I Is Connes' Model of the Higgs Field a Wormhole?

[arivero]

As this week the great topic is the Einstein-Rosen bridge and duals of it, I wonder what happened with the interpretation of Connes NCG model as two sheets of spacetime whose separation is measured by the higgs field. How is it different of a ER bridge?

 

[mitchell porter]

whose separation is measured by the higgs field

I didn't know that part, could you give a reference?

 

[arivero]

It was a sort of lore until the model become unified in a single algebra instead of having one in each sheet. But I see it is still mentioned as late as in 2010: https://arxiv.org/abs/1004.0464 for example (stress boldface mine)

"he reduction from the natural symmetry group SU(2)xSU(2)xSU(4) to U(1)xSU(2)xSU(3) is a consequence of the hypothesis that the two layers of space-time are finite distance apart but is non-dynamical"

Surely the intuition for the initial Connes-Lott models come from noticing that the mass terms for fermions connects the L and R sides, and then questioning if both sides could be living in different sheets of the universe. But I do not remember any connection with Rosen and Einstein paper.

 

[arivero]

Some links (will extract quotes later)

https://celebratio.org/Connes_A/article/842/
https://arxiv.org/abs/hep-th/0512169v3
https://lipn.univ-paris13.fr/~banderier/Seminaires/Slides/Martinetti_02_2011_1.pdf and https://www.newton.ac.uk/files/seminar/20060907163017001-150166.pdf
https://math.ucr.edu/home/baez/week83.html

 

[mitchell porter]

Since Peter Woit usually disallows my comments, I'll post here what I posted on his blog, regarding the claim of a wormhole in a quantum computer:

"The real issue here seems to be the nature of the wormhole/entanglement continuum alleged by ER=EPR. The original ER=EPR paper said that all entanglement is in some sense a wormhole. Now apparently the idea here is to mimic certain properties of a (theoretical) *traversable* wormhole, but in as minimal a way as possible. As I understand it, there was definitely physical entanglement created, not just simulation of it. But is that enough to say there was a wormhole? And if it was enough, doesn’t that mean that wormholes are created by many far more mundane acts of entanglement? For example, is a traversable wormhole created whenever quantum teleportation is carried out? Hopefully the paper addresses questions like these…"

 

[mitchell porter]

Some further thoughts on ER=EPR, and @arivero's original question...

There is a 2013 paper, "The holographic dual of an EPR pair has a wormhole". I am going to give a very rough description of how it may work. I can't vouch for these details at all, but this is what I picked up...

We are in AdS/CFT. The CFT is a gauge theory. We are looking at a particle and an antiparticle that are entangled, and which are coupled to the gauge field, and which are radiating energy.

In AdS, this is dual to a string shaped like one of those Shinto shrine gateways to nowhere... ⛩... but not with the double bar at the top. I just mean it has two vertical segments and a horizontal segment. The horizontal segment is a meson-like connection between particle and antiparticle. The vertical segments correspond to the energy radiating away.

If you look at the metric on this string, you find that the segments are causally isolated from each other. Something that happens on one segment, cannot propagate onto the other segments. Each corner of the string (the endpoints of the horizontal part) are therefore similar to event horizons within the 1-dimensional space of the string, and the horizontal part is the 1-dimensional analogue of a non-traversable wormhole.

I emphasize that I may have small or even crucial details wrong! But this is the impression I got so far - that for a single Bell pair, this is how you can get a "wormhole" - if there is a "double horizon" metric on the dual string, then the segment between the horizons is the "wormhole".

Another interesting paper was mention on Matt Strassler's blog, "Holography and Compactification". The idea here is that the extra dimensions of string theory, which are normally positively curved, may have slices of AdS in some places.

The lessons I draw from these two papers - at least if one is working in the context of string theory - are (1) the geometric dual of a general entangled state may have some of these AdS slices built in (2) if you want to know whether there is a wormhole structure, look at the metric, both the background space and on the strings and branes.

Now - back to the Connes model of the Higgs field... I will say, first of all, that obtaining the Higgs from a gap between parallel branes, is a known approach within string theory. I think some of the very first braneworld models, by the Spanish school (Quevedo et al) in the early 2000s, work this way. A stack of M+N coincident branes implements a SU(M+N) gauge field, it is higgsed to SU(M) x SU(N) by separating this into a stack of M branes and a stack of N branes, the distance between the parallel stacks is the Higgs vev, and a Higgs boson is an open string stretching between the two parallel stacks.

So if we mapped the Connes approach onto this template, and in the light of lesson (2) above... we might say: For a Higgs boson to be a (non-traversable) wormhole, it's not enough for it to just be a string connecting two parallel sheets of space; the metric along the string needs to have that "double horizon" structure.

For now, that's the closest I can come, to giving a reasoned answer. :-)