This article is within the scope of WikiProject Physics, a collaborative effort to improve the coverage of
Physics on Wikipedia. If you would like to participate, please visit the project page, where you can join
the discussion and see a list of open tasks.PhysicsWikipedia:WikiProject PhysicsTemplate:WikiProject Physicsphysics articles
This is always a
globalinternal symmetry of any quantum field theory with fermions and corresponds to a rotation by 2Ï€
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 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 2Ï€ is correct. A fermion behaves like one side of a
rotor sandwich. When the axes are rotated by 2Ï€, 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)Collin237reply
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
This article is within the scope of WikiProject Physics, a collaborative effort to improve the coverage of
Physics on Wikipedia. If you would like to participate, please visit the project page, where you can join
the discussion and see a list of open tasks.PhysicsWikipedia:WikiProject PhysicsTemplate:WikiProject Physicsphysics articles
This is always a
globalinternal symmetry of any quantum field theory with fermions and corresponds to a rotation by 2Ï€
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 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 2Ï€ is correct. A fermion behaves like one side of a
rotor sandwich. When the axes are rotated by 2Ï€, 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)Collin237reply
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