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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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Non-linear sigma-models on curved worldsheet

I am studying nonlinear sigma-models and topological twists using E.Witten's article "Mirror manifolds and topological field theory" (https://arxiv.org/abs/hep-th/9112056), as well as "Mirror symmetry"...
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Why don't we use Fermi-Dirac statistics for deriving Richardson equation for thermionic emission?

Why does the Richardson-Dushman equation need Maxwell-Boltzmann statistics instead of Fermi-Dirac statistics to model thermionic emission, though only fermions are our only interest in those emissions?...
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Why can't a Weyl Fermion have mass?

My understanding was that a particle may have mass if there is a quadratic term in the fields without derivatives. For a single left-handed Weyl fermion, the following expression is lorentz invariant, ...
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Understanding Periodic and Anti-periodic boundary condition for Jordan-Wigner transformation

In the study of spin chains with periodic boundary condition ($S_{N+1}=S_{1}$) when one applies Jordan-Wigner transformation to map the spin chain to spinless fermion chain, one needs to make sure in ...
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109 views

What if there were only 1 generation of fermions?

The number of fermion generations has many effects beyond the mere existence of "3 copies of fermions," including determining scatterings, constraining CP violation, influencing electroweak symmetry ...
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Pauli's exclusion principle in elementary particles

Elementary particles such as Quarks obey Pauli's exclusion principle since they exist in three colors (RBG). Where as electrons which is also elementary that does not have any color quantum numbers ...
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42 views

Anticommutation relations for fermionic fields imply that Hamiltonian / Lagrangian can at most be linear?

Fermionic field operators do obey anticommutation relations, so for a chosen Field operator (and the field momentum), we have: $$ \{\Psi_a, \Psi_b\} = \{\pi_a, \pi_b\}= 0 $$ with the $\Psi_a$ being ...
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41 views

Generalized commutator/anticommutator via phase factor

We know that the commutator between two operators $A$ and $B$ reads $[A,B]_{-}=AB - BA$, while the anticommutator reads $[A,B]_+=AB + BA$. I am wondering if someone has ever used a generalized ...
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Chiral preferences of photon polarization modes

Is there any experimental evidence that both photon (circular polarization) modes do indeed couple equally to both left- and right-handed fermion currents ($j_L,j_R$) ? I know it is a basic ...
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28 views

Expressions of $\varepsilon^0_F$ and $p^0_F$ [on hold]

A perfect gas of $N$ fermions of chemical potential $\mu$ is enclosed in an enclosure of volume $V$ at temperature $T$. We denote by $ g(p) dp$ the number of microstates accessible to fermions having ...
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33 views

Matrix representation of the CAR for the fermionic degrees of freedom

The canonical anticommutation relations (CAR) for a fermionic degree of freedom can be written as follows: $$ a^2 = \left( a^{\dagger} \right) ^2 = 0, $$ $$ a a^{\dagger} + a^{\dagger} a = 1. $$ ...
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42 views

Alpha decay and Fermi-Dirac vs Bose-Einstein statistics

This article on wikipedia on alpha decay states: One curiosity is why alpha particles, helium nuclei, should be preferentially emitted as opposed to other particles like a single proton or neutron ...
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72 views

How can multiple fermions combine to form a boson? [duplicate]

I understand that composite particles with integer spin form a boson. For example a helium nucleus is a boson because it has 2 protons and 2 neutrons. If all of the components on their own are ...
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113 views

Formal identity involving fermion propagator in quantum field theory

I'm studying from here: Roberto Soldati - Field Theory 2. Intermediate Quantum Field Theory (A Next-to-Basic Course for Primary Education) I'm trying to understand and prove an equality at page 52, ...
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Swap fermion with boson?

I wonder what actions/factors/terms show up when you swap a fermion and boson that are tensored together in second quantisation. It would suffice for me if someone could give me the name of such an ...
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35 views

Deriving the Heat capacity from Fermi-Dirac statistics

I was watching the lectures on Solid state physics by Steve Simon (Oxford). He was explaining how to find Heat capacity of metal due to electrons from Fermi-Dirac statistics. You can write the total ...
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Partition function of a two level system

So I have a system made up of $N$ indistinguishable fermions that interact with each other. Also, the system is made up of two energy levels, and their gap is $D$. Therefore, trying to write down ...
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29 views

Double Sommerfeld Expansion

Consider the Fermi-Dirac expansion for an arbitrary function $f(\epsilon)$: $I(f)=\int_0^\infty d\epsilon\frac{f(\epsilon)}{e^{\beta(\epsilon-\mu)}+1}$ The large $\beta$ expansion of this quantity ...
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48 views

Fermion parity operator

Fermion parity operator is defined as $$ \hat{\mathcal{Q}}=\exp(i\pi\sum_j \hat{n}_j) = (-1)^{\sum_j \hat{n}_j} $$ And also if $\sum_j \hat{n}_j = \sum_j c^{\dagger}_{j}c_j=N $ is constant then it ...
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112 views

A question about quarks and quantum chromodynamics

Penrose writes the following on pg 648 of his book "Road to Reality" How can we treat quarks as real particles, if they have the wrong spin-statistics relation? The way that this problem is dealt ...
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35 views

Strange material, Bydrogen

Let's take a hypothetical scenario in which you manage to make a meta material in which the bulk of the mass is made of by a central negative nucleus and that has a single spin 1 boson whizzing around ...
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Interaction between electrons obeying Pauli's exclusion principle

For fermions having half spin, obeying Pauli's exclusion principle, we know that for states with different spin states, the particles (say electrons) are distributed in such a way that if the first ...
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What's the momentum-space vacuum wave-functional of a fermion?

In the Schrödinger picture, the field eigenstates of a real scalar field $\hat\phi(\mathbf x)$ with $\mathbf x \in\mathbb R^3$ are the states $\hat\phi(\mathbf x)|\phi\rangle=\phi(\mathbf x)|\phi\...
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Normal-Ordering of operators of two different types of fermions

The normal-ordering of creation and annihilation operators is defined such that the creation operators are put to the left, with a minus sign for each permutation of two operators necessary to do this....
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98 views

Why are Grassmann variables the classical limit of fermions?

In many texts the anti-commutation relations for fermions are given as $$\{ \bar{\psi}^\alpha (\vec{x}), \psi^\beta(\vec{y}) \} = \delta^{\alpha\beta} \delta(\vec{x} - \vec{y})$$ $$\{ \psi^\alpha (\...
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How is the number density operator for Dirac fermions defined?

In my studies of quantum field theory I stumbled across the question of how to define the number operator for Dirac fermions? I.e. the operator that counts the total number of particle + antiparticles ...
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What's the space of eigenvalues/field configurations for a fermion?

In the Schrödinger picture of quantum field theory, the field eigenstates of a real scalar field $\hat\phi(\mathbf x)$ with $\mathbf x \in\mathbb R^3$ are the states $\hat\phi(\mathbf x)|\phi\rangle=\...
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Fermion commutation with two quantum numbers

I am trying to understand the second quantization formalism. Let's say we have a system of fermions (e.g. electrons) with spin in an array of quantum dots. The creation and annihilation operators $c^{\...
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1answer
42 views

The overlap of two Slater determinant states

Suppose I have two fermionic number states in different bases, with the same particle number $N$ - call them $|\Psi\rangle$ and $|\Phi\rangle$. In the position basis, I can write the many-body ...
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40 views

Anticommutation relation of Grassmann numbers

Let $c,c^*$ be the fermion annihilation/creation operators and $\xi,\xi^*$ denote Grasssmann numbers where $$|\xi\rangle = \exp(-\xi a^*)|0\rangle$$ is the coherent state. Then why is it true that $$ \...
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115 views

What's the ground state wave-functional of a fermion?

The vacuum state, free field wave-functional of a scalar field $\hat\phi(x)$ in the Schrödinger representation of quantum field theory is $$\begin{array}{cl} \Psi_0[\phi] &= C\prod_k e^{-\omega(k)...
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34 views

Density of levels in Fermi-Dirac statistics

We know that $$ u=\int_{-\infty}^{+\infty} D(\mathcal{E})\mathcal{E}f(\mathcal{E})d\mathcal{E} $$ where $D(\mathcal{E})$ is called density of levels per unit volume. My textbook (Kittel) says that: $...
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How to construct general multiparticle states that respect fermionic or bosonic symmetry?

Background: The arena is fixed particle number nonrelativistic quantum mechanics. The state space is $$ \mathbf{H}(1)=\mathcal H\otimes\mathcal S, $$ where $\mathcal H$ is an "orbital" state space ($L^...
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Fermionic thermal density matrix

Usually to describe the density matrix of a system at finite temperature, we use the Euclidean path integral $$\rho[\psi_1,\psi_2] = \int _{\psi_1}^{\psi_2}\mathcal D \psi e^{-S_{E}[\psi]}, $$ where $...
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52 views

Second quantisation for fermions

I am trying to build a model for reactions on a lattice in the Doi-Peliti formalism. Suppose there exists a lattice of $N$ sites indexed by $i$. Each site can be either occupied or unoccupied. ...
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46 views

Lorentz invariant but baryon number violating operators from a single fermion field?

In a theory of a single fermion, is it possible to write down a Lagrangian that violates the global U(1) symmetry (e.g. baryon number) but that is Lorenz invariant? I'm wondering because the only ...
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How are anticommuting fields ($\eta \chi = -\chi \eta$) “forced upon us” by representation theory of $SO(d-1,1)$?

I would like to know if anticommuting fields (which physicists use as fermions) emerge naturally from the spin representation theory of $SO(d-1,1)$. Is the fact that spinor fields anticommute a ...
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Entangled states and separable states

B Two electrons in the same orbital is clearly an entangled quantum state since it is not a tensor product: $$|\psi\rangle=\frac{1}{\sqrt{2}}(|\uparrow\rangle \otimes|\downarrow\rangle-|\downarrow\...
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Is Supersymmetry really swapping fermions with bosons?

I've been studying supersymmetry for the last few months, and while I can do some mathematics with the Wess-Zumino model (show the Lagrangian is invariant under a susy transformation, find the Noether ...
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Density of states for fermions for temperatures other than absolute zero

Let's say I wanted to calculate the density of states for $\mathrm{2D}$ Fermions with an energy dispersion of $E_F=ck^6$ at $T=0$ The process would be quite mathematically simple. I'd take the ...
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67 views

Why is the Jacobian factor for fermionic variables different from that for bosonic ones?

In Srednicki's textbook Quantum Field Theory, Section 77 discusses anomalies and the path integral for fermions. The path integral over the Dirac field is defined to be \begin{equation} Z(A) \equiv \...
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51 views

Spacial Wavefunction Symmetries and Identical particles

I was reading this and it mentions in the 3-electron section, that for a spacial wave function to be symmetric under fermion swapping, it must be a function of even parity. Similarly for anti-symmetry ...
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Connection between $2n$ real fermions and $SO(2n)$

In section 11.4 of "Basic Concepts of String Theory" by Blumenhagen et al, they say: Consider a system of $2n$ two-dimensional real fermion (...) transforming as a vector of $SO(2n)$. I guess they ...
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21 views

Transformation of fermionic creation and annihilation operators

How do the creation and annihilation operators of Dirac fermions transform under a Lorentz transformation whose axis is not parallel with the axis of spin quantization?
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68 views

Bound on fermions in a finite volume?

The Pauli Exclusion Principle says that two or more identical fermions cannot occupy the same quantum state within a quantum system simultaneously. However, I'm wondering if we could potentially pack ...
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256 views

Difference between fermions and bosons in Statistical Mechanics

I am an undergraduate student in Physics and Mathematics. I am now preaparing for my final exam in Statistical Mechanics and I would like some help in a particular point. So here it goes: In the ...
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What is the definition of functions of Grassmann numbers?

I understand there are some relevant questions, but none of them solves my issue. From Atland and Simons (Condensed Matter Field Theory), the definition of functions of Grassmann numbers are defined ...
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1answer
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Anticommutativity of an anticommutator of supercharges

In this paper, equation 38 gives the ${\cal N}=2$ Super-Poincare (extended with the central extension $\mathcal{Z}$). The anticommutation relation of the two different supercharges is given as: $$\{Q^...
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Legal values of spin-1/2 field can take: $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, .. (Grassmann)?

For the spin-1/2 fermion field $\psi$, we may choose it to be a spinor which needs to be Grassmann variable but can also be complex $\mathbb{C}$ Grassmann. (Dirac or Weyl spinor/fermion) We can ...
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145 views

Fermi Dirac distribution derivation

Does anybody understand how my lecturer is normalising the probability distribution at the end to achieve the Fermi Dirac distribution? I don’t understand how he gets 0x1 or the denominator at all.