- #1

latentcorpse

- 1,444

- 0

So I have a few questions about a string theory course I am taking, although I guess the questions are largely on indices/QFT stuff!

(i) Consider the expression

[itex]\bar{\psi}^a \gamma^\mu \partial_\mu \psi^b \eta_{ab}[/itex]

we were going to take the transpose of this and it was said that the transpose operation only acts on Dirac spinor indices. I thought Dirac spinor indices were the indices attached to the Dirac spinors, [itex]\psi[/itex]. However, we calculated

[itex] (\bar{\psi}^a \gamma^\mu \partial_\mu \psi^b \eta_{ab})^T= - \eta_{ab} \partial_\mu \psi^b^T \gamma^\mu^T C^T \psi^a[/itex]

i.e. the index on the [itex]\gamma^\mu[/itex] must also be a spinor index as it has been transposed as well. So my question is, what is the actual definition of a spinor index???

(ii) We then showed that

[itex] (\bar{\psi}^a \gamma^\mu \partial_\mu \psi^b \eta_{ab})^T=-\eta_{ab} \partial_\mu \bar{\psi}^b \gamma^\mu \psi^a[/itex]

and that

[itex](\bar{\psi}^a \gamma^\mu \partial_\mu \psi^b \eta_{ab})^\dagger=+\eta_{ab} \partial_\mu \bar{\psi}^b \gamma^\mu \psi^a[/itex]

i.e. that [itex] (\bar{\psi}^a \gamma^\mu \partial_\mu \psi^b \eta_{ab})^T = - (\bar{\psi}^a \gamma^\mu \partial_\mu \psi^b \eta_{ab})^\dagger[/itex]

why does this allow us to conclude that [itex]\bar{\psi}^a \gamma^\mu \partial_\mu \psi^b \eta_{ab}[/itex] is real?

(iii) Why does taking the transpose of two anticommuting objects introduce a minus sign?

i.e. if A,B anticommute then why does [itex](AB)^T=-BA[/itex]?

(iii) Does the Polyakov action describe a bosonic particle?

(iv) Why does teh equation of motion [itex] ( - \partial_\tau + \partial_\sigma ) \psi_-^a=0[/itex] imply that

[itex]\psi_-^a=k^a(\sigma + \tau)[/itex] i.e. a left moving wave?

I tried substituting it in and I get [itex]-\dot{h}(\sigma + \tau} + h'(\sigma + \tau)[/itex] where dot is tau derivative and ' is sigma derivative. Why should this thing equal zero though?

(v) Given that [itex]\delta \psi^a = \gamma^\nu \epsilon \partial_\nu X^a[/itex] we showed

[itex]\delta \bar{\psi}^a = \delta ( {\psi^a}^T C ) = \partial_\nu X^a ( \epsilon^T {\gamma^\nu}^T C) = - \partial_\nu X^a ( \epsilon^T C \gamma^\nu C^{-1} C ) = - \partial_\nu X^a \bar{\epsilon} \gamma^\nu[/itex]

I agree with all of this except I don't understand why the [itex]\partial_\nu X^a[/itex] at the front isn't also transposed?

(vi) Consider the following manipulations:

[itex]\bar{\psi}^a \gamma^\mu \partial_\mu ( \gamma^\nu \epsilon \partial_\nu X^b ) \eta_{ab}[/itex]

where [itex]\epsilon[/itex] is a constant anticommutating Majorana spinor that generates the supersymmetry.

We can write this as

[itex]\bar{\psi}^a \gamma^\mu \partial_\nu ( \gamma^\nu \epsilon \partial_\mu X^b ) \eta_{ab}[/itex]

This is using the fact that since gamma and epsilon are constant the only contributing term is the one with teh double derivative and then we can use commutativity of partial derivatives to swap [itex] \mu[/itex] and [itex]\nu[/itex]

Then (since this is found in the supersymmetry action, it is going to be getting integrated over the string worldsheet), we integrate by parts to get:

[itex]-\partial_\nu \bar{\psi}^a \gamma^\mu \gamma^\nu \epsilon \partial_\mu X^b \eta_{ab}[/itex]

I get where this term comes from but what happened to the surface term?

Then this becomes

[itex] - ( \partial_\mu \bar{\psi}^b \gamma^\nu \gamma^\mu \epsilon)^T \partial_\nu X^a \eta_{ab}[/itex]

I do not follow this last step AT ALL!

Thanks very much for any help!

(i) Consider the expression

[itex]\bar{\psi}^a \gamma^\mu \partial_\mu \psi^b \eta_{ab}[/itex]

we were going to take the transpose of this and it was said that the transpose operation only acts on Dirac spinor indices. I thought Dirac spinor indices were the indices attached to the Dirac spinors, [itex]\psi[/itex]. However, we calculated

[itex] (\bar{\psi}^a \gamma^\mu \partial_\mu \psi^b \eta_{ab})^T= - \eta_{ab} \partial_\mu \psi^b^T \gamma^\mu^T C^T \psi^a[/itex]

i.e. the index on the [itex]\gamma^\mu[/itex] must also be a spinor index as it has been transposed as well. So my question is, what is the actual definition of a spinor index???

(ii) We then showed that

[itex] (\bar{\psi}^a \gamma^\mu \partial_\mu \psi^b \eta_{ab})^T=-\eta_{ab} \partial_\mu \bar{\psi}^b \gamma^\mu \psi^a[/itex]

and that

[itex](\bar{\psi}^a \gamma^\mu \partial_\mu \psi^b \eta_{ab})^\dagger=+\eta_{ab} \partial_\mu \bar{\psi}^b \gamma^\mu \psi^a[/itex]

i.e. that [itex] (\bar{\psi}^a \gamma^\mu \partial_\mu \psi^b \eta_{ab})^T = - (\bar{\psi}^a \gamma^\mu \partial_\mu \psi^b \eta_{ab})^\dagger[/itex]

why does this allow us to conclude that [itex]\bar{\psi}^a \gamma^\mu \partial_\mu \psi^b \eta_{ab}[/itex] is real?

(iii) Why does taking the transpose of two anticommuting objects introduce a minus sign?

i.e. if A,B anticommute then why does [itex](AB)^T=-BA[/itex]?

(iii) Does the Polyakov action describe a bosonic particle?

(iv) Why does teh equation of motion [itex] ( - \partial_\tau + \partial_\sigma ) \psi_-^a=0[/itex] imply that

[itex]\psi_-^a=k^a(\sigma + \tau)[/itex] i.e. a left moving wave?

I tried substituting it in and I get [itex]-\dot{h}(\sigma + \tau} + h'(\sigma + \tau)[/itex] where dot is tau derivative and ' is sigma derivative. Why should this thing equal zero though?

(v) Given that [itex]\delta \psi^a = \gamma^\nu \epsilon \partial_\nu X^a[/itex] we showed

[itex]\delta \bar{\psi}^a = \delta ( {\psi^a}^T C ) = \partial_\nu X^a ( \epsilon^T {\gamma^\nu}^T C) = - \partial_\nu X^a ( \epsilon^T C \gamma^\nu C^{-1} C ) = - \partial_\nu X^a \bar{\epsilon} \gamma^\nu[/itex]

I agree with all of this except I don't understand why the [itex]\partial_\nu X^a[/itex] at the front isn't also transposed?

(vi) Consider the following manipulations:

[itex]\bar{\psi}^a \gamma^\mu \partial_\mu ( \gamma^\nu \epsilon \partial_\nu X^b ) \eta_{ab}[/itex]

where [itex]\epsilon[/itex] is a constant anticommutating Majorana spinor that generates the supersymmetry.

We can write this as

[itex]\bar{\psi}^a \gamma^\mu \partial_\nu ( \gamma^\nu \epsilon \partial_\mu X^b ) \eta_{ab}[/itex]

This is using the fact that since gamma and epsilon are constant the only contributing term is the one with teh double derivative and then we can use commutativity of partial derivatives to swap [itex] \mu[/itex] and [itex]\nu[/itex]

Then (since this is found in the supersymmetry action, it is going to be getting integrated over the string worldsheet), we integrate by parts to get:

[itex]-\partial_\nu \bar{\psi}^a \gamma^\mu \gamma^\nu \epsilon \partial_\mu X^b \eta_{ab}[/itex]

I get where this term comes from but what happened to the surface term?

Then this becomes

[itex] - ( \partial_\mu \bar{\psi}^b \gamma^\nu \gamma^\mu \epsilon)^T \partial_\nu X^a \eta_{ab}[/itex]

I do not follow this last step AT ALL!

Thanks very much for any help!

Last edited: