How? Why?RGauld said:And the SM is known to be effective.
I still think that this is a (too) bold claim; yes, there are indications that the SM cannot be fundamental, but I would not dare to claim that we know this for sure.RGauld said:And the SM is known to be effective.
It has long puzzled me that such unwanted degrees of freedom of the photon field (arising from the so-called "continuous spin" operators in the Poincare unirrep) should automatically hook up with local phase changes of the fermion field. The photon "continuous spin" operators are essentially just left over from the rotation group as we contract the invariant mass to zero. But a massive fermion retains physically non-trivial 3-rotation degrees of freedom. So it seems they're not the same degrees of freedom that are being matched up when we minimally-couple fermion and photon to obtain a gauge-invariant interaction. Something else seems to be going on, but I don't understand what.DarMM said:[...] massless spin-1 particles are not well suited to being described by fields (due to them have too few degress of freedom) however. Whenever we try the field has artificial unphysical extra degrees of freedom. The fact that these degrees of freedom can be altered without affecting the physics is known gauge symmetry.
But I doubt that nature "cares" whether we can/can't solve our full interacting theory by perturbation around a free theory.So basically in four dimensions an interacting theory must be a gauge theory, because only the interaction of massless spin-1 particles is infinity/divergence free.
The thing is however, that the extra degrees of freedom are not just unwanted, but unphysical. The photon is simply not an object suited to being described by fields, since it is a rep of the little group ##ISO(2)##, but fields can only naturally carry reps of the little group ##SO(3)##. Only constrained fields can described the photon. So, when one uses an unconstrained field like ##A_{\mu}##, then as you know, we get two extra nonphysical polarisations that cause unitarity to break down. This extra polarisations can then also interact with the electron, either through scattering, but more importantly, they form part of its Coloumb field.It has long puzzled me that such unwanted degrees of freedom of the photon field (arising from the so-called "continuous spin" operators in the Poincare unirrep) should automatically hook up with local phase changes of the fermion field.
It's nothing to do with solving the theory. Theories with divergences do not exist. So perhaps a clearer way to say it is that there doesn't exist a field theory in four dimensions that doesn't contain a massless spin-1 particle and hence has gauge symmetry in the field formalism.But I doubt that nature "cares" whether we can/can't solve our full interacting theory by perturbation around a free theory.
There seems to be an inconsistency in what you're saying: the extra polarisations are unphysical, but... the Coulomb field is physical.DarMM said:[...] So, when one uses an unconstrained field like ##A_{\mu}##, then as you know, we get two extra nonphysical polarisations that cause unitarity to break down. This extra polarisations can then also interact with the electron, either through scattering, but more importantly, they form part of its Coloumb field.
Actually, this might sound odd, but the Coloumb fields aren't physical in the standard approach to QED.strangerep said:There seems to be an inconsistency in what you're saying: the extra polarisations are unphysical, but... the Coulomb field is physical.
It is perfectly derived in http://www.sciencedirect.com/science/article/pii/S0003491684710591DarMM said:Only on a restricted subspace ##\mathcal{H}_{phys}## of the space of states of the theory will the Coloumb field not contain these extra polarisation, however since the electron field is an operator over the whole space it picks up the extra gauge phase regardless.
A gauge theory is a type of mathematical framework used to describe the fundamental forces and interactions in the universe. It is based on the principle of gauge invariance, which states that the laws of physics should not change under certain transformations.
Fundamental theories, such as quantum field theory and general relativity, are often described using gauge theories. This is because gauge theories provide a way to mathematically understand and unify the fundamental forces of nature.
Gauge symmetry is a central concept in gauge theories, as it ensures that the theory is consistent and invariant under certain transformations. This allows for the prediction and understanding of physical phenomena, such as the behavior of particles and the forces between them.
No, not all fundamental theories are gauge theories. While many fundamental theories, such as the standard model of particle physics and general relativity, can be described using gauge theories, there are also other mathematical frameworks used to understand the universe.
Gauge theories are tested and validated through experiments and observations. By making predictions based on the mathematical framework of gauge theories, scientists can compare them to real-world data and confirm their accuracy. For example, the discovery of the Higgs boson in 2012 provided evidence for the validity of the standard model, which is described using gauge theories.