Gauge symmetry for spin 1/2 fields

In summary, there is no gauge symmetry for spin-1/2 fields because they are not related to spin-1 particles.
  • #1
Lapidus
344
11
Why is there no gauge symmetry for spin-1/2 fields?

Has gauge symmetry to be related to spin-1 fields/ particles?

thanks
 
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  • #2
Who told you that? SU(2) has a gauge symmetry to spin 1/2 fields.
 
  • #3
That does seem strange, that spin-1 fields seem to be all gauge bosons (transform in the adjoint representation), while spin-0 and spin-1/2 transform in fundamental representations.

But beyond this, the spin-1 fields, besides transforming in the adjoint representation, have an extra term that involves the derivative of the local group parameter. This term is missing from fields that transform in the fundamental representations. It's as if spin-1 fields are undergoing a local translation in addition to a local rotation.

I don't know the answers to these questions. Can you have a spin-1 field transforming in the fundamental representation, and have a covariant derivative for it that involves a fermion field as the gauge field?
 
  • #4
Thanks for the answering so far.

I assumed that there is no gauge symmetry for spin-1/2 fields, since it is never mentioned in the textbooks. Gauge symmetry just always comes into play when spin-1 particles/fields are introduced.
 
  • #5
You have to distinguish between the fundametal gauge potentials A and the field strength F.

First of all a fermion field transforms in the fund. rep., i.e.

[tex]\psi \to U\psi[/tex]

A gauge field strength transforms in the adjoint rep.

[tex]F \to UFU^\dagger[/tex]

Only the gauge field A transforms as

[tex]A \to U(A-id)U^\dagger[/tex]

which is strictly speaking not the adjoint rep. The adjoint rep. means that the elements are elements of an algebra; an algebra is - besides some other properties - a vector space. But you can't simply add gauge potentials to get a new one:

[tex]A^1 \to U(A^1-id)U^\dagger[/tex]
[tex]A^2 \to U(A^2-id)U^\dagger[/tex]

So

[tex](A^1 + A^2) \to U(A^1-id)U^\dagger + U(A^2-id)U^\dagger = U((A^1+A^2)-2id)U^\dagger [/tex]

But if you define

[tex]A=(A^1 + A^2) [/tex]

you would get

[tex]A \to U(A-id)U^\dagger = U((A^1+A^2)-id)U^\dagger [/tex]

You see the difference.

The geometrical reason is that A is not only a four-vector but a connection in the fibre bundle generated by the gauge group over spacetime, whereas fermions and F are just ordinary spacetime spinors, tensors and elements of a vector space the gauge group is acting on; you can add these vectors w/o violating the transformation law.

Btw. there are indeed covaraint derivatives both in the fund. rep. and in the adjoint rep.
 
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  • #6
Lapidus said:
Thanks for the answering so far.

I assumed that there is no gauge symmetry for spin-1/2 fields, since it is never mentioned in the textbooks.
Well, that's true. The massive spin 1/2 field doesn't possesses gauge invariance, just global U(1) invariance.

Lapidus said:
Gauge symmetry just always comes into play when spin-1 particles/fields are introduced.

Fields of spin 1, 3/2 and 2 have gauge invariance.
 
  • #7
dextercioby said:
Well, that's true. The massive spin 1/2 field doesn't possesses gauge invariance, just global U(1) invariance.
That's wrong!

The (massive) electron field in QED and the quark fields in QCD both have local gauge invariance, namly U(1) and SU(3), respectively. Only in chiral theories (like electro-weak theory) local gauge invariance of massive fermions is forbidden. But this problem is not solved via forbidding local gauge invariance but via demanding that the elementary fields are massless.

That means that all fermions of the standard modle have local gauge invariance U(1)*SU(2)*SU(3).
 
  • #8
tom.stoer said:
That's wrong!

No, it's true. The spin 1/2 field if massive is described by the Lagrangian density
[tex] \mathcal{L} = \frac{1}{2} \bar{\Psi}(x)\left(i\gamma^{\mu}\substack{\leftrightarrow \\ \partial_{\mu}}- m\right) \Psi(x) [/tex] (http://en.wikipedia.org/wiki/Dirac_field#Dirac_fields)

which possesses global U(1) symmetry (rigid).

If you wish to couple this (or a collection of this) to some gauge fields(s), then it will indeed <gauge> its global symmetries (=> QED or QCD as primary fundamental theories).
 
  • #9
What you are saying is trivial: if you use a Lagrangian w/o gauge symmetry then the Lagrangian has no gauge symmetry. You are using the wrong Lagrangian.

This has nothing to do whether the fermion field in your Lagrangian is massive or massless: your Lagrangian with massless fields (i.e. m=0) has still no local gauge symmetry. In any case you have to use the covariant derivative.
 
  • #10
I'm using what's been known for 70 years as the Lagrangian for massive spin 1/2 fields in a flat 4D space-time. It can't be wrong.

As for covariant derivative, well, it enters the picture only if you want the collection of electrons/positrons described by the density I wrote to interact with something.

I've just addressed this post <I assumed that there is no gauge symmetry for spin-1/2 fields, since it is never mentioned in the textbooks. Gauge symmetry just always comes into play when spin-1 particles/fields are introduced.>
 
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  • #11
You wrote:
dextercioby said:
The massive spin 1/2 field doesn't possesses gauge invariance, just global U(1) invariance.
I said that this has nothing to do with the mass of the fermion fields.

dextercioby said:
I've just addressed this post I assumed that there is no gauge symmetry for spin-1/2 fields, since it is never mentioned in the textbooks. Gauge symmetry just always comes into play when spin-1 particles/fields are introduced.
Yes, as I said, you have to introduce the covariant derivative.

What about the following:
dextercioby + tom said:
The kinetic term of the spin 1/2 field doesn't possesses gauge invariance (just a global invariance) w/o introducing spin 1 gauge fields.
Could you agree?
 
  • #12
tom.stoer said:
What about the following:

The kinetic term of the spin 1/2 field doesn't possesses gauge invariance (just a global invariance) w/o introducing spin 1 gauge fields.

Could you agree?

Yes, the same can be said about a mass term.
 
  • #13
dextercioby said:
the same can be said about a mass term.
No, the mass term already has local gauge invariance, but of course this is useless w/o the kinetic term. Or let's say it the other way round: introducing local gauge invariance changes the kinetic term, but not the mass term.
 
  • #14
tom.stoer said:
No, the mass term already has local gauge invariance, but of course this is useless w/o the kinetic term.

I don't understand. Perhaps you mean the mass term for fermions as it appears from the electroweak interaction via the Higgs boson. But this is not what I meant. I was speaking only about QED.

tom.stoer said:
Or let's say it the other way round: introducing local gauge invariance changes the kinetic term, but not the mass term.

The kinetic term is not changed by coupling to a single spin 1-gauge field. The lagrangian density altogether acquires the coupling term.
 
  • #15
The mass term in QED reads

[tex]m\bar{\psi}\psi[/tex]

It is invariant w.r.t. a local gauge trf., that means you can transform it via

[tex]\psi \to U\psi[/tex]

where U=U(x). But of course this is not reasonable w/o having introduced gauge symm. in the kinetic energy term. What I mean is that once you introduce gauge symm. via adding the gauge field this does not affect the mass term, neither in QED nor in QCD. In the electro-weak theory it's different because there is no mass term.

Regarding "the kinetic term is not changed by coupling to a single spin 1-gauge field." What I mean is that you replace the standard kinetic energy

[tex]\partial_\mu \to D_\mu[/tex]

which creates the coupling term. Yes, stricty speaking you are right, the kinetic energy remains unchanged and an additional coupling term is introduced.
 

1. What is gauge symmetry for spin 1/2 fields?

Gauge symmetry for spin 1/2 fields is a fundamental concept in quantum field theory. It refers to the mathematical framework used to describe the behavior of spin 1/2 particles, such as electrons, in a quantum field. Essentially, it means that the physical laws governing these particles are unchanged if certain transformations, known as gauge transformations, are applied to them. This symmetry is crucial in understanding the behavior of elementary particles and the forces between them.

2. How does gauge symmetry relate to the Standard Model of particle physics?

The Standard Model is the most widely accepted theory for describing the fundamental particles and forces in our universe. It is built upon the principles of gauge symmetry, specifically the principles of local gauge invariance. This means that the laws of physics are the same at every point in space and time, and that the particles and forces interact through the exchange of gauge bosons, which are particles associated with the various symmetries in the model.

3. Why is gauge symmetry important in quantum field theory?

Gauge symmetry is important because it provides a framework for understanding the fundamental interactions between particles and the forces that govern their behavior. It allows us to describe the behavior of particles in a consistent and mathematically elegant way. Additionally, it has been crucial in the development of the Standard Model and other theories in particle physics.

4. Is gauge symmetry a fundamental symmetry of the universe?

While gauge symmetry plays a fundamental role in our understanding of particle physics, it is not considered to be a fundamental symmetry of the universe. This is because it is a mathematical concept used to describe the behavior of particles, rather than an inherent property or law of the universe itself. However, its importance in our current understanding of the universe cannot be overstated.

5. What are some practical applications of gauge symmetry?

The principles of gauge symmetry have been applied in a wide range of fields, including quantum mechanics, electromagnetism, and particle physics. In addition to its theoretical importance, gauge symmetry has also led to practical applications such as the development of technologies like transistors and lasers. It also plays a crucial role in the development of new materials with unique properties, such as superconductors and topological insulators.

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