The strings 2011 conference has opend today in Sweden. You can watch videos of the talks live or recorded straight after, starting with the introduction by David Gross.

Gross asked the usual questions starting with the number one “What is String Theory?” and ending with number 11 “What will we learn from the LHC?”

He is a bit disappointed that the LHC has not found anything surprising yet, but he still holds high hopes for SUSY.

There have been promising new results in trying to solve large-N SUSY gauge theory which has beautiful mathematics: twistors, polytopes in the grasmanian for example. Since this theory is dual to string theory Gross thinks these discoveries could tell us about the fundamentals of string theory.

He goes on to mention entropic gravity which he said was also promising but he had a little smirk on his face when he said it and also implied that it is ambitious. There will be a talk from Verlinde later in the conference.

Apparently it is unfortunate that we seem to live in De Sitter space. The theories work much better in anti- De Sitter space. There are lots of questions but the most important product of knowledge is ignorance, then again it would be nice to have some answers he said at the end.

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Although I admire the technique and unification possibillities of String theory, I have ONE basic question :
“What is the origin of the vibration of strings”
or is this the childish trick of repeating the question “WHY” ?
Wilhelmus

The vibration is quantum mechanical. A classical string vibrates because there is tension and some applied energy is then given. In string theory these are given by the uncertainty principle, where even the lowest mode of the string (vacuum) vibrates.

We do not live in an anti-de Sitter spacetime. We may live on the boundary of an AdS spacetime. The boundary of the AdS is a conformally flat spacetime, which is anything conformally equivalent to Minkowski spacetime. This can include the de Sitter spacetime. The conformal transformation g_{ab} — > Ω^2g_{ab} means any line element is conformal as

ds^2 — > Ω^2ds^2 = Ω^2(du^2 – dx^2 – dy^2 – dz^2)

Then for du/dt = Ω^{-1} and Ω = Ω(t) we then have a time dependent conformal spacetime which is de Sitter for Ω(t) = Ω_0cosh(at).

Recent astronomical measurements of the radar distance and ruler distance between 2.x10^5 galaxies indicates that this dark energy which is the vacuum energy density driving the de Sitter expansion is independent of the rest of gravity. This is a short JPL press release on this

This is interesting, for the AdS_n ~ CFT_{n-1} gives an equivalence between the CFT on the boundary ∂AdS_n and the gravitation inside the AdS_n. Yet this in the “perfect world” is a conformal theory. Yet mass can break conformal symmetry, giving rise to Ricci terms and the rest. Consider the AdS_n space as a “ball,” with the sphere as its boundary, then mass can cause little dimples in the sphere. The tangent plane in this region then “probes” the interior of the AdS_n and there is the occurrence of local gravity on the boundary.

David Gross is always immensely fun and he’s clearly able to deliver unprepared talks well – but no doubt, I prefer his carefully prepared, smooth, fast enough talks.

The appraisal of emotions and counter-emotions about the LHC results so far are about as uninteresting as the negative LHC data themselves. And I suppose that what he said about Verlinde’s bullshit is the most negative he could say about a nominally “major topic” in the introduction – but I still find it positive. In particular, it doesn’t try to solve whether the hypothesis is actually right and it is clearly wrong. There is no gravitational entropy without event horizons.

I agreed with Gross’ point that the twistor uprising may be just about some special properties of a highly symmetric, integrable theory, not a genuine lesson about the whole theory – especially in the strongly coupled limit. Still, I can imagine that the twistor uprising does tell us something – this is impossible for the entropic gravity bullshit. So the comparison of how he approached these two things is unfortunate.

There are also topics of the last year that were not really mentioned at all – e.g. some AdS/hydrodynamics by Strominger and friends, new – and several other things I found cool even though they’re heavily understudied and undercited, partly because they’re not hyped in the way in which even David Gross hyped Verlinde’s entropic nonsense.

But it’s not “entropic” gravity any more, now it’s “adiabatic reaction” gravity. Something about statistics of eigenvalue crossings in a matrix model? I haven’t understood it.

Mitchell, based on my listening to parts of Verlinde’s PI talk, it’s still the same thing. He just claims that the entropic forces are adiabatic and don’t increase entropy which is of course nonsense.

It’s the whole point of an entropic force to push a system towards states of a higher entropy, so they’re irreversible. There is no “conservation law for entropy” in general situations; quite on the contrary, the entropy increases. Of course, if the entropic force is tiny, the entropy increase may be tiny as well, but if the entropic force should be as finite – or huge – as the force between the Sun and the Moon, or anything else, and if the force were entropic, of course that the increase of the entropy would have to be finite and large, too.

Moreover, the conservation laws for e.g. the Solar System have been classified for centuries and they include energy, angular momentum etc. – which are linked to symmetries of the system. The entropy is clearly not a function of those things, so the only thing to make the “gravitational entropy” conserved is to admit that it is zero for a Solar System. And indeed, it is zero for all systems that have no event horizons.

There’s a lot of deliberately amplified confusion. It’s true that things in quantum gravity may have entropy – but we know exactly what the entropy is whenever it is large. It’s A/4G where A is the total event horizon area. No horizons, no entropy. There is no gravitational entropy in the Solar System.

The Perimeter Institute talk by Verlinde at the end has a remarkably “bold” portion in which he claims that dark energy doesn’t exist, dark matter doesn’t exist, and all of them are connected in some vague ways to some ill-defined things apparently linked to entropic gravity – and even matrix models. People protest that it violates inflation, so Verlinde confirms that he disbeliefs not only dark energy and WIMP-like dark matter but also inflation and, in fact, the Big Bang itself.

The audience pretty much politely leaves him to say all those things without a glimpse of evidence that he has anything that can replace the established physics theories and explain the relevant phenomena in a different way. Hard to get rid of the impression that the audience feels that Verlinde with his money and power could influence themselves, too.

I am kind of sick of where the physics community is drifting. Don’t they have a high enough intelligence and tight enough spine to openly explain why it is crackpottery?

To my view the interpretation of entropic gravity in the form represented by Verlinde is fatally wrong because of the assumed non-existence of gravitons. Sabine Hossenfelder has abstracted the essential formulas for entropy and temperature leaving the interpretation open. I have proposed totally different interpretation of the formulas for temperature and entropy by assigning entropy and temperature to gravitons propagating along flux tubes defining gravitonic space-time sheets. Thermalization would happen for all particles, not only gravitations.

Situation would be analogous to thermal em radiation from Sun with temperature decreasing like 1/r^2. Entropy would be naturally proportional to the length of flux tube. This gives the basic formally apart from proportionality constant.

In zero energy ontology S-matrix must be replaced with its product with hermitian square root of density matrix and the lines of generalized Feynman graphs indeed become entropic:

There is also very beautiful connection with the basic problem of defining path integral. Exponent of Kahler action for preferred extremals contains two contributions: one from regions with Minkowskian signature and proportional to imaginary unit coming from the square root of metric determinant. Second contribution comes from Euclidian regions representing generalized Feynman graphs. This term guarantees the convergence and mathematical existence of the functional integral. The phase factor in turn gives connection to QFT and interference and defines Morse function required by TGD as almost topological QFT. A very close analogy with Floer homology emerges. Kahler and Morse find place in TGD world order as real and imaginary parts of the exponent of vacuum functional;-).

The presence of also real part in the exponent of vacuum functional is one aspect of entropic gravity in TGD framework.

Lawrence mentions the new results supporting the interpretation of
dark energy in terms of cosmological constant in GRT framework.

The GRT limit of TGD would correspond with simplest possible assumptions to Maxwell-Einstein gravity with both Minkowskian and space-like regions,which is of course something new. Also GRT limit would be almost topological QFT since action would reduces to Chern-Simons terms.

Reissner-Nordstrom metric would represent the simplest imaginable Minkowskian region. For the space-like regions CP_2 would represent idealization so that altogether standard model quantum numbers would be obtained also at the limit. Also family replication phenomenon since the topology for 2-D section 3-D light-like surfaces at which metric signature changes would be characterized by genus and third lowest genera are hyper-elliptic and therefore very special. In spinor sector one must introduce
8-D spinors by hand but this natural since limit is in question.

A highly non-trivial prediction is that black hole interiors are replaced with space-like regions. This condition follows by just perturbing the R-N horizon so that the g_rr component of the metric becomes finite and horizon becomes light-like surface.

In the Lagrangian formulation of GRT limit one must introduce cosmological constant in Euclidian regions in order to obtain cosmological constant characterizing the CP_2 as constant curvature space. The conjecture is that the average value of this cosmological constant equal to volume fraction of Euclidian regions (generalized Feynman graphs with 4-D lines) multiplying CP_2 cosmological constant corresponds to the observed one in the conceptual framework of GRT. Order of magnitude comes out correctly since the average is proportional to particle density.

The entropy force of gravity is a bound on the entropy, where its increase corresponds to the motion of the “screen.” The entropy kT = ∫F*dx leads to Newtonian gravity in an extension of holographic concepts. This entropy bound is a first law of thermodynamic general relativity bound given by a form of the Birkhoff theorem. The Birkhoff theorem tells us that on a Gaussian surface any spherical distribution of matter within the surface is equivalent to a black hole. The entropy is a measure of the redistribution of states, or the microstate distribution in a macrostate, where the macrostate corresponds to a coarse grained description of a black hole. For weak gravity this recovers Newtonian gravity or the Poisson equation ∂_i∂^iφ = 4πGρ, for φ the Newtonian gravity potential and ρ the density of matter in a volume. For a black hole the Gaussian surface within a string length of the event horizon is the stretched horizon in string holography. So the entropy, or in particular a change in entropy corresponds to the motion of the holographic screen, or equivalently the addition of microstates to the stretched horizon.

I am not sure why people seem to be tripped up by thinking entropy force of gravity implies the motion of a test mass in a spacetime will have some irreversible thermodynamics. The change in entropy is with the motion of the holographic screen or the stretched horizon, not with the motion of a particle. Overhauser and Collela did an interferometer experiment with neutrons in a crystal and found a gravitational analogous physics to the Aharonov-Bohm result. The dynamics was completely reversible, and no demolition of interference phase was recorded. I don’t think entropy force of gravity implies some irreversible dynamics of a particle in a gravity field, it tells us about the motion of a holographic screen and the information content of it.

My criticism of the entropy force of gravity hypothesis is more in line with whether this tells us much more than we already understood. This seems to be a way of casting the black hole laws of thermodynamics in a form with Gaussian surfaces or the Birkhoff theorem.

With respect to entropic gravity, there is experimental showing/indication that gravity is associated/related to nonlocal effect (quantum entanglement) see article in Progress of Phsyics:

[2] Gravity as a Manifestation of de Sitter invariance over a Galois Field
viXra:1104.0065 or http://arxiv.org/abs/1104.4647

I argue that the dS (or AdS) symmetry should be understood not such that the background spacetime is the dS (or AdS) space (since the notion of background spacetime is not physical) but that the operators describing a system satisfy the commutation relations of the dS (or AdS) algebra. Then the fact that Lambda > 0 indicates that the symmetry is dS rather than Poincare or AdS. Then the cosmological acceleration is a trivial kinematical effect in a system of two FREE bodies. So there is no need to involve dark energy or other fields for explaining this phenomenon.
The dS invariant theory has several unusual interesting properties, in particular:
1) Only fermions can be elementary (in particular, there is no SUSY).
2) There are no neutral elementary particles.
3) The electric charge and such additive quantum numbers as the baryon and lepton quantum numbers can be only approximately conserved.
As shown in [2], standard Newton’s law of gravity, gravitational red shift of light and the precession of Mercury’s perihelion also can be explained as pure kinematical effects in dS invariant theory of two FREE bodies if we assume additionally that the symmetry is over a Galois field rather than the field
of complex numbers.

you are probably familiar with the work of finnish mathematical physicists Paul Kustaanheimo. He was probably one of the first ones who worked with models of physics based on Galois fields and finite geometries.

Galois groups emerge in TGD framework from the notion of infinite prime in turn having a direct connection with supersymmetric arithmetic QFTs and would be naturally associated with finite geometries. Sequences of Galois groups would be assigned with polynomials as one solves their roots iteratively. Galois groups would be lifted to their braided variants and represented as symplectic flows at the light-like boundaries of causal diamonds (CDs- intersections of future and past directed light-cones) allowing contact structure at their light-like boundaries.

Discretization would not be fundamental but represented finite measurement resolution. At space-time level it would mean discretization for what I call partonic 2-surfaces at boundaries of CDs at which braid strands end. I have not been thinking in terms of finite geometries but they could correspond to orbits of Galois groups consisting of these points.

The orbits of points carrying quantum numbers would define braids of braids of braids of … assigned to the hierarchy of braided counterparts of Galois groups and this would give a connection to topological quantum field theories: TGD is indeed “almost” topological QFT- “almost topological” would mean algebraic geometric. One can assign homology and cohomology to this hierarchy of Galois groups with physical interpretation for boundary and co-boundary operations.

The matrix elements involving functional integral over infinite-D “world of classical worlds” (space of 3-surfaces) would reduce to discrete sums in finite measurement resolution with outcome controlled by Galois symmetries so that one could really calculate!

Thank you for your comments. I knew about Paul Kustaanheimo but unfortunately did not succeed in finding his works on Galois fields and finite geometries. Now I see that in principle his books are available and will try to read them.

We probably have considerably different preferences on the relation between finite vs. infinite in fundamental physics. You believe that “Discretization would not be fundamental but represented finite measurement resolution.” while at fundamental level physics should be based on “infinite” mathematics. On the contrary, I believe that sooner or later fundamental quantum physics will be based only on finite mathematics while “infinite” mathematics will be used only for approximations. I believe that one of the main problems of modern quantum physics is that it is based on mathematics developed mainly in the 19th century when people did not know about atoms and elementary particles. For example, the notion of infinitely small is based on the macroscopic experience that every macroscopic object can be divided by any number of parts. However, now it is clear that when we reach the level of atoms and elementary particles, we cannot divide matter anymore. Even this simple example shows that standard division and infinitely small can be only approximate notions. Analogously, the notions of continuity, space-time, standard geometry, topology etc. came from macroscopic experience and are not fundamental on quantum level. I discuss these problems in details e.g. in

Although I admire the technique and unification possibillities of String theory, I have ONE basic question :

“What is the origin of the vibration of strings”

or is this the childish trick of repeating the question “WHY” ?

Wilhelmus

The vibration is quantum mechanical. A classical string vibrates because there is tension and some applied energy is then given. In string theory these are given by the uncertainty principle, where even the lowest mode of the string (vacuum) vibrates.

We do not live in an anti-de Sitter spacetime. We may live on the boundary of an AdS spacetime. The boundary of the AdS is a conformally flat spacetime, which is anything conformally equivalent to Minkowski spacetime. This can include the de Sitter spacetime. The conformal transformation g_{ab} — > Ω^2g_{ab} means any line element is conformal as

ds^2 — > Ω^2ds^2 = Ω^2(du^2 – dx^2 – dy^2 – dz^2)

Then for du/dt = Ω^{-1} and Ω = Ω(t) we then have a time dependent conformal spacetime which is de Sitter for Ω(t) = Ω_0cosh(at).

Recent astronomical measurements of the radar distance and ruler distance between 2.x10^5 galaxies indicates that this dark energy which is the vacuum energy density driving the de Sitter expansion is independent of the rest of gravity. This is a short JPL press release on this

http://www.jpl.nasa.gov/news/news.cfm?release=2011-149

This is interesting, for the AdS_n ~ CFT_{n-1} gives an equivalence between the CFT on the boundary ∂AdS_n and the gravitation inside the AdS_n. Yet this in the “perfect world” is a conformal theory. Yet mass can break conformal symmetry, giving rise to Ricci terms and the rest. Consider the AdS_n space as a “ball,” with the sphere as its boundary, then mass can cause little dimples in the sphere. The tangent plane in this region then “probes” the interior of the AdS_n and there is the occurrence of local gravity on the boundary.

David Gross is always immensely fun and he’s clearly able to deliver unprepared talks well – but no doubt, I prefer his carefully prepared, smooth, fast enough talks.

The appraisal of emotions and counter-emotions about the LHC results so far are about as uninteresting as the negative LHC data themselves. And I suppose that what he said about Verlinde’s bullshit is the most negative he could say about a nominally “major topic” in the introduction – but I still find it positive. In particular, it doesn’t try to solve whether the hypothesis is actually right and it is clearly wrong. There is no gravitational entropy without event horizons.

I agreed with Gross’ point that the twistor uprising may be just about some special properties of a highly symmetric, integrable theory, not a genuine lesson about the whole theory – especially in the strongly coupled limit. Still, I can imagine that the twistor uprising does tell us something – this is impossible for the entropic gravity bullshit. So the comparison of how he approached these two things is unfortunate.

There are also topics of the last year that were not really mentioned at all – e.g. some AdS/hydrodynamics by Strominger and friends, new – and several other things I found cool even though they’re heavily understudied and undercited, partly because they’re not hyped in the way in which even David Gross hyped Verlinde’s entropic nonsense.

But it’s not “entropic” gravity any more, now it’s “adiabatic reaction” gravity. Something about statistics of eigenvalue crossings in a matrix model? I haven’t understood it.

Mitchell, based on my listening to parts of Verlinde’s PI talk, it’s still the same thing. He just claims that the entropic forces are adiabatic and don’t increase entropy which is of course nonsense.

http://pirsa.org/11060065/

It’s the whole point of an entropic force to push a system towards states of a higher entropy, so they’re irreversible. There is no “conservation law for entropy” in general situations; quite on the contrary, the entropy increases. Of course, if the entropic force is tiny, the entropy increase may be tiny as well, but if the entropic force should be as finite – or huge – as the force between the Sun and the Moon, or anything else, and if the force were entropic, of course that the increase of the entropy would have to be finite and large, too.

Moreover, the conservation laws for e.g. the Solar System have been classified for centuries and they include energy, angular momentum etc. – which are linked to symmetries of the system. The entropy is clearly not a function of those things, so the only thing to make the “gravitational entropy” conserved is to admit that it is zero for a Solar System. And indeed, it is zero for all systems that have no event horizons.

There’s a lot of deliberately amplified confusion. It’s true that things in quantum gravity may have entropy – but we know exactly what the entropy is whenever it is large. It’s A/4G where A is the total event horizon area. No horizons, no entropy. There is no gravitational entropy in the Solar System.

The Perimeter Institute talk by Verlinde at the end has a remarkably “bold” portion in which he claims that dark energy doesn’t exist, dark matter doesn’t exist, and all of them are connected in some vague ways to some ill-defined things apparently linked to entropic gravity – and even matrix models. People protest that it violates inflation, so Verlinde confirms that he disbeliefs not only dark energy and WIMP-like dark matter but also inflation and, in fact, the Big Bang itself.

The audience pretty much politely leaves him to say all those things without a glimpse of evidence that he has anything that can replace the established physics theories and explain the relevant phenomena in a different way. Hard to get rid of the impression that the audience feels that Verlinde with his money and power could influence themselves, too.

I am kind of sick of where the physics community is drifting. Don’t they have a high enough intelligence and tight enough spine to openly explain why it is crackpottery?

The question session was cut off just as it was getting interesting. I did not realise just how far away from standard cosmology he has gone, wow.

To my view the interpretation of entropic gravity in the form represented by Verlinde is fatally wrong because of the assumed non-existence of gravitons. Sabine Hossenfelder has abstracted the essential formulas for entropy and temperature leaving the interpretation open. I have proposed totally different interpretation of the formulas for temperature and entropy by assigning entropy and temperature to gravitons propagating along flux tubes defining gravitonic space-time sheets. Thermalization would happen for all particles, not only gravitations.

Situation would be analogous to thermal em radiation from Sun with temperature decreasing like 1/r^2. Entropy would be naturally proportional to the length of flux tube. This gives the basic formally apart from proportionality constant.

In zero energy ontology S-matrix must be replaced with its product with hermitian square root of density matrix and the lines of generalized Feynman graphs indeed become entropic:

http://tgd.wippiespace.com/public_html/articles/egtgd.pdf .

There is also very beautiful connection with the basic problem of defining path integral. Exponent of Kahler action for preferred extremals contains two contributions: one from regions with Minkowskian signature and proportional to imaginary unit coming from the square root of metric determinant. Second contribution comes from Euclidian regions representing generalized Feynman graphs. This term guarantees the convergence and mathematical existence of the functional integral. The phase factor in turn gives connection to QFT and interference and defines Morse function required by TGD as almost topological QFT. A very close analogy with Floer homology emerges. Kahler and Morse find place in TGD world order as real and imaginary parts of the exponent of vacuum functional;-).

The presence of also real part in the exponent of vacuum functional is one aspect of entropic gravity in TGD framework.

Lawrence mentions the new results supporting the interpretation of

dark energy in terms of cosmological constant in GRT framework.

The GRT limit of TGD would correspond with simplest possible assumptions to Maxwell-Einstein gravity with both Minkowskian and space-like regions,which is of course something new. Also GRT limit would be almost topological QFT since action would reduces to Chern-Simons terms.

Reissner-Nordstrom metric would represent the simplest imaginable Minkowskian region. For the space-like regions CP_2 would represent idealization so that altogether standard model quantum numbers would be obtained also at the limit. Also family replication phenomenon since the topology for 2-D section 3-D light-like surfaces at which metric signature changes would be characterized by genus and third lowest genera are hyper-elliptic and therefore very special. In spinor sector one must introduce

8-D spinors by hand but this natural since limit is in question.

A highly non-trivial prediction is that black hole interiors are replaced with space-like regions. This condition follows by just perturbing the R-N horizon so that the g_rr component of the metric becomes finite and horizon becomes light-like surface.

In the Lagrangian formulation of GRT limit one must introduce cosmological constant in Euclidian regions in order to obtain cosmological constant characterizing the CP_2 as constant curvature space. The conjecture is that the average value of this cosmological constant equal to volume fraction of Euclidian regions (generalized Feynman graphs with 4-D lines) multiplying CP_2 cosmological constant corresponds to the observed one in the conceptual framework of GRT. Order of magnitude comes out correctly since the average is proportional to particle density.

For details see

http://tgd.wippiespace.com/public_html/articles/egtgd.pdf .

The entropy force of gravity is a bound on the entropy, where its increase corresponds to the motion of the “screen.” The entropy kT = ∫F*dx leads to Newtonian gravity in an extension of holographic concepts. This entropy bound is a first law of thermodynamic general relativity bound given by a form of the Birkhoff theorem. The Birkhoff theorem tells us that on a Gaussian surface any spherical distribution of matter within the surface is equivalent to a black hole. The entropy is a measure of the redistribution of states, or the microstate distribution in a macrostate, where the macrostate corresponds to a coarse grained description of a black hole. For weak gravity this recovers Newtonian gravity or the Poisson equation ∂_i∂^iφ = 4πGρ, for φ the Newtonian gravity potential and ρ the density of matter in a volume. For a black hole the Gaussian surface within a string length of the event horizon is the stretched horizon in string holography. So the entropy, or in particular a change in entropy corresponds to the motion of the holographic screen, or equivalently the addition of microstates to the stretched horizon.

I am not sure why people seem to be tripped up by thinking entropy force of gravity implies the motion of a test mass in a spacetime will have some irreversible thermodynamics. The change in entropy is with the motion of the holographic screen or the stretched horizon, not with the motion of a particle. Overhauser and Collela did an interferometer experiment with neutrons in a crystal and found a gravitational analogous physics to the Aharonov-Bohm result. The dynamics was completely reversible, and no demolition of interference phase was recorded. I don’t think entropy force of gravity implies some irreversible dynamics of a particle in a gravity field, it tells us about the motion of a holographic screen and the information content of it.

My criticism of the entropy force of gravity hypothesis is more in line with whether this tells us much more than we already understood. This seems to be a way of casting the black hole laws of thermodynamics in a form with Gaussian surfaces or the Birkhoff theorem.

With respect to entropic gravity, there is experimental showing/indication that gravity is associated/related to nonlocal effect (quantum entanglement) see article in Progress of Phsyics:

http://ptep-online.com/index_files/2007/PP-09-03.PDF

This was also theorized earlier here:

http://www.neuroquantology.com/journal/index.php/nq/article/view/126

Huping

In my papers:

[1] Positive Cosmological Constant and Quantum Theory

Symmetry: Quantum Symmetry, Vol. 2(4), pp. 1945-1980; http://www.mdpi.com/2073-8994/2/4/1945/

or http://arxiv.org/abs/1007.2260

[2] Gravity as a Manifestation of de Sitter invariance over a Galois Field

viXra:1104.0065 or http://arxiv.org/abs/1104.4647

I argue that the dS (or AdS) symmetry should be understood not such that the background spacetime is the dS (or AdS) space (since the notion of background spacetime is not physical) but that the operators describing a system satisfy the commutation relations of the dS (or AdS) algebra. Then the fact that Lambda > 0 indicates that the symmetry is dS rather than Poincare or AdS. Then the cosmological acceleration is a trivial kinematical effect in a system of two FREE bodies. So there is no need to involve dark energy or other fields for explaining this phenomenon.

The dS invariant theory has several unusual interesting properties, in particular:

1) Only fermions can be elementary (in particular, there is no SUSY).

2) There are no neutral elementary particles.

3) The electric charge and such additive quantum numbers as the baryon and lepton quantum numbers can be only approximately conserved.

As shown in [2], standard Newton’s law of gravity, gravitational red shift of light and the precession of Mercury’s perihelion also can be explained as pure kinematical effects in dS invariant theory of two FREE bodies if we assume additionally that the symmetry is over a Galois field rather than the field

of complex numbers.

Dear Felix,

you are probably familiar with the work of finnish mathematical physicists Paul Kustaanheimo. He was probably one of the first ones who worked with models of physics based on Galois fields and finite geometries.

Galois groups emerge in TGD framework from the notion of infinite prime in turn having a direct connection with supersymmetric arithmetic QFTs and would be naturally associated with finite geometries. Sequences of Galois groups would be assigned with polynomials as one solves their roots iteratively. Galois groups would be lifted to their braided variants and represented as symplectic flows at the light-like boundaries of causal diamonds (CDs- intersections of future and past directed light-cones) allowing contact structure at their light-like boundaries.

Discretization would not be fundamental but represented finite measurement resolution. At space-time level it would mean discretization for what I call partonic 2-surfaces at boundaries of CDs at which braid strands end. I have not been thinking in terms of finite geometries but they could correspond to orbits of Galois groups consisting of these points.

The orbits of points carrying quantum numbers would define braids of braids of braids of … assigned to the hierarchy of braided counterparts of Galois groups and this would give a connection to topological quantum field theories: TGD is indeed “almost” topological QFT- “almost topological” would mean algebraic geometric. One can assign homology and cohomology to this hierarchy of Galois groups with physical interpretation for boundary and co-boundary operations.

The matrix elements involving functional integral over infinite-D “world of classical worlds” (space of 3-surfaces) would reduce to discrete sums in finite measurement resolution with outcome controlled by Galois symmetries so that one could really calculate!

The article “Motives and Infinite Primes” at

http://tgd.wippiespace.com/public_html/articles/motivesart.pdf

represents also these and other ideas related in detail.

Dear Matti,

Thank you for your comments. I knew about Paul Kustaanheimo but unfortunately did not succeed in finding his works on Galois fields and finite geometries. Now I see that in principle his books are available and will try to read them.

We probably have considerably different preferences on the relation between finite vs. infinite in fundamental physics. You believe that “Discretization would not be fundamental but represented finite measurement resolution.” while at fundamental level physics should be based on “infinite” mathematics. On the contrary, I believe that sooner or later fundamental quantum physics will be based only on finite mathematics while “infinite” mathematics will be used only for approximations. I believe that one of the main problems of modern quantum physics is that it is based on mathematics developed mainly in the 19th century when people did not know about atoms and elementary particles. For example, the notion of infinitely small is based on the macroscopic experience that every macroscopic object can be divided by any number of parts. However, now it is clear that when we reach the level of atoms and elementary particles, we cannot divide matter anymore. Even this simple example shows that standard division and infinitely small can be only approximate notions. Analogously, the notions of continuity, space-time, standard geometry, topology etc. came from macroscopic experience and are not fundamental on quantum level. I discuss these problems in details e.g. in

“Introduction into a Quantum Theory over a Galois field”

Symmetry: Quantum Symmetry Vol 2(4), pp. 1810-1845 (2010) http://www.mdpi.com/2073-8994/2/4/1810/

or http://arxiv.org/abs/1011.1076

Best regards, Felix.