# Is space-time discrete or continuum?

1. Apr 1, 2013

### Boy@n

Is there a way to know one way or another?

If smallest possible scale is Planck scale, does it mean that space-time is discrete where smallest possible step is Planck's length (PL) and smallest possible time is Planck's time (PT)?

If I move my hand from point A to point B, say 1m exactly in 1s, how many steps have I made? 1m/PL and for each step I spent PT? So, from that viewpoint my hand made those steps with the speed of light?

Last edited: Apr 1, 2013
2. Apr 1, 2013

### Chronos

Tests to date have failed to reveal evidence of quantized space time. One of the early predictions of this model was that high energy photons from distant sources should arrive at earth at different times than low energy photons. Another was there should be differences in the airy disc between high and low energy photons. Neither effect was detected. The notion is still entertained in some models of loop quantum gravity, but, observational evidence remains lacking.

3. Apr 1, 2013

### rbj

this is musing from an electrical engineer that knows something about discrete modeling of analog or continuous-time systems.

systems that are continuous are often described by continuous differential equations. if the diff eqs. are linear, there is a way (using Laplace Transform) to describe the system exactly and solve for a closed-form solution. but if the system has little non-linear components in them, sometimes the only way to understand the system is to simulate it with a discrete-time approximation. probably the simplest discrete-time approximation is Euler's forward method . now when programming a computer to simulate a physical system will involve turning those differential equations into difference equations, using, say the Euler method. in doing so you will compute dimensionless constants that will serve as coefficients in the discrete-time simulation. the sampling period and other time parameters will play a role in these dimensionless coefficients.

similar to cellular automata, you can take equations of physical interaction, like Maxwell's equations, and Schrodinger, and Einstein's field equation, and continuity equations for conserved quantities of physical stuff, and you can turn them into difference equations that will involve the sampling period (in the time dimension) of the simulation and cell size (in the 3 length dimensions).

now if you were to select a quantization unit in time and length that were essentially the Planck units, but defined so that these constants are removed from the above equations of physical law (which means these constants are set to 1): $4 \pi G = c = \hbar = \epsilon_0 = 1$, then when you turn the differential equations into simple difference equations (Eulers method is plenty adequate because the discrete time and cell width are virtually infinitesimally small). but because these constants of nature disappear from expressions of physical law, there are no arbitrary scaling constants in the discrete-time simulator. we don't have Nature taking this quantity of flux density and pulling this number ($\frac{1}{\epsilon_0}$) out of her butt and scaling it to convert it into field strength. their quantities are equal so then maybe they, the two physical quantities, are the same thing. maybe flux density is the very same thing as field strength in Maxwells equations as well as the discrete-time simulation of it.

but that happens (ditch the arbitrary scaling constants) only if you run your discrete-time, discrete-space simulator with discrete units of the rationalized Planck time and Planck length (where $4 \pi G = c = \hbar = \epsilon_0 = 1$ ).

because the Planck scale is soooo.... ridiculously tiny, there is no way human beings can ever hope to measure anything around that scale to observe any of this discrete phenomena. way smaller than anything in the atomic or subatomic scale. so i like to pretend it could be true. i don't think there is any way to find out one way or another.

4. Apr 1, 2013

### bcrowell

Staff Emeritus
Even if LQG is correct, LQG quantizes area and volume, not length. Length quantization is not possible because a given length can be arbitrarily contracted by Lorentz contractions.

5. Apr 2, 2013

### rbj

don't the Lorentz contractions also affect area and volume? how can it not?

6. Apr 2, 2013

### tom.stoer

It is not even clear whether all models based in discrete spacetime structure do predict deviations from continuum models as tested in these experiments. Afaik LQG as of today does not predict any new disperison relation.

7. Apr 3, 2013

### Naty1

Here are some views:

Space is discrete:

associated with Planck units which clearly implies a discreteness:

so says Leonard Susskind, THE BLACK HOLE WAR, page 154

The following is a paraphrase of an argument for the discreteness of spacetime, made by Smolin in his popular-level book Three Roads to Quantum Gravity.

The Bekenstein bound says there's a limit on how much information can be stored within a given region of space. If spacetime could be described by continuous classical fields with infinitely many degrees of freedom, then there would be no such limit. Therefore spacetime is discrete.

There is no distinction between continuous and discrete:

http://arxiv.org/abs/1010.4354

http://pirsa.org/09090005/
Good discussion in these forums here:

8. Apr 3, 2013

### Naty1

and one more perspective.....
There is a well known contradiction between relativity and Planck length:

[Wikipedia explains the contradiction nicely:

http://en.wikipedia.org/wiki/Double_special_relativity

Yet the energy density of empty space is believed to have a positive value and apparently this cosmological constant of

"...universal energy density would have the same value for all observers, no matter where or when they made their observations no matter how they moved."

(which is why Einstein called it a cosmological "constant")

9. Apr 3, 2013

### martinbn

Why can't you do the same for a given area by arbitrarily contracting one side?

10. Apr 4, 2013

### tom.stoer

There is a common misconception regarding area-quantization in LQG. The area-operator with discrete spectrum is not a physical Dirac-observable, so it can't be used to classify physical states; or the other way round: its eigenstates carrying discrete area are not physical states. So physical states which are solutions to all three constraints Gauß G, Diffeomorphism D and Hamiltonian H have not yet been constructed (due to H!) and could very well carry continuous area!

Look at a discrete basis like the harmonic oscillator states |n>. They can be used for every problem constructed in an L2[-∞,+∞] Hilbert space. But the conclusion that discrete basis states |n> with discrete energy n+1/2 do exist, does not mean that every physical problem constructed in this Hilbert space must have discrete energy levels (the situation is even more complicated b/c in the case of LQG the operator used to construct the eigenstates is not an observable like energy in the harmonic oscillator case)

So what one has to do is to construct a physical observable "area" and calculate it's spectrum. I do not know whether this has already been achieved, but I guess the Erlangen / Thiemann has done some relevant work, especially for physical constuctions getting rid of diffemorphism invariance using "dust fields" or "obsever fields".

Last edited: Apr 4, 2013
11. Apr 4, 2013

### tom.stoer

Global Lorentz covariance is not a symmetry of general relativity; observers classified according to SR
are not well-defined in GR. Taking diffeomorphism invariance into account everything fits nicely. The strange idea is that 'discrete spacetime' is often treated as equivalent with fixed lattice-like structure. This is wrong. All approaches using discrete models like LQG, CDT do not rely on a fixed structure; of course the is no fixed structure! The structure is a) arbitrary due to diffeomorphism invariance and therefore locally obsever-dependent and b) subject to renormalization and therefore unphysical.

I think that this is not correct. This can be seen quite easily if the cosmological constant becomes part of the energy-momentum tensor Tab as "dark energy"; of course this term is observer dependent! This is trivial b/c T is subject to local coordinate transformations, and these do affect the metric term cc * gab as well. The reason why nobody cares about this is that in cosmology one always uses one fixed reference frame, namely a single solution to the Einstein field equations with highest degree of symmetry. But if you consider a local observer moving with some speed v relative to a fixed background metric (i.e. DeSitter space with cc) this observer would observe a different T', i.e. a different (cc * gab)'. Of course the value of cc is not affected, but the local energy density cc * g00 is.

Last edited: Apr 4, 2013
12. Apr 4, 2013

### phinds

In previous discussions on this forum it has been stated that if time IS quantized, then the quanta is MUCH less than the Plank Time. I do not say that with any authority, I'm just repeating what I've heard here and I do not recall the rationale for that assertion.

13. Apr 5, 2013

### julian

On the issue of quantized geometry and Lorentz contractions..

the basic argument is that one cannot have quantized values as we should be able to reduce them continuously further by performing a Lorentz transformation. The flaw in this argument is that we are not dealing with classical quantities, but rather quantum observables. The resolution then follows from the fact that the length operator, L, in the original frame does not commute with the length operator, L', in the transformed frame: they do not have simultaneous eigenstates - an eigenstate of L is not an eigenstate of L', rather it is a quantum superposition of eigenstates of L'. The eigenvalues of L' will be the same as the (discrete) eigenvalues of L: it is the expectation value of L' that will be Lorentz contracted in a continuous manner.

Last edited: Apr 5, 2013
14. Apr 5, 2013

### julian

There was some controversy about eigenvalues being discrete or continuous:

"Are the spectra of geometrical operators in Loop Quantum Gravity really discrete?" by Dittrich and Thiemann - http://uk.arxiv.org/pdf/0708.1721.

Rebuked by Rovelli in "Comment on "Are the spectra of geometrical operators in Loop Quantum Gravity really discrete?" by B. Dittrich and T. Thiemann" - http://uk.arxiv.org/pdf/0708.2481.

15. Apr 5, 2013

### tom.stoer

Another common misconception is the idea that discrete spacetime or eigenvalues will necessarily violate local Lorentz / Diffeomorphism invariance. Of course this need not be the case b/c
- these symmetries can be recovered in a continuum limit / renormalization flow (like lattice gauge theory)
- violations can be restricted to the unphysical sectors
- discrete eigenvalues for angular momentum in QM do not violate rotational invariance

16. Apr 5, 2013

### julian

17. Apr 5, 2013

### julian

Yep, by exactly the same argument I gave above.

18. Apr 5, 2013

### Naty1

Rovelli, from the above paper, seems to conclude:

19. Apr 6, 2013

### akdude1

well technically theres no definite answer some physicists say that time & motion don't exist that every thing that has ever happened is simply all existing I would suggest Julian Barbour's "The End of Time" if you want to pursue this idea further its a good book

20. Apr 6, 2013

### Boy@n

I am considering to buy this book (ebook if possible). I've read some reviews, saw his website (with ideas) and description of his paper on Shape Dynamics.

All of it looks very interesting and convincing... Anyone else wanna comment on the ideas presented by Barbour, and on his book?

Last edited: Apr 6, 2013