- #1

LAHLH

- 409

- 1

Hi,

If we start with [tex] \psi^{\dag}\bar{\sigma}^{\mu}\chi [/tex] and take its Hermitian conjugate:

[tex] \left[\psi^{\dag}\bar{\sigma}^{\mu}\chi\right]^{\dag}=\left[\psi^{\dag}_{\dot{a}}\bar{\sigma}^{\mu\dot{a}c}\chi_{c}\right]^{\dag} [/tex]

I'm basing this on Srednicki ch35 (p219 in my edition). His next line is,

[tex] =\chi^{\dag}_{\dot{c}}(\bar{\sigma}^{\mu a\dot{c}})^{*}\psi_{a} [/tex]

I can't understand how he's managed to use the transpose part of the dagger, to change which index of [tex]\sigma[/tex] is dotted. At first I thought well the index on [tex] \chi [/tex] has gone from [tex]c[/tex] to [tex]\dot{c} [/tex] so maybe he's just relabelling dummies, but then this of course wouldn't explain why the tranpose part has been used and we're left only with *.

I might have expected,

[tex] =\chi^{\dag}_{\dot{c}}(\bar{\sigma}^{\mu a\dot{c}})^{\dag}\psi_{a} [/tex]

[tex] =\chi^{\dag}_{\dot{c}}(\bar{\sigma}^{\mu \dot{c}a})^{*}\psi_{a} [/tex]

Could anyone spell it out for me? thanks

If we start with [tex] \psi^{\dag}\bar{\sigma}^{\mu}\chi [/tex] and take its Hermitian conjugate:

[tex] \left[\psi^{\dag}\bar{\sigma}^{\mu}\chi\right]^{\dag}=\left[\psi^{\dag}_{\dot{a}}\bar{\sigma}^{\mu\dot{a}c}\chi_{c}\right]^{\dag} [/tex]

I'm basing this on Srednicki ch35 (p219 in my edition). His next line is,

[tex] =\chi^{\dag}_{\dot{c}}(\bar{\sigma}^{\mu a\dot{c}})^{*}\psi_{a} [/tex]

I can't understand how he's managed to use the transpose part of the dagger, to change which index of [tex]\sigma[/tex] is dotted. At first I thought well the index on [tex] \chi [/tex] has gone from [tex]c[/tex] to [tex]\dot{c} [/tex] so maybe he's just relabelling dummies, but then this of course wouldn't explain why the tranpose part has been used and we're left only with *.

I might have expected,

[tex] =\chi^{\dag}_{\dot{c}}(\bar{\sigma}^{\mu a\dot{c}})^{\dag}\psi_{a} [/tex]

[tex] =\chi^{\dag}_{\dot{c}}(\bar{\sigma}^{\mu \dot{c}a})^{*}\psi_{a} [/tex]

Could anyone spell it out for me? thanks

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