Yeah, it's physics knowledge but it's not really all that difficult. Conventionally, momentum means it tends to keep going: i.e., it takes a force to slow it down or speed it up. Angular momentum has to do with the same phenomena except one is talking about rotation. It takes torque to speed up or slow down rotation. In Newtonian physics, momentum (often represented by the letter “p”) is given by [imath]\vec{p}=m\vec{v}[/imath] and angular momentum is given by [imath]\vec{L}=\vec{r}\times \vec{p}[/imath]. Note that, in a rotational system, velocity is equal to [imath]r \omega[/imath] where omega is the angular velocity. Think of a skater in a spin; as they draw their arms in they are reducing r thus constant angular momentum requires their angular velocity (their spin) to increase.Okay, hmmm... I suppose I should have more physics knowledge to understand exactly what you mean (I don't understand why angular momentum of an entity is given in such and such manner in QM, or what it means exactly), but I also suppose it is not important at this stage...(?)

At any rate, angular momentum is a vector quantity (normally thought of as pointing along the axis of rotation). Thus the angular momentum of the skaters right hand (given he is at the origin of the coordinate system) as he looks in the x direction is given by y times the momentum of his hand in the x direction. The angular momentum of his left hand is -y times the momentum of his hand in the x direction which is exactly the negative of the momentum of his right hand so the two add together. Converting these representation into quantum mechanics (where momentum in the x direction is given by [imath]\frac{\partial}{\partial x}[/imath]), what we are talking about is [imath]y\frac{\partial}{\partial x}[/imath].

When you mess with this mathematically, you will discover that the x, y and z components of angular momentum anticommute with one another. This is exactly what lies behind the idea of “spin” of elementary particles. As I said, we will get to that issue when I derive Dirac's equation. These alpha and beta operators give rise to both “spin” phenomena and electromagnetic phenomena. (Just put this in to keep you interested )

The alpha operator is a vector operator: i.e., [imath]\vec{\alpha}_i = \alpha_{ix}\hat{x}+\alpha_{iy}\hat{y}+\alpha_{iz}\hat{z}+\alpha_{i\tau}\hat{\tau}[/imath]. Each one of those alpha operators is a unique alpha operator; there are four operators associated with each vector alpha associated with a particular xThe chosen "particular specific operator" was denoted as [imath]\alpha_{qx}[/imath], I suppose if it was explicitly stated it could be, say, [imath]\alpha_{3x}[/imath], i.e. it just refers to one specific x in the input arguments.

_{i}. When we set that constraint that [imath]\sum_i\vec{\alpha}\vec{\Psi}=0[/imath], it means that each component of that vector sums to zero.

That x refers to the fact that we are summing over all of the alpha operators associated with the x axis; we are not summing over x. The letter q simply stands for the index of a particular alpha operator. I use a different letter because I cannot use i as the expression is already being summed over all i and I am simply multiplying by a single arbitrary alpha operator (of course in the sum over i , that same operator will occur once but I don't know where because I haven't told you the value of q).But if it's a single specific index, then I don't understand what does it mean to "sum over q", i.e. how does one do a "sum over one specific x"? I'd expect to see just one term in that sum.

It begins life as a simple index; it doesn't serve as a summation index until I decide to sum over that index.... "q" refers to a specific index, but then on the other hand it also refers to a summation index? Where am I getting it wrong?

You are absolutely correct . I have edited the thing to put it in and given you credit for spotting it.(I suspect it's missing by accident since your rearrangement implies it was supposed to be there... if I know my math at all... which I may not )

This is exactly the same as the earlier sum over q. Initially k & l are just indices referring to some specific beta operator. They do not become summation indices until I decide to sum over them.I think I understand that too now, except for the little strangeness regarding k & l referring to specific elements first, but then them being used to perform a sum where they appear as summation indices suddenly... If I didn't know to look at them as a summation indices suddenly (from all our conversations), I'd see a very short explicit sum in my mind :/

Hopefully I have cleared it up. If not, let me know and I will make another attempt.Yeah, I think I have a pretty decent idea of this step, I just hope you can still clear out my uneasyness with those "sums over specific indices" or how should I put it... :I

Sorry I have been so slow, I have been spending a lot of time following the economic crises. Our retirement account has shrunk by about fifteen percent since last year; a bit better than the DOW but still a troubling issue. I keep saying that I am not really concerned with the “value” of my holdings but rather with the return. What I am really afraid of is a collapse of earning or massive inflation, effectively the same thing. I suspect Christmas will be the breaking point. If the US gets through Christmas without a complete collapse I will be a happy man.

How are things in Finland?

Have fun -- Dick