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18

Great, important question. Here's the basic logic: We start with Wigner's Theorem which tells us that a symmetry transformation on a quantum system can be written, up to phase, as either a unitary or anti-unitary operator on the Hilbert space $\mathcal H$ of the system. It follows that if we want to represent a Lie group $G$ of symmetries of a system via ...


13

After the answers by joshphysics and user37496, it seems to me that a last remark remains. The quantum relevance of the universal covering Lie group in my opinion is (also) due to a fundamental theorem by Nelson. That theorem relates Lie algebras of symmetric operators with unitary representations of a certain Lie group generated by those operators. The ...


9

Antiparticles naturally arise when studying the Dirac equation within quantum field theory. Recall that we may expand a Dirac spinor field as a plane wave, namely, $$\psi= \sum_{s=1}^2 \int \frac{\mathrm{d}^3 p}{(2\pi)^3} \frac{1}{\sqrt{2E_{p}}} \left[ b^s_p u^s(p)e^{ipx}+c^{s\dagger}_p v^s(p)e^{-ipx}\right]$$ and similarly for the conjugate field. Notice ...


8

Indistinguishableness of particles is formulated in QM in terms of the total wave function symmetry. Then the wave functions can be symmetric or antisymmetric on their arguments. In case of antisymmetry ($\psi(x_1,x_2) = -\psi(x_2,x_1))$ the particles are called fermions and $\psi(x_1,x_2=x_1) = 0$. They say "the particles cannot occupy the same state". It's ...


8

The exchange interaction is, on the contrary, one of the approximate low-energy consequences of the Pauli exclusion principle and the identical character of the particles. The fundamental justification of these facts is offered by quantum field theory. Relativity requires observables to be linked to regions of spacetime - so that they don't communicate over ...


8

For the partition sum, you have so sum $e^{-E}$ ($T=1$) over all possible eigenstates of the system where $E$ is the energy of the corresponding state. Two bosons can be in the 10 states $|kl\rangle$, with $1\leq k \leq l \leq 4$ where we accounted for the degeneracy by introducing an additional state with $E_4 =2E$. The corresponding partition sum reads ...


8

Just in view of the double universal covering provided by $SU(2)$, $SO(3)$ must a quotient of $SU(2)$ with respect to a central discrete normal subgroup with two elements. This is consequence of a general property of universal covering Lie groups: If $\pi: \tilde{G} \to G$ is the universal covering Lie-group homomorphism, the kernel $H$ of $\pi$ is a ...


7

Majorana fermions are by definition fermions which are their own antiparticles, i.e. the do have spin and it's 1/2. An introduction to these fermions can be for example found here: http://arxiv.org/abs/0806.1690. In contrast bosons are their own antiparticles, e.g. photons, i.e. one does not need a "Majorana-boson" definition. Now, one has to say that these ...


7

How to obtain this braiding matrix from Non-Abelian Chern-Simon theory? To obtain braiding matrix $U^{ab}$ for particle $a$ and $b$, we first need to know the dimension of the matrix. However, the dimension of the matrix for Non-Abelian Chern-Simon theory is NOT determined by $a$ and $b$ alone. Say if we put four particles $a,b,c,d$ on a sphere, the ...


6

The classic place to start would be the book "PCT, Spin & Statistics, and All That", by R.F.Streater and A.S.Wightman. The spin statistics theorem can be proved as rigorously as you likely can want in the context of the Wightman axioms. The difficulty with this statement relative to your question is that we cannot prove that interacting fields satisfy ...


6

How about $\Psi(x_1,x_2) \equiv (x_1 -x_2)e^{-(x_1-x_2)^2}$?


6

BebopButUnsteady answer is obviously correct. Actually most wave functions cannot be written as single Slater determinants and certainly no energy eigenstates of N-fermion systems (atom, molecules or solids) interacting by two-body potentials (like the Coulomb potential). The Slater determinant is just a nice simple approximation to ground state ...


6

I write below the statement of the mentioned theorem which assumes, as hypotheses, the validity of so-called "Wightman axioms" in the four dimensional Minkowski spacetime. You see that there is nothing imposed by hand. It is actually a no-go theorem. Quantizing free fields, it establishes in particular that the standard choice is the only possible. ...


5

You are right. In a space-time with one time dimension and $D$ spatial dimensions, finding possible different statistics is equilalent to look at the fundamental group (first homotopy group) of $SO(D)$ For $D=1$, the fundamental group is trivial. For $D=2$, the fundamental group is $\mathbb{Z}$. For $D>=3$, the fundamental group is $\mathbb{Z}_{2}$ ...


5

The spin-statistics thing isn't a problem, it is a theorem (a demonstrably valid proposition), and it shouldn't be addressed, it should be understood and celebrated. The Higgs field gives us interactions between chiral fermions and the Higgs, $yh\cdot \chi_\alpha\eta^\alpha$ which produces mass terms $m \chi_\alpha\eta^\alpha$ if the Higgs field has a ...


5

On the deepest level, particles are indistinguishable if and only if they have the same quantum numbers (mass, spin, and charges). However, in statistical mechanics one ofte studies effective theories where there are additional means of distinguishing particles. Two important examples: In modeling molecular fluids, two atoms on the same molecule are ...


5

I will firstly point out some apparent misconceptions in the question and subsequently I will explain what goes wrong when quantizing a theory of integer spin fields or particles with anticommutators, and vice versa. First, if we quantize a real Klein-Gordon field using anticommutators, the Hamiltonian is going to vanish (or be a field independent ...


5

Let me start from the beginning, also making explicit what I understand from user35388's answer. Consider a system of a couple of identical particles. Their states are pictured by normalized wavefunctions $\psi(x,y)$. However this representation is not one-to-one (this holds for every quantum system): states are wavefunctions up to phases. That is $\psi$ ...


5

The way Shankar addresses the problem (pg. 278) is by introducing an "Exchange Operator" $P_{1,2}$, which would swap your two particles as follows: $P_{1,2} |\xi_1, \xi_2 \rangle = |\xi_2, \xi_1 \rangle$ I like the operator notation because it makes it clear (to me, at least) that applying the operator twice is just the identity operator, since swapping ...


5

I'd like to add to Josh's answer, because he didn't really explain what a universal covering group is. Essentially, a space $T$ is a covering space of another space $U$ if, for an open subset of $U$, there's a function $f$ that maps a union disjoint open subsets of $T$ to the subset of $U$. Or, more simply worded, pick a piece of your space $U$, and I'll ...


4

Both fractional/non-Abelian statistics and fractional charges come from the same origin: long-range entanglements. This is why fractional/non-Abelian statistics common for fractional charges. One way to realize long-range entanglements is through the string-net liquid phase of a pure bosonic model. The ends of strings in string-net liquid are non-local and ...


4

The (unitary) "phase" factor for non-Abelian anyons satisfies the (non-Abelian) Knizhnik-Zamolodchikov equation: $$\big (\frac{\partial}{\partial z_{\alpha}} + \frac{1}{2\pi k} \sum_{\beta \neq \alpha} \frac{Q^a_{\alpha}Q^a_{\beta}}{z_{\alpha} - z_{\beta}}\big )U(z_1, ....,z_N) = 0 $$ Where $z_{\alpha}$ is the complex plane coordinate of the particle ...


4

Please, let me first refer you to the original paper by Leinaas and Myrheim where the existence of anyonic statistics was first predicted before its actual discovery. All the ingredients of the understanding of the special properties of the two dimensional case already exist in this old paper, however, I'll try to cast it in a more modern terminology: By ...


3

If you put a non-Abelian anyon and its anti particle on a sphere, then moving the non-Abelian anyon around its anti particle only induces an Abelian phase. Also, twisting a non-Abelian anyon by 360$^\circ$ only induces an Abelian phase as well, which define the (fractional) spin of the non-Abelian anyon.


3

Antisymmetric wave functions instantly imply the Pauli exclusion principle – essentially because $\psi(x,x)=-\psi(x,x)=0$, to write the concept schematically – which implies that the occupation numbers are $N=0,1$ and statistical physics is therefore inevitably governed by the Fermi-Dirac statistics which may be derived from Boltzmann/statistical physics for ...


3

Suppose we assume that an object's statistics depend only on its spin and not on whether the object is composite or fundamental. This assumption seems natural, since if it failed, it would be too good to be true -- it would give us a way of finding out about the internal structure of any particle, at all scales, without having to build particle accelerators. ...


2

First of all, the Standard Model doesn't treat bosonic fields as classical. They're quantum mechanical i.e. non-classical, they're just not anticommuting or Grassmann-odd. Second, a consistent theory just requires the relationship between spin and statistics, see e.g. the http://en.wikipedia.org/wiki/Spin-statistics_theorem Combining integer spin with ...


2

If you think anti-commuting field is too arbitrary and do not like anti-commuting field, you may ask do we really need to use anti-commuting field to describe fermions? Can a theory with only bosons have fermionic excitations emerging at low energies? The answer is yes! So we do not need to use anti-commuting fields to describe fermions, and this is true in ...



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