Anomalies in gravity and how we quantize

Disclaimer: What I will relate in this post is the result of my research during the last year and specially the results of my last paper, uploaded as a preprint to with reference arXiv:1307.2060 1. Apart from the self-promotion, I think this is interesting since it can give people an insight on how short the steps in science are and specially how a theoretical physicist work is.

When talking about quantum field theory, one of the most important properties to study is the conservation of physical quantities, physical observables that always take the same value regardless of what happens. The most intuitive example of such a situation is momentum conservation. When the behavior of a bunch of particles is the same regardless of where they are or how far are them of each other, that means that there is no possible acceleration induced in the system by particle interaction or, in other words, that the velocity of any particle is always the same and momentum conservation follows immediately from this fact.

This argument can be refined and stated in the form of a mathematical theorem saying that whenever there is a symmetry, a particular situation that does not change the physical behavior of a system (like changing the distance of the particles in the example before), there is a quantity which is conserved. This asseveration lays in the deep core of Quantum Field Theory and conserved quantities are actually the pillars above we construct every physical theory. For example, Quantum Electrodynamics, the theory that explain how electric and magnetic fields behave, is constructed upon the fact that the electric charge is conserved and General Relativity, Einstein’s theory of gravitation, can be almost uniquely derived under the assumption that energy, in a particular relativistic form called Energy-Momentum tensor, is conserved.

Therefore, conserved quantities are strongly important and, when studying very high energy physics, it is cherished believed that a symmetry called scale invariance should play an important role in any quantum theory. The support for this idea is based in the equilibrium between kinetical energy and potential energy for a physical system. Normally, any system has both kinds of energy, the first of them depending upon velocity (or momentum in a general setting) and the second one given in terms of relevant physical quantities such as masses or charges of different particles. While we increase the energy of a system (by heating it, for example) it can be showed that the only part of the full energy that increases is the kinetic energy, being the potential energy the same (or slow varying) in any moment. Because of this, when the total energy is very high, one can neglect the potential term and assume that physical systems only have kinetic energy. Thus, all physical scales drop out (since they are contained in the potential energy) and the theory is said to be scale invariant. This has, of course, physical meaning. When a scale is present, imagine for example a single mass of a particle, the physics below and above this scale is different. In our example, if the energy is above the mass of the particle, this can decay into other things; but this is not possible if the energy is below the mass scale. Then, what dropping all the scales and therefore scale invariance means is that physics behaves in the same way for any value of the energy.

This fact is very important when considering a quantum theory since all the problems possibly caused by the quantization behavior are gonna be contained only on the high energy regime. But if the theory is scale invariant, this means that high energy behavior cannot be different from los energy behavior and therefore the quantization process is quite trivial if the theory is classically (low energy) well behaved.

The reason for us to focus on scale invariance is that it can be show that when a theory of gravity has this kind of symmetry, which for gravitational theories upgrades to a more general symmetry named Weyl invariance, the cosmological constant is not dynamical and its value is given in terms of initial conditions and does not requires the introduction of some exotic dark energy or quintessence of any type. This means that a successful quantum theory of gravity incorporating this symmetry would be not only a well behaved theory but also would solve the problem of the cosmological constant.

The main problem that arises here is that when one considers gravity, scale(or Weyl) invariance can easily be anomalous. Even if the corresponding conserved quantity is indeed conserved in the classical theory, it is not in the full quantum theory!!! Of course, if this happens, the theory is completely useless for the purposes we just mentioned here and we have to throw it to the garbage can.

Figure 1.The decay of pions into photons is the most known example in which an anomaly (here called chiral anomaly) produces a physical measurable effect. The rate of decay is highly enhanced from what we would expected if there was no anomaly at all. | Credit: Wikimedia Commons

Motivated by other recent work we performed over the last year 2 (arXiv) , we thus tried to understand when Weyl symmetry is anomalous or not and what is need to avoid the anomaly. In order to do that, we turned our attention to the simplest theory of gravity we can construct that also enjoys Weyl invariance, a theory in which a single scalar field, named dilaton, interacts with gravity and that we dub Tautological Weyl Gravity (TWG).

The reason for this name is that the Weyl symmetry of this theory is indeed tautological, in the sense that TWG can be regarded just as General relativity (which is not Weyl invariant) with a change of variables (simplifying, we wrote the graviton in a different way in the equations) so when reverted, Weyl symmetry results to be spurious and do nothing.

Since it is a tautological symmetry one should think that it is not possible that it could be anomalous, because it can always be reabsorbed in just a mathematical redefinition and besides that, anomalies have physical measurable effects that spurious symmetries should not cause. However, what we have showed is that even if it is a put-by-hand feature, quantization is not consistent with it and it is anomalous indeed!!!

This surprising fact has implications for the understanding of how quantization has to be performed when having Weyl invariant gravitational theories. The usual approach is that, when computing quantum corrections, we have to use some process named dimensional regularization in order to get rid of infinities that appear in the computations but do not have any real physical meaning.

Our result imply that when dealing with gravity, this process has to be slightly modified in a particular way that has an unexpected physical meaning. As we stated in the introduction of this post, General Relativity is constructed upon some special form of conservation of energy, which is related to certain symmetry called diffeomorphism invariance. What we found is that if we do not want the spurious symmetry to be anomalous, one has to get rid of this diffeomorphism symmetry. There is, quantization of gravity should unavoidably lead to a lack of symmetry, whether there is a loss of diffeomorphism invariance down to what is called transverse diffeomorphism invariance only or a loss of Weyl invariance, even if it is a tautological one. It is impossible to conserve both symmetries in the quantization process.

And even more surprising, at least for me, is the fact that the theory of gravity that enjoys both transverse diffeomorphism invariance and Weyl symmetry is unique and is named Unimodular Gravity, a theory that is classically identical to General Relativity (thus satisfying all experimental tests of gravity performed worldwide) but that has this relevant difference in the quantum regime.

However, as with everything in science, we have to be careful with our asseverations. We are not saying that Unimodular Gravity is a better theory of gravity than General Relativity. What we have showed instead is that when dealing with gravity at short distances, we have to be careful of how we apply the quantization process and that Unimodular Gravity is a good test ground for it.

In science, only few people like Newton or Einstein are able to sit upon giant’s shoulders and thus walk giant steps. The rest of us we are condemned to short walks, little results that continuously add in order to achieve important revelations about our universe. But the important thing is that it works, that is science.

Figure 2. Newton could sit upon giant’s shoulders. We definitely have to take the long path to knowledge. | Credit: Wikimedia Commons


  1. Enrique Alvarez & Mario Herrero-Valea (2013). Pseudo Weyl invariance is still anomalous, arXiv:
  2. Álvarez E. & Herrero-Valea M. (2013). No conformal anomaly in unimodular gravity, Physical Review D, 87 (8) DOI:

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