DAMTP 93/67

Implications of Conformal Invariance for Quantum Field Theories in

H. OSBORN

DAMTP, University of Cambridge, Silver St.,

Cambridge CB3 9EW, England.

ABSTRACT

Recently obtained results for two and three point functions for quasi-primary operators in conformally invariant theories in arbitrary dimensions d are described. As a consequence the three point function for the energy momentum tensor has three linearly independent forms for general d compatible with conformal invariance. The corresponding coefficients may be regarded as possible generalisations of the Virasoro central charge to d larger than 2. Ward identities which link two linear combinations of the coefficients to terms appearing in the energy momentum tensor trace anomaly on curved space are discussed. The requirement of positivity for expectation values of the energy density is also shown to lead to positivity conditions which are simple for a particular choice of the three coefficients. Renormalisation group like equations which express the constraints of broken conformal invariance for quantum field theories away from critical points are postulated and applied to two point functions. Talk presented at the XXVII Ahrenshoop International Symposium.

Conformal field theories in two dimensions have proved to be a remarkably fruitful area of investigation in recent years, motivated initially by applications to string theory but also of interest for systems in statistical physics at their critical points. In two dimensions there are a very large class of explicitly constructed non trivial conformal field theories which have a rich mathematical structure. In some sense in two dimensions conformal field theories appear to be dense in the space of all quantum field theories, under perturbation by relevant operators then there usually appears to be a ‘nearby’ conformal field theory defining an infra-red fixed point of the renormalisation group beta functions and to which the theory flows when the length scale is taken to infinity [1]. According to the Zamolodchikov -theorem [2] the value of the Virasoro central charge decreases from its initial value to that for the conformal field theory defined by the critical point describing the massless fields in the large distance limit, at least for unitary theories. The value of represents a measure of the degrees of freedom in the conformal theory and its decrease corresponds intuitively to a kind of ‘coarse-graining’ in the limit of large distance scales [3].

Such a detailed description of conformal field theories does not extend to higher dimensions. Nevertheless for field theories in which the trace of the energy momentum tensor may be expanded in a basis of scalar operators , , where are the beta functions of the various couplings, then at a fixed point defined by vanishing beta functions the trace of the energy momentum tensor is zero and the theory enjoys conformal invariance. The conserved charges have the form where so that, as described later, corresponds to an infinitesimal conformal transformation. In general scale invariance does not imply conformal invariance but for most quantum field theories it may be expected to hold at any renormalisation group fixed point [4]. However for the detailed knowledge of non trivial conformal field theories is much less explicit. Non gaussian fixed points may be investigated in the expansion for and and also in the expansion for some models when [5]. In three dimensions conformal field theories should correspond to the different universality classes related to the possible critical points of realistic statistical physical systems. In four dimensions conformal field theories are apparently rather sparse, based on the triviality of the continuum limit of lattice quantum field theories, although possible exceptions should be given by supersymmetric gauge theories and also by large QCD, with gauge group , where the number of fermions in the fundamental representation is fine tuned to ensure the leading beta function coefficient is negative but (there is then a perturbatively accessible infra-red fixed point with ).

With the motivation of exploring further the consequences of conformal invariance in A. Petkou and I [6] have attempted to establish a basis for further work by deriving simple conformally invariant forms for the two and three point functions for arbitrary spin operators and in particular apply these to the energy momentum tensor which plays an essential role in any quantum field theory. Although the conformal group is finite dimensional, being isomorphic to in the Euclidean case, there are significant complications as compared with due to the basic representations of the spin group acting on the quantum fields being no longer one dimensional.

The group of conformal transformations is defined by coordinate transformations such that

For an infinitesimal transformation we may write

Except for the general solution of (2) has the form

representing infinitesimal translations, rotations, scale transformations and special conformal transformations. The corresponding Lie algebra is easily determined from

Besides conformal transformations connected to the identity it is important to consider also inversions where

The whole conformal group may be generated just by combining inversions with translations and rotations. Geometrically the conformal group maps circles to circles (or straight lines) so any three distinct points may be moved to any other three points.

The quantum fields are assumed to transform under conformal transformations as a simple extension of their transformations under translations and rotations where

with belonging to some finite dimensional representation of . For a general conformal transformation as in (1) we may define an rotation depending on by

and then we take for quasi-primary fields the transformation rule [7]

where is the scale dimension of the field. Infinitesimally (8) is equivalent to

for and where are the generators of in the appropriate representation so that . It is straightforward to verify that .

To give a general expression for two point functions for quasi-primary fields it is convenient for each field to define a conjugate field transforming as

Then if the representation of to which belong is irreducible we may write

where is an overall constant scale factor and is the matrix belonging to defined according to (7) corresponding to the inversion (5),

To verify (11) it is sufficient, since translations and rotations are trivial, to use

As an illustration [8] we consider a vector operator and the traceless energy momentum tensor where application of (11) gives

where the dimensions of and are respectively and , as is necessary for conservation. For free scalar fields and free Dirac fermions with arbitrary

where is the dimension of the Dirac matrices in dimension while in four dimensions for free vector fields. and

The conformally covariant form for three point functions is also relatively simple when written in terms of an analogous group theoretic expression [6,9],

where

and is a homogeneous group invariant function satisfying

To see how the required conformal transformation properties of (16) follow from (18) it is important to note that , as defined in (17), transforms as a vector, and also

It is not difficult to see that (16) gives for the leading term in the short distance operator product expansion

for . Conversely using conformal invariance we may derive (16) directly from (20) [10]. If we consider an inversion through a point , so that , etc, and then let so that we get

which in the limit gives just (16) using the form (11) for the two point function

Additional constraints arise from the imposition of conditions such as conservation equations. Applying (16) to the case when one of the operators is a current of dimension so that we may write

then requires

Similar conditions arise from .

Using this formalism it is easy to recover, for instance, old results for the conformally invariant form for three point functions involving three vector currents [11]. As a new application we considered determining the possible conformally invariant forms for the three point function for the energy momentum tensor. Applying (12) gives

where is symmetric and traceless on each pair of indices , and , and is also symmetric under and is homogeneous of degree . There are 8 independent forms, compatible with transforming covariantly under rotations, satisfying these conditions while the conservation equation

provides 5 independent relations. It is also necessary to impose the condition

to ensure that the expression (24) is completely symmetric. However the relations so obtained are equivalent to those obtained from the conservation condition (25) so there remain in general 3 linearly independent forms for the conformally invariant energy momentum tensor three point function.

There is no obvious natural basis for the three independent forms but it is possible to specify the general form for in terms of particular components for some convenient choice of . The conformal invariant expressions become particular simple when lie on a straight line (it is possible to use the collinear configuration as a starting point for a general analysis). In consequence if we let , where denotes the reflection of through the plane , then for , we may take

where are parameters which determine the general expression or equivalently the form of . For is irrelevant and while for there is the restriction so that respectively there are only one, two linearly independent conformally invariant forms in these dimensions.

For free field theories these coefficients can be routinely calculated. For general scalars and fermions give

while when and

Clearly these free field theories realise the full range of possibilities in this case [12]. Restrictions are obtained by using Ward identities. These may be derived by extending the quantum field theory to curved space with metric and postulating

Taking functional derivatives gives relations between and point functions, for example

Restricting to flat space, since then , gives

and also we may obtain less trivial identities relating two and three point functions. The additional terms in the Ward identities for the three point function involving and arise from the singularities in when and . From (24) it is necessary and sufficient to require

To verify (33) requires careful regularisation of so that it is a well defined distribution. It is convenient to use the ideas of differential regularisation of extracting derivatives which reduces the degree of the singularity as so that it is integrable on [13]. The resulting distribution is arbitrary up to terms but these are fixed by imposing (33) which then entails

As is well known [14] in two and four dimensions there are contributions to the trace of the energy momentum tensor depending on the spatial curvature even for field theories which are conformal on flat space. For four dimensional conformal theories

In this case there are then extra purely local terms in the trace identity linking the two and three point functions

where is the projection operator onto 4 index tensors with the symmetries of the Weyl tensor (). From (36) and the usual renormalisation group equation it is easy to derive

which relates to the scale of the two point function

Clearly (36) provides an additional condition on the energy momentum tensor three point function involving when . Although a direct derivation is at present lacking by considering the results for free fields using (29) we may obtain

An important issue is whether there are any positivity constraints on the energy momentum tensor three point function. From positivity of the two point function in unitary theories it is necessary that the scale factor . As pointed out by Cappelli and Latorre [15] it is possible to obtain stronger conditions by making further assumptions on the matrix elements of the energy momentum tensor. If we define as the antilinear conjugation operator combined with reflection in the plane , , and for denoting some function of the fields restricted to then the requirement of reflection positivity for Euclidean theories is just . The requirement of positive energy density may then be translated after analytic continuation to the Euclidean regime into

This is of course stronger than the requirement that the Hamiltonian operator should have a positive spectrum. Since and then from (27) this requires

From (28) and (29) these conditions are satisfied for free theories for and also from (34) they imply . Unfortunately (39) indicates that there is no consequence that contrary to known results for free theories (). Perhaps a more complete analysis and/or imposition of stronger conditions such as quantum analogues of the energy conditions in classical general relativity [16] may give more information.

The above results may be relevant in discussing possible critical points in quantum field theories. Of course in physically realistic quantum field theories conformal invariance is broken by renormalisation effects arising from the need of some short distance cut off even if the classical theory is conformally invariant. As is well known the condition of scale invariance is transmuted into the renormalisation group or Callan-Symanzik equation where there are anomalous dimensions and also the function, reflecting the induced scale dependence of the couplings, is necessary. Since conformal invariance is much stronger than simple scale invariance it is interesting to discuss if there is any corresponding implications in quantum field theories. The problem with conformal transformations as compared with scale transformations is that, as is easily seen in (2) and (3) for , they lead to an effective local rescaling of the metric. However on curved space it is possible to formulate a local renormalisation group equation if besides the metric local -dependent couplings are introduced [17]. Although additional counterterms involving derivatives of the couplings are necessary renormalisability is unaffected (these counterterms are straightforward to calculate for standard theories to one and two loops and are related to additional local pieces in renormalisation group equations for local operators and their products [18]). For theories with a single dimensionless coupling , like QCD, then the basic local renormalisation group equation and also the equation expressing invariance under diffeomorphisms have the schematic forms

where is the vacuum self energy depending on the metric and coupling . Clearly for non constant consistency demands a non constant . For conformal transformations these equations may be combined and restricted to flat space giving

where now satisfies (2) and hence has the explicit form shown in (3).

Such equations may be extended to -point functions for arbitrary operators . For simplicity we consider only the application to two point functions for operators which are quasi-primary in the conformal limit when the corresponding broken conformal invariance equation may be written as

It is easy to check that the full generator corresponding to a conformal transformation appearing in (44) satisfies the required Lie algebra . At a fixed point, with constant, this reduces to the condition for conformal invariance, with dimensions , which was solved earlier as in (11). In general for constant and also (44) implies rotational and translational invariance for while for scale transformations where is constant then the usual renormalisation group equation is recovered

In general (44) requires -dependent couplings which is not very useful. However additional information may be obtained by considering an expansion in derivatives of at some convenient point. For scalar fields then we may thus write, with ,

where are just scalar functions of . If we take and hence then (44) gives the exact relation for terms of zeroth order in

Using the dependence of on this result is equivalent to (45) again together with the additional relation

Of course (48) implies that at a critical point must vanish if which coincides with the well known consequence of conformal invariance that the two point function must be zero if the operator dimensions are different. However this relation also has content away from any critical point since is perturbatively calculable. For instance knowing at loops is sufficient to give at loops if and is non zero at one loop. It is also possible from (44) to derive a standard renormalisation group equation for when we obtain

which is easily seen to be compatible with (45) and (48). Note that for which is necessary for symmetry under .

It is also of interest to consider the case of the two point function for a vector operator . In this case we may write for local couplings