0
votes
1answer
48 views

Fermion propagator is not a Grassmann-odd object?

Is the following differentiation correct: $$ \frac{\delta}{\delta\eta\left(z\right)}\int d^{4}yS_{F}\left(z-y\right)\eta\left(y\right) = S_F\left(z-z\right)$$ where $\eta$ is a Grassmann-valued ...
0
votes
0answers
14 views

dual variables for lattice fermions

I am quite familiar with duality transformations for lattice spin systems (i.e. systems with global $O(n)$ symmetry) and pure gauge systems (i.e. local $SU(n)$). However, after searching for a bit, I ...
3
votes
1answer
102 views

Path integral as a functional determinant

In Peskin and Schroeder on pg. 304, the authors call the fermionic path integral: \begin{equation} \int {\cal D} \bar{\psi} {\cal D} \psi \exp \left[ i \int \,d^4x \bar{\psi} ( i \gamma_\mu D^\mu - m ...
1
vote
1answer
153 views

A few simple questions about Grassmann numbers: commutation relations and derivatives

I'm trying to learn about Grassmann numbers from the book "Condensed Matter Field Theory" by Altland and Simons, but I am currently encountering some difficulties. I have several smaller questions ...
5
votes
1answer
201 views

Four Fermion Interactions

Given an action with a term like \begin{equation}S_{I}\sim \int\int (\psi^{\dagger}\psi)V(\psi^{\dagger}\psi)\end{equation} How do you evaluate this with a Fermionic path integral? I know the fields ...
1
vote
1answer
159 views

Grassmann fields according to Peskin and Schroeder

On page 301 in Peskin and Schroeder, they claim that a Grassman field $\psi(x)$ may be decomposed as $$\psi(x) = \sum_i c_i \phi_i(x),$$ where the $c_i$ are Grassmann numbers and the $\phi_i$ are ...
6
votes
1answer
359 views

What is the value of a quantum field?

As far as I'm aware (please correct me if I'm wrong) quantum fields are simply operators, constructed from a linear combination of creation and annihilation operators, which are defined at every point ...
1
vote
3answers
160 views

Constructing Supersymmetric Lagrangians

It is a very trivial doubt but somehow I am not able to figure it out. While constructing a supersymmetric lagrangian we always even number of fermionic fields. One reason is of course the product ...
1
vote
1answer
176 views

dimensional analysis of Grassmann integration/differentiation

There is another paradox that I need to resolve: The Berezin integration rules for Grassmann odd variables give the same result as differentiation: If $f=x+\theta\psi$ is a superfunction, the ...
1
vote
0answers
178 views

Number of Grassmann generators for Dirac field?

How many Grassmann generators are sufficient for the description of a Dirac spinor in 4 dimensions? i.e. The Dirac field is a map to $\Lambda_N$, the space of supernumbers with $N$ real Grassmann ...
8
votes
3answers
730 views

Grassmann paradox weirdness

I'm running into an annoying problem I am unable to resolve, although a friend has given me some guidance as to how the resolution might come about. Hopefully someone on here knows the answer. It is ...
1
vote
1answer
134 views

Superspace Uncertainty Principle

Do the "operator for translations in superspace" and the "position in superspace operator" follow an uncertainty principle? How "real" is superspace? Aside from being weird (and possibly just a ...
5
votes
1answer
253 views

Grassmann Variables Representation?

It might be a silly question, but I was never mathematically introduced to the topic. Is there a representation for Grassmann Variables using real field. For example, gamma matrices have a ...
3
votes
1answer
318 views

Basic Grassmann/Berezin Integral Question

Is there a reason why $\int\! d\theta~\theta = 1$ for a Grassmann integral? Books give arguments for $\int\! d\theta = 0$ which I can follow, but not for the former one.
2
votes
1answer
324 views

What are Grassmann (even/odd) numbers used in superalgebras?

Are Grassmann numbers a concept of graded Lie algebras or is something specific to superalgebras? What are they (i.e: how are they defined, important properties, etc.)? Is there a reasonable ...