# Diagonal part of the configuration space of two indistinguishable quantum particles

Why is the configuration space of two indistinguishable particles given by $\frac{M^n-\Delta}{S_n}$? My question is about the $\Delta$.

(Notation: $M$ is the configuration space of 1 particle. $M^n$ is the product space. $\Delta$ is the diagonal part : If $X=(x_i)_{1\leq i\leq n}\in M^n$, $X\in\Delta$ if $x_i=x_j$ for any two indices $i\neq j$. $S_n$ is the symmetry group on $n$ objects.)

I understand the mathematical convenience of removing $\Delta$, but what is the physical reasoning for saying that particles cannot sit on each other?

I looked at Laidlaw and DeWitt, they only say:

[...]Whether or not two point particles can simultaneously occupy the same point in space is not a question that we wish to settle here[...]

Leinaas and Myrrheim removes $\Delta$ saying that these are singular. But real particles like bosons can in fact sit on top of each other.

• According to Leinaas and Myrrheim the configuration space is singular at $\Delta$. They remove $\Delta$ to remove the singularity. Jul 29 '15 at 7:32
• Please add links for the papers you cite. I'm not familiar with either of the papers but managed to find the Leinaas and Myrrheim paper by Googling. I have yet to locate the Laidlaw and DeWitt paper. Jul 29 '15 at 7:34

Some time ago I have asked a similar question; although I have accepted one of the answers, It did not satisfy my main interest concerning the physics of the case when the diagonal $\Delta$ is not removed.

Now I have some more information that I can share with you.

Most of the authors give two reasons for the removal of the diagonal:

1. If we include the diagonal $\Delta$ , then the configuration space becomes an orbifold (The diagonal includes fixed points of the permutation group). (But, there is no conceptual problem in quantizing orbifolds).
2. After the removal of the diagonal, the allowed quantizations for $d>2$ are only of bosonic and fermionic statistics in agreement with experiment ($d$ is the dimension of the single particle configuration space).
3. Some authers also mention the impenetrability of the particles as a justification.

My new understanding is based on:

1. In a recent work N.P. Landsman shows for $d>2$ that although in the case when the diagonal is not removed, there are quantizations corresponding to parastatistics, However, (for $d>2$) all these quantizations can be reformulated as bosonic or fermionic quantizations with internal degrees of freedom.

2. Landsman postpones his treatment of the cases $d=1,2$ for a future work, but there is an older work by Bourdeau and Sorkin: (When can identical particles collide?) arguing that in the case $d = 2$ the discarding of the diagonal $\Delta$, removes legitimate quantization possibilities.