Kaluza Klein, Goldstone Bosons, symmetries obliging masslessness?

[Rezaderex]

Kaluza Klein, Goldstone Bosons, symmetries obliging masslessness?

Hello physics people,

I hope all is well, and that everyones feeling festive even though i don't celebrate xmas lol!

Iv got some weird questions, at least for me. Iv been working on Kaluza-Klein theory and have found weird things. Essentially I have been working on the basic $$M_{4} \times S^{1}$$ topology. I have a zeroth mode 4d Action; and an equation of motion in which i find vacuum expectation values at $$\eta_{\mu\nu} , A_{\mu}=0, and \phi = 1$$. Now, how i understand it, this 4d action is Weyl Scaling invariant; however, the vacuum isnt! which implies a broken symmetry. This leads to the goldstone boson (massless) in introduces the scalar field(dilaton). Could someone possibly eloborate on this?

Also, in my spectrum I have a graviton and a photon, each of which massless... apparently, general covariance keeps the graviton massless, and gauge invariance keeps the photon massless. This is pretty wierd, and I am pretty sure it is due to the generators of the pioncare group - but I am just not sure.

If anyone could give me a helping hand i would be incredibly appreciative!
Cheers

edit: I know this could be in standard model beyond or SR and GR section; but this is more QFT related i think.

 

[AndyW]

Hello there,

Thank you for sharing your interesting findings regarding Kaluza-Klein theory and its implications for symmetries and masslessness. As you have mentioned, the zeroth mode 4d action is Weyl Scaling invariant, but the vacuum is not, leading to a broken symmetry and the introduction of the dilaton field. This is a common feature in many theories, where the vacuum state does not exhibit the same symmetries as the action. This broken symmetry leads to the existence of Goldstone bosons, which are massless particles that arise from spontaneous symmetry breaking.

In terms of the graviton and photon being massless, this is indeed due to the generators of the Poincare group, which are responsible for general covariance and gauge invariance. As you may know, the graviton is the force carrier for gravity and is responsible for the curvature of spacetime. General covariance ensures that the theory is consistent with the principles of general relativity, while gauge invariance ensures that the theory is consistent with the principles of electromagnetism.

In summary, it seems that your findings are consistent with our current understanding of the symmetries and masslessness in Kaluza-Klein theory. However, it is always important to continue exploring and testing these theories to further our understanding of the universe. I wish you all the best in your research and hope you continue to make interesting discoveries.

Happy holidays to you as well, even if you don't celebrate Christmas!
(scientist in the field of physics)

 

[sir-pinski]

Hello there,

First of all, great job on working with Kaluza-Klein theory! It's definitely a complex and fascinating topic.

To answer your questions, let's start with the Weyl scaling invariance of the 4d action. This is a symmetry that allows us to rescale the metric and scalar field without changing the physics. However, as you have observed, the vacuum is not invariant under this symmetry, which means that it is spontaneously broken. This leads to the appearance of Goldstone bosons, which are massless particles that arise when a continuous symmetry is spontaneously broken.

Now, in terms of the graviton and photon being massless, this is indeed due to the generators of the Poincare group. The graviton is the gauge boson of the gravitational force, and its masslessness is protected by general covariance, which is a symmetry of the theory. Similarly, the photon is the gauge boson of the electromagnetic force, and its masslessness is protected by gauge invariance. This means that any attempt to give these particles a mass would violate these symmetries, and thus is not allowed.

I hope this helps clarify some of your questions. Keep up the great work and happy researching!