# How to understand permutations of particles in Quantum Mechanics?

I'm studying identical particles in Quantum Mechanics and I'm having a hard time to understand the idea of permutations of particles from a mathematical standpoint.

From one intuitive point of view it is quite simples: we have two identical particles and we label them arbitrarily as $$1,2$$. Permuting the particles them means permuting the labels, so that the particle once labeled $$1$$ is now $$2$$ and the particle once labeled $$2$$ is now $$1$$.

Now, mathematically things are more complicated. If the description of each particle alone is given by the state space $$\mathcal{E}$$, it seems at first, that for the two particle system the state space should be $$\mathcal{E}\otimes \mathcal{E}$$.

I know that later on we see that it is a subspace of that, but just to make my point clear, what is important is, it seems that the first $$\mathcal{E}$$ that appears is for particle $$1$$ and the second $$\mathcal{E}$$ that appears is for particle $$2$$.

Now, I'm reading Cohen's book and regarding that the author states the following:

Consider a system composed of two particles with the same spin $$s$$. Here it is not necessary for these two particles to be identical: it is sufficient that their individual state spaces be isomorphic. Therefore, to avoid problems which arise when the two particles are identical, we shall assume that they are not: the numbers (1) and (2) with which they are labeled indicate their natures. For example, (1) will denote a proton and (2), an electron.

We choose a basis, $$\{|u_i\rangle\}$$, in the state space $$\mathcal{E}(1)$$ of particle (1). Since the two particles have the same spin, $$\mathcal{E}(2)$$ is isomorphic to $$\mathcal{E}(1)$$, and it can be spanned by the same basis. By taking the tensor product, we construct, in the state space $$\mathcal{E}$$ of the system, the basis:

$$\{|1: u_i; 2: u_j\}$$

Since the order of the vectors is of no importance in a tensor product, we have

$$|2:u_j;1:u_i\rangle = |1:u_i;2:u_j\rangle.$$

However note that:

$$|1:u_j; 2:u_i\rangle \neq |1:u_i; 2:u_j\rangle, \quad \text{if} \ i\neq j.$$

The permutation operator $$P_{21}$$ is then defined as the linear operator whose action on the basis vectors is given by:

$$P_{21}|1:u_i;2:u_j\rangle = |2:u_i;1:u_j\rangle = |1:u_j;2:u_i\rangle.$$

Now I must confess this doesn't make any sense to me. What this notation $$|1:u_i;2:u_j\rangle$$ means? To say particle $$1$$ is at $$|u_i\rangle$$ and particle $$2$$ is at $$|u_j\rangle$$ is the same as saying that the system is at the state $$|u_i\rangle\otimes |u_j\rangle$$. But I can't have any idea on what this notation he uses means.

So, how to understand this piece of text the author says? How to understand his notation, and specially, how $$P_{21}$$ is rigorously defined. I really can't understand how:

$$|1:u_i;2:u_j\rangle \to |1:u_j;2:u_i\rangle$$

is any different than

$$|u_i\rangle \otimes |u_j\rangle \to |u_j\rangle \otimes |u_i\rangle.$$

So how do we understand this notation and the action of this operator from a mathematical standpoint?

Let us label the state spaces clearly as $\mathcal{H}_1$ and $\mathcal{H}_2$ for the first and second particle respectively and denote the canonical isomorphism sending a state in $\mathcal{H}_1$ of the first particle to the exact same state of the second particle by $\phi : \mathcal{H}_1\to\mathcal{H}_2$. Let us further denote the canonical "flip isomorphism" of the tensor product as $\mathrm{flip}: \mathcal{H}_1\otimes\mathcal{H}_2\to\mathcal{H}_2\otimes\mathcal{H}_1, v\otimes w\mapsto w\otimes v$.
Then $\lvert 1:u_i;2:u_j\rangle$ is the element $u_i\otimes u_j\in\mathcal{H}_1\otimes\mathcal{H}_2$ and $\lvert 2:u_j;1:u_i\rangle$ is its image $u_j\otimes u_i$ under $\mathrm{flip}$ in $\mathcal{H}_2\otimes \mathcal{H}_1$.
In contrast, $\lvert 1:u_j;2:u_i\rangle$ is the image under the exchange map $$P : \mathcal{H}_1\otimes \mathcal{H}_2\to\mathcal{H}_1\otimes\mathcal{H}_2, v\otimes w\mapsto \phi^{-1}(w)\otimes \phi(v).$$ Unlike $\mathrm{flip}$, which is a map between two different (but isomorphic) spaces, $P$ is an automorphism of $\mathcal{H}_1\otimes\mathcal{H}_2$. It has eigenvalues 1 and -1, whose eigenspaces are the spaces of symmetric and antisymmetric tensors, respectively.