- #1
Tomsk
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Hi, I'm trying to teach myself a bit about spinors, mainly from reading about geometric algebra. There is something that I can't figure out though. According to GA, spinors are elements of the even graded subalgebra, so scalars, bivectors and so on. But the electromagnetic field is a bivector, but surely it's not a spinor... so how does that make sense?
Also: When they talk about even graded elements being spinors, are they talking about spin 1/2, 1, 3/2... or something else?
The other thing I'd like to ask about is this unclear section on wikipedia:
I didn't really understand the bit about Clebsch-Gordon decomposition but I really don't get how [itex]\bar{\phi}\psi[/itex] can be considered as a scalar and/or a vector. I can see why it would be a scalar but not a vector as well. (BTW the latex \bars don't seem to be showing up)
These are probably fairly basic things... I'm pretty new to it all!
Also: When they talk about even graded elements being spinors, are they talking about spin 1/2, 1, 3/2... or something else?
The other thing I'd like to ask about is this unclear section on wikipedia:
http://en.wikipedia.org/wiki/Spinor#ConsequencesConsequences
There are many far-reaching consequences of the Clebsch-Gordan decompositions of the spinor spaces. The most fundamental of these pertain to Dirac's theory of the electron, among whose basic requirements are
* A manner of regarding the product of two spinors [itex]\bar{\phi}\psi[/itex] as a scalar. In physical terms, a spinor should determine a probability amplitude for the quantum state.
* A manner of regarding the product [itex]\bar{\phi}\psi[/itex] as a vector. This is an essential feature of Dirac's theory, which ties the spinor formalism to the geometry of physical space.
I didn't really understand the bit about Clebsch-Gordon decomposition but I really don't get how [itex]\bar{\phi}\psi[/itex] can be considered as a scalar and/or a vector. I can see why it would be a scalar but not a vector as well. (BTW the latex \bars don't seem to be showing up)
These are probably fairly basic things... I'm pretty new to it all!