Questions tagged [grassmann-numbers]

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Integration by parts in functional field integrals

In QFT one often encounters expressions of the form $$ \displaystyle Z = \int \mathscr{D}(\bar{\psi}, \psi) \, \mathrm{exp} \, S [ \bar{\psi}, \psi ] $$ with complex-valued (in the bosonic case) or ...
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63 views

About constants in fermionic path integral in Peskin and Schroeder

I am confused by fermionic path integral used in Peskin and Schroeder. Equation (9.69) gives $$\Big(\prod_n\int d\bar{\theta}_nd\theta_n\Big)e^{-\bar{\theta}_iM_{ij}\theta_j}=\det M\tag{9.69}$$ But ...
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52 views

(Anti)commutation of creation and annhilation operators for different fermion fields

The Fourier expansion of the fermion field operator is such that $$ \hat\psi=\int\!d^3p\,\left[ f_b(p)\hat b(p) +f_d(p)\hat d^\dagger\!(p) \right] ~~, $$ for some sufficiently complicated $f_b$ and $...
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75 views

Fermionic Version of the effective Action

For a scalar field theory one introduces the partition function with external sources $$ Z[j] = \int \mathscr{D} \varphi \, \exp \left( -S[\varphi] + \int j \, \varphi \right) \text{,} $$ the analogon ...
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20 views

How to get rid of the oscillatory behaviour in the fermionic functional?

The Gaussian fermionic integral is evaluated to be $$ I = \int \prod_{i,j}d\theta^*_i d\theta_j \exp\left(-(\theta^*_iM_{ij}\theta_j+c^*_i\theta_i+\theta^*_ic_i)\right)=N \det M e^{c^*_iM_{ij}^{-1}c_j}...
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110 views

What is a fermionic field theory?

Let $\mathscr{H}$ be a Hilbert space and $\mathscr{H}^{n}$ be the associated $n$-fold tensor product of this Hilbert space. I'll skip the mathematical details in what follows, but my approach follows ...
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36 views

Coherent states of Fermi Operators

I am currently following the book of Lowell.S.Brown. In the book, we construct: $$|\zeta\rangle = e^{\alpha^{\dagger} \zeta} |0\rangle \tag{2.4.38}$$ Now, to show that the states so constructed are ...
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34 views

Dimensional reduction with supermanifolds and gravity

When having compact dimensions (I guess it is not true with supermanifolds), the gravitacional constant gets diluted in extra dimensional space: $$G_N(4d)=G_N(Dd)/V(X)$$ However, I presume that the ...
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62 views

How do fermionic operators transform?

In quantum mechanics, if we have an operator $\Omega$, then under the transformation $T$, with infinitesimal generator $G$ (i.e. $T(\epsilon)=1-i\epsilon G + \ldots$), then operator transforms as $$\...
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64 views

Fermion Determinant

When we calculate fermion determinant for either Majorana or Weyl spinors, why do we get an extra factor of half as the coefficient of the determinant?
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45 views

Ward identity of QED - whether the fields are all $c$-number fields

I am following Sidney Coleman's lectures of Quantum Field Theory. At the end of ch.32, he derived the Ward identity for the 1PI generating functional $\Gamma[\psi,\bar{\psi},A_{\mu}]$ for QED: \begin{...
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72 views

Becker, Becker, Schwarz: “String Theory and M Theory” Exercise 5.3

Supersymmetric transformation: $$ \begin{align} \delta\Theta^{Aa} =& \varepsilon^{Aa}, \tag{5.3} \cr \delta X^\mu =& \bar{\varepsilon}^A\Gamma^{\mu}\Theta^A. \tag{5.4} \end{align} $$ The ...
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67 views

Right vs Left Derivatives

Let $\theta$ be a fermionic quantity and $f(\theta)=f(0)+\theta\frac{\partial f}{\partial\theta}=f(0)+\frac{\partial_r f}{\partial\theta}\theta$. Under a variation $\theta\mapsto\theta+\delta\theta$ ...
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41 views

Sign of pair of Dirac spinor bilinear

I don't understand the following statement: Any pair of Dirac spinors verifies $(\bar{\Psi}_1\Psi_2)^\dagger=\bar{\Psi}_2\Psi_1$ and it is valid for both commuting and anti-commuting (Grassmann-valued)...
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35 views

D'Alembert Operator on Fermionic Field in Path Integral

I am learning the Faddeev–Popov path integral formlism with Schwartz's QFT textbook. In the section 25.4.2 "BRST invariance", I came across the Lagrangian as: $$\mathcal{L}=-\frac{1}{4} F_{\mu \nu}^{2}...
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How do we construct the matrix representation for three Grassmann numbers? [duplicate]

I want to know how we find or construct the matrix representations for Grassmann numbers. For example, we can see from https://en.wikipedia.org/wiki/Grassmann_number: Grassmann numbers can always be ...
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What do Grassmann-valued terms in operators really mean?

I've read (for example, ch. 5 of Piers Coleman's book on Many Body Physics), that a simple general formulation of a fermionic driven harmonic oscillator problem is: $$H = H_0 + V(t)$$ $$H_0 = E_0 c^\...
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131 views

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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70 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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137 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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54 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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106 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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52 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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16 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
93 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 $$(x^\mu, \theta_\alpha, \bar{\theta}_\alpha)$$ ...
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462 views

Is fermion mass imaginary instead of real?

This seems to be an absurd question, but bear 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
63 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
40 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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211 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 ...
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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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154 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 ...
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258 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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34 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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61 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
67 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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35 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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203 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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153 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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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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42 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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52 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 $$ \...
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105 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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182 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 $q_3$ ...
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184 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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155 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 \...
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1answer
125 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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55 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^...
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1answer
101 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
234 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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144 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 ...