Talk:(−1)F

This article was nominated for deletion on 22 October 2020. The result of the discussion was keep.

π or 2π?[edit]

The article says:

This is always a global internal symmetry of any quantum field theory with fermions and corresponds to a rotation by 2π

Could it have meant π rather than 2π? That's what I'd expect a multiplication by −1 to be. Michael Hardy (talk) 23:18, 22 June 2008 (UTC)[reply]

See Spin#Spin and the Pauli exclusion principle and [1] (Anyon is a generalizatin of ferminon and bosons for twi dimensions. Phase

\theta 2\pi s

may have intermediate values. Exchange makes factor exp(i θ), monodromy (that's what we are talking about here) exp(2 i θ). So I think you'r right: Exchange(!) makes for fermions a factor -1.

Generally, I do not know why we need this F. The factor simply is (-1)^(2s) with spin quantum number s. So why not choose F = 2s? --Ernsts (talk) 19:05, 29 March 2009 (UTC)[reply]

The 2π is correct. A fermion behaves like one side of a rotor sandwich. When the axes are rotated by 2π, each rotor gets multiplied by -1. I don't know what this "actually means". I'm not sure anyone does. 166.137.14.113 (talk) 17:33, 4 April 2015 (UTC)Collin237[reply]

Two-pi is correct, this is just the conventional double-covering of the rotation group by SU(2). The minus sign is from the Pauli exclusion principle. F can be thought of as an operator: it just counts the number of fermions. For sufficiently simple systems (e.g. simple harmonic oscillator) it is just the number operator. For example, it is just the number of electrons in a box. Just count them. Nothing deeper than that. That's "what it is supposed to mean". The difficulty (why "no one knows what it means") is that in field theory, you have an uncountably-infinite number of these oscillators, and you don't know how to give a mathematically rigorous definition. The physicists are thrilled to hand-wave their way through all this. Mathematicians, not so much. (One more rigorous approach I learned about yesterday is abstract Wiener space. A middle ground is the industry pursuing deformation quantization. The epsilon of deformation theory is what physicists call h-bar: Planck's constant. It's quite the royal mind-bender. Which is why you need people as smart as Witten to figure it out.)

Regarding the AfD deletion discussion from 2020, this article could be expanded to book length; there's plenty to say about it, given ongoing attempts to give it mathematical rigor. Short term improvements would be to tie it into the Atiyah-Singer index theorem and to work in the 1990's (?) by Dan Freed at UTexas that computes (-1)^F for topological solitons. Freed has proven that the spectral asymmetry holds in a broad class of solitons with the chiral anomaly in them. I don't know the details. 67.198.37.16 (talk) 22:51, 29 May 2024 (UTC)[reply]