# Calculations with Weyl Spinor Indices in QFT

• nicouh
In summary, Nico found that he was unable to solve a problem involving spinors due to lack of knowledge regarding them. He researched and found that he knew how to handle the gamma matrices, but had no clue what the indices meant, why they were sorted the way they were, and how the product of two matrices was written using indices. He plans on continuing his search for a solution when he has more time and asks for advice from his interviewer.
nicouh

## Homework Statement

The task is to show the invariance of a given Lagrangian (http://www.fysast.uu.se/~leupold/qft-2011/tasks.pdf" ), but my problem is just in one step (which i got from Peskin & Schröder, page 70) which i can not reproduce due to my lack of knowledge regarding spinors.

The step i am talking about is 3.147 in the attached picture or written out:

## Homework Equations

$C \bar \psi \psi C = (-i \gamma^0 \gamma^2 \psi)^T (-i \bar \psi \gamma^0 \gamma^2)^T$
$= -\gamma^0_{ab}\gamma^2_{bc} \psi_c \bar \psi_d \gamma^0_{de} \gamma^2_{ea}$
$= \bar \psi_d \gamma^0_{de} \gamma^2_{ea} \gamma^0_{ab} \gamma^0_{bc} \psi_c$
$= -\bar \psi \gamma^2 \gamma^0 \gamma^0 \gamma^2 \psi$
$= \bar \psi \psi$

## The Attempt at a Solution

Well, i browsed through Wikipedia, Google and Friends, but did not find anything.
I know how to handle the gamma matrices (like their commutation relations or $(\gamma^0)^2 = 1$) .
I have just no clue what these indices mean, why they are sorted the way they are ("ab bc cd de" instead of e.g. "ab cd ef gh") and how $\psi^T$ gets converted to $\psi$.

Thanks!
Regards,
Nico

#### Attachments

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It's been years since I've done any QFT, and I don't have any books here with me right now, so what I write might be completely wrong.

First, consider some linear algebra. If $A$ is a square matrix, then $\vec{w} = A \vec{v}$ becomes

$$w_i = \sum_j A_{ij} v_j$$

in component form. It looks like the matrix multiplications have been written in component form with the summation symbols omitted, which is why there are repeated indices.

Next, is it true that for a scalar that

$$C \bar{\psi} \psi C = \left( C \bar{\psi} \psi C \right)^T ?$$

Can you use this to do the calculation more simply, taking into account that fermion fields anticommute?

George Jones said:
First, consider some linear algebra. If $A$ is a square matrix, then $\vec{w} = A \vec{v}$ becomes

$$w_i = \sum_j A_{ij} v_j$$

in component form. It looks like the matrix multiplications have been written in component form with the summation symbols omitted, which is why there are repeated indices.

At i thought it was the standard matrix multiplication, too. But as you wrote it, it already begins with an index. So its
$$w_i = \sum_j A_{ij} v_j$$ instead of
$$w = \sum_j A_{ij} v_j$$, which just confuses me.

George Jones said:
Next, is it true that for a scalar that

$$C \bar{\psi} \psi C = \left( C \bar{\psi} \psi C \right)^T ?$$

Can you use this to do the calculation more simply, taking into account that fermion fields anticommute?

Yeah, i think so, too. Since its a scalar transposing it shouldn't change anything :P
But i haven't come really far doing that task. I did others first, but ended up at the same spot with the same problem eventually. The same thing even appeared a few times more now.

nicouh said:
At i thought it was the standard matrix multiplication, too. But as you wrote it, it already begins with an index.

If A and B are matrices, how is the product AB written using indices?

ah, i think i got it.
yeah, since (as i just mentioned in my previous post :P) the end product is a scalar, it makes sense to write it this way and it really is just a index notation for a vector-many matrices-vector product. a normal scalar product of 2 vectors would have the same features when looked at it written out in components.
im kind of taking a break right now, but when i start again and still get stuck, i might will ask you further questions :P
thanks!

## 1. What are Weyl spinor indices in QFT?

Weyl spinor indices in QFT refer to mathematical objects used to represent the spin of particles in quantum field theory. They are used to describe the behavior of fermions, such as electrons, quarks, and neutrinos, which have half-integer spin. Weyl spinors are complex, two-component spinors that transform under the Lorentz group.

## 2. How are Weyl spinor indices used in QFT calculations?

Weyl spinor indices are used in QFT calculations to represent the spin states of particles and to construct the Lagrangian density, which describes the dynamics of a quantum field. They are also used in the Feynman rules for calculating scattering amplitudes in particle interactions.

## 3. What is the significance of Weyl spinor indices in QFT?

The use of Weyl spinor indices in QFT allows for calculations and predictions to be made about the behavior of particles at the quantum level. They are an essential tool for understanding the fundamental interactions and symmetries of particles and their associated fields.

## 4. Are there any limitations to using Weyl spinor indices in QFT calculations?

While Weyl spinor indices are a powerful tool in QFT calculations, they are limited in their ability to describe particles with integer spin, such as photons and gluons. In addition, the use of Weyl spinors assumes that particles are massless, which may not always be the case in reality.

## 5. How do Weyl spinor indices relate to other mathematical objects in QFT?

Weyl spinor indices are closely related to other mathematical objects used in QFT, such as Dirac spinors, Lorentz tensors, and gauge fields. They are also connected to the concept of chirality, which describes the asymmetry of a particle's spin with respect to its direction of motion.

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