I've always been fascinated by Einstein and Rosen's attempt to describe particles as tiny non-traversable wormholes ( The Particle Problem in the General Theory of Relativity ). Subsequently Wheeler also spent some time attempting to describe charged particles as electric flux trapped within such wormholes (“charge without charge” as he called it).
This is all in line with a viewpoint dating back (as far as I'm aware) to Clifford. In 1890 Clifford gave a talk from which today only the abstract survives:
I hold as fact
(1) That small portions of space are in fact of a nature analogous to little hills on a surface which is on the average flat; namely, that the ordinary laws of geometry are not valid in them.
(2) That this property of being curved or distorted is continually being passed on from one portion of space to another after the manner of a wave.
(3) That this variation of the curvature of space is what really happens in that phenomenon that we call the motion of matter, whether ponderable or etherial.
(4) That in the physical world nothing else takes place but this variation subject (possibly) to the law of continuity.
More recently I've been reading a paper by Gibbons and Hawking and for a compact spacetime $M$ we have a closed space-like boundary (one past one present/future) $ ∂ M$ . Such a space with k-wormholes has a topology:
$$\partial M = \#_{k} (S^2 \times S^{1})$$
Which we can choose to write in terms of cosets:
$$\partial M = \#_{k} ({SU(2)\over{U(1)}} \times U(1))$$
The group acting transitively here is $ G=SU( 2 ) × U( 1 )$ , with point stabilizer $H=U( 1 )$ . It follows that $G/H$ is a homogenous Model space (Klein geometry) for $ ∂ M $.
In this situation then we have a principle G-bundle over $ ∂ M$ with structure group $SU( 2 ) × U( 1 )$ and a reduction to an $H$ sub-bundle $U( 1 )$ (This can also be found from Thurston's Geometrization conjecture).
My understanding (and please correct me if I'm wrong) is that classically, Such a reduction of the structure group is equivalent to spontaneous symmetry breaking of $ G$ due to the presence of a $G/H=( SU( 2 ) × U( 1 ) )/ U( 1 )$ valued Higgs field* together with a $U( 1 )$ valued “matter/energy” field.
This situation should remind the reader of something...
A few more notes:
At this point we could try to proceed to formulate 3+1 General relativity in terms of tetrads and connections reflecting the above topology and consequent geometric structures (and construct a Lorentz cobordism). I've no doubt issues would arise regarding wormhole stability/collapse; however thus far we've only considered single “snapshots” in time.
From empirical data we know the spatial universe is expanding in time in a conformal manner, I imagine this would enlarge the above group somewhat.
I would make a guess here (shot in the dark really) that the cosmological scale parameter would play a critical role in the individual terms when written out. The hole thing (pun intended) is deeply reminiscent of Wheelers attempts at charge without charge and mass without mass in his “Geometrodynamics” (of which I'm obviously a huge fan).
I'm sure I'm not the first person to waltz down this particular path, can anyone comment on the above construction or has seen it (or something like it) elsewhere? At the very least, It seems fascinating that the very structures that had been posited to be particles by Einstein, also just happen to naturally carry an $SU( 2 ) × U( 1 )$ symmetry breaking to $ U( 1 )$ . I'd be interested to find research of a similar nature.
Finally, the above setup also reminds me of a topological quantum field theory in the spirit of Witten. I would deeply appreciate comments, criticisms, or simply direction toward similar research ideas. Thank you
*The decomposition $ {SU(2)\times U(1)\over U(1)}$ is not unique; however the various possibilities are discussed thoroughly here.
Q&A (4871)
