[info] Mathematician suggests extra dimensions are time-like

Wed Apr 18 09:14:07 UTC 2007

http://www.physorg.com/news96027669.html

The analytical structure underlying the spinorial theory can be represented
visually. The structure is a Xi-transform, which moves between the three
spaces in the directions given by the bendings of the upper case Greek letter
Xi. The distorted squares represent the wave operator. The product of a wave
operator and a Xi transform, taken in any order, is zero. Image credit: Erin
Sparling.

In a recent study, mathematician George Sparling of the University of
Pittsburgh examines a fundamental question pondered since the time of
Pythagoras, and still vexing scientists today: what is the nature of space
and time? After analyzing different perspectives, Sparling offers an
alternative idea: space-time may have six dimensions, with the extra two
being time-like.

Sparling’s paper, which was published in the Proceedings of the Royal Society
A, lays the groundwork for his theory. He explains how spatial dimensions
contain positive signs (e.g., Pythagoras’ 3D space is expressed as the sum of
the squares of the intervals in three directions, x, y, and z). Minkowski’s
time-like dimension, on the other hand, combines these three dimensions with
the square of time displacement, which contains an overall negative sign.

“In three dimensions, the formula reads s2 = x2 + y2 + z2,” Sparling
explained to PhysOrg.com. “Our standard spacetime has four dimensions, but
the formula has a critical minus sign: s2 = x2 + y2 + z2 - t2. The Lithuanian
Hermann Minkowski invented this idea, which was published just six weeks
before he died. Indeed, [Sir Roger] Penrose, for one, says that special
relativity was not a finished theory until Minkowski's famous Raum und Zeit
[‘Space and Time’] paper.”

Up until now, Sparling explains, most theories concerning extra dimensions
have dealt with space-like rather than time-like dimensions, which results in
a “hyperbolic” rather than an “ultra-hyperbolic” geometry. However, Sparling
notes that there are no a priori arguments for a hyperbolic geometry, and he
looks into the possibility of a “spinorial” theory of physics, where six
dimensions of space-time arise naturally.

“In general dimensions, we say that the space-time is hyperbolic if there is
only one minus sign in the formula for s2,” he said. “So, for example, in the
ten dimensions of superstring theory, there are nine spatial dimensions with
plus signs and one minus sign. Only in that situation is there a clear-cut
distinction between the future and the past.”

[Mathematician suggests extra dimensions are time-like] “In my case, I am led
to the conclusion that the ordinary four dimensional space-time extends
naturally into six dimensions: the four dimensional space is hyperbolic as
usual, but in the surrounding space there are equal numbers (3 each) of space
and time dimensions, so the formula for s2 reads something like s2 = x2 + y2
+ z2 - t2 - u2 - v2, where u and v represent the new time variables. I call
this structure a (3, 3)-structure (mathematicians call it ultra-hyperbolic).”

Space-Time is Spinorial

Sparling’s spinorial theory is based on Einstein’s general relativity and
Elie Cartan’s triality concept, which can link space-time with two twistor
spaces. Twistor spaces are mathematical spaces used to understand geometrical
objects in space-time landscapes. Sparling explains spinors in the following
way:

“In physics, the idea of a spinor stems from the finding that spectral lines
of atoms seem to behave as if the angular momentum of the particles radiating
photons was in half-integral units of the quantized spin (whose size is
determined by Planck's constant). This was fully explained by Dirac's famous
theory of the electron, which led him to successfully predict the existence
of the positron.”

Some spinorial particles include the electron, muon, tau, proton, neutron,
quarks, neutrinos, and all their anti-particles, which are called fermions
and have half-integer spins. There are also non-spinorial particles, called
bosons, such as the photon, graviton, pion, mesons, the W and Z bosons, the
Higgs, (if it exists) and so on, which have an integer spin, Sparling
explains.

“The key difference between spinors and non-spinors is their behavior under
rotations: typically, non-spinorial (integer-spin) particles return to their
initial value under a 360-degree (or 2π-radian) rotation; however, the
spinorial (half-integer-spin) fermions actually change sign under a
360-degree rotation, requiring a full 720-degree rotation to get back to
their initial values. This is completely foreign to our naive idea of how
rotations work, and yet it is a basic part of reality.

“Consider this analogy: if you take a plate and hold it in one hand
horizontally whilst twisting it under your arm backwards through 360 degrees,
your arm ends up in the air after one rotation, and it needs another 360
degree rotation to get it back to the beginning,” he said.

Twistors, then, are a special kind of spinor first introduced by Penrose
(Sparling was a PhD student of Penrose). In Sparling’s theory, the two
twistor spaces are each six-dimensional, forcing space-time to also have six
dimensions, in accordance with Cartan’s unifying triality. Because the
twistor spaces’ geometry is ultra-hyperbolic, the extra dimensions are
time-like.

“My work has three six-dimensional spaces which at one level are on an equal
footing and which are bound together by a new transform, which I call the
Xi-transform,” Sparling said. “Two of these spaces can be understood at the
space-time level as twisters. Then the third space can be given a space-time
interpretation, but only if we have two extra dimensions: so it is the
requirement of symmetry between the spinor spaces and the space-time that
dictates that the extra dimensions be there.”

A Harmonious Concinnity

While the concepts of twistor theory and spinors have been previously
investigated as an alternative to space-time, Sparling explains how his new
proposal is slightly different because it’s not a complete replacement of
space-time. Rather, the guiding principle of his idea is that of a harmonious
combination of three entities, or a “trinity.” Each part of the theory
reinforces the other parts.

“If one accepts that there are these three spaces [space-time and two twistor
spaces] that are central to my theory, one looks for a theory which unifies
them; this would be the ‘concinnity’,” he explained. “An indicator that there
might be such a theory comes from the theory of Jordan algebras, which
naturally unifies the three spaces into a twenty-seven dimensional whole,
called an exceptional Jordan algebra.” Sparling’s student Philip Tillman and
ex-students Dana Mihai, Devendra Kapadia and Suresh Maran also played a
significant role related to this work.

“A second indicator is that there are two radically different descriptions of
massless particles, such as the photon: the standard one uses Fourier
analysis in space-time and another uses twistor theory and sheaf cohomology,”
he added. “The mathematical formalisms used in these two different
descriptions are so different that it is simply amazing that they are
describing the same basic physics. The concinnity would provide an
explanation for this. This would then unify twistor theory, space-time theory
and string theory—this is very tentative, however.

“A very interesting aspect is that Newton fought strongly against the idea of
the trinity (in a religious context),” Sparling noted. “It is ironic that I
am invoking that very same idea in the context of gravity: perhaps Newton saw
that the concept could be used in physics, but because he could not think of
such a use he rebelled strongly against it (of course, I have no evidence for
this!).”

Although the theory is not definitive, Sparling explains that several major
ideas in current physics would likely play a role (such as condensed matter
physics, category theory, non-commutative geometry, string theory, and the
structure of superfluids). Such connections might also point the direction to
a unified theory, though currently speculative.

“My work can be seen as a strong antidote to the present air of pessimism
surrounding modern fundamental physics,” Sparling said. “As is well-known,
string theory has been roundly criticized for its lack of predictive power.
String theorists have been reduced to an absurd reliance on the anthropic
principle, for example. Here I have a clear-cut prediction, which goes
against the common wisdom, which gives experimenters a target to go for:
first find the extra dimensions, then decide their signature (a very tough
homework assignment!). Of course I could be proved wrong, but the effort to
decide is surely worthwhile.

“Actually, in the area of philosophy, I am in opposition to string theory,”
he said. “It is a top down theory: dream up something that works in some high
dimension and then try to finagle some way of reducing to fit in with the
lower-dimensional theory. My approach is bottom up: take the existing
four-dimensional theory seriously and try to build up from it. This is very
tough to do. Hopefully my ideas work. Note that my work only constitutes a
possible beginning at a more inclusive theory.”

Sparling continues to explore the ideas of this 6-D time-like spinorial
theory of space-time, with support from a workshop at the BIRS Institute in
Banff, Canada, and ideas from philosophers including Alexander Afriat, Steve
Awodey, Jonathan Bain and Rita Marija Malikonyte-Mockus. He predicts that
experimental investigations in the near future—such as the Large Hadron
Collider—might uncover the extra dimensions.

Citation: Sparling, George A. J. “Germ of a synthesis: space-time is
spinorial, extra dimensions are time-like.” Proc. R. Soc. A.
doi:10.1098/rspa.2007.1839.

Copyright 2007 PhysOrg.com.