Understanding the Incompatibility of Discrete Space and Time with Relativity

[bobie]

Relativity and continuum

I have read contrasting things about the issue.
Can you explain briefly why the assumption of discrete space and time is absolutely incompatible with relativity?

What is the main problem? Lorenz invariance or what?
Thanks

 

[bobie]

I have read that although the main feature of this theory is discreteness, yet the assumption that space and time is discrete is absolutely incompatible.

I read about HUP, the field equations etc...
Can you explain what are the real problems to consider them discrete.

I read about integration , differential equations etc , is it time the real problem?
Can we imagine a world where space is discrete and time is not?
Can't we consider the continuum a continuum of discrete building blocks?

 

[atyy]

The main feature of QM is not its discreteness. The discreteness of some quantities like energy levels in atoms is a secondary aspect particular to some quantum systems. The analogy here is like a violin string, which is continuous, yet has discrete harmonics due to its being tied down at both ends.

However, one can largely imagine that the matter we see is made from a discrete lattice, where the lattice spacing is fine enough so that we cannot see it with current experiments. There is one difficulty with this view, which is whether chiral interactions can be put on the lattice, which I think is still being researched. You can read more about this point of view in http://www.staff.science.uu.nl/~hooft101/lectures/basisqft.pdf:

"Often, authors forget to mention the first, very important, step in this logical procedure: replace the classical field theory one wishes to quantize by a strictly finite theory. Assuming that physical structures smaller than a certain size will not be important for our considerations, we replace the continuum of three-dimensional space by a discrete but dense lattice of points. ...

If this lattice is sufficiently dense, the solutions we are interested in will hardly depend on the details of this lattice, and so, the classical system will resume Lorentz invariance and the speed of light will be the practical limit for the velocity of perturbances. If necessary, we can also impose periodic boundary conditions in 3-space, and in that case our system is completely finite. Finite systems of this sort allow for 'quantization' in the old-fashioned sense: replace the Poisson brackets by commutators. Note that we did not (yet) discretize time ..."

At the end of that article, 't Hooft discusses taking the lattice spacing to zero, which at the moment cannot be done, because of the "Landau pole" of QED. He then says the problem is probably not so important on its own, because the Landau pole is above the Planck scale, and one would presumable need to solve the problem of quantum gravity too.

 

[Dale]

I am not aware that it is incompatible, just there is no experimental evidence for it.

Do you have a reference asserting the incompatibility?

 

[haael]

the assumption that space and time is discrete is absolutely incompatible.

The main problem with space and time quantization arises when we try to divide spacetime in a "crystal" manner. That is, regullary. Such quantization explicitly violates Lorenz invariance and also translation invariance.

One solution is to attempt a "glass" quantization. If the quanta are aligned irregulary, there is no preferred point, direction and frame.

 

[bhobba]

I have read that although the main feature of this theory is discreteness, yet the assumption that space and time is discrete is absolutely incompatible.

You might like to investigate Lattice Field Theory
http://en.wikipedia.org/wiki/Lattice_field_theory

Thanks
Bill

 

[jtbell]

Note: two threads have been merged.

 

[bhobba]

I am not aware that it is incompatible, just there is no experimental evidence for it.

There is no incompatibility, at least as far a I am aware.

But it is often thought stuff like virtual particles etc may simply be an artefact of taking the continuum limit in QFT and then using perturbation theory.

Evidently they do not occur in non perturbative approaches in lattice theory.

Thanks
Bill

 

[atyy]

Lattice gauge theory is typically not exactly Lorentz invariant. However, since we have no experimental proof of exact Lorentz invariance, lattice gauge theory is consistent with observations.

One form of "spacetime" discreteness that is compatible with Lorentz invariance is causal set theory. http://arxiv.org/abs/0709.0539

 

[Barry911]

I thought that the trend was toward a Leibnezian universe i.e. S-T independent and wholly relational. It would
seem that a fundamental description of the universe must exist at the level of the Planck length. At this
level (10 minus 16th below the quantum level of fermions) there would exist correlations and correlation
vertices (misleadingly termed spin nodes the units of which are h, the quantum of action. Q.M. is relevant
at this level but cannot be thought of as the fundamental level of the universe. At this level almost all is speculation and all interpretation premature.

thanks,

Barry911

 

[Barry911]

About discreteness:
With Bell's theorem and the experimental results of Alain Aspect the concept of discreteness seems to have
lost any real meaning for me. However I cannot accept the idea of action at a distance Big Al's "locality"
remains valid.

 

[bobie]

I thought that the trend was toward a Leibnezian universe i.e. S-T independent and wholly relational. 1

Can you expand on that or give me some useful links?
I read that when calculus was discovered there was a debate and Leibniz supported discrete integration and calculus but Newton's view prevailed.
Are you referring to that? what is a S-T independent universe?
Thanks

 

[bhobba]

Can you expand on that or give me some useful links?

He is just chewing the fat about some conjectures floating about.

We have zero idea at this juncture if space-time is in some manner discreet or we can always apply the methods of the calculus which assume a continuum amongst other things.

The kinds of questions you are asking relate to mathematical analysis:
http://math.univ-lyon1.fr/~okra/2011-MathIV/Zorich1.pdf

In particular the so called completeness or least upper bound axiom.

Thanks
Bill

 

[Barry911]

some conjectures floating about??
Space-time independence! hardly a "conjecture" Newton's universe contained such notions as "action
at a distance", "the rigid ether" and the idea that somehow both space and time are absolutes.
Space - time independence was established by Albert Einstein and is an essential part of Special
relativity theory. It is also essential to invariance in relativity theory.
I thought that this was a physics not a math forum.

 

[Barry911]

bobie:
Leibniz was a true genius and co-discoverer of Diff and Integral calculus. But he was much more.
Remarkably, he offered the insight that absolute space and time are unacceptable in any notion of the
universe. Further he felt that continua were impossible. He therefore posited the notion of a universe
of fundamental elements with correlations (?)
I would agree with bhobba that there are some really strange conjectures floating about e.g. Maldecena
and his "holographic universe", "the landscape", "string theory" (uh-oh now I'm in trouble) but certain
speculations seem more substantial than others i.e.- Lee Smolin's loop quantum gravity requires space-time
independence and relational structure. Roger Penrose also speculates on the validity of Smolin's theory and
offers the addition of "spin-networks" mathematical objects at the Planck level". I can't help but believe that
"quasi-discrete" is the best we can do since Bell demonstrated essential quantum correlations.

Respectfully,

Barry911 (911...the car not the emergency)

 

[bhobba]

some conjectures floating about??

The conjecture bit is if space-time is a continuum or not.

Space-time independence! hardly a "conjecture"

It's obvioius, in view of the context of this post, that's not what I was referring to. What I was referring to is:

At this level almost all is speculation and all interpretation premature.

I thought that this was a physics not a math forum.

When we model something using real numbers we subsume its axioms such as the LUB axiom.

Thanks
Bill

 

[Barry911]

I find an essential bias in most "conjectures" i.e. if one can fit a mathematical model to an assumption
the assumption becomes real physics as opposed to "speculations" about physics. e.g.- in a misguided
effort to conserve information in an absolute sense (What would Amy Noether have said?) Juan Maldecena
came up with a mathematical model ADS/CFT its interesting but irrelevant. I have in my old age become
more and more impressed with the insights of the truly great physicists: Einstein, Heisenberg, Schrodinger
et.al. Special Relativity was almost entirely conceptual and Einstein,s derivation of the Lorentz transformations
was independent of the ether (not so Lorentz) and followed his realization of the invariance (in S.R.) of the
constant C. I'm (I would guess like most of you just a very interested amatuer . I majored in physics in
college as a pre-med but went on to become a surgeon). I really like math but I love Physics. In my retirement
it has been my one great pleasure. I would guess that I resent a smattering of incomplete mathematical
renderings as being acceptable but all serious efforts at gauging "concepts" being unacceptable.
1. Mathematics though essential to Physics is not physics.
2. There are however certain mathematical structures that have great "physicality" i.e.- Euclidean, Riemannian,
Lobachevskian geomentry, Tensors and Tensor densities that the mathematician Liv Lieberman described
as the "facts for physics", Cartan's Spinors and perhaps Penroses's Twistors?? However, remember that
Newton's conceptual dynamics preceded his discovery (invention?) of the Calculus. Heisenberg independently
invented a form of matrix algebra that satisfied his concept of duality, just because he needed it! Einstein was
of course completely dismissive of mathematics in 1905 "unnecessary pretension" . He of course changed his
views after discovering Riemannian geometry and Tensors. However it took enormous insight on his part to
recognize this missing formalism as the "correct" mathematics.
Well I have been recently upbraided by the forces that be for insulting a member in response to his allegation
of my "chewing the fat about a few conjectures floating around" I have received 3 "bads" (Is big brother going
to come to my house and blast me with his gamma laser? uh-oh.
In any case I do sincerely apologize to those I've offended.
Barry911

 

[Barry911]

bhobba:
We don't model QM with real numbers?
Barry

 

[Barry911]

bhobba:
Since when can we assume a continuum in the fundamental theorem of calculus!
By its very nature it is not continuous cf.-Leibniz. What I'm trying to say is that the concept of the
infinitesimal "ds or dt" cleverly avoids continua by the definition "ds is the smallest number that is
not 0" i.e. teen-weeny but discrete.
Respectfully
Barry

 

[atyy]

@Barry911 and bhobba, the Leibniz thing has nothing (or very little) to do with calculus. It's a point of view advocated by Smolin http://arxiv.org/abs/hep-th/0507235, and does not have anything necessarily to do with fundamental discreteness.

 

[Barry911]

Dear Sci-advisor:
My response about the lack of a continuum was just a response to bhobba's implication that calculus
must be continuous.
Barry

 

[atyy]

Dear Sci-advisor:
My response about the lack of a continuum was just a response to bhobba's implication that calculus
must be continuous.
Barry

Not what you mentioned in post #10?

I thought that the trend was toward a Leibnezian universe i.e. S-T independent and wholly relational. It would
seem that a fundamental description of the universe must exist at the level of the Planck length. At this
level (10 minus 16th below the quantum level of fermions) there would exist correlations and correlation
vertices (misleadingly termed spin nodes the units of which are h, the quantum of action. Q.M. is relevant
at this level but cannot be thought of as the fundamental level of the universe. At this level almost all is speculation and all interpretation premature.

thanks,

Barry911

 

[Barry911]

Read Leibniz!
I know he hasn't published lately but he remains relevant!

Barry911

 

[bhobba]

We don't model QM with real numbers?

I don't remember commenting on that one way or the other.

Exactly what context are you alluding to?

But yes we do.

Thanks
Bill

 

[bhobba]

Since when can we assume a continuum in the fundamental theorem of calculus!

Again I never commented on the use of any specific axiom in the proof of any particular theorem.

But since you asked - here is the proof:
http://math.berkeley.edu/~ogus/Math_1A/lectures/fundamental.pdf

Continuity is used throughout the proof, and even in the statement of the theorem itself.

My comment had to do with the rigorous treatment of calculus, which makes use of the LUB axiom (or equivalent axioms), sometimes called the completeness axiom, or more relevant to the terminology used in this thread, also sometimes called the continuum property of real numbers.

Thanks
Bill

 

[bobie]

I am not aware that it is incompatible, just there is no experimental evidence for it.
Do you have a reference asserting the incompatibility?

I do not know if it is allowed to give a link to other forums, but I have copied the following statement by a famous scientist (Lubos Motl) which seems to contrast with almost everything stated in this thread. How do you comment that?

The main reason why physics isn't building on the assumption that the time is discrete is the fact that such an assumption is demonstrably incorrect. Physics is a natural science, a process of learning how Nature actually does work, not a movement to irrationally and indefensibly claim that there are some "cons" or "pros" about some arbitrary philosophical positions how Nature should work.

Time has to be described as a continuous variable because the Lorentz symmetry, the symmetry underlying relativity, a pillar of physics, is a symmetry continuously transforming continuous time and continuous space.
Moreover, all the evolution equations – equations of motion in classical mechanics, field equations in classical field theory, and/or Schrödinger's, Heisenberg's, or other equations governing any quantum mechanical theory – are differential equations for functions of time that, as David H said, couldn't work if time failed to be continuous. In quantum mechanics, one would really have to sacrifice any agreement for the experiment (by making the time really discrete) or to sacrifice unitarity because all generic enough quasi-continuous but not continuous transformations of the Hilbert space would fail to be unitary.

So the Planck time is the minimum duration beneath which time surely behaves differently and the everyday life statements about the time break down or cease to hold. But what replaces them is certainly not a naive picture of a discrete time that is counted by integers like apples.