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No photon mass renormalization?


Bob_for_short
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In QFT the mass of a particle is understood as a pole in a propagator. For the electron this is not protected. (If this is the right terminology). It gets shifted by loops.

 

I remember the old book by Nash [1] being quite clear about renormalisation.

 

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[1] Charles Nash, Relativistic Quantum Fields, Academic Press (February 11, 1979)

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What is meant by protected and not protected.

 

 

Renormalisation of QED (and other theories) is dependant on gauge symmetry. Classical gauge symmetry requires the photon to be massless. It is not at first obvious if the gauge symmetry survives quantisation (regulariation), if it does not the theory would be inconsistent. However, it can be shown that there is no quantum anomaly and that QED is gauge invariant.

 

You can think of the gauge invariance as fixing the mass of the photon to zero. Any quantum effects cannot "push it" away from zero. It is protected.


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It brings up an interesting question: what is then renormalized in photon?

 

In QED you still have wave function renormalisation of A. (I'd need to re-read my QFT books to say much more.)

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You can think of the gauge invariance as fixing the mass of the photon to zero. Any quantum effects cannot "push it" away from zero. It is protected.

 

The photon mass is zero because of Maxwell equations for the field strengths E and B. Of course it is so in terms of four-vector potential too.

 

Do "quantum effects" (you mean interaction) push the electron mass? It is numerically m_e in any order of perturbation theory, isn't it?

In QED you still have wave function renormalisation of A. (I'd need to re-read my QFT books to say much more.)

It is a coefficient, not a photon feature.

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The photon mass is zero because of Maxwell equations for the field strengths E and B. Of course it is so in terms of four-vector potential too.

 

One is interested in the Ward-Takahashi identities which expresses if the symmetry holds "quantum mechanically".

 

Do "quantum effects" (you mean interaction) push the electron mass? It is numerically m_e in any order of perturbation theory, isn't it?

 

Isn't that what renormalistion is all about?

 

The "bare mass", the pole in the propagator without interactions is not the same as the pole when interactions are present. These interactions "dress" the mass.

 

Renormalisation then allows us to get the right answer. The "dressed mass" is the physical thing as it is what we will measure.

 

Renormalisation is really the statement that we do not need to consider higher energy/momentum scales to get a handle on the theory at low scales, say at the scales of the LHC.

 

It is a coefficient, not a photon feature.

 

True, but it shows that renormalisation is required.

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One is interested in the Ward-Takahashi identities which expresses if the symmetry holds "quantum mechanically".

 

Again, it is a consequence of the original equations of theory which is gauge invariant by construction. WT identity is an identity, something like a = a. It's a banality, not a fundamental feature, I would say.

Isn't that what renormalistion is all about?"

 

Yes but for photon mass they say "it is protected" by this or that as if it was not the sequence of the theory basics. The electron mass is also "protected" then.

The "bare mass", the pole in the propagator without interactions is not the same as the pole when interactions are present. These interactions "dress" the mass.".

So what is the mass values before and after renormalizations? We do not have any other value but m_e.

Renormalisation is really the statement that we do not need to consider higher energy/momentum scales to get a handle on the theory at low scales, say at the scales of the LHC.

I do not think so. After renormalizations there is no trace of the cut-off or any other regularization parameter. QED is not an effective theory.

 

Renormalizations are necessary with an interaction term containing a self-action. Renormalizations remove the self-action "effects". In particular, the self-action terms contain a "perturbation" of a kinetic nature so such terms modify the original masses. If one uses the right experimental values as the original masses, the perturbatibe corrections are not necessary so they are discarded (= renormalizations).

 

With a potential interaction term no corrections to the masses arise and no renormalizations are necessary, in my opinion.

Edited by Bob_for_short
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So what is the mass values before and after renormalizations? We do not have any other but m_e.

 

What one does is calculate the observed mass of the electron, [math]m_{e}[/math] as you call it, in terms of the parameter in the action [math]m[/math], say. Formally, you can think of [math]m[/math] as being divergent. Regularisation and renormalisation allows a finite [math]m_{e}[/math] to be defined.

 

I did not want to imply that QED is an effective theory as such. To regulate it introduce a mass cutoff [math]M[/math]. What I mean is that one hopes that physics at a very high scale does not effect the physics at the scales we are interested in as we consider [math]M \rightarrow \infty[/math]. If so, then it stands a chance of being renormalisable, as we know QED is renormalisable.

 

After all, as we approach the Planck scale we expect QED to not hold.

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What one does is calculate the observed mass of the electron, [math]m_{e}[/math] as you call it, in terms of the parameter in the action [math]m[/math], say. Formally, you can think of [math]m[/math] as being divergent. Regularisation and renormalisation allows a finite [math]m_{e}[/math] to be defined.

 

In order to calculate the Compton effect or e-p scattering in the first Born approximation we use m = m_e. We constructed QED with the observable m_e, m_f, and e. It is our interaction term that gives divergences, not the original parameters. So the renormalizations are modification of the interaction term jA, namely, removing the self-action effects (=corrections to masses and charges). In the end we have results of another, renormalized theory with another interaction term.

 

I did not want to imply that QED is an effective theory as such. To regulate it introduce a mass cutoff [math]M[/math]. What I mean is that one hopes that physics at a very high scale does not effect the physics at the scales we are interested in as we consider [math]M rightarrow infty [/math]. If so, then it stands a chance of being renormalisable, as we know QED is renormalisable.

 

You know, it is a too speculative reasoning about physics at short distances because we judge about it from our classical notions, namely from a point-like electron. We speak of "vacuum polarization" that "screens" too singular potential, etc. At the same time the electron is very tightly bound with the quantized electromagnetic field so it is not point-like but smeared quantum mechanically. So there is no singularity at r = 0.

 

 

A point-like nucleus bound in an atom creates a positive charge cloud with no singularity at the atomic center. You may find the corresponding effective potential in my paper in Fig. 1 and Fig. 4, see http://www.springerlink.com/content/h3414375681x8635/?p=78336560cf3d4e3f98e6fb8eac587340π=0 (available also in arXiv).

 

After all, as we approach the Planck scale we expect QED to not hold.

 

Yes, the physical predictions of QED will be different at short distances but the theory as a model does need any scale distance to be renormalizable.

Edited by Bob_for_short
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