Homotopy classes in the path integral

Following the answer to this question about the role of homotopy classes in path integrals, it seems reasonable to me that, when calculating the propagator using the path integral formulation, we should first do the integral separately for paths within the same homotopy class of the configuration space, and then add together these contributions with some weight factors. Mathematically this would look something like

$$ K \, \sim \sum_{\alpha \in \pi_{1}(X)} \chi(\alpha) K^{\alpha}, $$

where $\pi_{1}(X)$ is the fundamental group of the configuration space $X$, $K^{\alpha}$ is the partial amplitude associated with the contributions from all paths within the homotopy class $\alpha$, and the $\chi(\alpha)$ are some weights to be determined.

Now, in their 1970 paper, Laidlaw and DeWitt purport to show that the weights $\chi$ must form a scalar unitary representation of the fundamental group $\pi_{1}(X)$. The proof is not too long, but I won't include it here for the sake of brevity.

Fundamental group of configuration space

For $n$ indistinguishable particles with hardcore interactions in $d$ dimensions, the configuration space is

$$ X = Y(n,d)/S_{n}, $$

where $S_{n}$ is the permutation group, quotiented out because the particles are indistinguishable, and $Y(n,d)$ is the set of all $n$-tuples of vectors in $\mathbb{R}^{d}$ such that no two vectors coincide, i.e.

$$ Y(n,d) = \{ y = (\mathbf{x}_{1},\dots,\mathbf{x}_{n}) : \mathbf{x}_{i} \in \mathbb{R}^{d} \hspace{0.5em} \text{and} \hspace{0.5em} \mathbf{x}_{i} \neq \mathbf{x}_{j} \}. $$

It is well known that the fundamental group of this configuration space is different in $d=2$ compared with higher dimensions, namely

$$ \pi_{1}(X) = \begin{cases} B_{n}, \quad d=2 \\ S_{n}, \quad d>2 \end{cases} $$

where $B_{n}$ is the braid group.

Anyonic statistics

Following Laidlaw and DeWitt's claim that the weights must form a scalar unitary representation of $\pi_{1}(X)$, we note that $S_{n}$ has only two one-dimensional (unitary) representations: the trivial representation $\chi(\alpha) = 1$, and the sign representation $\chi(\alpha) = \pm 1$ depending on the sign of the permutation $\alpha$. The former case corresponds to bosons, while the latter corresponds to fermions. Hence for $d>2$ these are the only possibilities.

However, for $d=2$, we have $\pi_{1}(X) = B_{n}$, which has a whole family of one-dimensional unitary representations parametrized by a single angle $\theta$ as

$$ \chi(\alpha) = e^{i \theta W(\alpha)}, $$

where $W(\alpha)$ is the winding number of the braid $\alpha$. This shows that in 2 dimensions we can get Abelian anyons, namely particles which acquire a phase $\theta \in [0,2\pi]$ as they are moved round each other.

However, it is also well known that in $d=2$ we can also get non-Abelian anyons, which in some sense corresponds to taking the weights to be elements of a non-commutative representation of $\pi_{1}(X) = B_{n}$.

I am aware of how to get non-Abelian statistics for a system of quasiparticles with some energy degeneracy by looking at the non-Abelian Berry phase. However, it seems to me that Laidlaw and DeWitt's result that the weights should be from a one-dimensional rep of the fundamental group limits us to Abelian statistics in the context of the path integral.


How do non-Abelian exchange statistics appear in the path integral formulation, and is this consistent with Laidlaw and DeWitt's result?


Although it is not the only way of description, multiple component or vector valued wave functions can be used to describe quantum systems with internal degrees of freedom. Even more generally, the wave functions can be sections of complex vector bundles $V \rightarrow Q$, where $Q$ is the configuration space.

The reasoning leading to the DeWitt-Laindlaw homotopy theorem holds also in the multiple component case, which can be very easily shown as follows (I am following here Horvathy Morandi and Sudarshan):

In order to be able to be projected to the configuration space $Q$, a vector valued wave function (section of a vector bundle) on the universal covering space $\bar{Q}$ of the configuration space $Q$ must satisfy:

$$\bar{\psi}(g \bar{q}) = U(g) \bar{\psi}(\bar{q})$$

where $\bar{q} \in \bar{Q}$, and $g\in \pi_1(Q)$ is used to translate between points in $\bar{Q}$ projected to a single point $ q \in Q$. The reason for this is that the different points $ g \bar{q}$ correspond to different coordinate patches on $Q$, and the matrices $U(g)$ become transition functions on the fibers the vector bundle $V$. Physically, the transition functions are required to be unitary in order to preserve probability, mathematically, it is because the structure group of a complex vector bundle can be reduced to the unitary group of its rank.

From the energy representation of the propagator on $\bar{Q}$:

$$\bar{K}( \bar{q}_a, t_a, \bar{q}_b, t_b) = \sum_n e^{i E_n (t_b-t_a)} \psi_n(\bar{q}_a) \psi_n(\bar{q}_b) ^{\dagger}$$ We deduce: $$\bar{K}( g_a\bar{q}_a, t_a, g_b\bar{q}_b, t_b) = U(g_a) \bar{K}( \bar{q}_a, t_a, \bar{q}_b, t_b) U(g_b)^{\dagger}$$ Identifying the wave function on $Q$ with the wave function on $\bar{Q}$, restricted to the fundamental domain, then from the relations: $$\psi(q_a, t_a) = \int_{\bar{Q}} d\mu(\bar{q}_b) \bar{K}( q_a, t_a, \bar{q}_b, t_b) \bar{\psi}(\bar{q}_b, t_b) \quad= \sum_{g \in \pi_1(Q)} \int_Q d\mu(\bar{q}_b) \bar{K}( q_a, t_a, g q_b, t_b)U(g)^{\dagger} \psi(q, t_b)$$ From which the propagator on $Q$ can be recognized: $$K (q_a, t_a, q_b, t_b) = \sum_{g \in \pi_1(Q)} K_g(q_a, t_a, q_b, t_b)U(g^{-1})$$ With: $$K_g(q_a, t_a, q_b, t_b) : = \bar{K}(q_a, t_a, g q_b, t_b)$$

Now, any unitary matrix can be written as a holonomy of a Lie algebra valued connection:

$$U(a) = \mathrm{P}(e^{i\int _{\gamma = [a]} A})$$

Since for any closed path in the simply connected space $\bar{Q}$, this holonomy should vanish, then the connection $A$ must be flat. Please see for example the following lecture note by Olivier Guichard , on the bijection between homotopy group representations and flat bundles.

Therefore, non-Abelian flat bundles classify the different DeWitt-Laindlaw homotopy factors in the non-Abelian case (generalizing the Abelian case). Now, we know that flat bundles also parametrize the gauge equivalent space of solutions of the Chen Simons theory. Thus one way to include these homotopy factors in the action is to couple the theory to a Chern-Simons term. This was explicitly done in the following work by Oh.

As I have mentioned, multicomponent wave functions are only one of the ways to describe internal degrees of freedom. Another possibility is to formulate the theory on a coadjoint orbit. In this case, the vector valued wave functions are retrieved after the quantization of the coadjoint orbit into a Hilbert space carrying an irreducible representation. The advantage of this method is that we can work with scalar wave functions but now the configuration space becomes a fiber bundle over $Q$ whose fibers are coadjoint orbits. This choice was actually taken by Oh in the above article when he added the internal degrees of freedom by coupling the system to $S^2$ which is the coadjoint orbit corresponding to $SU(2)$.

It is worthwhile that the resulting flat connections are solutions of the Knizhnik- Zamolodchikov equations centered at the vortices in the anyon locations.


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