particle physics - Is the right-handed electron really an $SU(2)$ singlet?

In the Standard Model, neutrinos are assumed to be massless, and the right-handed neutrinos thus do not exit. Is this the reason that the right-handed electron is regarded as an $SU(2)$ singlet? However, to be more precise (going beyond the Standard Model), neutrinos do have masses. This implies that right-handed neutrinos must exist. Then, is it more reasonable to think a right-handed electron and a right-handed neutrino to be an $SU(2)_{R}$ doublet? If this is true, the left-handed $SU(2)_{L}$ doublet $(\nu_{e}, e^{-})_{L}$ and the right-handed $SU(2)_{R}$ doublet $(\nu_{e}, e^{-})_{R}$ are symmetric, and the Yukawa interactions between the Higgs field and these lepton doublets will be entirely different from that of the Standard Model.

Answer

Many of the above misconceptions would not have been enunciated and built up to an impossible crescendo if only a good summary of the SM were consulted before taking off on their avalanche.

The standard model is a tight gauge theory of charged (V-A) interactions and neutral current interactions of subtler chirality constraints. So it is simply not true that

In the Standard Model, neutrinos are assumed to be massless, and the right-handed neutrinos thus do not exit.

When the SM was written down, no neutrino masses were known, so right-handed neutrinos were not included in it. Nevertheless, the electron did get a mass by a suitable higgs coupling to the left-handed electron multiplied by a right-handed SU(2)_L-singlet electron. This is the celebrated "second job of the higgs", and theorists cheered when they appreciated its genius.

As a result, instantly in 1968, any theorist would reassure you that, ipso facto, the neutrino could easily get a mass with a very analogous coupling of the neutrino to a conjugate higgs doublet, if there were a reason to accommodate such. (In fact, in the extension to quarks, both up and down type quarks got their masses via SU(2)_L-singlet right-chiral components.) They simply stuck to Occam's razor and left neutrino masses an open question, skipping them until discovery. It is doubly important to understand that the discovery of neutrino masses did not upset or reverse absolutely anything about the structure and logic of the SM.

(Indeed, to hype up the surprising neutrino mass discovery, sensationalist self-promoting sides kept on insisting they went "beyond the SM", but this only betrayed they had never fully appreciated its symmetry structure and logic and had no compunction about confusing the impressionable: this question may be one of their victims. So "within" or "beyond" the SM are mere meaningless words expressly tossed in to confuse the confusable, by isolating those in the know. There comes a time when opinionatedness may well be less damaging than its avoidance.)

While the full nature of neutrino masses is still open, an SM Dirac mass for them analogous to the electron mass is not excluded.

Now, having reassured full left-chiral invariance at low, energies, one might dream about inter-connecting the right-handed leptons with fanciful hypothetical interactions, truly beyond the SM: there have been models putting the electron and the neutrino in right-handed doublets, but they involve superheavy W analogs in a different gauge group SU(2)_R, and completely different exotic couplings visible at much, much higher energies than accessible at present, and I assume you are not asking about them: extant experimental bounds reassure us that no such things are faintly visible or even arguable at the standard energies of HEP today. The PDG provides exclusion limits for right-handed charged currents.

My strong urge is to suspend speculation of beyond-the-SM worlds unless a full admiring appreciation of the perfect-fitting-puzzle nature of the SM were achieved, fitting together tons of low (ours) energy data. To answer the question, then, the right-handed e is incontrovertibly an SM gauge singlet and there is absolutely no evidence or hint it might be part of an SU(2)_R doublet.

acoustics - Is sound a Nambu-Goldstone mode?

The usual sound exists in solids, liquids, and gases, as a long-wavelength excitation with linear dispersion. Can its presence be attributed to the spontaneous breaking of some symmetry? In other words, is it a Goldstone mode of some symmetry?

fluid dynamics - Liquid analog of the Black Hole (Part 2)

Similar question has been asked but did no get good answers.

We know that black hole can confine light, but can we create an ocean black hole - which can confine surface water waves or sound waves in water?

Definition:

• What is a static water flow which can CONFINE water waves?


• Is there a configuration confining only surface waves?



• Is there a configuration confining volume waves?

Description:

Inside a confined region any water density perturbations can not propagate outside.

PS:

• Overall volume of water in the "tank" can not change so the solution when the water is removed faster then the waves travel is ruled out.


• There possibly can be additional objects presented in the water "tank".


• It can be some unsual liquid or gas, whith additional "new" properties.

Thanks for your answers.

quantum mechanics - Scattering vs bound states

Why are these states called as such, and how do they differ? I vaguely understand that when $E > 0$ you obtain a scattering state, but when $E < 0$ you have a bound state.

Answer

These terms apply when you're solving the Schrodinger equation with a potential that goes to zero at large distances. In this situation, the solutions with $E<0$ have the property that $\psi$ dies away to zero for large distance. So the particle is, with high probability, guaranteed to be in a confined region (not at large distance). So those are bound states.

The solutions with $E>0$, on the other hand, do not die away to zero at large distances -- instead, they go like $e^{ikr}$ where $k=\sqrt{2mE}/\hbar$. So these solutions represent particles that have high probability to be arbitrarily far away. Physically, they are useful when describing particles that start far away, approach the scattering center, and end up far away again. Hence the name "scattering states."

quantum field theory - Fock space vs. wavefunctionals

There are at least two representations of the Hilbert spaces of quantum field theory. For a scalar field, we have

The Fock space representation, such that every state is represented as the Fock space of 'one particle' representations,

$$\mathcal H = \bigoplus_{n = 0}^\infty \text{Sym}(\mathcal H_1^{\otimes n})$$

with $\mathcal H_1^0 = \mathbb C$ and $\mathcal H_1 = \mathcal L^2(\Bbb R^3)$.

The wavefunctional representation, such that every state is represented by a functional on the space of configurations of the field on $\Bbb R^3$ which is square integrable with respect to some measure

$$\mathcal H = \mathcal L^2(\mathcal F[C(\Bbb R^3)], d\mu[\phi])$$

Since those two representations describe the same theory, I am guessing that there is some isomorphism between the two, but what would it be?

general relativity - What is meant when it is said that the universe is homogeneous and isotropic?

It is sometimes said that the universe is homogeneous and isotropic. What is meant by each of these descriptions? Are they mutually exclusive, or does one require the other? And what implications rise because of this?

Answer

A homogeneous cosmology is one in which there are no "special" places in the universe: at a given instant in time, the universe appears the same at every location (on large enough spatial scales).

An isotropic cosmology is one in which there are no "special" directions: at a given instant in time, the universe appears the same in every direction (again, on sufficiently large spatial scales).

Together, they form the Cosmological principle.

As pointed out by Brian Hooper, these symmetries (when applied to physical laws) give rise to the conservations of linear and angular momentum as a result of Noether's theorem.

In addition, the cosmological principal is important for the physical interpretation of observational data, and not only because it is a generally unspoken assumption when using physical laws tested on Earth to model distant objects (galaxies, quasars, etc.) For example, it supports the interpretation of the Hubble diagram as the result of the expansion of the universe as opposed to evidence that the Earth (or someplace "nearby") was at the center of a very big conventional explosion. After all, in a conventional explosion, the fragments that travel the farthest are those that had the highest velocity, so some time after the explosion, faster moving fragments are further away from the center. If, however, observers on all fragments see the same density of galaxies and relationship between velocity and distance in all directions, then this model doesn't work.

electric current - Why does vacuum have a nonzero characteristic impedance towards electromagnetic radiation?

On Wikipedia, the impedance of free space $Z_0$ is defined as square root of the ratio of the permeability of free space $\mu_0$ to the permittivity of free space $\epsilon_0$, i.e. $$Z_0 = \sqrt{\mu_0 / \epsilon_0} \, .$$ The value is approximately 377 Ohms.

Now impedance is described as an impeding effect to flow of something, it makes more sense for electric current travelling in a wire where the characteristic impedance of the line (as the line consists of capacitance and inductance per unit length) prevents the flow of AC/DC flow.

Why in the world does free space have a characteristic impedance? That makes no sense to me. Wires makes a lot of sense, but free space having 377 ohm of impedance is too much and not clear why such a value exists.