1 Department of Physics, Aristotle University of Thessaloniki,
2 Department of Civil Engineering, Technological Education Institute of Serres,
A novel approach in the semiclassical interaction of gravity with a quan-
tum scalar field is considered, to guarantee the renormalizability of the energy-momentum tensor in a multi-dimensional curved spacetime. According to it, aself-consistent coupling between the square curvature term R2 and the quan-tum field is introduced. The subsequent interaction discards any higher-orderderivative terms from the gravitational field equations, but, in the expense,it introduces a geometric source term in the wave equation for the quantumfield. Unlike the conformal coupling case, this term does not represent anadditional ”mass” and, therefore, the quantum field interacts with gravity ina generic way and not only through its mass (or energy) content.
Key-words: Quantum gravity - semiclassical theory - gravity quintessence
In the last few decades there has been a remarkable progress in understanding
the quantum structure of the non-gravitational fundamental interactions (Nanopou-los 1997). On the other hand, so far, there is no quantum framework consistentenough to describe gravity itself (Padmanabhan 1989), leaving string theory as themost successful attempt towards this direction (Green et al 1987, Polchinsky 1998,Schwarz 1999). Within the context of General Relativity (GR), one usually resortsto perturbations’ approach, where string theory predicts corrections to the Einsteinequations. Those corrections originate from higher-order curvature terms arising inthe string action, but their exact form is not yet being fully explored (Polchinsky1998). ∗Presented at the International Workshop on Cosmology and Gravitational Physics, 15 - 16
December 2005 Thessaloniki, Hellas (Greece), Editors: N. K. Spyrou, N. Stergioulas and C. Tsagas
A self-consistent mathematical background for higher-order gravity theories was
formulated by Lovelock (1971). According to it, the most general gravitationalLagrangian reads
where λm are constant coefficients, n denotes the spacetime dimensions, g is thedeterminant of the metric tensor and L(m) are functions of the Riemann curvaturetensor Rijkl and its contractions Rij and R, of the form
δj1.j2mRi1i2 .Ri2m−1i2m
where Latin indices refer to the n-dimensional spacetime and δj1.j2m
eralized Kronecker symbol. In Eq (2), L(1) = 1R is the Einstein-Hilbert (EH)
Lagrangian, while L(2) is a particular combination of quadratic terms, known as theGauss-Bonnett (GB) combination, since in four dimensions it satisfies the functionalrelation
µν Rµν + RµνκλRµνκλ
corresponding to the GB theorem (Kobayashi and Nomizu 1969). In Eq (3), Greekindices refer to four-dimensional coordinates. Introducing the GB term into thegravitational Lagrangian will not affect the four-dimensional field equations at all. However, within the context of the perturbations’ approach mentioned above, themost important contribution comes from the GB term (Mignemi and Stewart 1993). As in four dimensions it is a total divergence, to render this term dynamical, onehas to consider a higher-dimensional background or to couple it to a scalar field.
The idea of a multi-dimensional spacetime has received much attention as a can-
didate for the unification of all fundamental interactions, including gravity, in theframework of super-gravity and super-strings (Applequist et al 1987, Green et al1987). In most higher-dimensional theories of gravity, the extra dimensions are as-sumed to form, at the present epoch, a compact manifold (internal space) of verysmall size compared to that of the three-dimensional visible space (external space)and therefore they are unobservable at the energies currently available (Green et al1987). This so-called compactification of the extra dimensions may be achieved, ina natural way, by adding a square-curvature term (RijklRijkl) in the EH action ofthe gravitational field (M¨uller-Hoissen 1988). In this way, the higher-dimensionaltheories are closely related to those of non-linear Lagrangians and their combinationprobably yields a natural generalization of GR.
In the present paper, we explore this generalization, in view of the renormalizable
energy-momentum tensor which acts as the source of gravity in the (semiclassical)interaction between the gravitational and a quantum matter field. In particular:
We discuss briefly how GR is modified by the introduction of the renormalizable
energy-momentum tensor first recognized by Calan et al (1970), on the rhs of thefield equations. Introducing an analogous method, we explore the correspondingimplications as regards a multi-dimensional higher-order gravity theory. We find
that, in this case, the action functional, describing the semi-classical interaction ofa quantum scalar field with the classical gravitational one, is being further modifiedand its variation with respect to the quantum field results in an inhomogeneousKlein-Gordon equation, the source term of which is purely geometric (∼ R2).
Conventional gravity in n−dimensions implies that the dynamical behavior of the
gravitational field arises from an action principle involving the EH Lagrangian
where, Gn = GVn−4 and Vn−4 denotes the volume of the internal space, formed bysome extra spacelike dimensions. In this framework, we consider the semi-classicalinteraction between the gravitational and a massive quantum scalar field Φ(t, x)to the lowest order in Gn. The quantization of the field Φ(t, x) is performed byimposing canonical commutation relations on a hypersurface t = constant (Isham1981)
[Φ(t, x) , Φ(t, x )] = 0 = [π(t, x) , π(t, x )]
[Φ(t, x) , π(t, x )] = iδ(n−1) (x − x )
where, π(t, x) is the momentum canonically conjugate to the field Φ(t, x). Theequal-time commutation relations (5) guarantee the local character of the quantumfield theory under consideration, thus attributing its time-evolution to the classicalgravitational field equations (Birrell and Davies 1982).
In any local field theory, the corresponding energy-momentum tensor is a very
important object. Knowledge of its matrix elements is necessary to describe scat-tering in a relatively-weak external gravitational field. Therefore, in any quantumprocess in curved spacetime, it is desirable for the corresponding energy-momentumtensor to be renormalizable; i.e. its matrix elements to be cut-off independent (Bir-rell and Davies 1982). In this context, it has been proved (Callan et al 1970) that thefunctional form of the renormalizable energy-momentum tensor involved in the semi-classical interaction between the gravitational and a quantum field in n−dimensions,should be
where, the semicolon stands for covariant differentiation (∇k), ✷ = gik∇i∇k is thed’ Alembert operator and
Tik = Φ,iΦ,k − gikLmat
is the conventional energy-momentum tensor of an (otherwise) free massive scalarfield, with Lagrangian density of the form
It is worth noting that the tensor (6) defines the same n−momentum and Lorentzgenerators as the conventional energy-momentum tensor.
It has been shown (Callan et al 1970) that the energy-momentum tensor (6) can
be obtained by an action principle, involving
[f (Φ)R + Lmat] −gdnx
where, f (Φ) is an arbitrary, analytic function of Φ(t, x), the determination of whichcan be achieved by demanding that the rhs of the field equations resulting from Eq(9) is given by Eq (6). Accordingly,
ik − 2 ikR = −8πGnΘik = −2fik + 2f; ik − 2gik✷f )
To lowest order in Gn, one obtains (Callan et al 1970)
Therefore, in any linear Lagrangian gravity theory, the interaction between a quan-tum scalar field and the classical gravitational one is determined through Hamilton’sprinciple involving the action scalar
On the other hand, both super-string theories (Candelas et al 1985, Green et al
1987) and the one-loop approximation of quantum gravity (Kleidis and Papadopou-los 1998), suggest that the presence of quadratic terms in the gravitational action is apriori expected. Therefore, in connection to the semi-classical interaction previouslystated, the question that arises now is, what the functional form of the correspondingrenormalizable energy-momentum tensor might be, if the simplest quadratic curva-ture term, R2, is included in the description of the classical gravitational field. Toanswer this question, by analogy to Eq (9), we may consider the action principle
1(Φ)R + αf2(Φ)R2 + Lmat dnx = 0
where, both f1(Φ) and f2(Φ) are arbitrary, polynomial functions of Φ. Eq (13) yields
ik + 2F; ik − 2gik✷F + αgikf2(Φ)R2
where, the function F stands for the combination
F = f1(Φ) + 2αRf2(Φ)
For α = 0 and to the lowest order in Gn (but to every order in the coupling constantsof the quantum field involved), we must have
2 ikR = −8πGnΘik
where, Θik [given by Eq (6)] is the renormalizable energy-momentum tensor firstrecognized by Calan et al (1970). In this respect, we obtain f1(Φ) = f(Φ), i.e. afunction quadratic in Φ [see Eq (11)]. Furthermore, on dimensional grounds regard-ing Eq (14), we expect that
and, therefore, αRf2(Φ) ∼ Φ2, as well. However, we already know that R ∼ [Φ], asindicated by Whitt (1984), something that leads to f2(Φ) ∼ Φ 1 and in particular,
In Eq (18), the coupling parameter α encapsulates any arbitrary constant that maybe introduced in the definition of f2(Φ). Accordingly, the action describing the semi-classical interaction of a quantum scalar field with the classical gravitational one upto the second order in curvature tensor, is being further modified and is written inthe form
nΦ2)R + αR2Φ + Lmat dnx
is the so-called conformal coupling parameter (Birrell and Davies 1982). In this case,the associated gravitational field equations (14) result in
2 ikR = −8πGn (Θik + αSik)
The rhs of Eq (21) represents the ”new” renormalizable energy-momentum tensor. Notice that, as long as α = 0, this tensor contains the extra ”source” term Sik. Inspite the presence of this term, the generalized energy-momentum tensor still re-mains renormalizable. This is due to the fact that, the set of the quantum operators{Φ, Φ2, ✷Φ} is closed under renormalization, as it can be verified by straightforwardpower counting (see Callan et al 1970).
Eq (22) implies that the quadratic curvature term (i.e. pure global gravity) acts
as a source of the quantum field Φ. Indeed, variation of Eq (19) with respect toΦ(t, x) leads to the following quantum field equation of propagation
✷Φ + m2Φ + ξnRΦ = αR2
that is, an inhomogeneous Klein-Gordon equation in curved spacetime. It is worthpointing out that, in Eq (19), the generalized coupling constant α remains dimen-sionless (and this is also the case for the corresponding action) only as long as
In fact, [R] ∼ [Φ] n−2 and, therefore, f2 ∼ Φ2 n−4
n−2 . In order to render the coupling constant α
dimensionless, one should consider n = 6. Hence, f2 ∼ Φ only in six dimensions.
thus indicating the appropriate spacetime dimensions for the semi-classical theoryunder consideration to hold, without introducing any additional arbitrary lengthscales. Summarizing, a self-consistent coupling between the square curvature term R2
and the quantum field Φ(t, x) should be introduced in order to yield the ”correct”renormalizable energy-momentum tensor in non-linear gravity theories. The sub-sequent quadratic interaction discards any higher-order derivative terms from thegravitational field equations, but it introduces a geometric source term in the waveequation for the quantum field. In this case, unlike the conventional conformal cou-pling (∼ RΦ2), the quantum field interacts with gravity not only through its mass(or energy) content (∼ Φ2), but, also, in a more generic way (R2Φ).
This research has been supported by the Greek Ministry of Education, through thePYTHAGORAS program.
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