# Quantizing unimod GR and solving the cosmo constant problem

1. May 27, 2009

### marcus

Interest is gathering around unimodular General Relativity, so we should get some reference links together to facilitate following the research excursion into that area.
Einstein came up with unimodular GR around 1919, shortly after he proposed ordinary GR.
Here are some recent (and not-so-recent) papers on it:

http://arXiv.org/abs/hep-th/0501146
http://arXiv.org/cits/hep-th/0501146
Can one tell Einstein's unimodular theory from Einstein's general relativity?
Enrique Alvarez

http://arXiv.org/abs/hep-th/0702184
http://arXiv.org/cits/hep-th/0702184
Unimodular cosmology and the weight of energy
Enrique Álvarez, Antón F. Faedo
23 pages, Phys.Rev.D76:064013,2007
"Some models are presented in which the strength of the gravitational coupling of the potential energy relative to the same coupling for the kinetic energy is, in a precise sense, adjustable. The gauge symmetry of these models consists of those coordinate changes with unit jacobian."

http://arxiv.org/abs/0801.4477
Time and Observables in Unimodular General Relativity
Hossein Farajollahi
Gen.Rel.Grav.37:383-390,2005
"A cosmological time variable is emerged from the hamiltonian formulation of unimodular theory of gravity to measure the evolution of dynamical observables in the theory. A set of constants of motion has been identified for the theory on the null hypersurfaces that its evolution is with respect to the volume clock introduced by the cosmological time variable."

http://arXiv.org/abs/0809.1371
http://arXiv.org/cits/0809.1371
Troubles for unimodular gravity
Bartomeu Fiol, Jaume Garriga
17 pages

Some earlier papers:
http://arxiv.org/abs/hep-th/9911102
A small but nonzero cosmological constant
Y. Jack Ng, H. van Dam (University of North Carolina)
Int.J.Mod.Phys. D10 (2001) 49-56
"Recent astrophysical observations seem to indicate that the cosmological constant is small but nonzero and positive. The old cosmological constant problem asks why it is so small; we must now ask, in addition, why it is nonzero (and is in the range found by recent observations), and why it is positive. In this essay, we try to kill these three metaphorical birds with one stone. That stone is the unimodular theory of gravity, which is the ordinary theory of gravity, except for the way the cosmological constant arises in the theory. We argue that the cosmological constant becomes dynamical, and eventually, in terms of the cosmic scale factor R(t), it takes the form $$\Lambda(t) = \Lambda(t_0)(R(t_0)/R(t))^2$$, but not before the epoch corresponding to the redshift parameter $$z \sim 1$$.

W. G. Unruh, ”A Unimodular theory of canonical quantum gravity,” Phys. Rev.
D 40, 1048 (1989);

M. Henneaux and C. Teitelboim, “The cosmological constant and general covariance,” Phys. Lett. B 222, 195 (1989).

The most recent paper on unimod GR is this one:
http://arxiv.org/abs/0904.4841
The quantization of unimodular gravity and the cosmological constant problem
Lee Smolin
22 pages
(Submitted on 30 Apr 2009)
"A quantization of unimodular gravity is described, which results in a quantum effective action which is also unimodular, ie a function of a metric with fixed determinant. A consequence is that contributions to the energy momentum tensor of the form of the metric times a spacetime constant, whether classical or quantum, are not sources of curvature in the equations of motion derived from the quantum effective action. This solves the first cosmological constant problem, which is suppressing the enormous contributions to the cosmological constant coming from quantum corrections. We discuss several forms of uniodular gravity and put two of them, including one proposed by Henneaux and Teitelboim, in constrained Hamiltonian form. The path integral is constructed from the latter. Furthermore, the second cosmological constant problem, which is why the measured value is so small, is also addressed by this theory. We argue that a mechanism first proposed by Ng and van Dam for suppressing the cosmological constant by quantum effects obtains at the semiclassical level."

For another approach to the cosmo constant problem, that doesn't use unimodular, there's a recent paper by Afshordi
http://arxiv.org/abs/0807.2639

Last edited: May 27, 2009
2. May 27, 2009

### marcus

A birdseye view of the situation could be like this:

The cosmo constant problem is the biggest problem in physics right now (except maybe for reconciling GR theory of Geometry with quantum theory, and they are related.)

Unimod GR gives the same classical results, equations of motion, as Ordinary GR.
But Unimod does not couple to vacuum energy!

So switching to Unimod solves the primary CC problem: about this huge Vacuum Energy a big constant energy density that Ordinary GR says should be crumpling up space.
Unimod says it has no effect. CC is not a problem for Unimodular General Relativity.

So far there is no satisfactory quantization of Unimod. So it is an attractive problem to do that.

Now non-string Quantum Geometry-type QG is progressing along nicely now, with two path integral approaches (spacetime geometry treated like a particle path from initial to final space-geometry)---both SpinFoam and CausalDynamicalTriangulations approaches are making steady progress.

So from Smolin's point of view it must look as if he is free to scout out something entirely different which will eventually, if successful, use the Quantum Geometry path integrals of SF/CDT.

Exploring Unimodular will open up a bunch of research problems of how do you apply, say, SpinFoam path integral to the Unimodular case.

Last edited: May 27, 2009
3. May 28, 2009

### tom.stoer

Seems that Smolin is looking for different approaches to solve the cosmo constant problem:

http://arxiv.org/abs/0903.5303
Disordered Locality as an Explanation for the Dark Energy
Chanda Prescod-Weinstein, Lee Smolin
"We discuss a novel explanation of the dark energy as a manifestation of macroscopic non-locality coming from quantum gravity, as proposed by Markopoulou. It has been previously suggested that in a transition from an early quantum geometric phase of the universe to a low temperature phase characterized by an emergent spacetime metric, locality might have been "disordered". This means that there is a mismatch of micro-locality, as determined by the microscopic quantum dynamics and macro-locality as determined by the classical metric that governs the emergent low energy physics. In this paper we discuss the consequences for cosmology by studying a simple extension of the standard cosmological models with disordered locality. We show that the consequences can include a naturally small vacuum energy."

4. May 28, 2009

### marcus

Tom, it is pretty remarkable, two very different approaches to the CC problem presented in two months. I see that Smolin is one of the invited speakers at the 60th birthday Abhayfest that happens in a week or so from now. June 4-6.
http://igc.psu.edu/events/abhayfest/program.shtml
And he has chosen to present the Unimodular paper. I find it a lot more interesting than disordered locality, so I'm glad he's talking about that one and I hope he sticks with it for a while and gets some colleagues and graduate students working on it.

Unimodular GR also addresses the QG "problem of time" and it connects to something he is writing with Roberto Unger, a book I believe.

5. May 28, 2009

### tom.stoer

Why do you think it's more interesting?

Of course he is not very specific in the "disordered locality" paper as he already mentions in the introduction - but rather generic as this concept could apply to different QG theories.

What I find fascinating is that it seems that spin networks introduce new concepts like disordered locality (macroscopic scales) and topological excitations or braids (microscopic states).

What I find even more fascinating is that there are a lot of new approaches to quantum gravity (even if the string guys claim that there is nothing else but string theory :-)

What is the "problem of time"? Following Rovelli's ideas I would say that there is no problem of time on the fundamental level. It's like looking for the surface of water while studying atoms - you want find it as the surface is an emergent phenomenon.

6. May 30, 2009

### marcus

I have to give a partially irrational answer. This will not be persuasive reasons why you should find unimodular QG more interesting. Disordered locality is a lovely, possibly fertile, idea. My interest in unimodular is motivated partly by my personal psychology, not on clear rational grounds.

I am more excited by unimodular because of the reference [15] in the paper. I have been hearing about this book for more than a year. Now they are calling it "The reality of time..."

It may be wrong, but it could have the potential to play the devil with Carlo Rovelli's arrangement of ideas. It has a kind of adversarial energy which I like. One reason Smolin's contribution is always valuable is that his creativity is driven by an instinct for shaking the established ways of thinking.

Rovelli's viewpoint is to a certain extent becoming established. He has built a very fine structure where time has no fundamental existence. Time is merely emergent. I know that personally this has influenced me and formed how I think to a considerable extent. Unimodular could pose an effective challenge by directly confronting this.

In standard GR there is no preferred idea of time. The universe and its laws do not evolve, they simply are as they are. Existence is the eternal Given. If one takes standard GR seriously then there is, as you say, no problem of time because it is merely an appearance.

But we have the psychological experience of time as a progression of present moments and we have also the faint suspicion that the universe and its laws evolve---a paradoxical idea that never completely goes away. The suspicion that nothing is finally given. Evolution requires time. It is time, perhaps.

What could promote this suspicion and confront the more established Rovellian idea is to say "you took seriously the wrong version of GR, you learned the wrong lesson, you should take seriously what unimodular GR teaches, and it contains the seed of a preferred global time."

If this idea can even get to first base it has the ability to disturb things. Or so I think.

Last edited: May 30, 2009
7. May 30, 2009

### marcus

Tom,
I learn a lot from your comments and I would be quite happy to be disillusioned about unimodular, if you have some critical points about it.

But in the meantime I should probably give a page reference for the "seeds of a preferred time" idea.

For example look on page 7, equation (25)

"...is a time coordinate that measures the total four-volume to the past of that surface..."

That is, he is quantifying the past. Time is the quantity of past.

Just a brief brushing against the idea, on a formal level, nothing rigorously spelled out, but suggestive. And then he implicitly comes back to this idea again later on pages 14 and 16 where he re-introduces the term "Vol" for a global spacetime volume.

8. May 30, 2009

### marcus

Lee is on PIRSA with this paper.
I'll watch the video.
http://pirsa.org/09050091/

He goes further than in the paper. Implementing quantum unimodular gr with LQG.
Also there were a couple of slides about how to do it in spinfoam, but he skipped over those.

Also in the video talk he explains more about the new time.

Last edited: May 31, 2009
9. May 31, 2009

### marcus

This Pirsa talk is important so we should have a quick web-handle to get it whenever needed.
For now I just google "smolin unimodular pirsa" and it brings it up.

Unimodular Relativity presents a kind of paradox or contradiction. Sometimes it is good to take in a contradiction and try to assimilate or learn from it. One could, alternatively, immediately reject Smolin's gambit with UR because it contradicts the mainstream LQG of Rovelli and others, in which there is no time.

Personally I had found Rovelli's arguments persuasive, that fundamentally there is no time but time is rather a superficial appearance. (For what it's worth, Einstein already said in 1916 that space and time are deprived of the last remnant of physical or objective reality.)
We should review the argument and take it seriously because there is apparently a strong contradiction.

There is a very good summary of Rovelli's timeless viewpoint here, the Chapter One of Oriti's new book. The Cambridge University Press website allows us to browse the whole chapter.
http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=9780521860451

==excerpt==
...Much has been written on the fact that the equations of nonperturbative quantum gravity do not contain the time variable t. This presentation of the “problem of time in quantum gravity”, however, is a bit misleading, since it mixes a problem of classical GR with a speciﬁc quantum gravity issue. Indeed, classical GR as well can be entirely formulated in the Hamilton-Jacobi formalism, where no time variable appears either.

In classical GR, indeed, the notion of time diﬀers strongly from the one used in the special-relativistic context. Before special relativity, one assumed that there is a universal physical variable t, measured by clocks, such that all physical phenomena can be described in terms of evolution equations in the independent variable t. In special relativity, this notion of time is weakened. Clocks do not measure a universal time variable, but only the proper time elapsed along inertial trajectories. If we ﬁx a Lorentz frame, nevertheless, we can still describe all physical phenomena in terms of evolution equations in the independent variable x0, even though this description hides the covariance of the system.

In general relativity, when we describe the dynamics of the gravitational ﬁeld (not to be confused with the dynamics of matter in a given gravitational ﬁeld), there is no external time variable that can play the role of observable independent evolution variable. The ﬁeld equations are written in terms of an evolution parameter, which is the time coordinate x0, but this coordinate, does not correspond to anything directly observable. The proper time τ along spacetime trajectories cannot be used as an independent variable either, as τ is a complicated non-local function of the gravitational ﬁeld itself.

Therefore, properly speaking, GR does not admit a description as a system evolving in terms of an observable time variable. This does not mean that GR lacks predictivity.

Simply put, what GR predicts are relations between (partial) observables, which in general cannot be represented as the evolution of dependent variables on a preferred independent time variable. This weakening of the notion of time in classical GR is rarely emphasized: After all, in classical GR we may disregard the full dynamical structure of the theory and consider only individual solutions of its equations of motion. A single solution of the GR equations of motion determines “a spacetime”, where a notion of proper time is associated to each timelike worldline.

But in the quantum context a single solution of the dynamical equation is like a single “trajectory” of a quantum particle: in quantum theory there are no physical individual trajectories: there are only transition probabilities between observable eigenvalues. Therefore in quantum gravity it is likely to be impossible to describe the world in terms of a spacetime, in the same sense in which the motion of a quantum electron cannot be described in terms of a single trajectory...
==endquote==
http://arxiv.org/abs/gr-qc/0604045

I emphasized the word relations. It is relations among variables which are observable, and which are predicted by the theory. A mere quantity by itself is not meaningful. I still find Rovelli's argument here persuasive. There has been a progressive weakening of the concept of time, from Newton, to special rel, to general rel, to quantum general. I do see that a spacetime cannot exist any more than a classical trajectory can exist.

But UR seems to contradict this, by introducing something like time, that the universe seems to know about and according to which it evolves. I almost want to laugh at the idea: we have pushed time out the door and it comes back in through the window.

Last edited: May 31, 2009
10. Jun 2, 2009

### John86

Last edited by a moderator: Apr 24, 2017
11. Jun 2, 2009

### marcus

Last edited by a moderator: Apr 24, 2017
12. Jun 2, 2009

### MTd2

13. Jun 2, 2009

### marcus

I'm not expressing any antipathy at all, MTd2. I don't know how to respond to your post about Spin Networks and relationalism. It is not a question, it is a perception or viewpoint. I'm afraid there is a lot of interesting stuff that I don't answer because I'm not sure what to say. Please excuse my silence on these matters.

BTW Smolin's Unimodular Relativity (UR) gambit is interesting for several reasons--one because he is writing a book with the philosopher Unger developing these ideas. The current title is something like The Reality of Time (and the evolution of the laws of nature). So there is a coherent, momentum-building thrust in progress.

And it is also interesting because of Rovelli's stature and position. Rovelli has more philosophical depth and clarity than most other theoretical physicists. Much of string research has been wasted effort because it was superficially grounded. Many researchers did not consider fundamental issues of geometric background, the nature of space and time, but instead simply jumped in and started playing with differential manifolds. Even Riemann who invented manifolds was less naive.

So Rovelli, because of his relative clarity on philosophical issues has gained a commanding position. He knows what he is doing and his program is making progress.

The special significance of this UR gambit is that it profoundly challenges Rovelli's program at a basic level.