Wednesday, May 4, 2016

Progress on low spins

Transplanckian large spins decay into small spins under the dynamics
Fundamental (trans-Planckian) large spins are expected to decay into
many small spins under the dynamics. An explicit calculation showing
  this has been given now within group field theory. At the same time,
one can coarse grain the many small spins into few large spins to have
an effective continuum description. The precise relation between those
two, a priori distinct large spin regimes is so far unclear.  

The fundamentally important issue of low and high spins in the dynamics of LQG has been discussed in previous posts. In short, large geometries can be described using either many low or few large quantum numbers (SU(2) spins), but the respective dynamics needs to be interpreted with care. In particular, using large spins to describe continuum geometries requires to understand the renormalisation group flow of the theory. Most work in LQG has so far been in the context of large spins (without considering renormalisation), where calculations simplify drastically due to the availability certain asymptotic formulae for the SU(2) recouping coefficients with nice geometric interpretations. However, there seems to be some progress now on the low spin front. 

Monday, April 25, 2016

Is loop quantum gravity based on discretisations?

Sum of two spin networks for continuum limit of loop quantum gravity
Two spin networks, which can be interpreted as Hilbert space elements
describing truncations (or discretisations) of general relativity,
are summed. What is the physics of such states? 

This post is related to last week’s about criticism of loop quantum gravity and a comment I found in Sabine Hossenfelder’s blog. Saying that loop quantum gravity is based on discretisations quickly leads one to doubt that Lorentz invariance may be a property of LQG, as happened in the comment. So it seems worth to clear up this issue and precisely say in what context discretisations appear, in what context they don’t, and what this means for the physics that are described by LQG.

Thursday, April 21, 2016

Updates on some criticisms of loop quantum gravity

Criticism of loop quantum gravity: Lorentz violations, general relativity limit, ambiguities
Image from

In this post, we will gather some criticism which has been expressed towards loop quantum gravity and comment on the current status of the respective issues. The points raised here are the ones most serious in my own opinion, and different lists and assessments could be expressed by other researchers. 

Tuesday, April 12, 2016

Approaches to quantum gravity

Again as a part of a lecture series in preparation, I compiled a list of the currently largest existing research programmes aimed at finding a quantum theory of gravity (while unfortunately omitting some smaller, yet very interesting, approaches). A much more extensive account is given in here. For an historical overview, I recommend this paper.

Friday, March 25, 2016

How does loop quantum cosmology work?

Loop quantum cosmology big bounce vs Wheeler-de Witt big bang
The evolution of the expectation value of the volume v of the universe is plotted.
The blue and orange lines are the expanding and contracting branches in the
Wheeler-de Witt theory. The green curve follows from loop quantum cosmology
and exhibits a quantum bounce close to Planck density.

These days I have been working on lecture notes for an introductory lecture series on loop quantum gravity. An introductory article by Abhay Ashtekar introduced this subject via a discussion on loop quantum cosmology (LQC), where already many of the essential features of loop quantum gravity are present and can be studied in a simplified setting. I think that this is a very useful pedagogical approach and I also wanted to incorporate it into my lectures. I just finished my first draft of a lecture about this subject, focussing on a specific exactly soluble LQC model and its comparison to a similar quantisation using the Wheeler-de Witt framework. Mostly I follow the original paper, with some additional comments, slight rearrangements, and omission of more advanced material that is not necessary in an introductory course in my point of view. The current draft is available here, and comments are always welcome.

The short version goes as follows:

Tuesday, March 15, 2016

Yang-Mills analogues of the ambiguities in defining LQG geometric operators

Ambiguity in the geometric operators in loop quantum gravity from changing connection variables
Changing the connection variables underlying loop quantum gravity
also changes the geometric operators. They measure geometry with respect
to the metric encoded in the Yang-Mills electric field $E^a_i$.   
tl;dr: In some cases there are such analogues, but they are rather awkward.

A comment to a recent post on the current status of the issue of the spectra of geometric operators in loop quantum gravity raised the question of whether such ambiguities can also be found in Yang-Mills theory. The question if of course very interesting, however I am not aware of any reference commenting on it. So let me try.

Friday, March 11, 2016

Some comments on the properties of observable algebras

An example of non-locality in an observable algebra. Observables are
defined as fields at the endpoints of geodesics. Changing the metric
along the geodesic via $P^{rA}$, the field conjugate to some of the relevant
components of the spatial metric, leads to an apparent non-locality due to
a rerouting of the geodesic.

A recent post contained a talk of mine of a specific construction of observables in general relativity, where a physical coordinate system was specified by the endpoints of certain geodesics. The so constructed coordinates are simply Gau├čian normal coordinates and the usual relational construction of observables as fields at a physical point could proceed in the standard manner.

The motivation for the underlying paper came from AdS/CFT, where people were interested in constructing CFT operators corresponding to scalar fields in the bulk, localised at the endpoint of a geodesic. While the scalar fields were argued to commute in this construction, a contradictory statement was recently made in a perturbative calculation. To settle this question, we upgraded a calculation referring to spatial geodesics, based on this seminal work, to the case of spacetime geodesics considered in the AdS/CFT case.

The calculation which we performed illustrates nicely some points about the properties of observables which are often not spelled out, however seem of relevance for researchers interested in quantum gravity and should not be confused. In this post, we will gloss over global problems in defining observables, see e.g. this paper and references therein for a recent interesting discussion. See also this post for an earlier discussion of observables and the control we have on them.

1. Structure of the (sub-)algebra of observables
When we quantise, we always pick some preferred subalgebra of phase space functions that we want to quantise. It is well known that we cannot quantise all functions at the same time, as formalised in the Groenewold-van Hove theorem. Therefore, also in canonical quantum gravity such a choice has to be made, independently of whether we first quantise or first solve the constraints. We are interested here in the latter case, i.e. in the quantisation of a complete (= point separating on the reduced phase space) set of Dirac observables, or equivalently a complete set of phase space functions after gauge fixing the Hamiltonian and spatial diffeomorphism constraints, although similar statements should in principle also hold in the former case.
The precise nature of this choice will determine the properties of our observable algebra. In particular, examples can be given where
  1. the resulting algebra has a perfectly local structure (e.g. when employing dust to specify a reference frame),
  2. the resulting algebra is local, but some phase space functions not contained in the complete sub-algebra have non-local commutation relations
  3. the resulting algebra is non-local
Roughly, the following happens. If we localise some field with respect to some locally defined structure, such as the values that four scalar fields take at a point, then the algebra remains local, where local means that Poisson brackets will be proportional to a delta-distribution in the coordinates defined by the scalar field coordinates. Another such example is to use dust. We can however also localise some field with respect to a more global structure, such as the endpoint of a spatial geodesic originating from some observer or the boundary of the spatial slice. Then, phase space functions with a non-vanishing Poisson bracket with components of the metric specifying this geodesic will have non-local (in the coordinates specified by the geodesics) Poisson brackets with fields localised at the endpoints of geodesics, just because they change the geodesic which defines the point where the field is evaluated. However, it is still possible to pick a complete set of such observables which have local Poisson brackets among themselves if we pay attention not to include fields not commuting with the components of the metric determining the path of the geodesics. This is in fact what is happening here. For the last case, we simply locate our fields with respect to some extended structure that is specified using non-commuting phase space functions, e.g. a spacetime geodesic. This doesn’t mean that locality breaks down, it simply means that we are considering observables which are smeared over regions of spacetime, and two such regions associated to different localisation points might overlap.

2. Locality of the physical Hamiltonian
Another question is whether the physical Hamiltonian that one derives with respect to a certain choice of time will be non-local. Again, in the case of dust as a reference field, it is local. However, already in the second case above one finds that is non-local, simply because it contains the phase space functions which have non-local Poisson brackets. Since a generic Hamiltonian will contain all fields of the theory, it seems that one has to avoid relational observables with respect to non-local structures if one wants to obtain a local Hamiltonian. However, it should be stressed again that one still might have a complete local observable algebra, even though the Hamiltonian is non-local. A detailed example is given here.

3. Independence of spacetime boundaries
All arguments made above are independent of the structure of the spacetime at infinity. In particular, the math involved in constructing observables and computing their algebra, see e.g. here, here, and here, does not refer to infinity. Also, the fact that the on-shell Hamiltonian constraint is a boundary term is not of relevance here, as it generates evolution via its Hamiltonian vector field, which is non-vanishing on-shell. Once a clock field is chosen on the other hand, one obtains a physical Hamiltonian with respect to that clock which is non-vanishing on-shell. Therefore, as long as one can locally define a clock, one can also have evolution.