the gravitational interaction of f ( R ) theories of gravity in Palatiniformalism has on the stationary states of the Hydrogen atom. We show that the role of gravityin this system is very important for lagrangians f ( R ) with terms that grow at low curvatures,which have been proposed to explain the accelerated expansion rate of the universe. We find thatnew gravitationally induced terms in the atomic Hamiltonian generate a strong backreaction thatis incompatible with the very existence of bound states. In fact, in the 1 /R model, Hydrogendisintegrates in less than two hours. The universe that we observe is, therefore, incompatible withthat kind of gravitational interaction. Lagrangians with high curvature corrections do not lead tosuch instabilities. PACS numbers: 98.80.Es , 04.50.+h, 04.25.Nx
I. INTRODUCTION
The accelerated expansion rate of the universe [1] isone of the biggest puzzles that theoretical physics facesnowadays. Dark energy sources within the frameworkof General Relativity (GR) have been postulated as themissing element that could explain that phenomenon.On the other hand, modified theories of gravity havebeen proposed as an alternative to dark energy sources.Modified theories usually provide self-accelerated cosmicsolutions on purely geometrical grounds, making unnec-essary the introduction of unobserved exotic sources ofmatter-energy. The modification of the gravitationallaws is, however, a very delicate issue in which intuitionis not always a good guide. In fact, a modificationoriginally thought to affect the dynamics at largescales could end up having non-trivial effects at shorterscales. It is thus necessary to study the dynamics ofthe different gravitational theories in different regimesand identify their positive and negative aspects aimingat learning how to construct theories that exhibit thedesired properties. In this sense, theories of gravity inwhich the lagrangian is some function f ( R ) of the scalarcurvature R manifest many interesting properties andhave attracted much attention in the recent literature.The equations of motion of f ( R ) theories can bederived in two different ways depending on whether theconnection is seen as independent of the metric (Palatiniformalism) or as dependent of it (metric formalism). Inthe metric formalism, i.e., when the connection is theLevi-Civit`a connection of the metric, besides the metricone identifies an additional scalar degree of freedom,which turns the scalar curvature R into a dynamicalobject. The interaction range of this scalar field dependson the form of the lagrangian and can change dueto different reasons. When perturbation theory is ∗ Electronic address: [email protected] applicable, the first order approximation shows that, formodels of interest in the late-time cosmic acceleration,the interaction range changes driven by the cosmicexpansion[2]. The scalar field can be short-ranged forsome time (radiation and matter dominated eras) andthen turn into a long-ranged field, causing late-timecosmic acceleration. This type of lagrangian is ruled outby solar system experiments. Some f ( R ) lagrangians,however, cannot be treated perturbatively. Nonetheless,one can still define an effective mass or interactionrange for the scalar, which now depends on the localmatter density. The constraints on such lagrangians bylocal experiments have been discussed recently in theliterature[3].In this paper we will consider the other formulation of f ( R ) theories, namely, the Palatini formalism. Almostsurprisingly, allowing the connection to be determinedby the equations of motion does not introduce newdynamical degrees of freedom. In these theories, themetric turns out to be the only dynamical field, whichsatisfies second-order differential equations. As we willsee, the effect of the lagrangian f ( R ) is to change theway matter generates the metric by introducing on theright hand side of the field equations new matter termsthat depend on the trace of the energy-momentumtensor of the sources. In vacuum, the field equationsreduce (always and exactly) to those of GR with acosmological constant. For this reason, it has beenthought for some time that Palatini f ( R ) theories couldpass the solar system observational tests [4, 5] (seealso [6] for a discussion of this point). It has also beenshown that different choices of lagrangian f ( R ) are ableto accommodate several of the different cosmic eras ofthe standard cosmological model [7]. However, a morecareful analysis of the gravitational dynamics in thepresence of sources indicates that these theories mightbe in strong conflict with our understanding of themicroscopic [23] world [8, 9, 10] (the study of polytropicmatter configurations also points in this direction [11]).This aspect, together with the cosmological viabilityof different models, is of primary importance due tothe fact that almost all cosmological observations relayon the detection of electromagnetic radiation, which isintimately related to the quantum mechanical natureof atomic and molecular structure. In this work, weelaborate in this direction and study how the gravita-tional interaction in Palatini f ( R ) theories affects thenon-relativistic limit of the (one-particle) Dirac equa-tion. The analysis of various gravitationally-inducedcorrecting terms in the resulting Schrodinger-Pauliequation will provide us with solid arguments againstthe existence in the gravity lagrangian of correctingterms relevant at low cosmic curvatures. As we will see,the very low matter density (or curvature) scales thatcharacterize (infrared-corrected) modified lagrangianscan be reached near the zeros of the wavefunctions. Thiscauses a strong gravitational backreaction that makesunstable the stationary states of the Hydrogen atom. Inparticular, we find that the ground state disintegratesin a matter of hours. On the contrary, if the gravitylagrangian is modified by high curvature corrections, the backreaction effects are negligible and the atom remainsstable.The paper is organized as follows. In section II, we de-fine the action of f ( R ) theories in Palatini, derive the fieldequations, and discuss the metric generated by micro-scopic systems. We then introduce two illustrative mod-els, namely, f ( R ) = R + R R P and f ( R ) = R − µ R , whichwill help us compare the behavior of the metric whenthe GR action gets ultraviolet or infrared corrections, re-spectively. In section III we derive the non-relativisticlimit of Dirac’s equation starting from its curved space-time formulation. We then discuss the effects induced bythe modified Schrodinger-Pauli equation in the station-ary solutions of the Hydrogen atom. We conclude with asummary and discussion of the results obtained. In theAppendix we estimate the decay rate of the ground stateof the atom. II. THE THEORY
Let us begin by defining the action of Palatini theories S [ g, Γ , ψ m ] = 12 κ Z d x √− gf ( R ) + S m [ g µν , ψ m ] (1)Here f ( R ) is a function of R ≡ g µν R µν (Γ), with R µν (Γ)given by R µν (Γ) = − ∂ µ Γ λλν + ∂ λ Γ λµν + Γ λµρ Γ ρνλ − Γ λνρ Γ ρµλ where Γ λµν is the connection. The matter action S m de-pends on the matter fields ψ m , the metric g µν , whichdefines the line element ds = g µν dx µ dx ν , and its firstderivatives (Christoffel symbols). The matter action doesnot depend on the connection Γ λµν , which is seen as anindependent field appearing only in the gravitational ac-tion (this condition is not essential and can be relaxed atthe cost of introducing a non-vanishing torsion). Varying (1) with respect to the metric g µν we obtain f ′ ( R ) R µν (Γ) − f ( R ) g µν = κ T µν (2)where f ′ ( R ) ≡ df /dR . From this equation we see thatthe scalar R can be solved as an algebraic function of thetrace T . This follows from the trace of (2) f ′ ( R ) R − f ( R ) = κ T, (3)The solution to this algebraic equation will be denotedby R = R ( T ). The variation of (1) with respect to Γ λµν must vanish independently of (2) and gives ∇ ρ (cid:20) √− g (cid:18) δ ρλ f ′ g µν − δ µλ f ′ g ρν − δ νλ f ′ g µρ (cid:19)(cid:21) = 0 (4)where f ′ ≡ f ′ ( R [ T ]) is also a function of the matterterms. This equation leads toΓ λµν = t λρ ∂ µ t ρν + ∂ ν t ρµ − ∂ ρ t µν ) (5)where t µν ≡ φg µν , and φ ≡ f ′ ( R [ T ]) f ′ ( R [0]) is dimensionlessand normalized to unity outside of the sources ( T = 0).It is now useful to rewrite (2) adding and subtracting f ′ R ( T ) g µν ≡ f ′ t αβ R αβ (Γ) t µν to get f ′ G µν ( t ) = κ T µν − [ R f ′ − f ]2 φ t µν (6)where G µν ( t ) is the Einstein tensor associated to t µν .The equations of motion (6) for the auxiliary metric t µν are considerably simpler than those for g µν , R µν ( g ) − g µν R ( g ) = κ f ′ T µν − R f ′ − f f ′ g µν −− f ′ ) (cid:20) ∂ µ f ′ ∂ ν f ′ − g µν ( ∂f ′ ) (cid:21) ++ 1 f ′ [ ∇ µ ∇ ν f ′ − g µν (cid:3) f ′ ] , (7)because of the difficulty of dealing with the matterderivatives ∂f ′ ∼ ∂T and ∂ f ′ ∼ ( ∂T ) ∼ ( ∂ T ).Solving for t µν using the system (6) and then go-ing back to g µν via the conformal transformation g µν = φ ( T ) − t µν is a useful simplification that makesthe task of finding solutions much easier. Note thatthis fortunate circumstance is due to the fact that theconformal transformation completely cancels out thedisturbing derivatives of (7). Moreover, the conformalrelation between the two metrics puts forward the factthat the metric g µν receives two kinds of contributions:non-local contributions that result from integrationover the sources, which produce the term t µν , andlocal contributions due to φ ( T ), which depend on thelocal details of T at each space-time point. This localcontribution arises due to the independent characterof the connection and, to our knowledge, does notappear in any other metric theory of gravity, where theconnection is generally assumed to be metric compatible(Levi-Civit`a connection).The similarity between the field equations of GR and(6) suggests that for weak sources and reasonable choicesof lagrangian f ( R ) (those that lead to negligible cos-mological term R f ′ − f f ′ ), the right hand side of (6) issmall and, like in GR, t µν can be expressed as t µν ( x ) ≈ η µν + h µν ( x ), where | h µν ( x ) | ≪ h µν →
0. Onthe other hand, if our microscopic system is placed in anexternal gravitational field whose contribution to t µν isnot negligible, we can always take a coordinate system inwhich the metric at the boundaries of a box containingthe system (large as compared to the microscopic systembut small as compared to the range of variation of theexternal metric t µν ) becomes ∼ η µν . In both situations,the metric g µν becomes simply g µν ( x ) ≈ φ ( T ) − η µν (8)where φ ( T ) → auxiliarybox but might depart from unity within the sources,depending of the particular lagrangian chosen.If one trivializes the role of the local term φ ( T ), thenone finds that relative motion between particles is notvery much affected by the modified gravity lagrangian[12]. However, we find that the presence of φ ( T ) − infront of η µν is very important because the matter fieldsin (1) are coupled to g µν and, unlike in all other knownmetric theories of gravity, g µν only becomes locally η µν in regions where T = 0 exactly. Consequently, the φ ( T ) dependence of g µν induces new interactions andself-interactions between the matter fields [8], as willbe explained here in detail. Note that the presence ofthe (scalar) term φ ( T ) is physical and not a problemof choosing the wrong coordinate system, as criticizedin [13]. In fact, if one computes geometrical invariantssuch as [24] R αβγλ R αβγλ , various derivatives of φ ( T )appear and cannot be eliminated by choosing differentcoordinate systems because R αβγλ R αβγλ is a coordinateinvariant. Note also that the dependence of R αβγλ R αβγλ on the local energy-momentum distribution via φ ( T )tells us that the geometry might be subjected to mi-croscopic fluctuations driven by the fluctuations of T and modulated by the form of the gravity lagrangian[recall that φ ( T ) ≡ f ′ ( R [ T ]) /f ′ ( R [0])]. Therefore,lagrangians sensitive to low energy scales could lead tounnacceptable microscopic curvature fluctuations, whileothers could lead to more robust geometries which wouldonly fluctuate at very high energies. In this latter case,however, neglecting the contribution of h µν could notbe well justified (think for instance in the hypothetical production of black holes in particle accelerators). A. Two illustrative f ( R ) models. We will now study the behavior of the function φ ( T ) ≡ f ′ [ R ( T )] /f ′ [ R (0)] for two illustrative models.
1. Ultraviolet corrections: f ( R ) = R + R R P This model is characterized by a high energy/curvaturecorrection R /R P , where the subscript P stands for Planck scale . In this case, we find that R ( T ) = − κ T isthe same as in GR, and the function φ ( T ) is given by φ ( T ) = 1 − κ TR P (9)We thus see that only at very high matter/energy den-sities will the function φ ( T ) significatively depart fromunity.
2. Infrared corrections: f ( R ) = R − µ R This model was initially proposed in [14] within themetric formalism and is characterized by a low curvaturescale µ . In this case, we find that R ( T ) = − ( κ T + p ( κ T ) + 12 µ ) / | κ T | ≫ µ and tends to a constant R ∼ µ for | κ T | ≪ µ . Thefunction φ ( T ) is given by φ ( T ) = 1 − p /τ ] (10)Here τ ≡ − T /T c , T c ≡ µ /κ ≡ ρ µ , and ρ µ ∼ − g/cm represents the characteristiccosmic density scale of the theory, which triggers thecosmic speedup. It is easy to see that at high densities,as compared to ρ µ , φ ( T ) → /
4, whereas for ρ ≪ ρ µ we find φ ( T ) →
1. Note that we could have chosen thenormalization of φ ( T ) differently and in such a way thatat high densities φ ( T ) → φ ( T ) → /
3. In this latter case, however, the physicalmetric g µν would tend to η µν in vacuum. We findour first choice a more natural normalization (thougharbitrary anyway), since it makes g µν = η µν in vacuum. III. DIRAC EQUATION IN CURVED SPACE
It is well known [15, 16, 17] that the energy levels of aHydrogen atom falling freely in an external gravitationalfield (in GR) will be shifted in a very characteristic waydue to the interaction of the electron with the curvatureof the space-time. Though external fields in Palatini the-ories of gravity must also lead to this phenomenon, wewill focus here on a different aspect. We will study theeffect that the local energy-momentum densities have onthe non-relativistic limit of the Dirac equation due to thefactor φ ( T ) − appearing in (8).For the sake of clarity, let us briefly consider the differentcontributions that make up T , which generate the met-ric (8) seen by the system. The electromagnetic field,which is treated as classical, is traceless and, therefore,does not contribute to T . The atomic nucleus can bemodeled as point-like or as described by an extremely lo-calized wave-packet contributing with T N = − m N δ ǫ ( x ),where δ ǫ is some representation of the Dirac delta func-tion with spread ǫ centered at the origin, and m N is thenuclear mass. The motion of the electron is describedby the one-particle Dirac equation, which generalized tocurved space-time [15, 16, 17] can be derived from thefollowing action (the notation will be explained below) S m [ g µν , ψ ] = − Z d x √− g (cid:2) i ¯ ψλ µ D µ ψ − m ¯ ψψ (cid:3) (11)Upon variation of this action with respect to g µν one findsthe energy-momentum tensor associated to the electron,whose trace is given by [18] T e = − m ¯ ψψ (12)In summary, T = T N + T e = − m N δ ǫ ( x ) − m ¯ ψψ . A. Derivation of the non-relativistic limit
From the action (11), we can derive the curved space-time version of Dirac’s equation( iλ µ D µ − m ) ψ = 0 (13)Here λ µ = e µa γ a are the curved space Dirac matri-ces, which are related to the constant Dirac matrices { γ a , γ b } = 2 η ab by the vierbein e µa (recall that g µν = η ab e µa e νb ). The covariant derivative is given by D µ = ∂ µ + ieA µ + 12 w abµ Σ ab (14)with w abµ representing the spin connection, A µ is theelectromagnetic vector potential, and Σ ab = [ γ a , γ b ].Since, by construction, the matter action is not cou-pled to the connection Γ αµν , the spin connection w abµ must be defined in terms of the Christoffel symbols C λµν = g λρ ( ∂ µ g ρν + ∂ ν g ρµ − ∂ ρ g µν ) and the vierbein as w abµ = e aσ ∇ µ e σb = e aσ ( ∂ µ e σb + C σµλ e λb ). From (8) it iseasy to see that e aµ = φ − / δ aµ and e µa = φ / δ µa . After abit of algebra, (13) turns into[ iγ a ( ∂ a + ieA a − ∂ a Ω) − ˜ m ] ψ = 0 (15)where we have definedΩ ≡ (3 /
4) ln φ ( T ) (16)˜ m ≡ mφ − . (17) Even though (15) is not, in general, completely separabledue to the non-linearities introduced by the dependenceof T on ¯ ψψ , stationary solutions do exist. To find them,it is useful to write the equation in the form i∂ t ψ = Hψ [25] as follows i∂ t ψ = h ~α · ( ~p − e ~A + i~ ∇ Ω) + ( eA + i∂ t Ω) + ˜ mβ i ψ (18)Let us now focus on the positive energy solutions of thisequation. It is easy to see that taking ψ ( t, ~x ) = e − iEt ξ ( ~x )we have T e = − m ¯ ξξ , ∂ t Ω = 0, and (18) turns into Eξ = [ ~α · ~π + eA + ˜ mβ ] ξ (19)where we have used the shorthand notation ~π ≡ ( ~p − e ~A + i~ ∇ Ω). Denoting by η and χ the large and smallcomponents, respectively, of the Dirac spinor ξ = (cid:18) ηχ (cid:19) (20)we find the following relations χ = 1˜ m + E − eA ~σ · ~πη (21) Eη = (cid:20) ~σ · ~π m + E − eA ~σ · ~π + ˜ m + eA (cid:21) η (22) T e = − mη † (cid:20) I − ~σ · ~π †
1[ ˜ m + E − eA ] ~σ · ~π (cid:21) η (23)We will now proceed to compute the lowest-order non-relativistic limit. We first decompose the energy E in twoparts, E = m + E , where m is a constant of order ∼ m (to be discussed further below) and E ≪ m representsthe non-relativistic energy. We then expand assumingthat the rest mass is much larger than the kinetic andelectrostatic energies, ˜ m ∼ m ≫ |E − eA | , and retainterms only of order 1 /m . The above relations reduce to χ ≈ m + m ~σ · ~πη (24) E η ≈ (cid:20) m + m ( ~σ · ~π ) + ( ˜ m − m ) + eA (cid:21) η (25) T e ≈ − mη † (cid:2) I − O ( | ~π | /m ) (cid:3) η = − mη † η (26)The wavefunction of the electron is then identified with η , which to this order coincides with the positive energyFoldy-Wouthuysen bispinor [19]. From (26) we see thatthe non-linearities contained in ~π and ˜ m in (25) onlydepend on η . Expanding the operator ( ~σ · ~π ) we find E η = (cid:26) m + m [( ~p − e ~A ) − e~σ · ~B ] + eA (cid:27) η + (cid:26) m + m h i~σ ( ~ ∇ Ω × ~ ∇ ) − ie ( ~A · ~ ∇ Ω) (27)+ ~ ∇ Ω − | ~ ∇ Ω | + 2( ~ ∇ Ω · ~ ∇ ) i + ( ˜ m − m ) o η The first line of this equation is very similar to thewell-known non-relativistic Schrodinger-Pauli equa-tion (see (29) below). The only difference beingthe term 1 / ( ˜ m + m ). The second and third lines,however, represent completely new terms generatedby the Palatini gravitational interaction. When thegravity lagrangian is that of GR, φ ( T ) = 1, we recoverthe Schrodinger-Pauli equation if m is identified with m . IV. APPLICATION: THE HYDROGEN ATOM
To gain some insight on the role and properties of thevarious terms in (27), we will proceed as follows. We firstsolve (27) in the case of GR, f ( R ) = R, φ ( T ) = 1, whichis well known. Then we switch to a different gravitylagrangian (assuming that we have the ability to dothat) and study how the system reacts to that change.The reason for this is that in a general f ( R ) the metric issensitive to the local T µν via φ ( T ) − , and changes in themetric due to the matter distribution could react backon the matter equations. If the new interaction terms in(27) lead to small perturbations, then the initial wave-functions will be, roughly speaking, stable with perhapssmall corrections which could be computed using stan-dard approximation methods. If, on the contrary, theenergy associated to the gravitationally-induced terms islarge, that would mean that the original configuration isnot minimizing the modified Hamiltonian and, therefore,large modifications would be necessary to reach a newequilibrium configuration. Depending on the magnitudeof the reaction on the system, we could estimate whetherthe theory is ruled out or not.Let us first consider the f ( R ) model with ultravioletcorrections introduced in section II A 1. In this case, thefunction φ ( T ) = 1 − κ TR P can be expressed as φ ( T ) = 1 + 2[ ρ N ( x ) + ρ e ( x )] ρ P (28)where ρ N ( x ) = m N δ ǫ ( x ), ρ e ( x ) = mP e ( x ), P e ( x ) = η † ( x ) η ( x ) is the probability density, and ρ P ≡ R P /κ is a very high matter-density scale(Planck scale). Since the scale ρ P is much largerthan any density scale reachable by the electronwavefunction and even by the very peaked nu-clear wavefunctions ( ρ N /ρ P ∼ − ), we see thatΩ( T ) ≈ ρ N ( x )+ ρ e ( x ) ρ P and ˜ m ≈ m (1 − ρ N ( x )+ ρ e ( x ) ρ P )lead to strongly suppressed contributions (in fact, theyare much smaller than the corresponding Newtoniancorrections | h | = GM/c R N ∼ − ). Identifying m with the electron mass m , the leading order correctionsto the wave functions and the energy levels could becomputed by perturbation methods and would lead tovirtually unobservable effects. Let us now focus on the model with infrared correc-tions introduced in section II A 2. Expressing length unitsin terms of the Bohr radius ( a ∼ . · − m), wefind τ = ρ e ( x ) ρ µ = 10 P e ( x ), where we have intention-ally omitted the nuclear contribution (only relevant atthe origin) for simplicity. This expression for τ indicatesthat the electron reaches the characteristic cosmic den-sity, τ ∼
1, in regions where the probability density isnear P e ( x ) ∼ − . In ordinary applications, one wouldsay that the chance to find an electron in such regions isnegligible, that that region is empty. In our case, how-ever, that scale defines the transition between the highdensity ( τ ≫
1) and the low density ( τ ≪
1) regions.In regions of high density, we find that φ rapidly tendsto a constant, φ ∞ = 3 /
4, which leads to ˜ m = 2 m/ √ ~ ∇ Ω = 0. If we then identify m → √ m /
2, equation(27) reduces to the usual Schrodinger-Pauli equation E η = (cid:26) m [( ~p − e ~A ) − e~σ · ~B ] + eA (cid:27) η (29)This fact justifies the introduction of m above. Let usnow see what happens in regions of low density. In thoseregions, φ ( T ) tends to unity, ~ ∇ Ω = 0, and ˜ m → m as τ →
0. The mass factor dividing the kinetic term isnow a bit smaller ( m > m ) than in the high densityregion. But the mass difference ˜ m − m is not zero. Thisis a remarkable point, because ˜ m − m ≈ − . m isnegative and of order ∼ m , which represents a largecontribution to the Hamiltonian. To better understandthe effect of this term, it is useful to consider the groundstate, η (1 , , = e − r/a √ πa ⊗ | , s i , where | , s i represents anormalized constant bispinor. In this case, the transitionfrom the high density region to the low density regionoccurs at r ≈ a . In Fig.1 we have plotted the mostrepresentative potentials in dimensionless form V e = − x (30) V m = 2 m c a ~ h mφ − − m i (31) V Ω = (cid:20) x ∂ x Ω + ∂ x Ω − | ∂ x Ω | (cid:21) (32)where V e is the electrostatic potential generated by theproton, lengths are measured in units of the Bohr ra-dius, x = r/a , and energies in units of ~ m a ≈ . V Ω = ~ ∇ Ω − | ~ ∇ Ω | only contains the mostimportant contributions associated to Ω. In this case, V Ω represents a small transient perturbation. The massdifference V m , however, introduces a deep potential wellin the outermost parts of the atom that must have im-portant consequences for its stability. (Note that thiseffect is not an artifact of the non-relativistic approxi-mation, since it also occurs in the full relativistic theory(15)-(18) due to the density dependence of ˜ m ). In theinitial configuration of the atom, corresponding to GR,
22 24 26 28 30 32 x-0.4-0.3-0.2-0.10.10.2V @ x D Potentials for Η V_e + V_ W+ V_mV_e + V_ W V_e
Figure 1: Contribution of the different potentials in theground state. The solid line, which represents the sum ofall the potentials, tends to the constant value − . m (or − the wavefunction of the ground state is concentrated nearthe origin, where the attractive electric potential is morepowerful ( V e → −∞ ). As we switch on the 1 /R theory, adeep potential well of magnitude ∼ − . m appears inthe outer regions of the atom, where ρ e ( x ) . ρ µ , whichmakes the ground state unstable and triggers a flux ofprobability density (via quantum tunneling) to those re-gions. The half life of Hydrogen subject to this potentialcan be estimated using time dependent perturbation the-ory (see the Appendix) yielding τ ≡ ~ Γ ≈ · s (33)We thus see that the initial, stable configuration isdestroyed in a lapse of time much shorter than the age ofthe Universe, which is in clear conflict with experiments[26].Further evidence supporting the instability of theatom is found in the existence of zeros in the atomicwavefunctions in between regions of high density be-cause, obviously, before (and after) reaching ρ e ( x ) = 0the characteristic scale ρ e ( x ) ∼ ρ µ is crossed. The firstexcited state, η (2 , , = √ πa (1 − r a ) e − r/ a ⊗ | , s i ,has a zero at r = 2 a . The radial derivatives of φ ( T )at that point are very large and lead to very importantperturbations which overwhelmingly dominate overany other contribution (see Fig.2). The magnitude of V Ω = ~ ∇ Ω − | ~ ∇ Ω | at r = 2 a oscillates between 10 and − eV in an interval of only 2 · − a . Needlessto say that this configuration cannot be stable and thatstrong changes must take place in the wave function toreduce the energy of the system. Such changes shouldtend to reduce the magnitude of the density gradients( ~ ∇ Ω) to minimize the value of V Ω , which will likely leadto a rapid transition to the ground state, where V Ω issmall. One can easily verify that strong gradients ~ ∇ Ωalso appear at the zeros of all the η n, , wavefunctions, - - + - x-1.5 · -1 · -5 · · V @ x D V e + V W + V m for Η Figure 2: The different contributions in this plot are V e ∼ − V m ∼ − · , and V Ω ∼ ± . The y-axis is measured inunits of 13 . a . which generate large contributions V Ω in those regions.Furthermore, if one considers stationary states with l = 0, V Ω has important contributions not only at thezeros of the radial functions, but also at the zeros ofthe angular terms. Thus, the pathological behaviordescribed for the spherically symmetric modes getsworse for the l = 0 states. One thus expects the decayof these states into states with less structure (weakergradients) such as the ground state, which will laterdecay into the continuum. All this indicates that theexistence of bound states, with localized regions of highprobability density (where “high” means above the scale ρ µ ), are impossible in this theory because of the largegradient contributions V Ω and the deep potential well V m . V. SUMMARY AND DISCUSSION
In this work we have deepened into the effects thatthe matter-energy density dependence of the metric inPalatini f ( R ) theories has at microscopic scales. Inparticular, we have studied the effects that switchingfrom GR to a different gravitational interaction, such asthe f ( R ) = R + R /R P or f ( R ) = R − µ /R models, hason the stationary solutions of the Hydrogen atom. To doso, we started with the Dirac equation in curved spaceand computed its non-relativistic limit. Then we lookedat the contribution of the different new interactionterms appearing in the resulting (effective) Hamiltonian(27). We have found that the existence of bound statesin theories with infrared corrections is problematicfor several reasons. Firstly, due to the dependenceof the effective electron mass ˜ m = mφ − ( T ) on thematter-energy density T = − m ¯ ψψ , the effective massseen in the inner (high density) regions and the outer(low density) regions of the atom is not the same. Thisgenerates a potential well that triggers the tunneling ofprobability density from the inner parts to the outermostparts of the atom, which eventually disintegrates theatom. Secondly, wilder perturbations arise in thosepoints and directions in which the wavefunction haszeros. This is due to the contribution of terms like ~ ∇ Ωand | ~ ∇ Ω | when the characteristic scale ρ µ is crossed[27]. Minimizing the contribution of those terms wouldrequire a transition to states with less pronouncedgradients such as the ground state, which would latterdisintegrate into the continuum via tunneling.Though these instabilities have been discussed withinthe non-relativistic limit, we do not find any reason toattribute their existence to an artifact of this approxi-mation. In fact, the dependence of the mass on the localenergy density was already apparent in (15)-(18). Inaddition, derivatives of φ ( T ) appear in the term ∂ µ Ω( T ).Therefore, the relativistic description seems unable tocure the pathologies found in the non-relativistic limit.In addition, one can also check, by direct calculationof R αβγλ R αβγλ ∼ ( ∂ Ω) + . . . , that the space-timegeometry is strongly fluctuating and far from being flatin those regions where ρ e ∼ ρ µ .Our results are also likely to hold even in the case inwhich the spin connection in the matter action is keptindependent of the metric. In that case, the connectionhas a non-vanishing torsion, though the metric g µν (andhence the vierbein) is still conformally related to themetric t µν associated to the connection (see [20]), whichis the key to get terms of the form mφ − / and ∂ µ Ω.Though we have only analyzed in detail the infrared-corrected model f ( R ) = R − µ /R , the instabilities asso-ciated to the potential well ˜ m − m and the zeros of thewavefunction must be present in all gravity models sen-sitive to low curvature/energy-density scales. Since thematter, as we know it, would be unstable in those theo-ries, the cosmological models considered in that contextare empty of significance (see [8] for a list of references).On the contrary, models which introduce deviations fromGR at high curvatures, such as f ( R ) = R + R /R P , donot have any relevant effect on the atomic structure ifthe characteristic scale is sufficiently high. To reach andexcite the high energy-density scale one should deal withhighly localized wave-packets, which will surely requirethe consideration of quantum fields. The quantization ofthe matter fields then opens an exciting window to newphenomena. In fact, when ψ is seen as a quantum field,the function T appearing in (8) and (15)-(18) must beinterpreted as h T i , i.e., the quantum expectation valueof the operator ˆ T in a given state. The Hamiltonian ofthe theory then depends on the particular quantum stateunder consideration through the expectation value h T i .A direct consequence of this is that the time evolution ofthe states in the Hilbert space of the theory is nonlinear[21]. This highly non-trivial fact could be used to imposetight constraints on the form of the gravity lagrangian inPalatini theories via quantum experiments. In fact, webelieve that in order to guarantee the linear evolution ofquantum states, it could be necessary that the gravity lagrangian were exactly that of Hilbert-Einstein. Acknowledgments
The author thanks L.Parker, J. Navarro-Salas, andP.Singh for interesting discussions, R. Parentani forhis inquisitive criticisms on earlier versions of thismanuscript, and L. Smolin for his hospitality at thePerimeter Institute during the elaboration of this work.This work has been supported by Ministerio de Edu-caci´on y Ciencia (MEC) and by Perimeter Institute forTheoretical Physics. Research at Perimeter Institute issupported by the Government of Canada through Indus-try Canada and by the Province of Ontario through theMinistry of Research & Innovation.
Appendix
We briefly sketch here the computation of the halflife given in (33). Our calculation will be approximateand should provide a reasonable estimation of the or-der of magnitude of τ ≡ ~ / Γ. We will first assumethat the kinetic term − ~ ˜ m + m ∇ can be approximatedby − ~ m ∇ everywhere, even though ˜ m + m ≈ . m in the low-density regions (recall that ˜ m = m/ p φ ( T )becomes m when | T | → ∞ ). Secondly, we will neglectthe contribution of V Ω and will approximate V m ( r ) = m hq φ ( ∞ ) φ ( T ) − i by a step function of magnitude W S = m hq φ ( ∞ ) φ (0) − i ≈ − . m in the region r ≥ a andzero elsewhere (see Fig.1). The total potential (for l = 0)when the 1 /R interaction is turned on can thus be seenas V ( r ) = (cid:26) − Ze πǫ r if r ≤ a − . m if r > a (34)This way we have reduced our problem to that of aninitially stable bound state that becomes unstable anddecays into the continuum when the initial potential U ( r ) = − Ze πǫ r is transformed into V ( r ). This simplifiedscenario captures the essential features of our problem.The decay rate can be estimated using time-dependentperturbation theory. A simple and compact expressionfor the width Γ of a quasistationary state (which initiallywas a true bound state) is given by the following formula(see [22] for details)Γ = 4 ~ α mk | ψ ( R ) χ k ( R ) | (35)In our case, α = 1 /a , k = p m (0 . m c − | ǫ | ) / ~ , ǫ = − . eV , ψ ( R ) represents the radial part of thepartial wave expansion of the ground state evaluatedat R = 26 a , ψ ( R ) = √ a Ra e − R/a , χ k ( r ) representsthe outgoing continuum mode, and χ k ( R ) = a k √ a k ) .Putting these numbers in (35), we find (33), which im-plies that the ground state of Hydrogen in the Palatini version of the 1 /R theory would disintegrate in less thantwo hours. [1] J. L. Tonry et al., Astrophys. J. , 1 (2003); R. A.Knop et al.,
Astrophys. J. , 102 (2003).[2] G.J. Olmo,
Phys.Rev. D Phys.Rev. D
73 (2006) 064029,gr-qc/0507039; I. Navarro and K. Van Acoleyen, JCAP0702 (2007) 022,gr-qc/0611127; W.Hu and I.Sawicki,arXiv:0705.1158; T.Faulkner, M.Tegmark, E.F.Bunn,and Y.Mao, astro-ph/0612569.[4] D.N. Vollick,
Phys. Rev. D
68, 063510 (2003),astro-ph/0306630.[5] X. Meng, P. Wang,
Gen.Rel.Grav. ,1947,(2004);M.L. Ruggiero and L.Orio, JCAP
Gen. Rel. Grav. ,1891 (2005).[6] T.P. Sotiriou, Gen.Rel.Grav. (2006) 1407-1417; Ph.D.Thesis, arXiv:0710.4438 [gr-qc].[7] S.Fay, R. Tavakol and S. Tsujikawa, Phys.Rev. D
75, 063509 (2007); T.P.Sotiriou,
Phys.Rev. D
73 (2006)063515, gr-qc/0509029.[8] G.J. Olmo,
Phys. Rev. Lett. , 061101 (2007).[9] G.J. Olmo, Phys. Rev. Lett. , 261102 (2005).[10] E.E.Flanagan, Phys.Rev.Lett. , 071101 (2004).[11] E.Barausse, T.P.Sotiriou, and J.C.Miller, gr-qc/0703132(see also arXiv:0712.1141 and arXiv:0801.4852).[12] B.Li, D.F. Motta, and D.Shaw, arXiv:0801.0603.[13] K.Kainulainen et al., arXiv:0704.2729 .[14] S.M. Carroll, V. Duvvuri, M. Trodden and M.S. Turner, Phys. Rev. D
70, 043528 (2004)[15] L. Parker,
Phys. Rev. Lett. , 1559 (1980).[16] L.Parker, Phys.Rev. , 1922 (1980)[17] L.Parker and L.O. Pimentel, Phys.Rev. , 3180 (1982)[18] Birrel, N.D. and Davies, P.C.W. (1982). Quantum fieldsin curved space , Cambridge University Press, Cambridge,England.[19] L.L.Foldy and S.A. Wouthuysen,
Phys.Rev. , 29(1950).[20] D.N. Vollick, Phys.Rev. D
71, 044020 (2005);T.P.Sotiriou and S.Liberati,
Annals Phys. (2007)935-966. [21] T.W.B. Kibble,
Comm. Math. Phys. , 73 (1978); S.Weinberg, Annals of Physics , 336 (1989)[22] S.A.Gurvitz and G.Kalbermann,
Phys.Rev.Lett. (1987) 262; S.A.Gurvitz, Phys.Rev. A
38 (1988) 1747.[23] Note that we are assuming that the proposed f ( R ) mod-els are as fundamental as GR and, therefore, should pro-vide a consistent description of Nature in all experimen-tally accessible scales. To argue that the correcting termsof the lagrangian are only applicable in cosmic scalesshould come with a detailed explanation of why at differ-ent scales a different gravity lagrangian should be used.In the absence of such an explanation, we treat the f ( R )lagrangians as fundamental and check their predictionsat atomic scales, where GR predicts a virtually flat space-time structure.[24] Here R αβγλ is the Riemann tensor of the metric g µν .[25] The discussion and construction of the Hilbert space ofthe solutions of this equation lie beyond the scope of thispaper. However, we want to point out certain difficultiesrelated to the fact that the non-linearities induced by the ψ -dependence of T are in clear conflict with the super-position principle. Note also that the “Hamiltonian” H is not hermitian due to the imaginary term i ∇ Ω (thereare also other sources of non-hermiticity which also arisein pure GR and can be neglected when one focuses onthe one-particle sector of the theory [see [16] for moredetails]).[26] If to recover (29) in the low density region and to avoidthis external potential well we identify m with m , wethen find a potential barrier of magnitude ∼ +0 . m inthe interior of the atom, which makes extremely difficultthe capture of the electron by the atomic nucleus and isalso in conflict with observations.[27] These contributions represent extreme gradients thatcannot be counterbalanced by the electromagnetic inter-action to reach new equilibrium configurations. This con-flicts with the static configurationsstatic configurations