# Tag Info

## New answers tagged fermions

1

When normalized, $A$ is just equal to $1,$ so that $f(E)$ varies between $0<f(E)<1.$ Addendum for the edited question: The prefactor $\frac{2}{(2\pi\hbar)^3}$ crops up in the volume integration of density of states performed in k-space for the computation of number of states $N$ (i.e. all available energy states up to a certain maximum (fermi level) ...

2

The reason your logic fails is because $\psi$ is not simply a Grassmann variable; it is a four-component vector of complex Grassmann numbers (in four dimensions): $$\psi=\left(\begin{array}{c} \theta_1 \\ \theta_2 \\ \theta_3 \\ \theta_4 \end{array}\right)$$ With this knowledge, try computing $\bar{\psi}\psi\bar{\psi}\psi$ and ...

3

The argument is false in four dimensional space. The error is the assumption that you get one Grassman number per spinor. In fact, you get one Grassman number per spinor component! In 4d, spinors have multiple components. (Both Weyl spinors have 2 components, and Dirac spinors have 4.) In 1d space, this is a correct argument. In 2d, it is correct for ...

2

Are fermions non-local objects, in a sense in which gauge bosons are not? As far as I understand, the answer is definitely NO. Fermionic particles are local objects as bosonic ones. Based merely on the non-local form of the bosonized Jordan-Wigner fermions, one cannot conclude that fermions are non-local. Jordan-Wigner transformation, like any ...

-1

Okay, I think I have a semi-convincing picture of this in my head. Both of the other answers contain at least part of the story I wanted; I will put the whole thing here in hopes of feedback and that it is useful to someone else. As SM Kravec points out, fermionic parity is a non-local symmetry of a fermionic system. This suggests, as various people have ...

2

The degeneracy pressure is indeed due to one of the four fundamental forces, but it takes a bit of though to see why. If you put fermions into a box then their energy levels are quantised into the usual energy levels for a particle in a 3D box. So the first two fermions go into the ground state, then next two into a higher energy state and so on. Adding ...

1

A proof of the Nielsen-Ninomiya theorm, basically based on your quite ingenious suggestion is given in section 7 of Elias Kiritsis article on the topological properties of the Berry's phase.

5

The terminology of a mode of a free quantum field $\phi(x)$ comes from writing it as a Fourier transform, often also called mode expansion: $$\phi(\vec x) = \int \frac{\mathrm{d}^3 p}{(2\pi)^3}\frac{1}{\sqrt{2\omega_p}}\left(a(\vec p)\mathrm{e}^{\mathrm{i}\vec x\cdot\vec p} + b(\vec p)^\dagger\mathrm{e}^{-\mathrm{i}\vec x\cdot\vec p}\right)$$ where for a ...

2

The term mode is used to define a particular state of a system and may refer for instance to its spin, wavevector, polarisation, charge etc. If we wanted to create a boson at position $x$ with an up-spin and with wavevector $k$, we may use the field operator $\hat{a}^\dagger(x, k, \uparrow)$ on the vacuum state $\vert0\rangle$. The most clear distinction ...

0

Are fermions intrinsically non-local? Yes, definitely. It's quantum field theory, not quantum point-particle theory. An electron's field is what it is. And that field doesn't have a surface. From a great distance it will be swamped and undetectable, but there is no defineable place where that field stops. When one studies quantum mechanics of more ...

0

Basically, you need to use the Heisenberg-Lagrange-Hamilton approach...starting by the field Lagrangian that leads to Dirac wave equation; then you have to quantize the field by using a mode expansion in which the fermionic field is expressed in terms of the free solutions of the Dirac equation which as you know are spinor solutions. For instance, the ...

7

So what people mean by 'non-local' varies from context to context and person to person. Wen has a very particular meaning to this. 1) In fermionization in $D=1+1$ the Jordan-Wigner fermions are, in the bosonic language, operators supported over many sites. The emergent (mutual)-fermions in the toric code are also supported at the ends of strings. 2) ...

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