Freidel Girelli Livine: relativistic particles and DSR

[marcus]

http://arxiv.org/abs/hep-th/0701113
The Relativistic Particle: Dirac observables and Feynman propagator
Laurent Freidel, Florian Girelli, Etera R. Livine
14 pages

"We analyze the algebra of Dirac observables of the relativistic particle in four space-time dimensions. We show that the position observables become non-commutative and the commutation relations lead to a structure very similar to the non-commutative geometry of Deformed Special Relativity (DSR). In this framework, it appears natural to consider the 4d relativistic particle as a five dimensional massless particle. We study its quantization in terms of wave functions on the 5d light cone. We introduce the corresponding five-dimensional action principle and analyze how it reproduces the physics of the 4d relativistic particle. The formalism is naturally subject to divergences and we show that DSR arises as a natural regularization: the 5d light cone is regularized as the de Sitter space. We interpret the fifth coordinate as the particle's proper time while the fifth moment can be understood as the mass. Finally, we show how to formulate the Feynman propagator and the Feynman amplitudes of quantum field theory in this context in terms of Dirac observables. This provides new insights for the construction of observables and scattering amplitudes in DSR."

 

[CarlB]

The 5th dimension as proper time with particles massless in 5D giving rise to massive 4D particles? I've been trying to pedal this theory for 3.5 years now, but from an algebraic point of view:

The Proper Time Geometry
October 19, 2004
http://brannenworks.com/a_ptg.pdf

The Geometry of Fermions
December 11, 2004
http://brannenworks.com/a_fer.pdf

A Hidden Dimension, Cliord Algebra, and Centauro Events
July 25, 2005
A.) Proper Time as a Hidden Dimension
B.) Generalizing the Dirac Equation
C.) The Quantum Numbers of a Complexified Clifford Algebra
http://brannenworks.com/PHENO2005.pdf

This is great. They're working on the part of the problem I've been ignoring, the details of the continuous degrees of freedom position and momentum. I've been working on the details of the discrete degrees of freedom, that is, of what happens to spin when you add that hidden dimension. In their conclusion they write:

The case of the spinning particle deserves more attention. We have showed that the spin induces an extra non-commutativity of the space-time coordinates. This complicates the quantization of the algebra of observables and a full analysis of the 5d representation of the spinning particle is still under investigation.

Carl

 

[ccdantas]

What? They are going to higher dimensions?

Christine

 

[marcus]

What? They are going to higher dimensions?

Christine

don't worry Christine, it's all right

the key to this is Derek Wise's paper that we discussed earlier
and before that, the January 2005 paper of Freidel and Starodubtsev but you don't have to go back to that, everything you need is in Derek Wise paper with the hamster

going up to 5 dimensions is a trick to accommodate 4D deSitter spacetime, and DSR, and ultimately a curved momentum-space

also it is how the MacDow-Mansouri version of 4D Gen Rel works, M&M go to five dimensions to get "elbow room". but they are still talking a 4D reality.

please don't worry Christine, it really is just fine!
I promise

Willem de Sitter (1917) and Elie Cartan (1923) always wanted us to do this and they are smiling and nodding their heads when they see this.

By the way, how do you say the name of the prophet Elijah in Portuguese? In some language it is Elias, in other Elie, and in English it is Elijah.

 

[f-h]

Unless I'm mistaken (haven't looked at this yet), we're always on a 4-Manifold, the point is merely to replace the tangent bundle by a bundle with a 5D fibre to accommodate certain symmetry transformations in a geometric language, right?

 

[marcus]

I think maybe one can say that physicists usually do their field theory, their particles, their QFT on the tangent space of the real continuum.

the real continuum is bumpy and has no symmetry, but its tangentspace which is a local approximation has some symmetry

Freidel perhaps suspects that the real continuum has a Cartan geometry with deSitter spacetime as its tangentspace.
that is, a curved tangentspace that embodies accelerated expansion due to a cosmoconstant

Therefore it becomes necessary to rebuild all of QFT on deSitter instead of on flat Minko. The Minko spacetime not only does it not have accelerated expansion, it does not have any expansion at all! So it is a very flat stupid tangentspace, and usual QFT will always be wrong and incompatible with the real continuum as long as QFT lives on flat stinko Minko.

It takes a lot of guts to start to rebuild QFT on different realestate. but we know that Freidel is a kind of driven person who takes on hazardous projects.
I think this is part of it

 

[marcus]

Unless I'm mistaken (haven't looked at this yet), we're always on a 4-Manifold, the point is merely to replace the tangent bundle by a bundle with a 5D fibre to accommodate certain symmetry transformations in a geometric language, right?

I think you are right. "Certain symmetry" is the deSitter group.

 

[CarlB]

marcus,

Most of this stuff is over my head, and I suspect you're more familiar with it than any of us (no matter what you say). Is this the first paper you've seen concluding a 5th dimension is proper time?

They seem to have reached this from a quite complicted path. To me, it's the obvious way of making spacetime into something where QM is fully compatible with relativity.

During the development of QM, a series of little mysteries befuddled the original researchers, and then were ignored by their followers. When de Broglie first postulated "matter waves" he had the problem that their phase velocity always exceeded c. The issue is covered in Messiah's Quantum Mechanics, see Chapter II, section 3, which was written 1958, but is not mentioned much in later QM texts.

When you put in a small cyclic dimension for proper time, the result is that matter waves travel at speed c in 5 dimensions, it is only the projection down to 4 dimensions that exceeds c. The effect is similar to the fact that when waves roll in from the ocean, the speed at which they break along the beach always moves up or down the beach faster than the waves went on the ocean. I wrote this very simple and intuitive effect up here:
http://brannenworks.com/a_phase.pdf

Getting back to the beach analogy, the difficulty in analyzing matter waves in 4D arises from the fact that their speeds differ depending on the speed of the particle they represent. When you pass to the 5D case, the speeds of all your matter waves become identical, i.e. c, and they are much easier to deal with for distances small compared to the distance you associate with the proper time. I'm guessing that it is the circumference of the hidden proper time dimension that is the length scale (on the order of the Planck length) that is the preserved length in DSR.

And physics is very tightly interconnected. Once you change one of the foundation stones of physics, you produce cracks throughout the structure. Eventually you have to replace all the foundation, leaving not one stone standing on another. This will not be the last nor the most amazing paper that these guys have to write.

Putting proper time as a hidden dimension is a way of digging up some of the bodies buried under the foundations of physics and giving them a proper burial. As the very many other bodies get dug up, more little mysteries will be put in place.

Carl

 

[ccdantas]

please don't worry Christine, it really is just fine!
I promise

(...)

By the way, how do you say the name of the prophet Elijah in Portuguese? In some language it is Elias, in other Elie, and in English it is Elijah.

I'm not worried! Just wondering... eh eh
I'll put Derek's paper over my pile of "papers-to-read" -- from a first glance (and following up some discussions here recently), his paper did look clearly written and giving rise to an interesting study of Cartan geometry as "preparing the terrain" (as we say it here) for quantum gravity.

And Freidel et al. developments are worthy to follow!

BTW I'm not that "scared" about higher-dimensional theories, specially in the case here, when physical spacetime is still 4D anyway. Classical mechanics uses higher dimensional spaces all the time. And if nature turns out indeed to be higher-D, that is the way it is!

The prophet is called "Elias" in Portuguese.

Cheers
Christine

 

[marcus]

marcus,

Most of this stuff is over my head, and I suspect ...

Carl, I wrote you PM about this.

 

[Hans de Vries]

During the development of QM, a series of little mysteries befuddled the original researchers, and then were ignored by their followers. When de Broglie first postulated "matter waves" he had the problem that their phase velocity always exceeded c. The issue is covered in Messiah's Quantum Mechanics, see Chapter II, section 3, which was written 1958, but is not mentioned much in later QM texts.

Carl,

The de Broglie waves are solutions of the Klein Gordon equation. One can
draw an equivalent classical spring-mass system in the simple case of
the 1d real Klein Gordon equation which exhibits exactly this behavior of
a phase velocity which always exceeds c:

http://www.chip-architect.com/physics/Klein_Gordon.jpg

(The more complex version can also be implemented by spring-mass systems
but they get rather awkward.) The horizontal springs just represent the
classical wave equation here while the vertical springs implement the mass
term of the Klein-Gordon equation. A particle at rest corresponds with the
situation where all the masses are on a horizontal line moving up and down.
In this case the phase speed is infinite.My impression from the abstract of the paper is that they start with the
4d Klein Gordon propagator:

$$\frac{1}{E^2-p_x^2-p_y^2-p_z^2-m^2}$$

and then simply interpret the m^2 term as the "fifth dimension".

We interpret the fifth coordinate as the particle's proper time while the fifth moment can be understood as the mass.

The (rest) mass in momentum space leads quite logically to the proper
time in configuration space.

Well, This idea probably went through the head of a lot of different people.
I know I played with it, but, I'm not really that optimistic that one is doing
physics instead of just math. They make connections with DSR and the Sitter
space, I can't judge about that. As an exercise it might be quite interesting.Regards, Hans

 

[MeJennifer]

The Proper Time Geometry
October 19, 2004
http://brannenworks.com/a_ptg.pdf

Looks like the Lorentz ether theory with another name to me.

What am I missing?

So it is a very flat stupid tangentspace, and usual QFT will always be wrong and incompatible with the real continuum as long as QFT lives on flat stinko Minko.

Flat stinko Minko?

 

[CarlB]

MeJennifer,

The Lorentz ether theory is just special relativity plus a preferred reference frame, as far as I know. I assume that proper time is an actual hidden dimension, with a diameter on order of the Planck length (which provides the length scale that deformed special relativity preserves). One of the effects is that the Dirac equation gets another canonical basis vectors which bumps up the structure of the quantum states from a square to a cube. See
http://brannenworks.com/a_fer.pdf
for the details on how it is possible to count N hidden dimensions by assuming that the elementary particles are primitive idempotents of a Clifford algebra with 4+N canonical basis vectors. This is based on density operator theory.

A few of the people working on this believe that bringing proper time as a hidden dimension implies a preferred reference frame, and hence LET when you integrate out the hidden dimension. Which reminds me, when you intergrate out the hidden dimension, you end up with a U(1) symmetry that can be interpreted as changing the real Clifford algebra into a complex one.

Minko means Minkowski space.

This thing is fairly complicated and requires knowing a bunch of obscure stuff so it's hard to explain the interconnecting theories. What you have to do is come up with a reason it can't work, and then ask, because we've been working on this for 3 years and we've likely already thought out that problem and have a solution. The number of problems is about 30, so they're easy to find, but not easy to discuss in a short post. The standard way of doing physics is complicated, any new way of doing it (even if it makes physics very simple and intuitive) is going to be related to the standard way in a complicated fashion.

The connection to the usual GR is clear through the DSR paper. From the Clifford algebra side, there is a classic paper by Lasenby Gull and Doran showing that GR can be exactly accomplished on flat space (whch gets back to the comments about flat Minkowski space):

Gravity, gauge theories and geometric algebra
http://www.mrao.cam.ac.uk/~cjld1/pages/publications.htm

Now the above is a link to a web page that gives a bunch of other papers. Another set that are of interest to this discussion are the ones on "density operators". These are germane because the way one counts the number of hidden dimensions is by assuming that the particles are pure states of the density operator formalism instead of the spinor formalism.

Where my work differs from that of the Cambridge geometry group is that I assume a preon model. The above density operator links didn't come to my attention until fairly recently, and I haven't put references to it into my book, but it is also germane that the title of the book is "Operator Guide to the Standard Model":
http://www.brannenworks.com/dmaa.pdf

And the operator formalism, combined with the preon assumption, is a very powerful way of understanding the elementary particles. That's how I figured out how to rewrite the Koide mass formula as an eigenvalue equation, and how to rewrite it so that it works for the neutrinos. That neutrino mass formula is in the latest neutrino review paper by Smirnov and Mohapatra:
http://arjournals.annualreviews.org/doi/abs/10.1146/annurev.nucl.56.080805.140534?journalCode=nucl
and is referenced by Koide in several (not yet published) papers, most particularly:
http://www.arxiv.org/abs/hep-ph/0605074

So what you have here is the tying down of the other end of the Koide neutrino mass formula calculation.

 

[marcus]

Every quarter (every three months) I put together a list of what might be, or what "Beyond" forum people have indicated they think might be, the most important papers--most valuable to future research--that appeared that quarter.

Your suggestions are invited.

This one of Freidel Girelli Livine (FGL) that this thread is about is an obvious nominee.

What others come to mind?
===================

here are some past "most influential paper" listings and discussion
first quarter 2006
https://www.physicsforums.com/showthread.php?t=116791
second quarter 2006
https://www.physicsforums.com/showthread.php?t=124951
third quarter 2006
https://www.physicsforums.com/showthread.php?t=134513
fourth quarter 2006
https://www.physicsforums.com/showthread.php?t=149466
summarizing results
https://www.physicsforums.com/showthread.php?p=1193842#post1193842

==================

what seems at the moment like a good list for first quarter 2007 is

Freidel Girelli Livine
http://arxiv.org/abs/hep-th/0701113
The Relativistic Particle: Dirac observables and Feynman propagator

Sabine Hossenfelder
http://arxiv.org/abs/hep-th/0702016
Multi-Particle States in Deformed Special Relativity

Abhay Ashtekar
http://arxiv.org/abs/gr-qc/0702030
An Introduction to Loop Quantum Gravity Through Cosmology

E.E. Ita
http://arxiv.org/abs/gr-qc/0703052
Existence of generalized semiclassical Kodama states. I. The Ashtekar--Klein--Gordon model
http://arxiv.org/abs/gr-qc/0703056
Existence of generalized quantum Kodama states. II. The minisuperspace Ashtekar--Klein--Gordon model
http://arxiv.org/abs/gr-qc/0703057
Existence of generalized Kodama quantum states. III. A new approach to finite, full quantum gravity

Martin Bojowald
http://arxiv.org/abs/gr-qc/0702144
Singularities and Quantum Gravity