Register to reply 
Regarding the Clifford algebra and spinors 
Share this thread: 
#1
Feb1513, 01:45 PM

P: 102

Hello! I´m currently taking a course in RQM and have some questions for which I didnt get any satisfactory answers on the lecture. All comments are appricieted!
1. Is the gamma zero tensor some kind of metric in the space for spinors? When normalizing our solution to the Dirac equation it seems as we use them exactly as our metric in SR. 2. As I understood it the gamma matricies are Lorentz invariant tensors. However when taking the norm of a spinor describing a particle moving in lets say the zdirection we get a answer that depends on the energy (only one component of our 4momentum). In other words it seems as if our normalization is not invariant under Lorentz transformations and therefore not gamma 0? Thanks in advance! 


#2
Feb1513, 02:51 PM

Sci Advisor
Thanks
P: 4,160

For a Lorentz 4vector, x·x is not invariant. To get an invariant you have to insert a metric, x·η·x where η = (1, 1, 1, 1).
Likewise for spinors, ψ^{†}ψ is not an invariant, you must insert a 4x4 matrix that acts like a metric, and consider instead ψ†ηψ. Now if a Dirac spinor transforms like ψ → Lψ under a Lorentz transformation, then ψ^{†}ηψ → ψ^{†}L^{†}ηLψ, and so the condition that this quantity is a Lorentz scalar is L^{†}ηL = η. The solution is η = γ^{0}. 


#3
Feb1513, 07:33 PM

P: 102

Is it mathematically correct to actually call γ^{0} our metric in the space of spinors? Or is this just a smiliarity of how we use an metric? 


#4
Feb1613, 05:48 AM

P: 136

Regarding the Clifford algebra and spinors
Surely, if the electron is moving along the z axis then [itex]p_z = p[/itex] and then
[tex]1  (\frac{p}{(E + m)})^2 = \frac{E^2 + 2Em + m^2  p^2}{(E + m)^2} = \frac{2m}{(E + m)}[/tex] I'm no expert, but this doesn't look invariant to me. 


#5
Feb1613, 06:22 AM

Sci Advisor
Thanks
P: 4,160

The correctly normalized plane wave solution has an additional factor of ((E + m)/2m)^{½} in front. E.g. Bjorken and Drell vol I gives the plane wave solution as
ψ(p_{z}) = ((E + m)/2m)^{½}(p_{z}/(E + m), 0, 1, 0) Multiply your answer of 2m/(E + m) by this additional factor squared, and you get 1. 


#6
Feb1613, 07:26 AM

P: 1,020

for free particle spinors,the convention is
u^{}u=2mS[itex]\dagger[/itex]S,where S is two component spinor normalized by S[itex]\dagger[/itex]S=1. 


#7
Feb1613, 10:01 AM

P: 136

Thanks Bill



#8
Feb1613, 12:39 PM

P: 102

Yeah I agree with you, but dont you see the problem I´m pointing at? :)
1) Our metric (γ^{0}) defines an inner product which gives us a norm, ψ^{2} = ψ^{†}ηψ. 2)When I calculate the norm of a spinor it is dependent of its energy. 3) Energy is not a Lorentz scalar so the norm is not invariant under Lorentz transformations. So it seems to me that our metric is not Lorentz invariant. But I assume that there is some misunderstanding in my logic.. 


#9
Feb1613, 02:14 PM

Sci Advisor
Thanks
P: 4,160

The full set of wavefunctions for given p consist of two positive energy solutions u_{r}(p) and two negative energy solutions v_{r}(p) where r = ±1 is the spin coordinate. The normalization is: u_{r}(p)u_{s}(p) = δ_{rs} v_{r}(p)v_{s}(p) =  δ_{rs} u_{r}(p)v_{s}(p) = v_{r}(p)u_{s}(p) = 0 


#10
Feb1713, 06:39 AM

P: 102

Yeah, Im with you now. Dont know why but I regareded your additional normalization factor as a dimensionless constant which only normalizes for one inertial frame. :)



Register to reply 
Related Discussions  
Why are spinors interesting, from a Clifford algebra perspective  Linear & Abstract Algebra  7  
Understanding spinors and Clifford product  General Physics  1  
Spinors in d dimensions and Clifford algebra  High Energy, Nuclear, Particle Physics  1  
Action of Cliffordelements on vectors & spinors  Differential Geometry  1  
Clifford algebra isomorphic to tensor algebra or exterior algebra?  Linear & Abstract Algebra  2 