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Hi, I am a string theorist and a publicist.


Oct
12
comment The rebound height of a mass on a trampoline
The tilde means "is proportional to", $y\sim x$ means $y=kx$ for a constant $k$. I am not sure whether I am able to say whether the picture is "right" or "wrong" without further captions.
Oct
12
comment Jarlskog Invariant and its mathematical origin
But the standard parameterization isn't the only parameterization. If the matrix is complex in the standard parameterization, it doesn't mean that it can't be brought to a real form.
Oct
12
comment Jarlskog Invariant and its mathematical origin
I don't understand your comment, user. Your comment is exactly equivalent to the original question and my answer was written in order to answer this question, so why do you ask again? Obviously, CP-violation is present if and only if the matrix cannot be brought into a real form. So even if $s_{13}=0$ but some of the other sines and cosines entering $J$ are zero so that $J=0$, then it is possible to bring the matrix into a real form. The transformation needed to do so is different than those you probably have in mind but it exists.
Oct
11
comment Group theoretical reason that Gluons carry charge and anticharge
Just a correction. The 2-dimensional rep of $SU(2)$ is not real. You need complex numbers but it is equivalent to its complex conjugate - we say that it is pseudoreal or quaternionic.
Oct
9
comment Group theoretical reason that Gluons carry charge and anticharge
Dear Jakob, I think that these are basic questions about group theory and group theory in physics. You find them on first pages of every introductory text about these matters. Take e.g. Howard Georgi, Lie algebras in particle physics, or anything simpler. It doesn't really make sense to answer your questions because you're effectively asking about all the basics of group theory and to answer, one would have to effectively reproduce a whole textbook on these matters because you seem to be starting from scratch.
Oct
9
comment Relative Minus signs from different Feynman Diagrams
Thanks for good news, Jakob!
Oct
9
comment Relative Minus signs from different Feynman Diagrams
Do you understand that $(cd)^\dagger = d^\dagger c^\dagger$ but $(cd)^\dagger \neq c^\dagger d^\dagger$, for example? It's not hard to see that one of your two final matrix elements equals $+1$ and the other is $-1$, and a minute of calculation using $(AB)^\dagger = B^\dagger A^\dagger$ and the anticommutation etc. is enough to see which is which.
Oct
9
comment Relative Minus signs from different Feynman Diagrams
I don't understand how you may misunderstand these things. They differ because one of them has $cd$ somewhere inside and the other has $dc$. Because $cd=-dc$, these two matrix elements are obviously equal to minus each other, aren't they? You may only fail to see why they are minus themselves if you are completely sloppy about all the signs and all the ordering - but indeed, that's a bad starting point to become sure about similar minus sign issues.
Oct
9
comment Group theoretical reason that Gluons carry charge and anticharge
Take the polarizations $J_z$ of the $SU(2)=SO(3)$ generators. The 3-component vector has eigenvalues $-1,0,+1$ - that's for the combinations $L_x\pm i L_y$ and $L_z$ (the latter is the zero), respectively. But the smallest nontrivial representation has $J_z=\pm 1/2$ (this 2-colored "quark" is equivalent to its complex conjugate in this case). So you need to combine two of those to get $J_z=\pm 1$, so the W-bosons and the Z-boson also carry charges that may be obtained from the doublets (e.g. electron+neutrino).
Oct
9
comment Group theoretical reason that Gluons carry charge and anticharge
Dear Jakobh, there is no real difference between $SU(2)$ and $SU(3)$. You write the elements of the $SU(2)$ algebra as a "vector", a combination of Pauli matrices, but this "vector" is really a composite, not-the-smallest, representation. The smallest representation of $SU(2)$ is the 2-component spinor (the "true vector" of $SU(2)$), and the 3-dimensional representation is built from two copies of the 2-component spinors by the same way as the 8-dimensional adjoint of $SU(3)$ is built from the 3-dimensional fundamental rep.
Oct
9
comment Group theoretical reason that Gluons carry charge and anticharge
It's charge-anticharge because the gluon is in the adjoint representation of a non-Abelian group - that's why there are nonzero charges. The adj rep isn't trivial (singlets), so it transforms under itself. The adj rep is a "matrix", and the entries of the matrix are specified by $ij$, the row and column. Matrix $U_{ij}$ is multiplied by a vector $v_i$ on the right side, so if $v_i$ is a quark, the $j$ in $U$ contracts with i.e. annihilates the $j$-th color quark, i.e. carries the charge of the $j$-th antiquark, but $U_{ij}$ creates the $i$-th quark's charge instead.
Oct
8
comment History of the names “Feynman-gauge” & “Landau-gauge”. How arised & how settled?
If you only consider the free field, then the equations for all polarizations are just "box A mu is zero" regardless of xi. The indefinite Hilbert space is the same for each xi, too. The natural normalizatoin of some polarizations etc. may depend on xi but there's no way to choose "natural" if you don't want to consider propagators and interactions. So I don't know what you mean by the xi-dependence of the GB procedure of anything else. The answer to the procedure is just the space of states and it's the same for each xi.
Oct
7
comment Integrals and Legendre Polynomials
Klest: I don't believe that physicians know anything about the Legendre polynomials. ;-) By the way, what is $a$? It only appears on one side. Oh, I see, the result is independent of $a$, it's just scaling the orbit.
Oct
7
comment History of the names “Feynman-gauge” & “Landau-gauge”. How arised & how settled?
@Vinsanity - I find your question confusing because you seem to combine several things. First, the Lorenz gauge is a strict notion, either classical gauge or the "xi is infinity" limit of the R_xi gauges. Those are used for calculations of complicated diagrams with photon propagators and the physical results may be seen to be xi-independent but it's not "quite trivial" although the result may be justified more conceptually. However, the Gupta-Bleuler quantization is a treatment of the external photons which includes the unphysical polarizations and then says how to decouple those.
Oct
6
revised Relative Minus signs from different Feynman Diagrams
added 559 characters in body
Oct
6
answered Relative Minus signs from different Feynman Diagrams
Oct
6
comment Relative Minus signs from different Feynman Diagrams
Thanks, @DavidZ - I wasn't quite satisfied with my remark and wanted at least one more hint what is the stumbling block... Maybe I see it now.
Oct
6
comment Relative Minus signs from different Feynman Diagrams
Do you understand why, for anticommuting $c_i$ variables, $12 c_1 c_2 + 5 c_2 c_1 = 12 c_1 c_2 - 5 c_1 c_2 = 7 c_1 c_2$? If you do, then I can't understand how you could misunderstand the thing you are asking about.
Oct
6
comment Relative Minus signs from different Feynman Diagrams
You need to bring the states to the same order because the two contributions to the amplitude must be standardized in the same way, otherwise you would be adding apples with oranges. One may calculate the amplitude for a fixed well-defined (including the sign) initial state and final state. If your other calculation calculates a different term but the final and/or initial state differ by an odd number of exchanges of fermions, then the initial and final states have to be changed to the same form as the previous term which produces a minus sign.
Oct
4
comment why non orthogonal states are indistinguishable?
You probably meant that non-orthogonal states are not mutually exclusive - motls.blogspot.com/2014/07/… - for two different but non-orthogonal states in quantum mechanics, there is always some probability given by the squared inner product (in absolute value) that one state will fully emulate the other, and some probability that it will not. You can't "prove" such things by pure linear algebra - it is a claim about physics so you need some postulates of physics (Born's rule).