Majorana Path Integral: Deriving VEVs of Barred/Unbarred Fields

[LAHLH]

Hi,

By analogy with scalar field case, Srednicki leads us to $$Z_0 (\eta)=\int \mathcal{D}\Psi \exp{\left[i\int\,\mathrm{d}^4x (\mathcal{L}_0+\eta^{T}\psi)\right]}$$ for a Majorana field.

I was expecting something different, like maybe: $$Z_0 (\eta)=\int \mathcal{D}\Psi\exp{\left[i\int\,\mathrm{d}^4x (\mathcal{L}_0+\eta^{T}\psi+\eta\psi^{T})\right]}$$ at least.

or even: $$Z_0 (\eta)=\int \mathcal{D}\Psi \mathcal{D}\bar{\Psi}\ \exp{\left[i\int\,\mathrm{d}^4x (\mathcal{L}_0+\bar{\eta}\psi+\eta\bar{\psi})\right]}$$ (why a transpose on eta now and not a bar?)

I mean how would derive VEV's of a a product of barred and unbarred fields like $$\langle 0|T\Psi_{\alpha}(x)\bar{\Psi}_{\beta}(y)|0\rangle$$ as he computes at the end of ch42, with the above.

Thanks

 

[xepma]

Because it is the path integral for a Majorana field. The third path integral you wrote down would apply to a Dirac field. The second has no meaning. The first (and Srednicki's one) is the one that applies to a Majorana field.

In a nutshell, a Majorana field is a more "basic" object than a Dirac field. A Dirac field is the (complex) sum of two Majorana fields. There is no notion of a "barred" majorana field -- barring the Majorana field operator gives you back the same operator!

 

[LAHLH]

But there is a notion of barred for Majorana, it is just that we have the Majorana condition too: $$\bar{\Psi}=\Psi^{T}\mathcal{C}$$, and in fact in ch42 Srednicki actually calculates the VEVs of various products of Majornana barred fields etc

I know from the previous chapters that a Majorana field is more basic as you say, and a Dirac field is like two of these objects, and in some sense this is why a Majorana field is analogous to a real scalar field, whereas the Dirac is analogous to a complex scalar field. But nevertheless you can bar Majorana fields and calculate various VEV products of them, and I can't see how one would do this from such a path integral.

 

[LAHLH]

Could anyone say anything further about this?

Specifically I'm just wondering how one could calculate something like Srednicki's 42.19: $$i\langle 0|T\Psi_{\alpha}(x)\bar{\Psi}_{\beta}(y)|0\rangle =S(x-y)_{\alpha\beta}$$
which is for Majorna fields, from the functional integral Srednicki gives above.

thanks

 

[Avodyne]

You first compute with all psi's, and then change as many as you want to psibar's using the Majorana relation between psi and psibar. With Majorana fields (and sources), the barred fields are redundant, and therefore it is best to avoid using them.

 

[LAHLH]

I see, makes sense. Thanks alot.