# Is the "particle number" of "electrons" well defined in Wen's string-net theory of elementary particles?

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According to professor Wen's string-net theory, electrons can be viewed as the elementary excitations of string-net objects. Just like the phonons and magnons are the elementary excitations of crystals and spin systems, respectively.

But as we know, the "particle number" of phonons and magnons are not conserved since the statistical ensembles of crystal and spin-system are both canonical ensembles, so the"particle number" of phonons and magnons are not well defined. Of course there are no chemical potentials $\mu$ for them.

So I want to know whether the "particle number" of "electrons" is well defined in Wen's string-net theory? And what kind of statistical ensemble does the string-net system belong to,a canonical ensemble or a grand canonical ensemble?

Thank you very much.

This post imported from StackExchange Physics at 2014-03-09 08:48 (UCT), posted by SE-user K-boy
asked Mar 1, 2013
welcome to physics.se You should give a link or at least a reference to a publication on this theory. In principle if a theory has fermions and bosons then there should be no problem with the electrons being countable, as is the same in QFT. It is bosons that are unlimited in particle number

This post imported from StackExchange Physics at 2014-03-09 08:48 (UCT), posted by SE-user anna v
Thank you,I have added a reference. I think fermions can also be unlimited in particle number, e.g., when we deal with one dimensional spin-1/2 Heisenberg chain, we can use Jordan-Wigner transformation to fermionize the Hamiltonian, in this case, the elementary excitations(fermions) are unlimited in particle number.

This post imported from StackExchange Physics at 2014-03-09 08:48 (UCT), posted by SE-user K-boy
Here is a link for the preprint arxiv.org/abs/cond-mat/0407140

This post imported from StackExchange Physics at 2014-03-09 08:48 (UCT), posted by SE-user anna v
What distinguishes fermions from bosons is that the fermion number is conserved (lepton number conservation) and that two fermions cannot occupy the same energy level. In contrast there is no limit to the number of bosons that can occupy an energy level .By "particle number' do you mean "lepton number'?

This post imported from StackExchange Physics at 2014-03-09 08:48 (UCT), posted by SE-user anna v
Oh,maybe I didn't express the phrase clearly.Yeah, your point of view is right in particle physics.But in condensed matter physics, fermions and bosons are not limited to "elementary particles" in particle physics, instead, they always appear as quasiparticles(elementary excitations) in various quantum systems,like phonons. So the "particle number" I mentioned here is not limited to "lepton number". As I mentioned previously, the elementary excitations in one dimensional spin-1/2 Heisenberg chain are just quasiparticles, not any kind of leptons in particle physics.

This post imported from StackExchange Physics at 2014-03-09 08:48 (UCT), posted by SE-user K-boy
Charge is conserved, but the number of electrons isn't. This is true both in Wen's theory and (so far as we know) in the actual universe. In Wen's theory, every particle has an antiparticle (which might be itself) and you can always create a particle-antiparticle pair out of the vacuum.

This post imported from StackExchange Physics at 2014-03-09 08:48 (UCT), posted by SE-user Peter Shor
@ Peter Shor ，thanks for your reply. So do you mean that in Wen's theory, the chemical potential $\mu$ for electrons is not $0$ as in our conventional theories of electrons, right ?

This post imported from StackExchange Physics at 2014-03-09 08:48 (UCT), posted by SE-user K-boy

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