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Questions tagged [grassmann-numbers]

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What are Grassmann numbers in field theory?

nI'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 ...
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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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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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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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49 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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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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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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59 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 ...
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101 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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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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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 $\eta^2 = 0$. ...
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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 ...
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1answer
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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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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 ...
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78 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 ...
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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 ...
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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 $\partial^2$ ...
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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 \...
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1answer
138 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{\...
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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 ...
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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 $S$...
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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 ...
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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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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 = ...
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1answer
50 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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72 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 ...
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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 ...
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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 ...
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187 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) ...
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108 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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119 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^{(...
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2answers
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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 ...
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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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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 \...
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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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78 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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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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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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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 ...
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Dirac Fields and Derivatives (Am I gaining extra minus signs?)

I've given myself a severe headache jumping between East/West Coast sign conventions; I have picked up an extra minus sign and could do with a hand. I am currently using $\eta=\textrm{Diag}[-,+,+,+]$ ...
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Derivation of Schwingers action principle from Heisenberg Equation and CCR - Why does it work with Anticommuting variations?

In the Book "Quantum Field Theory I" by Manoukian, in section 4.3, from what I understood, he derived the quantum-action-principle of Schwinger only by using unitary time-evolution of the field ...
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366 views

Question on Wick's theorem for fermions

I have a guilty suspicion this should be obvious. What is the difference between these two expectations taken over the same measure ($\int \mathrm{d}\mu(\bar\psi,\psi)\exp{\sum \bar\psi A\psi}$ for ...
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Does d/d(spinor) anticommute with a spinor?

Weyl spinors anticommute (see, e.g. Why isn't the anticommutativity of spinors sufficient as "spin-statistics-theorem"?). If we consider the derivative, with respect to a Weyl spinor $\...
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1answer
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Ambiguity of Time Derivative of Superfunctions

I think that there is an ambiguity for defining the time derivative of a superfunction on the phase space of pseudo-classical mechanics of Grassmann numbers. Let $\xi$ be a Grassmann odd number. Its ...
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1answer
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How to make this short explanation to path integral approach to QED correct?

Edit: as suggested by @Slereah the following explanation of the path integral approach to QED misses gauge fixing and ghost terms to be correct. So, how can this explanation be made correct? What ...
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1answer
219 views

Link between the Grassmann algebra and spinors

What is the exact link between spinors and the Grassmann algebra? I'm pretty sure there's one, based on the following: The Berezin integral in path integrals is done over the Grassmann algebra of $\...
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138 views

Transpose Noether current from $U(1)$ symmetry of the free Dirac field

When I read through the notes of a particle physics script there is the following identity that I don't understand $$\left(\psi^T {\gamma^\mu}^T \bar{\psi}^T \right)^T = -\bar{\psi}\gamma^\mu \psi.$$ ...
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3answers
224 views

What is the definition of $\delta^{m|n}$ and $\delta^{m}$?

I am reading the paper. What is the definition of $\delta^{m|m}$ and $\delta^{m+k}$ in (1.1) and (1.3) on pages 2,3? Are they some kind of delta function?
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646 views

Why are Grassmann fields never classical?

I see this statement in many QFT books (e.g. Altland & Simons' Condensed Matter Field Theory) but the author never explains why. Can you briefly explain why Grassmann fields never have a ...