**Tim Koslowski, Perimeter Institute**

**Title:**Effective Field Theories for Quantum Gravity form Shape Dynamics

PDF of the talk (0.5Mb) Audio [.wav 31MB], Audio [.aif 3MB].

By Astrid Eichhorn, Perimeter Institute

Gravity and Quantum Physics have resisted to be unified into a common theory for several decades. We know a lot about the classical nature of gravity, in the form of Einstein's theory of General Relativity, which is a field theory. During the last century, we have learnt how to quantize other field theories, such as the gauge theories in the Standard Model of Particle Physics. The crucial difference between a classical theory and a quantum theory lies in the effect of quantum fluctuations. Due to Heisenberg's uncertainty principle, quantum fields can fluctuate, and this changes the effective dynamics of the field. In a classical field theory, the equations of motion can be derived minimizing a function called the classical action. In a quantum field theory, the equations of motion for the mean value of the quantum field cannot be derived from the classical action. Instead, they follow from something called the effective action, which contains the effect of all quantum fluctuations. Mathematically, to incorporate the effect of quantum fluctuations, a procedure known as the path integral has to be performed, which, even within perturbation theory (where one assumes solutions differ little from a known one), is a very challenging task. A method to make this task doable is the so called (functional) Renormalization Group: Not all quantum fluctuations are taken into account at once, but only those with a specific momentum, usually starting with the high-momentum ones. In a pictorial way, this means that we "average" the quantum fields over small distances (corresponding to the inverse of the large momentum). The effect of the high-momentum fluctuations is then to change the values of the coupling constants in the theory: Thus the couplings are no longer constant, but depend on the momentum scale, and so we more appropriately should call them running couplings. As an example, consider Quantum Electrodynamics: We know that the classical equations of motion are linear, so there is no interaction between photons. As soon as we go over to the quantum theory, this is different: Quantum fluctuations of the electron field at high momenta induce a photon-photon interaction (however one with a very tiny coupling, so experimentally this effect is difficult to see).

The question, if a theory can be quantized, i.e., the full path integral can be performed, then finds an answer in the behavior of the running couplings: If the effect of quantum fluctuations at high momenta is to make the couplings divergent at some finite momentum scale, the theory is only an effective theory at low energies, but not a fundamental theory. On the technical side this implies that when we perform integrals that take into account the effect of quantum fluctuations, we cannot extend these to arbitrarily high momenta, instead we have to "cut them off" at some scale.

The physical interpretation of such a divergence is that the theory tells us that we are really using effective degrees of freedom, not fundamental ones. As an example, if we construct a theory of the weak interaction between fermions without the W-bosons and the Z-boson, the coupling between the fermions will diverge at a scale which is related to the mass scale of the new bosons. In this manner, the theory lets us know that new degrees of freedom - the W- and Z-boson - have to be included at this momentum scale. One example that we know of a truly fundamental theory, i.e., one where the degrees of freedom are valid up to arbitrarily high momentum scales, is Quantum Chromodynamics. Its essential feature is the ultraviolet-attractive Gaussian (one that corresponds to a free, non-interacting theory) fixed point , which is nothing but the statement that the running coupling in QCD weakens towards high momenta, which is called asymptotic freedom (since asymptotically, at high momenta, the theory becomes non-interacting, i.e., free).

There is nothing wrong with a theory that is not fundamental in this sense, it simply means that it is an effective theory, which we can only use over a finite range of momenta. This concept is well-known in physics, and used very successfully. For instance, in condensed-matter systems, the effective degrees of freedom are, e.g., phonons, which are collective excitations of an atom lattice, and obviously cease to be a valid description of the system on distance scales below the atomic scale.

Quantum gravity actually exists as an effective quantum field theory, and quantum gravity effects can be calculated, treating the space-time metric as a quantum field like any other. However, being an effective theory means that it will only describe physics over a finite range of scales, and will presumably break down somewhere close to a scale known as the Planck scale (10^-33 in centimeters). This implies that we do not understand the microscopic dynamics of gravity. What are the fundamental degrees of freedom, which describe quantum gravity beyond the Planck scale, what is their dynamics and what are the symmetries that govern it?

The question, if we can arrive at a fundamental quantum theory of gravity within the standard quantum field theory framework, boils down to understanding the behavior of the running couplings of the theory. In perturbation theory, the answer has been known for a long time: (in four space-time dimensions) instead of weakening towards high momenta, the Newton coupling increases. More formally, this means that the free fixed point (technically known as Gaussian fixed point) is not ultraviolet-attractive. For this reason, most researchers in quantum gravity gave up on trying to quantize gravity along the same lines as the gauge theories in the Standard Model of particle physics. They concluded that the metric does not carry the fundamental microscopic degrees of freedom of a continuum theory of quantum gravity, but is only an effective description valid at low energies. However, the fact that the Gaussian fixed point is not ultraviolet-attractive really only means that perturbation theory breaks down. Beyond perturbation theory, there is the possibility to obtain a fundamental quantum field theory of gravity: The arena in which we can understand this possibility is called theory space. This is an (infinite dimensional) space, which is spanned by all running couplings which are compatible with the symmetries of the theory. So, in the case of gravity, theory space usually contains the Newton coupling, the cosmological constant, couplings of curvature-squared operators, etc. At a certain momentum scale, all these couplings have some value, specifying a point in theory space. Changing the momentum scale, and including the effect of quantum fluctuations on these scales, implies a change in the value of these couplings. Thus, when we change the momentum scale continuously, we flow through theory space on a so-called Renormalization Group (RG) trajectory. For the couplings to stay finite at all momentum scales, this trajectory should approach a fixed point at high momenta (more exotic possibilities such as limit cycles, or infinitely extendible trajectories could also exist). At a fixed point, the values of the couplings do not change anymore, when further quantum fluctuations are taken into account. Then, we can take the limit of arbitrarily high momentum scale trivially, since nothing changes if we go to higher scales, i.e., the theory is scale invariant. The physical interpretation of this process is, that the theory does not break down at any finite scale: The degrees of freedom that we have chosen to parametrize the physical system are valid up to arbitrarily high scales. An example is given by QCD, which as we mentioned is asymptotically free, the physical interpretation being that quarks and gluons are valid microscopic degrees of freedom. There is no momentum scale at which we need to expect further particles, or a possible substructure of quarks and gluons.

In the case of gravity, to quantize it we need a non-Gaussian fixed point. At such a point, where the couplings are non-vanishing, the RG flow stops, and we can take the limit of arbitrarily high momenta. This idea goes back to Weinberg, and is called asymptotic safety. Asymptotically, at high momenta, we are "safe" from divergences in the couplings, since they approach a fixed point, at which they assume some finite value. Since finite couplings imply finiteness of physical observables (when the couplings are defined appropriately), an asymptotically safe theory gives finite answers to all physical questions. In this construction, the fixed point defines the microscopic theory, i.e., the interaction of the microscopic degrees of freedom.

As a side remark, if, being confronted with an infinite-dimensional space of couplings, you might worry about how such a theory can ever be predictive, note that fixed points come equipped with what is called a critical surface: Only if the RG flow lies within the critical surface of a fixed point, it will actually approach the fixed point at high momenta. Therefore a finite-dimensional critical surface means that the theory will only have a finite number of parameters, namely those couplings spanning the critical surface. The low-momentum value of these couplings, which is accessible to measurements, is not fixed by the theory: Any value works, since they all span the critical surface. On the other hand, infinitely many couplings will be fixed by the requirement of being in the critical surface. This automatically implies that we will get infinitely many predictions from the theory (namely the values of all these so-called irrelevant couplings), which we can then (in principle) test in experiments.

Two absolutely crucial ingredients in the search for an asymptotically safe theory of quantum gravity are the specification of the field content and the symmetries of the theory. These determine which running couplings are part of theory space. They are the couplings of all possible operators that can be constructed from the fundamental fields respecting the symmetry have to be included. Imposing an additional symmetry on theory space means that some of the couplings will drop out of it. Most importantly, the (non)existence of a fixed point will depend on the choice of symmetries. A well-known example is the choice of a U(1) gauge symmetry (like that in electromagnetism) versus an SU(3) one (like the one in QCD). The latter case gives an asymptotically free theory, the former one does not. Thus the (gauge) symmetries of a system crucially determine its microscopic behavior.

In gravity, there are several classically equivalent versions of the theory (i.e., they admit the same solutions to the equations of motion). A partial list contains standard Einstein gravity with the metric as the fundamental field, Einstein-Cartan gravity, where the metric is exchanged for the vielbein (a set of vectors) and a unimodular version of metric gravity (we will discuss it in a second). The first step in the construction of a quantum field theory of gravity now consists in the choice of theory space. Most importantly, this choice exists in the path-integral framework as well as the Hamiltonian framework. So, in both cases there is a number of classically equivalent formulations of the theory, which differ at the quantum level, and in particular, only some of them might exist as a fundamental theory.

To illustrate that the choice of theory space is really a physical choice, consider the case of unimodular quantum gravity: Here, the metric determinant is restricted to be constant. This implies, that the spectrum of quantum fluctuations differs crucially from the non-unimodular version of metric gravity, and most importantly, does not differ just in form, but in its physical content. Accordingly, the evaluation of Feynman diagrams in perturbation theory in both cases will yield different results. In other words, the running couplings in the two theory spaces will exhibit a different behaviour, reflected in the existence of fixed points as well as critical exponents, which determine the free parameters of the theory.

This is where the new field of shape dynamics opens up important new possibilities. As explained, theories, which classically describe the same dynamics, can still have different symmetries. In particular, this actually works for gauge theories, where the symmetry is nothing else but a redundancy of description. Therefore, only a reduced configuration space (the space of all possible configurations of the field) is physical, and along certain directions, the configuration space contains physically redundant configurations. A simple example is given by (quantum) electrodynamics, where the longitudinal vibration mode of the photon is unphysical (in vacuum), since the gauge freedom restricts the photon two have two physical (transversal) polarisations.

One can now imagine how two different theories with different gauge symmetries yield the same physics. The configuration spaces can in fact be different, it is only the values of physical observables on the reduced configuration space that have to agree. This makes a crucial difference for the quantum theory, as it implies different theory spaces, defined by different symmetries, and accordingly different behavior of the running couplings.

Shape dynamics trades part of the four-dimensionsional, i.e., spacetime symmetries of General Relativity (namely refoliation invariance, so invariance under different choices of spatial slices of the four-dimensional spacetime to become "space") for what is known as local spatial conformal symmetry; which implies local scale invariance of space. This also implies a key difference in the way that spacetime is viewed in the two theories. Whereas spacetime is one unified entity in General Relativity, shape dynamics builds up a spacetime from "stacking" spatial slices (for more details, see the blog entry by Julian Barbour.) Fixing a particular gauge in each of the two formulations then yields two equivalent theories.

Although the two theories are classically equivalent for observational purposes, their quantized versions will differ. In particular, only one of them might admit a UV completion as a quantum field theory with the help of a non-Gaussian fixed point.

A second possibility is that both theory spaces might admit the existence of a non-Gaussian fixed point, but what is known as the universality class might be different: Loosely speaking, the universality class is determined by the rate of approach to the fixed point, which is captured by what is known as the critical exponents. Most importantly, while details of RG trajectories typically depend on the details of the regularization scheme (this specifies how exactly quantum fluctuations in the path integral are integrated out), the critical exponents are universal. The full collection of critical exponents of a fixed point then determines the unversality class. Universality classes are determined by symmetries, which is very well-known from second order phase transitions in thermodynamics. Since the correlation length in the vicinity of a second-order phase transition diverges, the microscopic details of different physical systems do not matter: The behavior of physical observables in the vicinity of the phase transition is determined purely by the field content, dimensionality, and symmetries of a system.

Different universality classes can differ in the number of relevant couplings, and thus correspond to theories with a different "amount of predictivity". Thus classically equivalent theories, when quantized, can have a different number of free parameters. Accordingly, not all universality classes will be compatible with observations, and the choice of theory space for gravity is thus crucial to identify which universality class might be "realized in nature".

Clearly, the canonical quantization of standard General Relativity in contrast to shape dynamics will also differ, since shape dynamics actually has a non-trivial, albeit non-local, Hamiltonian.

Finally, what is known as doubly General Relativity is the last step in the new construction. Starting from the symmetries of shape dynamics, one can discover a hidden BRST symmetry in General Relativity. BRST symmetries are symmetries existing in gauge-fixed path integrals for gauge theories. To do perturbation theory requires the gauge to be fixed, thus yielding a path-integral action which is not gauge invariant. The remnants of gauge invariants are encoded in BRST invariance, so it can be viewed as the quantum version of a gauge symmetry.

In the case of gravity, BRST invariance connected to gauge invariance under diffeomorphisms of general relativity is supplemented by BRST invariance connected to local conformal invariance. This is what is referred to as symmetry doubling. Since gauge symmetries restrict the Renormalization Group flow in a theory space, the discovery of a new BRST symmetry in General Relativity is crucial to fully understand the possible existence of a fixed point and its universality class. Thus the newly discovered BRST invariance might turn out to be a crucial ingredient in constructing a quantum theory of gravity.

Fig 1: Electron fluctuations induce a non-vanishing photon-photon coupling in Quantum Electrodynamics.

The question, if a theory can be quantized, i.e., the full path integral can be performed, then finds an answer in the behavior of the running couplings: If the effect of quantum fluctuations at high momenta is to make the couplings divergent at some finite momentum scale, the theory is only an effective theory at low energies, but not a fundamental theory. On the technical side this implies that when we perform integrals that take into account the effect of quantum fluctuations, we cannot extend these to arbitrarily high momenta, instead we have to "cut them off" at some scale.

The physical interpretation of such a divergence is that the theory tells us that we are really using effective degrees of freedom, not fundamental ones. As an example, if we construct a theory of the weak interaction between fermions without the W-bosons and the Z-boson, the coupling between the fermions will diverge at a scale which is related to the mass scale of the new bosons. In this manner, the theory lets us know that new degrees of freedom - the W- and Z-boson - have to be included at this momentum scale. One example that we know of a truly fundamental theory, i.e., one where the degrees of freedom are valid up to arbitrarily high momentum scales, is Quantum Chromodynamics. Its essential feature is the ultraviolet-attractive Gaussian (one that corresponds to a free, non-interacting theory) fixed point , which is nothing but the statement that the running coupling in QCD weakens towards high momenta, which is called asymptotic freedom (since asymptotically, at high momenta, the theory becomes non-interacting, i.e., free).

There is nothing wrong with a theory that is not fundamental in this sense, it simply means that it is an effective theory, which we can only use over a finite range of momenta. This concept is well-known in physics, and used very successfully. For instance, in condensed-matter systems, the effective degrees of freedom are, e.g., phonons, which are collective excitations of an atom lattice, and obviously cease to be a valid description of the system on distance scales below the atomic scale.

Quantum gravity actually exists as an effective quantum field theory, and quantum gravity effects can be calculated, treating the space-time metric as a quantum field like any other. However, being an effective theory means that it will only describe physics over a finite range of scales, and will presumably break down somewhere close to a scale known as the Planck scale (10^-33 in centimeters). This implies that we do not understand the microscopic dynamics of gravity. What are the fundamental degrees of freedom, which describe quantum gravity beyond the Planck scale, what is their dynamics and what are the symmetries that govern it?

The question, if we can arrive at a fundamental quantum theory of gravity within the standard quantum field theory framework, boils down to understanding the behavior of the running couplings of the theory. In perturbation theory, the answer has been known for a long time: (in four space-time dimensions) instead of weakening towards high momenta, the Newton coupling increases. More formally, this means that the free fixed point (technically known as Gaussian fixed point) is not ultraviolet-attractive. For this reason, most researchers in quantum gravity gave up on trying to quantize gravity along the same lines as the gauge theories in the Standard Model of particle physics. They concluded that the metric does not carry the fundamental microscopic degrees of freedom of a continuum theory of quantum gravity, but is only an effective description valid at low energies. However, the fact that the Gaussian fixed point is not ultraviolet-attractive really only means that perturbation theory breaks down. Beyond perturbation theory, there is the possibility to obtain a fundamental quantum field theory of gravity: The arena in which we can understand this possibility is called theory space. This is an (infinite dimensional) space, which is spanned by all running couplings which are compatible with the symmetries of the theory. So, in the case of gravity, theory space usually contains the Newton coupling, the cosmological constant, couplings of curvature-squared operators, etc. At a certain momentum scale, all these couplings have some value, specifying a point in theory space. Changing the momentum scale, and including the effect of quantum fluctuations on these scales, implies a change in the value of these couplings. Thus, when we change the momentum scale continuously, we flow through theory space on a so-called Renormalization Group (RG) trajectory. For the couplings to stay finite at all momentum scales, this trajectory should approach a fixed point at high momenta (more exotic possibilities such as limit cycles, or infinitely extendible trajectories could also exist). At a fixed point, the values of the couplings do not change anymore, when further quantum fluctuations are taken into account. Then, we can take the limit of arbitrarily high momentum scale trivially, since nothing changes if we go to higher scales, i.e., the theory is scale invariant. The physical interpretation of this process is, that the theory does not break down at any finite scale: The degrees of freedom that we have chosen to parametrize the physical system are valid up to arbitrarily high scales. An example is given by QCD, which as we mentioned is asymptotically free, the physical interpretation being that quarks and gluons are valid microscopic degrees of freedom. There is no momentum scale at which we need to expect further particles, or a possible substructure of quarks and gluons.

In the case of gravity, to quantize it we need a non-Gaussian fixed point. At such a point, where the couplings are non-vanishing, the RG flow stops, and we can take the limit of arbitrarily high momenta. This idea goes back to Weinberg, and is called asymptotic safety. Asymptotically, at high momenta, we are "safe" from divergences in the couplings, since they approach a fixed point, at which they assume some finite value. Since finite couplings imply finiteness of physical observables (when the couplings are defined appropriately), an asymptotically safe theory gives finite answers to all physical questions. In this construction, the fixed point defines the microscopic theory, i.e., the interaction of the microscopic degrees of freedom.

As a side remark, if, being confronted with an infinite-dimensional space of couplings, you might worry about how such a theory can ever be predictive, note that fixed points come equipped with what is called a critical surface: Only if the RG flow lies within the critical surface of a fixed point, it will actually approach the fixed point at high momenta. Therefore a finite-dimensional critical surface means that the theory will only have a finite number of parameters, namely those couplings spanning the critical surface. The low-momentum value of these couplings, which is accessible to measurements, is not fixed by the theory: Any value works, since they all span the critical surface. On the other hand, infinitely many couplings will be fixed by the requirement of being in the critical surface. This automatically implies that we will get infinitely many predictions from the theory (namely the values of all these so-called irrelevant couplings), which we can then (in principle) test in experiments.

Fig 2: A non-Gaussian fixed point has a critical surface, the dimensionality of which corresponds to the number of free parameters of the theory.

Two absolutely crucial ingredients in the search for an asymptotically safe theory of quantum gravity are the specification of the field content and the symmetries of the theory. These determine which running couplings are part of theory space. They are the couplings of all possible operators that can be constructed from the fundamental fields respecting the symmetry have to be included. Imposing an additional symmetry on theory space means that some of the couplings will drop out of it. Most importantly, the (non)existence of a fixed point will depend on the choice of symmetries. A well-known example is the choice of a U(1) gauge symmetry (like that in electromagnetism) versus an SU(3) one (like the one in QCD). The latter case gives an asymptotically free theory, the former one does not. Thus the (gauge) symmetries of a system crucially determine its microscopic behavior.

In gravity, there are several classically equivalent versions of the theory (i.e., they admit the same solutions to the equations of motion). A partial list contains standard Einstein gravity with the metric as the fundamental field, Einstein-Cartan gravity, where the metric is exchanged for the vielbein (a set of vectors) and a unimodular version of metric gravity (we will discuss it in a second). The first step in the construction of a quantum field theory of gravity now consists in the choice of theory space. Most importantly, this choice exists in the path-integral framework as well as the Hamiltonian framework. So, in both cases there is a number of classically equivalent formulations of the theory, which differ at the quantum level, and in particular, only some of them might exist as a fundamental theory.

To illustrate that the choice of theory space is really a physical choice, consider the case of unimodular quantum gravity: Here, the metric determinant is restricted to be constant. This implies, that the spectrum of quantum fluctuations differs crucially from the non-unimodular version of metric gravity, and most importantly, does not differ just in form, but in its physical content. Accordingly, the evaluation of Feynman diagrams in perturbation theory in both cases will yield different results. In other words, the running couplings in the two theory spaces will exhibit a different behaviour, reflected in the existence of fixed points as well as critical exponents, which determine the free parameters of the theory.

This is where the new field of shape dynamics opens up important new possibilities. As explained, theories, which classically describe the same dynamics, can still have different symmetries. In particular, this actually works for gauge theories, where the symmetry is nothing else but a redundancy of description. Therefore, only a reduced configuration space (the space of all possible configurations of the field) is physical, and along certain directions, the configuration space contains physically redundant configurations. A simple example is given by (quantum) electrodynamics, where the longitudinal vibration mode of the photon is unphysical (in vacuum), since the gauge freedom restricts the photon two have two physical (transversal) polarisations.

One can now imagine how two different theories with different gauge symmetries yield the same physics. The configuration spaces can in fact be different, it is only the values of physical observables on the reduced configuration space that have to agree. This makes a crucial difference for the quantum theory, as it implies different theory spaces, defined by different symmetries, and accordingly different behavior of the running couplings.

Shape dynamics trades part of the four-dimensionsional, i.e., spacetime symmetries of General Relativity (namely refoliation invariance, so invariance under different choices of spatial slices of the four-dimensional spacetime to become "space") for what is known as local spatial conformal symmetry; which implies local scale invariance of space. This also implies a key difference in the way that spacetime is viewed in the two theories. Whereas spacetime is one unified entity in General Relativity, shape dynamics builds up a spacetime from "stacking" spatial slices (for more details, see the blog entry by Julian Barbour.) Fixing a particular gauge in each of the two formulations then yields two equivalent theories.

Although the two theories are classically equivalent for observational purposes, their quantized versions will differ. In particular, only one of them might admit a UV completion as a quantum field theory with the help of a non-Gaussian fixed point.

A second possibility is that both theory spaces might admit the existence of a non-Gaussian fixed point, but what is known as the universality class might be different: Loosely speaking, the universality class is determined by the rate of approach to the fixed point, which is captured by what is known as the critical exponents. Most importantly, while details of RG trajectories typically depend on the details of the regularization scheme (this specifies how exactly quantum fluctuations in the path integral are integrated out), the critical exponents are universal. The full collection of critical exponents of a fixed point then determines the unversality class. Universality classes are determined by symmetries, which is very well-known from second order phase transitions in thermodynamics. Since the correlation length in the vicinity of a second-order phase transition diverges, the microscopic details of different physical systems do not matter: The behavior of physical observables in the vicinity of the phase transition is determined purely by the field content, dimensionality, and symmetries of a system.

Different universality classes can differ in the number of relevant couplings, and thus correspond to theories with a different "amount of predictivity". Thus classically equivalent theories, when quantized, can have a different number of free parameters. Accordingly, not all universality classes will be compatible with observations, and the choice of theory space for gravity is thus crucial to identify which universality class might be "realized in nature".

Clearly, the canonical quantization of standard General Relativity in contrast to shape dynamics will also differ, since shape dynamics actually has a non-trivial, albeit non-local, Hamiltonian.

Finally, what is known as doubly General Relativity is the last step in the new construction. Starting from the symmetries of shape dynamics, one can discover a hidden BRST symmetry in General Relativity. BRST symmetries are symmetries existing in gauge-fixed path integrals for gauge theories. To do perturbation theory requires the gauge to be fixed, thus yielding a path-integral action which is not gauge invariant. The remnants of gauge invariants are encoded in BRST invariance, so it can be viewed as the quantum version of a gauge symmetry.

In the case of gravity, BRST invariance connected to gauge invariance under diffeomorphisms of general relativity is supplemented by BRST invariance connected to local conformal invariance. This is what is referred to as symmetry doubling. Since gauge symmetries restrict the Renormalization Group flow in a theory space, the discovery of a new BRST symmetry in General Relativity is crucial to fully understand the possible existence of a fixed point and its universality class. Thus the newly discovered BRST invariance might turn out to be a crucial ingredient in constructing a quantum theory of gravity.