Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [fermions]

Fermions are particles with an intrinsic angular momentum (i.e. spin) equal to a "half integer" number of fundamental units: $\frac{(2n+1)}{2} \hbar$ for integer $n$. Fermions are required to be in a quantum state that is globally anti-symmetric, which leads to the Pauli Exclusion Principle barring identical fermions from occupying the same quantum state.

0
votes
0answers
36 views

Symmetry acting on a complex fermion operator

Suppose $S$ is a $\mathbb{Z}_2$ symmetry operator, i.e. $S^2=1$, acting on the fermion $c_{n}$ via $$S c_{n} S^{-1} = \sum_{m} U_{nm} c_{m}$$ and I am interested in $S$ is both linear or anti linear, ...
1
vote
1answer
43 views

Bogoliubov transformation for fermionic Hamiltonian

I have the Hamiltonian $H=\sum\limits_k [Ab^{\dagger}_{k}b_{k} + B(b^{\dagger}_kb^{\dagger}_{-k}+b_{k}b_{-k})]$, where $b^{\dagger}_k$ and $b_k$ are fermionic creation and annihilation operators. ...
0
votes
1answer
34 views

Unphysical degrees of freedom for the Weyl spinor?

I am attempting to solve the Weyl equation: $$\bar\sigma^{\mu}\partial_{\mu}\phi=0$$ Where $\bar\sigma^{\mu}=(-1,\vec{\sigma})$ in my convention, and $\phi$ is a two component Weyl spinor. I consider ...
0
votes
0answers
24 views

Reality of Dirac kinetic term

The Dirac kinetic term is $$\mathscr{L}_{\text{ferm}}=-i\bar{\psi}\gamma^\mu D_\mu\psi$$ where $\bar{\psi}\equiv \psi^\dagger \gamma^0$. Here I've assumed the mostly plus metric, so $\left(\gamma^0\...
1
vote
0answers
50 views

Feynman $i\epsilon$-prescription for fermion propagator via path integrals

In Section 9.4 of S. Weinberg's book "The quantum theory of fields" it is shown how to get the Feynman $i\epsilon$-prescription in the propagator of a free scalar field using path integrals and ...
1
vote
0answers
34 views

Product of Fermionic annihilation and creation operators

I have a bunch of fermions with annihilation $c_i$ and creation $c_i^\dagger$ operators. The index $i$ corresponds to different fermions. I'm interested in calculating the product $c_1^{\dagger} \Pi_{...
2
votes
0answers
26 views

How does the spin connection affect the dynamics of a fermion in curved space?

Consider a massless right-handed Majorana fermion in curved spacetime. Without any other fields present, the Lagrangian density is (I believe) the following: $$ \mathcal{L}_{\psi} = \sqrt{g}i\bar{\...
0
votes
2answers
62 views

Wavefunction Basis Formalism

If we have two electrons and two possible states $| 1 \rangle$ and $| 2 \rangle$, a possible state, as I understand, could be $\frac{1}{\sqrt{2}}(| 1 \rangle | 2 \rangle - | 2 \rangle| 1\rangle)$. ...
4
votes
5answers
129 views

Why are matter fields predominantly fermions?

Apart from the Higgs, all matter is made up of fermions. Is there any obvious reason why that is?
1
vote
0answers
86 views

Lagrangian for fermions

I was trying to understand last term in the Lagrangian. $$\mathcal{L} =- \frac{1}{4} F_{\mu \nu}(x)F^{\mu \nu}(x) - \frac{1}{2} \alpha\Big ( \partial_\mu A^\mu(x)\Big)^2 +\sum_{f} \overline{\Psi}(x) ...
0
votes
0answers
54 views

What is the QFT state with two distinguishable fermions present?

I want to describe a system with two non-interacting and definitely different fermions, say an electron neutrino, $\nu_e$, and an electron, $e^-$. The state describing a single electron is given ...
0
votes
1answer
37 views

Is the tight binding model an effective free fermion model?

The tight-binding Hamiltionian has the form $$H=-t\sum_i\left(c_i^\dagger c_{i+1} + c_{i}c_{i+1}^\dagger\right)$$ But does this mean that it can be represented in the form of free fermion modes?
2
votes
0answers
28 views

Why can the spin operator be written as a product of fermions?

I was studying the Hubbard model, where we define the spin operator $\vec{S} = \frac{1}{2} c^\dagger \vec{\sigma} c$, where the creation and annihilation operators are both vectors of the form $c^\...
0
votes
0answers
23 views

Vector-like Representation of fermions

In the literature, they often extend the Standard Model by adding a so-called vector-like fermion which is a multiplet invariant under $SU(2)_L\times U(1)$. The left- and right-handed components of ...
0
votes
1answer
25 views

General formulation of time reversal symmetry action on fermions

I'm wondering about a general way to define the action of time reversal on a fermion field $\psi$. From a few sources I've read (e.g. appendix A of Witten's paper on fermion path integrals), it seems ...
0
votes
0answers
32 views

Energy-momentum tensor of the Dirac field

I'm trying to compute the energy momentum tensor for the dirac field $$\mathcal{L}=\bar\psi(i\gamma_\mu\partial^\mu-m)\psi $$$$T^{\mu\nu}=\frac{\partial\mathcal{L}}{\partial(\partial_\mu\psi)}\partial^...
2
votes
1answer
33 views

Are the so-called representations of the Lorentz group actually all representations of it?

Fermionic fields change sign under a rotation by $2\pi$. However, in $SO^+\left(1,3\right)$ a rotation by $2\pi$ is the identity. For any representation $R$ of $SO\left(1,3\right)$ then we have $$R\...
1
vote
0answers
48 views

A proof that Heisenberg's and Euler Lagrange's equations are equivalent in QFT [closed]

I asked this before (link, link) but I think people didn't understand what I was asking, so I am going to try again . Thanks for everyone that helped so far. In QFT, Heisenberg's equation is ...
2
votes
0answers
26 views

How does binding energy change as more fermions interact?

The subject is few-body quantum mechanics. Given a system of $N$ identical fermions (spin 1/2) interacting through pairwise potentials $V_{ij}$, how does the binding energy change between $N$ and $N+1$...
1
vote
1answer
49 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} ...
2
votes
0answers
19 views

How to set the number of fermions in the whole system in fermionic-DMRG program?

In infinite DMRG (density matrix re-normalization group) algorithm, we increase size of super-block by two sites in each iteration. How do we set number of fermions in the system? Let's say we want to ...
0
votes
0answers
38 views

Time reversal for fermionic fields

I have some doubts about the way we apply time reversal to Dirac's Lagrangian in QFT. Looking for the transformed field, $\psi^t(x)$, I've found sources (see below) that claims: $$\psi^t(x) = \gamma^...
2
votes
1answer
53 views

Volume per $k$-state

We're talking about Fermi-energies for the first time, for $N$ spin $\frac{1}{2}$ particles in a 3D box, and she writes down that $$2 \times \frac{1}{8} \times \frac{4}{3} \pi k_f^3 = Nq \times \frac{\...
0
votes
1answer
49 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 ...
0
votes
1answer
79 views

How do Fermions with $0$ weak hyper charge couple to electro-weak force?

If you have two fermions (with spin $\pm \frac{1}{2}$) that form a weak-isospin doublet and their respective right-handed fermions which are weak-isospin singlets, what would it imply if the doublet ...
3
votes
1answer
96 views

Canonical Quantisation vs the Dirac Field, why does it even work?

Using the "Dirac Prescription", we can preserve the format of a differential equation in its QM form. If we define the canonical variables s.t. they have the same commutation relations times $i$ as ...
2
votes
1answer
72 views

What is difference between fermions and spins?

A spin model i.e. $H_s = \sum_i^{L-1} S_i^x\cdot S_{i+1}^x$ can be written in matrix form as following $$H_s = \big(S_1^x \otimes S_2^x \otimes I_3^2 \otimes I_4^2\otimes \cdots\otimes I_{L-1}^2\big)...
0
votes
0answers
24 views

Do anyonic statistics only arise from spatial degrees of freedom?

Elementary texts on quantum mechanics justify the existence of fermions and bosons using the simple argument that if we have a state of two indistinguishable particles $|a,b \rangle$, where $a$ and $b$...
1
vote
0answers
72 views

How to deal with fermionic operators in density matrix renormalization group (DMRG)?

Let we have 1D Hubbard model with spinless fermions $$H = -t\sum_i^{L-1} \big(c_i^\dagger c_{i+1}+c_{i+1}^\dagger c_i\big) +V\sum_i^{L-1} n_i n_{i+1}$$ Though this model can be mapped onto XXZ ...
0
votes
0answers
7 views

Will more than one composite boson can stay in the same energy state if constituent fermions has moderate entanglement?

Let say we consider two distinguishable fermions(bi-fermions) in compact form. The case when both fermions are existing as free fermions, they will obey Pauli exclusion principle. In other case if ...
0
votes
1answer
61 views

Anti-commutation relations in annihilation operators

It is claimed that $$\{c_\alpha,c_\beta \} = c_\alpha c_\beta + c_\beta c_\alpha = 0$$ where $c_\alpha$ and $c_\beta$ are the fermionic annihilation operators in second quantization. Why is that ...
3
votes
1answer
68 views

Does the massless fermion in $2+1$ dimensions suffer from gauge anomaly?

In Fermion Path Integrals And Topological Phases Witten showed that for a massless Dirac fermion in $2+1$ dimensions $$S[\bar{\psi},\psi]=\int d^{3}x\bar{\psi}iD\!\!\!\!/_{A}\psi,$$ where $A$ is a $...
0
votes
1answer
55 views

Is $c^{\dagger}|\psi_N \rangle= \sqrt{N+1}| \psi_N \rangle $ or $c^{\dagger}|\psi_N \rangle= | \psi_{N+1} \rangle $ in case of fermions

For the $N$ fermion state, when we apply creation operator $c^{\dagger}$, should we write the factor $\sqrt{N+1}$ with the resultant state, like $c^{\dagger}|\psi_N \rangle= \sqrt{N+1}| \psi_N \rangle ...
1
vote
1answer
50 views

Spinning Particles in Background Gauge Fields

A simple model for a spinning particle is $$L=m\int dt\left(\dot{x}^{2}-\frac{i}{2}\psi\dot{\psi}\right)$$ with SUSY algebra $\delta x=-i\epsilon\psi$ and $\delta\psi=-\epsilon\dot{x}$, where $\...
4
votes
2answers
77 views

Grassmann's variables under integration

If $\eta$ is a Grassmann variable, due to invariance under translations we get that, $$\int d\eta\ \eta = 1 \tag1$$ Nevertheless, for being Grassmann's, $\eta$ satisfies $\eta^2 = 0$. ...
2
votes
0answers
21 views

Higher form fermionic conserved currents

Higher form conserved currents have already been defined, such as those seen in Klebanov and Polyakov's work in 2002. There, the authors studied the $\text{AdS}_4$/CFT correspondence -- more ...
0
votes
3answers
103 views

Wave function of a system of two identical fermions

In N. Zettili's 'Quantum Mechanics Concepts and Applications' [chapter 8, solved problem 8.3], we have to find wave function and ground state energy of a system having two identical fermions and in ...
3
votes
1answer
59 views

BCS theory for neutral fermions

As I understand it, Cooper pairs form between two fermions and are the cause of superconductivity. I was told by a teacher that the formation of Cooper pairs and BCS theory requires both fermions to ...
0
votes
1answer
66 views

Can a general many-body Hamiltonian with quadratic and biquadratic terms be diagonalized?

Can an arbitrary many-body hamiltonian in second quantization form with quadratic and biquadratic terms $$H=\sum_{v_1,v_2} \alpha_{v_1 v_2}\ c_{v_1}^{\dagger}c_{v_2}+ \sum_{v_1,v_2,v_3,v_4}\beta_{v_1 ...
1
vote
1answer
45 views

Is the spin-statistics theorem true for antifermions?

The spin-statistics theorem says that having a system of identical fermions, the total wavefunction is antisymmetric with respect to exchange of any two fermions. My question is, does this hold for ...
3
votes
0answers
59 views

How do I derive Pauli's exclusion principle with path integrals?

I am trying to prove Pauli's exclusion principle using path integrals. My starting point is the configuration space $\mathcal{C}$ for two indistinguishable particles in 3D: $$ \mathcal{C} = \{ \{x_1,...
1
vote
1answer
81 views

Can we ever “measure” a quantum field at a given point?

In quantum field theory, all particles are "excitations" of their corresponding fields. Is it possible to somehow "measure" the "value" of such quantum fields at any point in the space (like what is ...
5
votes
2answers
146 views

Intuition behind Pauli’s Principle

While reading about electronic structure of multi-electron atoms, Pauli’s Principle comes out to be a very important feature. But it feels very vague as little explanation is given about it. I mean ...
1
vote
2answers
57 views

Indistinguishable particles and symmetrization of wavefunction

For 2 indistinguishable particles, we take the wave function to be $$\psi\pm (r_1,r_2) = A[\psi_a (r1)\psi_b (r2) \pm \psi_b (r1)\psi_a (r2) ]$$ where fermions get a - sign and bosons get a + But, if ...
2
votes
1answer
41 views

How to derive the formula for the radius of a Fermi sphere?

I'm trying to figure out how the radius of a Fermi sphere $$p_F = \hbar (3 \pi^2 \frac{N}{V})^{1/3}$$ is derived from the formula $$dN_{spatial}=\frac{V \ d^3p}{\hbar^3}.$$ The solution states that I ...
2
votes
0answers
45 views

Regarding this interpretation of the pair correlation function

On some lecture notes of the statistical mechanics of Fermi systems I found the following, specifically regarding spin 1/2 systems: The correlation function $$ g_{\uparrow\uparrow}(s) \equiv \...
0
votes
0answers
19 views

The strong CP problem in Grand Unified Theories

The strong CP problem could, in principle, be solved by a massless up quark. Can this solution be embedded in Grand Unified Theories, such as $SO(10)$ (or $SU(5)$)? These theories, in particular $SO(...
2
votes
1answer
55 views

Quantum statistics from the (anti)commutation relations of the operators?

From a QFT point of view, the difference between bosons and fermions is that their creation/annihilation operators ($a^{\dagger}$, $a$ and $c^{\dagger}$, $c^{\dagger}$ respectively) obey the following ...
2
votes
1answer
91 views

The correct definition of Klein Factor

Klein factors are the operators which make sure that the anticommutation between the different species is correct during the bosonization procedure. According to this famous review by Jan Von Delft, ...
1
vote
1answer
67 views

Spin and many particle systems

I learned about many particle systems and second quantization recently. The Fock space of distinct particles with single particle Hilbert space $\mathcal H$ was defined to be $\bigoplus_{N=0}^\infty \...