Grassmann variables and functional derivatives

QED and has a question about the generating functional for fermionic fields. The functional is $$Z_0=e^{-i\int{d^4xd^4y \bar{J}(x)S(x-y)J(y)}}$$ where $$J(x)$$ and $$\bar{J}(x)$$ are Grassmann numbers. Einj wonders if the functional derivative follows the same rule as the ordinary derivative and if the order of $$J$$ and $$\bar{J}$$ matters. The response is that the same rules apply, but the order of $$J$$ and $$\bar{J}$$ may need to be adjusted depending on which derivative is used.
  • #1
Einj
470
59
Hi all! I'm sorry if this question has been already asked in another post...
I'm studying the path integrals formalism in QED. I'm dealing with the functional generator for fermionic fields. My question is:

The generating functional is:

$$Z_0=e^{-i\int{d^4xd^4y \bar{J}(x)S(x-y)J(y)}}$$

Where $$J(x)$$ and $$\bar{J}(x)$$ are Grassmann numbers.
When I have to extract Green function from the generating functional I have to perform, for example, a functional derivative rispect to J(z). Does the functional derivative follow the same rule as the ordinary derivative? Do I have to anticommutate $$\bar{J}$$ and $$J$$ before deriving??

Thaks to all
Einj
 
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  • #2
Normally yes, because you'd have to use either the left functional derivative or the right one constantly throughout. For example, if you use the right derivative and the Jbar is to the left, you'd have to shift it past J (that is bring it to the right) and then differentiate. So the answer is yes.
 
  • #3
Thank you very much! You have been great!

Einj
 

1. What are Grassmann variables?

Grassmann variables are mathematical objects used in the field of supersymmetry. They are anti-commuting variables, meaning that their multiplication is anti-symmetric and they square to zero. They are often represented by the symbols θ and θ̄ and are used to describe fermionic particles in physics.

2. How are Grassmann variables used in functional derivatives?

Grassmann variables are used in functional derivatives to describe the variation of functionals with respect to anti-commuting variables. This is important in supersymmetric theories, where functionals are often used to describe the interactions between bosons and fermions.

3. What is the relationship between Grassmann variables and differential forms?

Grassmann variables and differential forms have a similar algebraic structure, as both are anti-commuting objects. In fact, differential forms can be seen as a generalization of Grassmann variables, with Grassmann variables being the simplest type of differential form.

4. How do Grassmann variables behave under integration?

Grassmann variables are integrated using the Berezin integral, which is defined as the anti-commuting analogue of the Lebesgue integral. The Berezin integral is a linear functional that assigns a complex number to a function of anti-commuting variables, and it satisfies the properties of an integral, such as linearity and the fundamental theorem of calculus.

5. What are some applications of Grassmann variables and functional derivatives?

Grassmann variables and functional derivatives have a wide range of applications in theoretical physics, particularly in the field of supersymmetry. They are used to formulate supersymmetric quantum field theories and to study the interactions between bosons and fermions. They also have applications in quantum mechanics, differential geometry, and mathematical physics.

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