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?

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?

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

  • [deleted]

Thanks Garrett!

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

4 months later
  • [deleted]

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?

8 months later
  • [deleted]

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 one of language, but that's not all.

I do agree that it is a good thing to ask: "What is a fermion, really?". Don't ever take any concept for granted.

And I do agree that the answer to "What is a fermion, really?" has to do with looking at graded structures.

But just as there are many groups out there but not every group one comes across is a gauge group, there are many graded structures out there and not every such is a BRST structure.

It should be clear that identifying BRST ghosts with physically observable particles is a contradiction in terms -- unless you actually only have some aspects of BRST formalism in mind but are secretly discarding others.

I think you are really essentially just looking at the graded algebra aspect of BRST. Which is good and fine. But shoudn't be addressed as "BRST".

I think there are two important sources of graded structures in physics: those coming from groupoidal structures and those coming from super-structures.

The BRST complex is, really, the (Chevalley-Eilenberg differential algebra of the) Lie algebroid version of what is called the "action groupoid" of the gauge group acting on the space of (on-shell) fields.

It is a fact deeply rooted in math -- essentially the Dold-Kan theorem -- that linearized (higher) categories correspond to graded complexes: the degree k of the complex knows about the tangent space to the space of k-morphisms of some n-category.

This is how N-gradings enter physics. And the BRST complex is the central example.

But all geometry that is out there also has a super-version. This introduces a Z_2-grading on top of everything else.

Now, it is a simple standard observation that a symplectic structure on an odd vector space is the same as a symmetric bilinear form. Hence Clifford algebra is precisely the Heisenberg algebra of an odd configuration space.

Just as L^2-spaces, the space of states of bosonic quantum theory, are the unqique irreps of the ordinary bosonic Heisenberg algebra, spinor reps are the reps of the odd-graded Heisenberg algebra.

That's what spinors really are: elements in an odd space of states. (Where "odd" means: graded by the nontrivial element in Z_2!) In the target space field theory context that you are looking at mostly, you see the "second quantized" version of spinors, where they no longer look like elements in a space of states, but as fields themselves. But that's just the magic of second quantization.

Now, the difference between Z_2 graded super structures and N-graded higher-order structures can at times be small, since lots of phenomena survive the passage through the homomorphism from N to Z_2.

This is the reason why spinorial phenomena can be seen also in BRST-like contexts. But that shouldn't make us go as far as saying that "physical fermions are ghosts". Because that's a contradiction in terms. It cannot possibly be true if both "physical fermions" and "ghosts" are used in the standatd sense.

Don't get me wrong. I think that your general approach to fermions (as far as I remember it) is all right. I just think that it is wrong to advertize it as being about BRST.

When you start thinking about quantizing your "exceptially theory of everything", or whatever gauge theory you have, you'll start wanting to use BRST-quantization of all the fields in your theory, including the fermions you have. They will have even (bosonic) ghosts attached to them.

9 months later
  • [deleted]

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.

3 years later
  • [deleted]

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.

3 months later
  • [deleted]

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

    2 years later
    • [deleted]

    Dear Amrit,

    Greetings!

    Higgs boson has been found. I do have a question, though: Can the Higgs boson actually assign a specific value of a rest mass, say, that of an electron? Or it merely explains why non-photon particles have rest masses?

    Because by my model of a combined spacetime 4-manifold of {(t + ti, x + yi, y + zi, z + xi)}, the column of (0, i) in one of the three Pauli matrices becomes (0, 0, 1) and a free electron-wave spins firstly from (x, y, z) = (-1, 0, 0) = W with momentum direction (0, 1, 0), then secondly to (0, 1, 0) = N with momentum direction (1, 0, 0), and thirdly to (1, 0, 0) = E with momentum direction (0, i) = (0, 0, 1); i.e., the spin changes to a perpendicular plane so that the energy wave spins fourthly from E to (0, 0, 1) = T, and finally from T back to E. Since the angular momentum of the spin around the first semi-circle, W to N to E, is not the same as that around the second semi-circle, E to T to W, the electron-wave must stop at the intersection point E, thus the origin of a rest mass.

    By varying the angle between the two semi-circles from the above 90, to 60, to 30, and to 0 degrees, respectively corresponding to (0, ½ + sqrt3/2 i), (0, sqrt3/2 + i/2), and (0, 1), we model the spinning wave motions of the three generations of the up quark, the down quark, and neutrino. As such, they all must have rest masses.

    Best,

    Gregory L. Light

    4 years later
    Write a Reply...