On the trace anomaly and the energy-momentum conservation of quantum fields at D=2 in classical curved backgrounds

On the trace anomaly and the energy-momentum conservation of quantum fields at D=2 in classical curved backgrounds

M. Alves; J. Barcelos-Neto

Instituto de Física, Universidade Federal do Rio de Janeiro, RJ 21945-970, Brazil

ABSTRACT

We study the conformal symmetry and the energy-momentum conservation of scalar field interacting with a curved background at D = 2. We avoid to incorporate the metric determinant into the measure of the scalar field to explain the conformal anomaly and the consequent energy-momentum conservation. Contrarily, we split the scalar field in two other fields, in such a way that just one of them can be quantized. We show that the same usual geometric quantities of the anomaly are obtained, which are accompanied by terms containing the new field of the theory.

1 Introduction

The interaction between quantum fields and classical gravity has been intensively studied for a long time [1]. One of the main motivations is that this procedure is considered to be the first initial step to understand the full quantum theory including the gravitatonal field itself. It resembles the old formalism of dealing with quantum fields interacting with the classical external electromagnetic background before the advent of the quantum electrodynamics.

However, even though these are aparently similar procedures, the former is much more involved. For example, the concept of particle is not well defined. Consequently, the definition of |inñ and |outñ states cannot be done in a clear way and the definition S-matrix becomes meaningless. So, instead os particles, it is the energy-momentum tensor that plays the fundamental role, due to its local nature and being the source of the curvature in the general relativity theory. There is an important property that the energy-momentum tensor has to satisfy, the (covariant) divergenceless, which corresponds to the conservation of energy and momentum of the theory. This is a kind of symmetry that cannot be modified by quantum corrections.

On the other hand, we have an interesting symmetry related to massless theories, called conformal symmetry, which means, in a broader sense, the absence of scales. This symmetry is manifested in the traceless of the energy-momentum tensor. Contrarily to previous case, it is not necessarily kept in the quantum scenario (trace anomaly). This occurs because the quantum formalism naturally introduces scale parameters in order to deal with infinities during the regularization procedures.

In this paper, we consider a quantum massless scalar field interacting with a classical curved background. We mention that this theory exhibits the conformal symmetry, where for D > 2 it is necessary to couple the scalar field to the classical curvature (nominimal coupling) [1]. The particular case of D = 2, that is the subject of the present paper, has an interesting feature. The conformal symmetry is verified without necessity of coupling the scalar field to the curvature, because there is no conformal transformation for it. This result may lead to a wrong conclusion that there is no trace anomaly for scalar fields at D = 2 because the absence of conformal transformation for them leads to an invariance of the corresponding measure in the path integral formalism.

The above reasoning and conclusion, which there is no trace anomaly for D = 2, cannot be true because it is accompanied by an unpleasant absence of energy and momentum conservation (after quantum corrections). A more carefull study of the conformal symmetry shows that the conformal anomaly does actually exist and the energy and momentum are actually conserved, as they should be [2, 3]. We mention that this problem can be circumvented by splitting the of the action as and incorporating each one of these factors to the scalar field [2]. In this way, the new scalar field acquires a convenient conformal transformation [4], whose noninvariance of the measure renders the expected trace anomaly and the energy-momentum conservation.

The purpose of the present paper is to display a different alternative of dealing with this problem. We avoid to incorporate any factor involving the metric tensor to the scalar field because, since we intend to work with the path integral formalism, this would be inconsistent with the initial assumption that the gravitational field is classical. Our proposal consists in splitting the scalar field in a product of two fields with different conformal transformations. The classical conformal symmetry is not modified, but the measure of one of them is. We show that this leads to a trace anomaly, that has the same geometrical terms of the usual case, plus other ones related to the new field. We also show that, eventhough more involved, the energy momentum conservation is also achieved quantically.

Our paper is organized as follows. In Sec. II we brief discuss how this this problem can be solved by incorporating a factor inovolving the metric tensor into the scalar field. Eventhough this is a section review, we follow the lines that we shall used into the next section, where our formalism is presented. We left Sec. IV for some concluding remarks.

2 Trace anomaly and energy-momentum conservation

Let us start from the action

The classical energy-momentum tensor is

which has the following properties

where is the covariante derivative and is the Laplace-Beltrami operator, .

Expression (2.3) tell us that the theory exhibits the conformal symmetry. The conformal transformation for the metric tensor is and, consequently, is conformally invariant. This means that the conformal symmetry is verified at D = 2 without any transformation for the scalar field f. The meaning of expression (2.4) is that the energy and momentum are conserved. It is opportune to mention that this expression was obtained by using the equation of motion for f.

In the quantum scenario (with a classical background metric) the energy momentum tensor can be obtained by means of the vacuum functional as

where

Since the scalar field does not change under conformal transformation, we have that the measure [df] also remains invariant. Consequently, the semiclassical expression for the energy-momentum tensor reads

The curved background is considered to be classical, so we may have

It is not possible to conclude that expression (2.9) is zero because the equation of motion cannot be used in the quantum scenario. Of course, the results above do not merit confidence because there is no reason to believe that energy and momentum are not conserved after quantum effects are taken into account.

To circumvent this problem, the action (2.1) can be rewriten as

where

Of course, the classical T_mn is precisely the previous one given by (2.2) (which can be rewritten in terms of F) and the action, S given by (2.10), is still conformally invariant. The conformal transformation for the new scalar field F is

However, for the vacuum functional, the measure [dF] is not invariant under conformal transformation. In a general way, we have [5]

where A(x) is a badly divergent quantity that can be regularized by means of the zeta function technique leading to [6]

The last step of the expression above is restrict to D = 2, and the coefficient a₁(x, x¢) is related to heat kernel expansion. The notation [a₁(x)] means [a₁(x)] = lim_{x¢® x}a₁(x, x¢). These coefficients can be obtained by means of a recursion relation that depends on the kind of operator that acts on the field [7]. For the present case, , where R is the Ricci scalar curvature

Now, the corresponding energy-momentum tensor obtained by means of expression (2.6), which we shall denote by , is not traceless. The trace áT^m _mñ can be directly obtained by

This result embodies the trace anomaly. Since in two spacetime dimensions we have the identity , one may say that the full expression for the energy-momentum tensor áñ should be

where áT_mnñ is the one given by (2.7) (it is indifferent to write it in terms of f or F). Acting the covariant derivative in both sides of the expression above, we get

Expanding the field f in terms of eigenfunctions of the operator , one can show that [2]

and the quantity áf fñ can be regularized and leads to [2]

So,

as it should be.

3 Alternative procedure

Now, instead of incorporating the factor to the scalar field, we go in a opposite direction by splitting the field f as

where q and j are considered to be two independent quantities with the following conformal tranformations

The field j remains quantum, but q can be quantum or not. It is important the field q appears in a exponential term and, consequently, with a conformal transformation like (3.3). This is so because, in the hypothesis that q is also quantum, its corresponding measure, [dq], remains unchanged (the jacobian is trivial). In the developments which follow, we shall consider q classical. At the end, we briefly talk on the possibility of q being quantum.

Replacing f given by (3.1) into the initial expression for S, (2.1), we have

We have just done a change of variables and, consequently, there is no changing into the classical case. But, in the path integral, the measure [dj] is not invariant under conformal transformation. Considering q classical, se have the vacuum functional

Now, the coefficient [a₁], related to the operator that is acting on j, is

So, the trace anomaly reads

Since the field q is considered to be classical, one may infer that the expression for the energy momentum tensor áñ is given by

where áT_mnñ is the same one as given by (2.7), with f replaced by e^qj, i.e.

Now, let us verify the consistency with respect the energy and momentum conservation. First, we consider Ñ^máT_mnñ. This can be done by directly acting the operator Ñ^m on áñ, given by the expression (3.9), or by using (2.9), where we should replace f by e^qj [notice, however, that this replacement cannot be done into (2.18), because just part of f is quantum]. We thus have

For the second term of (3.8) we have

Now, since just j is quantum, we can write

Considering the action involving j and with the term e^2q factorized, as well as the renormalization factor we are using into (2.14), we can obtain the regularized quantities

Finally, since j is a scalar quantity and á¶_nj²ñ is an average involving all directions aleatory, and the same occurs with á¶_rj¶_njñ, we have that these quantities are null. Replacing all these results into (3.10) and (3.11), we get

which express the consistency with the energy-momentum conservation.

In the case that q is also quantum, the measure [dq] does not change under conformal transformation, but the problem is much more involved and difficult to solve. Just formally, one may write that the trace anomaly reads

From which one cannot either infer an expression similar as the second term of (3.8) or try to regularize it because the bad divergencies occuring in the exponential e^2q.

4 Conclusion

In this paper we have study the problem of conformal anomaly and the energy-momentum conservation for a quantum scalar field interacting with a classical curved background in the spacetime dimension D=2. We have considered the scalar field as split in two other scalar fields, and kept just of them quantum. This procedure is in opposite direction to what is done in literature, where a factor containing the metric determinant is absorbed by the scalar field, leading to a new field with a convenient conformal transformation. We have shown that our procedure is consistent with the geometric terms of the usual treatment of the conformal anomaly and also consistent with the expected result that energy and momentum should be conserved after quantum effects are taken into account.

Acknowledgments

This work is supported in part by Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq and Fundação Universitária José Bonifácio - FUJB (Brazilian Research Agencies) One of us, J.B.-N. has also the support of PRONEX 66.2002/1998-9.

References

[1] For a general review see N.D. Birrell and P.C.W. Davies, Quantum fields in curved space, Cambridge (1982), and references therein.

[2] K. Fujikawa, U. Lindström, N.K. Nielsen, M. Rocek, and P van Nieuwenhuizen, Phys. Rev. D37, 391 (1988).

[3] Fujikawa K Quantum Gravity and Cosmology ed H Sato and I Inami (Singapore: World Scientific)(1982).

[4] M. Alves and C. Farina, Class Quantum Grav. 9, 1841 (1992).

[5] K. Fujikawa, Phys. Rev. Lett. 44, 1733 (1980).

[6] See, for example, S. Hawking, Commun. Math. Phys. 55, 133 (1977) and references therein.

[7] B.S. DeWitt, Dynamical theory of groups and fields, Les Houches: Relativiry, groups and topology (1963). Editors: B.S. DeWitt and C. DeWitt; B.S. DeWitt, The spacetime approach to quantum field theory, Les Houches: Relativity, groups and topology (1983). Editors: B.S. DeWitt and R. Stora; S.M. Christensen, Phys. Rev. D14, 2490 (1976).

[8] W. Kummer, H. Liebl, and D.V. Vassilevich, Mod. Phys. Lett. A12, 2683 (1997); S. Hawking and R. Boussos, hep-th/9705236; J. S. Dowker, hep-th/9802029; S. Ichinoise and S.Odintsov, hep-th/9802043.

Received on 10 November, 2003

Publication Dates

• Publication in this collection
  01 Sept 2004
• Date of issue
  June 2004

History

• Received
  10 Nov 2003
• Accepted
  10 Nov 2003