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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(Anti)commutation of creation and annhilation operators for different fermion fields

The Fourier expansion of the fermion field operator is such that $$ \hat\psi=\int\!d^3p\,\left[ f_b(p)\hat b(p) +f_d(p)\hat d^\dagger\!(p) \right] ~~, $$ for some sufficiently complicated $f_b$ and $...
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Fermionic Version of the effective Action

For a scalar field theory one introduces the partition function with external sources $$ Z[j] = \int \mathscr{D} \varphi \, \exp \left( -S[\varphi] + \int j \, \varphi \right) \text{,} $$ the analogon ...
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Transforming a Left-Handed Spinor into a Right-Handed Spinor

In pg. 125-126 of Symmetry and the Standard Model book, it was stated that for a spinor $\psi_L$ in the left-handed representation and a spinor $\psi_R$ in the right-handed representation, we can ...
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Is spin associated with rotations or boosts?

EDIT: It seems that I made an error and it was $S^{ij}$ that was used after all. I will not delete the question though because even though it is erroneous, the answer given below is rather insightful. ...
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Show that the divergence of the energy-momentum tensor of Dirac is 0

I've to show that $\partial_\mu T_{\mu \nu}^\psi=0$ where $T_{\mu \nu}^\psi$ is the energy-momentum tensor of Dirac field and using the Dirac equations, $$i\partial_\sigma \gamma^\sigma \bar{\psi}=-m\...
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Bosonisation of two non-interacting Fermions

Assume we have 2 sets of non-interacting fermions which I show by $\psi^{\pm}$ and $\chi^{\pm}$ where we have $\left< \psi^{+}(z) \psi^{-}(0) \right>=\frac{1}{z}$ and similar for $\chi$. Now we ...
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1answer
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Anti-symmetrized total tensor of two anti-symmetric tensors

Suppose we would like to anti-symmetrize a tensor $$T^{\mu_1, \mu_2,\ldots, \mu_n} = G^{[\mu_1, \mu_2,\ldots, \mu_r]} H^{[\mu_{r+1},\ldots, \mu_n]},$$ where $G$ and $H$ are anti-symmetric. One could ...
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Spin part of the angular momentum from the Lagrangian

For fermions of spin $1/2$ the angular momentum has following form: $$ \mathcal{J}_z = \int d^{3}x \ \psi^{\dagger} (x) \left[i(- x \partial_y + y \partial_x) + i\sigma^{xy} \right] \psi(x) $$ Here ...
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1answer
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Fermionic vacuum energy from Hamiltonian

I have some questions about some commentsthat Zee makes treating this problem in Sec II.2 of his QFT book. The Hamiltonian density of a spin-1/2 field is $$ \mathcal{H}=\bar\psi(i\vec\gamma\cdot\vec\...
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Commutation relations of derivatives of fermionic fields from the commutation relations of the original fields

I have a general question regarding such type of calculations, but let me start with a concrete question. Consider the $bc$- free fermion CFT so that $b(z)$ and $c(z)$ are free fermions with OPE, $$b(...
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Lagrangian for massless fermions and associated currents

My question is based on the pdf in this link On page 8, the authors consider an example of massless fermions. I follow through most of the steps in here except for equation 41, which I believe should ...
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Making indistinguishable particles distinguishable 2.0

An extended discussion of the selected best answer to the following SE question. In part 2 of the answer concerning 'far-apart' particles, I don't see how the equality can follow in the first integral....
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Permutation of Identical Fermions - Spatial and Spin Decomposition

As far as I know, fermions are the particles which exhibit antisymmetric states: $\hat{P}\left|n_1\right>\otimes\left|n_2\right> = -\left|n_2\right>\otimes\left|n_1\right>$. Often times, ...
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Might we deduce the masses of standard model fermions from symmetry principles alone?

There are 24 fermions in the standard model (if we consider left/right and particle/anti-particle as the same type). Thus there are 24 masses to find. (Yes, it is generally considered quarks of ...
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Jordan-Wigner transformation on a circle and spin structures?

Is there an analog of the Jordan-Wigner transformation between fermion algebra on a circle and a Pauli algebra? For example, the continuum analog of bosonization of "compact boson $\...
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1answer
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Partial derivative of Dirac Lagrangian with respect to derivatives of fields

Why is $\frac{\partial\mathcal{L}}{\partial(\partial_\nu \bar{\psi})} = 0$, for the Dirac Lagrangian $\mathcal{L} = \bar{\psi}(i \gamma^\mu \partial_\mu - m)\psi$? This comes up in deriving the ...
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General expression of time-ordered thermal average Green's function does not reproduce non-interacting limit (Fetter ch. 31 Eq. (31.24))

Hi I am going through Fetter's Quantum Theory of Many-Particle Systems Dover Edition. In ch. 31 he computed the relation between $\bar{G}(\mathbf{k},\omega)$, ${\bar{G}}^{R}(\mathbf{k},\omega)$ and $\...
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Phase density representation two-site Hubbard Hamiltonian with Fermions

I'm looking for the two-site Fermi-Hubbard Hamiltonian in phase density representation in linearized form and I don't know how to derive it. I then want to derive the equations of motion from that. ...
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Why is every electron in the universe not entangled with every other electron?

According to the principles of identical particles, the wavefunction of a collection of fermions must be antisymmetric and such a state is entangled. Doesn't this mean that any given electron in the ...
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Boundary (Anti)Periodic conditions and fermion partition functions

The path integral with antiperiodic fermions (Neveu-Schwarz spin structure) on a circle of circumference $\beta$, in a theory with Hamiltonian $H$, has partition function $$ \rm{Tr} \exp(−\beta H)$$ ...
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What would happen if the Pauli exclusion principle did not exist?

There can be never two fermions in exactly the same state, which is known as Pauli’s exclusion principle, but infinitely many bosons. I read in the book saying that if Pauli's exclusion principle does ...
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1-loop Vacuum energy of a fermion field

Following the method by Peskin and Shroesder 11.4 Trying to calculate the vacuum energy of a fermion. If my method is correct so far the next step is to find gamma function , the formula I have for ...
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Using Grassmann variables in describing all the states in the system given by two identical fermions on a circle

We are asked to find the all the states in the system given by two identical fermions on a circle only in terms of the Grassmann formalism. Furthermore, we have to show that $\lim_{n \to \infty} N(E)/...
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Ground state of 3 identical fermions laying over a circumference

We are asked to show that the ground state in a system of 3 free identical fermions that are on a circle of length, 2$\pi$, is equivalent to $\sin(q_1-q_2)+\sin(q_2-q_3)+\sin(q_3-q_1)$ up to some ...
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String theory allows chiral gauge coupling

In Polchinski String Vol 1: Chiral gauge couplings. The gauge interactions in nature are parity asymmetric (chiral). This has been a stumbling block for a number of previous unifying ideas: they ...
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Why does the fermion action involve derivatives of the metric?

I've noticed that the bosonic actions can be written in terms of bosonic fields, their derivatives and the metric tensor e.g. $$\sqrt{g}g^{\mu\nu}\partial_\mu \phi \partial_\nu\phi$$ and $$\sqrt{g}g^{\...
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Action of fermionic operators of various species on Fock states

My question is closely related to this and this questions. However, instead of asking about the commutation relation between the operators, I would like to ask about their action of the Fock states. ...
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Calculating the residue as part of Matsubara summation

On page no. $166$ of "Many-body quantum theory in condensed matter physics" by Henrik Bruus & Karsten Flensberg, while explaining the summation of Matsubara frequency, the following ...
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Quantum factorization approximation for first order Coulomb energy

I'm working through "Advanced Quantum Mechanics" by Franz Schwabl, and he uses this G-correlation function to estimate the first order correction to the ground state energy in a Coulomb ...
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1answer
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A quadratic Hamiltonian is a model of independent particles - why?

I'm reading some notes on the Anderson Hamiltonian: $$ H=\sum h_i c_i^\dagger c_i -q\sum_{\langle i,j\rangle}(c_i^\dagger c_j+c_j^\dagger c_i)$$ Where the $c_i/c_i^\dagger$ are fermionic annihilation/...
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Expectation value of time-evolved number operator for ground state Coulomb system

I'm going through "Advanced Quantum Mechanics" by Franz Schwabl, and he calculates the electron energy levels from the Coulomb interaction in a perturbative way (section 2.2.3). In the ...
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Are combined fermion wavefunctions still antisymmetric after wavefunction collapse?

If we have two electrons in a state $|\psi\rangle=\frac{1}{\sqrt2}[|\uparrow\downarrow\rangle+|\downarrow\uparrow\rangle]$ and we measure the spin of the first electron to be up, does the wavefunction ...
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Physical look to supersymmetry

A supersymmetry is a symmetry that transforms fermions into bosons and bosons into fermions. What's the physical interpretation of it? I could notice a symmetry between the boson sector and fermion ...
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1answer
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What is a fermionic field theory?

Let $\mathscr{H}$ be a Hilbert space and $\mathscr{H}^{n}$ be the associated $n$-fold tensor product of this Hilbert space. I'll skip the mathematical details in what follows, but my approach follows ...
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1answer
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Why is the anti-commutation relation $\lbrace \psi_a(x), \bar{\psi}_b(y) \rbrace = 0$ enough to ensure causality?

In quantum field theory, it is crutial that two experiments can not effect each other at space-like seperation. Thus $[\mathcal{O}_1(x), \mathcal{O}_2(y)] = 0 $ if $(x-y)^2 < 0$. For the Klein-...
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Is there a unique way to construct the overall spatial wavefunction for identical particles?

While studying the quantum mechanics of $N$ identical particles, I stumbled upon formulas for generalizing the spatial wavefunction for bosons: $$\psi(x_1,...,x_N)=\frac{1}{\sqrt{N!\prod_\alpha N_\...
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Counting number of states for fermions

I have a system of $N$ fermions that can occupy $M$ single particle states, and $M$ is much larger than $N$, $M \gg N$. Since only a single or no fermion can be in a particular state, the number of ...
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Virtual fermions vs exclusion principle

How QED eliminates the cases when in loop corrections two fermions get created with the same momenta and spin state? Is it done by the ladder operators? Edit: the two fermions are in two distinct ...
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Experimental evidence of Pauli principle for non-bound fermions

I mean electrons in atoms/molecules/solid bodies not count here. I heared of an experiment measuring the degeneracy pressure of a fermion gas.
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Which term in QED Lagrangian represents the muon?

I often heard that electron-muon scattering is a QED process. So I suppose that muon is a field in QED. However, when looking at the QED Lagrangian, it is basically F^2 + psi D_A psi where D_A is the ...
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How fermion doublers cause practical issues?

These days I learn about the lattice gauge theory, and in particular learned when one naively discretizes the fermion action, doublers, superfluous poles for a propagator, emerge. I wonder what issue ...
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How many fermions can at most occupy a single Landau level of a two-dimensionalsystem?

The $n$th Landau level is given by $\psi_{n,m}=f(z)({\phi^{+}})^{n}({\bar{\phi}^{+}})^{m}\psi_{0,0}={\bar{z}}^{n}{z^{m}}e^{-z\bar{z}}$, where $f(z)$ is a normalization condition. In general, the ...
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Can single fermion get converted into single boson?

Can a fermion get converted into a boson?
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Why the electrons in a insulator fill up the valance band exactly?

The classic picture of the band structure for an insulator is a filled valance band below $E_F$ and empty conduction band above $E_F$. This picture seems to me that the electron density is fixed. For ...
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Eigenstates/Eigenfunctions of 2 non-interacting spin 1/2 particles

For my homework I have to find the every eigenstate for a non-interacting 2 fermion particle system with spin 1/2. The problem goes like this: I have a total Hamiltonian $H(1,2)=h(1)+h(2),$ such that ...
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Why should all species of elementary fermions have the same spin angular momentum?

Is there any deeper motivation behind the fact that all elementary fermions have the same exact amount of spin angular momentum ($\frac{\sqrt3}{2}\hbar$ total or $\frac12\hbar$ projected) or is it ...
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2answers
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How do fermionic operators transform?

In quantum mechanics, if we have an operator $\Omega$, then under the transformation $T$, with infinitesimal generator $G$ (i.e. $T(\epsilon)=1-i\epsilon G + \ldots$), then operator transforms as $$\...
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1answer
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How does a boson interact with a fermion?

We have antisymmetric wavefunctions for a fermionic systems and symmetric wavefunctions for bosonic systems that give us a hint that quantum states can be occupied by a single fermion(or none) while ...
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Would a highly degenerate cosmic neutrino background affect fusion reactions that proceed via the weak interaction?

This might seem like a really strange question, but here's my reasoning. A proton-proton fusion reaction proceeds in two steps: \begin{align*} p + p + \text{1.25 MeV} &\rightarrow {}^2_2\mathrm{He}...
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Is a degenerate Fermi gas with high Fermi energy an eigenstate of the number operator?

A degenerate Fermi gas with a small Fermi energy relative to the particle mass is roughly an eigenstate of the particle number, i.e., there are 2 fermions per Compton wavelength in the volume.  This ...

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