Questions tagged [grassmann-numbers]

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Partition function for quantum Ising model

I have hamiltonian for fermionic field as $${\cal H}_F=E_0+\int dx[\frac{v}{2}(\Psi^\dagger\frac{\partial \Psi^\dagger}{\partial x}-\Psi\frac{\partial \Psi}{\partial x})+\Delta\Psi^\dagger\Psi]\tag{1}$...
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63 views

Grassmann confusion

I am trying to calculate $$\mathcal{\bar{D}}_\dot{\alpha}y^\mu= \left(\bar{\partial_\dot{\alpha}}+i\theta^\alpha\sigma^\mu_{\alpha\dot{\alpha}}\partial_\mu\right)\left(x^\mu+i\theta^\beta\sigma^\mu_{\...
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2answers
95 views

SUSY $\mathcal{N}=1$ algebra

Given the definitions $$ P_\mu= -i\partial_\mu $$ $$ Q_\alpha=-i(\partial_\alpha-(\sigma^\mu\bar{\theta})_\alpha\partial_\mu) $$ $$ \bar{Q_\dot{\alpha}}=+i(\bar{\partial}_\dot{\alpha}-({\theta}\sigma^...
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1answer
60 views

Superspace integrals

I understand the definition of the Grassmann Integral, which is as follows: $$ \int{d\theta}=0,\;\;\;\;\;\; \int{d\theta}\;\theta=1 \tag{4.17} $$ with $$\int{d\theta}=\frac{\partial}{\partial\theta}.\...
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35 views

Why there's no missing determinant in Gaussian integration with Grassmann variables?

This integral appears in Ashok p. 82-83. We have the integral $$ I = \int \prod_{i,j}d\theta^*_id\theta_j e^{-(\theta_i^*M_{ij}\theta_j + c_i^*\theta_i + \theta_i^*c_i)}, $$ and if the inverse of $...
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1answer
45 views

What is the complex conjugate of two fermionic fields coupled? $(\bar{\psi} \chi)^{\ast} =$ ____?

Suppose $\psi$ and $\chi$ are fermionic fields, and suppose I want to calculate the hermitian conjugate of the operator $\bar{\psi} \chi$ $$ h.c.\ \mathrm{of}\ \bar{\psi} \chi = (\bar{\psi} \chi)^{\...
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1answer
42 views

Boundary conditions of fermionic coherent states path integral

Given the algebra of a fermionic oscillator $$ \{\hat{a},\hat{a}^\dagger \}=1\,, \qquad \hat{a}^2=(\hat{a}^\dagger)^2=0, $$ with coherent states $ \hat{a}|\xi\rangle=\xi|\xi\rangle $, let's ...
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11 views

Boundary conditions of fermionic path integrals

When considering path integrals with grassmann variables, as stated on page 159 of Quantization of gauge systems - Henneaux, Teitelboim we only have one boundary condition, since the equations of ...
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1answer
76 views

What is the meaning of a Grassmann variable?

I don't seem to understand the concept of a Grassmann-variable. When studying superspaces and superfields I am told that the coordinates being used are $$(x^\mu, \theta_\alpha, \bar{\theta}_\alpha)$$ ...
3
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1answer
239 views

Is fermion mass imaginary instead of real?

This seems to be an absurd question, but bare with me. In quantum field theory, the Dirac fermion mass Lagrangian term reads $$ m\bar\psi \psi = m(\bar\psi_L \psi_R + \bar\psi_R \psi_L) = m(\psi_L^\...
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1answer
46 views

Chiral Symmetry and Charge Algebra

I'm following Section 5.1 in Cheng and Li's Particle Physics book and I am having trouble reproducing some of the commutation relations. The axial current is given by $$ (J_A^a)^\mu = \bar{\psi}_\...
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1answer
35 views

Are there cases where the use of the Grassmann variables simplifies computations in the usual bosonic analysis?

When one introduces complex numbers and complex analysis one can then use the new machinery to solve some real-analysis problems. A lamppost example is computing integrals via residues. I think I've ...
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1answer
165 views

$(\psi_L^\dagger \psi_R)^\dagger \neq (\psi_R^\dagger \psi_L)^\dagger$ ? What is the transpose for spinors?

The dirac mass term in terms of Weyl spinors is $$\psi_L^\dagger \psi_R + \psi_R^\dagger \psi_L.$$ My understanding is that both terms are necessary to form a hermitian term. Naively, if you take ...
3
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1answer
65 views

Supercharge in $\mathcal{N}=1$ supersymmetric quantum mechanics and Noether's theorem

Consider the $0+1$ dimensional Lagrangian $$L=\frac{1}{2}\dot{X}^2(t)+i \psi(t) \dot{\psi}(t).\tag{1.24}$$ Essentially this the Lagrangian of a particle moving in one dimension, $X$, with an ...
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25 views

Mass term with charge conjugation and fermion transposition

I am trying to convince myself that a $G_{ab}[\overline{(\psi^c_a)_R}(\psi^c_b)_L-\overline{\psi_{aR}}\psi_{bL}]$, where $G_{ab}$ is antissymetric, is just as good a mass term as $-\bar{\psi}\psi$ (...
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1answer
104 views

Is the expectation value of a Fermi field operator a Grassmann number?

It's often noted that Bosonic fields result from quantizing classical field theories defined on a regular numbers, whereas Fermionic fields arise when quantizing a classical field theory defined on ...
3
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1answer
156 views

What exactly are “Grassmann-valued fields”?

Peskin & Schroeder define a Grassmann field $\psi(x)$ as a function whose values are anticommuting numbers, that can be written as : [p.301 eq. 9.71] $$\psi(x) = \sum\psi_i \phi_i(x),\tag{9.71}$$ ...
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1answer
31 views

Expand superspace function into component form

In 2D (1,1) superconformal field theory, the invariant "distance" between two points $Z_1=(z_1,\theta_1)$ and $Z_1=(z_1,\theta_1)$ in superspace is $$Z_{12}=z_1-z_2-\theta_1\theta_2.$$ My question ...
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52 views

Gamma Matrices as nonstandard numbers, and Grassman Numbers

I'm in the process of exploring the Dirac equation and its forms and consequences, and as such have just been initiated into the theory of spinors and their accompanying formalism. One of the things ...
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1answer
51 views

Anticommutation relations for fermionic fields imply that Hamiltonian / Lagrangian can at most be linear?

Fermionic field operators do obey anticommutation relations, so for a chosen Field operator (and the field momentum), we have: $$ \{\Psi_a, \Psi_b\} = \{\pi_a, \pi_b\}= 0 $$ with the $\Psi_a$ being ...
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39 views

Question about Majorana fermion current conservation of problem 3.4(d) in Peskin's QFT book?

I have some trouble proving the current conservation mentioned in problem 3.4(d) of Peskin&Schroeder's QFT book. Relevant sections of the problem: (b). Does the Majorana equation $$ i\bar\...
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25 views

Swap fermion with boson?

I wonder what actions/factors/terms show up when you swap a fermion and boson that are tensored together in second quantisation. It would suffice for me if someone could give me the name of such an ...
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1answer
145 views

On the king's way to Grassmann numbers

I have two questions on Grassmann numbers. In particular, their naive representation using "huge matrices". Apparently, the anticommutation relations $\{ \xi_i, \xi_j \} = 0$, $\{ \xi_i, \xi^{*}_j \...
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1answer
123 views

Why are Grassmann variables the classical limit of fermions?

In many texts the anti-commutation relations for fermions are given as $$\{ \bar{\psi}^\alpha (\vec{x}), \psi^\beta(\vec{y}) \} = \delta^{\alpha\beta} \delta(\vec{x} - \vec{y})$$ $$\{ \psi^\alpha (\...
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165 views

What's the space of eigenvalues/field configurations for a fermion?

In the Schrödinger picture of quantum field theory, the field eigenstates of a real scalar field $\hat\phi(\mathbf x)$ with $\mathbf x \in\mathbb R^3$ are the states $\hat\phi(\mathbf x)|\phi\rangle=\...
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37 views

Fermion commutation with two quantum numbers

I am trying to understand the second quantization formalism. Let's say we have a system of fermions (e.g. electrons) with spin in an array of quantum dots. The creation and annihilation operators $c^{\...
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46 views

Anticommutation relation of Grassmann numbers

Let $c,c^*$ be the fermion annihilation/creation operators and $\xi,\xi^*$ denote Grasssmann numbers where $$|\xi\rangle = \exp(-\xi a^*)|0\rangle$$ is the coherent state. Then why is it true that $$ \...
2
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1answer
76 views

Expanding superfields: inconsistency of notation?

If I have a wavefunction of a fermion field $\Psi[\psi]$ I can expand it like so about some vacuum: $$\Psi[\psi] = \Psi_0[\psi]( a + \int a(x)\psi(x)dx+\int a(x,y)\psi(x)\psi(y)dxdy+...)$$ Now all ...
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1answer
158 views

Representing quaternionic algebra with creation and annihilation operators?

The paper "Quantized Grassmann variables and unified theories" says given creation and annihilation operators $b$ and $b^\dagger$ one can represent quaternionic imaginary units $q_1$, $q_2$ and $q_3$ ...
4
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1answer
131 views

What's the ground state wave-functional of a fermion?

The vacuum state, free field wave-functional of a scalar field $\hat\phi(x)$ in the Schrödinger representation of quantum field theory is $$\begin{array}{cl} \Psi_0[\phi] &= C\prod_k e^{-\omega(k)...
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1answer
98 views

Why is the Jacobian factor for fermionic variables different from that for bosonic ones?

In Srednicki's textbook Quantum Field Theory, Section 77 discusses anomalies and the path integral for fermions. The path integral over the Dirac field is defined to be \begin{equation} Z(A) \equiv \...
2
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1answer
79 views

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
34 views

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^...
2
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1answer
95 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 \...
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2answers
152 views

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
115 views

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
24 views

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 ...
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1answer
49 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
65 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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1answer
44 views

$R$-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 ...
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1answer
46 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
106 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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58 views

“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 ...
3
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1answer
311 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
66 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
65 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
58 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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39 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}(\...
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1answer
118 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 ...
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1answer
371 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 ...