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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.

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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 ...
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What does sub-bosonic behavior of composite boson means? [on hold]

Let say we consider two distinguishable fermions(bi-fermions) in compact form. This pair of fermions make a composite boson. Now we develop formalism to check either our composite bosns is like a pure ...
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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 ...
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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 $...
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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 ...
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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 $\...
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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 $\eta^2 = 0$. ...
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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 ...
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62 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 ...
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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 ...
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1answer
60 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 ...
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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 ...
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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,...
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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 ...
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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 ...
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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 ...
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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 ...
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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 \...
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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(...
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1answer
49 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 ...
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How to show second quantized Hamiltonian is Hermitian?

Consider fermionic Hamiltonian: $$H= i A \sum_k c_{-k} c_{k} + c_{-k}^{\dagger} c_{k}^{\dagger} $$ with annilinting operator $c_{k}$, creating operator $c_{k}^{\dagger}$, wavevector k and constant A....
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1answer
68 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, ...
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1answer
58 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 \...
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1answer
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Fermi-Hubbard model: adiabatically change tunnelling

Consider the 3D Fermi-Hubbard model in a cold-atom setting (harmonic confinement, $\epsilon_i$): $ H = - t \sum_{\langle i, j\rangle, \sigma} c^{\dagger}_{i, \sigma}c_{j,\sigma} + U\sum_{i}n_{i,\...
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Parity Anomaly and Gauge Invariance

In Fermionic Path Integral and Topological Phases, Witten shows that in $2+1$ dimensions, the free massless Dirac fermion suffers from parity anomaly. To be specific, he shows that it is impossible to ...
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1answer
61 views

Meaning of the subscripts $L,R$ for the two component Weyl spinors $\phi_{L,R}$

For a Dirac spinor $\psi$, its chiral projections are $\psi_{L,R}$ are defined as $$\psi_{R,L}=\frac{1}{2}(1\mp\gamma^5)\psi.\tag{1}$$ Acting with the chirality operator $\gamma^5$, we find $$\gamma^5\...
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1answer
144 views

Intepreting Fermions as Differential Forms?

In this paper on path-integral quantization of Chern-Simons theory, on page 434 (equation 4.17), the authors used fermions to interpret wedge product and contractions of differential forms. Let $M$ ...
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1answer
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How can the derivation of the energy of an electron in a Fermi gas using the Heisenberg uncertainty principle be made rigorous?

When modeling a large number of non-interacting identical fermions in a potential well of volume $V$ as a harmonic oscillator and assuming the Pauli exclusion principle, it is easily seen that the ...
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1answer
43 views

How to calculate the spin of an atom [duplicate]

If given an atom say ${^{108}_{47}Ag}$, what is the systematic way to determine its spin so that one knows whether it is a boson or a fermion?
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Fermi Energy For platinum

I have been conducting an experiment based on the photoelectric effect and have calculated my work function as 1.15eV which is afar lower than what we expect for platinum. Then i found out that the ...
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How can I draw the energy bands for the first and second zones of Brillouin? Is it conductor or insulator?

I want to draw the energy ($E$) diagrams for a simple cubic cell of parameter $a$, where each atom provides two electrons for the almost free electron levels for planes [100], [110] and [111]. I ...
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1answer
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When exactly do identical fermions interact?

For the case of $N$ identical fermions in a three-dimensional box, the Pauli Exclusion Principle necessitates that the overall wavefunction of the system is antisymmetric. No two fermions can occupy ...
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Equations of motion for a Weyl spinor in the context of SUSY

I'm learning supergravity from the textbook of Antoine Van Proeyen (this is from page 114). Suppose I'm given a Lagrangian $$ \mathcal{L} = - \partial^{\mu} \bar{Z} \partial_{\mu} Z - \bar{\chi} \...
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1answer
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Auxiliary Grassmann variables in supergeometry

I was reading on super geometry and how it is used to model fermions and supersymmetry in classical field theory. In texts like [1] or [2] the authors introduced auxiliary Grassmann odd variables to ...
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1answer
67 views

Dirac propagator causality

I was studying the Dirac propagator and came across an excelent article which includes all the derivation, and interestingly we can conclude that the anticommutator is zero for space-like intervals. ...
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How to calculate correlation functions for fermionic operators?

In the paper by Peschel (2003) https://arxiv.org/pdf/cond-mat/0212631.pdf How does one derive the following relation: $$ \langle c_{n}^\dagger c_{m}^\dagger c_{k}c_{l}\rangle = \langle c_{n}^\...
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1answer
54 views

Correction to the fermion propagator

Given the Lagrangian $$\mathscr{L}=\bar{\psi}\left(i\partial\!\!\!/-m\right)\psi +\frac{1}{2}\left(\partial\phi\right)^2- \frac{1}{2}M^2\phi^2 - g\bar{\psi}\psi\phi^2,$$ calculate the propagator ...
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2answers
106 views

Two-point Green for Free Dirac Fields

I am trying to compute the $2$-point Green function $\tau_2(x,y)$ for free Dirac fields. The corresponding formula for $\tau_2(x,y)$ is given by $$\tau_2(x,y) = -\frac{\delta^2}{\delta\eta_x \delta \...
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Do distinguishable fermions obey the Pauli exclusion principle?

We know that fermions are identical particles and obey Pauli exclusion principle. But what is meant by distinguishable fermions? Does that mean, like proton and electron both are fermions but they are ...
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1answer
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Are there examples of nondegenerate Fermi gases?

A degenerate Fermi gas is an ensemble of fermions with very low interactions and at temperatures that are low enough (lower than Fermi temperature). Most of the examples in the literature are about ...
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How can we calculate the imaginary part of a fraction that has a term $i0_+$ in the denominator?

I have recently started dealing with thermal field theory for fermions and I am faced with a paper that, at some point, tries to calculate the imaginary part of a fraction that looks like: $$\frac{1}{...
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How do Feynman diagrams and loop quantum gravity fit together?

Feynman diagrams are a good way of calculating effects in quantum electrodynamics on a constant background space-time. Spin foams (LQG) are conjectured to be a good way of calculating quantum effects ...
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1answer
54 views

Anticommutation relation different specie/type of fermion

Suppose we have two distinct fermions, say $X$ is Dirac, $Y$ is Majorana, part of different irreps of some Gauge group (e.g. SM). Alternatively, consider a lepton $l=l_L+l_R$ and a Majorana neutrino $...
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1answer
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How do I know that gauge fields are bosons?

QED and the Dirac equation have field operators $\psi$ interact with a gauge field $A^{\mu}$. We identify $\psi$ as a fermionic field and $A^{\mu}$ as a gauge boson - the photon. Do we or can we ...
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How are supersymmetry transformations even defined?

I am just starting to read about supersymmetry for the first time, and there is something bothering me. Supersymmetry transformations transform between bosonic fields and fermionic fields, but I don't ...
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1answer
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Why are fermion masses considered free parameters of the standard model rather than Yukawa coupling constants?

From what I understand, fermion masses in the standard model are due to Yukawa couplings to the Higgs field. Nevertheless, when listing free parameters of the standard model it is the fermion masses ...
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1answer
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Spin of electron [duplicate]

While reading 'The Universe in a Nutshell' by Stephen Hawking, I came across the example of cards and how he used it to explain concept of spin and fermions and bosons. There he defined 'Spin' as ...
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Does an electron have a frequency (and hence an energy)?

The formulation is provocative, the question is similar to the question here. There I can follow the question, but not the answers, which for me imply that an electron in a momentum eigenstate does ...
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1answer
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A naive question on the eigenvalues of fermionic operators?

Let $A$ be a fermionic operator which is a product of odd number of fermion operators or a summation of them, say $A=C_{i_1}^{\dagger}\cdot \cdot\cdot C_{i_m}^{\dagger}C_{j_1}\cdot \cdot\cdot C_{j_n}...
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Quantum Field interaction transferred via “exchanging fermions” [duplicate]

In Standard Model every fundamental interaction is described by means of exchanging gluons of particular kind. It is very natural as gluons has spin with values given by inteegers, and can share the ...