Fermion Definition and 112 Threads

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. Paul Colby

    B In QFT, do Fermion fields belonging to distinguishable particles commute or anti commute?

    In the standard model fermion field components collect quite a few labels. The basic fermion field has 4 components that obey anti commutation relations. If one has two types of fermions, say electrons and muons. Do these commute or anti commute? Same question for other labels like gauge group...
  2. T

    I How to Show \(\psi_{i}\ket{\xi} = \xi_{i}\ket{\xi}\)?

    I came across the following formula (2.68) in di Francesco's CFT book for a fermionic coherent state: $$\ket{\xi} = e^{\psi^{\dagger}T\xi}\ket{0}$$ where##\ket{\xi} = \ket{\xi_{1},...,\xi_{n}}##, ##\xi_{i}## is a Grassman number, ##T## is some invertible matrix, and ##\psi^{\dagger}## is the...
  3. G

    A Problem evaluating an anticommutator in supersymmetric quantum mechanics

    I am trying to reproduce the results of a certain paper here. In particular, I'm trying to verify their eqn 5.31. The setup is N = 4 gauge quantum mechanics, obtained by the dimensional reduction of N = 1 gauge theory in 4 dimensions. ##\sigma^i## denotes the ith pauli matrix. ##\lambda_{A...
  4. phos19

    I Fermi energy for a Fermion gas with a multiplicity function ##g_n##

    I ran across the following problem : Statement: Consider a gas of ## N ## fermions and suppose that each energy level ## \varepsilon_n## has a multiplicity of ## g_n = (n+1)^2 ##. What is the Fermi energy and the average energy of this gas when ## N \rightarrow \infty## ? My attempt: The...
  5. S

    I Propagator of massless Weyl field

    I have this Lagrangian for a free massless left Weyl spinor, so it’s just the kinetic term, that can be written embedding the field into a larger Dirac spinor and then taking the left projector in this way: $$i \bar{\psi} \cancel{\partial} P_L \psi$$ Srednicki says that the momentum space...
  6. S

    I Noether currents for a complex scalar field and a Fermion field

    For a complex scalar field, the lagrangian density and the associated conserved current are given by: $$ \mathcal{L} = \partial^\mu \psi^\dagger \partial_\mu \psi -m^2 \psi^\dagger \psi $$ $$J^{\mu} = i \left[ (\partial^\mu \psi^\dagger ) \psi - (\partial^\mu \psi ) \psi^\dagger \right] $$...
  7. S

    I 1-loop Fermion mass correction in toy EFT

    Where does the ##m## in ##(3.2)## come from? It doesn’t seem to enter anywhere in Feynman rules for the given diagram
  8. P

    A Weyl Fermion in an infinite well

    Hello everyone, I have a problem with bounds states of the 1D Weyl equation. I want to solve the Dirac equation ##−i\hbar \partial _x\Psi+m(x)\sigma _z \Psi=E\Psi## with the mass ##m(x)=0,0<x<a##, ##m(x)=\infty,x<0,x>a##. ##\Psi=(\Psi_1,\Psi_2)^T## is a two component spinor. Outside the well...
  9. The black vegetable

    A How to find the gamma function for a fermion vacuum energy calculation?

    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 gamma fuctions doesn't match this equation. Can anyone help with the next step? Starting with the...
  10. Andrew3

    A question about the derivation of Fermion Quantization in QFT

  11. G

    A Fermion mass terms in Peskin and Schroeder's book

    I'm currently looking at how fermion masses are produced via the Higgs mechanism in "An Introduction to Quantum Field Theory" by Peskin and Schroeder. It all makes a lot of sense and I've been fine with it so far, but I ended up getting stuck on something that's driving me nuts. I feel silly...
  12. RicardoMP

    A Trace of a product of Dirac Matrices in a Fermion loop

    I'm working out the quark loop diagram and I've drawn it as follows: where the greek letters are the Lorentz and Dirac indices for the gluon and quark respectively and the other letters are color indices. For this diagram I've written...
  13. Physics4Funn

    A Another question about a Causal Fermion System

    What are the specific objections to Felix Finster's Casual Fermion System besides "many objections" and "very exotic, and very, very far from mainstream"? The comment in the summary above says forget about the Dirac sea. I am sorry, but CFS is an extension of the Dirac sea idea written in...
  14. C

    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...
  15. F

    I Scattering of a scalar particle and a Fermion

    Hello everyone, I am working on the following problem: I would like to determine the invariant Matrix element of the process ##\psi\left(p,s\right)+\phi\left(k\right)\rightarrow\psi\left(p',s'\right)+\phi\left(k'\right)## within Yukawa theory, where ##\psi\left(p,s\right)## denotes a fermion...
  16. Demystifier

    A Bosonization Formula and its Effects on Fermion Number

    Shankar, in the book "Quantum Field Theory and Condensed Matter", at page 328 writes the famous bosonization formula in the form $$\psi_{\pm}(x)=\frac{1}{\sqrt{2\pi\alpha}} e^{\pm i \sqrt{4\pi} \phi_{\pm}(x)}$$ and then writes: "This is not an operator identity: no combination of boson operators...
  17. Jamister

    A How can infrared divergences in the fermion propagator be cured in QED?

    Summary: how to cure infrared divergences in fermion propagator in QED? In calculating the fermion propagator in QED, we identify Ultraviolet and Infrared divergences. the Ultraviolet divergences solved by regularization, but I don't understand how to treat the Infrared divergences. Infrared...
  18. hilbert2

    A Exploring the 2-D Ising Model & Its Continuum Limit

    I've recently been reading about the 2-dimensional Ising model and its continuum limit from several sources, including https://webhome.weizmann.ac.il/home/fnfal/papers/Ising/lecture1.pdf https://webhome.weizmann.ac.il/home/fnfal/papers/Ising/lecture2.pdf As far as I understood it, the state...
  19. hideelo

    Composite Fermion Approach to FQHE

    I am following David Tong's notes on the Quantum Hall Effect (https://arxiv.org/abs/1606.06687). One of the approaches he takes to the FQHE is the composite fermion approach (Section 3.3.2). There are two things I am struggling with. First of all he says that a vortex is something around which...
  20. J

    I Some questions about electrons and the Fermi energy

    Hello ,evreyone.I have two questions about fermi energy. 1,Can I claim that 'fermi energy ' play the role of chemical potential? 2,I have learned from thermal physics that only electrons near fermi level can conduct in metals.How can electrons behave like this? I can't figure out why only...
  21. YoungPhysicist

    B Can liquid helium conduct heat infinitely fast?

    Can liquid helium is superfluidity state conduct heat infinitely fast? I thought I have seen this is a paper somewhere a long time ago, but now I am not sure about that.
  22. Spinnor

    I A Fermion interacts with the Higgs field

    In a blog post of Matt Strassler we are told about the top quark, "when the Higgs field is not zero, its presence, and the fact that it has a direct interaction with the top-left and the top-right, forces the top-left to convert over to a top-right, and back again. How often does this happen...
  23. YoungPhysicist

    B How could a Majorana fermion exist?

    Acoording to the internet, majorana fermions are particles which its antiparticle is itself. But shouldn't particles and antiparticles annihilate each other? Then how could such particle exist or being predicted?
  24. A

    I Recovering Fermion States in New Formalism?

    Hi, I just started a book on QFT and one of the first things that was done was switch from labeling states with their individual particles and instead label states by the number of particles in each momentum eigenstate. In addition, some "algebras" (not sure if they qualify by the mathematical...
  25. M

    I Commuting observables for Fermion fields?

    In nonrelativistic QM, we usually describe the Hilbert space by choosing a complete set of commuting observables, so that the set of states that are eigenstates of all the observables can be used as a basis. For instance, the "wavefunction" is the state as expressed in terms of "states" with...
  26. J

    I Causal Fermion System and revival of Dirac Sea

    If the Higgs Field could exist with constant 246GeV across all of space. How come the Dirac Sea couldn't exist? If the Universe can easily accommodate Higgs Field.. why not Dirac Sea for all particles. Also how does the Dirac Sea of bosons work? Like W+, W-? Any idea? I was asking about the...
  27. N

    I Magnetic moment of a massless charged Fermion?

    The spin magnetic moment of a charged, spin-1/2 particle is $$g \frac e {2m} \frac \hbar 2$$ where g is the g-factor (2 for any particle in tree-level approximation, 2.00231930436182 for electron), e is charge m is mass ##\frac \hbar 2## is spin But with zero mass this expression does not make...
  28. V

    I Is the Fermion number operator squared equal to itself?

    What the title says. Acting on a fermionic state with the number operator to a power is like acting with the fermionic operator itself. Does this allow us to define ## \hat{n}^k=\hat{n} ##? Or is there any picky mathematical reason not to do so?
  29. L

    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...
  30. N

    I Eigenvalues of Fermionic field operator

    Hello, I'm a bit confused about the eigenvalues of the second quantized fermionic field operators \psi(x)_a. Since these operators satisfy the condition \{\psi(x)_a, \psi(y)_b\} = 0 the eigenvalues should also anti-commute? Does this mean that the eigenvalues of \psi(x)_a are...
  31. S

    A Feynman rule for closed fermion loop in QED

    One of the Feynman rules of QED is the following: For a closed fermionic loop, the Feynman rule is to start at an arbitrary vertex or propagator, follow the line until we get back to the starting point, multiply all the vertices and the propagators in the order of the line, then take the trace...
  32. S

    A Bar on a fermion field, arrows on fermion lines and particle-antiparticle nature of a fermion

    This question is about the use of bar on a fermionic field in a Lagrangian, the use of arrows on external fermion lines and the particle-antiparticle nature of a fermion. For illustration of my question, I will use the following the charged-current interaction of the Standard model...
  33. S

    A Flow of charge on fermion propagator

    The momentum-space fermion propagator in the free Dirac theory is given by The arrow on the fermion propagator is said to represent the flow of charge. How can we derive this statement quantitatively from the Dirac Lagrangian? What is the quantitative form of the charge being referred to here?
  34. kiwaho

    A Photon number needless conservation, consolidation possible?

    We know lepton conservation law, that means multiple neutrinos can not be consolidate to big single neutrino. But photon is boson not lepton, no need of conservation, does that mean it is possible to combine or fuse a bunch of photons into ONE big photon, or say, more energetic photon, i.e...
  35. Safinaz

    I Fermion Self-Energy: Calculation and Analysis

    Hi all, In Peskin's book, Chapter 7, the self energy of electron has been calculated. In Equation (7. 28) ##p\!\!\!/## set to equal the mass of the electron ## m_0 ##. What if I calculate the self energy of a massless fermion mediated by a loop of another massless fermion and a scalar, like the...
  36. K

    I The Higgs Field and Fermion Generations: Stability and Mass

    if there were no higgs field, would the second and third generation of fermions, such as the top quark, be exactly the same mass as first generation? is the coupling between the top quark and the higgs field the sole reason the top quark is heaviest SM particle? is there a reason the top...
  37. B

    B Exploring Fermion Fields: Experimental Evidence for Matter Waves and Fields

    It is natural to make quantum fields out of electromagnetic wave or even the weak force. But is there experimental proof for the need to create fermion fields like separate electron fields for the electron.. quark fields for the quarks.. When de Broglie tried to propose matter wave from analogy...
  38. L

    How Does the Elastic Collision of Fermions Determine Gas Pressure?

    Homework Statement For a gas of N fermions of mass m confined in a volume V at a temperature ##T<E_F/kB##, consider the quantity ##<n_p>/V## as you would a classical distribution f(p,q) in the system phase space. Show that the impulse transfer of the elastic collisions of the particles with the...
  39. L

    Finding 2D Fermion Gas U/N with Temperature & Area

    Homework Statement For a gas of N fermions with mass M in 2D in a region of area A in thermal equilibrium at temperature T, we are asked to find ##U/N## in fuction of ##T## and ##a=A/N##. The attempt at a solution I used ##U=\sum(<n_i>\epsilon_i) = \sum(\exp(\beta(\mu-\epsilon_i))\epsilon_i...
  40. C

    A Time-ordering fermion operators

    If A and B are fermionic operators, and T the time-ordering operator, then the standard definition is T(AB) = AB, if B precedes A = - BA, if A precedes B. Why is there a negative sign? If A and B are space-like separated then it makes sense to assume that A and B anticommute. But...
  41. S

    Anticommutation relations Fermion creation and annihilation

    Homework Statement This problem is from Lahiri and Pal (2nd edition) Exercise 1.4: Suppose in a system there are operators which obey anticommutation relations ##[a_{r},a^{\dagger}_{s}]_{+}\equiv a_{r}a^{\dagger}_{s}+a^{\dagger}_{s}a_{r}=\delta_{rs}## and ##[a_{r},a_{s}]_{+}=0,## for...
  42. M

    Confusion about Slater Determinants

    Consider a system of 2 identical fermions. $$\psi_{k_1,k_2}(x_1,x_2,m_1, m_2) = \langle x_1\,x_2\,m_1\,m_2\mid \psi \rangle$$ According to what I have read we can construct a state with the right antisymmetry properties by $$\psi_{k_1,k_2}(x_1,x_2,m_1, m_2) =...
  43. Finny

    What are the prospects for the Causal Fermion System?

    Wikipedia: The theory of causal fermion systems is an approach to describe fundamental physics. It gives quantum mechanics,general relativity and quantum field theory as limiting cases and is therefore a candidate for a unified physical theory. It seems possible this might be progress. A step...
  44. Ravendark

    Second functional derivative of fermion action

    Homework Statement [/B] Consider the following action: $$\begin{align}S = \int \mathrm{d}^4 z \; \bar\psi_i(z) \, (\mathrm{i} {\not{\!\partial}} - m)_{ij} \, \psi_j(z)\end{align}$$ where ##\psi_i## is a Dirac spinor with Dirac index ##i## (summation convention for repeated indices). Now I would...
  45. S

    Why chiral fermions don't exist in odd dimensions?

    In four dimensions, left and right chiral fermion can be written as \psi_L= \begin{pmatrix} \psi_+\\ 0 \end{pmatrix},\qquad \psi_R= \begin{pmatrix} 0\\ \psi_- \end{pmatrix}, respectively, where \psi_+ and \psi_- are some two components spinors(Weyl spinors?). In this representation, the...
  46. L

    For the Lagrangian of fermion masses, how do I understand?

    Hello, everyone. I have a one question which is related to the fermion masses. If you see my latex mathematics, you can know what I want to say. Here, L means SU(2) left-handed lepton doublets and R means SU(2) right-handed lepton singlets. So I am too much confusing to understand this...
  47. E

    Explaining the Absence of Band Gaps in Superconductors

    Hi all, I am currently writing a report about superconductors, and am currently reading about how the band gap shows that single electrons are not the charge carriers responsible for superconductivity. However, I was confused when I read that electrons are fermions and as such there are no band...
  48. L

    How Should Spinors Be Applied in Physics Homework Involving Fermions?

    Homework Statement Homework EquationsThe Attempt at a Solution I've started from writing out the amplitude. Here I know that fermion has definite helicity so I can't sum over spins but I should input explicit forms of spinors. Am I correct? How to do this? I would be grateful for helping me...
  49. X

    Interacting Fermion System Commutation

    Problem Question My question isn't an entire homework problem, but rather for a certain mathematical step in the problem which I assume to be very simple. The problem is dealing with interacting fermion systems using second quantization formulas. I am essentially following my notes from class...
  50. M

    How Do Fermion Commutation Relations Affect Current Operators in 2D Spacetime?

    Homework Statement Consider left-handed fermions in two spacetime dimensions ##(t,x)##: ##\psi_L=\frac{1}{2}(1-\gamma_5)\psi_D## with ##J_0^\epsilon(t,x)=\psi_L^+(x+\epsilon)\psi_L(x-\epsilon)##. (a). Use canonical equal-time anti-commutation relations for fermions to compute...
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