# What is Fermions: Definition and 176 Discussions

In particle physics, a fermion is a particle that follows Fermi–Dirac statistics and generally has half odd integer spin: spin 1/2, spin 3/2, etc. These particles obey the Pauli exclusion principle. Fermions include all quarks and leptons, as well as all composite particles made of an odd number of these, such as all baryons and many atoms and nuclei. Fermions differ from bosons, which obey Bose–Einstein statistics.
Some fermions are elementary particles, such as the electrons, and some are composite particles, such as the protons. According to the spin-statistics theorem in relativistic quantum field theory, particles with integer spin are bosons, while particles with half-integer spin are fermions.
In addition to the spin characteristic, fermions have another specific property: they possess conserved baryon or lepton quantum numbers. Therefore, what is usually referred to as the spin statistics relation is in fact a spin statistics-quantum number relation.As a consequence of the Pauli exclusion principle, only one fermion can occupy a particular quantum state at a given time. If multiple fermions have the same spatial probability distribution, then at least one property of each fermion, such as its spin, must be different. Fermions are usually associated with matter, whereas bosons are generally force carrier particles, although in the current state of particle physics the distinction between the two concepts is unclear. Weakly interacting fermions can also display bosonic behavior under extreme conditions. At low temperature fermions show superfluidity for uncharged particles and superconductivity for charged particles.
Composite fermions, such as protons and neutrons, are the key building blocks of everyday matter.
The name fermion was coined by English theoretical physicist Paul Dirac from the surname of Italian physicist Enrico Fermi.

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1. ### Fourier transform of t-V model for t=0 case

To compute the Fourier transform of the ##t-V## model for the case where ##t = 0##, we start by expressing the Hamiltonian in momentum space. Given that the hopping term ##t## vanishes, we only need to consider the potential term: $$\hat{H} = V \sum_{\langle i, j \rangle} \hat{n}_i \hat{n}_j$$...
2. ### I Why do all fermions have the same spin 1/2?

We are taught that all fermions have spin ##\frac{1}{2}##, short for spin angular momentum ##\frac{\hbar}{2}##, which can be added to the orbital angular momentum. Considering spin is a kind of angular momentum, it must be dependent on the mass (or moment of inertia) of the particle. However...
3. ### A Question about commutator involving fermions and Pauli matrices

Suppose ##\lambda_A## and ##\bar{\lambda}_A## are fermions (A goes from 1 to N) and ##\{ \lambda_{A \alpha}, \bar{\lambda}_B^{\beta}\} = \delta_{AB}\delta_{\alpha}^{\beta}##. Let ##\sigma^i## denote the Pauli matrices. Does it follow that ##[\bar{\lambda}_A \sigma^i \lambda_A, \bar{\lambda_B}...

6. ### I Two hydrogen atoms and Pauli's exclusion principle

Hello, I recently came across the following (apparent, I hope) paradox: suppose we have two H atoms. Now, a hydrogen atom is made up of one proton and one electron (fermions), so it is a boson. Then one could have two hydrogen atoms which are in the exact same state (including position). This...
7. ### Finding Probability of Two-Identical-Particle System in a Given State

Problem: A system contains two identical spinless particles. The one particle states are spanned by an orthonormal system ##|\phi_k>##. Suppose that particle states are ##|\phi_i>## and ##|\phi_j>## (##i \neq j##). (a) Find the probability of finding the particle in the state ##|\xi>## and...
8. ### A Grassmann number: a more rigorous treatment?

Most textbooks on fermionic path integral only briefly introduce Grassmann numbers. However, I want a more systematic treatment to feel comfortable about this approach. For illustration, I have several examples here. Example 1: Consider a system with only one state, how to calculate ##\langle...
9. ### A Three generations of Fermions from octonions Clifford alegbras

Quanta has this article, The Peculiar Math That Could Underlie the Laws of Nature New findings are fueling an old suspicion that fundamental particles and forces spring from strange eight-part numbers called “octonions.”...
10. ### Show that an expected value of a vacuum state is equal to 1

\langle \phi_0| \hat{c}_{-k \downarrow} \hat{c}_{k \uparrow}\hat{c}^\dagger_{k \uparrow}\hat{c}_{-k \downarrow}|\phi_0\rangle = \\ \langle \phi_0| - \hat{c}_{k \uparrow} \hat{c}_{-k \downarrow} \hat{c}^\dagger_{k \uparrow}\hat{c}_{-k \downarrow}|\phi_0\rangle = \\ \langle \phi_0| \hat{c}_{k...
11. ### A Cohomology and fermions in supersymmetry

Hello, I have been looking at some differential geometry and watching Hirosi's video lecture online: At 1:03:00, I found that they claimed that there is a correspondence between the Hibert space of the symmetric Hamiltonian and the cohomology of the manifold. I am super new to the subject and...
12. ### Proof involving exponential of anticommuting operators

For ##N=1##, I have managed to prove this, but for ##N>1##, I am struggling with how to show this. Something that I managed to prove is that $$\langle\psi |b_k^\dagger=-\langle 0 | \sum_{n=1}^N F_{kn}c_n \prod_{m=1\neq k, l}^N \left(1+b_m F_{ml}c_l \right)$$ which generalizes what I initially...
13. ### I Sgr A* better DM halo/core of 56KeV Fermions than BH

A team in Italy has been studying the paths of stars affected by Sgr A* and finds that Sgr A* is better modeled by a galactic halo and core of 56KeV Dark Matter fermions by a Black Hole. Their paper is available in arxiv which also reports that it has been accepted for publication in the...
14. ### A Fermions, Monopoles & Quantum Hall: New Advances

Two papers today: https://arxiv.org/abs/2103.13639 Missing final state puzzle in the monopole-fermion scattering Ryuichiro Kitano, Ryutaro Matsudo [Submitted on 25 Mar 2021] https://arxiv.org/abs/2103.13574 Quark-Gluon Plasma and Nucleons a la Laughlin Wei Lu [Submitted on 25 Mar 2021]...
15. ### I Decay time of different types of bosons, hadrons and fermions

I'm interested in knowing where can i find the information on decay time of (possibly every?) different type of bosons, hadrons and fermions, which is available to the public (tiletles of books, articles, ...). Any suggestions or ideas?
16. ### B Fermions, Pauli and antisymmetry

How did Pauli determine his exclusion principle? Was it based on how he posited electron shells filled? Is the fact that fermions are antisymmetic a mathematical solution to make the principle work with quantum theory?
17. ### Fermions, Bosons, and nonidentical particles in a 1-d oscillator

I'm having a hard time understanding how to treat fermions, bosons, and distinguishable particles differently for this problem. To the best of my understanding, I know that my overall state for bosons must be symmetric, and because they're spin-0, this means there's only one coupled spin state...
18. ### Annihilation operators of two different types of Fermions

IfA=cd, where c and d are annihilation operators of two different types of Fermions, then {A,A°}is? A.1+n1+n2 B.1-n1+n2 C.1-n2+n1 D.1-n1-n2 Where,n1 and n2 are corresponding number operator, A° means A dagger or creation operator,as the particles are fermions they will obey anti-commutation I think
19. ### B How are Fermions formed from energy?

We all know that matter is formed from energy. I am wondering how that is done. I am assuming there is some smallest unit of energy, which I will call a "Planck Energy Block" or "Planck Block". I am also assuming these Planck Blocks may be arranged in different ways to form fermions. I do...
20. ### A Spin change of Fermions and quantum energy spectrum

Okay i was reading abrikosov's book and he said since in QM spin only changes by integer values boson excitiation happens one at a time and fermion ALWAYS appears or disappears in pairs. but isn't change from a spin up to spin down 1/2 to -1/2? or i had the wrong convention which |1/2| shouldve...
21. ### A Non interacting Fermions satisfy the Pauli exclusion principle

This question is more a question I'd ask in a chat rather than formally on paper/forum. If we take the free electron model, the electrons are considered as non interacting. It is essentially a 1 particle problem where the potential is constant through space. The electrons are not perturbed at...
22. ### Second Quantization - Quasiparticles

(Simplified version of Baym, Chapter 19, Problem 2) Calculate, to first order in the inter-particle interaction V(r-r'), the energy of an N+1 particle system of spin-1/2 fermions with on particle of momentum p outside an N-particle Fermi sea (quasiparticle state). The answer should be expressed...
23. ### A What is best way to start learning DMRG for Fermions?

I want to learn the density matrix renormalization group (DMRG) method in traditional formalism (not MPS). While there are many good introductory level articles available for bosonic (and spin) systems, I have not encountered any introductory level article which deals with fermionic systems i.e...
24. ### Ground state and 1st excited state energy of 3 Fermions

Homework Statement So in my problem, there's a given of 3 non interacting fermions in a harmonic well potential. I already got the wavefunction but i have problems in obtaining the ground state energy and its 1st excited state energy for 3 fermions (assuming they are non interacting and...
25. ### Density-density correlation function for spinless Fermions

Homework Statement Consider the groundstate of a one-dimensional, non-interacting system of spinless fermions. Let ##a^†(x)## and ##a(x)## be the creation and annihilation operators for a fermion at the point ##x##, so that the density operator is ##n(x) = a^†(x)a(x)##. Show that the...
26. ### A Grassmann numbers and Fermions

Hello! I am a bit confused about fermions in QFT when they are considered grassmann numbers. If you have 2 grassmann numbers ##\theta_1## and ##\theta_2##, something of the form ##\theta_1\theta_2\theta_1\theta_2## gives zero. However, a term in a QED lagrangian of the form...
27. ### I Masses of Fermions, string theory, Higgs mechanism

Take the first family of fundamental fermions, u, d, e-, and ν. The u and d are more massive than the e- and the e- is more massive than the ν. The u and d interact via 4 forces, the e- interacts via 3 forces, and the ν interacts via 2 forces. The fermions that interact via the most forces are...
28. ### A Why is the partition for Fermions a sum of Boltzman factors?

The partition function should essentially be the sum of probabilities of being in various states, I believe. Why is it then the sum of Boltzmann factors even for fermions and bosons? I've never seen a good motivation for this in literature.
29. ### A Is there an Expression for Entropy of Fermions or Bosons?

Is there an expression similar to the Sackur-Tetrode equation that describes the statistical entropy of fermions or bosons, maybe for the electron gas in a metal or the photon gas in a cavity?
30. ### B What happens to Fermions during Beta decay radiation?

I recently started learning about quarks and leptons and was wondering what happens to the fermions (specifically the quarks and leptons) during a beta decay. How is the electron/positron created and what causes the up quarks and down quarks to change flavours? If this is a bad question please...
31. ### A Commutation relations for bosons and fermions

For the free boson, the field operators satisfies the commutation relation, $${\varphi}_{x'}{\varphi}_{x} - {\varphi}_{x}{\varphi}_{x'} = 0$$ at equal times. While the fermions satisfies, $${\psi}_{x'}{\psi}_{x} + {\psi}_{x}{\psi}_{x'} = 0$$ at equal times. I interpret ##{\varphi}_{x}## and...
32. ### Fermion Pressure at Room Temperature

Homework Statement I have to find the mean Energy $<E>$ and pressure of a system of N fermions with spin 1/2. The energy per particle is \varepsilon = \frac{p^2}{2m}. \endu{equation} Homework Equations The relevant equations are the degeneracy of the system: ...
33. ### Problem Proving a Spinor Identity

Homework Statement Given the spinors: \Psi_{1}=\frac{1}{\sqrt{2}}\left(\psi-\psi^{c}\right) \Psi_{2}=\frac{1}{\sqrt{2}}\left(\psi+\psi^{c}\right) Where c denotes charge conjugation, show that for a vector boson #A_{\mu}#; A_{\mu}\overline{\Psi_{1}}\gamma^{\mu}\Psi_{2} +...
34. ### Ground state of 3 noninteracting Fermions in an infinite well

In Zettili's Quantum Mechanics, page 477, he wants to determine the energy and wave function of the ground state of three non-interacting identical spin 1/2 particles confined in a one-dimensional infinite potential well of length a. He states that one possible configuration of the ground state...
35. ### I In quantum statistics, inhibition/enhancement factors

These ideas come from the book Quantum Physics by Eisberg and Resnick (specifically ch11), can anyone explain what the inhibition factor and enhancement factors are in a little more detail? I do not understand what the book is trying to explain, and I can't seem to find these anywhere online...
36. ### I Mirror fermions / mirror families. How does it work?

From time to time we have some minor threads mentioning real vs complex representations of fermions, chiral theories, etc and how a loophole is to use mirror generations, but I do not remember some detailed discussion of how does it work. For starters, do we need an even number of...
37. ### Center of mass system of two interacting identical Fermions

Homework Statement Consider the center of mass system of two interacting fermions with spin 1/2. a) What is the consequence of the Pauli exclusion principle on the two-particle wave function? b)Let S1 and S2 be the spin operators of the two individual fermions. Show that the operators...
38. ### A Virtual Fermions and Pauli Principle

Hi guys, Do virtual particles, when they are fermions, obey Pauli exclusion principle as real fermions do? More specifically, what I am wondering is the following: Fermion fields would have some energy at every point in spacetime due to the uncertainty principle. Now, is it possible for the...
39. S

### A How many generations and fermions are there in the Standard Model Group?

Dear All The standard Model Group is SU(3)*SU(2)*U(1) i know that there is 3 families and each family contain 16 fermion. I am trying to guess the number of families and fermions from the representation of the concerned group. For example the SO(10) How many generations we have and how may...
40. ### A Hubbard model diagonalization in 1D K-space for spinless Fermions

I am trying to diagonalize hubbard model in real and K-space for spinless fermions. Hubbard model in real space is given as: H=-t\sum_{<i,j>}(c_i^\dagger c_j+h.c.)+U\sum (n_i n_j) I solved this Hamiltonian using MATLAB. It was quite simple. t and U are hopping and interaction potentials. c...
41. ### I Fermions Bosons vertices in SM - but no SUSY

In the Standard Model fermions interact via exchanges of massless (virtual) spin-1 particles. Fermions are turned into a boson. How is that different from the SUSY transformation that turns fermions into bosons?
42. ### A Symmetrisation of wave function for fermions

The wave function for fermions has to be anti-symmetric with respect to exchange of positions of electrons, but what if it depends on wave vector as well. Does they have to be exchanged as well, in other words, for two-electron system what is correct Ψ(r1,k1,r2,k2) = - Ψ(r2,k1,r1,k2) or...
43. ### Retrocausality interpretations for fermions

This question is about experiments involving entangled electrons or any other fermion for that matter. I’ll get to that in a sec. I’ve been interested in understanding interpretations that have retrocausality. (TIQM by Cramer, Wheeler–Feynman absorber, time symmetric by Price) It’s easy to...
44. ### Need help finding fermion energies and probabilities

<Moved from a technical forum, therefore no template> For two non-interacting fermions confined to a 1d box of length L. Construct the antisymmetric wave functions (Slater determinant) and compare ground state energies of two systems, one in the singlet state and the other in the triplet state...
45. ### I Why does topology matter in determining fermions and bosons?

A straightforward argument for showing that indistinguishable particles in 3D can either be bosons or fermions goes as follows. Consider the wavefunction of identical particles 1 and 2 at positions \psi (\vec{r}_1, \vec{r}_2). If we swap these particles around then this just becomes \psi...
46. ### I Exploring the Reason Behind SU(3)xU(1) Group Acting on Dirac Fermions

This is a companion question to https://www.physicsforums.com/threads/why-su-3-xsu-2-xu-1.884004/ Of course the Higgs mechanism over the standard model produces this low-energy group, SU(3)xU(1), which acts on Dirac fermions (this is, no Left-Right asymmetry anymore). Is there some reason...
47. ### A ##SU(2)## doublets, Majorana Fermions and Higgs

Say ##L## and ##L^{c}## are a pair of ##SU(2)## doublets (electroweak-charge fermions) and ##N_{1}## and ##N_{1}^{c}## are a pair of neutral Majorana fermions. Say that these fermions couple to the Higgs via Yukawa coupling and have vector masses ##M_0## and ##M_1## respectively...
48. ### A Help with Feynman diagrams involving Majorana Fermions

Hello everyone, I am trying to compute the ΔF=2 box diagrams in SUSY with gluinos. The relevant diagrams are the following: I want to use the Dirac formalism and NOT the Weyl one. So, the only reference that I have for Feynman rules with Majorana spinors is the old but good SUSY review from...
49. ### B Collision of identical fermions

What happens if I shoot a fermion at another identical fermion at rest? For example, do the fermions stick together, or do they bounce? Let's ignore gravity, electro-magnetism, weak force and strong force. Edit: We are considering the Pauli exclusion principle.
50. ### Fermions in infinite square well in compact geometry

Homework Statement The global topology of a ##2+1##-dimensional universe is of the form ##T^{2}\times R_{+}##, where ##T^{2}## is a two-dimensional torus and ##R_{+}## is the non-compact temporal direction. What is the Fermi energy for a system of spin-##\frac{1}{2}## particles in this...