Is our universe a mix of both continuous and discrete elements?

[Naty1]

Two recent posts/threads asking are spacetime and frequency continuous got me wondering: Do we live in a continuous or a discrete universe? Or a mix of both?
Your thoughts appreciated.

(Do we have experimental evidence anything is continuous...I can't think of anything..also, is this question essentially butting up against quantum vis relativity contradictions and so has no decent answer yet? )


(threads are:
Is Spacetime smooth, in the quantum physics forum:
https://www.physicsforums.com/showthread.php?p=2261914&posted=1#post2261914

and Is there a limit to frequency,in the general physics forum:
https://www.physicsforums.com/showthread.php?p=2259041#post2259041)


a few inputs for consideration:

Seems like big bang and cyclic models are discontinuous affairs starting the universe,
Seems like a lot of experimental evidence suggest discrete: all those particles in the standard model;
Relativity sure seems continuous itself;
the quantum nature of light (photons) strongly suggests EM waves are discrete;
electron "orbits" are discrete,
Pauli exclusion principle seems discrete,
and Planck minimums and maximums don't seem a continuous concept;
Quantum field theory in the standard model reflects lots of particles...all discrete, both matter and forces,
"it from bit" (John Wheeler) seems discrete,
Holographic theory seems discrete (bit per plack area),
electric charge is discrete (1/6.25 x 10¹⁸ coulombs),
all the nuclear constituents are particles, hence discrete;
dimensions we live it appear discrete (3+1) but some emergent theoretical models posit fractal and partial dimensional origins;
gravitons if they exist seem discrete;
Confinement of a particle in a box leads to a quantization of its allowed energy eigenvalues.

"...increasing the energy of an electromagnetic wave by one quantum energy step is the same as adding a single photon" Leonard Susskind

 

[Civilized]

In the two PF threads you linked, I have belabored the point that all the successful theories of 20th century physics are based on a continuous smooth spacetime. I won't bother repeating that point in this thread, instead I will bring in new information, and won't argue dogmatically for smooth physics: in fact, if I have time I will go into detail about why discreteness is important in physics for many observables like energy, angular momentum, and the existence of quanta (particles), although I will not necessarily say anything favorable about discrete spacetime, except that it is remotely conceivable.

Do we have experimental evidence anything is continuous...I can't think of anything

It will always be impossible to show that the mathematical definition of continuity directly applies to spacetime, since mathematical continuity requires smoothness on arbitrary small scales. This means that if we have verified smoothness up to istances as small as $10^{-n} m$ meters, then we still have not verified it at $10^{-(n+1)} m$ meters, and so the mathematical definition is not satisfied.

In this mathematical sense you are right that we have "no evidence", but keep in mind that mathematical definitions can never be shown to apply to the world with absolute certainty, and so most physicist think that verifing continuity down to $10^{-6} m$ is better than verifying continuity down to $10^{-5} m$, even if a mathematician or philosopher could easily be skeptical and say that this proves nothing. Although the philosopher's skepticism is difficult to attack logically, I can defeat it, but that's another thread, so let's keep the spirit of physics in this thread and say that $10^{-6} m$ accuracy is better than $10^{-5} m$ accuracy.

One relatively direct way to test spacetime on small scales is by measuring micro-gravity and looking for deviations. So far no deviations have been found down to the micrometer scale: http://www.npl.washington.edu/eotwash/experiments/shortRange/sr.html" I consider these kinds of test to be the most direct measurements of spacetime, although electron microscopes etc routinely probe smaller distances.

An important theoretical constraint on discrete spacetime has to do with a notion that I will intuitively describe as 'compounding' (technically I mean relevant operators in the renormalization group). Initially we may think of small scale discreteness as only causing small scale effects. But in physics the opposite often occurs, as we decrease the scale the effects become even more visible. To understand this it helps to think of chaos theory, where complex systems are extremely sensitive to initial conditions and small perurbations, and after running for a long enough time to initially identical systems can become as disparate as night and day. The bottom-line about 'compounding' is that it strongly constrains discrete theories based on physics we have already observed.

is this question essentially butting up against quantum vis relativity contradictions and so has no decent answer yet?

Well, we know there is at least one consistent theory of quantum gravity in which spacetime is continuous: superstring theory. So far there aren't any consistent theories of quantum gravity with discrete spacetime, but that could be a historical accident.

Seems like big bang and cyclic models are discontinuous affairs starting the universe

The cyclic models are definitely continuous, after all the solution to the Freidman equations of cosmology for a universe with positive curvature and large positive energy density are http://mathworld.wolfram.com/Cycloid.html" , continuous curves.

If you view the link on cycloids, you will see that the 'big crunch' events are not mathematically smooth, although they are mathematically continuous. I'm not sure if you know the difference between smoothness and continuity, I would be happy to explain, but the big bang and big crunch are continuous without being smooth. There is no mathematical sense in which they are discrete (I can explain the precise meaning of this term as well, if desired).

Seems like a lot of experimental evidence suggest discrete: all those particles in the standard model;

Yes, families of massive particles have discrete masses, this is correct! Understanding this is called the mass gap problem, and the http://www.claymath.org/millennium/Yang-Mills_Theory/"

Keep in mind that quantum fields are continuous, scroll down on this page to see some of the state of the art on localizing particles observationally: http://en.wikipedia.org/wiki/Squeezed_coherent_state"

Relativity sure seems continuous itself;

Yes, both special and general relativity are deeply rooted in smooth spacetime.

the quantum nature of light (photons) strongly suggests EM waves are discrete;

The Wigner functions in the wikipedia link above literally applies to photons in an EM wave. The thing that is discrete is their energy, not spacetime itself.

Pauli exclusion principle seems discrete,

Obviously fermion spin is discrete, but the exclusion principle also applies to the spatial wavefunctions for two identical fermions with the same spin, and these spatial wave functions are continuous.

and Planck minimums and maximums don't seem a continuous concept;

Planck units are combinations of fundamental constants of QM, relativity, and gravity. Beyond this, everything else is theoretical, and there is no doubt that bizarre physics is going on at this scale, but at present we just don't know enough to say that spacetime is discrete on that scale.

confinement of a particle in a box leads to a quantization of its allowed energy eigenvalues.

A better word would be 'discretization', since 'quantization' has a precise meanig in physics that is different from what you think. For example, momentum and position are quantized in QM but a particle can have its momentum and position be any real number with respect to some coordinate system, these variables are continuous.

It is a general feature of QM that particles in stable bound states, such as in a box, or in a harmonic oscillator, or in a hydrogen atom, will have discrete energy levels, while unstable scattering and decay states will have continuous energy levels.

 

[atyy]

An open string on the guitar only resonates at integer multiples of the fundamental. Is the guitar string discrete or continuous?

 

[malawi_glenn]

An open string on the guitar only resonates at integer multiples of the fundamental. Is the guitar string discrete or continuous?

Is this the 1 000 000$ question in "Who wants to be a physicist?"

 

[Naty1]

Here's an interesting, if somewhat windy comment:

As Robert Mills (of the Yang-Mills theory of fundamental
interactions) says, “The only way to have a consistent relativistic theory is to treat all the particles of nature as the quanta of fields…. Electrons and positrons
are to be understood as the quanta of excitation of the electron-positron field, whose ‘classical’ field equation, the analog of Maxwell’s equations for the EM field, turns out to be the Dirac equation, which started life as a relativistic version of the single-particle Schroedinger equation.”8

good grief! (I am again reminded why I did not choose math as a major...)
any way, it's from

http://physics.uark.edu/hobson/pubs/07.02.TPT.pdf

Does this apply to QCD and QED in the standard model??

 

[malawi_glenn]

what does apply?

 

[Naty1]

Civilized, great input...I'll have to think about some...

It will always be impossible to show that the mathematical definition of continuity

I disagree...that is what has always been thought about most theories and we have found smart ways and new technology...but I take your point...Leonard Susskind may have some "work arounds"...I'll see if I can find any hints...

The cyclic models are definitely continuous...you will see that the 'big crunch' events are not mathematically smooth, although they are mathematically continuous

I may have gotten carried away! I do believe that's correct.

Keep in mind that quantum fields are continuous

I don't think it's especially relevant...(See my prior post from Robert Mills, for example)

be back later..

 

[malawi_glenn]

yeah but is a guitar string quantized or discrete?

 

[apeiron]

Do we live in a continuous or a discrete universe? Or a mix of both?

There is another metaphysical option here that may be worth considering as a grounding to physical expectations.

Instead of either/or it could be both - and both in a special "limits" sense. That is asymptotic. Continuous and discrete would both be extremes that would be approached infinitely and infinitesimally closely, but never actually reached.

So on the global scale, the world would be continuous. On the most local scale, it would be discrete. Or rather it would approach both these extremes on both these scales without ever completely being either/or.

Of course, mathematically, we can then model the world in terms of these limits states. The metaphysics can say one thing, the physics apparently another. Which is what I feel has happened with GR/QM for example.

An analogy might be the number line. We have the apparent paradox that we can add up to a continuous quantity (infinity) by adding up in discrete steps. Well I'm with the ancient greeks in saying metaphysically, we can only approach infinity via a discrete counting process, not arrive at it. But mathematically, we can then simplify in a useful way by acting as if the limit state is indeed reached.

The paradox applies the other way too, though much less often mentioned. So we just assume the discreteness of the number 1. But really, saying 1 implies we mean 1.000... (and not for example, 1.0000...1). So we can approach the idea of a discrete integer infinitesimally closely by constraining the continuous number line. Metaphysically, we never reach the state of perfect constraint where we can be sure we have exactly 1. But mathematically, we can chuck away the final asymptotic uncertainty and just get on with using the construct.

So I think this is the spirit to approach the question in.

Metaphysically, it seems obvious that reality tends towards both the continuous and the discrete. Reality encompasses both diametrically opposed and mutually contradicting (dichotomous) extremes. This then sets up the nagging question of which is the more fundamental.

A good way out of this - metaphysically - is to say both are equally fundamental. They just actually exist in opposite directions. Opposite directions of scale. So go large and all seems continuous and connected. Go small and all breaks up into discrete and local. Reality is a "mix" in this way.

But then mathematically, we find it more effective to separate the mixture. We jump each nearly state to a fully broken state. The continuous becomes the Continuous. The discrete becomes the Discrete. (And the mixed can then become the Mixed in more complicated modelling if we desire).

So what I am arguing here is that too much confusion can be created by asking for evidence of "what reality really is" so we can then model it in "true fashion".

The physics does not have to exactly reflect the metaphysics. But if the dichotomous story is "true" we should expect two complementary viewpoints on the world to emerge in our modelling.

The two viewpoints will seem irreconcilable because each is defined by being exactly what the other is not. To be discrete is precisely to not be continuous, and vice versa. How can you map one back on to the other?

But if we can step back and see the one world from which these two modelling extremes emerged - as moves in opposing directions of scale - then irreconcilable mathematics becomes reconciled metaphysics.

[Ought to add that I faced exactly the same foundational issue in neuroscience - is what the brain does to produce awareness a digital or an analog process? A binary metaphysical choice that continues to create huge confusion in mind theorising. Of course, neuroscience is much further away from having effective theories in terms of either modelling choice. And indeed, a Mixed model may be what is necessary.]

 

[Fra]

To question the continuum models is good I think.

Except for the two threads you mention, there is another one where I expressed some opinons before

(1) Are there evidences of a discrete space-time
https://www.physicsforums.com/showthread.php?t=306445

(2) Dark energy as inertia of information-reference?
https://www.physicsforums.com/showthread.php?t=270765
Goto post #14

If if you question the continuum as a scientific inquiry, then the questions becomes entangled with "is information discrete" or is evidence discrete. you might ask yourself that, and consider a finite bounded observer, what this observer can distinguish.

I do not think it a coincidence that all the headache we see from divergences and infinities are seen in continuum models. It is as if this continuum itself contains a non-physical redundancy the we can not tame properly.

The question is, if you remove the mathematical redundancy, what kind of structure do we get left?

/Fredrik

 

[atyy]

How do we decide continuous/discrete in say electroweak theory, where the calculations all treat spacetime as continuous, but the theory does not have a formal continuum limit (because it is neither asymptotically free like QCD nor asymptotically safe)?

 

[Fra]

The generic way in which I imagine that some future theory will handle all interactions is that the interactions themselves, as action properties of material observers, emerge along with spacetime as evolving relations between matter.

For me at least, the idea of reconstructing the spacetime continuum, from the physical information (in which the continum is merely a special limit) must go hand in hand with emergence of interactions.

No spacetime, no interactions and vice versa.

I personally think that the question of howto deal with some of the technical issues in the standard model, might not be treated just by considering the same interaction lagrangians ontop of whatever replaces spacetime. Since the lagrangians somehow loose meaning if spacetime isn't what we think, the only remedy is to reconstruct also the actions and matter.

/Fredrik

 

[ccdantas]

Given the metaphysical nature of the question, I think one would have to consider first a method of analysis in order to ascertain whether this is a false problem (or not). By itself, this is exceedingly difficult question. Given the operation of our minds, we envision models at extreme (and mutually exclusive) directions -- discrete vs. continuous. But it does not mean that those are the only possible articulations of reality.

I suggest a reading of Deleuze, https://www.amazon.com/dp/0942299078/?tag=pfamazon01-20, first chapter.

 

[Count Iblis]

If we were really analog, then we would not be formally describable. It is not good enough to look at whether or not space-time is discrete. What matters is if the universe admits a formal description in terms of a finite amount of information. This amounts to saying that all processes in the universe are computable.

It is known that classical mechanics is not formally describable in general. Classical mechanics admits the existence of a so-called "rapidly accelerating computer". This is a mechanical computer that accelerates such that the duration of a clock cycle is half of the duration of the previous clock cycle. If the first clock cycle takes one second, then in a time span of 2 seconds you can pack an infinite number of clock cycles.

Such a computer can then "solve" the halting problem merely by running an algorithm for an infinite number of steps in a finite amount of time. It can thus be used to find the solution of mathematical problems that are not decidable. E.g. if the Riemann hypothesis is true but no proof of that exists (if it were false then a proof would exist, as you could then point out that zero that is not on the critical line), then the rapidly accelerating computer could check if all the zeroes are on the critical line one by one in a finite amount of time. It could even go through all proofs using Hilbert's proof checking's algorithm and tell you that no proof of this fact exists.

 

[Hans de Vries]

An open string on the guitar only resonates at integer multiples of the fundamental. Is the guitar string discrete or continuous?

The discrete state is a simplifying abstraction allowed by the semi-
stationary nature of the continuous function.

Quantization is per definition a simplified abstraction. Expressions
using quantized states generally violate Special Relativity. This does
not mean that they don't lead to the right answers but they ignore
the underlying relativistic processes, which are often unknown.
For example the amplitude.

$$\langle \psi_1 | \psi_2 \rangle$$

violates Special Relativity. The integral over space is not allowed since
it leads to space-like outside-the-light-cone dependencies.


Regards, Hans

 

[Naty1]

Wow, great replies by many...Thank you so much. I must consider a number of new thoughts!

These are the type replies I hoped this forum would provide... so as for all who are interested to understand better. As I may have posted in another thread, I never even thought much about discreteness of spacetime until someone asked the " is frequency continuous" question. After reading perhaps 15 or so popular contemporary books from world reknowned physicsts who repeatedly remark on it's discreteness, I never even thought there would be an "objection".

In any case my own mind remains open...

NOTE:
Chapter 16 of Roger Penrose's book, THE ROAD TO REALITY, titled The Ladder of Infinity seems to address at least some issues related to this discussion...perhaps without clear and firm resolution. 16.2 is "A Finite or Infinite Geometry for Physics?" and 16.5, "Puzzles in the Foundations of Mathematics" might even be disturbing for some... If anyone can comment on Penrose's perspectives that would be fascinating. The math discussion, including sets, classes and (16.7) "Sizes of Infinities in Physics" is rather advanced and I know enough to know what I don't know.

 

[apeiron]

Given the metaphysical nature of the question, I think one would have to consider first a method of analysis in order to ascertain whether this is a false problem (or not). By itself, this is exceedingly difficult question. Given the operation of our minds, we envision models at extreme (and mutually exclusive) directions -- discrete vs. continuous. But it does not mean that those are the only possible articulations of reality.

This was my point. Metaphysics always breaks down into dichotomies. We then have to ask is this just because humans are simple minded or whether this is in fact a deeper reflection of how things really are?

This is the area I have been studying the past decade so I do feel qualified to offer some answers.

Dichotomies are suspicious beasts because we think perhaps reality is more complex and reducing to pairs of alternatives is over-simplifying. But the key lies in the fact that dichotomies are formed by mutual exclusion. One extreme is everything the other is not. It is its antithesis. The symmetry is broken and you have instead maximal asymmetry.

The question then is where do you go from the dichotomy. One familiar metaphysical response is to say we have two extremes, one must be valid, the other false. Or one must be fundamental, the other emergent. This is monadism - the search for a single essence. So in this case, the expectation that either continuity or discreteness will prove to be the primal view.

An allied response is dualism. Now this accepts both extremes as fundamental. But there are two essences. So we have the dualism of substance and form, or body and mind. A double monadism.

The alternative path (one far less frequently trodden) is instead to accept both extremes (as limit states only) and then heal the divide via their fruitful interaction. So two extremes may be produced, but the story is then in the "thirdness" (a Peircean term) of how they interact. This leads on to a hierarchy theory approach in which you have levels in dynamic interaction.

So everyone lands up in dichotomies (because that is the logical result of trying to divide a complex reality into its simplest possible categories or features). And then you have to chose either to make one extreme fundamental, or go triadic and model two extremes in interaction. Or take the unsatisfactory path of dualism in which you believe in two fundamentals, but see no possible causal connection between them.

Just think of all the dichotomies we accept as useful. Symmetry-asymmetry. Algebra-geometry. Space-time. Substance-form. Local-global. Atom-void. Particle-wave. Local-nonlocal. Signal-noise.

And recall what Bohr said about dichotomies (or complementarities as he called them). Paraphasing - when you find that the absolute negation of an obvious truth also seems obviously true, then you know you have arrived at a profound level of modelling.

As you say, a crucial question is whether dichotomisation is just epistemology, a habit imposed on reality by inadequate human minds? Or whether it is ontic, in fact also the way reality forms itself?

 

[friend]

For me at least, the idea of reconstructing the spacetime continuum, from the physical information (in which the continum is merely a special limit) must go hand in hand with emergence of interactions.

No spacetime, no interactions and vice versa.

I personally think that the question of howto deal with some of the technical issues in the standard model, might not be treated just by considering the same interaction lagrangians ontop of whatever replaces spacetime. Since the lagrangians somehow loose meaning if spacetime isn't what we think, the only remedy is to reconstruct also the actions and matter.

/Fredrik

We know that the field equations of General Relativity can be derived from the least action principle of the Hilbert-Einstein action integral. And we put this into the Feynman path integral to get QG. But if we could justify the path integral from first principle and discern the Hilbert-Einstein within that, then we would know that the spacetime metric IS quantized.

 

[Civilized]

Here is a http://motls.blogspot.com/2005/11/discrete-physics.html" that articulates a lot of points I agree with. I will give his main points in bold and my thoughts on each:

Both discrete and continuous mathematics matter

The classical guitar string is a great example, the string itself is can be treated as continuous, and the frequencies of the normal modes are discrete.

Discrete and continuous concepts are related

For example the Reimann hypothesis which connects the primes with smooth functions in the complex plane, or the entire branch of mathematics that treats the topology of manifolds, which seeks to classify smooth structures using (typically) discrete invariants.

Crackpots are almost always discrete

Here by crackpots I believe Motl doesn't mean the extended sense e.g. anyone whose research he doesn't like, but rather the utter and total cranks who we have all seen, e.g. work alone, barely use math, off the charts on the Baez crank scale, etc. I do agree that most really terrible cranks do use discrete concepts, but it probably mostly has to do with there not having learned calculus at all.

Modern history of continuous dominance

It would be difficult to argue against the statement that continuous methods have dominated fundamental physics for the last two centuries. In cases like the guitar string, the continuous description is fundamental and the discrete modes are only in the solutions to the equations; this is the same story as QM. In other words, we start with a continuous description and sometimes derive discrete solutions, this is very different than starting from a discrete description.

I think Lubos illustrates the point well in this paragraph:

The "old" quantum theory with its model of the Hydrogen atom from Niels Bohr became a childish game when the "new" quantum mechanics was constructed. The correct picture that explains the quantization of energies may be defined in terms of a continuous wavefunction. Shockingly enough, the eigenfunctions of a Hermitean operator may form a discrete set. Also, there still existed discrete particles. Nevertheless, it turned out that they were again manifestations of a discrete spectrum of a completely continuously defined operator in quantum field theory where it acts on a completely continuous Hilbert space.

There are considerably more points, and I suggest anyone who is interested to read the post, it is not one of his more mean-spirited rants, and it is mostly full of physics content. The latter section of his rant is dedicated to debunking the "we just invented digital computers => the universe might be a digital computer" line of thought. A good read.

 

[apeiron]

Modern history of continuous dominance

It would be difficult to argue against the statement that continuous methods have dominated fundamental physics for the last two centuries. In cases like the guitar string, the continuous description is fundamental and the discrete modes are only in the solutions to the equations; this is the same story as QM. In other words, we start with a continuous description and sometimes derive discrete solutions, this is very different than starting from a discrete description.

Continuous methods could be what works best for modelling, yet still leave the ontological question open. And a good grasp of the metaphysics would be what helps to understand the current limits of models.

So for instance, you quote Motl on QM and the continuity of the wavefunction. Yet the wavefunction is precisely a powerful model because it (as completely as possible) excludes the discrete from its domain. The discreteness being the issue of observer collapse which happens at some point in spacetime.

It is not that using "pure continuity" to create a model is bad. It is just that we can expect it then to exclude some matching aspect of the "purely discrete". This then has to be inserted somehow back into the picture to make it complete.

The guitar string analogy, while very appealing, indeed useful, is also flawed in that it too must assume discreteness. A guitar string must be pegged at some distinct point. In fact at both ends. So all of the string is continuous, but it only gives well behaved harmonics due to it being discrete in its boundary.

Imagine for example a string that is only vaguely pinned down at its ends. It can no longer be plucked crisply.

To take this discussion in another direction, I would suggest a better analogy is fractal phenomena like critical opalescence, cantor dust, and other "is it continuous/is it discrete" examples of geometry.

Fractals are an example of what I would call Mixed - the Continuous and the Discrete expressed in interaction over all scales.

 

[Naty1]

Reading the above posts and thinking more about my own question makes me think I inadvertently stumbled into the relativity quantum mechanics contradictions. So the experimental results we obtain depend on the questions we ask.

If anyone can interpret Penrose's Chapter 26 analyses that could be insightful if not conclusive. (post #16)

a few comments,observations,questions:

Civilized posted (#2)

Well, we know there is at least one consistent theory of quantum gravity in which spacetime is continuous: superstring theory. So far there aren't any consistent theories of quantum gravity with discrete spacetime, but that could be a historical accident.

I thought superstring theory was background dependent, that is, a spacetime continum is an input to the model. Are there any string theories with background independence?

I thought loop quantum gravity arrived at a (background independent) discrete spacetme.

Malawi posts (#8) :

yeah but is a guitar string quantized or discrete?

well unless the periodic table has become continuous, it's discrete...but I wonder if that could in general be discerned from the "quantized" vibration modes? maybe not: aperion in post #20 comments on boundries at the pegged ends of the string...so we tease out discreteness from that constraint.

Aperion posts (#9)

Metaphysically, it seems obvious that reality tends towards both the continuous and the discrete. ... This then sets up the nagging question of which is the more fundamental...A good way out of this - metaphysically - is to say both are equally fundamental.

I don't like to admit its a "good" way but it may end up being the best we can do...I like the "nagging" part!

The physics does not have to exactly reflect the metaphysics

No, but isn't that the ideal objective? I mean we want to be able to describe and predict what we observe, but don't we aspire to more?

Fra (post #10) Thanks for the additional thread references!...looks like I posted in #1 early on then either was away or just lost track...both interesting! will study.

Iblis post #14:

What matters is if the universe admits a formal description in terms of a finite amount of information...It is known that classical mechanics is not formally describable in general.

Not sure I understand the broader point, but so far it seems finite amount of information resides in a finite space.

Hans post #15

The discrete state is a simplifying abstraction allowed by the semi-
stationary nature of the continuous function. Quantization is per definition a simplified abstraction.

That can be true of course, but it does not seem universally so.

friend post #18:

We know that the field equations of General Relativity can be derived from the least action principle of the Hilbert-Einstein action integral.

I'd love to see a thread devoted to alternative derivations of relativity...approaches based on other theories and what, if any, insights theey might offer...Susskind mentioned on in THE BLACK HOLE WAR and untuil I reread the book I'll probably never find it...

aperion post #20

It is not that using "pure continuity" to create a model is bad. It is just that we can expect it then to exclude some matching aspect of the "purely discrete"

I am afraid that's a big piece of our puzzle. good thought.


again, my thanks for the replies...lots to read and maybe even learn a little

 

[Count Iblis]

I would still argue that there isn't a shred of evidence that the universe is really continuous, because then the universe would not be formally describable. To describe the state of a system in some finite volume requires a finite amount of information...

 

[apeiron]

I would still argue that there isn't a shred of evidence that the universe is really continuous, because then the universe would not be formally describable. To describe the state of a system in some finite volume requires a finite amount of information...

This is again making the confusion between our models and the reality which we model.

It is true that formal models seem to depend on discrete techniques. Co-ordinates, measurements, etc. But note the trick. Physics does thing like measure a start and stop point - discrete measurements to plug into an equation - then relies on the continuity of the path inbetween. So the models discard the presumed continuity of the world, condensing it as a function that can map one discrete state on to a second discrete state.

The formal models of physics are actually based on a crisp dichotomisation to make a mixed reality describable.

 

[Lambda3]

I thought superstring theory was background dependent, that is, a spacetime continum is an input to the model. Are there any string theories with background independence?

I thought loop quantum gravity arrived at a (background independent) discrete spacetme.

I'm fairly sure that there aren't any background independent string theories. Loop quantum gravity does arrive at a background independent theory. In Lee Smolin's book The Trouble With Physics, he says, "One would like to do better and show that the discreteness of space and time is a consequence of putting the principles of quantum theory and relativity theory together. This is what loop quantum gravity accomplishes."
He also says, "Loop quantum gravity is the same idea but developed in a completely background-independent theory."

 

[Civilized]

I want to clarify the mathematical meaning of discrete and continuous, and I'll try to keep it simple.

Let S be a set of points on the real number line $\{ p_1, p_2, p_3, ... \}$. The set S can be either finite or infinite.

We say that S is discrete if for each point $p_i$ in S there is a line segment $l_i$ containing $p_i$, such that none of the line segments covering the points in S overlap each other. A mathematician would say that the intersection of the line segments $\{ l_1, l_2, l_3, ... \}$ is an empty set.

The prototypical discrete set is the integers. Intuitively, a set of points is discrete if each of the points has some 'elbow room', in other words there are no points that have infinitely many arbitrarily close neighbors.

Of course, there is nothing special about the points being on a line, you could have discrete sets of points in any dimension, except in 2d you would cover the points with disks instead of line segments, and in 3d you would cover the points with balls, and so on, and then say that if all the points are covered with balls and none of the balls over lap, then the set is discrete.


Now, what do we mean by continuous? In 1d we mean that it is as "full of points" as a straight line is, or that (infinitely) close up it looks just like a straight line. Put another way, any continuous one-dimensional object is a line or circle that has been twisted up and bent and tied in knots or such. Therefore a line is the prototypical continuous set in one-dimension, and spheres, donuts, pretzels, in 2d, solid balls and so on in 3d, etc.

In mathematics, the adjectives "discrete" and "continuous" only apply to sets of points.

These definitions agree with apeiron that fractals are neither discrete nor continuous, but share some properties of both.

To describe the state of a system in some finite volume requires a finite amount of information...

I don't see why you think this should be true, is it just intuition?

Loop quantum gravity does arrive at a background independent theory.

That is an overstatement, I would say that LQG "hopes to" or "is attempting to" develop a background independent theory.

String theory is the first theory we have in physics that constrains the dimension of spacetime enough to be predictive. I understand that there are alternative approaches in which "background independence" means starting without spacetime, just using a discrete graph of edges and vertices, labels, maybe a lattice, and having the thing come to equilibrium and finding spacetime to emerge in a sense. As far as I know, nothing like this has succeded in making predictions or matching known observations.

By the way, in string theory "background independence" has a completely different meaning. We say that a formulation is background dependent when it is given as a perturbation of a solution to classical Einstein gravity. A non-perturbative string theory is a fully quantum theory of gravity, not just a perturbation of classical gravity, and this is what string theorist mean by "background independent."

 

[Count Iblis]

The entropy of a system of finite volume is a finite number if the energy is finite. Even if you allow for the energy to be arbitrarily high, the number of possible states in the volume is bounded via the holographic principle.

 

[Civilized]

The entropy of a system of finite volume is a finite number if the energy is finite.

This is true if and only if the phase space is finite-dimensional, and so it does not include systems with an infinite number of degrees of freedom, such as (dynamic) continuous spacetime. Therefore the argument is circular.

 

[Count Iblis]

This is true if and only if the phase space is finite-dimensional, and so it does not include systems with an infinite number of degrees of freedom, such as (dynamic) continuous spacetime. Therefore the argument is circular.

It may be that gravitational effects that will cause black holes to appear if you go to high enough energies (i.e. to short enough length scales) is a manifestation of Nature being formally describable and hence not a real continuum.

 

[Civilized]

formally describable and hence not a real continuum.

Why can't nature be both formally describable and be a continuum?

After all, the equation 5x - 2x = 3x is a formal description of an infinite amount of information, and in fact most of physics consists of formal descriptions of infinite amounts of information, such as a trajectory x(t) = t^2. The point is that using mathematics we can and do give finite formal descriptions that contain an infinite amount of information, and we are especially good at doing this when the infinite amount of information is "continuously" connected because it's domain is continuous e.g. differential equations, which are certainly the heart of classical physics and continue to play a central role in quantum physics, allow compact expressions to capture a staggeringly infinite amount of information.

 

[Count Iblis]

The alleged "infinite amount of information" isn't really there. With a finite number of bits you cannot possibly define a structure than needs an infinite amount of information to be fixed. I'll defer to logicians to make rigorous statements on this topic, but my understanding is that the reason why things like the continuum hypothesis are undecidable is precisely because with only finite number of axioms there is no way you could precisely define the uncountable continuum.

This is also mentioned here:

http://www.earlham.edu/~peters/courses/logsys/low-skol.htm

The proof for LST relies on the fact that there are at most countably many descriptions of anything, viz. names, sentences, paragraphs, books... There are at most countably many strings of symbols (when the strings are finite in length). This fact is easily proved by arithmetizing the alphabet of our language of description. Each wff then becomes a natural number. Since there are only countably many natural numbers, there are at most countably many wffs to do the describing.

One countable model that is always available for inspection, if only to demystify LST a bit, is the interpretation in which the terms of the language are assigned to their own tokens, or to the typographic strings which express them. We've seen that by arithmetization there are at most countably many such strings. Hence, even if the intended interpretation of the marks on paper refers to the uncountably many real numbers, one obvious alternate interpretation refers only to the countably many marks on paper that comprise the system.

As the typographic interpretation of S shows, the interpretations with merely countable models will be (or may as well be) non-standard. The "meaning" of the marks changes at the same time as the domain. The theorem which in the intended interpretation made some assertion about uncountable reals speaks of something entirely different in the countable LST models.

Remember, a formal system "about the reals" is really a system of wffs of some formal language. The language is inherently uninterpreted. We might give its symbols some interpretation, and on that intended interpretation we may say that the system is "about" the reals. LST asserts that every consistent first-order theory can be intepreted as being "about" some set of things no more numerous than the natural numbers, even if we thought it was --indeed, even if under another interpretation it is-- "about" the uncountable reals.

LST does not deny the possibility that some system S might have uncountable models alongside the critical countable model. In that sense, S might "succeed". LST qualifies this success by giving S other models whether it wanted them or not. How does this detract from S's success? In only one way: S is thwarted if it aspired to capture its intended domain unambiguously or uniquely.

 

[Count Iblis]

It is actually easy to construct a counterexample.

Suppose that using a huge computer you simulate a planet with mathematicians and physicsits living on it. The computer computes everything down to the atomic level, so it would be able to reproduce most of the mathematical and physics results right until the 19th century. You would thus expect to see mathematicians developing calculus based on the uncountable reals being developed by your digital mathematicians.

But the uncountable reals do not exist at all in their digital world! Their world is even less than countable, it is finite. All possible states the computer can be in, can be specified using a finite number of bits.

So, what is really ging on is that people can represent phenomena in their world using some abstract rules which involves manipulating finite bistrings. But an interpretation about "uncountable reals existing" does not have to be correct.

The only way you could prove that the continuum really exists is by constructing a machine that produces results that are not formally describable, e.g. the so-called "rapidly accelerating computer" I wrote about earlier in this thread.

 

[Naty1]

Roger Penrose has some interesting insights in THE ROAD TO REALITY, my version is 2004.
He does not resolve the issues here but perhaps his insights offer some perspective:

Chapter 16:

There are some...who would prefer a universe..that is finite in extent..only finitely divisible..so that a fundamental discreteness might begin to emerge at the tinest levels...(it's distinctly unconventional..but not inherently inconsistent. In the early days of quantum mechanics... a hope was not realized by future developments...that the theory was leading to a picture of discreteness at the tinest levels. In the successful theories of our present day we take spacetime as a continum even when quantum concepts are involved and ideas that involve small scale discreteness must be regarded as 'unconventional'...It appears, for the time being at least, we have to take the use of the infinite seriously.

Then follows several sections involving "Puzzles in the foundation of mathematics" sets,classes, Godels Theorem which seems inconclusive and introduces further analyses in later chapters.

In 31.1, regarding the Holographic conjecture/Principle, Penrose says:

A reason for hoping (Maldacena's) ADS/CFT is true appears to be that it might provide a handle on what a a string theory could be like, without resorting to the usual pertebative methods with all the severe limitations such methods have.

Section 33.1 is also interesting: "Theories where geometries have discrete elements" followed by his own quite different "twister theory".

 

[Civilized]

I'll defer to logicians to make rigorous statements on this topic, but my understanding is that the reason why things like the continuum hypothesis are undecidable is precisely because with only finite number of axioms there is no way you could precisely define the uncountable continuum.

No, there is no problem with defining the continuum with a finite number of axioms. The continuum hypothesis does not pertain to this at all.

The real numbers are the unique complete totally ordered field; students are exposed to the rigorous construction as senior undergraduates or beginning graduates.

The continuum hypothesis is the claim that the cardinality of the real numbers is equal to the cardanality of the power set (set of all subsets of) the natural numbers. The continuum hypothesis is undecidable in ZFC. There is no problem with any of these things being rigorously defined in a finite axiomatic setting.

The Löwenheim-Skolem Theorem

First of all, countable != discrete. The rational numbers Q are countable, but in fact every point has infinitely many arbitrarily close neighbors!

Furthermore, the set of analytic functions from Q to Q is countable, and so smooth physics could be done entirely over the countable rationals if desired.

Suppose that using a huge computer you simulate a planet with mathematicians and physicsits living on it. The computer computes everything down to the atomic level, so it would be able to reproduce most of the mathematical and physics results right until the 19th century.

Here you go again assuming the world can be simulated by a finite digital computer. What is the basis for this assumption? If the world were a continuum, then in principle I could encode all the information on the internet into a scratch on a rod.

So, what is really ging on is that people can represent phenomena in their world using some abstract rules which involves manipulating finite bistrings.

Mathematics is much more than symbolic manipulation, and a comuter which contains only the finite-bit strings of various mathematical papers without containing the semantic meaning of these has not captured all of the information.

The point is that these finite-bit strings refer to an infinite number of possibilities.

The only way you could prove that the continuum really exists is by constructing a machine that produces results that are not formally describable, e.g. the so-called "rapidly accelerating computer" I wrote about earlier in this thread.

You can't prove anything about the physical world, ever. But you saying we can't prove continuum physics is a world a part from saying that continuum physics cannot possibly be the case because of some information perspective.

 

[Count Iblis]

No, there is no problem with defining the continuum with a finite number of axioms. The continuum hypothesis does not pertain to this at all.

According to Chaitin it does.

 

[Count Iblis]

The real numbers are the unique complete totally ordered field; students are exposed to the rigorous construction as senior undergraduates or beginning graduates.

Yes, but that doesn't mean that real numbers defined in this way make any physical sense. Almost all real numbers are uncomputable and as far as we know the physical world is computable.