Questions tagged [grassmann-numbers]

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Fermionic propagator [closed]

Given the fermionic generating functional $$Z[\eta]=\ det^{\frac{1}{2}}(K_{ij})e^{-\frac{i}{2}\eta_{i}G^{ij}\eta_{j}},\tag{1}$$ where $$G^{ij}=K^{-1}_{ij}$$ is the Green function of our theory, then ...
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2 answers
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Proving a Grassmann integral identity

How to prove the following identity $$ \begin{align} \int {\rm d} \eta_{1} {\rm d} \bar{\eta}_{1} \exp\left(a \left(\bar{\eta}_{1}-\bar{\eta}_{0}\right)\left(\eta_{1}-\eta_{0}\right) + b \left(\bar{\...
Faber Bosch's user avatar
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Trace formula in Grassmann algebra

From Grassmann algebra, we know the following relation \begin{equation} \mathrm{Tr} e^{-a\hat{c}^{\dagger}\hat{c}} =1+e^{-a} \end{equation} Now, how to prove the following generalized results? \begin{...
Santanu Singh's user avatar
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How does this term in the Majorana mass not vanish?

This is classical field theory. In the Majorana mass term, we have the expression $$\nu_L^T\sigma_2\nu_L \tag{1}$$ where the left-handed spinor field $\nu$ has a Grassmann-valued amplitude, i.e., $\nu ...
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Grassmann variables and orthogonality of coherent fermionic states

Let a coherent fermionic state $$ \left|\phi\right> := \left|0\right> + \left|1\right> \phi,\tag{0} $$ where $\phi$ is a Grassmann number (i.e. it anticommutes with other Grassmann numbers). ...
Gabriel Ybarra Marcaida's user avatar
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1 answer
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Confusion about whether a fermion field and its conjugate as an Grassmann number

I'm confused about what "a Grassmann-odd number" really means and how does it apply to fermions. In some text, it says that "if $\varepsilon \eta+\eta \varepsilon =0 $, then $\...
Errorbar's user avatar
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How to tell if a composite boson field should be real or complex?

Let's say I have a system with two species of fermions, $f$ and $c$, where $f$'s are neutral but $c$'s are charged. Each of these has its own $U(1)$ related to particle-number conservation. Now, if I ...
dumbpotato's user avatar
1 vote
1 answer
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Fierz idendity (supersymmetry)

So basically I have two Fierz identities involving spinors: $$\psi^a \psi^b = -\frac{1}{2} \epsilon^{ab} \psi \psi$$ And $$\overline{\psi}^{\dot{a}} \overline{\psi}^{\dot{b}} = \frac{1}{2} \epsilon^{\...
LSS's user avatar
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Why reasonable observables are made of an even number of fermion fields?

On Michele Maggiore book on QFT (page 91) is stated, out of nothing, that "observables are made of an even number of fermionic operator" and similar sentences is in Peskin book (page 56). Is ...
Andrea's user avatar
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What is the action of fermionic Hamiltonian $\mu_1 n_1 + \mu_2 n_2 + U n_1 n_2$

Problem Consider a Hamiltonian \begin{equation} H(c^\dagger, c) = \mu_1 c_1^\dagger c_1 + \mu_2 c_2^\dagger c_2 + U c_1^\dagger c_1 c_2^\dagger c_2\,, \end{equation} where $c_i$ are fermionic ...
Michał Jan's user avatar
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Dirac Lagrangian in Classical Field Theory with Grassmann numbers

The concept of the Grassmann number makes me confused. It is used to describe fermionic fields, especially path integral quantization. Also, it is used to deal with the classical field theory of ...
Jaeok Yi's user avatar
5 votes
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174 views

How the supercharge operators act on superfields in quantum mechanics, and the adjoints of supercharges?

I'm watching this lecture on introductory Supersymmetry (Clay Cordova, 2019 TASI lecture 2 on Supersymmetry). My question relates to the first 20minutes or so. The lecturer is introducing Superfields ...
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Total derivative of Grassmann variables

From page 21 of "Conformal Field Theory" by Di Francesco, Mathieu, and Sénéchal, the free Fermion Lagrangian is given by: $$L=\frac{i}{2}\psi_i T_{ij}\dot{\psi}_j-V(\psi)$$ Where the $\psi$ ...
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Operator representation in Fermionic Fock space

The representation of any operator $F$ in the fermionic Fock space in terms of displacement operators as - \begin{equation} F = \int d^2{\bf{\xi}}~f(\xi) D(-\xi) \end{equation} where $f(\xi)$ is the ...
Santanu Singh's user avatar
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Legendre transform involving fermions, sign issues? [duplicate]

Given a Lagrangian, to switch to a Hamiltonian, we do a Legendre transform. Suppose the Lagrangian has fermions, say a term like $\frac{i}{2}(\bar{\psi} \dot{\psi} - \dot{\bar{\psi}} \psi)$, then I ...
Gleeson's user avatar
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Positive definiteness of Dirac hamiltonian?

In David Tong's notes, he says that the Hamiltonian (4.92) is positive definite ( see page 112 of chapter five ). Here is equation (4.92) from chapter four. $$ E = \int d^3 x \ T^{00} = \int d^3 x \ \...
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If a Grassmann number can represent a fermionic wave function, how does it represent the w.f. amplitude in space, and maybe time too?

There's a clear account in https://en.wikipedia.org/wiki/Grassmann_number of how Grassmann numbers $z$ can be used to represent fermionic wave functions, constructed from the exterior algebra $Λ(v)$ ...
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Why are physical states not eigenstates of BRST charge?

In many texts in quantum field theory or string theory, it is stated that the BRST charge $Q$ must annihilate physical states because the states are required to be BRST invariant. Since $Q$ generates ...
Chang Hexiang's user avatar
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1 answer
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Path Integral Measure Transformation as $(DetU)^{-1}$

The path integral measure transforms as $D\Psi\rightarrow (DetU)^{-1}D\Psi$ for fermions, with $DetU=J$ the Jacobian. I am referring to Peskin and Schroeder's Introduction to Quantum Field Theory, ...
Archie C's user avatar
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Literature on Representation Theory of Graded Lie Algebras

I am currently studying 'advanced' representation theory, including topics like super-Lie algebras. I've come across various gradings (excluding the $\mathbb{Z}_2$ grading), such as how to select odd ...
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Cubic Grassmann term

Consider a Grassmannian action with a cubic interaction as follows $$L_{int} = -i(\theta_1\theta_2\dot{\theta}_3-\dot{\bar{\theta}}_3\bar{\theta}_2\bar{\theta_1})$$ where $\theta_1, \theta_2, \theta_3$...
Dr. user44690's user avatar
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Are representations of (bosonic) Lie groups over Grassmann variables well understood?

When one studies representations of (bosonic) Lie groups in physics, whether dealing with spacetime symmetries or gauge symmetries, it is often left implicit whether the representations are over real ...
Craig's user avatar
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Berezin Integration, confirming an measure is invariant

I am working through the Mirror Symmetry book, available here. I already had a question about an earlier part of the same Exercise 9.2.1 on page 157: We are given the following action with one boson ...
Gleeson's user avatar
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A Question of Pseudo-Classical Mechanics of Grassmann-Odd Variables [duplicate]

I would like to understand the following problem: You have a classical fermion in one dimension. It has no mass, and no interactions. One can write its action as follows: $$S=\int_{\mathbb{R}}dtL(\...
Valac's user avatar
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A Question of Pseudo-Classical Mechanics of Grassmann-Odd Variables

I would like to understand the following problem: You have a classical fermion in one dimension. It has no mass, and no interactions. One can write its action as follows: $$S=\int_{\mathbb{R}}dtL(\...
Valac's user avatar
  • 2,903
2 votes
1 answer
121 views

Confirming an action is invariant under a supersymmetric transformation

I am studying chapter 9 of the book Mirror Symmetry, available here. My question is relating to page 156/157 where Supersymmetry is being introduced for the first time in QFT in 0-dimensions. We are ...
Gleeson's user avatar
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Schwartz's Quantum field theory (14.100)

I am reading the Schwartz's Quantum field theory, p.269~p.272 ( 14.6 Fermionic path integral ) and some question arises. In section 14.6, Fermionic path integral, p.272, $(14.100)$, he states that $$ ...
Plantation's user avatar
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1 answer
113 views

Why does fermion have the expansion with Grassmann-numbers?

I learn the chiral anomaly by Fujikawa method. The text book "Path Integrals and Quantum Anomalies, Kazuo Fujikawa", in the page 151, says that …one can define a complete orthonormal set $\{...
s.h's user avatar
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Explicit form of Noether current for Majorana equation

I am looking for an explicit form of the Noether current for the Majorana equation.
MKO's user avatar
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Strange definition of the fermion number operator in Polchinski

In Polchinski's exposition of the RNS formalism for the superstring (String Theory: Volume II, chapter 10), in page 8, he mentions the worldsheet fermion number operator, which he calls $F$. He then ...
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1 answer
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Fermionic path integral with boundary

Given a path integral: $$K(\eta,\xi) = \int\limits_{\psi(0)=\eta}^{\psi(1)=\xi} e^{\int_0^1\dot{\psi}(t)\psi(t) dt} D\psi\tag{1}$$ where $\psi(t)$ are a real Grassmann fields. I get two answers ...
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0 answers
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References to lattice supermanifolds

Do you have any references (textbooks and/or internet links) to lattice supermanifolds or, more generally, discrete superspaces?
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0 answers
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Supersymmetry transformation superstring theory [closed]

in the study of worldsheet supersymmetry which im using Becker/Becker/Schwarz we introduce the idea of superspace by introducing a coordinate $\theta^{A}$ in addition to our $\sigma^{\alpha}=(\sigma,\...
Spooooonnnzzz's user avatar
1 vote
1 answer
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Fermionic Parity operator or number operator are not conserved after Bogoliubov transformation

I have a number operator $a^\dagger a + b^\dagger b$, where $a^\dagger$ and $b^\dagger$ are fermion operators. If a unitary transformation $U$ is performed, the number operator is written in the new ...
lsdragon's user avatar
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1 answer
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Classical fermions, where are they?

Context: Studying the path integral formulation of QFT I stumbled upon a fairly simple statement: when doing loop expansions of a partition function: $$Z[\eta ; \bar{\eta}] = \int [d\psi][d\bar{\psi}]...
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4 votes
1 answer
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How exactly is spinor with Grassmann variables as component defined?

I'm reading K. Muller's Introduction to Supersymmetry about spinor representation. He said that the components of a spinor are Grassmann variables. I understood Grassmann variables as follows. For a ...
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1 answer
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Free fermion OPE

In Di Francesco's Conformal Field Theory, the propagator for the free Majorana fermion theory is given by $$ \langle{\psi(z) \psi(w)}\rangle = \dfrac{1}{2\pi g} \dfrac{1}{z-w}$$ and the energy-...
phenolphthalein's user avatar
5 votes
1 answer
200 views

Weinberg's path integral for fermions in Volume 1

In sec. 9.5 of Weinberg's QFT 1, he introduces operators $Q$ and $P$ satisfying $$ \{Q,P\}=i \tag{9.5.1} $$ $$ \{Q,Q\}=\{P,P\}=0 \tag{9.5.2} $$ and eigenstates $|q\rangle$: $$ Q|q\rangle =q|q\rangle \...
dennis's user avatar
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When does the spinor need to be in a Grassmann variable?

Follow the closed question When does the spinor need to be in a grassmann variable? -- Does the spinor in the spinor representation of the space-time symmetry Lorentz space-time symmetry, like $so(1,...
Марина Marina S's user avatar
1 vote
1 answer
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Calculating a Gaussian-like path integrals with Grassmann variables and real variables

I want to compute the following path integral $$Z[w] = \frac{1}{(2\pi)^{n/2}}\int d^n x \: \prod_{i=1}^{n}d\overline{\theta}_id\theta \: \exp{\left(-\overline{\theta}_i \partial_j w_i(x)\theta_j -\...
Geigercounter's user avatar
6 votes
2 answers
291 views

Grassmann parameter in supersymmetry

Let's consider a free Wess-Zumino Lagrangian given by $$\mathcal{L} = \partial^{\mu}\overline{\phi}\partial_{\mu}\phi + i\psi^{\dagger}\overline{\sigma}^{\mu}\partial_{\mu}\psi\tag{1}$$ Whose ...
user avatar
2 votes
1 answer
130 views

Bosonic representation of delta function for Grassmann-even quantity

Suppose I have 2 Grassmann scalars $\theta$ and $\bar{\theta}$ and form the bosonic quantity $X = \bar{\theta}\theta$. Is there a purely bosonic representation of the delta function $\delta(X - \...
eric's user avatar
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0 answers
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Spin of covariant derivatives in 3D $\mathcal{N}=2$ superspace

Consider the 3D $\mathcal{N}=2$ superspace covariant derivatives $D_\alpha$ and $\bar D_\beta$, which have the following anti-commutation relations $$\{ D_\alpha, \bar D_\beta \}=-2i \gamma^\mu_{\...
TaeNyFan's user avatar
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2 votes
1 answer
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What are self-interacting fermions?

There're a bunch of models of fermions with quartic self-interactions. There's an introduction from this wikipedia page. For example, one can construct the Soler model of self-interacting Dirac ...
Valac's user avatar
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2 votes
1 answer
186 views

Gaussian integral identity over Grassmann numbers

I am reading Zee's Quantum Field Theory in a Nutshell. In the section about Grassmann numbers, there is an identity:$$\int dx\int dy\,e^{yAx}=\det A\tag{II.5.13}$$ where $x=(x_1,x_2,\dots,x_N),y=(y_1,...
rioiong's user avatar
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1 answer
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Why can the commutator of a general expression be replaced by the anti-commutator in the $bc$ CFT theory?

Polchinski states in his equation 2.6.14 (in his book String Theory Vol. 1, Introduction to the Bosonic String) that for charges $Q_1$ and $Q_2$ the following equation holds, where $j_i$ is the ...
kalle's user avatar
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2 answers
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Grassmann numbers for fermions in QFT

I'm studying the Grassmann variables from Polchinski's string theory textbook appendix A. On page 342, In order to follow the bosonic discussion as closely as possible, it is useful to define states ...
IGY's user avatar
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Symmetries and equations of superspheres and other superspaces

What are the symmetries and the most studied/most standard examples of superspaces? I include exceptional superalgebras and infinite-dimensional spaces. Bonus: Do quantum groups apply in the above ...
riemannium's user avatar
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3 votes
1 answer
96 views

Contour integral for commutator of fermionic fields

Suppose we have primary fields $A$ and $B$ which have the OPE, $$A(z) B(w) = \frac{1}{z-w} = -B(z)A(w), \quad |z| > |w|,\tag{1}$$ so they have fermionic statistics. Now I was curious how this would ...
user2062542's user avatar
1 vote
0 answers
84 views

Gaussian integral over Grassmann numbers

I'm trying to evaluate a Gaussian integral over Grassmann numbers but not sure if I've made a mistake. What I want to evaluate is \begin{equation} \left(\prod^N_i\int d\theta^*_i d\theta_i\right)\...
Physics440's user avatar

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