Is Spacetime Countable? by Sean Gryb

Sean,

the "number of degrees of freedom" of a standard QFT is discrete. Not continuous. This is the entire point of the Fock space construction. The one-particle space describes, obviously, a finite number of degrees of freedom, and so the two-particle state space, and so the n-particle space state, so you can associate an integer number to each degree of freedom, which is your definition of a system with a finite number of degrees of freedom.

ciao, carlo

Dear Sean,

Very good essay that, in my opinion, that tackles the ambitious task of unifying QM and GR in the right way.

My topic has to do with "dessins d'enfants" and contextuality. In the wikipedia paper you have a picture of the triangulated hyperbolic plane.

http://en.wikipedia.org/wiki/Dessin_d'enfant

In his book "The icosahedron and the solutions of equations of the fifth degree" (Dover, NY, 1956), Klein's antcipates the Belyi functions for platonic solids by using the stereographic projection.

My essay is here

http://www.fqxi.org/community/forum/topic/1789

I don't look at scale invariance, I don't look at the large scale universe but strangely there are closely related mathematical objects at the starting point.

Good luck,

Michel

Hello Sean

Richard Feynman in his Nobel Acceptance Speech (http://www.nobelprize.org/nobel_prizes/physics/laureates/1965/feynman-lecture.html)

said: "It always seems odd to me that the fundamental laws of physics, when discovered, can appear in so many different forms that are not apparently identical at first, but with a little mathematical fiddling you can show the relationship. And example of this is the Schrodinger equation and the Heisenberg formulation of quantum mechanics. I don't know why that is - it remains a mystery, but it was something I learned from experience. There is always another way to say the same thing that doesn't look at all like the way you said it before. I don't know what the reason for this is. I think it is somehow a representation of the simplicity of nature."

I too believe in the simplicity of nature, and I am glad that Richard Feynman, a Nobel-winning famous physicist, also believe in the same thing I do, but I had come to my belief long before I knew about that particular statement.

The belief that "Nature is simple" is however being expressed differently in my essay "Analogical Engine" linked to http://fqxi.org/community/forum/topic/1865 .

Specifically though, I said "Planck constant is the Mother of All Dualities" and I put it schematically as: wave-particle ~ quantum-classical ~ gene-protein ~ analogy- reasoning ~ linear-nonlinear ~ connected-notconnected ~ computable-notcomputable ~ mind-body ~ Bit-It ~ variation-selection ~ freedom-determinism ... and so on.

Taken two at a time, it can be read as "what quantum is to classical" is similar to (~) "what wave is to particle." You can choose any two from among the multitudes that can be found in our discourses.

I could have put Schrodinger wave ontology-Heisenberg particle ontology duality in the list had it comes to my mind!

Since "Nature is Analogical", we are free to probe nature in so many different ways. And you have touched some corners of it.

Best

Than Tin

Dear Dr. Gryb,

interesting essay. I like the idea of shape dynamics. One point is especially interesting. You wrote about scale invariance and I think this point of view has something to do with my approach using exotic smoothness. In my essay, I also discuss the question about the countability of the spacetime. I came to the same conclusion, in particular I do not put the smooth manifold away. There is another possibillity to kill the infinities (I believe). All relevant structures in 3D and 4D are very rigid, for instance one has Mostow rigidity for hyperbolic 3-manifolds etc.

I certainly have to read your other papers.

Good luck for the contest

Best

Torsten

Dear S. Gryb,

The reduction of dimensions like the use of meter defined in S. R. simultaneity and time as nothing inherently more than the ticks of hands pointing to positions on a dial indicate a departure form dimension of some if not all quantities by a type of foreshadowing.

Thankful for the good read,

Amos.

Dear Sean,

Please pardon my straying from subject matter. As I am not a professional physicist and just for me to be clear and learn from the experts, especially those with a relationist leaning: Is it being implied by the relational view of space and as suggested by Mach's or other relational principle that what decides whether a centrifugal force would act between two bodies in *constant relation*, would not be the bodies themselves, since they are at fixed distance to each other, nor the space in which they are located since it is a nothing, but by a distant sub-atomic particle light-years away in one of the fixed stars in whose reference frame the *constantly related* bodies are in circular motion? Or in which other relational reference frame can such circular motion be described?

You can reply me here or on my blog. Much appreciated. And please pardon my naive view of physics.

Accept my best regards,

Akinbo

Dear Sean,

You propose to trade the scale-invariance of special relativity to explain the real universe in a way that its expansion is not influenced by any form of matter and energy. In fact, there already exists one such theory - the Milne's model which is a kinematic theory based on the cosmological principle.

It has recently been shown [Phys. Scripta, 5, 055901, (2013)] that this theory explains all the cosmological observations without even requiring the speculative dark matter, dark energy and inflation. Moreover, this theory evades the long-standing problems of the standard cosmology. It would not be correct to say, as is generally (mis)believed, that the Milne model represents an empty universe.

As the Milne's model appears as a homogeneous, isotropic solution of equations R^{ik}=0 (see my essay in the present contest), and as equations R^{ik}=0 present an scale-invariant theory, it appears that your holy grail is the theory I have presented! I would appreciate your comments on my essay.

Best Regards.

___Ram

Dear Sean,

One single principle leads the Universe.

Every thing, every object, every phenomenon

is under the influence of this principle.

Nothing can exist if it is not born in the form of opposites.

I simply invite you to discover this in a few words,

but the main part is coming soon.

Thank you, and good luck!

I rated your essay accordingly to my appreciation.

Please visit My essay.

Dear Dr. Gryb,

A clear and nicely illustrated essay (the pictures help). If I understand correctly you are proposing scale invariance as a guiding principle to give a small scale structure for space-time which is smooth rather than foamy or discrete. This is done in the context of dS space-time rather than Minkowski space-time. This is a nice idea and may have some connection with an article by Aldrovandi and Pereira "Is Physics Asking for a New Kinematics?" Int.J.Mod.Phys.D17:2485-2493,2009. The abstract reads "It is discussed whether some of the consistency problems of present-day physics could be solved by replacing special relativity, whose underlying kinematics is ruled by the Poincare' group, by de Sitter relativity, with underlying kinematics ruled by the de Sitter group. In contrast to ordinary special relativity, which seems to fail at the Planck scale, this new relativity is "universal" in the sense that it holds at all energy scales." Anyway the two features of using dS and being valid at all energy scales was what made me think there might be a connection.

Also there may be *experimental* support for a smooth space-time structure at the smallest scales. About a year ago the Fermi satellite timed three gamma ray photons which arrived essentially at the same time after traveling a great distance from a gamma ray burst. Since the photons had slightly different energies one would expect *if* space-time were discrete, foamy etc. that the different energy photons would disperse due to interactions with the space-time fine structure. Since they didn't, within limits this implies that space-time is smooth within some limits. The original paper is "Bounds on Spectral Dispersion from Fermi-detected Gamma Ray Bursts", R. J. Nemiroff, et al. arXiv:1109.5191 [astro-ph.CO]

Finally if your fundamental theory is scale invariant at some point to connect with GR and the SM at low energy you need to introduce a scale somehow. Any ideas along these lines? Or in fact the cosmological constant does have a scale to it although I;m not sure if this will do.

Anyway a clever idea.

BEst,

Doug

Hi Sean,

Within so many essays it is very hard to find something so interesting as your essay.

We do not agree in 100% but we are very close. My key concept for the unification in physics is scale invariant metric. I have even proposed a simple spin experiment to find out if that metric exists.

Despite the differences between our views (we could discuss them if you read my concept) your essay deserves the highest rating.

Best regards

Dear Sean,

You are proposing the opposite as what I think is the essence of reality, but your proposition is so full of good basic assumptions that I read it twice, also you explain some mathematical techniques very clearly, even for a layman like me. You even handed me some tools to explain my own perceptions, thank you for that.

regarding the process that are leading to your conclusions :

Scale Invariance : Only the shape is important not the size and any transformation does not have impact on the basic shape. You could see it as a fractal, this goes from infinity(small) to infinity(large) and it is impossible to determine what is small(shape) and what is big(shape), but is forms all kind of "realities". However I think that this kind od scale invariance is a mathematical formulation that leads to a primal stamp and since every scale is available you can find at any scale a "reality" that is in conformity with the original stamp.

Singularities : I agree that a singularity is dimensionless, it is a point, anything with "dimension(s)" is causal. a singularity is so a dimensionless quantity which is scale independent. Which can also been interpreted that a singularity combines the infinite small and large , it is both the ultimate Zero and Everything (both are scale independent).

You introduce the dS hyperboloid as the base of perception of space-time with two-d spheres as representation of "snapshots" of the universe. The axes of the hyperboloids are not represented but are present, so I wonder what is happening to your 2d spheres when we go in both directions to infinity, I am just feeling that the spheres will be longer spheres but become ellipsoids and in infinity but forgive me if I am wrong, I am not a mathematical pro. My perception is that the NOW can be represented as the cross-point of two axes, as a torus, and this cross-point is moving in the direction of t. This cross-point is a singularity that can be seen as the "Eternal Now moment".

Of course this is also an "assumption" or a "postulate" like every hypothesis it has "conventions" that are agreed upon by human beings.

My opinion is that scale invariance and the mathematical layer you are developing to reach this through conformal mathematical transformations which lead you to the conclusion that the whole shebang is a continuum without any limits is just a (very fine) mathematical projection that works like the stamp I described above, a stamp that is only valid for a certain scale of our experienced reality.

The Escher drawing you are ending with gives a good image of your perception but it is just a stereographic projection.

I enjoyed very much your essay and rated it high, if you read mine I am sure you will not agree but anyway here is the [link:fqxi.org/community/forum/topic/1810]link[:link]

respectfully

Wilhelmus

PS and good luck in my native country (I now live in France but lived 53 years in Holland)

Dear Sean

By the time I finished reading your well-written and illustrated essay I could understand what makes scale invariance attractive to you. The technicalities of your approach are beyond me, but I feel that if one starts from the point of view of particle physics with its very absolute value of (h) scale invariance would apply from the small to the large - but would it apply in the other direction?

I wonder how scale invariance would apply in Loop Quantum Gravity, or in an ordered lattice Universe such as my Beautiful Universe Theory BU also found here. I feel that at some minimum scale invariance does not apply in these discrete theories.

For example the measurement of the Gravitational Constant G would seems scale invariant in a continuous physics - but when it comes down to the attraction (and/or repulsion) between individual nodes of an ordered lattice - such as in BU - G (and hence the Planck scale) would be quite different.

With best wishes

Vladimir

Dear Physician,

Mass ratio of neutron/proton is fundamental in physics.

Maybe you do not have time to read essay of unknown authors. I encourage you, therefore, allow you to comment on the very essence of following formula:

[math]\gamma= 2^{(cy+p+3t)/(2+2a^{2}m)}=1.0013784192[/math]

Where mathematical constant are:

[math]2\pi=6.2831853, t=log(2\pi,2)=2.6514961295, cy=e^{2\pi}= 535.4916555248 [/math]

Physical constants:

[math]a=1/\alpha=137.035999074, \mu=1836.15267245,m=log(\mu,2)=10.8424703056

p=cy/2-(\mu/a+1)/(\mu/a+2)-1=265.8107668189

[/math]

An important physicist said it was a coincidence, or perhaps just a curiosity. Perhaps you feel the same. My opinion is opposite. I think that in terms of, such a significant relationship physicist should have an attitude.

Greetings Branko

Sean,

I just read your essay, and I must say that it is one of the very finest I've read. As a non-physicist, I was quite impressed with your exceptionally smooth writing style and the fact that, in my opinion, you took the time to explain your thoughts, concepts, and formulas in such a comprehensible manner. This made it so much more enjoyable for me to read. Also, great credit to your graphic artist - truly incredible graphics as well. I gave you the highest rating possible.

Best to you in the future. You would make (in my humble opinion) an outstanding instructor/teacher because of your remarkable ability to convey complex information that is comprehensible to the average person, but I suspect you will be very successful at whatever you choose to do.

Sincerely,

Ralph

Dear Sean,

While the title and the beginning sentences (countability) in your paper gave the impression that you would address some aspects of relation ebtween matter and information, it finally was addressing the relation between quantum gravity and scale invariance. I find only a tenous connection, subject to interpretation, with the original question in discussion - do you have an explanation of the connection between what you wrote and the relation between matter and information?

Thanks.

Unnikrishnan

Having read so many insightful essays, I am probably not the only one to find that my views have crystallized, and that I can now move forward with growing confidence. I cannot exactly say who in the course of the competition was most inspiring - probably it was the continuous back and forth between so many of us. In this case, we should all be grateful to each other.

If I may, I'd like to express some of my newer conclusions - by themselves, so to speak, and independently of the logic that justifies them; the logic is, of course, outlined in my essay.

I now see the Cosmos as founded upon positive-negative charges: It is a binary structure and process that acquires its most elemental dimensional definition with the appearance of Hydrogen - one proton, one electron.

There is no other interaction so fundamental and all-pervasive as this binary phenomenon: Its continuance produces our elements - which are the array of all possible inorganic variants.

Once there exists a great enough correlation between protons and electrons - that is, once there are a great many Hydrogen atoms, and a great many other types of atoms as well - the continuing Cosmic binary process arranges them all into a new platform: Life.

This phenomenon is quite simply inherent to a Cosmos that has reached a certain volume of particles; and like the Cosmos from which it evolves, life behaves as a binary process.

Life therefore evolves not only by the chance events of natural selection, but also by the chance interactions of its underlying binary elements.

This means that ultimately, DNA behaves as does the atom - each is a particle defined by, and interacting within, its distinct Vortex - or 'platform'.

However, as the cosmic system expands, simple sensory activity is transformed into a third platform, one that is correlated with the Organic and Inorganic phenomena already in existence: This is the Sensory-Cognitive platform.

Most significantly, the development of Sensory-Cognition into a distinct platform, or Vortex, is the event that is responsible for creating (on Earth) the Human Species - in whom the mind has acquired the dexterity to focus upon itself.

Humans affect, and are affected by, the binary field of Sensory-Cognition: We can ask specific questions and enunciate specific answers - and we can also step back and contextualize our conclusions: That is to say, we can move beyond the specific, and create what might be termed 'Unified Binary Fields' - in the same way that the forces acting upon the Cosmos, and holding the whole structure together, simultaneously act upon its individual particles, giving them their motion and structure.

The mind mimics the Cosmos - or more exactly, it is correlated with it.

Thus, it transpires that the role of chance decreases with evolution, because this dual activity (by which we 'particularize' binary elements, while also unifying them into fields) clearly increases our control over the foundational binary process itself.

This in turn signifies that we are evolving, as life in general has always done, towards a new interaction with the Cosmos.

Clearly, the Cosmos is participatory to a far greater degree than Wheeler imagined - with the evolution of the observer continuously re-defining the system.

You might recall the logic by which these conclusions were originally reached in my essay, and the more detailed structure that I also outline there. These elements still hold; the details stated here simply put the paradigm into a sharper focus, I believe.

With many thanks and best wishes,

John

jselye@gmail.com

Dear Sean,

You wrote: "However, there is an even more basic reason for wanting fundamental scale invariance in your theory: only dimensionless quantities have objective meaning." I am not sure to agree or disagree, however, in KQID, Qbit is that dimensionless objective meaning in itself as Planck's matrix of all matter and Maxwell's infinite being that has unlimited storage capacity thus, no bit/qbit needs to be deleted, thus no entropy per Laudauer's principle will be generated.

You wrote: "We will now add these two postulates to Einstein's original postulates of Special Relativity: 1. The Laws of physics should take the same form for any inertial observer.2

2. The speed light is defined to be a finite constant, c, for all inertial observers."

Wonderful! Similarly, KQID has the same assumptions.

You wrote: "Just as in Special Relativity, we will assume the existence of idealized rods and clocks which can be perfectly synchronized." KQID relativity ψτ(iLx,y,z, Lm) Multiverse has flexible c-timerod that measures distance, area and volume in terms of time. Time contracts so does length contracts. In contrast, time contracts, the mass as well as the energy increases.

You wrote: "The first ingredient is an assumption of simplicity. If true, then a holistic picture of the world is, at least in principle, possible."

Yes, KQID has "holistic picture" of Existence and has done the theorical framework just to do this.

KQID has contextuality through KQID Ouroboros Equations of Existence that combines Newton, Maxwell, Planck, Boltzmann, Lorentz,Einstein, Laundauer, Wheeler , Feynman, Ssusskind, Hooft, Wilczek, Bousso and others. The Ouroboros Equations mean each interpretation involves every beginning to every ending. Similarly, everything we do involve the Ouroboros action or totality of any action. Nature is such unbelievable phenomena that we are just now starting to peek into its secret that is shockingly simple in the beginning but infinitely complex in the ending that per KQID every absolute digital time ≤ 10^-1000seconds. Interestingly, the mechanism is also simple. See my essay Child of Qbit in time.

Furthermore, KQID has done that no other theories that I know has done so far:

First, KQID Qbit is (00,1,-1) which is singularity Qbit Multiverse in zeroth dimension at absolute zero temperature that computes and projects Einstein complex coordinates (Pythagoras complex triangles or Fu Xi's gua or Fibonacci numbers!) onto the 2D Minkowski Null geodesic and then instantaneously into the 3D in Lm, our Multiverse timeline to allows Existence to move around 360 degree and its arrows of time as you described below. As you also agree below, no information is ever deleted. See my essay Child of Qbit in time. KQID is the only theory out there that can calculate the dark energy of our Multiverse ≤10^-153Pm/Pv and the minimum bits as the lower bound ≥ 10^153 bits in our Multiverse. KQID is the only theory that I knows here that proves bit = it, and KQID calculates Sun lights into Sun bits; calculates electron, proton and neutron in terms of bits; set up equivalent principle of bits with energy and matter. Therefore, Wheeler's it from bit and bit from it. Correct me if I am wrong. Furthermore, KQID is the only theory in this universe has the mechanism on how Holographic Principle works. Also answer the mother of all questions, the why, how and what Existence.

Pythagoras famously summarized: "All things are numbers." KQID rephrase it that all thing are one Qbit: Qbit is all things and all things are Qbit. Thus, Wheeler's it from bit and bit from it because bit = it.

You are definitely a serious original thinker. Let us share our thoughts.

Best wishes,

Leo KoGuan

Why dis you stop answering the posts on your thread ?

holidays ?

best regards

Wilhelmus

Hi Sean,

It was nice meeting you at the foundations conference in Munich. It was also really nice reading your essay. I really liked you analysis, and the subject matter is very interesting to me. I have a couple of comments which I'd like to hear your thoughts on.

The first one is mostly a remark: you mentioned a few times that you want to think of the cosmological constant as what drives cosmic expansion. This is exactly what motivated my PhD thesis, and at one point I picked up Eddington's 'Expanding Universe' and found that it was really the idea he had in mind, too. Throughout the 1920s, guys like him and Weyl had been thinking about cosmic expansion in terms of this 'cosmical repulsion' due to Lambda; and in 1924, Weyl even wrote a dialogue between 'Petrus' and 'Paulus' (Saints Peter and Paul), who presented Einstein's views and his own, respectively. The dialogue presents an obvious reaction to the postcard that he got from Einstein in 1923 (dated something like a week prior to Einstein posting his retraction to what he originally wrote about Friedman's paper, which he now called "correct" and "clarifying"), where he famously said "If there is no quasi-static world, then away with the cosmological term." The interesting thing in Weyl's paper, though, is that he didn't know what the hell Einstein was talking about, because he had never heard of Friedman's "correct" and "clarifying" paper, so when he received the post card he clearly though Einstein was saying he wanted to go back to special relativity, which obviously makes no sense if the Universe is expanding. 'Paulus' aka Weyl replied to 'Petrus' aka Einstein at this point in the dialogue, by saying something like, 'but all the spiral nebulae are receding, so the Universe is probably expanding, so dS space-time is the best bet that we know of'.

So anyway, guys like Weyl and Eddington had this idea of a 'cosmical repulsion' that drives expansion in their minds throughout the 1920s. And in Eddington's 'Expanding Universe' (1933), he did not hide the fact that he was really really upset that Einstein, after pretty much remaining silent on the issue of expansion, and not saying anything to anyone about Friedman's paper throughout the 1920s (everyone else learned about it in like 1930 or 1931, after Hubble confirmed the expansion with his redshift-distance relation in 1929), chose Friedman's solution with no cosmological constant (in the Einstein-de Sitter 1932 paper), where the cosmical repulsion can play no fundamental role in explaining the cosmical expansion. The reason is that near to the big bang, the cosmical repulsion is negligible, so the expansion is, to begin with, always less than it was, with Lambda driving accelerated expansion later on.

I do think there is a way around this last problem, which is what I wrote about in my last essay. There are more details about all of this there, if you're interested.

Okay, that's point one, and my comment's already getting long. I think point two is more interesting, though, so I'm still going to post it. It has to do with the motivation for the dS hyperboloid, which you presented in an interesting way in your essay. I wanted to discuss and alternative approach, though, which I think makes fewer assumptions, and provides a really neat way of thinking about this cool geometry that you've presented such an interesting analysis of.

So: what's the one aspect of nature that seems to be physically the most important? Symmetry, right? Symmetry that can be broken, but which is fundamental in nature. So, to begin I want just to assume that reality has an underlying metric, and that it has maximal symmetry. The only other thing I want to require, since relativity indicates that it's a property of nature, is Lorentzian signature of that metric. I don't want to assume that from the outset, though, but will only pick the solution with this property out of the list of possibilities derived from the assumption of maximal symmetry.

The maximally symmetric spaces can all be recovered as constant curvature hypersurfaces of Euclidean space, so begin by writing down the induced metric, [math]ds^2=\sum_{\mu=0}^{4}dx_{\mu}^2,[/math] [math]\sum_{\mu=0}^{4}x_{\mu}^2 =\alpha^2.[/math] This can be re-written in just four coordinates by arbitrarily solving for one of the five, as in [math]x_0=\pm\sqrt{\alpha^2-\sum_{i=1}^{4}x_{i}^2 },[/math] and the result can be neatly written as [math]ds^2=d\mathbf{x}^2\frac{(\mathbf{x}\cdot{d}\mathbf{x})^2}{\alpha^2-\mathbf{x}^2},[/math] where [math]\mathbf{x}=(x_1,x_2,x_3,x_4)[/math] is a real vector, and alpha is now the spherically symmetric space's *intrinsic* "radius of curvature". We can require this x to be real, since we began in E^5, but there's actually no reason that x_0 can't be imaginary, since it's fictitious anyway; i.e., the 4D maxiamlly symmetric spaces are still described in terms of a real Euclidean basis. From this line-element, we can pretty much just read off the components of the metric tensor in this basis, [math]g_{ij}=\frac{1}{\alpha^2-\mathbf{x}^2}\cdot[\alpha^2-(\mathbf{x}^2-{x_i}^2)],~\mathrm{if}~i=j,[/math] [math]g_{ij}=\frac{1}{\alpha^2-\mathbf{x}^2}\cdot{x_i}{x_j},~\mathrm{if}~i\neq{j}.[/math] The eigenvalues of the tensor are all positive except one, which is given by [math]\lambda=\frac{\alpha^2}{\alpha^2-\mathbf{x}^2}.[/math]

The closed sphere is described by setting alpha^2>0 and alpha^2>x^2; hyperbolic space, by setting 0>alpha^2 and x real; and dS space, by setting alpha^2>0 and x^2>alpha^2. If alpha=0, the metric space is degenerate.

So it turns out that just in case alpha^2 is positive (by the way, alpha^2=1/Lambda), and only when x^2>alpha^2 (dS space) does the metric have Lorentzian signature. The two points that are of further interest are that Lambda>0 is what we actually observe anyway, and that this space (actually, all of them) satisfies the vacuum Einstein equation R_ab=Lambda*g_ab.

I think it's pretty cool that with no reference to GR, and with no prior requirement that the metric should be Lorentzian, but only by requiring maximal symmetry to begin with, and an embedding in Euclidean space, it can be shown that dS space is the only one that does have a Lorentzian metric, and the space satisfies the vacuum Einstein equation.

I'm interested to know what you think of all this, because I think there is a lot of overlap in what we're considering.

Best of luck in the contest--and really, a great job done on the essay!

Daryl