Questions tagged [grassmann-numbers]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
4 votes
0 answers
35 views

Path integral on many-body quantum mechanics

Suppose $\mathscr{H}$ is a Hilbert space describing a one-particle quantum system and $\mathcal{F}(\mathscr{H})$ is its associated Fock space, which is used to describe a many-body quantum system. Let ...
user avatar
  • 519
1 vote
0 answers
64 views

Definition of a determinant Peskin&Schroeder

In page 514 of Peskin&Schroeder we are given the definition of a determinant as $$ \det\left(\frac{1}{g}\partial_\mu D^\mu\right)=\int{\cal{D}cD\bar{c}\exp\left[i\int {d^4x\bar{c}(-\partial^\mu D_\...
user avatar
2 votes
1 answer
50 views

What does it exactly mean by right and left functional derivatives?

In BV formalism of the gauge theory, we need to compute the right / left functional derivatives of the actions that include fermions. I do not quite see what it means by that. For example, let us ...
user avatar
  • 1,049
1 vote
1 answer
52 views

Anti-Symmetry of Dirac Operator

In his paper Fermion Path Integrals And Topological Phases, Witten states “Whenever one has a theory of fermions, the quadratic part of the fermion action is always antisymmetric by virtue of fermi ...
user avatar
  • 113
1 vote
0 answers
44 views

Feynman rules in Grassmann variables [duplicate]

I'm given the following 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}...
user avatar
  • 109
2 votes
1 answer
39 views

Complex valued Grassmann variables $(\theta \eta)^* $, $(\theta \eta)^T$ and $(\theta \eta)^\dagger$

Since hermitian conjugation and complex conjugation are serious issues in a QFT lagrangian with Grassmann variables, see here and here. Let us try to go to the bottom. We start by accepting the ...
user avatar
9 votes
2 answers
661 views

Path integral for complex scalar field

I am taking a QFT course which focuses on the path integral formulation. At a certain point, I was confused because we saw that, when integrating over complex Grassmann fields for fermions, we defined ...
user avatar
  • 180
1 vote
1 answer
64 views

Poisson bracket to quantum commutator for Grassmann-valued coordinates

In Dirac's Principles of Quantum Mechanics (pg. 86 eq. 5), the quantum commutator is motivated by looking for a bracket that satisfies the same properties of the Poisson bracket. When deriving the ...
user avatar
  • 101
4 votes
1 answer
111 views

How do we trace over subregions in a fermionic QFT?

Bosonic Case In a bosonic QFT, the Hilbert space associated to a surface $\Sigma$ is the appropriate space of wavefunctionals on $\Sigma$. Hence, if $\Sigma=\Sigma_1 \sqcup \Sigma_2$, we find that the ...
user avatar
3 votes
1 answer
109 views

Interaction with odd number of fermionic fields?

Can there exist an interacting part of the Hamiltonian with odd number of fermionic operators? In other words, can we have a vertex which couples an odd number of fermions (there can also be 1, 2, or ...
user avatar
  • 1,515
4 votes
0 answers
41 views

Where are the multi-instantons in Supersymmetric QM?

Instantons can be used to find non perturbative corrections to ground state energies. However, the way in which they are used seems to me to be very different between the two common toy models of the ...
user avatar
  • 10.9k
1 vote
1 answer
65 views

Requirement of Jordan-Wigner string in creation operator on Fock state

Our lecture notes described the action of the particle creation operator on a fermionic Fock state: $$c_l^\dagger |n_1 n_2...\rangle = (-1)^{\sum_{j=1}^{l-1}n_j}|n_1 n_2 ... n_l+1 ...\rangle.$$ I am ...
user avatar
  • 2,282
1 vote
1 answer
73 views

Jacobi identity of the anti-bracket

I'm currently reading a volume 2 of Weinberg's QFT, and am puzzled by the Jacobi identities of the anti-bracket.  The anti-bracket is defined using the anti-field $\chi^n$ and $\chi_n^{‡}$ as follows $...
user avatar
  • 681
2 votes
1 answer
63 views

Why do the electron and position creation operators anti-commute?

I am learning QFT and is baffled by a minor problem. The electron and the positron should be distinguishable, as they have different charges. So why do their creation operators anti-commute? They ...
user avatar
  • 957
0 votes
2 answers
115 views

Do I run into trouble if I interpret the fermionic field operator as a linear combination of a real and an imaginary part?

As some other questions on this website suggest, I have a really hard time with the fermionic field operator $\psi(x)$. I'd like to come to terms with this blockade. It serves as the smallest building ...
user avatar
  • 5,499
2 votes
0 answers
70 views

How to justify $\bar \psi(x) \psi(x) \bar\psi(y)\psi(y)\Rightarrow -\bar \psi(x) \psi(y) \bar\psi(y)\psi(x)$ in path integral?

Consider an interaction term of the form $$(e\int dx^4\bar \psi(x) \psi(x))(e\int dy^4 \bar\psi(y)\psi(y))$$ where the generating function was $$\bar\eta(x)\psi(x)+\bar\psi(x)\eta(x)\Rightarrow \int ...
user avatar
2 votes
1 answer
110 views

Berezin integral of a Grassmann field

Consider a time dependent Grassmann field i.e. $\theta(t)$. Now, consider the following Berezin integral $$\int [\mathcal{D}\theta] ~\prod_{t}\dot{\theta}\tag{1}$$ where $\dot{\theta}$ time derivative ...
user avatar
  • 810
0 votes
2 answers
61 views

Expanding the Following Grassmann Function

I am reading Altland and Simons, and they perform the following step in a calculation I am not sure how to perform. If $a_i$ and $\bar{\eta}_i$ are Grassmann variables, then $$e^{-\sum_i a_i \bar{\eta}...
user avatar
  • 544
2 votes
1 answer
223 views

Proof involving exponential of anticommuting operators

Problem: On page 23 of the book "Quarks, gluons and lattices" by Creutz, he defines a state $$\langle\psi|=\langle 0|e^{bFc}e^{\lambda b^\dagger G c^\dagger}$$ where $\lambda$ is a number, $...
user avatar
2 votes
1 answer
123 views

Partition function in SYK Model

In SYK model, we have the partition function for $N$-interacting fermions as \begin{equation} z=\int d^{N} \psi \exp \left(\imath^{q / 2} \sum J_{a_{1} a_{2} \ldots a_{q}} \psi_{a_{1} a_{2} \ldots a_{...
user avatar
1 vote
0 answers
61 views

Majorana Fermions and the Divergence of Currents

I am working on exercise 3.4d in Peskin and Schroeder's Introduction to Quantum Field Theory. In part c I found the Dirac Lagrangian density to be $$L = i\chi_1^\dagger \bar{\sigma}^\mu \partial_\mu \...
user avatar
1 vote
0 answers
40 views

Pfaffian for Weyl Symbol approach to path integrals

In M. Henneaux and C. Teitelboim, Quantization of Gauge Systems. Princeton University Press, 1992. the authors claim that integrating the $\xi$'s out of this expression for the Weyl symbol of the time ...
user avatar
  • 3,344
1 vote
1 answer
47 views

Conventions for graded wedge product in supergeometry

There are two conventions for the graded exterior product on superspace (see https://ncatlab.org/nlab/show/signs+in+supergeometry): $$\alpha \wedge \beta = (-1)^{pq+|\alpha||\beta|}\beta\wedge\alpha \;...
user avatar
1 vote
2 answers
261 views

Using Grassmann variables on fermionic theories

I think the best way to put my question is the following: what (fermionic) theories make use of Grassmann variables? Let me clarify my question a little further. I remember some discussions in quantum ...
user avatar
  • 487
1 vote
2 answers
94 views

Logarithm of Grassmann numbers

What is $\log \theta$ where $\theta$ is a Grassmann number such that $\theta^2 = 0$. How does one then look at its logarithm? Does it even make sense?
user avatar
  • 810
3 votes
2 answers
136 views

Integration with complex Grassmann numbers

I have a question about a convention from Peskin & Schroeder, namely that $$\int d\theta^{*}\, d\theta \, (\theta \theta^*) = 1,$$ where $\theta$ and $\theta^*$ are independent Grassmann numbers. ...
user avatar
1 vote
2 answers
70 views

Coefficients of the power of a sum of fermionic operators?

Consider the sum of $N$ fermionic operators $f_1+f_2+\cdots+f_N$, where the $f$‘s anti-commute, i.e. $\{f_i,f_j\}=0$. What is the expression for $(f_1+f_2+\cdots+f_N)^N$? Would the $f$’s be normal ...
user avatar
1 vote
1 answer
85 views

Gaussian integral with respect to Grassmann variables

Let $A$ be an antisymmetric matrix of even dimension $n$ and $\theta$ be a column vector consisting of $n$ Grassmann variables $\theta_i$. Then the solution of the integral $$\int d\theta_1\dots d\...
user avatar
  • 155
0 votes
1 answer
43 views

Dimensionality of Grassmann numbers

I have noticed in a few texts ( Introduction to Supersymmetry by Harald J. W. Muller-Kirsten, Armin Wiedemann for instance) state that Grassmann variables that are used in superfields have dimension $\...
user avatar
1 vote
2 answers
214 views

Meaning of the fermion path integral?

I'm trying to understand fermion fields with the Feynman integral. Is there an explicit matrix representation of the Grassmann numbers used in the field integral? Is there a Grassmann-valued measure ...
user avatar
1 vote
1 answer
83 views

How do I prove that the product of chiral superfields is itself a chiral superfield?

I am currently learning about $\mathcal{N}=(2,2)$ supersymmetry and have come up against what is probably a really silly question. The $\mathcal{}N=(2,2)$ superspace consists of bosonic coordinates $x^...
user avatar
2 votes
1 answer
150 views

Right derivative of Grassmann number and associated anti-commutation relation

I am reading chapter 3 of Anomalies in quantum field theory by Reinhold Bertlmann and I found one statement that I don't know how to prove. First of all he defined the right derivative on the ...
user avatar
  • 691
3 votes
1 answer
178 views

Where do fermionic coherent states live?

Although there have been a couple of questions on fermionic coherent states, I don't think any has answered the question "on what space do fermionic coherent states live?", or at least not ...
user avatar
  • 1,598
1 vote
0 answers
204 views

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 ...
user avatar
  • 141
2 votes
1 answer
132 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 ...
user avatar
  • 1,703
0 votes
1 answer
106 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 $...
user avatar
0 votes
2 answers
128 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 ...
user avatar
  • 141
1 vote
1 answer
82 views

Showing that the variation in given action leads to Majorana equation

This is a question on mathematics rather than the physics. It is based on QFT Q3.4, part b on Peskin and Schroeder. My confusion stems from the fact that we are consider $\chi(x)$ to be a classical ...
user avatar
  • 127
1 vote
0 answers
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}...
user avatar
2 votes
1 answer
207 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 ...
user avatar
  • 487
1 vote
0 answers
43 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 ...
user avatar
  • 357
1 vote
0 answers
40 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 ...
user avatar
  • 5,373
2 votes
2 answers
132 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 $$\...
user avatar
  • 767
0 votes
2 answers
321 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?
user avatar
1 vote
1 answer
70 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{...
user avatar
  • 691
2 votes
1 answer
128 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 ...
user avatar
2 votes
1 answer
329 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 $\theta\mapsto\theta+\delta\theta$ ...
user avatar
  • 3,344
1 vote
1 answer
83 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)...
user avatar
0 votes
1 answer
58 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}...
user avatar
0 votes
0 answers
33 views

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 ...
user avatar
  • 1

1
2 3 4 5