Question

Understanding the Dirac spinor representation $(1/2,0) \bigoplus (0,1/2)$ of the Lorentz group?

Answer 1

I read that the generators of the Lie algebra in this representation are

Jk=(12σk0012σk)Jk=(12σk0012σk)

(Rotations) and

Kk=(−i2σk00i2σk)Kk=(−i2σk00i2σk)

(Boosts) Since u∈SU(2)u∈SU(2) is parametrized by

exp(−θkσk)=(eiθ1cos(θ2)eiθ3sin(θ2)−e−iθ3sin(θ2)e−iθ1cos(θ2))exp⁡(−θkσk)=(eiθ1cos⁡(θ2)−e−iθ3sin⁡(θ2)eiθ3sin⁡(θ2)e−iθ1cos⁡(θ2))

It would seem

exp(iakJk)=(exp(iak2σk)00exp(iak2σk))exp(iakJk)=(exp(iak2σk)00exp(iak2σk))

=⎛⎝⎜⎜⎜⎜⎜⎜⎜⎜e−a12cosh(a22)−ie−a32sinh(a22)00iea32sinh(a22)ea12cosh(a22)0000ea12cosh(a22)iea32sinh(a22)00−ie−a32sinh(a22)e−a12cosh(a22)⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟=(e−a12cosh(a22)iea32sinh(a22)00−ie−a32sinh(a22)ea12cosh(a22)0000ea12cosh(a22)−ie−a32sinh(a22)00iea32sinh(a22)e−a12cosh(a22))

Which I find odd since a boost would be of the same format but with trigonometric instead of hyperbolic functions-which seems backwards since rotations are parametrized with 3 spherical angles and boosts with 2 spherical angles and 1 hyperbolic angle. Did I mess up my algebra or is there an explanation for this that I'm missing

Answer 2

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✔️✔️Understanding the Dirac spinor representation $(1/2,0) \bigoplus (0,1/2)$ of the Lorentz group

I read that the generators of the Lie algebra in this representation are

exp(−θkσk)=(eiθ1cos(θ2)eiθ3sin(θ2)−e−iθ3sin(θ2)e−iθ1cos(θ2))exp⁡(−θkσk)=(eiθ1cos⁡(θ2)−e−iθ3sin⁡(θ2)eiθ3sin⁡(θ2)e−iθ1cos⁡(θ2)) It would seem exp(iakJk)=(exp(iak2σk)00exp(iak2σk))exp(iakJk)=(exp(iak2σk)00exp(iak2σk))

=⎛⎝⎜⎜⎜⎜⎜⎜⎜⎜e−a12cosh(a22)−ie−a32sinh(a22)00iea32sinh(a22)ea12cosh(a22)0000ea12cosh(a22)iea32sinh(a22)00−ie−a32sinh(a22)e−a12cosh(a22)⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟=(e−a12cosh(a22)iea32sinh(a22)00−ie−a32sinh(a22)ea12cosh(a22)0000ea12cosh(a22)−ie−a32sinh(a22)00iea32sinh(a22)e−a12cosh(a22))



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