Maybe there is no grand unified theory?


by Warp
Tags: grand, theory, unified
julian
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#19
Dec17-12, 09:04 PM
P: 289
Einstein's equation

[itex]R_{\mu \nu} - {1 \over 2} g_{\mu \nu} R = T_{\mu \nu}[/itex]

tells us how geometry is coupled to matter. Since matter is in a quantum superposition, isnt geometry as well?
LastOneStanding
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#20
Dec17-12, 09:11 PM
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Whom is your question directed to, julian? That is in fact one the points of tension between QM and GR. The curvature tensor is purely classical and we don't have a well-defined notion for a 'superposition of curvature'. However, QM says it should be coupled to something that is in a superposition. Hence, one of the incompatibilities between QM and GR is the question of what the gravitational field looks like for a particle small enough to have quantum effects dominate.
bapowell
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#21
Dec17-12, 09:18 PM
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Quote Quote by julian View Post
Einstein's equation

[itex]R_{\mu \nu} - {1 \over 2} g_{\mu \nu} R = T_{\mu \nu}[/itex]

tells us how geometry is coupled to matter. Since matter is in a quantum superposition, isnt geometry as well?
This is the motivation behind so-called semi-classical treatments of gravitational phenomena, in which the stress-energy tensor is replaced by its quantum expectation value, [itex]\langle T_{\mu \nu} \rangle[/itex], but the geometric side of the Einstein Equations remain classical. This approach, while only approximating the full quantum theory of gravitation, is quite powerful. Hawking used it to discover that black holes radiate, and it is central to the derivation of the temperature anisotropies in the CMB.
julian
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#22
Dec17-12, 09:23 PM
P: 289
The origin question was can GR and QM be independent theories...and I'm saying they are incompatible.
julian
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#23
Dec17-12, 09:24 PM
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Quote Quote by bapowell View Post
This is the motivation behind so-called semi-classical treatments of gravitational phenomena, in which the stress-energy tensor is replaced by its quantum expectation value, [itex]\langle T_{\mu \nu} \rangle[/itex], but the geometric side of the Einstein Equations remain classical. This approach, while only approximating the full quantum theory of gravitation, is quite powerful. Hawking used it to discover that black holes radiate, and it is central to the derivation of the temperature anisotropies in the CMB.
Really the right hand side is an operator. Attempts are made to replace the right hand side by an expectation value but an iterative procedure must be implemented to define the expectation values. It has been shown that the iteration does not converge in general, whence we must quantise the gravitational field...see p.g. 5-6 of Thiemann's book.
bapowell
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#24
Dec17-12, 09:37 PM
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Quote Quote by julian View Post
Attempts are made to replace the right hand side by an expectation value but an iterative procedure must be implemented to calculate expectation values. It has been shown that the iteration does not coverge, whence we must quantise the gravitational field...see p.g. 5-6 of Thiemann's book.
Of course. Hence how I said that this approach approximates the full quantum theory of gravititation (but I'm unclear on the non-convergence issue. I'm not an expert on stress-tensor renormalization by any means, but I was under the impression that there were finite representations of [itex]\langle T_{\mu \nu}\rangle[/itex]. In other words, aren't those dangerous UV modes Thiemann mentions integrated out?). Saying that GR and QM are incompatible isn't new -- this was recognized long ago, and forms the basis of the OP's question. It seems you already know the answer to the question you asked.
julian
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#25
Dec17-12, 09:45 PM
P: 289
People talk about singularities and how GR and QM must merge in certain circumstances. But if you take QM to apply at all levels (which I do!) then they are always incompatible.
julian
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#26
Dec17-12, 09:59 PM
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Hello Bapowell

The iteration is unstable in general. It was Robert Wald and Flanagan [gr-qc/960252] who showed this.

"In other words, aren't those dangerous UV modes Thiemann mentions integrated out?"

I'm not sure how to answer it, I'm not an expert either.

But what I do know is that Thiemann shows in his book that a canonical quantization of GR (with full backreaction of matter on the quantum gravitational field) is finite.
bapowell
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#27
Dec17-12, 10:21 PM
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Thanks for the reference julian. I'll have a look.
julian
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#28
Dec17-12, 10:38 PM
P: 289
Hi bapowell

I think people take they view that quantum mechanics stops at some length scale, beyond which matter is classical, and then put it into Einstein's equations. I'm of the opinion that there is no artificial boundary between classical and quantum...
PAllen
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#29
Dec17-12, 11:00 PM
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Quote Quote by julian View Post
Hi bapowell

I think people take they view that quantum mechanics stops at some length scale, beyond which matter is classical, and then put it into Einstein's equations. I'm of the opinion that there is no artificial boundary between classical and quantum...
That may be (and I agree), but as a practical matter, there is enormous range of validity to the limits of measurement for Newtonian mechanics, Maxwell's equations, and GR. As a practical matter, one must try to understand the boundaries of when classical theories are good enough, unless you want to waste endless effort calculating trivial results.
atyy
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#30
Dec17-12, 11:15 PM
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Quote Quote by julian View Post
The origin question was can GR and QM be independent theories...and I'm saying they are incompatible.
Here is a discussion of GR as a quantum theory.

http://arxiv.org/abs/1209.3511
lpetrich
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#31
Dec18-12, 07:39 PM
P: 514
So a Grand Unified Theory is one that includes all the Standard Model without including gravity, and a Theory of Everything one that also includes gravity.

I don't think that I want to split hairs over this issue.
bapowell
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#32
Dec18-12, 08:41 PM
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Quote Quote by lpetrich View Post
So a Grand Unified Theory is one that includes all the Standard Model without including gravity, and a Theory of Everything one that also includes gravity.
Traditionally, yes, that's the correct distinction. GUTs are meant to unify the strong, weak, and electromagnetic forces only.
lpetrich
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#33
Dec18-12, 10:17 PM
P: 514
I remember once discovering a pattern in the discovery of the fundamental constituents of matter, a pattern that involves several stages:
  1. Discovery of a few entities. Category may be poorly defined or not even properly recognized.
  2. Discovery of many entities. Well-defined category.
  3. Discovery of regularities among the entities.
  4. Discovery of underlying simplicity and the causes of the regularities.
These entities have gone through that cycle, with all but the Standard Model completing it:
  1. Atoms and chemical elements
  2. Atomic nuclei
  3. Hadrons
  4. Standard-Model particles
The details.

Atoms and chemical elements

The idea of chemical elements goes back to antiquity, with Greek earth, air, fire, water and Chinese earth, wood, metal, fire, water. However, chemical elements became a well-defined category with Antoine-Laurent de Lavoisier, and atomism became rigorous with John Dalton.

By the middle of the 19th cy., Dmitri Mendeleev proposed his Periodic Table of Elements, complete with predictions of missing members of that table. Those members were later found, and they had the properties that DM had predicted for them.

With the discovery of electrons, nuclei, and quantum mechanics, and the development of quantum chemistry, these entities entered the fourth stage.

Atomic nuclei

Discovered in 1909, they quickly skipped through the first stage to reach the second stage, and they entered the third stage around 1920, when Ernest Rutherford proposed the existence of a "neutral proton". This particle was discovered in 1932, and named the neutron, bringing nuclei into the fourth stage.

Hadrons

The first hadrons discovered were protons and neutrons, around 1920 and 1932. I'm counting protons by Ernest Rutherford's recognition of them; a hint of them goes back about a century more to Prout's hypothesis. When they were the only two known strongly-interacting particles, hadrons remained stuck in the first stage. That began to change in 1947 with the discovery of the pion, and by the 1950's, physicists had discovered a big zoo of strongly-interacting particles, bringing hadrons into the second stage.

But regularities soon became evident, and in 1964, Murray Gell-Mann, George Zweig, and Yuval Ne'eman proposed the quark model. It had some oddities, like quarks in baryons being symmetric despite having spin 1/2, and the non-observation of free quarks. But by the early 1970's, physicists had discovered evidence that protons are composite, and that some of the "partons" in them have the properties expected of quarks. They also developed a theory of the quark's interactions, QCD. It put quarks in baryons into an antisymmetric color state, resolving the symmetry discrepancy.

With this and other evidence, like quark and gluon jets and the success of lattice QCD, hadrons entered the fourth stage.

Standard-Model particles

The first stage can go back a long way, depending on what one wants to count as first hints of the photon and electron. Visible light? Electrostatic effects? Magnetic effects? Electric shocks from electric fish?

A complication along the way is that for some decades, hadrons seemed as elementary as electrons, muons, neutrinos, and photons. In fact, in the 1960's, a "bootstrap model" of hadrons used to be popular, depicting them all as fully elementary. But the success of the quark model made it evident that hadrons were composite and not quite elementary, and that quarks and the gluon were instead elementary.

The term "Standard Model" was coined in the early 1980's or thereabouts, but some physicists were already speculating about Grand Unified Theories.

The unbroken Standard Model has some regularities in its particles' gauge-field quantum numbers, even if not in their masses, and one can find GUT's that fit all the Standard-Model particles into a small number of multiplets. So it's still in the third stage.
lpetrich
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#34
Dec19-12, 01:27 AM
P: 514
As to Standard-Model particles, there are some interesting regularities in their gauge interactions. I'll concern myself with the unbroken SM here, since that is what we must account for. Quantum numbers:

(QCD multiplicity, weak-isospin multiplicity, weak hypercharge)

The gauge particles are all in the adjoint representations of their gauge-symmetry groups, so that does not really tell us very much.
QCD: gluon: (8,1,0)
WIS: W: (1,3,0)
WHC: B: (1,1,0)

The Higgs particle is a single doublet: (1,2,-1/2)

The left-handed and right-handed elementary fermions:
Quark: (3,2,1/6) - (3*,2,-1/6)
Up: (3*,1,-2/3) - (3,1,2/3)
Down: (3*,1,1/3) - (3,1,-1/3)
Lepton: (1,2,-1/2) - (1,2,1/2)
Neutrino: (1,1,0) - (1,1,0)
Electron: (1,1,1) - (1,1,-1)

If you are starting to suspect some patterns, you are not alone. In fact, there is an interrelationship that I rediscovered; I don't know who originally discovered it.

Weak isospin works like 3D angular momentum, with overall quantum number WIS and multiplicity 2*WIS + 1. There is a "spin parity" that is conserved in rep products. Integer spins have parity 0, half-odd spins parity 1, and they add modulo 2.

QCD has a similar quantum number, "triality". It's more complicated to calculate, so I'll give its values for the reps mentioned here:
1 (scalar) -- 0
8 (adjoint) -- 0
3 (fundamental) -- 1
3* (fund. conjugate) -- 2
Trialities add modulo 3.

With QCD triality and WIS parity:
Quark: (1,1,1/6) - (2,1,-1/6)
Up: (2,0,-2/3) - (1,0,2/3)
Down: (2,0,1/3) - (1,0,-1/3)
Lepton: (0,1,-1/2) - (0,1,1/2)
Neutrino: (0,0,0) - (0,0,0)
Electron: (0,0,1) - (0,0,-1)

Now to the weak hypercharge. After some experimenting, one finds a simple formula:

WHC = (integer) + (1/2)*(WIS parity) - (1/3)*(QCD triality)

One can use (WIS) instead of (1/2)*(WIS parity), and it will work just as well.

One can get this formula from some GUT's, like Georgi-Glashow.
rodsika
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#35
Dec19-12, 05:28 PM
P: 275
OP, the reason it should be related is very simple.

EFE simply tells how the geometry must be curved by the influence of mass/stress/energy. It doesn't show how the mass is connected to geometry... which occurs in very small scale. So without knowing how mass is connected to geometry. It's just like believing in Magic like telling children about TV getting images and not explaining how.. but only that pushing channel and volume buttons can change the images (like EFE).
lpetrich
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#36
Dec19-12, 09:18 PM
P: 514
It isn't as magical as it might seem when one considers the Lagrangian:

L = R/(16*pi*GN) + L(nongravitational)

The first term is the Einstein-Hilbert term.

Doing a variation by the metric gives Einstein's field equation G = 8*pi*GN*T.


That aside, there isn't much that suggests a connection between gravity and everything else that's known, no readily-apparent pattern that the graviton fits into alongside some other elementary particles.


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