Notation relating to gamma matrices

[vertices]

Hi

My QFT course assumes the following notation for gamma matrices:

$$\gamma ^{\mu_1 \mu_2 \mu_3 \mu_4} = {\gamma ^ {[\mu_1}}{\gamma ^ {\mu_2}}{\gamma ^ {\mu_3}}{\gamma ^ {\mu_4 ]}}$$

what does the thing on the right hand side actually mean? Its seems to be a commutator of some sort.

 

[tiny-tim]

Hi vertices!

(have a gamma: γ and a mu: µ )

It means you add every possible permutation (just as in γ^µ_[1µ_2] you'd add every possible permutation, but there'd only be two of them! ).

 

[vertices]

Hi vertices!

(have a gamma: γ and a mu: µ )

It means you add every possible permutation (just as in γ^µ_[1µ_2] you'd add every possible permutation, but there'd only be two of them! ).

Thanks tiny-tim:)

So for example, would I be right in saying that:

γ^µ₁µ₂=γ^µ₁γ^µ₂+γ^µ₂γ^µ₁?

 

[Ben Niehoff]

Tiny-Tim forgot to tell you that you also must multiply each term by the sign of its permutation. That is, you must take a totally-antisymmetrized sum. You might also have to divide by N afterward, depending on the convention you're using (where N is the number of gamma matrices being antisymmetrized).

 

[ismaili]

Thanks tiny-tim:)

So for example, would I be right in saying that:

γ^µ₁µ₂=γ^µ₁γ^µ₂+γ^µ₂γ^µ₁?

Actually, the usual definition for the square bracket, like $$\gamma^{\mu\nu} \equiv \gamma^{[\mu} \gamma^{\nu]}$$ is the anti-symmetrization of the indices. For this case I just mentioned,
$$\gamma^{\mu\nu} = \gamma^\mu\gamma^\nu - \gamma^\nu\gamma^\mu$$
And for example,
$$\gamma^{\mu\nu\rho} \equiv \gamma^{[\mu}\gamma^{\nu}\gamma^{\rho]} = \frac{1}{3!}\left( \gamma^\mu\gamma^\nu\gamma^\rho - \gamma^\mu\gamma^\rho\gamma^\nu + \gamma^\nu\gamma^\rho\gamma^\mu -\gamma^\nu\gamma^\mu\gamma^\rho +\gamma^\rho\gamma^\mu\gamma^\nu -\gamma^\rho\gamma^\nu\gamma^\mu \right)$$

 

[vertices]

Tiny-Tim forgot to tell you that you also must multiply each term by the sign of its permutation. That is, you must take a totally-antisymmetrized sum. You might also have to divide by N afterward, depending on the convention you're using (where N is the number of gamma matrices being antisymmetrized).

sorry can I ask a stupid question: what is the 'sign' of a permutation?

 

[vertices]

Actually, the usual definition for the square bracket, like $$\gamma^{\mu\nu} \equiv \gamma^{[\mu} \gamma^{\nu]}$$ is the anti-symmetrization of the indices. For this case I just mentioned,
$$\gamma^{\mu\nu} = \gamma^\mu\gamma^\nu - \gamma^\nu\gamma^\mu$$
And for example,
$$\gamma^{\mu\nu\rho} \equiv \gamma^{[\mu}\gamma^{\nu}\gamma^{\rho]} = \frac{1}{3!}\left( \gamma^\mu\gamma^\nu\gamma^\rho - \gamma^\mu\gamma^\rho\gamma^\nu + \gamma^\nu\gamma^\rho\gamma^\mu -\gamma^\nu\gamma^\mu\gamma^\rho +\gamma^\rho\gamma^\mu\gamma^\nu -\gamma^\rho\gamma^\nu\gamma^\mu \right)$$

can i ask you how the the sign of each term is determined? How do you decide which term is positive and which term is negative?

thanks.

 

[vertices]

Aahh I've figured it out i think - cyclic permutations are positive; anticyclic permutations are negative. A bit like the levi-civita thingy..

 

[Ben Niehoff]

Aahh I've figured it out i think - cyclic permutations are positive; anticyclic permutations are negative. A bit like the levi-civita thingy..

This works for 3 indices or less. For N indices, you check whether it is an even or odd permutation of {1,2,3,4,...,N}. That is, you count the number of pairwise interchanges required to bring the indices back to numerical order. If it takes an even number of switches, the term gets a plus sign; otherwise, a minus sign.

 

[vertices]

This works for 3 indices or less. For N indices, you check whether it is an even or odd permutation of {1,2,3,4,...,N}. That is, you count the number of pairwise interchanges required to bring the indices back to numerical order. If it takes an even number of switches, the term gets a plus sign; otherwise, a minus sign.

interesting. thanks for pointing this out Ben.