r/math • u/Necessary-Wolf-193 • 7d ago

Non-commutative geometry and quantum mechanics

https://hidden-phenomena.com/articles/quantum
85 Upvotes

95% Upvoted

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18

u/Tomba_Pls 7d ago

If you enjoy non-commutative geometry and its relation to physics, I'd really recommend looking into spectral triples. It has been a few years since I took a course on non-commutative geometry, but from what I remember spectral triples are able to extend Gelfand duality to subclasses of Riemannian spin bundles.

What I thought is beautiful about this, is that these spin bundles "feel" much more geometric, compared to the compact Hausdorff spaces. But still you are able to turn them into purely functional analysis data in some recoverable way (with similar implications for physics).

12

u/SouthernStatus7004 6d ago

While I am a big fan of NCG (I even use it in research), I feel like it is mandatory to tell students that there has been virtually no progress on connecting pseudo-Riemannian spaces to the NCG dictionary, hence the connection to actual physics is very iffy. Additionally, it has very much fallen out of favour in the physics community itself by a wrong prediction of the Higgs bozon's mass by Connes.

This is not a jab at NCG, but I always feel very uncomfortable when people don't mention the flaws (related to physics).

3

u/DarthMirror 6d ago

Can you give me a reference to read about this wrong prediction?

9

u/SouthernStatus7004 6d ago edited 6d ago

It should be noted that they provided a correction to the prediction after the measurement.

The original claim can be found in: https://arxiv.org/abs/hep-th/0610241

The post-measurement correction can be found in: https://link.springer.com/content/pdf/10.1007/JHEP09(2012)104.pdf104.pdf)

Connes himself on Woit's blog discussing the problem: https://www.math.columbia.edu/~woit/wordpress/?p=9110#comment-225080

For those interested in NGC physics, I wholeheartedly recommend van Suijlekom's book (terrifically nice guy)! https://link.springer.com/book/10.1007/978-3-031-59120-4

2

u/Migeil Operator Algebras 6d ago

Saved this comment, thanks for the references!

3

u/XunitaryX Noncommutative Geometry 6d ago

I can’t speak for the particle physics stuff, but Noncommutative Geometry seems to still have merits for topological insulators and superconductors at the very least.

1

u/wolf-among-sheeps 4d ago

the last equation ("[t, d/dt] = 1") is backwards; it should be "[d/dt, t] = 1"