# Fixed Points of Quantum Gravity

1. Dec 13, 2003

Staff Emeritus
A very nice short paper,

http://arxiv.org/abs/hep-th/0312114

giving an analytical (Wilson renormalization group) development of the fixed points of the quantum gravity models as previously suggested by numerical studies.

Conclusions,

1) At very short distances, the coupling runs and gravity should become weaker as the strong force does.

2) If the fixed point research is confirmed, then quantum gravity is renormalizable.

2. Dec 14, 2003

### marcus

I've done a brief bit of grubbing around in the references to find the main roots of the paper and have come up with

Martin Reuter
Nonperturbative Evolution Equation for Quantum Gravity
http://arxiv.org/abs/hep-th/9605030
(Physic Review D, 1998)

Martin Reuter
Newton's Constant Isn't Constant
http://arxiv.org./hep-th/0012069
he calls it a "brief pedagogical introduction"
it is a short paper and describes the "running" of G
by analogy with other field theories

found other papers too but no time to type in the titles
it would be good to have some background (can you fill in some?)
on this line of research
I have seen papers by Lauscher and Reuter and Saueressig
as well as the authors of the one you mentioned that seems
to be a school of quantizing GR in a direct head-on way
that just persists in trying to overcome the problems
with renormalization that surfaced, what, 30 or more years ago?
and they seem to be making a bit of progress. can you clarify

3. Dec 14, 2003

### marcus

another paper in this research line, this one's abstract
gives an idea of the common approach I think. It is a long paper though, 99 pages. Published in Physics Review D in 2002

Oliver Lauscher and Martin Reuter
"Ultraviolet Fixed Point and Generalized Flow Equation of Quantum Gravity"
http://arxiv.org/hep-th/0108040

I quote the abstract with a key sentence bold:

"A new exact renormalization group equation for the effective average action of Euclidean quantum gravity is constructed. It is formulated in terms of the component fields appearing in the transverse-traceless decomposition of the metric. It facilitates both the construction of an appropriate infrared cutoff and the projection of the renormalization group flow onto a large class of truncated parameter spaces. The Einstein-Hilbert truncation is investigated in detail and the fixed point structure of the resulting flow is analyzed. Both a Gaussian and a non-Gaussian fixed point are found. If the non-Gaussian fixed point is present in the exact theory, quantum Einstein gravity is likely to be renormalizable at the nonperturbative level. In order to assess the reliability of the truncation a comprehensive analysis of the scheme dependence of universal quantities is performed. We find strong evidence supporting the hypothesis that 4-dimensional Einstein gravity is asymptotically safe, i.e. nonperturbatively renormalizable. The renormalization group improvement of the graviton propagator suggests a kind of dimensional reduction from 4 to 2 dimensions when spacetime is probed at sub-Planckian length scales."

an odd twist here: these people arent even using Ashtekar-Sen 1986 "new variables"----they are stubbornly persisting in the effort to quantize the METRIC the original form of the gravitational field in hoary old 1915 GR.

This was what Wheeler-DeWitt were trying to do, and all those people Pauli, Dirac, Feynmann, in the Fifties and Sixties or even earlier, Rovelli has a fascinating detailed history of the effort to quantize GR in an appendix at the end of his book "Quantum Gravity"

According to Rovelli's timeline, in 1973 it was found by t'Hooft that GR was not renormalizable (divergences in GR with matterfields that could not be made to go away) and this was confirmed by others, Veltmann, Deser, ...
By 1975, says Rovelli, it was generally accepted that GR was non-renormalizable.
This was one of the things, I gather, which led people to take up with string theory (beginning around 1984) and abandon the attempt to quantize General Relativity as such.

And also one of the things that motivated people taking up with "new variable" and the LQG approach---since they couldnt seem to quantize the metric itself---beginning around 1986 or soon after.

But here are Lauscher and Reuter and others including the author of the paper you just flagged still grinding away at the problem! No strings, no loops, no monkeybusiness: just quantize the metric please

selfAdjoint, want to give a thumbnail sketch of what the story is about asympt. safety, general practice of renormalizing as applies here, what they mean by fixed point---what is the flow that has a fixed point?---or any other background.

4. Dec 14, 2003

Staff Emeritus
The best introduction to "RG, fixed points, and all that" is by
John Baez .

Nonperturbative asymptotic safety is an idea of Stephen Weinberg's, dating back to the early days of Wilson's Renormalization Group (1976). Here is a brief discription, from a paper (hep-th/0305208) that isn't relevant to quantum gravity, Weinberg's safety idea concerns gauge theories. The quopte will also explain your questions about the "old" quantum gravity, I think.

Nevertheless, perturbative nonrenormalizability does not constitute a “no-go” theorem. Despite this tarnish, theories can be fundamental and mathematically consistent down to arbitrarily small length scales, as proposed in Weinberg’s “asymptotic safety” scenario
[3]. It assumes the existence of a non-Gaußian (=nonzero) UV fixed point under the renormalization group (RG) operation at which the continuum limit can be taken. The theory is “nonperturbatively renormalizable” in Wilson’s sense. If the non-Gaußian fixed
point is UV attractive for finitely many couplings in the action, the RG trajectories along which the theory can flow as we send the cutoff to infinity are labeled by only a finite number of physical parameters. Then the theory is as predictive as any perturbatively
renormalizable theory, and high-energy physics can be well separated from low-energy physics without tuning (infinitely) many parameters.

So all is not lost if perturbative methods have shown a theory (like early quantum gravity) to be nonrenormalizable, It may yet turn out to be nonpertubatively renormalizable!

5. Dec 22, 2003

### dlgoff

Is there a simple (classical way of looking at it) equation for the strong force like that of gravity (F=Gm1m2/r^2)? If so, could these be put together somehow?

6. Dec 23, 2003

### jeff

No:

The dynamical structure of classical theories doesn't depend on the energy scales characteristic of given processes.

But the need for renormalization prescriptions in QFT - unnecessary in classical physics - allowing perturbation theory at small coupling to remain valid at high energies requires the couplings (and other parameters, but these have no classical correspondence) scale with energy in a way having no analog in classical physics.

Last edited: Dec 23, 2003
7. Dec 23, 2003

### dlgoff

So why do I hear the strong force holding quarks together becomes stronger as the "distance" between them increases and when they are "near" to eachother the force approaches zero? Sounds like there should be some way of setting up an equation of force vs. distance.

8. Dec 24, 2003

### jeff

There is, just not in the sense you're dreaming of. The magnitude of a coupling constant determines the strength of the associated interactions. For example, newton's constant G in his gravitational force law familiar from high school physics is rather small and thus gravity is weak in comparison to, say, the electromagnetic interaction.

But couplings in quantum theories must in general acquire a dependence on energy, or equivalently by the uncertainty principle, on distance, whose origin is purely quantum theoretic. The alteration of the conventional relation between interaction strength (i.e., force) and distance due to this is described by the renormalization group equations giving this dependence and these have no classical counterparts.

These RG equations give the asymptotic behaviour of couplings at high energies (i.e., short distances). For theories with a single type of interaction, the associated couplings can exhibit four possible types of behaviour. One of these has the coupling flowing to zero as energies become arbitrarily large. This is known as asymptotic freedom. But the reasons why some theories are asymptotically free can't be explained by appeals to classical analogy.

The flip side of asymptotic freedom is infrared slavery - the increase in strength of the strong force at low energies. However, since $g$ grows as energy increases, it's no longer possible to compute the RG equations using perturbation theory. This complicates discovering why free quarks aren't observed. This property, called confinement, though plausible given the scaling properties in QCD of the strong coupling and the failure to observe free quarks, is yet to be proven and thus fully understood (note that asymptotic freedom doesn't imply confinement: Any non-abelian yang-mills theory can be asymptotically free without necessarily exhibiting confinement. In fact, this is the case with electroweak theory). Permanent confinement is a rather bizarre concept since in elementary physics forces decrease with distance. There is one system in which confinement happens and which can be described in reasonably transparent terms, but we don't know how valid for the strong interaction these ideas are.

So consider a magnetic monopole in a superconductor. A quantized amount of magnetic flux comes out of the monopole, but according to the Meissner effect, a superconductor expels magnetic flux. Thus a single magnetic monopole can't live inside a superconductor. Now consider an antimonopole a distance R away. The magnetic flux coming out of the monopole can flow into the antimonopole, forming a tube connecting them and obliging the superconductor to give up being a superconductor in the region of the flux tube. It's then no longer energetically favourable for the field to be constant everywhere; instead it vanishes in the region of the flux tube. The energy cost of this arrangement grows with R so it costs more and more energy to increase the separation of monopoles from antimonopoles. The result is the confinement of monopoles inside a superconductor. This is a popular mode of thought in the context of confinement of the QCD analogs of the aforementioned magnetic monopoles and antimonopoles, namely quarks and antiquarks. In fact the idea of quarks and antiquarks being connected by a flux tube was one of the original concepts motivating string theories.

Although your question was about the strong force, there is something simple and intuitive - though still not classical - one can say about the electromagnetic force in the context of QED since this theory isn't asymptotically free with the electromagnetic coupling growing with increasing energy. The idea is that this scaling is closely related to the polarization of the vacuum due to a given electric charge it surrounds.

The quantum vacuum surrounding a charge consists of virtual electron-positron pairs popping into and out of existence. These pairs form "dipoles" oriented with positrons nearer than electrons to the charge. The effect is that the charge is shielded at larger distances so that photons interact more strongly with the charge the closer they are to it.

Last edited: Dec 27, 2003
9. Dec 25, 2003

### dlgoff

OK. Since you can't move from the classical to the quantum in a possible quantum gravity conection to the strong force, is it possible to look at these RG equations to find at what energy the strong force coupling constant may coincide with that of gravity; that is with trying to find a conection to G at short distances.

Thanks Jeff and Merry Christmass

10. Dec 25, 2003

Staff Emeritus
Jeff, would it be at all possible to motivate the original Gell-Mann & Low paper that described running in a sort of pre-wilson way? Rather than trying to explain the whole RG approach (which isn't very intuitive even in the advanced textbooks).

11. Dec 26, 2003

### jeff

The method of what eventually (and misleadingly) came to be known as the renormalization group was originated in work by gell-mann and low to address a failure of perturbation theory in QED at high energies associated with the way naive renormalization prescriptions gave rise to the appearance in amplitudes of factors (familiar to anyone who’s taken an introductory course in QFT) of &alpha;ln(q&sup2;/m&sup2;) in which q, me and &alpha; are respectively internal momentum, electron mass and fine structure constant. The problem is that at high energies such factors will obviously diverge even for small &alpha; rendering perturbation theory useless.

The key idea was to introduce coupling constants g&mu; defined at a sliding scale &mu; so that by choosing &mu; to be of the same order as the energy E typical of the process in question, the factors ln(E/&mu;) were rendered harmless. Perturbative methods then remain valid as long as g&mu; remains small.

In particular, given g&mu;, amplitudes could be calculated perturbatively at energy &mu;+d&mu;, and then used to calculate g&mu;+d&mu;. By integrating the resulting differential equation the coupling constants at the scale of interest could be related to the coupling constants as conventionally defined.

I’ll sketch the basic idea in terms of an amplitude A(E, g, m) depending on an overall energy scale E, various dimensionless coupling constants and masses represented collectively by g and m with it’s dependence on other dimensionless quantities understood. If A(E,g,m) has dimensionality [mass]D, then A(E, g, m) = EDA(1, g, m/E) so that, naively, in the limit E &rarr; &infin; we have

A(E, g, m) &rarr; EDA(1, g, m/E).

However A(E, g, m) is usually formally divergent, so to calculate, a cutoff &Lambda; is introduced yielding the regulated quantity A(&Lambda;; E, g, m). Then the conventional renormalization prescription to bury &Lambda; in higher order corrections – and this is the point - yields a renormalized quantity AR(E, g, m) that doesn’t have this simple power law behaviour but instead the factor ED is accompanied by powers of ln(E/m).

As discussed above, we introduce a renormalized coupling g(&mu;) that depends on a sliding energy scale &mu; which (at least for &mu; >> m) has no dependence on the scale m of the masses of the theory. Then A(E, g ,m) may be expressed as a function of g&mu; & &mu;, instead of gE. Dimensional analysis shows such functions may be written as

A(E, g&mu;, m/E, &mu;/E) = EDA(1, g&mu;, m/E, &mu;/E).

But &mu; is completely arbitrary so we can take &mu; = E giving

A(E, g&mu;, m, &mu;) = EDA(1, gE, m/E, 1).

Now, since gE doesn’t depend on m for m << E, there are no large logarithms, so perturbation theory may be used to calculate A(E, g&mu;, m, &mu;) in terms of gE as long as gE itself remains sufficiently small. In particular, to any finite order of perturbation theory, as E &rarr; &infin;, A(E, g&mu;, m, &mu;) has the asymptotic behaviour

A(E, g&mu;, m, &mu;) &rarr; EDA(1, gE, 0, 1).

Reminding ourselves that the point of the RG as it was originally introduced was to allow calculation at high energies E perturbatively, let’s see how the preceding considerations allow us to calculate gE. Consider a regulated but unrenormalized amplitude A(&Lambda;; s, g, m) with cut-off &Lambda; and in which s is a collective coordinate for the squares of all external momenta. Working to some given order in g, the conventional renormalized coupling gR is defined by

gR &equiv; A(&Lambda;; 0, g, m),

that is, gR &equiv; g&mu;=0. Then the renormalized amplitude is defined by

AR &equiv; A(s, gE, m).

Now we define

g&mu; &equiv; A(&Lambda;; &mu;, g, m)

which in terms of the renormalized coupling is

g&mu;=A(&mu;, gE, m).

Then for a real scalar field &phi; with lagrangian density

$L=-\frac{1}{2}\partial_{\nu}\phi\partial_{\nu}\phi-\frac{1}{2}m^2\phi^2 \frac{1}{24}g\phi^4$,

this is to second order in the renormalized coupling

(1) $g_\mu=g_R+\frac{3g_R{}^2}{32\pi^2}\int_{0}^{1}dx\ln{[1+\frac{\mu^2x(1-x)}{m^2}]}+O(g_R^2)$.

The point is that this formula is reliable only if the correction term is smaller than gR; that is, only if |gRln(&mu;/m)| << 1: If this were the case for &mu; &cong; E, we wouldn’t need RG methods since ordinary perturbation theory would remain valid.

Thus, rather than working directly with large &mu;, we must instead proceed in stages: g&mu; may be calculated in terms of gR as long as &mu;/m isn’t much larger than unity; then g&mu;’ may be calculated in terms of g&mu; as long as &mu;’/&mu; is not much larger than unity, and so on up to gE.

Instead of discrete stages, this may be done continuously. Dimensional analysis gives the relation between g&mu;’ and g&mu; as

g&mu;’ = G(g&mu;, &mu;’/&mu;, m/&mu;).

Differentiating with respect to &mu;' and then setting &mu;' = &mu; yields the differential equation

&mu;dg&mu;/d&mu;=&beta;(g&mu;, m/&mu;)

where

(2) &beta;(g&mu;, m/&mu;) &equiv; [&part;G(g&mu;, z, m/&mu;)/&part;z]z=1.

For &mu; << m we have

&mu;dg&mu;/d&mu; = &beta;(g&mu;, 0) = &beta;(g&mu;)

which is the gell-mann-low form of the callan-symanzik equation. We are to calculate gE by integrating this equation with initial value gM at some scale &mu; = M chosen large enough that for &mu; &ge; M, the masses m may be neglected compared with &mu;, but not so large that ln(M/m) is too big to perturbatively calculate gM in terms of the conventional renormalized coupling constant gR. The solution may be formally written

$\ln{(E/M)}=\int_{g_M}^{g_E}dg/\beta(g)$

as long as &beta;(g) doesn’t vanish between gM and gE.

The preceding results don’t rely on perturbation theory, but the functions G and &beta; must usually be calculated perturbatively. As an example, consider (1), but where the bare coupling g is expressed in terms of g&mu; instead of gR. This gives

$g_{{\mu}'}=g_\mu-\frac{3g_\mu^2}{32\pi^2}\int_{0}^{1}dx\ln{[\frac{m^2+\mu^2x(1-x)}{ m^2+\mu’^2x(1-x)}]}+O(g_R^3)$.

Then (2) gives

$\beta(g_\mu,m/\mu)= \frac{3g_\mu^2}{16\pi^2}\int_{0}^{1}dx\ln{[\frac{\mu^2x(1-x)}{ m^2+\mu^2x(1-x)}]}+ O(g_R^3 )$.

which for &mu; >> m is

&beta;(g&mu;) = 3g&mu;&sup2;/16pi&sup2; + O(gR&sup3;).

A couple of closing remarks:

Firstly, I pointed out in a previous post in this thread that in the case of QED - studied by gell-man & low - the growth of the coupling with increasing energy can be understood (at least at energies that aren't too high) in terms of shielding of a bodies charge at larger distances by the vacuum polarization it gives rise to.

Secondly, to avoid large logarithms, operator normalizations must also scale (though this scaling behaviour depends generally on that of the couplings as well). But the case that gell-man & low studied was QED which is special in that the scaling of the coupling completely determines that of the operator normalization. In any event, these normalizations satisfy RG equations somewhat analogous to the one's satisfied by couplings.

Last edited: Dec 30, 2003
12. Dec 27, 2003

### dlgoff

original Gell-Mann & Low paper? Thanks for your help.

13. Dec 28, 2003

### jeff

It's possible, but before we reach that energy, we'd reach the energy at which the fundamental non-gravitational interactions unify, and then we'd proceed to integrate gravity at some still higher energy.

14. Dec 29, 2003

### dlgoff

I'm not sure I understand. Do you mean that the EM interaction will unify with the strong force before gravity (since gravity is so week)? Has there been any success in doing this?

Thanks Jeff

Last edited: Dec 29, 2003
15. Dec 29, 2003

### jeff

Basically, yes, but the EM and weak interactions have been successfully united to form electroweak theory (a few guys got the nobel prize for this). The EW interaction should then be united at higher energies with the strong force achieving so-called "grand unification", the most often discussed model of which is based on the group SU(5). In any event, the theories describing the EW together with the strong interaction is called the standard model.

16. Dec 29, 2003

### dlgoff

Jeff,

Thank you for posting the Gell-Mann & Low work. I'm still trying to go through it but some of the graphics are coming up as μ (square). I'm not sure how to get my brower to display properly.

added text: I found and correct my display problem. Now I can study the paper. Thanks again.

Sincerely

Last edited: Dec 29, 2003
17. Dec 30, 2003