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.

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  1. G

    Computing path integral with real and Grassmann variables

    The first step seems easy: computation of the $\theta$ and $\overline{\theta}$ integrals give $$Z[w] = \frac{1}{(2\pi)^{n/2}}\int d^n x \: \det(\partial_j w_i(x)) \exp{\left(-\frac{1}{2}w_i(x)w_i(x)\right)}.$$ From here, I tried using that $$\det(\partial_j w_i (x)) = \det\left(\partial_j w_i...
  2. T

    A Grassmann number: a more rigorous treatment?

    Most textbooks on fermionic path integral only briefly introduce Grassmann numbers. However, I want a more systematic treatment to feel comfortable about this approach. For illustration, I have several examples here. Example 1: Consider a system with only one state, how to calculate ##\langle...
  3. J

    Proof involving exponential of anticommuting operators

    For ##N=1##, I have managed to prove this, but for ##N>1##, I am struggling with how to show this. Something that I managed to prove is that $$\langle\psi |b_k^\dagger=-\langle 0 | \sum_{n=1}^N F_{kn}c_n \prod_{m=1\neq k, l}^N \left(1+b_m F_{ml}c_l \right)$$ which generalizes what I initially...
  4. JD_PM

    Showing an identity involving Grassmann variables

    All we should need for this problem are the basic rules for the Grassmann algebra \begin{equation*} \{ \theta_i, \theta_j\} = 0, \quad \theta^2_i=0 \end{equation*} \begin{equation*} \int d\theta_i = 0, \quad \int d\theta_i \ \theta_i = 1 \end{equation*} Starting from left to right...
  5. K

    A Grassmann numbers and Fermions

    Hello! I am a bit confused about fermions in QFT when they are considered grassmann numbers. If you have 2 grassmann numbers ##\theta_1## and ##\theta_2##, something of the form ##\theta_1\theta_2\theta_1\theta_2## gives zero. However, a term in a QED lagrangian of the form...
  6. U

    Functional Derivative with respect to Dirac Spinors

    Homework Statement I am currently working on an exercise list where I need to calculate the second functional derivative with respect to Grassmann valued fields. $$ \dfrac{\overrightarrow{\delta}}{\delta \psi_{\alpha} (-p)} \left( \int_{x} \widetilde{\bar{\psi}}_{\mu} (x) i \partial_{s}^{\mu...
  7. N

    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. M

    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. marir

    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. Ravendark

    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...
  11. K

    Grassmann Integration: Clarifying Notation in "hep-th/0108200

    Hi, everyone! I am trying to understand notation of this textbook http://arxiv.org/abs/hep-th/0108200 page 8, formulas 2.1.4 and 2.1.5 $$\int d \theta_\alpha \theta^\beta=\delta_\alpha^\beta$$ this could be found in any textbook the weird that from the above formula follows $$\int d^2...
  12. MathematicalPhysicist

    Do you think that Grassmann used to smoke grass?

    Do you think that Grassmann used to smoke grass? :-) http://www-history.mcs.st-andrews.ac.uk/Biographies/Grassmann.html
  13. L

    Integer Cohomology of Real Infinite-Dimensional Grassmann Manifold

    I can't seem to find on the web a site that gives the Z cohomology of the infinite dimensional Grassmann manifold of real unoriented k planes in Euclidean space. I am interested in computing the Bockstein exact sequence for the coefficient sequence, 0 -> Z ->Z ->Z/2Z -> 0 to see which...
  14. Einj

    Grassmann variables and functional derivatives

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

    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. S

    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. maverick280857

    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. 3

    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. F

    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. F

    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. K

    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. S

    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. L

    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. C

    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. R

    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. S

    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. pellman

    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. P

    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. J

    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. C

    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. B

    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