Can Extra Dimensions Explain Mass Generation Beyond the Higgs Mechanism?

[arivero]

The Higgs mechanism allows to give masses to massless particles. In extra dimensions there is another mechanism

to understand the basic idea (see also a discussion by Palla), suppose that in a 4+n dimensional theory we have a massless spin one half fermion. It satisfies the 4+n dimensional Dirac equation
$$\kern+0.25em /\kern-0.80em D \psi =0$$
or explicitly
$$\sum_{i=1}^{4+n} \gamma^i D_i \psi=0$$
This Dirac operator can be written in the form
$$\kern+0.25em /\kern-0.80em D^{(4)} \psi +\kern+0.25em /\kern-0.80em D^{(int)}\psi =0$$
where
$$\kern+0.25em /\kern-0.80em D^{(4)} \equiv \sum_{i=1}^{4} \gamma^i D_i$$ is the ordinary four-dimensional Dirac operator, and $$\kern+0.25em /\kern-0.80em D^{(int)}\equiv \sum_{i=5}^{4+n} \gamma^i D_i$$ is the Dirac operator in the internal space of n compact dimensions.
The expression (above) immediately shows that the eigenvalue of $$\kern+0.25em /\kern-0.80em D^{(int)}$$ will be observed in practice as the four-dimensional mass.

Any hint about earlier uses of this idea?

 

[selfAdjoint]

I believe Lisa Randall uses this in he "Warped Passages" extra dimensional theory. She expects to see massive new particles at LHC that will be projections or "shadows" of particles moving in the higher dimensional bulk, and states that their massis will be projections of their higher simensional motion. The bulk in her theory has AdS geometry which may factor in too.

 

[arivero]

will be projections or "shadows" of particles moving in the higher dimensional bulk, and states that their massis will be projections of their higher simensional motion.

Hmm, should such particles run at lightspeed in the full manifold, but to appear slower in the 3+1 dimensional one?

If so, what about helicity?

 

[garrett]

It's a reasonable approach. I'm not sure where it first arose, but it seems very natural once you have Kaluza-Klein theory and the Dirac equation on the scene. The two main problems with it I know of are: the masses tend to be too big when the internal space is small, and you get an infinite spectrum (often called the Kaluza-Klein tower) of masses corresponding to the harmonics of the internal space. Also, a curious fact about the Dirac operator: it has no zero eigenvalues for a compact space.

Along with a third problem -- the difficulty of producing chiral symmetry breaking -- these facts lead to the demise of Kaluza-Klein in the 80's.

 

[bananan]

It's a reasonable approach. I'm not sure where it first arose, but it seems very natural once you have Kaluza-Klein theory and the Dirac equation on the scene. The two main problems with it I know of are: the masses tend to be too big when the internal space is small, and you get an infinite spectrum (often called the Kaluza-Klein tower) of masses corresponding to the harmonics of the internal space. Also, a curious fact about the Dirac operator: it has no zero eigenvalues for a compact space.

Along with a third problem -- the difficulty of producing chiral symmetry breaking -- these facts lead to the demise of Kaluza-Klein in the 80's.

Do these criticisms apply to string theory? I was of the understanding that string theory is a Kaluza-Klein theory

 

[selfAdjoint]

Hmm, should such particles run at lightspeed in the full manifold, but to appear slower in the 3+1 dimensional one?

If so, what about helicity?

Dunno. Read the book. Randall (and let's not forget her colleague, Sundrum) has the questions all worked out so I assume she has that one answered too. But I haven't read, or retained at a deep enough level to give you the answer.

 

[arivero]

the masses tend to be too big when the internal space is small, and you get an infinite spectrum (often called the Kaluza-Klein tower) of masses corresponding to the harmonics of the internal space.

I guess Randall-Sundrum do not have problems about the mass being to big if they expect extra dimensions at a TeV. But the issue of the tower of masses sound unphysical to me. Sure, there are harmonic in the compact dimensions, but in what way are we expected to observe them? At low energy? Because beyond the compactification scale, the compactification picture does not apply.

 

[arivero]

On second view the argument seems a bit falacious. Suppose we have an operator
$$A \equiv A_1 \otimes 1 + 1 \otimes A_2$$ acting on a Hilbert space $$H_1 \otimes H_2$$. Ok, I agree that if $$|v_1>$$ is an eigenvector of $$A_1$$ and $$|v_2>$$ is an eigenvector of $$A_2$$, then $$|v_1> \otimes |v_2>$$ is an eigenvector of $$A$$. But does it work in the inverse direction? Can any eigenvector of A to be decomposed in this way?

Moreover, when building the higher dimensional Dirac operator I believe to remember that one does a small twist on gamma5, ie one builds
$$\kern+0.25em /\kern-0.80em D^{(4)} \otimes 1 + (\gamma^1\gamma^2\gamma^3\gamma^4) \otimes \kern+0.25em /\kern-0.80em D^{(int)}$$

 

[CarlB]

On second view the argument seems a bit falacious. Suppose we have an operator
$$A \equiv A_1 \otimes 1 + 1 \otimes A_2$$ acting on a Hilbert space $$H_1 \otimes H_2$$. Ok, I agree that if $$|v_1>$$ is an eigenvector of $$A_1$$ and $$|v_2>$$ is an eigenvector of $$A_2$$, then $$|v_1> \otimes |v_2>$$ is an eigenvector of $$A$$. But does it work in the inverse direction? Can any eigenvector of A to be decomposed in this way?

I think that the basic problem here is that you are bringing in an unnecessary formalism. All that is needed to give the mass interaction is the usual physical assumption that one can write the wave function as a sum over products of eigenfunctions.

Before I got into Euclidean relativity and Clifford algebra, I messed around with this. The way it's done is strictly as a differential equation solution. By which I mean that the mass shows up in a completely classical manner.

 

[nnunn]

will be projections or "shadows" of particles moving in the higher dimensional bulk, and states that their massis will be projections of their higher dimensional motion.

Hmm, should such particles run at lightspeed in the full manifold, but to appear slower in the 3+1 dimensional one?

Paul Wesson elaborates on this in his 2006 "Five-Dimensional Physics"
-- Classical and Quantum Consequences of Kaluza-Klein Cosmology http://www.worldscibooks.com/physics/6029.html"

Wesson (p77, section 3.4)

The idea that massive particles on timelike geodesics in 4D are on null paths in 5D, is in fact quite feasible, ...

Nigel

 

[arivero]

Paul Wesson elaborates on this in his 2006 "Five-Dimensional Physics"
l

I will look for the book.

Indeed in momentum space it is trivial to see it: we have on one hand
$$E^2 - \sum_{i=1}^3 p_i^2 =m^2$$
and on the whole
$$E^2 - \sum_{i=1}^{3+n} p_i^2 =0$$
Thus we can set the (square of the) momentum in the internal coordinates equal to the (sq) mass we see in the external 3+1 world.

$$m^2=\sum_{i=4}^{3+n} p_i^2$$

The puzzling thing is that we haven't got full working Lorentz transformations anymore: if we want the mass to be a constant, as we know it happens in our world (it is the rest mass), then we need to ask separately for a invariance in the extra $$p_i$$ coordinates. We can not mix uncompactified and compactified momentum, and this excludes mixing with the time coordinate (uncompactified it is).

 

[arivero]

I think that the basic problem here is that you are bringing in an unnecessary formalism. All that is needed to give the mass interaction is the usual physical assumption that one can write the wave function as a sum over products of eigenfunctions.

Yeah, the problem is that I did not brough up the necessary formalism. really I have built a orthogonal basis and, seen this, then the result (for the straigh tensor product) follows trivially. You are perhaps right that to cope with infinite dimensional spaces is perhaps better to keep with the differential equation setup.

On the other hand, the point about the use of a "twisting" via chirality is better seen in the tensor decomposition.

 

[CarlB]

The puzzling thing is that we haven't got full working Lorentz transformations anymore: if we want the mass to be a constant, as we know it happens in our world (it is the rest mass), then we need to ask separately for a invariance in the extra $$p_i$$ coordinates. We can not mix uncompactified and compactified momentum, and this excludes mixing with the time coordinate (uncompactified it is).

Yes. From my point of view, this implies a preferred rest frame. The natural conclusion is that the invariance you're seeing in the non compact dimensions must be an accidental symmetry only. You can put the whole thing on a foundation that is fully consistent and simple, if you follow through with this. To get it to work, you really do have to chase through the foundations and modify everything. You'll end up deriving invariance as an approximation. It took me about a year to work it out from one end to the other, but I'm a slow old guy.

And by the way, momentum space is the way to do the problem.

 

[CarlB]

The idea that massive particles on timelike geodesics in 4D are on null paths in 5D, is in fact quite feasible, ... (book by Wessen)

Does Wessen reference Almeida? For example:
http://www.arxiv.org/abs/physics/0410035

There is so much stuff done along this line (especially in "Euclidean Relativity") that amounts to reinvention of the wheel. I noticed that Almeida doesn't refer to Wessen in the above.

 

[arivero]

There is so much stuff done along this line ... that amounts to reinvention of the wheel. I noticed that Almeida doesn't refer to Wessen in the above.

It seems that the "Klein-" past in Klein-Gordon equation is derived exactly (year 1926) in this way, as a wave equation without mass but in five dimensions. So the only new thing is to do it with the Dirac equation -obviously unavailable in 1926-, or with Rarita-Schwinger (??).

Colateral thought: usually it is said that the compactification scale gives us, or is equal to, the Planck mass, thus Newton Constant. But if you think that Newton Constant is a more fundamental element, then it can be said that the compatification scale gives you... the Planck constant! This is argued also by Klein in his 1926 paper, related to the ideas of De Broglie where the Planck constant is usually related to a kind of internal vibration of the particles.

 

[nnunn]

Does Wesson reference Almeida? For example: http://www.arxiv.org/abs/physics/0410035. There is so much stuff done along this line (especially in "Euclidean Relativity") that amounts to reinvention of the wheel. I noticed that Almeida doesn't refer to Wesson in the above.

In:http://www.arxiv.org/abs/physics/0601194, "Can physics laws be derived from monogenic functions?", Almeida quotes Wesson http://www.arxiv.org/abs/gr-qc/0507107:

The implication of this for particles is clear: they should travel on null 5D geodesics. This idea has recently been taken up in the literature, and has a considerable future. It means that what we perceive as massive particles in 4D are akin to photons in 5D.[3]

[3] P. S. Wesson, In defense of Campbell’s 2005, gr-qc/0507107.

Looks like Prof. Wesson has been working on this a while. I think he and his colleagues were responsible for that approach to the nature of matter that involves extending "spacetime" (ST) to "spacetimematter" (STM).

Nigel