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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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1answer
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Anticommutativity of an anticommutator of supercharges

In this paper, equation 38 gives the ${\cal N}=2$ Super-Poincare (extended with the central extension $\mathcal{Z}$). The anticommutation relation of the two different supercharges is given as: $$\{Q^...
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1answer
78 views

Show: $\langle n \vert \psi \rangle \langle \psi \vert n \rangle = \langle -\psi \vert n \rangle \langle n \vert \psi \rangle$ [closed]

The book (Altland and Simons, Condensed Matter Field Theory, Ch. 4.2) I am reading makes use of the identity \begin{equation} \langle n \vert \psi \rangle \langle \psi \vert n \rangle = \langle -\psi \...
3
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2answers
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Is the derivative with respect to a fermion field Grassmann-odd?

Fermion fields anticommute because they are Grassmann numbers, that is, \begin{equation} \psi \chi = - \chi \psi. \end{equation} I was wondering whether derivatives with respect to Grassmann numbers ...
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1answer
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Why is supersymmetry a continuous symmetry?

Supersymmetry feels like a discrete symmetry to me, since the fermions are turning into bosons, and vice versa. I understand there is an infinitesimal parameter involved in the transformations, but I ...
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0answers
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Vanishing partition function [duplicate]

I am currently stuck with the following partition function Let the action be $$S(X, \psi^1, \psi^2) = \frac{1}{2} (\partial h)^2 - \partial^2h\psi^1 \psi^2 ,$$ where $h$ is a real function of the ...
2
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1answer
37 views

Stationary phase method for theories with both bosons and fermions

I am wondering if there exists a method to compute path integrals using the stationary phase method for theories with both bosons and fermions. (I am aware of such a method for theories with bosons ...
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2answers
50 views

Grassmann-even action

I am currently studying supersymmetric quantum mechanics with the help of the book Mirror Symmetry by Kentaro Hori (and others). On page 155 where they introduce Grassmann variables they say that the ...
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0answers
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$R$-Symmetry of gauge field

Suppose $V$ is a superfield scalar under R-transformations. This means that under an R-transformation $V\mapsto V'$ where $V'(x,\theta,\bar{\theta})=V(x,e^{-iK}\theta,e^{iK}\bar{\theta})$. What is ...
0
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1answer
37 views

Anti-Commutator of derivatives of Grassmann variables

How do I evaluate the anti-commutator of $\frac{\partial}{\partial\chi}$ and $\frac{\partial}{\partial\eta}$ when both $\chi$ and $\eta$ are Grassmann variables?
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1answer
56 views

What are Grassmann numbers in field theory?

I've been struggling with the use of Grassmann numbers in QFT e.g. Peskin and Schroeder. They are introduced as "numbers" whose product is antisymmetric, and associative (this isn't said, but used in ...
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“Data structure” for a fermion field

I am understanding the path integral formalism of fermion fields. Most textbooks told me that grassmannian integration is only algebaric notation. It shouldn't be understood in a Lebesgue Integral ...
2
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1answer
102 views

Change of variables in path integral measure

In fermion's path integral we have a measure that you can write, in terms of the Grassmann variables $\psi, \bar{\psi}$ as $$ D\bar{\psi}D\psi, \quad \psi(x) = \sum_n a_n\phi_n(x), \quad \bar{\psi}(x)...
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1answer
42 views

Identity Involving Grassmann Variables and Pauli Matrices

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{\...
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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{...
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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} ...
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0answers
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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}(\...
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1answer
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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 ...
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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 ...
3
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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 ...
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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, $$\...
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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 $\eta^2 = 0$. ...
3
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1answer
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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 ...
2
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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 ...
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1answer
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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 ...
2
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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 ...
3
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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 [1] or [2] the authors introduced auxiliary Grassmann odd variables to ...
0
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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 $\partial^2$ ...
2
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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 \...
2
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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{\...
0
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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 ...
4
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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 $S$...
4
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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 ...
4
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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 ...
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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 = ...
2
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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 ...
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1answer
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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 ...
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0answers
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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 ...
1
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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 ...
4
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0answers
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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) ...
2
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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. ...
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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^{(...
3
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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 ...
2
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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;~~...
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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 \...
2
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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 ...
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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 (\...
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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 ...
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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 $...
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1answer
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Can Grassmann-number variations of operators be represented by operators?

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 ...