# Complex valued Grassmann variables $(\theta \eta)^*$, $(\theta \eta)^T$ and $(\theta \eta)^\dagger$

Since hermitian conjugation and complex conjugation are serious issues in a QFT lagrangian with Grassmann variables, see here and here. Let us try to go to the bottom.

We start by accepting the Grassmann variable has the property $$\eta \theta = -\theta \eta$$. In Peskin & Schroeder eq 9.65, they wrote for complex Grassmann variable

It is convenient to define complex conjugation to reverse the order of products, just like the Hermitian conjugation of operators: $$\text{ P&S eq 9.65: }(\theta \eta)^* = \eta^* \theta^*= -\theta^* \eta^*. \tag{1}$$

## question 1: why do we define complex conjugation $$*$$ this way? why $$(\theta \eta)^* = \eta^* \theta^*= -\theta^* \eta^*$$?

Note that this complex conjugation $$*$$ is not yet hermitian conjugate $$\dagger$$, see below.

Follow P&S below eq.9.56, the complex valued Grassmann variables $$\theta =\theta_1 + i\theta_2, \quad \theta^* =\theta_1 - i\theta_2$$ $$\eta =\eta_1 + i\eta_2, \quad \eta^* =\eta_1 - i\eta_2$$

Then we derive that $$(\theta \eta)^*=((\theta_1 + i\theta_2)(\eta_1 + i\eta_2))^* =((\theta_1\eta_1 -\theta_2\eta_2) + i(\theta_1\eta_2 +\theta_2\eta_1))^* =(\theta_1\eta_1 -\theta_2\eta_2) - i(\theta_1\eta_2 +\theta_2\eta_1)$$ To make comparisons,
$$\theta^* \eta^* =(\theta_1- i\theta_2)(\eta_1 - i\eta_2) =(\theta_1\eta_1 -\theta_2\eta_2) - i(\theta_1\eta_2 +\theta_2\eta_1).$$

Thus I showed that $$(\theta \eta)^* = +\theta^* \eta^*. \tag{2}$$

So P&S equation (1) contradicts with my (2). Why is that?

## question 2: Does it make sense to define tranposed on Grassmann variable? do we have $$(\theta \eta)^T = \pm \eta^T \theta^T= \pm \eta \theta= \mp \eta \theta$$?

What is the plus or minus sign here?

## question 3: Does it make sense to define hermitian conjugate on Grassmann variable? if we can combine the above to get $$(\theta \eta)^\dagger =$$?

• "Since hermitian conjugation and complex conjugation are serious issues in a QFT lagrangian with Grassman variables" <--- it absolutely is not a serious issue in any way. You are having trouble with it and that's fine, but there's no reason to claim that any serious issue exists. Mar 29 at 17:06
• There is a good answer to your first question on the math stackexchange here. The answers to the others follow a similar logic. Mar 29 at 17:07
• I thought hermitian conjugation and complex conjugation are serious issues always for quantum mechanics? should they be something deep in quantum mechanics? thanks I voted up Mar 29 at 17:10
• Related: physics.stackexchange.com/q/695933/2451 and links therein. Mar 29 at 17:12
• @МаринаMarinaS it’s absolutely not. It maybe a difficult subject to learn for young students but it’s perfectly well defined. No issue exists at all! Mar 29 at 17:13

Complex conjugation for complex numbers satisfies $$a^* a > 0$$. We would like to choose our definition for complex conjugation so that $$\eta^*\eta>0$$. Let's assume this is true for $$\eta$$ and $$\theta$$. Then, suppose that $$(\eta \theta)^* = \epsilon \eta^* \theta^*$$ Then, $$(\eta \theta)^* (\eta\theta) = \epsilon \eta^* \theta^* \eta \theta = - \epsilon ( \eta^* \eta ) ( \theta^* \theta)$$ It follows that this is positive if and only if $$\epsilon=-1$$. Thus, the correct definition for complex conjugation is $$(\eta\theta)^* = \theta^*\eta^*$$ The mistake you made in doing your P&S calculation is that you assumed $$(\theta_1\eta_2)^* = + \theta_1\eta_2$$ but this sign is not true.
Now, let's talk about transpose and hermitian conjugate of Grassmann valued matrices. These are matrices such that each element $$A_{ij}$$ of the matrix is a Grassmann number. Then, $$[(AB)^T]_{ij} = (AB)_{ji} = A_{jk} B_{ki} = (A^T)_{kj} (B^T)_{ik}$$ At this stage, we have to exchange $$A$$ and $$B$$. But these are Grassmann valued, so we have a sign $$[(AB)^T]_{ij} = - (B^T)_{ik} (A^T)_{kj} = - (B^TA^T)_{ij} \qquad \implies \qquad (AB)^T = - B^T A^T$$
The conjugation properties for matrices follows similarly $$[(AB)^*]_{ij} = (A_{ik} B_{kj})^* = B_{kj}^* A_{ik}^*$$ To bring it back into matrix multiplication form, we need to exchange the elements. So $$[(AB)^*]_{ij} = - A_{ik}^* B_{kj}^* = - (A^*B^*)_{ij} \quad \implies \quad (AB)^* = - A^*B^*.$$
Hermitian conjugation is a complex conjugation + transpose and you can use the results above to derive $$(AB)^\dagger = + B^\dagger A^\dagger$$.
• This trick $(𝜂𝜃)∗(𝜂𝜃)=𝜖𝜂∗𝜃∗𝜂𝜃=−𝜖(𝜂∗𝜂)(𝜃∗𝜃)>0$ is very neat, where did you learn this from? +1 Mar 29 at 17:19