I am currently studying some basic materials about Majorana spinor 1 form.
I know usual (0-form) Majorana spinor very well(i guess). They satisfy the ψc=ψ for majoran spinor ψ and from them we can do some computation as follows
¯λχ=¯χλ¯λγμχ=−¯χγμλ
Deriving above formula one use
ψc=ψ and
λaχb=−χbλa. and so on.
Here come back to the problem, for Majorana spinor 1 form, only surviving form of majorana spinor is known as ¯ψγaψ≠0 and ¯ψγabψ≠0.
From deriving above things i got stucked.
I think my main problem is the lack of concept for Majorna spinor 1 form
First usual spinor 1 form that i know is
A=Aμdxμ
In that sense i can do for Majorana spinor 1 form
ψ=ψμdxμ
What i want to know is for 1-form Majorna spinor case is still grassman property holds?
i.e
Ψχ=−χΨ
or it only holds for coefficient ψμ, χμ for one form? i.e for Ψ=ψμdxμ, χ=χμdxμ
ψμχν=−χνψμ
I know for the wedge product of p and q form gives
w(p)∧w(q)=(−1)pqw(q)∧w(p)
What i really want to do is following derivation for Majorna spinor 1 form
¯Ψγ5Ψ=0
¯Ψγ5Ψ=Ψa(Cγ5)abΨb=ΨaΨb(Cγ5)ab=(−?)ΨbΨa(Cγ5)ab
Does the − sign holds?
I can do the other leftover derivation except above parts.
If you know something please let me learn from you.
This post imported from StackExchange Physics at 2015-05-14 21:01 (UTC), posted by SE-user phy_math