# What is Grassmann: Definition and 31 Discussions

Hermann Günther Grassmann (German: Graßmann, pronounced [ˈhɛʁman ˈɡʏntɐ ˈɡʁasman]; 15 April 1809 – 26 September 1877) was a German polymath, known in his day as a linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mathematical work was little noted until he was in his sixties.

View More On Wikipedia.org

7. ### I Eigenvalues of Fermionic field operator

Hello, I'm a bit confused about the eigenvalues of the second quantized fermionic field operators \psi(x)_a. Since these operators satisfy the condition \{\psi(x)_a, \psi(y)_b\} = 0 the eigenvalues should also anti-commute? Does this mean that the eigenvalues of \psi(x)_a are...
8. ### Grassmann Integral into Lagrange for scalar superfields

I have a more philosophical question about the interpretation of a mathematical process. We have a chiral superscalarfield shown as partiell Grassmann Integral and transform it into a lagrange. where S and P are real components of a complex scalarfield and D and G are real componentfields of...
9. ### Order or Grassmann, vector fields and tensors

Hello. There is one thing I can not find the answer to, so I try here. For instance, writing a general superfield on component form, one of the terms appearing is: \theta \sigma^\mu \bar{\theta} V_\mu My question is if one could have written this as \theta \bar{\theta} \sigma^\mu V_\mu ...
10. ### Second functional derivative of fermion action

Homework Statement [/B] Consider the following action: \begin{align}S = \int \mathrm{d}^4 z \; \bar\psi_i(z) \, (\mathrm{i} {\not{\!\partial}} - m)_{ij} \, \psi_j(z)\end{align} where ##\psi_i## is a Dirac spinor with Dirac index ##i## (summation convention for repeated indices). Now I would...

15. ### I have never seen a Grassmann Number

Is there a way to represent Grassmann Numbers from previously known mathematical entities? Something like when it is said, for "C", that z=x+i*y and i^2=-1 or that z = [a -b; b a] with the usual rules of matrix sum and multiplication? It is pretty strange to me that it is so hard to find...
16. ### How do I get the 1st fundamental form on Grassmann Manifold

Consider G(n,m), the set of all n-dimensional subsapce in ℝ^n+m. We define the principal angles between two subspaces recusively by the usual formula. When I see "Differential Geometry of Grassmann Manifolds by Wong", http://www.ncbi.nlm.nih.gov/pmc/articles/PMC335549/pdf/pnas00676-0108.pdf I...
17. ### Query about Grassmann Variables in Lewis Ryder's book

Query about Grassmann Variables a + b\eta Hi, What does a + b\eta where a and b are c-numbers and \eta is a Grassmann number, really mean? At first sight, this does not seem like a valid thing to do, as this can't be interpreted as a regular composition and sum in Grassmann space...
18. ### Grassmann variables and Weyl spinors

I just started studying supersymmetry, but I am a little bit confused with the superspace and superfield formalism. When expanding the vector superfield in components, one obtains therms of the form \theta^{\alpha}\chi_{\alpha}, where \theta is a Grassmann number and \chi is a Weyl vector. I...
19. ### Understanding the Grassmann Integral: Exploring the Logic Behind ∫dθ θ = 1

I tried putting this in the math forum, but got no response: I can find various derivations of ∫ dθ = 0 which are satisfactory, but none of ∫dθ θ =1. Cheng and Li says it's just a normalization convention, of course that assumes that the integral is finite. Is this just a matter of...
20. ### Question about Grassmann Integral

I can find various derivations of ∫ dθ = 0 which are satisfactory, but none of ∫dθ θ =1. Cheng and Li says it's just a normalization convention, of course that assumes that the integral is finite. Is this just a matter of definition, or is there a better reason that that? And would...
21. ### Grassmann number and anti-commuting c-number?

People call grassmann numbers anti-commuting c-numbers, but from what I see grassmann numbers look like operators in many aspects, so what is the feature that distinguishes a grassmann number and an operator.
22. ### Integral over Grassmann variable in the holomorphic representation

Homework Statement Show that \int d\theta e^{\theta(\xi-\eta)}=\delta(\xi-\eta), where all of the above variable are Grassmann numbers. All this is in the holomorphic representation, where for some generic function: f(\theta)=f_0 + f_1\theta Homework Equations How do I arrive at that...
23. ### Grassmann Algebra: Derivative of $\theta_j \theta_k \theta_l$

If \{ \theta_i \} are a set of Grassmann numbers then what is \frac{\partial}{\partial \theta_i} ( \theta_j \theta_k \theta_l) I know that \frac{\partial}{\partial \theta_i} ( \theta_j \theta_k ) = \delta_{ij} \theta_k - \theta_j \delta_{ik} - we need this to be the case becuse if we set...
24. ### BRS operator, ghost fields, Grassmann numbers

Hi, I am learning what is going on with ghost fields in string theory. Does anyone know where I can find the basics for the brs (brst) operator? Specifically I need help with the following calculation. I have for the action of the brs operator on a ghost field and on the metric s \xi^\alpha =...
25. ### Grassmann Numbers & Commutation Relations

If you have a Grassman number \eta that anticommutes with the creation and annihilation operators, then is the expression: <0|\eta|0> well defined? Because you can write this as: <1|a^{\dagger} \eta a|1>=-<1| \eta a^{\dagger} a|1> =-<1|\eta|1> But if \eta is a constant, then...
26. ### Solve Quadratic w/ Grassmann Variables & NxN Matrix

Can anyone advise on the following : Complete the square for this expression : (theta)*i Bij (theta)j +(theta)*i(eta)i +(eta)*i(theta)iwhere theta and eta are Grassmann variables and Bij an NxN matrix with indices i,j=1,...,N. Many thanks
27. ### Delta function for grassmann numbers?

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...
28. ### Gaussian Integral Identity with Grassmann Numbers

Hi, I read the chapter "Anticommuting Numbers" by Peskin & Schröder (page 299) about Grassmann Numbers and now I would like to prove \int d \bar{\theta}_1 d \theta_1 ... d \bar{\theta}_N d \theta_N e^{-\bar{\theta} A \theta} = det A \theta_i are complex Grassmann Numbers...
29. ### How to make given numbers grassmann

At high school age I had trouble with complex numbers, because there was no rigor definition given to them, but instead only the property i^2=-1, and then we were supposed to calculate with it. This lead to somewhat mystical interpretations of imaginary unit sometimes, until I figured out the...
30. ### Grassmann Numbers: Generator Symbols & More

i was not sure where to put this topic since I don't know which subject of math grassmann math constitutes. Is there an actual grassman number or is it symbolically represented by generators?
31. ### Learn About Grassmann Manifolds: Intro, Charts, Atlas

I need to following subjects about GRASSMANN MANIFOLDS,what do I? 1)introduction(together with details) 2)charts,atlas(together with details) 3)depended subjects