Questions tagged [grassmann-numbers]
The grassmann-numbers tag has no usage guidance.
317
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Wick's theorem proof for fermions
This is my first question. I am trying to prove Wick's theorem for fermions. I am currently using Peskin and Schroeder's discussion, which is in section 4.7 (and uses section 4.3, where Wick's ...
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Should Faddeev-Popov ghosts (in path integral) belong to infinite-dimensional Grassmann (Banach) algebra?
Many years ago, I had the opportunity to study gauge theories based on the book by Faddeev and Slavnov. The Faddeev-Popov ghost fields in the path integral for the Yang-Mills (YM) theory did not ...
- 101
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$2\pi$-rotation of fermionic states vs. fermionic operators
Given a fermionic state $|\Psi\rangle$, a $2\pi$ rotation should transform it as
\begin{equation}
|\Psi\rangle \quad\to\quad -|\Psi\rangle \,,
\end{equation}
On the other hand, given a fermionic ...
- 707
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53
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Trace formula for fermionic variables
I am using Bravyi's paper "Lagrangian representation of fermionic linear optics" and one formula that stumbled me is the trace formula in Eq. (15) in the picture below:
I do not see how to ...
2
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97
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Commuting/anticommuting properties of fermionic ghost fields in BRST Quantization
I was reading the paper "Batalin-Vilkovisky analysis of supersymmetric systems" (by Laurent Baulieu and others). I am struggling to understand how commutation/anticommutation relations of ...
- 126
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N=4 Supersymmetric Ward identity
(This question pertains to exercise 4.13 of Elvang and Huang's textbook (which used to be lecture notes). This is not for a class, just to learn some new tools for work).
Consider the expansion of the ...
- 779
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Grassmann numbers and fermion creation and annihilation operators
Reading Fradkin's book on Condensed Matter Physics, I encountered Grassmann numbers. In the following $\hat\Psi$ and $\hat\Psi^\dagger$ are the fermion annihilation and creation operators whereas $\...
- 145
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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 ...
- 21
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2
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396
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Proving a Grassmann integral identity
How to prove the following identity
$$
\begin{align}
\int {\rm d} \eta_{1} {\rm d} \bar{\eta}_{1} \exp\left(a \left(\bar{\eta}_{1}-\bar{\eta}_{0}\right)\left(\eta_{1}-\eta_{0}\right) + b \left(\bar{\...
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Trace formula in Grassmann algebra
From Grassmann algebra, we know the following relation
\begin{equation}
\mathrm{Tr} e^{-a\hat{c}^{\dagger}\hat{c}} =1+e^{-a}
\end{equation}
Now, how to prove the following generalized results?
\begin{...
- 187
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48
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How does this term in the Majorana mass not vanish?
This is classical field theory. In the Majorana mass term, we have the expression
$$\nu_L^T\sigma_2\nu_L \tag{1}$$
where the left-handed spinor field $\nu$ has a Grassmann-valued amplitude, i.e., $\nu ...
- 2,772
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2
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97
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Grassmann variables and orthogonality of coherent fermionic states
Let a coherent fermionic state
$$
\left|\phi\right> := \left|0\right> + \left|1\right> \phi,\tag{0}
$$
where $\phi$ is a Grassmann number (i.e. it anticommutes with other Grassmann numbers). ...
- 1,925
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101
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Confusion about whether a fermion field and its conjugate as an Grassmann number
I'm confused about what "a Grassmann-odd number" really means and how does it apply to fermions.
In some text, it says that "if $\varepsilon \eta+\eta \varepsilon =0 $, then $\...
- 368
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61
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How to tell if a composite boson field should be real or complex?
Let's say I have a system with two species of fermions, $f$ and $c$, where $f$'s are neutral but $c$'s are charged. Each of these has its own $U(1)$ related to particle-number conservation.
Now, if I ...
- 59
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Fierz idendity (supersymmetry)
So basically I have two Fierz identities involving spinors:
$$\psi^a \psi^b = -\frac{1}{2} \epsilon^{ab} \psi \psi$$
And
$$\overline{\psi}^{\dot{a}} \overline{\psi}^{\dot{b}} = \frac{1}{2} \epsilon^{\...
- 980
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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 ...
- 625
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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 ...
- 191
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462
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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 ...
- 79
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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 ...
- 213
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77
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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 $\psi$ ...
- 147
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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 ...
- 187
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75
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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 ...
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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.
$$ E = \int d^3 x \ T^{00} = \int d^3 x \ \...
- 523
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0
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If a Grassmann number can represent a fermionic wave function, how does it represent the w.f. amplitude in space, and maybe time too?
There's a clear account in https://en.wikipedia.org/wiki/Grassmann_number of how Grassmann numbers $z$ can be used to represent fermionic wave functions, constructed from the exterior algebra $Λ(v)$ ...
- 1,097
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Why are physical states not eigenstates of BRST charge?
In many texts in quantum field theory or string theory, it is stated that the BRST charge $Q$ must annihilate physical states because the states are required to be BRST invariant. Since $Q$ generates ...
- 1,063
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Path Integral Measure Transformation as $(DetU)^{-1}$
The path integral measure transforms as $D\Psi\rightarrow (DetU)^{-1}D\Psi$ for fermions, with $DetU=J$ the Jacobian.
I am referring to Peskin and Schroeder's Introduction to Quantum Field Theory, ...
- 57
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Literature on Representation Theory of Graded Lie Algebras
I am currently studying 'advanced' representation theory, including topics like super-Lie algebras. I've come across various gradings (excluding the $\mathbb{Z}_2$ grading), such as how to select odd ...
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42
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Cubic Grassmann term
Consider a Grassmannian action with a cubic interaction as follows
$$L_{int} = -i(\theta_1\theta_2\dot{\theta}_3-\dot{\bar{\theta}}_3\bar{\theta}_2\bar{\theta_1})$$
where $\theta_1, \theta_2, \theta_3$...
- 1,208
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Are representations of (bosonic) Lie groups over Grassmann variables well understood?
When one studies representations of (bosonic) Lie groups in physics, whether dealing with spacetime symmetries or gauge symmetries, it is often left implicit whether the representations are over real ...
- 1,133
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Berezin Integration, confirming an measure is invariant
I am working through the Mirror Symmetry book, available here.
I already had a question about an earlier part of the same Exercise 9.2.1 on page 157:
We are given the following action with one boson ...
- 213
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A Question of Pseudo-Classical Mechanics of Grassmann-Odd Variables [duplicate]
I would like to understand the following problem:
You have a classical fermion in one dimension. It has no mass, and no interactions. One can write its action as follows:
$$S=\int_{\mathbb{R}}dtL(\...
- 2,943
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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:
$$S=\int_{\mathbb{R}}dtL(\...
- 2,943
2
votes
1
answer
139
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Confirming an action is invariant under a supersymmetric transformation
I am studying chapter 9 of the book Mirror Symmetry, available here. My question is relating to page 156/157 where Supersymmetry is being introduced for the first time in QFT in 0-dimensions.
We are ...
- 213
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1
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Schwartz's Quantum field theory (14.100)
I am reading the Schwartz's Quantum field theory, p.269~p.272 ( 14.6 Fermionic path integral ) and some question arises.
In section 14.6, Fermionic path integral, p.272, $(14.100)$, he states that
$$ ...
- 317
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1
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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 $\{...
- 129
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0
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36
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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.
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1
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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 ...
- 917
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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 ...
user84158
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0
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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,\...
1
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1
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283
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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 ...
- 337
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354
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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:
$$Z[\eta ; \bar{\eta}] = \int [d\psi][d\bar{\psi}]...
- 1,809
4
votes
1
answer
170
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How exactly is spinor with Grassmann variables as component defined?
I'm reading K. Muller's Introduction to Supersymmetry about spinor representation. He said that the components of a spinor are Grassmann variables.
I understood Grassmann variables as follows. For a ...
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Free fermion OPE
In Di Francesco's Conformal Field Theory, the propagator for the free Majorana fermion theory is given by
$$ \langle{\psi(z) \psi(w)}\rangle = \dfrac{1}{2\pi g} \dfrac{1}{z-w}$$
and the energy-...
- 351
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Weinberg's path integral for fermions in Volume 1
In sec. 9.5 of Weinberg's QFT 1, he introduces operators $Q$ and $P$ satisfying
$$
\{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 \...
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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,...
- 593
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161
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Calculating a Gaussian-like path integrals with Grassmann variables and real variables
I want to compute the following path integral
$$Z[w] = \frac{1}{(2\pi)^{n/2}}\int d^n x \: \prod_{i=1}^{n}d\overline{\theta}_id\theta \: \exp{\left(-\overline{\theta}_i \partial_j w_i(x)\theta_j -\...
- 168
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2
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329
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Grassmann parameter in supersymmetry
Let's consider a free Wess-Zumino Lagrangian given by
$$\mathcal{L} = \partial^{\mu}\overline{\phi}\partial_{\mu}\phi + i\psi^{\dagger}\overline{\sigma}^{\mu}\partial_{\mu}\psi\tag{1}$$
Whose ...
user366054
2
votes
1
answer
160
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Bosonic representation of delta function for Grassmann-even quantity
Suppose I have 2 Grassmann scalars $\theta$ and $\bar{\theta}$ and form the bosonic quantity $X = \bar{\theta}\theta$. Is there a purely bosonic representation of the delta function $\delta(X - \...
- 300
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Spin of covariant derivatives in 3D $\mathcal{N}=2$ superspace
Consider the 3D $\mathcal{N}=2$ superspace covariant derivatives $D_\alpha$ and $\bar D_\beta$, which have the following anti-commutation relations
$$\{ D_\alpha, \bar D_\beta \}=-2i \gamma^\mu_{\...
- 4,256
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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 ...
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