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.

The canonical formalism of LQG is most suited to discuss this question, since one has a direct relation to general relativity as the theory which is quantised to obtain LQG. In particular, one really quantises continuum general relativity and the Hilbert space that one obtains thus has to be interpreted as the continuum Hilbert space. However, the elementary excitations on this Hilbert space, comparable to particles on a Fock space, are discrete quanta of geometry. They are the quantum operators corresponding to certain holonomies and fluxes that one can compute in the classical theory. A finite number of such holonomies however only captures a limited amount of the continuum geometry that is roughly equivalent to a discretisation of GR. Therefore, the (simplest) quantum states that one usually discusses in LQG, which also form a basis of the Hilbert space, have a certain interpretation as discrete geometries. This however doesn’t mean that there cannot exist any continuum states in this Hilbert space, since we can of course take arbitrary linear combination of basis elements, and thus, morally speaking, arbitrary superpositions of lattices. Gaining control over such states and understanding their physics is currently the main open problem in LQG. A computation in a simplified toy model explicitly showed that such quantum geometries can be consistent with local Lorentz invariance.

In conclusion, the Hilbert space of LQG contains elements which can be interpreted as discrete quantum geometries. In different approaches to LQG, these states are given a more or less fundamental status which might suggest that one is really dealing with a discretisation. However, one always needs to keep in mind that one can arbitrarily superpose such states and that the state of quantum geometry that correctly describes our universe at all scales should be searched for in a suitable continuum limit.

(As a short remark, this does not mean that states containing only a few quanta of geometry cannot be useful when one is considering certain truncations of the theory, such as cosmology. However, one then has to interpret those states in a coarse grained setting and properly understand the renormalisation group flow which relates the dynamics of continuum states to those of coarse grained few quanta states.)


  1. When i benefit your existing document. It might be top notch to examine everyone reveal with text throughout the coronary heart in conjunction with lucidity due to this vital difficulty may very well be easily viewed. idm crack

  2. This particular looks for example certainly superb. Most of these slight tips are made using availablility of cosmetic foundation interest. I like to him or her appreciably. idm

  3. I loved your article!! It has given me plenty to think about moving forward on my social media journey!Dashlane Crack

  4. It’s not my first time to pay a quick visit this web
    site, i am visiting this site dailly and obtain fastidious data from here

    anytrans crack
    adaware pro security
    dolby atmos apk

  5. Great, submit, very informative.
    I ponder why the opposite experts of this sector don’t realize this.
    You must proceed your writing.
    I’m confident, you have
    a great readers’ base already!
    filmora crack
    transmac license key
    synfig studio crack
    cubase pro crack with keygen free download

  6. This article is indeed a satisfying article that can help new Internet users.
    Who needs help with blogging?You can also visit my website. I hope to help you
    cyberlink powerdirector
    windows 7 loader with activator 2020
    avira antivirus pro crack
    google earth pro crack


  7. Sincerely, I am so glad I found your site, I accidentally found you while searching on google
    for something else, I am still here and I want to congratulate you on a nice post and
    a nice blog all over (love the theme/design too),
    I don't have time to watch anywhere in the minute, but I noticed and added
    your RSS feeds, so if I have time, I will return to
    read on, keep up the good work.
    stopzilla antivirus crack
    amcap crack serial key

  8. It is very interesting, you are an experienced blogger. I added your feed and I hope you continue your search
    great posts. We also shared your site on social networks!
    express vpn crack

  9. optimum! I want to do an internship while editing the website, how can I subscribe?
    For blog sites? This account helped me close a deal.
    I am a little familiar with your post, it provides a clear and bright concept.
    drivermax pro crack
    tubedigger crack

  10. This is really helpful and helpful information.
    I'm glad you just shared this helpful information
    site. Protect us like this. Thanks and sharing.

    pubg pc crack
    netlimiter pro
    aiseesoft blu ray player crack
    sketchup pro crack