In short, Wilczek proposed a model consisting of attractively coupled particles on an Aharonov-Bohm ring, and found a solution to the corresponding Schrödinger which (for non-zero Aharonov-Bohm flux) consists of a rotating lump, thus breaking symmetry with respect to time translation.

However, Wilczek did not bother to prove that the solution he has found is really the ground state, i.e., that there exist no other states of lower energy.

Furthermore, one can show that, if correct, Wilczek result would lead to several paradoxical (unphysical) consequences. In particular, the classical behavior would not be recovered in the limit of infinitely large interactions, as it should, and the system would be able to radiate energy while being in its ground state, thereby violating the principle of energy conservation.

The true ground state can be found, is perfectly stationary, and does not lead to any unphysical paradox; this is discussed in detail in the following Comment: http://arxiv.org/abs/1210.4128

Patrick BRUNO
ESRF

So, the symmetries of a spacetime cylinder are the same as the symmetries of an ordinary cylinder in space.

I once read an incredibly detailed book on crystallography, which classified the things like ‘space groups’ for cylinders. These are important for structures like nanotubes:

So, this math is already out there, waiting for people who want to build spacetime crystals using rings of atoms.

I wonder if a similar thing happens with spacetime-groups.

Marco

I might as well take this excuse to quote Wikipedia about ‘space groups’:

In n dimensions, an affine space group, or Bieberbach group, is a discrete subgroup of isometries of n-dimensional Euclidean space with a compact fundamental domain. Bieberbach (1911, 1912) proved that the subgroup of translations of any such group contains n linearly independent translations, and is a free abelian subgroup of finite index, and is also the unique maximal normal abelian subgroup. He also showed that in any dimension n there are only a finite number of possibilities for the isomorphism class of the underlying group of a space group, and moreover the action of the group on Euclidean space is unique up to conjugation by affine transformations. This answers part of Hilbert’s 18th problem. Zassenhaus (1948) showed that conversely any group that is the extension of Zⁿ by a finite group acting faithfully is an affine space group. Combining these results shows that classifying space groups in n dimensions up to conjugation by affine transformations is essentially the same as classifying isomorphism classes for groups that are extensions of Zⁿ by a finite group acting faithfully.

It is essential in Bieberbach’s theorems to assume that the group acts as isometries; the theorems do not generalize to discrete cocompact groups of affine transformations of Euclidean space. A counter-example is given by the 3-dimensional Heisenberg group of the integers acting by translations on the Heisenberg group of the reals, identified with 3-dimensional Euclidean space. This is a discrete cocompact group of affine transformations of space, but does not contain a subgroup Z³

Since space groups are subgroups of the isometries of Euclidean space it may seem odd to classify them up to conjugation by affine transformations (which aren’t necessarily isometries). The reason is that if we have something like a triclinic crystal:

we don’t want to say its space group changes whenever we change the angles slightly… though of course it does change, by getting suddenly bigger, when some of these angles take the value 90°. Then we say the crystal isn’t triclinic anymore: it belongs to some other, more symmetrical crystal system.

Organic chemistry is topologically 2-D, less an epsilon. Molecules’ skeletal Schlegel diagrams are planar non-crossing in about 36 million cases. So-called “K_5″ molecules, about a dozen catalogued, are more interesting. Excluded from that catalog, for reasons of noisome cleverness, are [6.6]chiralane and its analogues,

http://www.mazepath.com/uncleal/schwartz.png
Terahedral core, octahedral periphery, point group T (not T_h or T_d).
http://www.mazepath.com/uncleal/schwart3.png
The five central carbon atoms are inescapably chiral, but are utterly unnamable as to hand, even in principle.
http://www.berezin.com/3d/3dprism.htm

For stereogram easy viewing.

The same strategy allows creation of planar carbon atoms (e.g., sp^2 hybridized in acyclic olefins) that are chiral centers (no S_n improper symmetry axes) in 3-D. Neither the ACS nor IUPAC were amused. Party poopers.

But there are basic questions about higher-dimensional chemistry that need to be settled before we start having have fun. For example: in 4 dimensions, instead of potential, electrostatic forces have a potential

Classically a force with a potential has strange properties, first discovered by Newton.

But for chemistry we need quantum mechanics! It turns out that if the attractive force between charged particles blows up sufficiently intensely as they get close, atoms are quantum-mechanically unstable, so chemistry doesn’t exist.

Mathematically, the question is this: is the operator

self-adjoint in 4 dimensions? I used to know this stuff, but I forget now, and the relevant book is still packed away—I just got back from Singapore. I remember that in 3 dimensions,
the operator

is self-adjoint for , while for it’s only self-adjoint when the number is smaller than (or equal to?) some constant. Chemistry in 3 dimensions works fine because it uses .

I taught a course about this once and you can see proofs of some of these facts here:

• John Baez, Quantum Theory and Analysis, around page 55.

The beautifully delicate borderline case can be found in the book I have packed away somewhere:

• Reed and Simon, Methods of Modern Mathematical Physics, Vol. 2: Fourier Analysis, Self-Adjointness, Academic Press, 1978

But the exact power at which

ceases to be self-adjoint depends strongly on the dimension of space, and I’m forgetting the exact formula. The answer will settle whether chemistry based on electrostatic attraction of oppositely charged particles can exist in 4d space.