Idempotents and involutions and a feeling of purpose?

Me and Quantum Field theory, or really advanced quantum physics in general, just aren’t getting along right now. The subject itself really  is fascinating and whenever I have come across something which I’ve found strange or difficult in the past it has always pushed me to want to learn more about it. I think like most students I am in a constant pursuit of that lovely intellectual high, the feeling of pure and genuine understanding that can ambush you at any time when you’re trying to think deeply or rigorously and which fills you with a strange joy and excitement which a piece of text or an abstract idea has no right to give you. But for some reason that feeling has eluded me for almost 4 months now and has instead been substituted for the stale and annoying:

“I don’t get it…”

Ignorance or lack of knowledge has never itself ever caused me any distress. Things you do not know are simply things you have not learnt yet and things you cannot learn are not going to hold any relevance. Time and energy spent are the principal factors when it comes to learning anything but I have felt so drained of energy lately dredging through archaic formulas and regurgitating statements such as “the carriers of the strong interaction are the gluons” feeling no personal connection to it. And when I feel drained of energy or that the end goal is obscure that of course just feeds back into a loop and slows progress even further. This really is petty stuff but if you don’t feel that deep connection you to the stuff you’re supposed to spend 50% of your time thinking about then life certainly does lose some of it’s direction and I like having a sense of direction.

I need my flashes of insight to function but right now and the path I usually try to take to get to it are to try and delve deeper into the mathematical side. It will probably turn out to be a time-hole that further messes up my studies but I’m going to have to look into algebras again, in particular Clifford algebras in order to try to understand what the hell Dirac spinors are in a more fundamental way. (Then maby it will make more sense). What gets me so excited today is that I can see the outline already or a tiny parcel of insight that I feel a tingleing will pay off shortly.

In group theory I have already come across the terms idempotent elements and involution before but since it had just been in passing I had dismissed them as simply useful labels without much significance to the stuff I was interested in. An idenpotent element p is after all just an element which when composed with itself yields itself and an involution j  is just one which when composed with itself gives the multiplicative identity.

p^2 = p, \qquad j^2 = 1

What I had not though of was that in a unital algebra (a vector space with a multiplicative structure and a multiplicative identity) these things are related and for each idempotent $j$ one can define two corresponding idempotent elements

\frac{1}{2}(1 + j), \qquad \frac{1}{2}(1 - j)

Now firstly that is awesome because it is really easy for me to find involutions but harder to find idempotent operators but the thing is I have already seen this thing a thousand times before but suddenly all those disparate observations are collapsed into a single concept. Most importantly I recognise this relationship as precisely that of the chirality in QFT which will (likely) provide an explanation for the concept of left- and right handed spinors which in my available literature has been fairly axiomatic.

Oh well, I guess I need to work on figuring out what it is I want this blog but I think focusing on what flies into my head at 2 in the morning is a good start. One will have to work on the length in future entries.


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