LuboŇ° Motl

Czech Republic

motls.blogspot.com

Age: 40

Hi, I am a string theorist and a publicist.

Apr
15
awarded Good Answer
Apr
10
awarded Nice Answer
Apr
10
awarded Enlightened
Apr
4
comment Klein factors and Conformal Field Theory
Dear Costa, the right prefactor (normalization) is clearly needed for the right normalization of the commutator, but that's it. The essence of the operator is the same, however. It's like if you used $X=x(a/L)^n$ instead of $x$, then the commutator $[X,p]$ will be $(a/L)^n \cdot i\hbar$ instead of $i\hbar$. But the special point about the exponential operators is that only the (divergent) overall normalization is affected by the normal-ordering effects. At least I hope so.
Apr
4
answered Klein factors and Conformal Field Theory
Mar
29
comment If the earth left the solar system for interstellar space. How long would it take for atmosphere to freeze
You're right, thanks.
Mar
28
comment Selection Rules in electron spectroscopy
Dear Laurent, this is really the general situation. The orbital angular momentum does not commute with the full Hamiltonian, and neither does the spin, so they are not "good quantum numbers". It's still possible to diagonalize the Hamiltonian. In a calculation method, we may diagonalize the Hamiltonian without the L.S interaction, and that part commutes with L and S separately. If we can't find a useful "bulk" of the Hamiltonian that commutes with L,S separately, then L,S eigenstates are useless in any search for the H eigenstates, but H still has eigenstates and they may be found otherwise.
Mar
26
answered What is probability current in quantum mechanics?
Mar
20
awarded Nice Question
Mar
19
comment What's the $\ell$ in the Bicep2 paper mean?
The local peak from the primordial gravitational waves is normally expected around $\ell=90$. The spherical harmonic $Y_{90,90}$, for example, has the $J_z$ aligned "maximally vertically, so it is spinning maximally vertically among the $Y_{90,m}$ harmonics, and on this one, you see that the angular dependence contains $\exp(90 i\phi)$ which contains 90 maxima around the circle. So the "wavelength" of the component spans 360/90=4 degrees on the sky. The resolution has to be a bit better to actually "see" the shape of these waves.
Mar
17
comment What was the major discovery on gravitational waves made March 17th, 2014, in the BICEP2 experiment?
A more advanced essay on what the discovery means, by Prof Liam McAllister of Cornell: motls.blogspot.com/2014/03/…
Mar
13
comment Heisenberg's uncertainty principle - Planck's (reduced) constant divided by two or not?
Well, in these formulae, $\Delta x$ means the standard deviation $\sigma_x$ or whatever is your preferred other notation.
Mar
11
answered Can a D-brane be closed and contractible?
Mar
11
comment Isometries and Kaluza Klein theories
Not really. If you pick an "irrational" combination of two natural generators, $t_a+K t_b$ where $K$ is irrational, the group that this combined generator generates is isomorphic to $R^+$ and not $U(1)$ because you will never return quite to the same place - the periodicity disappears. But it's always possible to choose a Cartan (maximal commuting) subalgebra of commuting and compact generators isomorphic to $U(1)^\ell$. Any generator of a compact group may be written as a combination of some generators in some $U(1)^\ell$ subalgebra.
Mar
10
answered Isometries and Kaluza Klein theories
Mar
6
comment Why does $1+2+3+\dots = {-1\over 12}$?
Sure, it's a completely analogous sum. Just like $1+2+3+\dots$ quantifies the ground state energy in 1+1 dimensions (of a string), $1+1+1+\dots$ quantifies the ground state's charge (of a system of free fermions, for example). It's really the reason why the two degenerate states $|0\rangle$ and $c_0|0\rangle$ have ghost numbers $\pm 1/2$, for example.
Mar
5
comment Does limit $\hbar \rightarrow 0$ in Quantum Mechanics mean anything?
Raskolnikov, you meant $c\to\infty$, not $c\to 0$.
Mar
1
answered Majorana mass vs Dirac Mass
Feb
21
awarded Guru
Feb
19
comment How can a photon have no mass and still travel at the speed of light?
Dear @clicky-hacky-clicky-hacky, sorry for using a transliterated version of your name via Czech (orig: klikyháky, klikykáky). :-) I don't have the right keyboard to write your name and copy-and-paste would be mindless and disrespectful... If individual claims in a set of claims in physics imply each other so that you may find a loop that you call "circular reasoning", it surely doesn't mean that any of the claims is wrong. Instead, everything I write and everything I have written in my life, more or less, is true. In physics, some claims have to be assumed/extracted from observations.
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