How could spacetime become discretised at the Planck scale?

Question

I didn't have much luck getting a response to this question before so I have tried to reword and expand it a little:

In early 2010 I attended this inaugural lecture by string theorist- Prof. Mavromatos entitled 'MAGIC strings'. In it he proposes that some string theory models may violate Lorentz symmetry at the Planck scale resulting in a kind-of foamy spacetime that could be observed by differing arrival times of photons of different energies reaching us from distant astronomical sources. See http://www.kcl.ac.uk/news/events_details.php?year=2010&event_id=2178 or here for one of the papers: http://iopscience.iop.org/1742-6596/174/1/012016

Furthermore, in 'Cycle's of Time' that I read recently, Prof. Sir Roger Penrose mentions (page 203) that Wheeler and others have strongly argued that if we could examine spacetime at the Planck scale we would see a turbulent chaotic situation (from vacuum fluctuations of the quantum fields I suppose) or perhaps a discrete granular one. Penrose goes on to list some other approaches that may suggest how this discrete structure may manifest itself. Loosely transcribed these are: spin foams, casual sets, non-commutative geometry, Machian theories, twistor theory, [EDIT] loop quantum gravity, or strings and membranes existing in some higher-dimensional geometry...

I have studied some QM, introductory QFT and the Standard Model as well as some basic GR but I have no formal experience of string theory. My questions are therefore:

• What's involved with each of the above approaches? I.e. in what way does the spacetime become discretised? (Particularly in string theory)

• Are there any other popular(ish) approaches that should be added to the list?

• Supplementary query, with GR being a background-independent theory, I fail to see how one can end up with discretised spacetime without it being a pre-defined background onto which a theory of the dynamics would have to be 'bolted-on'??

Please forgive my ignorance if what I have said is misinformed, all comments and elucidations would be most welcome.

Answer 1

Let me try to address your questions, even though just the first one seems quite heavy in itself.

1. Spacetime Discreteness: let me give you links to references that are relevant to your questions, than i'll make some general comments. An Introduction to Spin Foam Models of Quantum Gravity and BF Theory; Spacetime in String Theory; The quantum structure of spacetime at the Planck scale and quantum fields; Meaning of Noncommutative Geometry and the Planck-Scale Quantum Group; Causal Sets: Discrete Gravity (Notes for the Valdivia Summer School); On the Origins of Twistor Theory — this should get you going. As for string theory and spacetime discreteness, let me say that, in a crude way, the $\alpha$ that appear in the Action in this link Superstring theory, called 'string tension', is basically what 'measures' this.

2. Other approaches not listed: i think your list is fairly complete. But, you didn't list Loop quantum gravity — maybe you were thinking of it, or maybe it fits in one of your named categories: i just thought i'd make it explicit.

3. GR discreteness: to me, this is a subtler question, in the sense that once you have discretized spacetime (for one reason or another), you should not expect that the other [geometric] structures remain 'continuous' — in fact, there's a whole branch of research dealing with 'quantum groups' and 'discretized' (or 'latticized': think computer simulations) theories. The point being that if you discretized all of your ingredientes, you still maintain a certain relation among them (e.g., discrete gauge symmetry, or $q$-gauge symmetry). The bottom line is that you can perfectly define a theory where all ingredients are properly 'discretized', and so it maintains its relevant features (recovering the continuum theory in some limit). As a side note, it's worth seeing that it's possible to discretize theories at the level of differential forms, à la Discrete Differential Forms, Gauge Theories, and Regge Calculus (PDF) (and similar constructions by several other folks). In this sense, many of the relevant properties are kept even after discretization (quite robust method).

I hope this can get this discussion started.

Answer 2

Qftme, I think that part of the problem you have with this question is the lack of a Uniform approach to these extra theories. You may have try the papers on their own; although there might be recent conference proceedings on the subject of Quantum Gravity, with a paper on each approach. You will see that the Penrose book does have references for these theories too. For example a paper on "Causal Sets as discrete spacetime" is arXiv:gr-qc/0309009.

Penrose's own approach (Twistors) itself derives from an older (non-relativistic) model of his called "Spin Networks". In the Penrose cases it is not (just) that spacetime is discrete but that spacetime is derived from another structure. In the "Spin Network" model this was a combinatoric construction, hence discrete. In Twistors it is a space called, appropriately enough "Twistor Space".

I am not sure whether Loop Quantum Gravity is on your list explicitly. There have been recent Stack discussions about this with emphasis on whether it is Lorentz invariant. Without wanting to go into the details of that discussion, just to say that a spacetime "box" of a certain size say $\hbar$ by $\hbar$ by $\hbar$ by $\hbar$ - a discretization of existing spacetime - will have a different size in a different lorentz frame. So is this simple approach consistent with Special Relativity? Different answers to that yield different approaches too.