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FQXi FORUM
February 9, 2012
Most of my research was motivated by an early frustration with what fermions are and where they come from, mathematically. A column of complex numbers, why? Ah, they're the irreducible spinor representation of the Lorentz group, or they're minimal left ideals of a Clifford algebra. But those aren't explanations of why they are, just what they are. And wait, they're not actually complex numbers, but complex Grassmann (anti-commuting) numbers? Why? Ah, they have to anti-commute to satisfy the spin-statistics theorem -- a step in the right direction, but not really satisfying.
Frustrated with this, every couple years I revise my notion of what and why fermions are. At this point my understanding may have gone pretty far afield of convention, but it's at the heart of my research so I'd like to put it out there and see what people think. Here's the basic idea:
When doing QFT with non-abelian Yang-Mills fields (describing all known forces) it is necessary to introduce ghost fields to properly restrict and account for gauge freedom. The Yang-Mills fields (principal bundle connections) we work with are Lie algebra valued 1-forms, and the ghost fields are Lie algebra valued Grassmann numbers. So, I can cook up some Yang-Mills fields with gauge freedom such that the ghosts act as spinors, algebraicly, and have the right dynamics (Dirac Lagrangian). My crazy idea, and my current understanding, is that these ghosts ARE the physical fermions.
In a nutshell: the fermions, represented by Grassmann spinors, are BRST ghosts corresponding to pure gauge degrees of freedom of a certain Yang-Mills theory.
Thoughts? Is this idea crazy enough to be true?
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Frustrated with this, every couple years I revise my notion of what and why fermions are. At this point my understanding may have gone pretty far afield of convention, but it's at the heart of my research so I'd like to put it out there and see what people think. Here's the basic idea:
When doing QFT with non-abelian Yang-Mills fields (describing all known forces) it is necessary to introduce ghost fields to properly restrict and account for gauge freedom. The Yang-Mills fields (principal bundle connections) we work with are Lie algebra valued 1-forms, and the ghost fields are Lie algebra valued Grassmann numbers. So, I can cook up some Yang-Mills fields with gauge freedom such that the ghosts act as spinors, algebraicly, and have the right dynamics (Dirac Lagrangian). My crazy idea, and my current understanding, is that these ghosts ARE the physical fermions.
In a nutshell: the fermions, represented by Grassmann spinors, are BRST ghosts corresponding to pure gauge degrees of freedom of a certain Yang-Mills theory.
Thoughts? Is this idea crazy enough to be true?
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Hello Garrett,
If you have time, can you explain this in more detail to non insiders like me? I don't understand how you can have ghosts that appear outside of the loops in Feynman diagrams.
This is what I know about ghosts:
You have some path integral that should be over physical field configurations but it is more convenient to integrate over potentials but then you get overcounting because of gauge symmetry. This can be solved by introducing ghost fields. In Feynman diagrams the ghost contributions will only appear in loops.
That's pretty much all I know about ghosts.
So, how does this "BRST ghost" differ from the Fadeev Popov ghost?
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If you have time, can you explain this in more detail to non insiders like me? I don't understand how you can have ghosts that appear outside of the loops in Feynman diagrams.
This is what I know about ghosts:
You have some path integral that should be over physical field configurations but it is more convenient to integrate over potentials but then you get overcounting because of gauge symmetry. This can be solved by introducing ghost fields. In Feynman diagrams the ghost contributions will only appear in loops.
That's pretty much all I know about ghosts.
So, how does this "BRST ghost" differ from the Fadeev Popov ghost?
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By your command,
Fadeev Popov ghosts and BRST ghosts are the same thing -- the BRST technique just adds some geometric flavor. So your understanding is fine.
When you say "introducing ghost fields" what this means is going back to the original Lagrangian and adding some fields and new terms to it. This is conventionally understood as a (necessary) mathematical trick. What I'm suggesting is that we take this mathematical trick seriously and consider these fields as being "really there." And that if we choose the right gauge fields to start with, some of these ghost fields behave exactly like our physical fermions.
Here's a one page description of the BRST technique on my wiki:
http://deferentialgeometry.org/#%5B%5BBRST%20technique%
5D%5D
Here's a more thorough introduction to the BRST technique:
Aspects of BRST Quantization
And here's a paper I wrote a year ago detailing how to use this to get fermions:
Clifford bundle formulation of BF gravity generalized to the standard model
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Fadeev Popov ghosts and BRST ghosts are the same thing -- the BRST technique just adds some geometric flavor. So your understanding is fine.
When you say "introducing ghost fields" what this means is going back to the original Lagrangian and adding some fields and new terms to it. This is conventionally understood as a (necessary) mathematical trick. What I'm suggesting is that we take this mathematical trick seriously and consider these fields as being "really there." And that if we choose the right gauge fields to start with, some of these ghost fields behave exactly like our physical fermions.
Here's a one page description of the BRST technique on my wiki:
http://deferentialgeometry.org/#%5B%5BBRST%20technique%
5D%5D
Here's a more thorough introduction to the BRST technique:
Aspects of BRST Quantization
And here's a paper I wrote a year ago detailing how to use this to get fermions:
Clifford bundle formulation of BF gravity generalized to the standard model
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I have a very different picture for fermions. I believe that
the space is a quantum liquid of long strings or string-nets (whose sizes are of order of universe). Under this assumption, fermion are ends of open strings and gauge bosons are the quanta of the collective waves in the string liquid. In other words Fermi statistics and gauge interaction are unified under this string-net picture. The string-net theory for fermions has a prediction that all composite fermions must carry gauge charge. The string-net theory for fermions
also explains why all fermions must carry half-odd-integer spins. For details see http://dao.mit.edu/~wen/pub/uni.pdf
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the space is a quantum liquid of long strings or string-nets (whose sizes are of order of universe). Under this assumption, fermion are ends of open strings and gauge bosons are the quanta of the collective waves in the string liquid. In other words Fermi statistics and gauge interaction are unified under this string-net picture. The string-net theory for fermions has a prediction that all composite fermions must carry gauge charge. The string-net theory for fermions
also explains why all fermions must carry half-odd-integer spins. For details see http://dao.mit.edu/~wen/pub/uni.pdf
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Dear Garrett,
Thank you very much for the trouble you took to understand and revise this problem. I like to inform you that the ghosts are not physical fermions due to the wrong relationship between spin and statistics for them. Actually, spin-statistics relationship corresponds to micro-causality which is also very important to obtain unitarity of free quantum field theories as well as nonfree QFTs. Some problems of quantum Yang-Mills theory is why there exsists nonequvalence of choosing gauge condition (there is a bad gauge choice in which gauge condition intersects some orbits (classes of equvalent Yang-Mills fields) more than one (Gribov problem)). Also, for the same theory in a particular gauge condition there are no ghosts at all. Why is that? Also, the IR sector of this theory is not well known, maybe pure quantum Yang-Mills theory possesses the mass gap (gluons have masses producing glueballs i.e. confinement of gluons). This corresponds to the Clay Institute Millenium Prize Problem.
Though, it seems that ghosts are very important for renormalization and unitarity of the quantum Yang-Mills theory as well as the Standard model, but they are not represented on-shell (in physical processes, as in scattering) (for these statements see the literature on no-ghost theorem and optical theorem).
Best regards,
aca
P.S. In the Standard model, as in the quantum Yang-Mills type theory, the choice of gauge condition is very important to prove its unitarity and renormalization and there is no unique choice of gauge condition in which you can prove both (see `t Hooft and Veltman`s papers). Why is that so?
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Thank you very much for the trouble you took to understand and revise this problem. I like to inform you that the ghosts are not physical fermions due to the wrong relationship between spin and statistics for them. Actually, spin-statistics relationship corresponds to micro-causality which is also very important to obtain unitarity of free quantum field theories as well as nonfree QFTs. Some problems of quantum Yang-Mills theory is why there exsists nonequvalence of choosing gauge condition (there is a bad gauge choice in which gauge condition intersects some orbits (classes of equvalent Yang-Mills fields) more than one (Gribov problem)). Also, for the same theory in a particular gauge condition there are no ghosts at all. Why is that? Also, the IR sector of this theory is not well known, maybe pure quantum Yang-Mills theory possesses the mass gap (gluons have masses producing glueballs i.e. confinement of gluons). This corresponds to the Clay Institute Millenium Prize Problem.
Though, it seems that ghosts are very important for renormalization and unitarity of the quantum Yang-Mills theory as well as the Standard model, but they are not represented on-shell (in physical processes, as in scattering) (for these statements see the literature on no-ghost theorem and optical theorem).
Best regards,
aca
P.S. In the Standard model, as in the quantum Yang-Mills type theory, the choice of gauge condition is very important to prove its unitarity and renormalization and there is no unique choice of gauge condition in which you can prove both (see `t Hooft and Veltman`s papers). Why is that so?
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Hi Garrett,
I remember we were talking about this by email years ago, when I was still a student -- and before you became kind of a web celibrity :-).
Concerning the topic of this thread here (to which I am arriving overly late, I know) allow me to say that I feel unsure about your proposed identification of physical fermions with BRST ghosts. To some extent the disagreement is just...
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I remember we were talking about this by email years ago, when I was still a student -- and before you became kind of a web celibrity :-).
Concerning the topic of this thread here (to which I am arriving overly late, I know) allow me to say that I feel unsure about your proposed identification of physical fermions with BRST ghosts. To some extent the disagreement is just...
view entire post
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I believe that the brain creates specific energys when using emotions that if strong enough may manifest itself in a place, and live off the energys around it. It would probably mimic the mind and body it came from. At least thats what i believe.
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Ghost! is the scared feeling when we are alone or empty minded . Yeh it is the strong eneygy that disturbs the mind & body too.
John
www.donjoaoresortgoa.com
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John
www.donjoaoresortgoa.com
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I have a quite different answer to the "what are fermions" question. It is part of my "cell lattice model" which allows to explain the particle content of the standard model, which can be found here. It associates a single scalar field on a spatial lattice with a doublet of staggered Dirac fermions, which are identified with electroweak pairs.
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Open letter to scientific community
Dear Sirs,
Higgs boson will be never discovered as it not exists.
Physicists as the CERN know that, what they are telling us this days in media is just to get money for searching for a black cat in a black room that is not there.
Our research shows mass has origin in energy density of quantum vacuum. Mass is an energy form of quantum vacuum in symmetry with diminished energy density of quantum vacuum. Presence of mass diminishes energy density of quantum vacuum respectively to the energy of a given mass. A given particle with a mass diminishes energy density of quantum vacuum, mass-less particle does not diminish energy of quantum vacuum. In order to explain mass of elementary particles this view does not require existence of the hypothetical boson of Higgs.
Yours Sincerely, Amrit Sorli, Space Life Institute
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Dear Sirs,
Higgs boson will be never discovered as it not exists.
Physicists as the CERN know that, what they are telling us this days in media is just to get money for searching for a black cat in a black room that is not there.
Our research shows mass has origin in energy density of quantum vacuum. Mass is an energy form of quantum vacuum in symmetry with diminished energy density of quantum vacuum. Presence of mass diminishes energy density of quantum vacuum respectively to the energy of a given mass. A given particle with a mass diminishes energy density of quantum vacuum, mass-less particle does not diminish energy of quantum vacuum. In order to explain mass of elementary particles this view does not require existence of the hypothetical boson of Higgs.
Yours Sincerely, Amrit Sorli, Space Life Institute
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