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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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Average number of particles in single particle state meaning

So I was reading Fermi-dirac statistics and there is something that bothers me. I don't fully understand this line: "For a system of identical fermions with thermodynamic equilibrium, the average ...
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Show: $\langle n \vert \psi \rangle \langle \psi \vert n \rangle = \langle -\psi \vert n \rangle \langle n \vert \psi \rangle$

The book (Altland and Simons, Condensed Matter Field Theory, Ch. 4.2) I am reading makes use of the identity \begin{equation} \langle n \vert \psi \rangle \langle \psi \vert n \rangle = \langle -\psi \...
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Is the derivative with respect to a fermion field Grassmann-odd?

Fermion fields anticommute because they are Grassmann numbers, that is, \begin{equation} \psi \chi = - \chi \psi. \end{equation} I was wondering whether derivatives with respect to Grassmann numbers ...
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Why is supersymmetry a continuous symmetry?

Supersymmetry feels like a discrete symmetry to me, since the fermions are turning into bosons, and vice versa. I understand there is an infinitesimal parameter involved in the transformations, but I ...
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What happens to the scattering cross section of two fermions with different spins at low temperature?

The scattering cross section for two identical fermions at low temperature is given by: $\sim 8\pi a_1^2(ka_1)^4$. I was wondering what happens if the two fermions have different spins - does it ...
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Does the size, and distance between, expanding ultracold gas clouds depend on the populations in each cloud?

Say you had some $^{87}Rb$ atoms at a few kelvin and you knew that the atoms were distributed among the magnetic sub-levels of a hyperfine manifold. You want to know the relative population of each by ...
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Using the Slater determinant to find the associated antisymmetric wavefunction

My lecture notes read: If there is one electron in the ground state, one in the first excited state, and one in the second excited state, why can we not instantly assume then, that: $$\phi_{n_i}(x_j)...
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Why can there be more than one electron in an energy level if electrons are fermions?

By the Pauli-Exclusion Principle, no two electrons can be in the same quantum state. So, how can both be in the same energy eigenstate? Atom orbitals certainly have more than one electron per energy ...
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There's only time, no space in Quantum Mechanics

In this lecture (44:23) Nathan Seiberg: Topics in 2+1 Dimensional Quantum Field Theories 2. Nathan Seiberg says there's no space in QM and therefore fermions have spin 0. This sounds pretty revolting ...
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Partitioning Fermionic Systems

When dealing with separate non-interacting systems in quantum mechanics (i.e. with distinct Hamiltonians acting on different Hilbert spaces like $\hat{H}=\hat{H}_1 + \hat{H}_2$), one can often write ...
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Stationary phase method for theories with both bosons and fermions

I am wondering if there exists a method to compute path integrals using the stationary phase method for theories with both bosons and fermions. (I am aware of such a method for theories with bosons ...
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How do I find this relation between total energy and fermi energy?

I am currently studying the Sommerfeld model. I understand how to find the fermi energy $\epsilon_F$ by using the density of states function but I am unsure how to answer this question about the total ...
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Single atom states with energy 0, probability and occupation number fermions and bosons

Two mutually non-interacting atoms are trapped in a double-well potential in equilibrium at a temperature $T$, such that an atom can only occupy two possible single- atom quantum states, $Ψa(x)$ and $...
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Pauli Exclusion and Black Holes [duplicate]

Pauli exclusion principle states that 2 identical electrons cannot be in the same state, where state includes a spacial component. I have heard that, in order to avoid being in the same state, in a ...
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Using Brillouin zones to write Hamiltonian and find momenta values

This is a homework problem that I have spent a large amount of time on..I will try my best to simplify the questions down to conceptual ones, but I will also write more or less that whole question ...
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How is Pauli's exclusion principle valid for electrons of two hydrogen atoms in ground state, having same spin?

Suppose we have two hydrogen atoms in the ground state with spin of both electrons pointing upwards. Then the two electrons are in the same state. This should be against the exclusion principle. Now ...
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Mass term in QCD is not hermitian? [duplicate]

Accordingly to the result $$ (\bar{\psi}\psi)^\dagger = (-1)\bar{\psi}\psi \tag1$$ coming from the fact that fields with and without bar anti-commutes, you can deduce that the QCD mass term $$ -m\...
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Pauli Exclusion Principle and Quantum States [closed]

We know that two identical fermions cannot be in the same state together because of the Pauli exclusion principle. My questions are: Can two bosons (for example, photons) be arbitrarily close ...
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CPT transformation for bilinears

In the page 5 of the document 'CPT Symmetry and Its Violation' by Ralf Lehnert (https://core.ac.uk/download/pdf/80103866.pdf), appears a discussion about how the spin-statistics theorem applies to the ...
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Fermion Superposition [closed]

In case of superposition of identical particles, we usually just add their amplitudes. For example, if we have several particles having the amplitudes of being in a particular quantum state $\psi_1, \...
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Probability of finding a particle in a two/three particle system

Let us consider a system of 2 identical particles, 1 and 2. Let, $ψ_a(1)$ is the amplitude of finding particle 1 at state $a$, and $ψ_a(2)$ is the amplitude of finding particle 2 at state $a$. Let N.F ...
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What are Grassmann numbers in field theory?

I've been struggling with the use of Grassmann numbers in QFT e.g. Peskin and Schroeder. They are introduced as "numbers" whose product is antisymmetric, and associative (this isn't said, but used in ...
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“Data structure” for a fermion field

I am understanding the path integral formalism of fermion fields. Most textbooks told me that grassmannian integration is only algebaric notation. It shouldn't be understood in a Lebesgue Integral ...
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Pauli Exclusion Principle and Identical Fermions

Pauli exclusion principle means no two identical fermions can be in the same quantum state. Does it mean, two electrons with the same spin cannot be in the same De Broglie Wavelength? Or, more ...
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Are particles that are moving at the fermi energy easier to remove if you model a nucleus as a fermi gas?

Due to the Pauli exclusion principle, the nucleons build up the energy states from the bottom up. When talking about binding energy per nucleon, you are talking about how much energy is needed to ...
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1answer
63 views

IIB Supergravity from worldsheet (super)conformal invariance of Green-Schwarz string

After reading this question How are low energy effective actions derived in string theory? I began to wonder what is the coupling of the string to the other sugra fields. In almost all textbooks ...
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Massive chiral fermions

The main question is: why does nobody care about massive chiral fermions? It is well-known that in QFT (in the axiomatic framework of Garding & Wightman) the quantum field transform according (...
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The importance of phase when defining fermions and bosons

In my lecture on indistinguishable particles, my lecturer is trying to illustrate to me the notion of particles being indistinguishable when considering that when we swap two particles in a box the ...
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0answers
28 views

Spin 1/2 Fermion With Both Majorana and Dirac Masses?

I will use Van der Waerden notation in the following: It is easy to construct both Majorana and Dirac mass terms in the lagrangian of a Dirac spinor with left-handed component $\phi_{\alpha}$ and ...
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2answers
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Is the partial derivative in the Dirac equation in curved space contracted with a tetrad?

The Dirac Equation in Curved spacetime makes a difference between Lorentzian indicies and Covariant indicies. In the equation we find a $\partial_\mu$. Is this actually $e^a_\mu\partial_a$ where $e$ ...
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Peskin equation on the treatment of chiral anomaly

In page 666 (it couldn't be other way - bad joke), chapter 19, the Eq. (19.73) claims (see properties of the $\phi_n(x)$ functions in this post: Change of variables in path integral measure): $$ \...
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Change of variables in path integral measure

In fermion's path integral we have a measure that you can write, in terms of the Grassmann variables $\psi, \bar{\psi}$ as $$ D\bar{\psi}D\psi, \quad \psi(x) = \sum_n a_n\phi_n(x), \quad \bar{\psi}(x)...
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Why should fermions vanish for a solution of a supergravity theory that preserves supersymmetry?

On p.249 of Freedman and Van Proeyen's Supergravity, the following is stated: "Given the action of a supergravity theory, it is generally useful to search for solutions of the classical equations ...
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Why a field theory containing only fermions does not show spontaneous symmetry breaking?

For a real scalar field $\phi$, a theory as simple as the $\phi^4$ theory, can exhibit the phenomenon of Spontaneous Symmetry Breaking (SSB). For a complex scalar field $\phi$, a theory as simple as ...
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How do fermions explicitly interact with curvature via the tetrad?

I am aware of the basics of the tetrad formalism and am clear on why bosonic fields do not have couplings to curvature via their covariant derivatives in a curved space Lagrangian i.e. why $\nabla_\mu\...
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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, ...
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91 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. ...
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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 ...
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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\...
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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 ...
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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_{...
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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{\...
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67 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 $\frac{1}{\sqrt{2}}(| 1 \rangle | 2 \rangle - | 2 \rangle| 1\rangle)$. ...
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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?
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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 ...
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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?
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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^\...
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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 ...
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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 ...
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106 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^...