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• Users: terra
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1. ### SU(2) lepton doublet conjugation rules

Ok, I think I got your point. Under an infinitesimal transformation ##\psi_{\nu,L} \to \psi_{\nu,L} \big( 1 + i \alpha_3 \big) + \psi_{e,L} \big( i \alpha_1 + \alpha_2 \big)##, ##\psi_{e,L} \to \psi_{e,L} \big( 1 - i \alpha_3 \big) + \psi_{\nu,L} \big( i \alpha_1 - \alpha_2 \big)## and I can...
2. ### SU(2) lepton doublet conjugation rules

Yes, but as you can see, I can't see how.
3. ### SU(2) lepton doublet conjugation rules

Ok. But for me, ##\ell^c_L## is just a symbol with components ##\ell_1## and ##\ell_2##, so I still can't see what tells me it doesn't transform appropriately.
4. ### SU(2) lepton doublet conjugation rules

Whoops. If ##\ell^c_L## is left-chiral, it should transform trivially, right? But I still don't see how does this help.
5. ### SU(2) lepton doublet conjugation rules

I had two suggestions. As I see it, my naive suggestion would transform exactly the same way, whereas the one multiplied with ##-i \sigma^2## would transform as ##\begin{pmatrix} -\psi^c_{e,L} \\ \psi^c_{\nu,L} \end{pmatrix} \to \begin{pmatrix} -\psi^c_{e,L} \big[ \cos(|\boldsymbol\alpha|) +...
6. ### SU(2) lepton doublet conjugation rules

Ok. I appreciate your help and sorry I can't follow your point. I'll go back to your previous reply. The components transform as ##\psi_{\nu,L} \to \psi_{\nu,L} \big[ \cos(|\boldsymbol\alpha|) + \sin(|\boldsymbol\alpha|) i \alpha_3 \big] + \psi_{e,L} \sin(|\boldsymbol\alpha|) (i \alpha_1 +...
7. ### SU(2) lepton doublet conjugation rules

My question was not about transforming ##\ell_L## under ##SU(2)## but under hermitian and charge conjugation and what is meant by the notation ##\bar\ell_L##. I'm sorry, but I still can't see how I see them from the transformation properties of ##\ell_L## under ##SU(2)##, as I don't know how is...
8. ### SU(2) lepton doublet conjugation rules

Assuming ##\psi_{\{\nu,e\}L}## are Dirac spinors for which ##\psi_L := P_L \psi## I have, in Weyl's representation defined by: \begin{align*} \gamma^0 &= \begin{pmatrix} 0 & \mathbb{1} \\ \mathbb{1} & 0 \end{pmatrix}, \gamma^i = \begin{pmatrix} 0 & \sigma^i \\ -\sigma^i & 0 \end{pmatrix}...
9. ### SU(2) lepton doublet conjugation rules

##\ell_L \to U \ell_L = \begin{pmatrix}\cos(|\boldsymbol\alpha|) \psi_{\nu,L} + \sin(|\boldsymbol\alpha|) \big[ i \alpha_3 \psi_{\nu,L} + (i \alpha_1 + \alpha_2) \psi_{e,L} \big] \\ \cos(|\boldsymbol\alpha|) \psi_{e,L} + \sin(|\boldsymbol\alpha|) \big[ (i \alpha_1 - \alpha_2) \psi_{\nu,L}- i...
10. ### SU(2) lepton doublet conjugation rules

Yes, that's what I meant. I also know the conjugation rules for Dirac spinors. I don't see where you are going, neither do I see how can I derive the transformation properties of these doublets from those of the Dirac spinors unless they follow 'trivially', that is ##\bar\ell = \begin{pmatrix}...
11. ### SU(2) lepton doublet conjugation rules

Which one? I'm sorry, but I can't follow. An infinitesimal ##SU(2)## transformation would read ##\mathbb{1} + i \alpha^a \sigma^a / 2 = \mathbb{1} + \begin{pmatrix} i \alpha_3 & i \alpha_1 + \alpha_2 \\ i \alpha_1 - \alpha_2 & - \alpha_3 \end{pmatrix}## (or something along those lines), right...
12. ### SU(2) lepton doublet conjugation rules

I have a left-handed ##SU(2)## lepton doublet: ## \ell_L = \begin{pmatrix} \psi_{\nu,L} \\ \psi_{e,L} \end{pmatrix}. ## I want to know its transformation properties under conjugation and similar 'basic' transformations: ##\ell^{\dagger}_L, \bar{\ell}_L, \ell^c_L, \bar{\ell}^c_L## and the general...
13. ### Lorentz transforming a momentum eigenstate

That's right, sorry, I was sloppy adding that. Weinberg shows that $$U(\Lambda) \Psi_{p,\sigma} = \sum_{\rho} D^{(j)}_{\rho \sigma}(W(\Lambda,p)) \Psi_{\Lambda p, \rho}$$ where ##W(\Lambda,p)## is such that ##W^{\mu}_{\nu} k^{\nu} = k^{\mu}## so such transformations form a little group for...
14. ### Lorentz transforming a momentum eigenstate

Let's take a quantum state ##\Psi_p##, which is an eigenstate of momentum, i.e. ##\hat{P}^{\mu} \Psi_p = p^{\mu} \Psi_p##. Now, Weinberg states that if ##L(p')^{\mu}\,_{\nu}\, p^{\nu} = p'##, then ##\Psi_{p'} = N(p') U(L(p')) \Psi_{p}##, where ##N(p')## is a normalisation constant. How to...
15. ### 2j+1 d representation for Poincaré group

I want to learn how to write down a particle state in some inertial coordinate frame starting from the state ##| j m \rangle ##, in which the particle is in a rest frame. I know how to rotate this state in the rest frame, but how does one write down a Lorentz boost for it? Note that I am not...
16. ### Non-leptonic hadron decays: preferred paths?

I see, thank you for your insight. I take that by the qcd scale you mean the scale at which the coupling constant diverges, and as such perturbation theory can't be used.
17. ### Non-leptonic hadron decays: preferred paths?

My background of some introductory courses in particle physics has left me with severe shortcomings. Say we start from a hadron, which decays purely to other hadrons. My question is this: through which interaction does the process take place? Is there a preferred interaction, and why/why not...
18. ### Cohen-Tannoudji on mutually exclusive (?) events

Thanks for the reply, yeah. Let ## a,b## be exclusive events. For classical physical things ## P(a\mathrm{ \, or \, }b)= P(a) + P(b)##. In quantum physics, however, we have ## A(a\mathrm{ \, or \, }b)= A(a) + A(b)## where ##A \in \mathbb{C}## is a 'probability amplitude' so that ## P(x)= A(X)...
19. ### Cohen-Tannoudji on mutually exclusive (?) events

I was looking at what Cohen-Tannudji has to say on compatibility of observables. Assumptions: ## A,B## are operators such that ##[A,B]=0 ## and we denote ## |a_i \,b_j\rangle## to be states for which ##A | a_i \, b_j \rangle= a_i | a_i \, b_j \rangle##, ##B | a_i \, b_j \rangle= b_j | a_i \...
20. ### About role of Planck constant in classical physics

I agree, now that you said it, my last example probably goes by the name 'semiclassical'. And one more note about your first question. The intuition I tried to give is more of a guideline, I think, to determine whether or not one should be aware of the commutation relations. I didn't really...
21. ### About role of Planck constant in classical physics

Let me try to answer your first question first. The essential thing, as you said, is to compare Planck's constant to other variables in your system. If other variables in your system are 'small enough', Planck's constant becomes larger in comparison. For example, let's look at the scale of the...
22. ### Volume elements of phase space

Possibly a last update. I think this problem is pretty much solved (or, well, solved in an alternative way). I still couldn't see ##\mathrm{d}/\mathrm{d}t \, \mathrm{d}\Gamma \, \rho= 0## from any equations. But, as the last messages show, I was confused about the definition of the probability...
23. ### Hamiltonian time dependence

I see. Thank you for explaining (and apologies for my slow replies). I see that energy and Hamiltonian coincide in the rotating frame, and are both conserved in that frame. About when ## H## is and isn't energy (in a classical system). One can write down the following. ## H= p_i \dot{q}_i - L##...
24. ### Volume elements of phase space

I do have some updates on this problem. I'm still not convinced with the above derivation. Maybe it works with some explanation, but I prefer something mathematically more straight-forward. However, I did find an alternative derivation, which, to my mind, goes more smoothly. As a reminder, we...
25. ### Hamiltonian time dependence

True, thanks for the correction (as you can see, at first I talked about quantum systems). I've heard this before, but I must admit I haven't thought about it that much. In your example the idea is to get the equation of motion for the ball in pipe's rest frame? The pipe doesn't rotate in its...
26. ### Hamiltonian time dependence

Hamiltonian gives the energy of a system. Let's discuss the case of pure states (where we have quantum states that can be written as vectors ## | \psi \rangle ##). Conservation of energy means that the (expectation value of) amount of energy does not change in time, i.e. ## \mathrm{d} \langle...
27. ### Volume elements of phase space

First, two definitions: let ## \varrho (M)## be the probability density of macro states ##M ## (which correspond to a subgroup of the phase space) and ## \mathrm{d} \Gamma ## be the volume element of a phase space. In my lecture notes, the derivation for continuity equation of probability...