Skip to main content
Physics

Return to Question

added 7 characters in body
Source Link
user26143
user26143
  • 6.5k
  • 3
  • 29
  • 54

I have a question about the spinor helicity formalism from arXiv:1308.1697

Denote the massless spin-1/2 fermions as Eqs. (2.10)-(2.11) in that paper $$v_+(p)= \begin{pmatrix} |p]_a \\ 0 \end{pmatrix} $$ $$v_-(p)= \begin{pmatrix} 0 \\ |p \rangle^{\dot{a}} \end{pmatrix} $$ $$\bar{u}_-(p)= (0, \langle p |_{\dot{a}})$$ $$\bar{u}_+(p)= ([p |^{a},0)$$ For real momenta, there is an identity in that paper $$ [k| \gamma^{\mu} |p \rangle^*= [p|\gamma^{\mu}|k \rangle \tag{2.33}$$

My question is, how to prove (2.33)? I know $$[p|^a=(|p \rangle^{\cdot{a}})^*, \langle p |_{\dot{a}} = (|p]_a)^* \tag{2.14}$$ for real momenta.

By using (2.14) I got $$[k| \gamma^{\mu} |p \rangle = ([p|)^* | \gamma^{\mu} (|k \rangle)^* $$, since $\gamma^{\mu*} \neq \gamma^{\mu}$$\gamma^{\mu*} \neq \gamma^{\mu}, \mu=2$, I still miss a complex conjugation...

I have a question about the spinor helicity formalism from arXiv:1308.1697

Denote the massless spin-1/2 fermions as Eqs. (2.10)-(2.11) in that paper $$v_+(p)= \begin{pmatrix} |p]_a \\ 0 \end{pmatrix} $$ $$v_-(p)= \begin{pmatrix} 0 \\ |p \rangle^{\dot{a}} \end{pmatrix} $$ $$\bar{u}_-(p)= (0, \langle p |_{\dot{a}})$$ $$\bar{u}_+(p)= ([p |^{a},0)$$ For real momenta, there is an identity in that paper $$ [k| \gamma^{\mu} |p \rangle^*= [p|\gamma^{\mu}|k \rangle \tag{2.33}$$

My question is, how to prove (2.33)? I know $$[p|^a=(|p \rangle^{\cdot{a}})^*, \langle p |_{\dot{a}} = (|p]_a)^* \tag{2.14}$$ for real momenta.

By using (2.14) I got $$[k| \gamma^{\mu} |p \rangle = ([p|)^* | \gamma^{\mu} (|k \rangle)^* $$, since $\gamma^{\mu*} \neq \gamma^{\mu}$, I still miss a complex conjugation...

I have a question about the spinor helicity formalism from arXiv:1308.1697

Denote the massless spin-1/2 fermions as Eqs. (2.10)-(2.11) in that paper $$v_+(p)= \begin{pmatrix} |p]_a \\ 0 \end{pmatrix} $$ $$v_-(p)= \begin{pmatrix} 0 \\ |p \rangle^{\dot{a}} \end{pmatrix} $$ $$\bar{u}_-(p)= (0, \langle p |_{\dot{a}})$$ $$\bar{u}_+(p)= ([p |^{a},0)$$ For real momenta, there is an identity in that paper $$ [k| \gamma^{\mu} |p \rangle^*= [p|\gamma^{\mu}|k \rangle \tag{2.33}$$

My question is, how to prove (2.33)? I know $$[p|^a=(|p \rangle^{\cdot{a}})^*, \langle p |_{\dot{a}} = (|p]_a)^* \tag{2.14}$$ for real momenta.

By using (2.14) I got $$[k| \gamma^{\mu} |p \rangle = ([p|)^* | \gamma^{\mu} (|k \rangle)^* $$, since $\gamma^{\mu*} \neq \gamma^{\mu}, \mu=2$, I still miss a complex conjugation...

Source Link
user26143
user26143
  • 6.5k
  • 3
  • 29
  • 54

An identity for spinor helicity formalism

I have a question about the spinor helicity formalism from arXiv:1308.1697

Denote the massless spin-1/2 fermions as Eqs. (2.10)-(2.11) in that paper $$v_+(p)= \begin{pmatrix} |p]_a \\ 0 \end{pmatrix} $$ $$v_-(p)= \begin{pmatrix} 0 \\ |p \rangle^{\dot{a}} \end{pmatrix} $$ $$\bar{u}_-(p)= (0, \langle p |_{\dot{a}})$$ $$\bar{u}_+(p)= ([p |^{a},0)$$ For real momenta, there is an identity in that paper $$ [k| \gamma^{\mu} |p \rangle^*= [p|\gamma^{\mu}|k \rangle \tag{2.33}$$

My question is, how to prove (2.33)? I know $$[p|^a=(|p \rangle^{\cdot{a}})^*, \langle p |_{\dot{a}} = (|p]_a)^* \tag{2.14}$$ for real momenta.

By using (2.14) I got $$[k| \gamma^{\mu} |p \rangle = ([p|)^* | \gamma^{\mu} (|k \rangle)^* $$, since $\gamma^{\mu*} \neq \gamma^{\mu}$, I still miss a complex conjugation...