Even in an election year, physics marches on. Physics is forever.
In this case it’s a fun little paper by Heywood Tam (a grad student here at Caltech) and me, arXiv:0802.0521:
We propose a new way to hide large extra dimensions without invoking branes, based on Lorentz-violating tensor fields with expectation values along the extra directions. We investigate the case of a single vector “aether” field on a compact circle. In such a background, interactions of other fields with the aether can lead to modified dispersion relations, increasing the mass of the Kaluza-Klein excitations. The mass scale characterizing each Kaluza-Klein tower can be chosen independently for each species of scalar, fermion, or gauge boson. No small-scale deviations from the inverse square law for gravity are predicted, although light graviton modes may exist.
This harkens back to the idea of a vector field that violates Lorentz invariance (which Ted Jacobson and friends have dubbed “aether,” appropriately enough), and in particular a vector that picks out a preferred direction in space. I explored this possibility last year in a paper with Lotty Ackerman and Mark Wise, and Mark recently wrote a followup with Tim Dulaney and Moira Gresham. (Our paper was detailed in the “anatomy of a paper” series, 1 2 3.)
There is an obvious problem with the notion of a vector field that violates rotational invariance by picking out a preferred direction through all of space — we don’t see any evidence for it! Physics as we have thus far experienced it seems pretty darn rotationally invariant. In my paper with Lotty and Mark, we sidestepped this issue by imagining that the vector was important in the early universe, and subsequently decayed away.
But there’s another way to sidestep the issue, pretty obvious in retrospect: have the vector point in a direction we don’t see! Extra dimensions are of course a popular theoretical construct, and once you make that leap you can ask what would happen if an unseen extra dimension contained a constant vector field. That would leave good old four-dimensional Lorentz invariance completely unbothered, so it’s not immediately constrained by any well-known experimental bounds.
So, beyond being fun and not ruled out, is it good for anything? The answer is: quite possibly. Heywood and I calculated what the influence of such a vector would be on other fields that propagated in a single extra dimension. In good old-fashioned Kaluza-Klein theory, momentum in the extra dimension can only take on discrete values (it’s quantized, in other words), and each kind of field breaks into an infinite “tower” of particles of different masses. The separation between different mass levels is just the inverse of the size of the extra dimension in natural units. What’s that? You insist upon seeing the equation? Okay, if the original mass of the field is m and the size of the extra dimension is R, we have a series of masses indexed by n:
Here, 
is Planck’s constant, c is the speed of light, and n is just a whole number that can be anything from 0 to infinity. So the effect of the compact fifth dimension is to give us an infinite set of four-dimensional particles, indexed by n, each with a different mass. Not a very big mass, unless the extra dimension is pretty small; separating the levels by about 1 electron volt requires a dimension that is about 1 micrometer across. We would certainly have noticed all those new particles unless the extra dimensions were considerably smaller than a Fermi (10-15 meters).
The interesting thing that Heywood and I discovered is that the effect of an aether field pointing in the extra dimension is to boost all of the mass levels in the Kaluza-Klein tower. There is a new set of coupling constants, αi for every kind of particle i, that tells us how strongly that particle interacts with the aether. The mass formula is modified to read
So if αi is huge, you could have a huge mass splitting even with an extra dimension that was pretty large. This gives a new way to hide extra dimensions — not just make them invisibly small (the old-school Kaluza-Klein method) or confine us to some thin brane (the new-school ’90s style), but to boost the effective masses associated with momentum in the new direction. And there is an obvious experimental test, if you were to find all of these new particles: unlike plain vanilla compactification, where the towers associated with each kind of field have the same mass splittings, here the splittings could be completely different for every kind of particle, just by choosing different αi‘s.
To be fair, this idea does not by itself suggest any reason why the extra dimensions should be large. To allow for a millimeter-sized dimension, the coupling αi has to be at least 1015, which any particle physicist will tell you is an unnaturally big number. But the aether at least allows for the possibility, which I think is worth exploring. Who knows, some clever young graduate student out there might figure out how to use this idea to solve the hierarchy problem and the cosmological constant problem, then we would discover aetherized extra dimensions at the LHC, and everyone would become famous.
Caltech? Not UW-Madison? You sure of that?
Oh. These genetics posts did seem kind of weird. I thought it was just me–turns out that you weren’t really you.
http://www.genetics.wisc.edu/faculty/profile.php?id=98
I hate this crap. When we start getting into scientists saying “I believe…” we have a problem. This is all great fun math gymnastics, but means nothing practicality wise as it is based on amazing calculations not observation, evidence, or anything resembling proof.
http://www.gdargaud.net/Humor/Pics/string_theory.png
Scientists used to look at something in nature that we had not understood, like light for instance. They would come up with ideas about how it would work, then go to observation to confirm and calculations for exactness. Now we go from calculations to theories and back again, veering so far from observable reality and the other steps, primarily confirmation before continuing, the result is nonsense. I know all of you mean well (I’m not saying all of these people are wrong) it’s just the process of coming up with raw theory is not science.
Maybe this is a way of deriving the axion particle. A compactification [itex]nhbar/Rc[/itex] for R large might create just the right particle mass for the CP-preserving field with QCD. In E_8 the two exceptional groups F_4 and G_2 or
$latex
e_8~=~f_4~+~g_2~+~26times 7
$
where f_4 contains the so(3,1) for gravity and the so(4) for weak interactions with so(4) = su(2) + su(2) for the left and right handed parts. CP violation leaves the left su(2), but we have no analogous handedness breaking in QCD. Then to balance everything out maybe large scale compactification in an S-duality sense leaves generates a tiny mass field —- the axion, and there is a dual field which compactifies on a tiny scale ~ near L_p. This would “balance” the role of CP violation in the gravitation plus weak F_4 sector with the hypercharge plus QCD G_2 sector.
Lawrence B. Crowell
Sean: “I would give better than 50-50 odds on the existence of extra dimensions…”
I would probably take up that bet if it was possible for it to be settled in my favour! Why do you have such great confidence in the existence of extra dimensions?
Spayced,
You do realize that most scientists do exactly what you want them to do? And in fact, so do most physicists?
Rhys, I don’t think that “better than 50-50 odds” translates easily into “such great confidence.” But I think that some version of string theory is probably correct, and string theory requires extra dimensions.
Regarding extra dimensions. the existing paradigm in fundamental physics is that things becomes simpler, more symmetric and more unique at short distances, and complexity arises in the process of symmetry breaking. If this continues we expect new degrees of freedom at shorter distances and new symmetries to relate them. Extra dimension is just a way of organizing those new degrees of freedom (as harmonics in some space) which works for a large class of models. The new symmetries are then rotations in those extra dimensions. Having new degrees of freedom without new symmetries I find less motivated, but it is a matter of taste.
Why am I suddenly getting a Hunter S. Thompson voice in my head?
“There is nothing in the world more helpless and compactified than a man in the depths of an aether binge. . . .”
OK, what about “quintessence”? Any way to work that in, how does it relate to dark energy in general etc? I get the impression, Q isn’t just a mere synonym for DE in general. I wish I’d thought to ask earlier, but anyone LMK if have good scoop.
This compactification scheme predicts the existence of light particles. If this is an unknown particle then maybe the axion is a candidate. Tight compactification, R very small in the above formula, would correspond to a massive particle — maybe dark matter.
Quintessence and dark energy may involve a phase of the vacuum state, or at least this is what I suspect. In fact dark matter might be local changes in this phase. So called phantom energy might also be such a phase, though its amplitude or probability in the universe I think is near zero.
Lawrence B. Crowell
Re: Blake Stacey, #33:
More Pynchon than Thompson, though your joke was nostalgic…:
“Against the Day”,
Page 566
In a dream…
This passage, describing Kit’s dream of Umeki and the message it conveys, pulls together many of the main themes of Against the Day, tying things together in a way that Pynchon seldom does, almost as if he’s providing a rather large piece of the puzzle to help the reader understand the novel:
“Deep among the equations describing the behavor of light, field equations, Vector and Quaternion equations, lies a set of directions, an intinerary, a map to a hidden space. Double refraction appears again and again as a key element, permitting a view into a Creation set just to the side of this one, so close as to overlap, where the membrane between the worlds, in many places, has become too frail, too permeable, for safety…. Within the mirror, with the scalar term, within the daylit and obvious and taken-for-granted has always lain, as if in wait, the dark intinerary, the corrupted pilgrim’s guide, the nameless Station before the first, in the lightless uncreated, where salvation does not yet exist.”
See references in Against the day also at:
Aether
55; 58; 132-133; 140; 306; 320; “sounds like light” 426; 458; 557; 565-66; 595; aka Akasa, 613; 620; aka Luminiferous aether ; Wikipedia article on Luminferous aether; 1911 Encyclopedia Britannica
Q-weapon
“Quaternion-ray weapons” 445; “intelligence of a Quaternionic Weapon, a means to unloose upon the world energies hitherto unimagined — hidden … ‘innocently,’ inside the w term.” 542; “Unfamiliar with the Tesla system and alarmed by the strengths of the electric and magnetic fields, de Decker’s people naturally conflated this with those recent rumors of a Quaternion weapon…” 549; “as if this mysterious Q-weapon were a common firearm and he hoping the seller would allow him a few courtesy shots [emphasis added]” 557; “alive in Woevre’s hands” 563; “Kit found himself alone with the enigmatic object, back inside the leather case. He slung it nonchalantly by its strap over one shoulder” 564; technical details, 564-565; 784;
Quaternions
130; In mathematics, quaternions are a non-commutative extension of complex numbers. They were first described by the Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. At first, quaternions were regarded as pathological, because they disobeyed the commutative law ab = ba. Although they have been superseded in most applications by vectors, they still find uses in both theoretical and applied mathematics, in particular for calculations involving three-dimensional rotations. James Clerk Maxwell first published his famous theory describing electricity and magnetism as a set of twenty equations, but he was later able to reformulate it as four equations using quaternions. Heaviside translated these into four vector equations, the form typically taught in basic physics classes today. Heaviside’s vector version is compatible with Einstein’s special relativity, but the quaternion form is not; 131; 156; 511; 525; Wars, 526, 548; 533-34; 538-39; Quaternionic Weapon, 542; 557; 564; 590; Wikipedia entry; Quaternions at MathWorld; Hamiltonian quaternions at PlanetMath; Finding the site of Hamilton’s inspiration (by mathematical physicist John Baez); Conspiracy-theory takes on mathematical history [1] [2] (by Tom Bearden, promotor of a dubious free energy machine); Primer on quaternions and their history
vacuum
60;
Vector
156; In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. More formally, a vector space is a set on which two operations, called (vector) addition and (scalar) multiplication, are defined and satisfy certain natural axioms. Vector spaces are the basic objects of study in linear algebra, and are used throughout mathematics, science, and engineering. 158; 165; vectorists, 534; and Monotony, 560; “in five dimensions” 675
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