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 ...

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Charge operator for Dirac spinor

In QED, the gauge transformation which acts upon a fermionic field $\psi$ is $$\psi'(x)= e^{i \alpha(x) Q}\psi(x)$$ where $Q$ is the charge operator. Most of the time it's just written as $$\psi'(x)= ...
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26 views

What is the number of fermions in Kitaev honeycomb model?

One way to solve the Kitaev honeycomb model: $$ H = J_x \sum_{\textrm{x links}, <ij> } \sigma^x_i \sigma^x_j + J_y \sum_{\textrm{y links}, <ij> } \sigma^y_i \sigma^y_j + J_z ...
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1answer
44 views

Fermions in a well

I have two identical fermions in an infinite potential well. They are non-interacting. How should I show that the first excited state is four-fold degenerate? Is the wavefunction just the ...
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2answers
76 views

A problem with the loosing identity by fermions and the order of applying operators

This question comes in consequence of another one. I want to stress a problem that none of the answers addressed it. For making my problem more understandable let me first remind a well-known state, ...
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2answers
122 views

Fock Space and fermionic annihilation & creation operators

I have been trying very hard to understand, I am reading Ballentine's book on this topic, but I need help: I realized that I don't understand how many particle states work with the creation & ...
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1answer
23 views

Fermi distribution and ideal gas

I was wondering about the following: If we have ideal gas particles, then $E \ge 0$, so one would expect that the state $E=0$ is occupied with probability one for low temperatures, but this is not ...
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0answers
44 views

Where in nature does a fermionic coherent state occur?

We see evidence of bosonic coherent states everywhere. Lasers and microwave circuits naturally condense into photonic coherent states and resonators do the same except with phonons. A coherent state ...
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2answers
49 views

Multivariable functions of Grassmann numbers

I'm trying to derive the closed form of the fermionic coherent state defined by the relation: $$ f_i|\vec{\eta}\rangle = \eta_i |\vec{\eta}\rangle \tag{4.10} $$ My book (Atland and Simons, Condensed ...
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2answers
112 views

Why do the $1/2$ factor appear in the Majorana mass Lagrangian?

In case of Dirac neutrino there is no $1/2$ factor in the mass Lagrangian but for Majorana type neutrino there is a half factor in the mass Lagrangian.
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2answers
19 views

What would influence two or many photons to collide and create an electron-positron fermion pair?

If there were billions of photons around in nothing - no gravity, no electromagnetic fields, etc., what (if anything) would cause the photons to interact in a way that would cause a collision ...
4
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1answer
58 views

Stress-energy tensor for a fermionic Lagrangian in curved spacetime - which one appears in the EFE?

So, suppose I have an action of the type: $$ S =\int \text{d}^4 x\sqrt{-g}( \frac{i}{2} (\bar{\psi} \gamma_\mu \nabla^\mu\psi - \nabla^\mu\bar{\psi} \gamma_\mu \psi) +\alpha \bar{\psi} \gamma_\mu ...
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1answer
40 views

Ladder operators evolution for fermions

For the free Dirac field we have $$ \psi(x) = \sum_s\int d\Omega_{m}\frac{1}{\sqrt{2}k_0}\left(b(\mathbf{k},s)u(\mathbf{k},s)e^{-ik\cdot x}+d^\dagger(\mathbf{k},s)v(\mathbf{k},s)e^{+ik\cdot ...
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4answers
309 views

Fermion vs. Bosons and particle vs. wave: is there a link?

I'm puzzled since several years on this basic aspect of quantum mechanics. Quantum theory is supposed to describe particle-wave symmetry of our world. It also describes our universe in term of bosons ...
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1answer
149 views

What do we mean when we say Bose condensation? And why can't fermions condense if they don't pair?

In common textbooks, we are told that bosons can condense in a single-particle state because of bose statistics and when the system undergoes a bose condensation, the bose field operator obtains a ...
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0answers
50 views

How to deal with coupled fermion boson operators?

I am a beginner in field theory and I have an exercise where I have a product of coupled fermion boson operators? $$ \hat{b_{l} }^{\dagger}\hat{c_{l^{'}} }^{\dagger}\hat{a_{q} }\hat{b_{l} ...
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1answer
38 views

Density depletion for Fermions

In my recent advanced statistical physics class, I read about the density depletion of Fermions, which are "defending" a given volume around them against other Fermions, while the exchange hole ...
1
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1answer
29 views

Can a Fermi gas and a Bose gas be both at the same pressure and temperature?

The title says it all: can a Fermi gas and a Bose gas be both at the same pressure and temperature? It comes from a quiz about statistical mechanics
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2answers
40 views

In which part of physics can we utilize the rotational antisymmetry of certain wave functions?

Just a quick question - I fail at Googling this topic since I do not remember clearly, in which case (e.g. for what type of particles) is the wave function antisymmetric in terms of spatial rotation ...
3
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2answers
76 views

Helium Nucleus as boson

I am rather perplexed with this fact that though Helium Nucleus is a boson, the particles insides it: protons and neutrons are essentially fermions. How the nucleus which is made by fermions can be a ...
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1answer
39 views

Fermions and Bosons

For fermions $$P-\frac{Nk_BT}{V}\geq 0 $$ and for bosons, $$P-\frac{Nk_BT}{V}\leq 0$$ What can we understand from these results.
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262 views

Modeling non-quantum objects (in finance, sociology etc) using fermionic fields?

Please provide (if any) applications of fermionic field theory in non-physics macro contexts (finance, sociology etc). I see only bosonic fields being used mostly. The only (minor) application of ...
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1answer
60 views

Supersymmetry transformations as coordinate transformations

Usually, a supersymmetry transformation is carried out on bosonic and fermionic fields which are functions of the coordinates (or on a superfield which is a function of real and fermionic ...
4
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1answer
266 views

What is the fundamental representation in field theory?

In field theory we associate to each Gauge theory a continuous group of local transformations (a Gauge group), and then we require\define fermion fields to be irreducible representations belonging to ...
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1answer
30 views

Derivation of Fermi level for T>0

I am working through the derivation of the Fermi level $ \mu_0$ for T>0. However, at one point in the notes I have, it states without any explanation that: $$ \int_0^\infty F'(\epsilon) ...
5
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1answer
120 views

Are the left-chiral and right-chiral yukawa couplings equal?

I guess another way to ask this is: Does the "physical electron" spend EQUAL time being a left-chiral and right-chiral fermion, on average? Clarification: The electron switches between (-1/2 T3, -1Y) ...
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2answers
54 views

Why don't we have particles whose wavefunctions are symmetric wrt one exchange operator and anti-symmetric wrt other exchange operator?

Consider a system with $n$ identical particles. Let the wavefunction of the system be $\psi(r_1,\ldots, r_2)$. Let $P_{a,b}$ represent the exchange operator which exchanges particle $a$ with particle ...
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1answer
28 views

does electron - positron pair annihilation occur when E is negative?

Suppose I release an electron-positron pair from rest at a distance of $r$. Then the particles attract each other and collide. The total energy $E$ is $$E = 2m_ec^2-\frac{e^2}{4\pi\varepsilon_0r},$$ ...
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69 views

Is there a soft Goldstino theorem?

For ordinary spontaneously broken symmetries, you can demonstrate relations between S-matrix elements with a soft goldstone emission and another S-matrix element without the emission. If I break ...
9
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2answers
215 views

Are all fermions massless at high temperatures?

According to the standard model, the electroweak symmetry is unbroken at high temperatures, and therefore all gauge bosons are massless then. But since fermions are said to acquire mass by a different ...
3
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2answers
724 views

Is deuterium a boson or a fermion?

I want to know if deuterium is a fermion or boson. Please give me a descriptive answer. I tried the formula that is the combination of protons and electrons which gives odd number but the answer is ...
0
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2answers
61 views

Transposition of spinors

Suppose we have two 4-components Dirac spinors, that is two non commuting objects, $\psi$ and $\chi$. We know that: $ \bar{\psi} \chi= - \chi \bar{\psi} $ $ \bar{\psi} = \psi^{+} \gamma_0 $ $+=T*$ ...
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4answers
680 views

Why do leptons and quarks mix?

Is the fact that weak eigenstates are not mass eigenstates completely arbitrary? Or is there a deeper reason for the existence of the PMNS and CKM matrices?
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4answers
790 views

What is the Difference between a Lepton and a Fermion?

As the Title Says: I am Wondering what the Difference between a Lepton and A Fermion is. I know they both have an ½ integer spin number e.g. a electron, an atom with an odd mass number such as ...
2
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1answer
62 views

How to calculate the speed of electrons in a metal

According to the Sommerfeld model, the electrons on the Fermi level has the relation $$ \epsilon_F=\frac{\hbar^2k_F^2}{2m_e}=\frac{1}{2}m_ev_F^2 $$ i.e. $\hbar k_F=m_ev_F$ with $k_F=(3\pi^2n)^{1/3}$ ...
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1answer
37 views

The one-loop contribution to a time ordered product of conserved currents

In two dimensions one can define for a Lagrangian describing free Dirac fermions with $N$ associated flavours by $$\mathcal{L}=i\bar{\psi}_i\gamma^\mu \partial_\mu \psi^i $$ and associate vector ...
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1answer
65 views

Normal ordering for a two fermion case

I am trying to understand how normal ordering works. I am considering a system of two photons, with $\hat{f}_i$ and $\hat{f}_i^\dagger$ being the annihilation and creation operators, respectively. I ...
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2answers
205 views

Why doesn't this multiplication of Grassmann variables give the expected result?

Would anyone be able to tell me how srednicki goes from step $(44.29)$ to $(44.30)$? Here is the paragraph: Now let us introduce the notion of complex Grassmann variables via $$\begin{align} ...
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0answers
83 views

Diamagnetism of a degenerate electron gas for weak fields

In the book "Statistical Physics, Part I ($3^{{\rm rd}}$ edition)" by Landau and Lifshitz, at $\S59$ when he treats the diamagnetic part of the magnetisation of a degenerate electron gas for weak ...
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0answers
60 views

What is really interacting in weak interactions?

Only particles with chirality $-1$ do interact weakly. The corresponding eigenstate in the Dirac basis is $ \Psi_L = \begin{pmatrix}f \\ -f \end{pmatrix} = \begin{pmatrix}u_r {\mathrm{e}}^{-imt} \\ ...
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0answers
34 views

Minimization of a quaradic-like expression when calculating the generating functional for free Dirac field

The generating functional for a free Dirac field is $$Z_0[\eta,\bar{\eta}]=\int D\bar{\psi}D\psi \mathrm{exp}\{i\int [\bar{\psi}(x)S^{-1}\psi(x)+\bar{\eta}(x)\psi(x)+\bar{\psi}(x)\eta(x)]dx\}$$ where ...
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0answers
76 views

Geometric interpretation of Grassmann variable

Grassmann variables were introduced to make path-integral formalism easier to handle fermionic (anti-commutating) fields. Mathematically they represent the exterior algebra of forms (or exterior ...
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5answers
2k views

“Velvet way” to Grassmann numbers

In my opinion, the Grassmann number "apparatus" is one of the least intuitive things in modern physics. I remember that it took a lot of effort when I was studying this. The problem was not in the ...
5
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4answers
148 views

Is there some special case where a fermion can mediate a force?

Looking at the comments of this questions Does the gravitino contribute to the gravitational interaction? and even considering that the answers here in this other question Why are all force particles ...
0
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1answer
62 views

Few particle fermion system wavefuction

Suppose I have 3 fermions($\left|\psi_1\right\rangle$, $\left|\psi_2\right\rangle$, $\left|\psi_3\right\rangle$) and a system with 3 states ( $\left|1\right\rangle$, $\left|2\right\rangle$, ...
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1answer
422 views
3
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201 views

Grassmann number representation for fermions

How one can simultaneously represent fermionic operators and its corresponding Grassmann variables, so that all the anticommutation relations between them and also states would take place? $$ ...
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0answers
67 views

Diffeomorphisms and the Dirac action

I have a question concerning fermions in curved space-time. Please read it to the end before suggesting the spin-connection and vierbein-based approach. I heard that there is a special way of ...
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3answers
962 views

Does black hole formation contradict the Pauli exclusion principle?

A star's collapse can be halted by the degeneracy pressure of electrons or neutrons due to the Pauli exclusion principle. In extreme relativistic conditions, a star will continue to collapse ...
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0answers
28 views

Mean-field approach to quantum phase transitions in Fermi systems

I have a basic confusion concerning the mean-field theory of quantum phase transitions in Fermi systems. Consider as an example the BCS theory of superconductivity in a Dirac fermion system, ...
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1answer
124 views

What causes Paulis Exclusion Principle?

Currently I'm taking an astrophysics class and has now come across electron degeneracy. As far as I understand, the reason why white dwarfs and such, does not collapse, is due to this, meaning that ...