Fluctuations of the Metric Tensor: On Fermion Propagators and on the Cosmological Constant

Conformal fluctuations of the metric tensor are considered. They give rise to a Planck scale size lower bound of the proper length. Infinit ies arising from the point-like character of particles are thus avoided, expressions related to fermion propagators are finite without the need to renormalize or regularize. The quark condensate, constituent quark mass, pion mass, and pion weak decay constant are calculated within the SU(2) Nambu--Jona-Lasinio model. The numerical values are consistent with known strong interaction physics if one assumes very small values of the current quark mass and effective coupling constant. The usual values of these quantities in hadron phenomenology can be interpreted as resulting from dressing of the very s mall bare quantities with the fluctuations. With respect to the cosmological constant it is shown that a cosmological constant term, which may initially exist in the gravitational action, is ”masked” by the fluctuations of the metric, i.e., it does not appear in the final Einstein equation. Instead, the fluctuations give rise to a cosmological constant themselves.


Introduction
The concept of a particle in quantum field theory is pointlike. This fact leads to the known divergent expressions which are usually tackled by renormalization or regularization techniques. Quantum gravity, however, introduces a coarse graining of spacetime at the Planck scale. In the present work an effective description of spacetime fluctuations is proposed. On the microscopic scale, of interest in quantum field theory, the resulting fu zziness of spacetime is shown to render usually d ivergent expressions finite, without the need for renormalizat ion or regularization. On a macroscopic scale, the fluctuations affect the behavior of the cosmological constant.
The coarse graining of spacetime indicates that the metric tensor has to be considered a quantum variable. The idea that quantum fluctuations of the metric can have the effect of a regulator is not new, see for instance [1] and references therein, and [2,3]. The expected effect is that propagators should be "smeared out" [4]. In o rder to preserve the light cone structure, a necessary condition for not violat ing causality at any instant of the variation of the met ric , only conforma l fluctuations [5] are studied here. The averaging over these fluct uatio ns results in a fu zziness of the light cone at the level of the Planck scale.
A full theory of quantum gravity is expected to require mo re than the quantization of the conforma l mode. Issues related to gauge-fixing and the isolation of the relevant degrees of freedom are not addressed here. The conformal mode should nevertheless refle ct the basic features of the effect of quantum gravity on the observables under consideration. Due to the conformality and the fact that the field representing the fluctuations is treated in lowest order, i.e. in classical appro ximation when deriving Einstein's equation fro m variat ion of the action, the present procedure is in this sense not fundamental but should be regarded as an effective approach.
The fluctuations give rise to a lower bound of the proper length, as should be expected also on heuristic grounds. Thus the infinit ies mentioned above are avoided without the need for regularization. The consequences for several physical quantities related to Green's functions are studied in the context of the Nambu--Jona-Lasinio model [6,7] using standard techniques [8][9][10][11], adapted to the case at hand. The intention here is to check the consistency of the values, with the implicit assumption that the quarks do behave point-like down to the Planck scale (but not beyond that point), and that strong interactions do not introduce another (much larger) regularizat ion scale.
An additional consequence of fluctuations of the metric is the emergence of a cosmo logical constant [12,13], like in other theories with scalar fields coupled to the metric (see e.g. [14] and references therein). This cosmo logical constant is independent of the one conjectured fro m vacuum fluctuations of quantum field theory (QFT), wh ich pose the well-known 120 orders of magnitude problem [15]. It is shown that in the present approach this problem is of no relevance due to a cancellation of terms in Einstein's equation.
The following section introduces the scalar field wh ich represents the fluctuations of the metric, and it is shown how they impose a lower bound on the proper length. Section 3 discusses fermion propagation in curved spacetime. In section 4 the formal exp ressions for the quark condensate and the quark mass are calcu lated, followed by the expressions for the pion mass and weak decay constant in section 5. Section 6 is devoted to the cosmological constant which arises from the fluctuations of the metric . Nu merical results and a discussion are presented in the final section.

Conformal Fluctuations of the Metric
This section summarizes and adapts some known results (see [5] for instance). The starting point of the present approach is a scalar field wh ich is to represent the quantum fluctuations of the generalized metric about its "classical" or "background" value ̅ . All "background" quantities and operators are denoted by overbars. In the sign convention applied here the Ricci tensor is calculated by the contraction = and all spacetime indices, Lat in or Greek, run fro m 0 to 3, with the signature (+ ---). A large part of the following discussion assumes flat background spacetime, so ̅ = η ij = di ag(1, −1, −1, −1), except for the section on the cosmological constant, as in cosmology ̅ is usually the Fried man-Le maître-Robertson-Walker metric, although this fact is not used explicitly. In most expressions, natural units with ħ = c = 1 are used.
The simp lest kind of metric variation is where the spacetime coordinate dependent function (x) is a scalar field representing the quantum fluctuations of the metric about the classical value. The fluctuation average (vacuum expectation value) is required to be < > = < ,m > = 0 (the co mma denotes partial derivative), meaning that the center of the fluctuations is the classical value and that there is no drift o f in spacetime, a fact consistent with energy conservation [13]. The field represents an additional, conformal degree of freedo m.
The full met ric is now inserted into the Hilbert action of General Relativity = ( 16 ) −1 ∫ d 4 �− .
(2) Due to the presence of the total scalar curvature is where � denotes the classical background scalar curvature, i.e. the usual one without fluctuations, and the semicolon denotes covariant derivative. In the special case of flat background spacetime ( � =0) wh ich is studied next, the action reduces to = 3 with the Planck length = √ (in natural units), being the gravitational constant. (The sign convention differs fro m the one used in an earlier version of this work [16]). Note that the resulting action is proportional to the usual scalar action, with the additional factor 3/ 4 2 .
The vacuum expectation value of the line element be- The quantity < 2 > is calcu lated formally as the limit → of the scalar propagator [17] but with the additional factor mentioned above appropriately included [5]: The quantity < 2 > diverges as ( 2 = ℏ/3 3 , written here in its fu ll form, will appear frequently in the following expressions). This means that the fluctuations of the metric impose a lower bound on the proper length. Any point-like object is "smeared out" at the level of the Planck length. Infinit ies arising fro m the point-like character of particles are thus avoided. Moreover, due to the presence of the fluctuations, the classical squared distance 2 is replaced by < 2 > = 2 + 2 (9) in the expressions considered here. This implies also that, after averaging over the fluctuations, the light cone is smeared out at the level of the Planck length.

Fermions in Curved Spacetime
When dealing with fermion propagators one has to take into account that any deviation of the met ric fro m Minkowski spacetime alters the Dirac equation. Its general form is [18,19] ( i ∇ k − ) Ψ = 0 , (10) where ∇ is the covariant derivative of a spinor, A rather lengthy calculation is needed to obtain the explicit form of the matrices Γ fro m (12). Inserting them back into the covariant derivative, the Dirac equation can be written for the case at hand as where the operator p;/ reads

Quark Condensate and Quark Mass
Passing now to the discussion of quantities containing a single fermion propagator, the quark condensate is known to be related to the fermion propagator by is the nu mber of flavors and Tr stands for the trace in flavor, color and Dirac spaces. The spacetime dependent propagator ( ) is obtained fro m its four-mo mentum dependent counterpart ( ) by a Fourier transform. The condensate becomes Here, tr is the Dirac trace and the constituent quark mass (see below), and the +i term has been omitted fro m the denominator, for simp licity of notation. Also implicit is the evaluation of the vacuum expectation value of the fluctuations. The mo mentum p;/A is the full one (16), and refers to the determinant of the metric tensor To calculate the vacuum expectation value of the fluctuating field, it is convenient to rewrite the denominator in terms of ̅ 2 . After some algebra one has (remembering the presence of the term i in the deno minator), the d 4 ̅ -integration can be performed, where use of (7) has been made to write the fluctuation average lim →0 < Φ 2 x 2 > = 2 .
(23) The remain ing integral evaluates to with 1 being the mod ified Bessel function. In the s mall argument limit, 1 behaves as the inverse of the argument, so one finally gets < � >= −3 −2 −2 .
(25) To proceed to the discussion of the constituent mass of the quark one needs an explicit interaction lag rangian. An often used effective chiral model lagrangian is due to Nambu and Jona-Lasinio [6] The modern version of the model in quantum chro modynamics (QCD) is reviewed for instance in [7]. The lagrangian of the simplest SU(2) version is written as = Ψ �� i ∇ − 0 �Ψ + [( Ψ � Ψ ) 2 − ( Ψ � 5 Ψ ) 2 ] , (26) where is the effective strong coupling constant, 0 is the current quark mass and Ψ is the quark field. As one can see, the quark fields are the only degrees of freedom in this model, the information on the gluons is residing in the constant . It has been shown in [20] that this type of lagrangian can be obtained from QCD by integrating out the gluonic degrees of freedom. It is reasonable to expect that gluonic degrees of freedom are unimportant at "high energy" where we know that quarks are asymptotically free. But how high is "high energy"? Asymptotic freedo m is observed in QCD where energy scales are measured in terms of GeV, but since in this work energ ies up to the Planck energy are considered, there is no guarantee that the present type of lagrangian can still be used, the gluons could become important again. Let me simp ly assume here the validity of this lagrangian down to the Planck scale and use it to study the consequences of the fluctuations, remembering the words of caution about its applicability. After Fierz symmet rizat ion in color, flavor and Dirac spaces the lagrangian acquires more terms which are not written exp licitly here, since they do not contribute to the quantities to be studied. The coupling constant is redefined by the presence of the exchange terms. Let me call this redefined value again .
Note that the model is non-renormalizable and is normally defined only together with some regularization procedure, for instance by using a cut-off typically in the order of 1 GeV, the strong interaction scale. In the present consideration there is no need to regularize, since the results stay finite. The "regularization" arises naturally fro m the fluctuations at the Planck scale.
Some co mments are in order regarding the -matrices appearing in this kind of quartic interaction lagrangian. The expression of the pseudoscalar-isovector term ( Ψ � 5 Ψ ) 2 stands actually for ( Ψ �( 5 ) + Ψ )( Ψ � 5 Ψ ) . Since 5 = i 0 1 2 3 , it carries the inverse fourth power of Φ: The adjoint is however so the contributions of Φ cancel in the lagrangian. (The same holds true for terms containing ). It should be noted also that Ψ � = Ψ + ̅ 0 contains only the flat spacetime ̅ 0 , in order to be compatible with current conservation ∇ being again the covariant derivative.
In principle, all the terms appearing in the Fierz symmet ric lagrangian have to be considered in the evaluation o f the constituent mass. As in the flat spacetime case, however, only the scalar term ( Ψ � Ψ ) 2 contributes, essentially since p;/ has the same ̅ -structure as p;/; -, wh ich cancels the contributions in the traces in the same way as in flat spacetime without fluctuations, with the exception of the vector term �Ψ � Ψ� 2 , see below. The constituent mass is related to the current mass 0 via the self-energy Σ: = 0 + iΣ( ) .
(33) The contribution wh ich does not vanish immediately by tracing is the vector term iΣ = i lim → Tr ( , ) .
After evaluating the trace, this is proportional to ̅ ∫ d 4 ̅ ( ̅ Φ 5 + 3 2 iΦ 2 Φ), so the first term vanishes right away since it is odd in ̅ in the integral, and the second because it is odd in the quantum fluctuations. Therefore, using (25), the constituent quark mass is obtained from This relation can be used to determine the value of the coupling, with the assumption ≫ 0 : = 2 2 /6 . (36)
It is worthwhile noting that, since integrals are finite in the present scheme, it is allowed to shift variables without introducing ambigu ities, contrary to the usual case of the infinite integrals appearing in normalizat ion or regularization procedures.
The pion weak decay constant in the chiral limit is written as This leads to 2 = −12i 2 ( 0 ) .

The Cosmological Constant
The Casimir effect is usually considered to be a proof for the existence of the quantum fluctuations of the vacuum required by QFT. Considering them as "real" in the sense of contributing to the energy density of the universe leads to the well-known problem of a cos mological constant 120 orders of magnitude bigger [15] than the one indeed observed [21]. What could be the impact of the metric fluctuations? The procedure to obtain the cosmological constant fro m the total action = + is described in [13], where the gravitational action did not contain a cosmological constant fro m the outset, and is the matter part in dust approximat ion. Here, the same procedure is followed, but starting instead from the action = Λ � + , where Co mpared to [13], the two equations contain now new terms proportional to Λ � , but when the second equation is used to eliminate the -dependence fro m the first, the new terms cancel exactly. The resulting Einstein equation is identical to the one previously obtained without the prior Λ � -term. It can again be cast in the form of an Einstein equation of the same form as in [13], is a consequence of the inclusion of in the generalized metric and not of the prior Λ � . The contributions of the QFT vacuum fluctuations (if that was the assumed origin of the term Λ � ) cancel in the present approach. The situation is remin iscent of the exp lanation of the Casimir effect put forward in [22], which does not prove nor require the existence of QFT vacuum fluctuations.

Numerical Results and Discussion
Neither the pion weak decay constant nor the quark condensate contain the effective coupling constant explicit ly. In this way, these quantities are model independent, as long as one regards the quarks as the only degrees of freedom. Both quantities are dependent indirectly on via the constituent quark mass. In order to be able to reproduce the experimental value of the pion weak decay constant = 93 MeV, it is necessary to assume a rather small constituent mass = 34.6 MeV. With the more acceptable value = 386 MeV, the decay constant becomes = 1011 MeV. The discrepancy may be due to the fact that expression (45) is an approximat ion in the chiral limit. It is interesting to realize that is only logarith mically dependent on the regularizat ion point. Using the Planck length as the covariant cutoff in [8] yields very similar results.
The quark condensate, although finite, evaluates to the unusually large value < � >= ( 5.5 × 10 15 MeV ) 3 , for the choice = 386 Me V. As discussed for instance in [8], the quark condensate is not renormalization invariant, but 0 < � > is. In fact, the Gell-Mann--Oakes--Renner relation -0 < � >= 2 2 indicates that the current quark mass must be very small in the present situation, of the order of 0 = 10 −39 MeV, instead of the usual few MeV. We can imagine that the "bare" current quark mass, appearing in the lagrangian, is successively dressed by the fluctuations of the metric to the few MeV current quark mass usually encountered in hadron physics, and then by the strong interaction, to finally yield the constituent quark mass . Th is is seen in (35), where both the Planck length and the effective coupling enter. To be compatible with (35), the coupling must be very small, = 10 −45 MeV −2 fro m (36). This can also be imagined as the bare value wh ich is then dressed by the fluctuations to give the more usual order of magnitude in the combinations ( Ψ � Ψ ) 2 etc. appearing in the lagrangian.
Turning next to the pion mass = 140 MeV, one finds that (41) is satisfied with the choices 0 = 6.1 × 10 −38 MeV for = 386 MeV, and 0 = 5.7 × 10 −39 MeV for = 34.6 MeV, respectively. Again, the smaller constituent mass seems to be preferred, as it co mpares well with the Gell-Mann--Oakes--Renner result. One should, however, not attribute too much value to this statement, since the model lagrangian used has to be taken with caution when approaching the Planck scale. As mentioned earlier, g luonic degrees of freedom may become increasingly important at very high energies again, since it is not clear if asymptotic freedom holds up to the Planck energy. It may not be allowed to simply use an effective coupling constant to represent strong interactions close to the Planck energy.
The next step along this line of investigation will be to consider a model lagrangian with gluonic degrees of freedom to assess their importance close to the Planck scale and to see whether the results presented here suffer significant changes.
No nu merical results are d iscussed with regard to the cosmological constant, as this is done in [13] and the intention here is to show what the effect of the fluctuations of the metric is on a prior presence of a cosmological constant in the action. The contribution of the prior term vanishes, but the metric fluctuations generate a cosmological constant themselves. This is a new and unexpected result. Work is in progress to evaluate whether the prior term cancels only in the dust approximation used here or also under more general conditions.
To summarize the results, no essential contradictions have been found in the present scheme, in wh ich conformal quantum fluctuations of the metric tensor introduce a fuzziness of point-like part icles at the scale of the Planck length. This fact avoids infinities without the need for renormalization or regularization. The physical values of quantities like the pion mass, pion weak decay constant and constituent quark mass attain reasonable values, if one assumes nonstandard (very small) values of the "bare" current quark mass and effective strong coupling constant. The standard values of the current quark mass and of the coupling constant can be interpreted as resulting from dressing of the bare quantities with the fluctuations of the metric, in a similar way as the constituent mass is obtained by dressing of the usual current quark mass with strong interactions.
In the context of the Einstein equation, the fluctuations mask any prior cosmo logical constant present in the gravitational action. This is an interesting and novel result worth being pursued further.