On the Foundational Assumptions of Modern Physics by Benjamin F. Dribus

Ben,

I have read several complaints and analyses about the voting system since this morning. I think that the others could have been anticipated both from the rules and the wide range of expertise allowed. Perhaps not all of it for first timers. This is the fourth year and those kinds of effects on the ratings are known and need to be learned. I accept them because if things were tightened up I might not be permitted to participate. My lowly finish this year was consistent with my finishes in each of the previous three years. Personally I don't think that it reflects the value of my work. But it does reflect how these contests rate my work. So be it. However, This one that you mention concerns me:

" I noticed yesterday that essays I knew I had already rated displayed the "do you want to rate this essay?" message at the top. A number of submissions went unrated by me at the end because I did not want to risk double-voting. I am sure others felt the same way, and this itself affected the process."

It shows that the system did not work as could have been expected. I care very much that the results be correct however they turn out. Your words tell me that it didn't happen that way this time, and, I think that nothing can be done to make it right. I think it has to be accepted knowing that the administrators will have it in mind for future contests. I am confident that high quality essays will be selected by the juding system as has been the case in the past contests.

James

Congratulations Ben!

It is good to see you with a top ranking. You had kind things to say about my essay, and I am pleased to report also being among the lucky few (assuming there are no more oscillations). Now you can impress your students with the fact that you ended the qualifying round scoring above a scientist who co-authored a book with Stephen Hawking. But I digress.

Yours was a very interesting essay, which demonstrated a deep understanding, and it will be intriguing to see what other ideas you have to share - and compare notes.

More later,

Jonathan

Thanks Ben,

Well it may be that the positions up to the last 24 hours are more indicative, but that's all in the margin in some ways anyway. My email address is on my thread but I forgot to put it on my essay, it's jonathan.kerr@to-gl.net . Have a well earned rest!

Best wishes, Jonathan

Dear Ben,

Thank you a lot of for your attentions and comments. Please accept my apology for delay.

As you correctly mentioned, we reviewed all non-local hidden variable models which simulated quantum singlet state by non-local hidden variables. We derived inequalities which are based on these models and showed that they violated by quantum correlation function.

Unfortunately, my essay has typos at first paragraph of page 5 (is equal to one...), however, equation 6 and fig. 3 are correct.

As you mentioned at your essay, there are some physical models which are based on the Extra Dimensions and compactification of extra dimensions. However, please pay attention that these models take place at high energy physics (about 10^{18} Gev). I can accept your opinion if you find rationalization for it. In other words, how space-time microstructure is changed at low energy physics (about 1 kev)?

I am ready to see your work at more detail.

Thank you in advance

Sincerely yours.

Akbar

Dear Ben,

My warmest congratultions to your first - and most well deserved - rating in this contest.I have spent a few days in rural southern Sweden, hardly within reach of the Internet. But today I have re-read the above conversation, and learnt more from it. Now, when everything is settled about the final essays, I look forward to the possibility of a more peaceful converstation, in which I have a slight chance to catch up.

Best regards!

Inger

Hello again Ben and Friends,

I'm taking up your invitation to post at any time, with due respect to John Baez, who is not my cousin in real life.

A story:

They say my uncle is crazy, and cousin John tells me some family members wanted to lock old Uncle Octonius up in the attic, but I think he is only eccentric because he's seen the universe, and knows its secrets. For years we thought he just wouldn't associate with the other family members at all, but somehow we worked out how to do it safely. You see; Octonius is very persuasive, and can make people do almost anything - so he can't be trusted, or rather no one person can ever see him alone. And when we send two, they always disagree on what was said. Therefore; we always visit Uncle Octonius in committees of three. But; the first time a group of us visited, he insisted that he must see all the family members - with equal frequency - and that there always be someone in common between any two visits. Luckily; this worked out, because there are seven of us.

The thing is; Octonius is incredibly wealthy and knows the secrets of the universe, but we were all so afraid of him that we never knew why he seemed so crazy. You see; he always liked to break the laws of algebra - or insist on things being backwards sometimes - whenever we tried to use the associative and commutative rules to simplify expressions for him. But we never understood why that was, until we attempted to rank ourselves - thinking that both the greatest and slightest within our family needed to be included, within each committee, to assure trust. Then Octonius explained that committees follow a rule that is non-commutative, and then if you include everyone at once things become non-associative, because there can be disagreements between members or committees - but there is also a hierarchy or ordering of and within any committee.

Though we are still not sure we can trust him, Uncle Octonius tells us this is as fair as it can be, and now he is teaching us the secrets of the universe. So who could complain? I'm glad cousin John didn't let the others lock him up in the attic, or we would never have learned of his vast wealth and untold secrets.

end of story

Jonathan

Dear Ben - first of all I want to thank you again for your supportive remarks on John and my submission! Thankfully, after checking my records, I found your essay in the list of ones I loved as well. I am happy to see that enough people also loved your essay. Honestly, the rating and voting was overwhelming to me, and I admire your thoroughness in this essay contest. Hopefully you win a prize, both for your actual essay submission as well as your engagement in the contest. It will be well deserved.

On a sidenote, after flying over the 300 or so essay titles, zipping through some 30 essays without actually reading it, there were some 10-15 essays that I actually read and felt competent voting on; with an average score somewhere in the 8's since - by selection - I chose only to read essays that already looked interesting on first sweep. It should be needless to say that I voted honestly, if overwhelmed by the mass of entries.

Regarding your causal metric hypothesis, there is a scenario that I would love to ask for your opinion on whether or not it may be compatible with your hypothesis. Let me attempt:

(1) What you describe as finite set members in your universe are representations of actual particles. Rather than placing particles into a somehow finite spacetime model, spacetime only exists (if emergent) at places where there is a particle. Each particle has well defined attributes, some of which are "location-like" parameters that are used to model when and how strong these particles interact; some others are properties that characterize strengths of interaction under the various fundamental forces (charge, mass). Just as there is no such thing as a continuous charge or mass between any two interacting particles, there is no such thing as continuous space or time between any two particles or interaction, either.

(2) Particle location parameters are coordinates in the similar sense to spacetime coordinates in General Relativity, they are unobservable in principle but model something ultimately observable. There is not necessarily a distinct time-like coordinate or three distinct space-like coordinates, though. Instead, the 4 (or more, but not much more) coordinates will eventually appear space-like or time-like under observation as defined in the next (3) and (4).

(3) Human bias evaluates physical forces by observing motion of electromagnetically bound objects (atoms, molecules). It makes us humans believe that there is a Lorentzian base manifold that other forces act upon. Accordingly, conventional physical law has electromagnetism as genuinely describable on Lorentzian base manifolds. What we use to call "observation" is in fact a projection of the entire space of particles and parameters into a smaller subspace; the causal set is projected into a more narrow parameter space of itself, where that subspace has: (a) one time-like coordinate, (b) one space-like coordinate, and (c) genuinely Minkowskian metric. However, this is not a genuine property of your universe, but merely an extraneous projection that mimics human experience.

(3a) Granted, it is a very useful projection since it would be hard to observe any kind of force in a lab that explicitly does not use atoms or molecules ...

(4) If you model interaction between any two particles, you do so by understanding all the properties of the two corresponding causal set members. Causality, and partial order, is defined by the effective change of properties of each set member: Interaction is unique and well defined, therefore, the change of parameters of your set members through the interaction are well defined, "causal" relations: The location parameters how strong interaction is based on the set geometry (warning: weasel word "geometry"!). Pairwise calculation between any two particles (any two set members) give you an effective physical interaction, a force so to speak, which in turn is modeled by modifying the particle's location parameters.

(4a) In the language of causal sets, I believe that this would mean that physical interaction may change the (partial) order of your set members. I am not sure, though.

(5) Projecting all such particle location parameters onto an overwhelmingly electromagnetic (Minkowskian) observer space results in the fundamental laws of nature we know today.

Do you think such a procedure is compatible with your causal metric hypothesis?

Best wishes, Jens

Dear All,

I am suffering a bit of a backlog of deep and important points raised by a number of different people here over the last two weeks. Each of these points requires a careful and somewhat involved response, to whatever extent I am intellectually capable of providing one!

This is midterm week at my university, and I have a throng of needy calculus students tugging at my coattails at present. It may well be the weekend before I am able to catch up on some of these communications. Some of the main priorities are the following:

1. The implications of experimental results constraining certain types of nonmanifold structure/covariance breaking.

2. The proper application of path summation in general contexts.

3. Discussing what "particles" might look like in view of the causal metric hypothesis.

4. Discussing some special algebraic structures of particular importance...

In the meantime, please feel free to continue posting such remarks here; you're contributing to my education! Since this thread is at the top of the list, it's a reasonably convenient place for discussion. Many of you know more about some of these issues than I do, so feel free to post remarks in response to others' comments. While I will do my best to answer things myself, I am a bit greedy in the sense that my greater interest at present is absorbing, black hole-like, what everyone else is saying. I will endeavor to give off a little Hawking radiation, however!

Finally, I am trying to compile a coherent email list; my email is bdribus@math.lsu.edu, and I'd appreciate hearing from any of you. I have already contacted a number of you who included email addresses on your essays. Take care,

Ben

The breakdown of the Lorentz symmetry is something which was advanced by loopvariable quantum gravity theorists. As Ben points out the observations of distant burstars has put considerable doubt upon these theories. Spacetime appears to be absolutely smooth down to scale of 10^{-45}cm or so. The graininess of spacetime that would result from violations of Lorentz symmetry should result in dispersion of light. Higher frequency light would interact more strongly with this graininess, so over billions of light years the subtle effect would be observed in different arrival time of light with different frequencies. This has failed to emerge in observations. The double relativity proposed by Smolin and Magueijo is an example of how this might occur in special relativity. This is a sort of Planck scale obstruction to the boost operations of special relativity.

We do not expect this on a number of grounds. If I were to accelerate a proton to Planck energy I would be surprised to find that I could not boost if further. In a thought experiment I could imagine boosting an identical apparatus to a gamma smaller than that of the proton. The proton might then come upon the apparatus, where in that frame I boost it to a high gamma to the Planck energy. This obstruction would then say that if I boost to the frame of the original apparatus this proton would be observed to have an energy equal to the Planck energy. This is an inconsistency.

The idea is somewhat enticing, but if there is something going on with this I would expect the physical world to somehow cancel the effect. In what I write below that is just what is proposed.

The paper by Smolin and Magueijo sets up a postulate on the relativity of inertial frames, the equivalence principle and the observer independence of the Planck scale of length and energy. A nonlinear Lorentz group is then proposed with the generator of the standard Lorentz rotation generator J^i defined as

[math]

L_{ij} = p_i{\partial\over{\partial p^j}} - p_j{\partial\over{\partial p^i}}, J^i = \epsilon^{ijk}L_{jk},

[/math]

where the boost generator modified with the dilaton operator

[math]

D = p^i{\partial\over{\partial p^i}}

[/math]

is

[math]

K^i = K^i_0 L_p p^i D.

[/math]

These satisfy the standard commutation relationships for the Lorentz algebra

[math]

[J^i, J^j] = \epsilon^{ijk}J_k, [J^i, K^j] = \epsilon^{ijk}K_k, [K^i, K^j] = \epsilon^{ijk}J_k.

[/math]

The addition of the dilaton operator means there is the inclusion of a p^i in the boost, which means the action is nonlinear in momentum space. The entire nonlinear Lorentz boost in the x direction then gives

[math]

p_0^\prime = {{\gamma(p_0 - vp_x)}\over{1 L_p(\gamma - 1)p_0 - L_p\gamma vp_x}}

[/math]

[math]

p_x^\prime = {{\gamma(p_x - vp_0)}\over{1 L_p(\gamma - 1)p_0 - L_p\gamma vp_x}}

[/math]

[math]

p_{y,z}^\prime = {{p_{y,z}}\over{1 L_p(\gamma - 1)p_0 - L_p\gamma vp_x}}.

[/math]

The effect of the operator U(p_0) on the element of the Lorentz group g = exp(ω_{μν}L^{μν}) defines the nonlinear representation of the Lorentz group by

[math]

{\cal G}[\omega_{\mu\nu}] = U^{-1}(p_0)gU(p_0) = U^{-1}(p_0)(1 \omega^{\mu\nu}L_{\mu\nu})U(p_0) \dots =

[/math]

[math]

1 \omega^{\mu\nu}{{L_{\mu\nu}}\over{1 - L_p^2p_0^2}} \dots = exp{\Big(\omega^{\mu\nu} L_{\mu\nu}\frac{1}{1 - L_p^2p_0^2}\Big)} = e^{\omega^{\mu\mu}M_{\mu\nu}[p_0]}.

[/math]

This modifies the structure of general relativity. Let the vector e^a, where the index a indicates an internal space direction, define a tetrad basis by e^a_μ = ∂_μe^a. The tetrad exhibits the nonlinear realization of the transformation according

to

CONTINUED:

[math]

e^a_\mu \rightarrow e^{a\prime}_\mu = {\cal G}[\omega_{\alpha\beta}, p^b_0]e^a_\mu,

[/math]

where p^b_0 = e^b_0. For e^{aμ} the transformation involves {\cal G}^{-1}[p_0]. Similarly the differential operator

[math]

D_\mu \rightarrow {\cal G}[e_0](\partial_\mu \omega^{a\nu}_\mu e^a_\nu),

[/math]

transforms locally under the nonlinear Lorentz group. This then gives

[math]

D_\mu e^a_\nu = {\cal G}[p_0](\partial_\mu e_\nu \omega^a_{\sigma\nu}){\cal G}[e_0] \big({\cal G}[p_0] \partial_\mu{\cal G}^{-1}[p_0] e^a\big)_\nu,

[/math]

which for the local nonlinear transformation written according to indices gives the connection coefficients

[math]

{\omega^{a\nu}}_\mu[p_0] = {{\cal G}_\alpha}^\nu[p_0]{\omega^{a\alpha}}_\beta{{\cal G}^{-1\beta}}_\mu[p_0] {{\cal G}_\alpha}^\sigma[p_0] \big(\partial_\mu{{{\cal G}^{-1\alpha}}_\nu[p_0]}\big) e^a_\sigma.

[/math]

There is then an additional connection term. For p_0L_p \le\le 1 these additional connection terms are correspondingly small. Define these additional connection terms

[math]

{\gamma^{a\mu}}_{\nu } = {\gamma^{a\mu}}_{\nu\sigma}e^\sigma = {{\cal G}^\mu}_\rho\partial_\nu{{\cal G}^{-1\rho}}_\sigma e^a

[/math]. The

curvature tensor is then

[math]

{R^{a\alpha}}_{\mu\beta\nu}[p_0] = {{\cal G}^\alpha}_\sigma[p_0] {R^{a\sigma}}_{\mu\rho\nu}{{\cal G}^{-1}_\beta}^\rho \partial_{[\beta}{\gamma^{a\alpha}}_{\mu\nu]} \epsilon^{abc}{\gamma^{b\alpha}}_{[\beta}{\gamma^c}_{\mu\nu]}.

[/math]

The standard curvature is homogeneously transformed by the nonlinear term, where the additional curvature in the last two terms is labelled as

[math]

{\rho^{a\alpha}}_{\mu\beta\nu}.

[/math]

This additional curvature is then some gravity field effect induced by this extreme boost. A particle boosted to near the Planck scale might be expected to experience the cosmological constant or curvature induced Einstein tensor G_{μν} = Λg_{μν}more strongly. A Lorentz contracted curvature with Gaussian curvature R has curvature radius 1/sqrt{R} and this would be contracted by the Lorentz factor γ, and so the curvature "amplified" by R --- > R/γ^2. I would then propose that the above curvature term. The cosmological constant is defined on the Hubble frame, which is due to the symmetry of the spacetime. The apparent "preferred frame" is not some violation of relativity. This highly boosted particle would then experience the cosmological curvature much more strongly.

This does not appear to solve the 123 orders of magnitude problem. The boosted cosmological constant is ~ 10^{76} times larger This boosted particle is interacting with the vacuum of the universe much more strongly, but it is still 47 orders of magnitude too small. In other words returning to the unboosted frame would suggest the cosmological constant would be 10^{47} times larger than it is. However, if we were to boost a Planck mass we might expect that it interacts more strongly with the vacuum. This might then increase this "boost factor" At this point I have no particular idea on how to proceed.

Cheers LC

Dear Benjamin Dribus,

Causal metric hypothesis is much applicable with the Tetrahedral-brane scenario of Coherently-cyclic cluster-matter paradigm of universe, in that time emerges with the eigen-rotational strings and a causality-effect continuum is expressional for an eternal universe and thus Causal cycles is descriptive with this paradigm. Causality of three-dimensional structures is the effect of tetrahedral-brane expressions by eigen-rotational strings, in that spacetime emerges from eigen-rotations of one-dimensional string-matter segments. Thus in this paradigm, the metric properties of spacetime is descriptive by the configuration space with string-length and time, in that the nature of spacetime is expressed differently.

With best wishes

Jayakar

Thank You Ben and good Sir Lawrence!

Good summary and references Ben, and explication of the territory Lawrence. Lorentz invariance vilolation (LIV) is likely a problem for for some causally structured theories like CDT, but as LC pointed out, it first came out as a prediction by some LQG folks, as a possible means of validating the loops approach. Greatly summarized, the Fermi and INTEGRAL results detected the near simultaneous arrival of both very high and lower energy radiation pulses from the same distant gamma ray burstar event. This does greatly constrain things, but as you pointed out Ben - it does not close the door on all causal approaches. And my conversations with a couple of LQG researchers, would indicate that their approach is not ruled out either - only constrained.

As I understand it; the very thing which makes causal dynamical triangulation (CDT) work is what makes it problematic, in terms of LIV. The timelike lines must line up at the boundaries of each simplex, as the simplicial fabric evolves, and so there is a local discrete arrow of time that has a particular direction in space. The CDT approach has a fixed clock as well, yielding a very definite grain, but Smolin and Markopoulou showed that a varying clock yields similar results that show evolving dimensionality. One of the CDT authors (Loll or Ambjorn) pointed out in correspondence that they don't think it is an exact model anyway, but rather a discrete simulation of the way the spacetime metric evolves.

My guess is the CDT folks did not start with enough degrees of freedom, to end up with an invariant model, and that's where I think the Octonions come in. That would greatly increase the options at the outset. I have lots of ideas of how that would come together. I'll have to continue my comments on the morrow, though, as it is already very late here. I've downloaded the papers you provided links to, and also a few by Sorkin and colleagues on topics relating to this discussion. Though I think a lot of the work on the Causal Sets approach is entirely sound, I tend to believe that we are looking for something similar to - but not exactly like - it. I need to do more reading though, to see how far that program (CauSets) has come along, before I comment further.

All the Best,

Jonathan