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Question 4: [6+6+4=16 Marks] Under a Lorentz transformation $\Lambda$, the Dirac field $\psi(x)$ transforms as $\psi(x) \rightarrow \tilde{\psi}(\tilde{x}) = S(\Lambda)\psi(s)$ were the 4x4 matrix $S(\Lambda)$ satisfies $S^{-1}(\Lambda)\gamma^\mu S(\Lambda) = \Lambda^\mu_{\nu}\gamma^\nu$ (a) Find how the following quantities transforms by applying the Lorentz transformation, where $\gamma^5 = i\gamma^0\gamma^1\gamma^2\gamma^3$ (i) $\bar{\psi}(x)\gamma^5\psi(x)$ (ii) $\bar{\psi}(x)\gamma^5\gamma^\mu\psi(x)$ (b) Also check that, by applying parity transformation, $\bar{\psi}(x)\gamma^5\psi(x)$ transforms as pseudo-scalar, while $\bar{\psi}(x)\gamma^5\gamma^\mu\psi(x)$ transforms as axial vector. (c) Show that the adjoint spinors $\bar{u}^{1,2}$ and $\bar{v}^{1,2}$ satisfy the equations $\bar{v}\gamma^\mu p_\mu + m = 0, \bar{u}\gamma^\mu p_\mu - m = 0$

          Question 4: [6+6+4=16 Marks]
Under a Lorentz transformation $\Lambda$, the Dirac field $\psi(x)$ transforms as
$\psi(x) \rightarrow \tilde{\psi}(\tilde{x}) = S(\Lambda)\psi(s)$ were the 4x4 matrix $S(\Lambda)$ satisfies $S^{-1}(\Lambda)\gamma^\mu S(\Lambda) = \Lambda^\mu_{\nu}\gamma^\nu$
(a) Find how the following quantities transforms by applying the Lorentz transformation,
where $\gamma^5 = i\gamma^0\gamma^1\gamma^2\gamma^3$
(i) $\bar{\psi}(x)\gamma^5\psi(x)$ (ii) $\bar{\psi}(x)\gamma^5\gamma^\mu\psi(x)$
(b) Also check that, by applying parity transformation, $\bar{\psi}(x)\gamma^5\psi(x)$ transforms as
pseudo-scalar, while $\bar{\psi}(x)\gamma^5\gamma^\mu\psi(x)$ transforms as axial vector.
(c) Show that the adjoint spinors $\bar{u}^{1,2}$ and $\bar{v}^{1,2}$ satisfy the equations
$\bar{v}\gamma^\mu p_\mu + m = 0, \bar{u}\gamma^\mu p_\mu - m = 0$

Question 4: [6+6+4=16 Marks]
Under a Lorentz transformation Λ, the Dirac field ψ(x) transforms as
ψ(x) →ψ̃(x̃) = S(Λ)ψ(s) were the 4x4 matrix S(Λ) satisfies S^-1(Λ)γ^μ S(Λ) = Λ^μνγ^ν
(a) Find how the following quantities transforms by applying the Lorentz transformation,
where γ^5 = iγ^0γ^1γ^2γ^3
(i) ψ̅(x)γ^5ψ(x) (ii) ψ̅(x)γ^5γ^μψ(x)
(b) Also check that, by applying parity transformation, ψ̅(x)γ^5ψ(x) transforms as
pseudo-scalar, while ψ̅(x)γ^5γ^μψ(x) transforms as axial vector.
(c) Show that the adjoint spinors u̅^1,2 and v̅^1,2 satisfy the equations
v̅γ^μ pμ + m = 0, u̅γ^μ pμ - m = 0

Added by Miriam C.

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