Questions tagged [grassmann-numbers]

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What is the definition of functions of Grassmann numbers?

I understand there are some relevant questions, but none of them solves my issue. From Atland and Simons (Condensed Matter Field Theory), the definition of functions of Grassmann numbers are defined ...
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I am trying to prove the following identity: $$\theta\sigma^{\mu}\bar{\theta}\theta\sigma^{\nu}\bar{\theta}=\frac{1}{2}g^{\mu\nu}\theta\theta\bar{\theta}\bar{\theta}$$ Where $\theta$ and $\bar{\... 1answer 61 views Symmetry of the Batalin-Vilkovisky (BV) antibracket operation Batalin and Vilkovisky define$^1$an operation they call antibracket which is $$(F,H) = \Big(\frac{\partial_r F}{\partial \Phi^A}\Big) \Big(\frac{\partial_l H}{\partial \Phi^* _A} \Big) - \Big(\frac{... 1answer 50 views Explicit quantization of free fermionic field The canonical quantization of a scalar field \phi(x) can explicitly be realized in the space of functionals in fields \phi(\vec x) (here \vec x is spacial variable) by operators \begin{eqnarray} ... 0answers 33 views A Naive Question about SUSY Variation I am following BUSSTEPP Lectures on Supersymmetry to learn supersymmetry. My simple question is the following. My Lagrangian for the Wess-Zumino model in 4D is$$\mathcal{L}=-\frac{1}{2}(\... 1answer 87 views A few questions about spinors and gamma matrices I am following BUSSTEPP Lectures on Supersymmetry and trying to show that the Wess-Zumino action is invariant under SUSY transformations. I encountered the following questions about spinors and gamma ... 1answer 103 views Transpose of fermion bilinears TL;DR When we take the transpose of two Grassmann-valued spinors (fermions), should we add a minus sign because we end up anticommutating the two spinors? More details. I'm studying the behavior of ... 1answer 107 views Canonical Quantisation vs the Dirac Field, why does it even work? Using the "Dirac Prescription", we can preserve the format of a differential equation in its QM form. If we define the canonical variables s.t. they have the same commutation relations times$i$as ... 1answer 273 views How does canonical quantization work with Grassmann variables? Every quantum field theory textbook I've encountered seems to have the same logical oversight, because of the particular order they cover topics. First, the books introduce the Dirac Lagrangian, $$\... 2answers 102 views Grassmann's variables under integration If \eta is a Grassmann variable, due to invariance under translations we get that,$$\int d\eta\ \eta = 1 \tag1$$Nevertheless, for being Grassmann's, \eta satisfies \eta^2 = 0. ... 1answer 70 views Intuition for the supertrace identity in supersymmetry In pretty much every introductory book/lecture notes I've come across, one finds the expression for the mass matrices for scalars, fermions and vector bosons for a generic Lagrangian, and simply ... 1answer 57 views Fermionic ghost path integral results in \delta function? This is related to a statement in pg 20 of hep-th/9408074 formula (2.39). Suppose$$\mathcal{L}\sim\frac{i}{\lambda^{\prime}}\bar{\eta}^xg_{ij}U_x{}^i\psi^j+\cdots \tag{2.35}$$where \bar{\eta} to ... 1answer 73 views What is the content of an occupied QFT fermionic state? A simple non-interacting quantum field is constructed by analogy to a harmonic oscillator, with \hat{x} & \hat{p} replaced by \hat{ \phi} & \hat{\pi} & with a separate oscillator ... 1answer 89 views Supersymmetry transformation: why does the Lagrangian transform as total derivative? There is something I don't understand at page 36 of these lecture notes (Author: Fiorenzo Bastianelli from the university of Bologna, title: Path integrals for fermions and supersymmetric quantum ... 1answer 79 views Auxiliary Grassmann variables in supergeometry I was reading on super geometry and how it is used to model fermions and supersymmetry in classical field theory. In texts like  or  the authors introduced auxiliary Grassmann odd variables to ... 1answer 76 views How to integrate by parts ghost fields in electrodynamics? When applying Faddeev-Popov method to electrodynamics in the Lorenz gauge we obtain the ghost action$$S=\int d^4xd^4y\bar\eta(x)\left(\partial^2\delta(x-y)\right)\eta(y),\tag{0}$$where \partial^2 ... 2answers 119 views Two-point Green for Free Dirac Fields I am trying to compute the 2-point Green function \tau_2(x,y) for free Dirac fields. The corresponding formula for \tau_2(x,y) is given by$$\tau_2(x,y) = -\frac{\delta^2}{\delta\eta_x \delta \... 1answer 159 views Plugging Majorana Spinor into Dirac Lagrangian does not give Majorana Lagrangian? This seems like it should be simple but somehow I do not see how. The Majorana Lagrangian can be written in terms of a left handed Weyl spinor$\psi_L$as $$\mathcal{L}_M= i \psi_L^\dagger \bar{\... 1answer 36 views Grassmann-odd extra dimensions and gravity Take a world with D=3+n space-time dimensions, where n are extra space-like dimensions. With extra-dimensional newton gravity$$F=G_N(D)\dfrac{Mm}{r^{2+n}}$$Can n affect IF the extra ... 1answer 87 views Can you quantize Grassmann-even superfields in the same fashion as Boson fields? In a related Phys.SE question about supersymmetric Lagrangian$$ \mathcal{L} = - \frac{1}{2} (\partial S)^2 - \frac{1}{2} (\partial P)^2 - \frac{1}{2} \bar{\psi} \partial\!\!\!/ \psi, $$the fields S... 2answers 343 views How are supersymmetry transformations even defined? I am just starting to read about supersymmetry for the first time, and there is something bothering me. Supersymmetry transformations transform between bosonic fields and fermionic fields, but I don't ... 1answer 137 views Why two different spinors are Grassmann quantities? In Rydberg Quantum Field Theory page 441 (this edition, unfortunately page 441 is not in the link) it says If \xi and \eta are Majorana spinors [...] and since \xi and \eta are Grassmann ... 2answers 137 views How to change a commutator of SUSY super-charges into an anti-commutator? I would like to understand an apparently rather simple calculation which checks the closure of the Supersymmetry algebra via the commutator of 2 supersymmetric variations of the type:$$\delta \phi = ... 1answer 53 views Could you get real space from Grassmann numbers? You can get a vector field from a pair of spinor fields with$A_\mu(x)=\psi(x) \gamma_\mu \overline{\psi}(x)$. Using this fact could you define a space-time vector in terms of Grasman numbers? Say ... 1answer 73 views Supersymmetry transformation of auxiliary scalar in Wess-Zumino model This question is related to my earlier question "Error bringing in the auxiliary scalar field in the Wess Zumino model". In equation (3.1.13) of "A Supersymmetry Primer", arXiv:hep-ph/9709356, the ... 0answers 78 views Coherent state representation of an operator in a Grassmann algebra I'm (still) working through the textbook Quantum Many-Particle Systems by Negele and Orland and want to show that the most general coherent state representation of an operator$A(\xi,\xi^*)$in a two ... 1answer 61 views Problems with anti-commutator between fermionic ladder operators I am trying to build the fermionic coherent state formalism in conformance with the grassmann conventions used in the book "Mirror Symmetry", relation (9.20), where the fermionic integration is ... 0answers 190 views Feynman rules for this perturbative expansion in Grassmann variables I'm given the integral $$Z[w] = \frac{1}{ (2 \pi)^{n/2}} \int d^n x \prod_{i=1}^n d \overline{\theta}_i d \theta_i \exp \left( - \overline{\theta}_i \partial_j w_i (x) \theta_j - \frac{1}{2} w_i(x) ... 1answer 115 views What is the Grassmann parameter \epsilon in the BRST transformation? Whenever I learn about anything involving fermions and the path integral, I get confused about Grassmann numbers. I'm currently following Weigand's notes, specifically the section on BRST symmetry. ... 2answers 139 views Majorana Flip Relations In the Supergravity book of Freedman et.al, which uses the signature (+,-,\dots,-), we have defined the charge conjugation matrix for general Clifford Algebra as (C\Gamma^{(r)})^T = -t_rC \Gamma^{(... 2answers 130 views Cauchy-Schwarz inequality for Grassmann Integrals? For square integrable functions f,g of a real variable, the Cauchy-Schwarz inequality states that$$ \left(\int f(x)g(x)\,dx \right)^2 \le \int f(x)^2\,dx \int g(x)^2\,dx. $$My question is: are ... 1answer 104 views Product of complex Grassmann numbers in higher dimensions If two numbers \eta and \xi anti-commute. i.e.,$$\eta\xi=-\xi\eta$$they are called Grassmann numbers. It immediately follows that$$\eta^2=\xi^2=0,$$and relations such as$$e^{a\eta}=1+a\eta;~~... 1answer 66 views Do half integer spin fields commute or anti-commute with spin integer fields? What are the fundamental commutation/anti-commutation relations between half integer and integer spin fields? For instance, in QED do we have \begin{equation} [\psi(x),A^{\mu}(y)]=0 \end{equation} or \... 1answer 94 views Supergravity action as a total integral, over 4 spacetime and 4 Grassmann coordinates Wess and Bagger, in their Supersymmetry and Supergravity, give the action for a global SUSY,${\cal N}=1$,$D=4$, Yang-Mills gauge model as an integral over the 4 spacetime coordinates and 4 Grassmann ... 1answer 82 views Variation of Fermionic Field Operator Suppose we have a Hamiltonian containing some interaction term $$V = \sum _{\sigma \sigma '\sigma ''\sigma '''}\iint d^3rd^3r'\hat{\psi }_{\sigma}^\dagger (\textbf{r})\hat{\psi }_{\sigma'}^\dagger (\... 2answers 365 views An Identity for a Gaussian Grassmann integral from Wikipedia I found this identity on Wikipedia:$$\int\exp\left[\theta^T A\eta+\theta^T J+K^T\eta\right]d\theta d\eta =\det A\exp\left[-K^TA^{-1}J\right],$$where the integration variables are Grassmann ... 1answer 280 views How to construct a supersymmetry algebra? Starting with the general notion of supersymmetry:$$Q| boson \rangle = | fermion \rangle \\ Q| fermion \rangle = | boson \rangle$$I want to construct the superalgebra relations. After applying$...
In my previous question, I asked about how to handle Grassmann-number variations of operators. I read a book that uses those variations $\delta \Phi = c \mathbb{1}$, with $c$ being a grassmann number ...