What are the key concepts of Loop Quantum Cosmology?

[Physics Monkey]

Hi everybody,

I would like to chat about the physics of loop quantum cosmology. Let me list a few broad topics I'm interested in understanding better:

1. What simplifications/approximations make loop quantum cosmology a solvable simplification of full loop quantum gravity?

2. How is the Hamiltonian constraint dealt with and how does time emerge?

3. Why is the semi-classical limit easier here (as I understand it, the general semi-classical limit is still an open question, but please correct me if I'm wrong)?

4. What is the physics intuition behind the various cosmological results of the theory?

Of course, I appreciate excellent pedagogical or review references if you know them, but I would primarily prefer to discuss the actual physics if anyone is interested/knowledgeable.

As a start, I want to comprehend the basic setup. My understanding is that loop quantum cosmology makes some symmetry assumptions that enable a reduction of the degrees of the freedom. If so, can we state what these are in a simple way?

 

[marcus]

I can't claim authoritative knowledge or the ability to give a satisfactory answer myself in each case, but I will try to give some tentative responses and hope others will also. First off, I want to confirm that these are all good well-posed questions that should be asked and discussed. I really like the list of questions.

I'll give some preliminary reactions.

As a start, I want to comprehend the basic setup. My understanding is that loop quantum cosmology makes some symmetry assumptions that enable a reduction of the degrees of the freedom. If so, can we state what these are in a simple way?

For starters make sure you understand that cosmology is based on the Friedman equations (1922-1923) which are derived from the full Gen Rel theory by making symmetry assumptions (homogeneity, isotropy). On average largescale uniformity of the distribution of matter, sameness in all directions at all locations.

Einstein enunciated that "cosmological principle" back around 1916, correctly expressed as an approximation.

Assuming such approximate uniformity leads to a radical simplification where there is geometry has only one dynamic variable, the spatial scalefactor a(t) which plugs into the Friedman metric. All the Friedman equations do is govern the evolution of the scalefactor a(t) in relation to the density and pressure rho(t) and p(t) of matter. Inflation, decelerating expansion, accelerating expansion, in some cases contraction, are all about a(t). The Hubble rate is defined H(t) = a'(t)/a(t).

Anyone unfamiliar with it can google "Friedmann equations" (spellings differ, the German version has two nn's)

The two Friedman equations and their associated metric have been the basis of cosmology for over 70 years.

The Friedman equations are classical and have a singularity at the outset of expansion, a failure and limitation. It was hoped since way back with Wheeler and DeWitt that quantizing the Friedman model would get rid of the singularity.

To answer your question the main simplification that LQC is based on is the same simplification that standard Friedman cosmology is based on and it does radically reduce the number of degrees of freedom.

This is only the bare beginnings of a response. I hope to get back to this later today. I will quote the questions you asked here for future reference:

1. What simplifications/approximations make loop quantum cosmology a solvable simplification of full loop quantum gravity?

2. How is the Hamiltonian constraint dealt with and how does time emerge?

3. Why is the semi-classical limit easier here (as I understand it, the general semi-classical limit is still an open question, but please correct me if I'm wrong)?

4. What is the physics intuition behind the various cosmological results of the theory?

Of course, I appreciate excellent pedagogical or review references if you know them, but I would primarily prefer to discuss the actual physics if anyone is interested/knowledgeable.​

We should not simply give pedagogical references, we should reply explicitly as best we can to each of these four question, but we can give links to pedagogy as well.

 

[marcus]

The LQC Hamiltonian was changed in 2006. Anyone interested in the reasons or the history can see Ashtekar's 2006 papers that refer to the "new dynamics", or "improved dynamics". At present I think the least confusing thing for a newcomer is to stick to papers where Ashtekar is (co-)author and which appeared 2006 or later.

One possible pedagogical link would be:
http://arXiv.org/abs/gr-qc/0702030
An Introduction to Loop Quantum Gravity Through Cosmology
Abhay Ashtekar
20 pages, 4 figures
(Submitted on 5 Feb 2007 (v1), last revised 15 May 2007 (this version, v2))
"This introductory review is addressed to beginning researchers. Some of the distinguishing features of loop quantum gravity are illustrated through loop quantum cosmology of FRW models. In particular, these examples illustrate: i) how 'emergent time' can arise; ii) how the technical issue of solving the Hamiltonian constraint and constructing the physical sector of the theory can be handled; iii) how questions central to the Planck scale physics can be answered using such a framework; and, iv) how quantum geometry effects can dramatically change physics near singularities and yet naturally turn themselves off and reproduce classical general relativity when space-time curvature is significantly weaker than the Planck scale."

As general background, here is a listing of all Spires Quantum Cosmology (QC) papers that have appeared after 2005. This includes other QC approaches besides LQC, including various string-inspired. But scanning down you will see a fair number of LQC ones, whose titles give a rough idea of recent directions in LQC research.

http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=dk+quantum+cosmology+and+date%3E2005&FORMAT=WWW&SEQUENCE=citecount%28d%29

 

[Physics Monkey]

For starters make sure you understand that cosmology is based on the Friedman equations (1922-1923) which are derived from the full Gen Rel theory by making symmetry assumptions (homogeneity, isotropy). On average largescale uniformity of the distribution of matter, sameness in all directions at all locations.

Thanks, this comment reminds me to mention something about the level of the discussion. Personally, I'm fine with gr, cosmology, and quantum mechanics at both an intuitive and technical level. However, I don't want to exclude anyone from the discussion. I would invite everyone to post at whatever level they see fit (as if I could stop you ). My own preference is to keep things intuitive at first as much as possible.

 

[atyy]

Symmetry, which is basically a background, ie. already there is some fixed spacetime. Since spacetime is already partially there, recovering classical spacetime in some limit is easier. At least that was the starting point, there's been some work since then trying to relax the symmetry requirements in LQC.

 

[marcus]

Symmetry, which is basically a background, ie. already there is some fixed spacetime. Since spacetime is already partially there, recovering classical spacetime in some limit is easier. At least that was the starting point, there's been some work since then trying to relax the symmetry requirements in LQC.

Atyy, you make a good point here. I would say also that in LQC "spacetime is already partially NOT there" at the outset. The spacetime geometry arises dynamically.

Just to clarify, as I'm sure you know, the term "background" in non-string QG discussions is shorthand for a fixed background spacetime metric---a pre-established 4D geometry.

LQC can be set up without using a pre-established spacetime geometry. The metric arises dynamically, as a solution to the equations.

This is what Loop people call background independence. LQC is background independent because it does not require a pre-established spacetime geometry.

(In string theory "background independence" apparently means something else. Considerable confusion arose at one point from different groups meaning different things by the term "background".)

The symmetries which you mentions are not 4D symmetries. They are spatial.
Matter approximately uniformly distributed throughout space at a given moment in time.
That still leaves a great deal of possible variation in spacetime geometry.

It's an obvious point but i thought I'd make it anyway, just in case others besides ourselves are reading the thread.

 

[marcus]

Personally, I'm fine with gr, cosmology, and quantum mechanics at both an intuitive and technical level...

Excellent! Maybe the Ashtekar introduction will suffice? Or it may not be the right thing. You may require more detail.
So far i haven't responded to your four questions. Time permitting I'll take a stab at it tomorrow.

 

[Physics Monkey]

Right, I think atyy and marcus have made an important point that needs to be understood. It seems to me that while the ultimate goal of LQG is to be background independent, in the case of LQC one has taken a step back and assumed some partial background (highly symmetric). The program is then to quantize the resulting reduced set of variables. Is this an accurate statement?

So as atyy suggested, I think perhaps the semiclassical analysis is easier because one starts half-way to the goal, so to speak. Is this an accurate statement?

On my end, I will be studying the ashtekar review to get some more sense of things.

Thanks for the replies so far.

 

[tom.stoer]

I think you are right, LQC selects a specific (highly symmetric) background.

 

[atyy]

Maybe you can also read section 5.1 and 7 of http://relativity.livingreviews.org/Articles/lrr-2008-4/ and tell us whether our handwaving had any merit.

 

[Physics Monkey]

Based on what I've read so far, the basic symmetry assumption seems to be that the triad specifying the three geometry depends on one variable only. In other words, knowledge of one component of the triad tells one the other two components. I think this is basically the assumption of some kind of homogeneous and isotropic quantum universe.

I think pre-LQC, this assumption reduced one to studying what was in essence the quantum physics of the scale factor (plus some scalar fields, say). I'm still not clear what the complete list of degrees of freedom is in the LQC case. I have the impression from the Ashtekar and Bojowald reviews that there may be more degrees of freedom than just the scale factor, but I'm not sure.

Any comments?

 

[tom.stoer]

In LGC w/o deformations there is one gravitational and one matter degree of freedom.

 

[Physics Monkey]

So it remains true in LQC that we are basically considering the quantum physics of the scale factor and the zero mode of some scalar field?

 

[marcus]

Of course in some recent papers they relax isotropy and, for example, they have three scale factors instead of just one . Some LQC also relax homogeneity, I forget how.

But the basic idea is just what you say----so far, two degrees of freedom or, to be more accurate a small number of degrees of freedom.

Maybe I'll get some links.
Well I looked and there are too many LQC papers now that have more than two degrees of freedom.

I will just give one as a sample and if anyone wants more just ask and I can get some. Here is a sample:

http://arxiv.org/abs/0812.1889
Magnetic Bianchi I Universe in Loop Quantum Cosmology
Roy Maartens, Kevin Vandersloot
(Submitted on 10 Dec 2008)
"We examine the dynamical consequences of homogeneous cosmological magnetic fields in the framework of loop quantum cosmology. We show that a big-bounce occurs in a collapsing magnetized Bianchi I universe, thus extending the known cases of singularity-avoidance. Previous work has shown that perfect fluid Bianchi I universes in loop quantum cosmology avoid the singularity via a bounce. The fluid has zero anisotropic stress, and the shear anisotropy in this case is conserved through the bounce. By contrast, the magnetic field has nonzero anisotropic stress, and shear anisotropy is not conserved through the bounce. After the bounce, the universe enters a classical phase. The addition of a dust fluid does not change these results qualitatively."

A Bianchi I universe has 3 scale factors.

There have been a bunch of LQC papers starting in or before 2007 that treated Bianchi universes, and thus they had those 3 DoF plus whatever matter DoF.
But this paper of Maartens Vandersloot is the first LQC paper I know that has both the Bianchi anisotropy extra DoF and also a magnetic field.

Side comment: Maartens is one of UK's top cosmologists. He is the director of the Institute of Cosmology and Gravitation at the University of Portsmouth ( http://www.icg.port.ac.uk/people/staff/maartens.html ) Invited editor of special journal issues. Invited speaker at QG and cosmo conferences. Research interest in LQC phenomenology, among other things.
Vandersloot is an Ashtekar PhD who got a Noether fellowship and went to Maarten's institute for postdoc. He would be someone to watch for research that gradually adds DoF and relaxes the LQC restrictions of isotropy and homogeneity.

 

[Physics Monkey]

I am forging through a dizzying array of indices in the Ashtekar paper, but I do think I understand the degrees of freedom. Basically, there is a single canonical pair in the geometrical sector (in the most symmetrical case), which Ashtekar calls c and p. This canonical pair determines the overall size of the connection and the conjugate triad.

Both Ashtekar and Bojowald agree that there is some sort of background structure. In this case, I think it amounts to the specification of the action of some symmetry group on space. This permits a reduction of the gravitational degrees of freedom to a finite set. I'm still trying to understand the details of the state space, and in particular, the somewhat unusual quantization that LQG uses.

The theory has a good semi-classical limit, like the older Wheeler-deWitt approach, because coherent states approximating classical geometries remain sharply peaked under the emergent time evolution coming from the Hamiltonian constraint. At least until space begins to get too small ...

More to come, does this sound reasonable so far?

 

[marcus]

Certainly sounds reasonable to me. Looking forward to hearing more.

 

[Physics Monkey]

Before dealing with the physical consequences of LQC for the big bang, etc. I want to understand the "emergence of time" completely. This issue can I think be addressed within the WdW framework and Ashtekar seems to agree. Maybe later we can go back and understand if LQC adds anything to this part of the story.

The variables are the following: the value of the massless scalar field zero mode $$\phi$$, its conjugate momentum $$p_{\phi}$$, the scale factor (or its log) $$\alpha = \ln{a}$$, and its conjugate momenta $$p_{\alpha}$$. There is an operator made of these variables called the Hamiltonian $$H$$, and the Hamiltonian is required to annihilate physical states $$H |\Psi \rangle = 0$$. This is the statement of timelessness of the theory.

Here is where things begin to become confusing in my opinion. The classical "time" evolution of this system is such that the momentum $$p_{\phi}$$ is conserved, and the value of the scalar field increases monotonically with "time". After introducing the crutch of this time evolution, we can remove it from the description by using the monotonically increasing value of $$\phi$$ along the classical trajectory as an internal time. Classically I don't see any problems with this.

Now in the quantum case, we should restrict ourselves to states annihilated by the Hamiltonian. Let's use a wavefunction $$\Psi(\alpha, \phi)$$ as in Ashtekar section II. In the second to last paragraph (right above the remark) Ashtekar talks about definining an "initial quantum state" at " 'time' $$\phi = \phi^*$$ " which is peaked about a specific value of $$p_{\phi}$$ and of the "volume of the universe" which is related to the scale factor. I don't understand this statement. A naive reading of this statement appears non-sensical since in ordinary quantum theory one can't prepare a state at a definite value of $$\phi$$ with a finite width for $$p_{\phi}$$. This is issue 1.

Suppose I forget about this sentence and just solve the WdW equation for $$\Psi$$. What is the meaning of this wavefunction? If I continue to interpret $$\phi$$ as the internal time, then is this saying the quantum state is a superposition of different times? More concretely, is $$|\Psi(\alpha,\phi) |^2$$ something like the probability density to find a universe of spatial size $$\alpha$$ and internal time $$\phi$$? In other words, does this state encode the entire "four dimensional history" to the extent that this makes sense? This is issue 2.

Another issue which really bothers me is the bizarre quantization scheme used in LQC and more generally in LQG. This is issue 3.

If anyone has any thoughts on these issues, I would be happy to hear them.

 

[marcus]

I should also have mentioned a more recent and complete development of LQC:
http://relativity.livingreviews.org/Articles/lrr-2008-4/
Sorry I neglected to do this!

If you do have a look at it, you'll notice that it is a bit longer and goes into greater detail. Section 6 describes "effective" solvable models. Differential equations which successfully imitate the system of difference equations. Conceivably the equations in section 6 might be familiar to you and provide additional insight.

You may already know the Living Reviews of Relativity series. They publish authoritative reviews online. The best review of full LQG is their Irr-2008-5 by Rovelli and so far the best review of LQC is IMHO this one (Irr-2008-4) by Martin B.

 

[Physics Monkey]

Thanks, marcus, I think someone (possibly you) already provided me with that link.

No takers on any of the other questions? I'm particularly interested in the wave function issue. Does anyone know of an interpretation besides the one I mentioned?

 

[marcus]

You might be interested: Ashtekar's latest paper offers a critique of LQG pointing out "rough edges" where things don't fit together smoothly, and then goes on to reformulate LQC along the lines of the new spinfoam model. I regret to say his critique of the current state of LQG does not directly address the questions you have asked in this thread, but it does provide some account of the parts of the theory that need the most work.

==sample quote from Ashtekar's latest paper, pages 3, 4==
...Furthermore, one can regard these SFMs as providing an independent derivation of the kinematics underlying LQG. The detailed agreement between LQG and the new SFMs [28, 29] is a striking development. There are also a number of results indicating that one does recover general relativity in the appropriate limit [32, 33].

Finally, the vertex amplitude is severely constrained by several general requirements which the new proposals meet. However, so far, the vertex amplitude has not been systematically derived following procedures used in well-understood field theories, or, starting from a well-understood Hamiltonian dynamics.

Therefore, although the convergence of ideas from several different directions is
impressive, a number of issues still remain. In particular, the convergence is not quite as seamless as one would like; some rough edges still remain because of unresolved tensions.

For example, the final vertex expansion is a discrete sum, in which each term is itself a sum over colorings for a fixed triangulation. A priori it is somewhat surprising that the final answer can be written as a discrete sum. Would one not have to take some sort of a continuum limit at the end? One does this in the standard Regge approach [30] which, as we indicated above, is closely related to SFMs. Another route to SFMs emphasizes and exploits the close resemblance to gauge theories. In non-topological gauge theories one also has to take a continuum limit. Why not in SFMs? Is there perhaps a fundamental difference because, while the standard path integral treatment of gauge theories is rooted in the smooth Minkowskian geometry, SFMs must face the Planck scale discreteness squarely?

A second potential tension stems from the fact that the construction of the physical inner product mimics that of the transition amplitude in Minkowskian quantum field theories. As noted above, in a background independent theory, there is no a priori notion of time evolution and dynamics is encoded in constraints. However, sometimes it is possible to ‘de- parameterize’ the theory and solve the Hamiltonian constraint by introducing an emergent or relational time a la Leibnitz. What would then be the interpretation of the spin-foam path integral? Would it yield both the physical inner product and the transition amplitude? Or, is there another irreconcilable difference from the framework used Minkowskian field theories?

There is a also a tension between SFMs and GFTs. Although fields in GFTs live on an abstract manifold constructed from a Lie group, as in familiar field theories the action has a free part and an interaction term. The interaction term has a coupling constant, λ, as coefficient. One can therefore carry out a Feynman expansion and express the partition function, propagators, etc, as a perturbation series in λ. If one sets λ = 1, the resulting series can be identified with the vertex expansion of SFMs. But if one adopts the viewpoint that the GFT is fundamental and regards gravity as an emergent phenomenon, one is led to allow λ to run under the renormalization group flow. What then is the meaning of setting λ = 1? Or, do other values of λ have a role in SFMs that has simply remained unnoticed thus far? Alternatively, one can put the burden on GFTs. They appear to be efficient and useful calculational schemes. But if they are to have a direct physical significance on their own, what then would the gravitational meaning of λ be?

Such questions are conceptually and technically difficult. However, they are important
precisely because SFMs appear to lie at a junction of several cross-roads and the recent
advances bring out their great potential. Loop quantum cosmology (LQC) provides a physi-
cally interesting yet technically simple context to explore such issues...

==endquote==
http://arxiv.org/abs/1001.5147