What is Gauge Invariance in QFT?

[izh-21251]

According to Steven Weinberg ('The quantum theory of fields', vol.1), the principle of gauge invariance stems from the fact, that one cannot build the 4-vector field from the creation/annihilation operators of massless bosons with spin >= 1.
This '4-vector field' ('vector potential'), if we build it, does not transform under the Lorentz transformations as a 4-vector.
However, to preserve using the '4-vector field' in the theory, one suggests to introduce the 'gauge equivalence' principle. This claims, that 'non-covariance' of the vector potential must not affect the invariance of the S-matrix (and thus the observables) under the Lorentz transformations.
In other words, one can consider the 'gauge transformations' as Lorentz transformations, which alter the form of vector-potential, leading to the appearance of 'gauge terms'. Any of 'gauge-equivalent' potentials, according to the Principle, must lead to the Lorentz-invariant theory with one and the same S-matrix.
At this point one should already speak about the family of Lee-algebras that are unitary-equivalent (or, say, 'gauge-equivalent'), so as to give the same S-matrix.
How was I surprised, after I had not found any inversigations, which deal with this things more profoundly as it was done by Weinberg.
I would be most grateful if anyone can help me clarify the topic.
Am I not misunderstanding something?
Thanks in advance,
Ivan

 

[strangerep]

Since no one else has answered, I'll try to say a few things. (I get the feeling a lot
of people are away at this time of year.) Possibly, this thread should be moved into
the "quantum physics" forum.

According to Steven Weinberg ('The quantum theory of fields', vol.1), the principle of gauge invariance stems from the fact, that one cannot build the 4-vector field from the creation/annihilation operators of massless bosons with spin >= 1.
This '4-vector field' ('vector potential'), if we build it, does not transform under the Lorentz transformations as a 4-vector.
However, to preserve using the '4-vector field' in the theory, one suggests to introduce the 'gauge equivalence' principle. This claims, that 'non-covariance' of the vector potential must not affect the invariance of the S-matrix (and thus the observables) under the Lorentz transformations.
In other words, one can consider the 'gauge transformations' as Lorentz transformations,

I believe that's not quite correct. One does not usually consider gauge transformations as
Lorentz transformations.

Actually, I'm not sure exactly what part of Weinberg you're referring to. Usually, one deals
with the non-covariance of the photon "field" by coupling it to a conserved current, which
I think is what Weinberg does (though it's been a while since I looked at it).

which alter the form of vector-potential, leading to the appearance of 'gauge terms'.
Any of 'gauge-equivalent' potentials, according to the Principle, must lead to the Lorentz-invariant theory with one and the same S-matrix.
At this point one should already speak about the family of Lee-algebras that are unitary-equivalent (or, say, 'gauge-equivalent'), so as to give the same S-matrix.

I presume you mean "scattering-equivalent".

How was I surprised, after I had not found any inversigations, which deal with this
things more profoundly as it was done by Weinberg. I would be most grateful
if anyone can help me clarify the topic. Am I not misunderstanding something?

I might not be the best person to help. But if no one else answers, you could try
quoting the relevant parts of Weinberg more explicitly, and I'll have a go... :-)

 

[meopemuk]

One does not usually consider gauge transformations as
Lorentz transformations.

Actually, I'm not sure exactly what part of Weinberg you're referring to. Usually, one deals
with the non-covariance of the photon "field" by coupling it to a conserved current, which
I think is what Weinberg does (though it's been a while since I looked at it).

Yes, that's how Weinberg makes the connection between Lorentz covariance and gauge transformations. I see his logic as follows:

1. Our goal is to develop a theory whose S-matrix is Lorentz-invariant.

2. One way to achieve that is to make sure that the interaction Hamiltonian density (the interaction Hamiltonian itself is obtained as a 4-integral of this density) transforms as a 4-scalar field with respect to (non-interacting) Lorentz transformations.

3. 4-scalars can be obtained as products of 4-vectors, 4-tensors, etc. However creation/annihilation operators of particles do not have such simple transformation laws. Their transformation laws involve Wigner rotations and other complications.

4. This problem can be solved by forming certain linear combinations (called quantum fields) of creation/annihilation operators. These linear combinations (in simple massive cases) do transform as 4-scalars, 4-vectors, 4-tensors, etc. So, in these simple cases the interaction Hamiltonian density can be formed as a simple product (or polynomial) of quantum fields.

5. This method doesn't work when mass=0, spin=1 particles (e.g., photons) are involved, because their quantum field does not transform as a 4-vector. The transformation law contains an additional term (which has the form of a 4-derivative, and can be "killed" by a "gauge transformation"), which spoils the relativistic invariance of products of fields.

6. Luckily, the above problem is absent in one particular case, i.e. when the photon's field is multiplied by a conserved current composed of electron fields. In this case, the product transforms as a 4-scalar, and it can be used to form the interaction operator in QED. This is the justification for using the minimal coupling interaction in quantum electrodynamics.

For me, this logic looks like a fine heuristic guess of the interaction operator. Once the interaction has been guessed we can calculate everything we need from it. However, I don't think that one should go as far as to claim (as all textbooks do) that there exists a fundamental "local gauge" symmetry of nature. I fail to see any value in (quite complex mathematical) studies of relationships between different gauges, etc.

 

[Bob_for_short]

There is one more approach to this problem. It arises in Lorentz non-invariant gauges, in the Coulomb gauge, for example. A simple Lorentz transformation introduces new terms in the new, transformed Hamiltonian so the latter looses the original “Coulomb” structure. A specific gauge transformation can be used to restore the Coulomb structure of the transformed Hamiltonian (K. Johnson, Ann. of Phys. 1960. V. 10. P. 536.).

Try to find the Polubarinov's review "Equations of Quantum Electrodynamics" on internet (in Russian), page 754.

 

[izh-21251]

I dare say, one should not try to find a way to excuse the need for gauge invariance, but to formulate a Lorentz-invariant theory.
Is it not true?

However, it is still some things that I don't understand.
Ok, as you said, in QED we have a family of scattering equivalent Hamiltonians, corresponding for different gauges.
Now, if one starts to apply the dressing procedure to these Hamiltonians, it will appear that he needs different dressing transformations in different gauges.
Thus, the renormalization procedures wiil be different. It means, I think, that, for instance, that the electron mass shift (in a 'dressing renorm program') will depend on the gauge (which means on the Lorentz-frame as well).
At the same time -- the dressed electron should have observable mass.

 

[izh-21251]

Thanks for links!

In fact, I read a number of books in QFT and in particular in QED...
I think there is nothing better than Weinberg's book yet.

By the way, let me provide an exact link to Weinberg's book:
It is
Steven Weinberg, "The Quantum Theory of Fields", vol.1. Cambridge University Press, 1995
The chapter, concerning massless spin-1 particles is 5.9.
I am quoting: 'In fact, we shall see in this section that the creation and annihilation operators for physical massless particles of spin >=1 cannot be used to construct all of the irreducible (A,B) fields that can be constructed for finite mass. This peculiar limitation on field types will lead us naturally to the introduction of gauge invariance.' (p. 246)

On p.250 he shows that appearance of the 'gauge terms' in vector potential is the result of the combined 'boost-rotation' tranformations in the plain, which is normal to the photon momentum. (see formula (5.9.22))
SO, the gauge terms appear due to Lorentz transformation.

 

[Haelfix]

You can have gauge invariance without lorentz invariance. There is no problem for instance writing down a classical theory with the former and not the latter. edited out for stupidity.

One of the ways of motivating gauge invariance in QFT is of course what Weinberg does, but its a bit of an extra consistency condition and a 'well what else can it be' type of argument.

Another way is to look at the Hamiltonian framework and notice that the first class constraints are the 'things' that generate Gauge transformations. So in trying to carry out Dirac's program, you are invariably led to needing the extra structure, completely independantly of lorentz invariance.

 

[humanino]

Take for instance electromagnetism.

I do not understand : Mawxell's equations are not only Lorentz invariant, their discovery launched Lorentz invariance itself into existence.

 

[Haelfix]

Absolutely correct... Excuse the brain fart. How about NRQCD as an example.

 

[humanino]

How about NRQCD as an example.

How about NRQED ?
It seems to me, in both cases, the physics is well approximated by Galilean invariance. So it would just be a limiting case where the breaking of Lorentz invariance is small. On the other hand, I fully agree that gauge invariance and Lorentz invariance are independent in principle.

 

[meopemuk]

On the other hand, I fully agree that gauge invariance and Lorentz invariance are independent in principle.

This would be true if you accept the gauge invariance as an independent fundamental principle of nature. Then you can define (relativistic or non-relativistic) quantum fields, postulate their local gauge invariance and so on (as in textbooks).

It would be more interesting to understand where the gauge invariance comes from? Does it follow from any deeper physical principle? "Who ordered it?" The beauty of the Weinberg's approach is that it does not assume the gauge invariance principle as a postulate. His postulate is that the S-matrix must be Poincare-invariant. (More exactly: the interacting theory must provide an unitary representation of the Poincare group in some Hilbert (or Fock) space), and Weinberg proves that a formally gauge invariant theory (QED) does satisfy this postulate.

The Poincare (or Lorentz) invariance plays a significant role in this proof. I haven't seen if somebody tried to repeat Weinberg's logic with the Galilei group. My guess is that the result would be less restrictive, i.e., it is much easier to build a Galilei-invariant interacting theory than a Poincare-invariant interacting theory. So, the idea of the gauge invariance could be easily lost in such a non-relativistic derivation.


Edit: In fact, Weinberg's approach does not prove the necessity of the local gauge invariance principle. It only proves that the interaction must have the form (photon field) times (conserved electron current). That's all we need to construct a viable interacting theory. The local gauge invariance principle is an additional construction, whose only role is to pseudo-explain the above form of interaction. If we follow the letter and spirit of Weinberg's approach, we should admit that the local gauge invariance is simply a formal mathematical trick, which may not have any relevance to real physics.

 

[genneth]

It would be more interesting to understand where the gauge invariance comes from? Does it follow from any deeper physical principle? "Who ordered it?"

Now this is a topic dear to my heart. First: what do we mean by a gauge theory? On a classical level, we seem to work exclusively with systems whose configuration space is the space of connections of some G-bundle on spacetime. Trying to put coordinates on it, we find that there is no simple, smooth mapping, and we're led to vector potentials and gauge transforms. So then the question becomes, why do we choose these configuration spaces? After all, they are a little artificial, and do not, a priori, seem special (ignoring renormalisability for the moment).

However, we should realize that the world is not classical. Quantum mechanically, gauge invariance is can be understood much in a broader sense. We distinguish carefully between the space V of quantum states, and the Hamiltonian H that act on it. A (proper) symmetry is an operater S: V->V such that [H,V] = 0. A gauge symmetry (thus, improper) is a symmetry G of a larger space V', such that V = V'/G, i.e. we define our physical Hilbert space by a quotient procedure. Such a procedure is general enough to encompass all the gauge theories usually considered in HEP, and in fact a vast number more. It may be too general to do anything useful with, but it's an important point of view. In fact, this is just a quantum version of the coordinate-choosing problem.

So maybe, we should be turning the question around. When is a classical (in the sense of usually seen) gauge theory a good low-energy description or effective theory? Is there any reason why these should be favoured? I can only offer one idea: classical gauge theories give massless excitations --- the gauge "symmetry" enforces it. Thus if you keep integrating out successive energies, eventually you'll only be left with the gauge field, and any other enforced zero-mass fields (Goldstone bosons, usually).

I note that in condensed matter studies, many systems produce gauge fields as part of their low energy behaviour. E.g. RVB states in cuprate superconductors, anyons in the fractional quantum hall effect, various spin liquids.

 

[humanino]

It would be more interesting to understand where the gauge invariance comes from?

There are proposals to link gauge theories and Lorentz invariance, such as string theory or non-commutative geometry. I certainly agree that those are interesting possibilities.

 

[Bob_for_short]

It would be more interesting to understand where the gauge invariance comes from? If we follow the letter and spirit of Weinberg's approach, we should admit that the local gauge invariance is simply a formal mathematical trick, which may not have any relevance to real physics.

The gauge invariance has no physical meaning at all. It is similar to the ink-color invariance.
There are gauge-invariant theory formulations that are free from this "puzzle".

 

[humanino]

There are gauge-invariant theory formulations that are free from this "puzzle".

Gauge fixing is unphysical, but the question bears on the choice of gauge group.

 

[Bob_for_short]

Gauge fixing is unphysical, but the question bears on the choice of gauge group.

You mean the choice of an interaction symmetry? It is dictated with the observable facts. One is not obliged to make this symmetry group a gauge group. The latter is one of the ways to "guess" the interaction term. There might be other ways.