# Bose and Fermi statistics in 1+1 spacetime?

• Spinnor
In summary, there are two types of elementary particles, fermions and bosons, which behave differently according to their statistical behavior in three or more dimensions. However, in 1+1 dimensional space, there is a mapping between bosons and fermions, and the Pauli Exclusion Principle can still operate. This mapping involves creating fermion operators from spin operators and has interesting topological properties. In 2D, anyons can also exist, which have any phase when exchanged and can be abelian or non-abelian. The FQHE is an example of a system where anyons are present. There is also speculation about the existence of anyons in 3D, but this is not yet confirmed.
Spinnor
Gold Member
Do we have Bose like and Fermi like particles (fields) in 1+1 dimensional spacetime, Fermi like particles (fields) that obey the Pauli Exclusion Principle?

For what space dimensions does the Pauli Exclusion Principle operate?

Thanks for any help!

Actually, the system is quite interesting in 1+1D. You can actually map bosons into fermions using the Jordan Wigner transformation. The simplest examples and the quantum Ising and XY models (the spatial lattice is a chain). What you do is create fermion operators from spin operators by introducing a "string", a product of spin operators preceeding the site you are on. This operator helps reproduce fermionic anti commutation relations. It is also highly nonlocal which I relates to very interesting topological properties of fermion ex citations in these systems. It's really a question of how you interpret the system. In some sense in certain situations bosons and fermions in 1+1D are actually the same thing! In the mapping, you see that sigma z (or whatever direction of the field in the quantum Ising model is written as (2n-1)/2 where n is the number operator. So we can interpret the direction of the spin as the presence or absence of a boson. There are these things called "hard core" bosons where only one boson occupies a site at a time (this is the situation which is favored) but they still have bosonic computation relations
Fermions can be interpreted as branch cuts which are analogous to spin domain walls. The conservation of fermion parity has important consequences when considering boundary conditions on a ring. Fermion parity is always conserved.

You can go from fermions to bosons by thinking of fluctuations in electron density as bosonic modes and doing something called bosonization for interacting electron systems. An interesting example is the FQHE edge states which are really like a chiral Luttinger liquid (they only go in one direction determined by the B field. You will find that if you describe density fluctuations as bosons, you can map the system from electrons to boson operators and reproduce the same degeneracies you see in the original electron spectrum! The FQHE is very interesting since you have a very physical picture using the hydrodynamic interpretation of edge states.

In 2D you can look at statistics by the phase you get exchanging particles. You get exp(2pi/n) where produces -1 for fermions +1 for bosons. However, you can also get anyons which actually can have any phase when exchanged. These can obey abelian or Nonabelian statistics. This happens when you consider braiding three or more particles where the order of exchange matters. Anyon quasiparticles appear in the FQHE. They can be abelian or Nonabelian but only abelian have been seen in experiment so far. Anyone have a lot of interesting topological properties.

I have heard you can possibly have anyone in 3D (my professor who actually was one of the main players in developing the theory of the FQHE may have mentioned this in class). However, I do not know how this could happen.

So overall, while this seems like a simple question, it is actually pretty profound. You should take a look at Field theories in condensed matter physics by Eduardo Fradkin and also at the papers he references.

vanhees71 and Spinnor

## 1. What is the difference between Bose and Fermi statistics?

Bose and Fermi statistics are two types of quantum statistics that describe the behavior of identical particles. The main difference between them is that Bose particles can occupy the same quantum state simultaneously, while Fermi particles cannot. This means that Bose particles exhibit a phenomenon known as Bose-Einstein condensation, where a large number of particles all occupy the same quantum state. On the other hand, Fermi particles follow the Pauli exclusion principle, which states that no two identical fermions can occupy the same quantum state.

## 2. How do Bose and Fermi statistics apply to particles in 1+1 spacetime?

In 1+1 spacetime, particles are described by two-dimensional quantum field theories, which are governed by Bose and Fermi statistics. In this context, Bose particles are described by bosonic fields, while Fermi particles are described by fermionic fields. The behavior of these particles is influenced by the statistics they follow, and can lead to phenomena such as Bose-Einstein condensation for Bose particles and the Fermi-Dirac distribution for Fermi particles.

## 3. What are some real-world applications of Bose and Fermi statistics in 1+1 spacetime?

Bose and Fermi statistics have many practical applications in various fields, such as condensed matter physics, quantum computing, and particle physics. In condensed matter physics, Bose-Einstein condensation is studied in systems such as superfluids and superconductors. In quantum computing, fermionic fields are used to represent qubits, the basic units of quantum information. In particle physics, the behavior of fundamental particles is described by Bose and Fermi statistics, and these concepts are crucial in understanding the properties of matter.

## 4. Can Bose and Fermi statistics be generalized to higher dimensions?

Yes, Bose and Fermi statistics can be extended to any number of dimensions. In 1+1 spacetime, particles are described by two-dimensional fields, but in 3+1 spacetime, particles are described by four-dimensional fields. The principles of Bose and Fermi statistics still apply, but the behavior of particles can differ in higher dimensions due to the increased number of possible quantum states.

## 5. How do Bose and Fermi statistics relate to the concept of spin?

Bose and Fermi statistics are closely related to the concept of spin, which is an intrinsic property of particles. Bosons, which follow Bose statistics, have integer spin, while fermions, which follow Fermi statistics, have half-integer spin. This relationship is known as the spin-statistics theorem, which states that particles with integer spin must follow Bose statistics, and particles with half-integer spin must follow Fermi statistics.

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