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Bose and Fermi statistics in 1+1 spacetime?

  1. May 11, 2015 #1
    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!
     
  2. jcsd
  3. May 11, 2015 #2
  4. May 13, 2015 #3

    radium

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    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.
     
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