# Delta function for grassmann numbers?

• pellman
In summary, the function f(x) = a+b\psi is a Dirac delta function for integrals over \psi when a and b are not complex numbers.
pellman
Claim: if $$\psi$$ is a variable grassmann number, then $$\delta(\psi)=\psi$$ is a Dirac delta function for integrals over $$\psi$$.

I'm not seeing this.

A general function of a grassmann number can be written $$f(\psi)=a+b\psi$$ because anti-commutativity requires $$\psi^2=0$$. According to the wikipedia article (which doesn't elaborate on why), integrals satisfy

$$\int 1 d\psi=0$$
$$\int \psi d\psi=1$$

Ok. So let's check.

$$\int\delta(\psi-\psi')f(\psi)d\psi$$
$$=\int(\psi-\psi')f(\psi)d\psi$$
$$=\int(\psi-\psi')(a+b\psi)d\psi$$
$$=a\int \psi d\psi + 0 -a\psi'\int d\psi-b\psi'\int\psi d\psi$$
$$=a - 0 -b\psi'$$
$$=f(-\psi')$$
$$\neq f(\psi')$$

Are one of my assumptions wrong?

The definition of $\delta$ is $\int_{-\infty}^{+\infty} \delta(x) f(x) \, dx = f(0)$.

Does that change of variable make sense for such integrals? Does it make sense for f?

Hurkyl said:
The definition of $\delta$ is $\int_{-\infty}^{+\infty} \delta(x) f(x) \, dx = f(0)$.

Does that change of variable make sense for such integrals? Does it make sense for f?

I don't know, Hurkyl. I'm not even sure what integration itself means here. I am just following the steps in a formal sort of way. Warren Siegel on page 48 of Fields has that $$\int d\psi'(\psi'-\psi)f(\psi')=f(\psi)$$ (the prime-unprimed reverse of what I have above.) But I get $$f(-\psi)$$ .

Never mind if we can call it the delta function. Siegel himself puts "anti-commuting delta function" in quotes.

I remember trying to read through that book; I stopped after finding that section extremely off-putting.

Anyways, one thing Siegel says in that section is if we write $f(\psi) = a + b \psi$, then a and b aren't necessarily complex scalars -- they can be commuting numbers or anticommuting numbers or whatever.

Assuming that we really do have
$$\int d\psi (a + b \psi) = b$$
whenever a and b don't involve psi, then if I take special care not to pass any variable through another I get

$$\int d\psi' (\psi' - \psi) f(\psi') = -\psi b + \int d\psi' \psi' a$$

which can be rearranged to

$$= f(\psi) - \{\psi, b\} + \int d\psi' [\psi', a]$$

of course, it can be rearranged into other things -- such as

$$= f(-\psi) - [\psi, b] + \int d\psi' [\psi', a]$$

Last edited:
I think I'm with you about that section. I'll just let it go and press on.

Thanks!

## 1. What is the delta function for grassmann numbers?

The delta function for grassmann numbers is a mathematical function that represents a generalized version of the Kronecker delta function for ordinary numbers. It is used in the context of Grassmann algebra, which is a mathematical framework that extends the concept of complex numbers to include anticommutative elements called grassmann numbers.

## 2. How is the delta function for grassmann numbers different from the ordinary delta function?

The delta function for grassmann numbers is different from the ordinary delta function in several ways. Firstly, it takes grassmann numbers as its arguments rather than ordinary numbers. Secondly, it is anticommutative, meaning that it switches sign when the order of its arguments is interchanged. Lastly, it satisfies a different set of properties and identities compared to the ordinary delta function.

## 3. What are the applications of the delta function for grassmann numbers?

The delta function for grassmann numbers has various applications in theoretical physics and mathematics. It is commonly used in the study of supersymmetry, which is a theoretical framework that describes the relationship between elementary particles with different spin. It also has applications in quantum field theory, where it is used to simplify calculations involving fermions, particles with half-integer spin.

## 4. How is the delta function for grassmann numbers defined?

The delta function for grassmann numbers is defined as a series of derivatives of the logarithm of a grassmann number. It is a function that takes two grassmann numbers as its arguments and returns a numerical value. The exact definition may vary depending on the field of study, but it is always defined in terms of the anticommutation relations of the grassmann numbers.

## 5. Can the delta function for grassmann numbers be generalized to higher dimensions?

Yes, the delta function for grassmann numbers can be generalized to higher dimensions. In fact, there are multiple ways to define a generalized delta function in higher dimensions, each with its own set of properties and applications. Some examples include the supersymmetric delta function, which is used in superspace formalism, and the higher-dimensional grassmann delta function, which is used in the study of higher-dimensional supersymmetric theories.

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