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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Most general 4-fermion EFT

In order to understand EFTs, I'm trying to work with an example: namely, the UV Yukawa theory that reduces to 4-fermion theory in the IR: $$\mathcal{L}^\text{UV}=\frac{1}{2}(\partial\phi)^2-\frac{1}{2}...
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Why are fermionic atoms less prevalent than bosonic ones?

Many atoms have no stable fermionic isotopes. Those that do typically have more stable bosonic isotopes than fermionic ones. Furthermore, the fermionic isotopes of most atoms are lower in natural ...
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Number of maxima of 1D many-body fermion density

Consider in quantum mechanics a single particle in one dimension, with a Hamiltonian $$ H_0(x) = -\frac{\hbar}{2m} \frac{d^2}{dx^2} + V(x). $$ It is well known that the $n$:th bound eigenstate $\psi_n(...
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Why is the anticommutation relation for the Dirac field between fields? [duplicate]

The commutation relation for neutral Klein Gordan field is $$[\phi(x,t),\pi(x',t)]=i\delta^3(x-x')$$ with all other commutators zero; The commutation relation for charged Klein Gordan field is $$[\phi(...
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Resonance level model: Commutator

As a small part of an exercise on the resonant level model (all fermionic (field-)operators, $\Psi(\vec{x}) = \sum\limits_{\vec{k}}e^{i\vec{k}\vec{x}}c_{\vec{k}} $, $V$ is a constant, $d$ and $c$ ...
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Diagonalizing a given Hamiltonian

The following Hamiltonian, which has to be diagonalized, is given: $H = \epsilon(f^{\dagger}_1f_1 + f_2^{\dagger}f_2)+\lambda(f_1^{\dagger}f_2^{\dagger}+f_1f_2)$ $f_i^{\dagger}$ and $f_i$ represent ...
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Pauli's principle on two-electron system

I just have been introduced to the axiom "defining" bosons and fermions, namely (for fermions): Consider a collection of $N$ identical particles moving in $\mathbb{R}^3$ and having a (half-...
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Why does normal-ordering ensure finiteness?

I will be using Jan von Delft's rigorous construction of bosonization/refermionization as an example, but I will try to explain my question in more general terms. Consider an (countably) infinite-dim (...
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Electron density via bosonization/refermionization

I'm currently trying to understand the rigorous construction of bosonization/refermionization via Jan von Delft. In the constructive approach, we consider a system on a finite $L$ circle and thus in ...
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What is the correct way of looking at the Dirac field?

All quantum fields are operators in QFT. However, the Dirac field operator $\hat{\psi}$ has the following difference with the scalar field operator $\hat{\phi}$: For the $\hat{\psi}$, it makes sense ...
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How to derive the equations of motion of finite temperature Green function?

I'm having a trouble deriving the equation of motion of a Green function. My understanding of the derivation is the following. Given a set of fermionic creation annihilation operators $$\{a_\alpha (\...
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Question regarding the degeneracy of energy states of Fermions

My professor during the lecture said exactly the following Let there be a system of non-interacting fermions. Since they are indistinguishable, they have the same Hamiltonian, and the single-particle ...
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Quantum counting statistics for fermions

A system of two identical particles which ay occupy any of three energy levels: $$ε_n=nε$$ $n=0,1,2$ The lowest energy state, $ε=0$ is doubly degenerate. The system is in thermal equilibrium at ...
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Does every fermion loop have two contributions?

Suppose I have a Feynman Diagram with a closed fermion loop. This introduces a negative sign, and a trace over the product of the propagators. Is the same Feynman diagram, but with the arrows in the ...
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Fermionic oscillator and reducible representation

Consider the fermionic oscillators $\{a, a^\dagger\} = 1$, $\{a, a\} = \{a^\dagger, a^\dagger\} = 0$. The commonly used irreducible representation is given by $|0\rangle$, $a^\dagger |0\rangle$ where $...
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Ideal Fermi gas close to $T=0 \rm K$

I am reading about the behaviour of ideal Fermi gas close to $T=0K$ from Kardar's Statistical Mechanics. In the paragraph which I have highlighted, we have the inyegral representation of $f_m^{-}(z)$....
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High temperature limit of ideal quantum gas

I am reading ideal quantum gas from Kardar's Statistical Mechanics. $VII.35$ is the representation of pressure, number density and energy density in the form of $f_m^{\eta}$. $z=e^{\beta\mu}$ where $\...
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Why do scalars and fermions have a different result in a Lagrangian?

Consider the Lagrangian for Yukawa theory: $$ \mathcal{L} =i\bar{\psi}\not{\partial}\psi- \bar{\psi}m_F \psi +\frac{1}{2} \partial_\mu \phi \partial^{\mu} \phi - \frac{1}{2}m_s^2 \phi^2 + \mathcal{L}_{...
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Partition function for the indistinguishable particles using symmetrization of states

In the derivation of the partition function for the the N particle ideal gas, the factor of $\frac{1}{N!}$ does not come naturally. We have to go for symmetrized and asymmetrized state. So, to derive ...
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How do you get the Jordan-Wigner commutation relations for spin-1/2 fermions?

I am currently trying to figure out how the answer is derived here. I understand that commuation is $[A,B] = AB - BA$. However I am confused how the $\exp(-i\varphi$) acts on the operators $c$ and $c^\...
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How the Zero-point Energy of the System containing 2 Fermions in 3 Micro-Energy States is 1?

If we distribute 2 Fermions $\mathrm{(A,A)}$ in 3 Micro-Energy States (0,$\epsilon$,$2\epsilon$), the confirmation is given by : $$ \begin{array}{|c|c|c|c|c|} \hline 0 & \varepsilon & 2 \...
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2 fermions in a box (infinite potential well)

I have 2 fermions in a box. I know that they are in the state: $$|\psi\rangle = {1 \over \sqrt2}\, (|1\rangle |2\rangle -|2\rangle|1\rangle)\,|+,+\rangle$$ If I hadn't spin, I could find wave ...
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Units in Actions (Functional Field Integrals)

When one rewrites the partition function of a grand-canonical ensemble (quantum version) as functional field integral $$ Z = \operatorname{Tr}_{ \mathscr{F}} \mathrm{e}^{ - \beta \left( H - \mu N \...
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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 ...
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About Einstein's sum rule and Dirac equation

I am studying the Dirac equation and I'm having some trouble about something that I think should be trivial. I'm working in a (1+1)-dimensional Minkowski spacetime with signature $(+, -)$, i.e., $ds^2=...
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Why do we have distinct generations of fermions, rather than a continuous distribution of fermions with increasing mass? [closed]

As far as I am aware, each generation of fermion varies only in mass, thus my question is why do we have only three distinct generations of fermion, instead of a continuous mass spectrum of particles ...
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$\mathcal{su}(3)$ irreps in the fermionic oscillator

Consider the fermionic oscillator, for $d=3$ degrees of freedom, with the Hamiltonian $$H=\frac{1}{2}\sum_{j=1}^3(a_{F_j}^{\dagger} a_{F_j}-a_{F_j} a_{F_j}^{\dagger})$$ Use fermionic annihilation and ...
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Origins and understanding hamiltonians for free fermions

I am starting to do some work on free-fermionic models, but I am having some problems understanding some things. My professor led me know that the hamiltonian for free fermions without mass in a ...
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Are black hole singularities fermions? [closed]

I was just wondering. I apologize if this is a dumb question. Is it possible that the mass of a black hole is converted into quantum energy that gets distributed across the universe uniformly so that ...
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Allowed k values for Spinless Fermi-Hubbard in 2D

Suppose we have the following Hamiltonian of spinless Fermi-Hubbard model: $\hat H = -t \displaystyle\sum_{<i,j>} (c^+_ic_{j} + h.c.)$ The Ground state of the free fermion Hamiltonian can be ...
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SUSY vs bosonization and fermionization

(New to the concepts.) From what was known SUSY described a theory consisted of both a boson and a fermion pair as a symmetric counter part. Bosonization and Fermionization on the other hand described ...
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Single-particle wavefunction in Slater determinant

The ground state of $N$ non-interacting fermions can be written using a Slater determinant as: $$ \Phi_{GS}(\textbf{r}_{1}, ..., \textbf{r}_{N}) = \frac{1}{\sqrt{N!}} \begin{vmatrix} \phi_{\mu_{1}}(\...
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Anticommutator of Fourier transform

Let \begin{equation} a_x = \int_{-\pi}^\pi \frac{\text{d}q}{2\pi}e^{iqx}a(q) \end{equation} \begin{equation} a_x^\dagger = \int_{-\pi}^\pi \frac{\text{d}q}{2\pi}e^{-iqx}a^\dagger(q) \end{equation} ...
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Why electrical holes can have spin? [duplicate]

As we know the electrical holes are the missing electrons. Electrons can definately have spin, but how can a hole has spin? Is the spin of holes depend on the other electrons? e.g. I remove an ...
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Are the definitions of chirality in continuum QFT and the Nielsen-Ninomiya theorem equivalent?

I have seen two definitions of chirality in quantum field theory: According to the Wikipedia article, chirality is defined as whether a particle transforms under a left- or right-handed ...
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Are there chiral fermions in $d=4$ Euclidean space?

In $d=4$ Euclidean space the spin group is $\operatorname{Spin}(4)\cong SU(2)\times SU(2)$ where the two $SU(2)$’s are independent (as opposed to Lorentzian signature where they conjugate into each ...
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Fermions in Euclidean vs Lorentzian Signature

We know that in Lorentzian signature, fermions are representations of \begin{equation} Spin(3,1)\cong SL(2,\mathbb{C})\cong SU(2)\times SU(2)^* \end{equation} where crucially left/right handed ...
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Writing fermionic correlation functions as a product of fermion propagators

I’m doing some work on Vafa-Witten and Weingarten type inequalities which realy on the positive definite path integral measure of vector-like gauge theories. In the original Weingarten paper , the ...
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Derivation of the Grand Canonical Partition Function for Fermions

Regarding the derivation on this page: http://lampx.tugraz.at/~hadley/ss2/fermigas/thermo/thermo.php I'm stuck with the summation over macrostates {$q$} being the same as the sum over microstates {$...
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Are paired electrons not bosonic?

A pair of electrons has spin 0 which makes any such system a boson rather than a fermion. The Pauli exclusion principle does not apply therefore to paired electrons and any such two electrons can ...
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Why neutron degeneracy pressure is much stronger than electron degeneracy pressure?

I am asking about Pauli Exclusion Principle stating that no two fermions can have the same 4 quantum numbers which is why we have periodic table of elements, but straight to my question: despite Pauli ...
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Fermi gas model of nucleus, specifically when the numberof neutrons does not equal Z

I am studying for a final for an introductory course in particle physics.(Undergraduate course) and while working though old final problems I found one part question which I could not figure out how ...
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Number of fermions in Fermi-Hubbard model

This is the Fermi-Hubbard model for a system of $L$ sites. $$\hat H = -t \displaystyle\sum_{i=1} ^L (c^+_ic_{i+1} + h.c.) +V\sum_{\langle i,j\rangle}^L n_in_j.$$ By looking at this equation, how do we ...
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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 ...
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Qualitative difference of excitations in Fermi VS Bose superfluids

Assume that we have an electrically neutral interacting gas (or liquid) of Bose or Fermi particles in a superfluid state. For simplicity, assume that the particles interact via an assigned central ...
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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 ...
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How do we show that for massless fermions, Helcity and Chirality align?

The Helicity operator of a representation of the Lorentz group is given by $$h = \varepsilon_{ijk}S^{jk}\frac{P^i}{|P|}$$ where $S^{\mu\nu}$ are the generators of the Lorentz group. In the $(\frac{1}{...
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Background field expansion of supersymmetric string action

For a reasearch project I am studying the paper by L. Alvarez-Gaumé, D. Freedman and S. Mukhi called "The Background Field Method and the UV Structure of the Supersymmetric Nonlinear $\sigma$ ...
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How do many-body fermionic operators transform under coordinate transformations?

In non-relaivistic many-body physics, sometimes called second quantisation, fermionic fields $\psi(x)$ obey the anti-commuation relations $$ \{ \psi(x),\psi^\dagger(y) \} = \delta(x-y), \quad \{ \psi(...
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Making sense of negative $\mu$ in the Fermi-Dirac distribution

I'm trying to make sense of the fact that, in the Fermi-Dirac distribution, we have that the chemical potential can have any positive or negative value, that is $$ -\infty<\mu<\infty $$ But at $...
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