Building SM Matter from Discrete Quantum Geometry

In summary: The model is inspired by recent work on neutrino oscillations which suggests that the three generations of fermions in the standard model are related by representations of the finite group A(4), the group of symmetries of the tetrahedron. This coupling is possible because the representation category of A(4) is a module category over the representation categories used to construct the EPRL model. The vertex operators which interchange vacua in the resulting quantum field theory reproduce the bosons and fermions of the standard model, up to issues of symmetry breaking which we do not resolve."He then goes on to say:"We are led to the hypothesis that physical particles
  • #1
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String field from Loop SpinfoamQG:how to build SM matter on discrete quantum geometry

note: EPRL is the current standard spinfoam formulation of Loop Quantum Gravity.

http://arxiv.org/abs/1201.0525
String Field Theory from Quantum Gravity
Louis Crane
(Submitted on 2 Jan 2012)
Recent work on neutrino oscillations suggests that the three generations of fermions in the standard model are related by representations of the finite group A(4), the group of symmetries of the tetrahedron. Motivated by this, we explore models which extend the EPRL model for quantum gravity by coupling it to a bosonic quantum field of representations of A(4). This coupling is possible because the representation category of A(4) is a module category over the representation categories used to construct the EPRL model. The vertex operators which interchange vacua in the resulting quantum field theory reproduce the bosons and fermions of the standard model, up to issues of symmetry breaking which we do not resolve. We are led to the hypothesis that physical particles in nature represent vacuum changing operators on a sea of invisible excitations which are only observable in the A(4) representation labels which govern the horizontal symmetry revealed in neutrino oscillations. The quantum field theory of the A(4) representations is just the dual model on the extended lattice of the Lie group E6, as explained by the quantum Mckay correspondence of Frenkel Jing and Wang. The coupled model can be thought of as string field theory, but propagating on a discretized quantum spacetime rather than a classical manifold.
15 pages

Excerpt from introduction:"In the last few years, a new development [1] [2] [3] [4] has largely resolved the problems of the old BC model [5] for quantum gravity. It is now a natural task to study extensions of the EPRL model which would include realistic matter fields. It would be extremely desirable to find an algebraic extension of the EPRL model which was essentially unique or at least had a small number of possibilities and which gave us the standard model..."

Excerpt from conclusions: "It was quite a surprise to have the string appear in this theory, which started from a completely different program. Rather than ending up with an almost infinite landscape as in Kaluza-Klein theories, we get an essentially unique theory, which relates fairly directly to the standard model.
Embedding the string field in a discrete model for spacetime removes the difficulties that beset string field theories in a continuum.
There is no longer the integration over worldsheet metrics which leads to bad behavior on moduli space in the Polyakov string; rather the dual models couple to the quantum geometry of the EPRL model itself."
 
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  • #3


"It is now a natural task to study extensions of the EPRL model which would include realistic matter fields. It would be extremely desirable to find an algebraic extension of the EPRL model which was essentially unique...and which gave us the standard model..."

So that's the program. I don't see any reason such a program couldn't succeed. He makes a start on it here and also describes what further work needs to be done.

EPRL is a formulation of quantum geometry that makes a lot of sense and seems to yield GR in the appropriate limit. Some people like to emphasize what still needs to be worked on such as the relation of EPRL dynamics to Hamiltonian dynamics. Or they worry about the convergence of this or that limit process.

That's fine and we shouldn't minimize the importance of attending to every possible imperfection. But Crane is saying that the quantum geometry is mature enough so that it is a "natural task" for some people to be thinking about putting standard matter into it.

I think that's quite reasonable. Some people can work on getting EPRL absolutely right and absolutely equivalent to some Hamiltonian formulation (presumably yet to be invented.) And meanwhile other people don't have to wait but can go ahead and think about putting standard matter into the EPRL geometry as it is, even if not yet in final form.

On page 11, at the beginning of section 12-Physical Consequences of the Model, Crane says:

"The proposal that physical particles correspond to different vacua of an underlying quantum field of tetrons is a major departure. Could it have observable consequences?"

So that's something to try to understand. A field with a discrete group of symmetries, living on a discrete geometry. In EPRL the dynamical evolution of geometry is represented by a labeled two-complex---analogous to a graph but in one higher dimension. The two complex ("foam") is BOUNDED BY A LABELED GRAPH that represents the geometric measurements we have made or will make---lengths, angles, areas, etc. and which implement the idea of a STATE of geometry. So a graph bounds a foam and the foam tells the amplitude of a certain transition, say from initial to final state.

Crane understands the structure I just described in terms of category theory and group representation theory---the catch is the labels, or how you understand the Hilbertspace of states associated with a given graph. He understands it in some way that allows him to talk about a "tetron" field defined on the spinfoam, i.e. on the spacetime: the evolving geometry which the spinfoam describes in a discrete fashion.

This sounds reasonable to me---like some math we could have expected to eventually be done---even though I don't yet understand it in detail. So the paper seems quite interesting.

Crane does not always show the arxiv number of his references, so to save time I will post them now and then.
Reference [10] refers to http://arxiv.org/abs/hep-ph/0211393
Reference [23] (Baez Huerta, note change of title) http://arxiv.org/abs/0904.1556

Some Crane-related FQXi pages: http://fqxi.org/grants/large/awardees/view/__details/2006/crane
http://fqxi.org/community/articles/display/127
 
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  • #4


I think the link between A4 and E6 is potentially of interest beyond the EPRL model.

A4 is the symmetry group of a tetrahedron when you don't allow reflections. Tetrahedra show up in EPRL as boundaries between 4-simplexes. That is, the space-time is represented as a complex of 4-dimensional hypertetrahedra which are joined to each other by 3-dimensional tetrahedral "hyperfaces", and Crane talks about the line from the center of one 4-simplex to the center of its neighbor, which will pass through one of these tetrahedral hyperfaces. Being a tetrahedron, the hyperface has an A4 symmetry, and the A4/E6 quantum duality applies to families of representations which are associated with the line passing through it.

But tetrahedra show up in lots of places in physics.

Crane also mentions that A4 symmetry might explain tribimaximal mixing for neutrinos. But the recent results from T2K experiment (nonzero theta13 mixing angle) indicate a strong deviation from tribimaximal mixing. There was a recent paper on the nuMSM which tried to explain the mixing angles (and lepton mass hierarchies) using a Q6 symmetry. That's the "dicyclic group" or "binary dihedral group" of order 12.
 
  • #5


mitchell porter said:
I think the link between A4 and E6 is potentially of interest beyond the EPRL model...
I believe you are right. Both you and Crane give reasons why it is that go beyond the nice fit with this one QG model.

==quote page 3==
A large body of recent work on grand unified models [17] inspired by the neutrino oscillations has focussed on symmetry groups which are a product of a finite discrete group with a Lie group; and A(4) × E6 seems to be the most successful of the possibilities...

Discrete symmetry as a fundamental symmetry of nature is quite a puzzle. In a continuum theory it would be hard to avoid domain walls. The conceptual foundation of the BC and EPRL models, in which spacetime is modeled by a simplicial complex and NOT by a manifold, is necessary to include discrete symmetry as in this paper...
==endquote==

I think that part of what we see in Crane's work and that of a number of quantum relativists more generally is a mathematical instinct to explore NON-MANIFOLD representations of spacetime geometry.

The urge to go beyond the smooth manifold picture of continuum---perhaps one could date it 1850, with Riemann---has already been simmering in the math community for many decades, surely going back to before 1970s. The sticking point has perhaps been with the conservatism of physicists.

But simply to have a quantum theory of spacetime geometry seems almost to necessitate a discrete representation. And now Crane is pointing to an additional justification, the broader interest of discrete symmetry in matter.
 
  • #6


"The sticking point has perhaps been with the conservatism of physicists."

I think that the sticking point until scientists started to consider them was that we didn't have the computational power to implement and apply discrete models, which couldn't be analyzed with the tools of mathematical analysis available in continuum models. Euler, Lagrange and Laplace were doing very fancy real and complex analysis in physics applications before George Washington became President of the United States in 1789. The Univac, the first mainframe computer, came on line in 1951, four years before Einstein died. Stanford physicists and engineers through the late 1960s were limited to slide rules, punch cards, French curves and graph paper. There still wasn't enough computing power available even in supercomputers to do things like interesting 4D Lattice QCD during Feynmann's lifetime. LQG is only possible in any even conceivably applicable version because of advances associated with massive computing power.
 
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  • #7


ohwilleke said:
"The sticking point has perhaps been with the conservatism of physicists."

I think that the sticking point until scientists started to consider them was that we didn't have the computational power to implement and apply discrete models,...

I see your point. Continuum math affords beautiful and elegant means of calculation. This could explain the enduring preference for that kind of model.
Now it seems to me that (rightly or wrongly) Crane is arguing that continuum geometry is unphysical on theoretical grounds. Here he has just been talking in the preceding section about the state sum models of EPRL and BC quantum gravity.

==quote page 4 http://arxiv.org/abs/1201.0525 ==
2. THE MATHEMATICAL STRUCTURE OF SPACETIME
The state sum models for quantum gravity should not be thought of as approximations corresponding to triangulations of an underlying smooth manifold [11]. In quantum Physics coupled to relativity, such a point of view is unphysical, because no information can be communicated from a sub-Planck scale region. The attempt to recover a continuum limit is the one unsuccessful part of the interpretation of the model, and is naive.
Although the replacement of manifolds by simplicial sets is a mathematically well understood point of view for geometry and topology [19] [20], it seems very foreign to physicists.
Let us take some simple region of spacetime and compare two descriptions of it, one as a subset of a smooth manifold, the other as a simplicial complex, thought of as a discrete combinatorial object, not a point set...
==endquote==
 
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  • #8


What does a discrete spacetime model do to the differential equations of quantum mechanics that differentiate with respect to continuous time and space?
 
  • #9


I think the reason for wanting a continuum limit is that without it the model may not be unique. This is the reason for Rovelli's summing = refining.

Does Crane do EPRL with or without the continuum limit that Rovelli and Smerlak discuss?
 
  • #10


friend said:
What does a discrete spacetime model do to the differential equations of quantum mechanics that differentiate with respect to continuous time and space?

First it should be clear that "differentiate with respect to continuous time and space" refers to coordinates only and is therefore not as physical as it may seem.

Second we know that topologically and even w.r.t. to differential topology there is (nearly) identity between topological manifolds, differential manifolds and piecewise linear manifolds. But piecewise linear manifolds and "triangulations" are again nearly identical (I don't want to be precise regarding the meaning of "nearly identical"). It may be the case that we don't need continuous differential structures at all.

Third: a discrete spacetime model will turn differential equations into difference equations (LQG and LQC do exactly this) and it may very well be that these are "physically equivalent".

Even the following may be true, namely that we are allowed to use continuous models which give us identical results. Let's make a very simple example: Assume we are small quanta living in a space of weakly coupled harmonic oscillators. Usually we observe discrete energy levels |n> and we are jumping from n to n+1 etc. we can calculate jumping rates, etc. using creation and annihilitation operators. Now we organize a conference and invite two quanta called deBroglie and Schrödinger and they present us their theory of waves ψ(x), wave equations etc. Of course we are rather shocked b/c nobody has ever observed continuous space x, all we know about are quantum jumps and probabilities. But then Schrödinger proves the equivalence of his equations with our description of the world in terms of creation and annihilation operators. We would still feel that his equations are strange, but we accept the math. We will even understand that in a so-called macrsoscopic world we are not aware of but to which Schrödinger is always referring to his strange description may be helpful.

That means that both continuous strings and discrete spinfoams can be correct descripton of nature (or let's be slightly more careful: may point towards reasonable pathways towards the construction of candidates for unified model of 'quantum matter and quantum geometry' ;-)
 
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  • #11


I like this story a lot!
tom.stoer said:
...Assume we are small quanta living in a space of weakly coupled harmonic oscillators. Usually we observe discrete energy levels |n> and we are jumping from n to n+1 etc. we can calculate jumping rates, etc. using creation and annihilation operators. Now we organize a conference and invite two quanta called deBroglie and Schrödinger and they present us their theory of waves ψ(x), wave equations etc. Of course we are rather shocked b/c nobody has ever observed continuous space x, all we know about are quantum jumps and probabilities. But then Schrödinger proves the equivalence of his equations with our description of the world in terms of creation and annihilation operators. We would still feel that his equations are strange, but we accept the math. We will even understand that in a so-called macroscopic world we are not aware of, but which Schrödinger is always referring to, his strange description may be helpful...
 
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  • #12


mitchell porter said:
A4 is the symmetry group of a tetrahedron when you don't allow reflections. Tetrahedra show up in EPRL as boundaries between 4-simplexes.

If he allowed the volume of the Tetrahedra to have reflections, since we are talking about volume as side of a 4-simplex, wouldn't we have half the symmetries that he wants to get the SM? This also would be a way of implement CPT symmetry.
 
  • #14


I have to wonder what a discrete spacetime would do to wave propagation. If there is an effect, then I suppose it would become obvious at cosmological scales.
 
  • #15


friend said:
I have to wonder what a discrete spacetime would do to wave propagation. If there is an effect, then I suppose it would become obvious at cosmological scales.

Tom already addressed the general issue. I assume the answer would depend on what kind of discreteness you are imagining and what mathematical models you are using.

Good things to keep in mind would, I suppose, be what Bohr said (quantum physics focuses not on what nature IS but on what we can say about it) and Rovelli's clarification (it's not what nature IS but how it responds to measurement).

A quantum model of geometry should embody the limitations of geometrical measurement. If we can only make a finite number of measurements and the readings we take of length angle area etc can only take on a discrete spectra of values then we expect the model to keep track of these limitations for us.

If you like you can picture a continuous continuum equipped a discrete geometry.

But that is not really on the main topic which is Crane's paper.
 
  • #16


atyy said:
...Does Crane do EPRL with or without the continuum limit that Rovelli and Smerlak discuss?

I don't know the answer to that, Atyy :biggrin:

He refers to some kind of continuum limit concern as naive. I don't know if he is specifically talking about their work or not.
 
  • #17


Basically what we're focusing on in this thread is the program sketched in the first few paragraphs of Louis Crane's 3 January 2012 paper http://arxiv.org/abs/1201.0525

These first few paragraphs lay out the basic idea, which is then elaborated in more detail in the next dozen or so pages.

==quote Crane 1201.0525==
In the last few years, a new development [1] [2] [3] [4] has largely resolved the problems of the old BC model [5] for quantum gravity. It is now a natural task to study extensions of the EPRL model which would include realistic matter fields. It would be extremely desirable to find an algebraic extension of the EPRL model which was essentially unique or at least had a small number of possibilities and which gave us the standard model, rather than some random collection of particles and fields.

During the same time frame, a series of delicate experiments [6] has given us a detailed picture of the oscillations of the neutrinos of the three generations (electron muon and tau neutrinos) into one another. The oscillations turned out to be much larger than the ones for quarks, and seem to be well approximated by a form called the tribimaximal matrix [7].

The neutrino masses indicate that the neutrino has a right handed component, called the sterile neutrino because it does not interact under any force. This changes the form of the three generations of fermions; each now has 16 Weyl fermions.

The new neutrino physics is crucial to the search for a unified theory; it is as if we have been working a jigsaw puzzle with a missing piece. If the proposal in this paper is correct, the missing piece was the most important one.

In explaining the tribimaximal matrix, a number of researchers have proposed that the particles appear in representations of the discrete group A(4) [8], the alternating group on 4 letters, and that the interactions with the Higgs particle be invariant under this symmetry. This reproduces the tribimaximal form very nicely.

The fact that A(4) has three one dimensional representations and one irreducible three dimensional representation allows the quark and lepton mixing matrices to be very different. Quarks and neutrinos get different A(4) representational labels in this approach.

The A(4) symmetry cannot be explained as a broken or residual symmetry from a Lie group [9], it can only be understood as a fundamental symmetry of nature. This lends itself much more naturally to coupling to a discrete model such as EPRL rather than to a continuum Yang Mills theory.

The EPRL model, like the BC model which preceeded it, is constructed from the representation categories of Lie groups, which are used to assign quantum geometrical variables to the faces of a four dimensional simplicial complex. The tensor structure of the representation categories is used in a natural way to construct the model. The EPRL model, in particular, uses the representation categories of SO(3) and SO(3,1), together with a functor connecting them [11].

The group A(4) which was interesting to the neutrino physicists is also the group of symmetries of the tetrahedron...

==endquote==

Here are some of Crane's previous papers that might help us understand this one:
http://arxiv.org/abs/1006.1248
Holography in the EPRL Model (this earlier one seems to me the more helpful of the two)
http://arxiv.org/abs/1105.6036
Discrete symmetry in the EPRL model and neutrino physics
 
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  • #18


marcus said:
Here are some of Crane's previous papers that might help us understand this one:
http://arxiv.org/abs/1006.1248
Holography in the EPRL Model (this earlier one seems to me the more helpful of the two)
http://arxiv.org/abs/1105.6036
Discrete symmetry in the EPRL model and neutrino physics

Thanks, it's very helpful to have the earlier papers! He answers my question in the first of them. He rejects Rovelli and Smerlak's proposal: "Summing over complexes ala GFT returns us to a formal divergent expression, which is tragic after finiteness has come to us in such a magical way."
 
  • #19


From your quote one sees that back in June 2010 Crane was not especially clairvoyant as to the future course of other people's research. But that's OK, nobody should be expected to be so. Of primary interest here are his own ideas, which are imaginative and in some cases I think original.

http://arxiv.org/abs/1112.2511
q-Deformation of Lorentzian spin foam models
Winston J. Fairbairn, Catherine Meusburger
(Submitted on 12 Dec 2011)
We construct and analyse a quantum deformation of the Lorentzian EPRL model. The model is based on the representation theory of the quantum Lorentz group with real deformation parameter. We give a definition of the quantum EPRL intertwiner, study its convergence and braiding properties and construct an amplitude for the four-simplexes. We find that the resulting model is finite.
12 pages, 2 figures, Proceedings of the 3rd Quantum Gravity and Quantum Geometry School (Zakopane, 2011), to appear in PoS
 
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  • #20


The effort to show a refinement limit for EPRL may actually NOT be misguided and may in fact be successful. I would not take Crane's word as gospel about the world of other people's research. But his own ideas about the structure of spacetime (e.g. page 4 section 2 of the current paper) are interesting and I would like to understand them better. Partially recapitulating, I'll quote the whole of section 2.

==quote 1201.0525 page 4==
2. THE MATHEMATICAL STRUCTURE OF SPACETIME
The state sum models for quantum gravity should not be thought of as approximations corresponding to triangulations of an underlying smooth manifold [11]. In quantum Physics coupled to relativity, such a point of view is unphysical, because no information can be communicated from a sub-Planck scale region. The attempt to recover a continuum limit is the one unsuccessful part of the interpretation of the model, and is naive.

Although the replacement of manifolds by simplicial sets is a mathematically well understood point of view for geometry and topology [19] [20], it seems very foreign to physicists. Let us take some simple region of spacetime and compare two descriptions of it, one as a subset of a smooth manifold, the other as a simplicial complex, thought of as a discrete combinatorial object, not a point set. Both of these have associated to them a differential graded complex, the differential forms for the manifold, and the cocycles for the complex.

If we compute the cohomology of the region using both complexes, we find there is an important difference: the differential forms can only give vector spaces, while the cocycles can be calculated using integer coefficients, and give finite group components to the cohomology as well. This is known as the torsion. Thus finite group invariants appear more naturally in the complex description.

Similarly, it is very easy to construct topological quantum field theories from the representations of finite groups on a simplicial complex, while a continuum Lagrangian is hard to imagine.

We interpret the discovery of discrete symmetry as a fundamental symmetry in neutrino Physics as an indication that the simplicial complex description is favored by nature.
==endquote==

It seems to me (at my current stage of appreciating his work) that Crane may be RIGHT about simplicial complex being a good math description of spacetime geometry to be using. I don't think it follows that one needs to scoff at physicists for wanting a unique limit of the transition amplitude under refinement. Of course the physicists want this and they are right to devote effort to attain it.

If I am not mistaken, in pragmatic terms, operationally, this is all "continuum limit" means in the simplicial state sum context. "Continuum limit" does not necessarily mean that Nature IS a smooth manifold or that the manifold structure is always the best for putting matter and geometry together! We are always concerned with how she responds to measurement, not with what she is. But in any case, whatever she is, there has to be a unique limit of the transition amplitude under refinement. So it is not naive to look for it and try to prove it exists.
 
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  • #21


I'm sure Crane must have known about the q-deformation stuff, since although there was no technical proof at that time, it was widely conjectured based on analogous earlier work. I think the q-deformation stuff deals with infrared divergences, whereas the proposed Rovelli-Smerlak limit is more like a "UV" divergence in spirit, even if not technically so.

OTOH, maybe Crane's proposal doesn't require the limit not to exist?
 
  • #22


References looked up, for convenience:
[7] Low Volkas http://arxiv.org/abs/hep-ph/0305243
[8] Babu Ma Valle http://arxiv.org/abs/hep-ph/0206292
[9] Adulpravitchai Blum Lindner http://arxiv.org/abs/0907.2332
[10] Ma http://arxiv.org/abs/hep-ph/0211393
[12] Etingof Ostrik http://arxiv.org/abs/math/0302130
[13] Frenkel Jing Wang http://arxiv.org/abs/math/9907166
[17] Albright http://arxiv.org/abs/0905.0146
[21] Altarelli http://arxiv.org/abs/0905.3265
[22] Babu Mathur http://inspirehep.net/record/23581 (KEK scan available)
[23] Baez Huerta (different title) http://arxiv.org/abs/0904.1556
[32] Lau http://arxiv.org/abs/math/0405576

More detail on Babu Ma Valle reference [8]:
http://arxiv.org/abs/hep-ph/0206292
K.S. Babu (Oklahoma State U)
Ernest Ma (UC Riverside)
J.W.F. Valle (University of Valencia, Spain)
 
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  • #23


Crane's reference [21] is to a 2009 Altarelli review of neutrino mixing. There is a 2011 Altarelli review which could be helpful if only because more recent:
http://arxiv.org/abs/1111.6421
The Mystery of Neutrino Mixings
Guido Altarelli
(Submitted on 28 Nov 2011)
In the last years we have learned a lot about neutrino masses and mixings. Neutrinos are not all massless but their masses are very small. Probably masses are small because neutrinos are Majorana particles with masses inversely proportional to the large scale M of lepton number (L) violation, which turns out to be compatible with the GUT scale. We have understood that there is no contradiction between large neutrino mixings and small quark mixings, even in the context of GUTs and that neutrino masses fit well in the SUSY GUT picture. Out of equilibrium decays with CP and L violation of heavy RH neutrinos can produce a B-L asymmetry, then converted near the weak scale by instantons into an amount of B asymmetry compatible with observations (baryogenesis via leptogenesis). It appears that active neutrinos are not a significant component of Dark Matter in the Universe. A long list of models have been formulated over the years to understand neutrino masses and mixings. With the continuous improvement of the data most of the models have been discarded by experiment. The surviving models still span a wide range going from a maximum of symmetry, with discrete non-abelian flavour groups, to the opposite extreme of anarchy.
27 pages, 2 figures. Lectures given at the 2011 Ettore Majorana International School of Subnuclear Physics, Erice, June 2011

==================
The central question is whether that particular (tri-bi-max) mixing matrix is the right one for neutrinos. Or is it just a coincidence that gives a good fit to the data? Mitchell (post #4) mentioned that there were some other candidates. Is A4 an actual symmetry of nature? Or is it just a coincidence that so far it seems like the A4 group fits?

Crane's proposal depends on an exact A4 symmetry being confirmed by neutrino experiments during the coming months. This suspected symmetry could be ruled out even sometime this year. On the other hand if it's confirmed, having faced a stringent test and risked falsification, that would be quite interesting.
==================

"MathWorld" enumerates the elements of A4--the even permutations on 4 letters.
(An even permutation is a permutation obtainable from an even number of two-element swaps.) For initial set 1,2,3,4, the twelve even permutations consist of the identity with zero swaps: (1,2,3,4)
and eleven others with two swaps:
(23)(34)→ (1,4,2,3)
(24)(34)→ (1,3,4,2)
(12)(34)→ (2,1,4,3)
(13)(23)→ (2,3,1,4)
(12)(14)→ (2,4,3,1)
(13)(12)→ (3,1,2,4)
(14)(34)→ (3,2,4,1)
(13)(24)→ (3,4,1,2)
(12)(24)→ (4,1,3,2)
(13)(34)→ (4,2,1,3)
(14)(23)→ (4,3,2,1)

If you think of A4 as the group of the tetrahedron then you can picture 1 as the top of a little threesided pyramid and then the first nontrivial one on the list is
(1,4,2,3) fix the top and twirl the base in cyclic permutation of 234→423
(1,3,4,2) same but twirl the other way, 234→342
(12)(34)→ (2,1,4,3) flip the tet so top trades with #2, rest is automatic.
(13)(23)→ (2,3,1,4) fix #4 and twirl
(12)(14)→ (2,4,3,1) fix #3 and twirl
(13)(12)→ (3,1,2,4) fix #4 and twirl the other way
(14)(34)→ (3,2,4,1) fix #2 and twirl
(13)(24)→ (3,4,1,2) another flip, top trades with #3, and so on...
we could organize the enumeration better but I think it's probably clear.
 
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  • #24


Louis Crane is making a risky conjecture involving the A4 group. It's falsifiable which should command respect--we will know a lot more about neutrinos by the end of this year or next and may be able to rule his A4 idea out, if it's wrong. But suppose the idea survives---what it suggests to me is that I ought to get more familiar with A4. No problem, it only has 12 elements. Eleven nontrivial plus the identity.

Picture a RIGHTHAND fist thumbs-up numbered tetrahedron. vertices numbered 1 thru 4---with 1 up and 234 around the base in the "positive" sense i.e. in the direction of your fingers.

For each vertex we have two tet symmetries which fix that vertex and do a cyclic roll on the other three. In each case the first listed is the one going in a righthand positive sense.

Fix 1 and do either (2,3,4) or the reverse (2,3,4)-1
Fix 2 and do either (1,4,3) or the reverse (1,4,3)-1
Fix 3 and do either (1,2,4) or the reverse (1,2,4)-1
Fix 4 and do either (1,3,2) or the reverse (1,3,2)-1

For example if you fix 2 and make that the thumb direction the positive sense rotation (with your fingers) is 1→4→3→1.
We could write these 8 permutations as Ri± where i = 1,2,3,4.

The other kind of tet symmetry is where you flip it over so that two vertices are exchanged and that automatically exchanges the other two. So there are only 3 possibilities. In effect you pick an edge to map end-to-end on itself and there are 6 edges, but they are paired, so 3 possible ways to flip the tetrahedron.

F12, F13, and F14

Each one is its own inverse. For example F12 swaps 1 with 2 and 3 with 4.
Have to go. If anyone has an online source for this elementary A4 stuff please post the link.

The group of the tetrahedron seems to be a minimal structure exhibiting "handedness" in the sense that the full permutation group on four letters has 24 elements and half of them turn the tet inside out. So if you restrict to the ones that don't turn it inside out you just have these 12 that we call A4---the "even" ones.
 
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  • #25


Marcus wrote:
"THE MATHEMATICAL STRUCTURE OF SPACETIME
The state sum models for quantum gravity should not be thought of as approximations corresponding to triangulations of an underlying smooth manifold [11]. In quantum Physics coupled to relativity, such a point of view is unphysical, because no information can be communicated from a sub-Planck scale region. The attempt to recover a continuum limit is the one unsuccessful part of the interpretation of the model, and is naive.

Although the replacement of manifolds by simplicial sets is a mathematically well understood point of view for geometry and topology [19] [20], it seems very foreign to physicists. Let us take some simple region of spacetime and compare two descriptions of it, one as a subset of a smooth manifold, the other as a simplicial complex, thought of as a discrete combinatorial object, not a point set. Both of these have associated to them a differential graded complex, the differential forms for the manifold, and the cocycles for the complex."

Is it something like MUH ?
Tegmark's sole postulate is: All structures that exist mathematically also exist physically. That is, in the sense that "in those [worlds] complex enough to contain self-aware substructures [they] will subjectively perceive themselves as existing in a physically 'real' world".[2][3] The hypothesis suggests that worlds corresponding to different sets of initial conditions, physical constants, or altogether different equations should be considered equally real.
http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis
 
  • #27


marcus said:
... But his own ideas about the structure of spacetime (e.g. page 4 section 2 of the current paper) are interesting and I would like to understand them better. Partially recapitulating, I'll quote the whole of section 2.

==quote 1201.0525 page 4==
2. THE MATHEMATICAL STRUCTURE OF SPACETIME
The state sum models for quantum gravity should not be thought of as approximations corresponding to triangulations of an underlying smooth manifold [11]. In quantum Physics coupled to relativity, such a point of view is unphysical, because no information can be communicated from a sub-Planck scale region...

Although the replacement of manifolds by simplicial sets is a mathematically well understood point of view for geometry and topology [19] [20], it seems very foreign to physicists. Let us take some simple region of spacetime and compare two descriptions of it, one as a subset of a smooth manifold, the other as a simplicial complex, thought of as a discrete combinatorial object, not a point set...
...
...
Similarly, it is very easy to construct topological quantum field theories from the representations of finite groups on a simplicial complex, while a continuum Lagrangian is hard to imagine.

We interpret the discovery of discrete symmetry as a fundamental symmetry in neutrino Physics as an indication that the simplicial complex description is favored by nature.
==endquote==
...

Please be clear about what Crane is saying in section 2!

He is not saying that Nature IS a simplicial complex or that her spacetime geometry IS some mathematical description.

He is discussing what mathematical DESCRIPTION is currently the MOST CONVENIENT.

Mathematics is a human language which evolves as humans interact with nature. It has no eternal "essence" any more than does English or any other living evolving language. At any given time the most convenient description in math language can be said to be the math description that is "favored by Nature".

The description favored by Nature---i.e. the most convenient---will of course change as the language evolves and acquires more possibilities of description to choose from. At no point should we say that Nature IS some math structure. We are monkeys and our languages have only been evolving for a few hundred thousand years. Mathematics is just one of our monkey languages.

Tegmark MUH is, I guess, XXXXXXX XXXXXXX and Hossenfelder pointed this out in the most gentle friendly way already in 2007. http://backreaction.blogspot.com/2007/09/imaginary-part.html Tegmark is off-topic here.
 
  • #28


Crane simply says that the discrete model is not an approximation of an underlying, better, continuous model for spacetime, but that the discreteness is something fundamental. He says that it makes no sense to look for a smooth structure below Planck length b/c any resolution is limited physically, not only by an approximation.

It's like saying that atoms are only a discrete approiximation of a continuous surface of water. Of course it's the other way round, the smooth surface is only an approximation of a fundamentaly discrete structure.

I think this has nothing to do with MUH at all.
 
  • #29


Thank you. I understand the difference between Crane and Tegmark now.
 
  • #30


"THE MATHEMATICAL STRUCTURE OF SPACETIME
The state sum models for quantum gravity should not be thought of as approximations corresponding to triangulations of an underlying smooth manifold [11]. In quantum Physics coupled to relativity, such a point of view is unphysical, because no information can be communicated from a sub-Planck scale region..."

As we discussed it before, the information can be both the discrete and continouos. Therefore the discrete space-time could be a topology created by the non-material information rather then a material vacuum (aether).
There is possible the non-locality and superposition in quantum decoherence , I think.
 
  • #31


I think that we should not pay too much attention on the interpretation of the model. It can very well be that a continuous model (a manifold) generates essentially discrete physics, i.e. discrete spectra of observables. Look at the harmonic oscillator in ordinary QM: you can discuss it entirely based on the discrete algebra of creation and annihilation operators acting on the 'discrete Hilbert space' spanned by |n>. The same is true for angular momentum.

If due to diff. inv. the continuous structure is 'washed away' and all that remains physically is discrete information, discrete spectra etc., then it doesn't mater whether you use continuous coordinates or a equivalent discrete model.


I think there are some research programs doing exactly that, namely Asymptotic Safety and Exotic Smoothness..
 
  • #32


tom.stoer said:
If due to diff. inv. the continuous structure is 'washed away' and all that remains physically is discrete information, discrete spectra etc., then it doesn't mater whether you use continuous coordinates or a equivalent discrete model.

You raise an interesting point. My question is how can you have discrete and diffeomorphism invariance at the same time. It seems to me that a discrete spacetime would be with respect to a frame of reference. But as soon as you switch to a different frame of reference, then the discretization with respect to that frame would be different from the discretization of a different frame. And if the two discretizations don't match up, then how can discretization be physical?
 
  • #33


You wouldn't a discrete structure and diffeomorphism invariance 'at the same time'; mathematically diffeomorphism invariance would reduce the smooth structure to a kind of discrete set set of equivalence classes of smooth spacetimes.
 
  • #34


CZES:
As we discussed it before, the information can be both the discrete and continouos. Therefore the discrete space-time could be a topology created by the non-material information rather then a material vacuum (aether).



Marcus previously posted this..which I liked a LOT:
[post+338459]

http://arxiv.org/abs/1010.4354
Spacetime could be simultaneously continuous and discrete in the same way that information can
Achim Kempf
(Submitted on 21 Oct 2010)
"There are competing schools of thought about the question of whether spacetime is fundamentally either continuous or discrete. Here, we consider the possibility that spacetime could be simultaneously continuous and discrete, in the same mathematical way that information can be simultaneously continuous and discrete. The equivalence of continuous and discrete information, which is of key importance in information theory, is established by Shannon sampling theory: of any bandlimited signal it suffices to record discrete samples to be able to perfectly reconstruct it everywhere, if the samples are taken at a rate of at least twice the bandlimit. It is known that physical fields on generic curved spaces obey a sampling theorem if they possesses an ultraviolet cutoff. Most recently, methods of spectral geometry have been employed to show that also the very shape of a curved space (i.e., of a Riemannian manifold) can be discretely sampled and then reconstructed up to the cutoff scale. Here, we develop these results further, and we here also consider the generalization to curved spacetimes, i.e., to Lorentzian manifolds."
 
  • #35


If as already noted, for example, information can be considered both analog and digital, continuous and discrete, if a continuous Schrodinger wave, can lead to discrete values for solutions, and if matter and light is described both as a wave and as discrete quanta, etc,etc,etc why should we be surprised if space and time present such duality?
 
<h2>1. What is "Building SM Matter from Discrete Quantum Geometry"?</h2><p>"Building SM Matter from Discrete Quantum Geometry" is a theoretical framework that aims to explain the fundamental particles and forces of the Standard Model of particle physics using discrete units of space and time known as quantum geometry.</p><h2>2. How does this theory differ from other theories of particle physics?</h2><p>This theory differs from other theories of particle physics in that it proposes that space and time are not continuous, but rather made up of discrete units. It also aims to unify the fundamental particles and forces of the Standard Model, which have been traditionally described by separate theories.</p><h2>3. What evidence supports this theory?</h2><p>Currently, there is no direct evidence to support this theory. However, it is based on mathematical models and theories such as loop quantum gravity and spin networks, which have shown promise in explaining the fundamental properties of space and time.</p><h2>4. How does this theory relate to the search for a theory of quantum gravity?</h2><p>This theory is closely related to the search for a theory of quantum gravity, as it attempts to reconcile the principles of quantum mechanics with the theory of general relativity. By incorporating discrete units of space and time, it offers a potential solution to the problem of infinities that arise in traditional quantum field theory.</p><h2>5. What are the potential implications of this theory?</h2><p>If this theory is proven to be accurate, it could have significant implications for our understanding of the fundamental nature of the universe. It could also potentially lead to new technologies and applications, as well as a deeper understanding of the origins of the universe and the laws that govern it.</p>

1. What is "Building SM Matter from Discrete Quantum Geometry"?

"Building SM Matter from Discrete Quantum Geometry" is a theoretical framework that aims to explain the fundamental particles and forces of the Standard Model of particle physics using discrete units of space and time known as quantum geometry.

2. How does this theory differ from other theories of particle physics?

This theory differs from other theories of particle physics in that it proposes that space and time are not continuous, but rather made up of discrete units. It also aims to unify the fundamental particles and forces of the Standard Model, which have been traditionally described by separate theories.

3. What evidence supports this theory?

Currently, there is no direct evidence to support this theory. However, it is based on mathematical models and theories such as loop quantum gravity and spin networks, which have shown promise in explaining the fundamental properties of space and time.

4. How does this theory relate to the search for a theory of quantum gravity?

This theory is closely related to the search for a theory of quantum gravity, as it attempts to reconcile the principles of quantum mechanics with the theory of general relativity. By incorporating discrete units of space and time, it offers a potential solution to the problem of infinities that arise in traditional quantum field theory.

5. What are the potential implications of this theory?

If this theory is proven to be accurate, it could have significant implications for our understanding of the fundamental nature of the universe. It could also potentially lead to new technologies and applications, as well as a deeper understanding of the origins of the universe and the laws that govern it.

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