Permutation operators

For a two-identical particles system we can define the permutation operator $P_{21}$$P_{21}$ as the linear operator whose action on the basis vector is given by:

$${\begin{aligned}P_{21}\left|u_{i}\right\rangle _{1}\left|u_{j}\right\rangle _{2}=\left|u_{i}\right\rangle _{2}\left|u_{j}\right\rangle _{1}\\P_{21}\left|u_{i}(1),u_{j}(2)\right\rangle =\left|u_{i}(2),u_{j}(1)\right\rangle \end{aligned}}$$

$${\begin{aligned}P_{21}\left|u_{i}\right\rangle _{1}\left|u_{j}\right\rangle _{2}=\left|u_{i}\right\rangle _{2}\left|u_{j}\right\rangle _{1}\\P_{21}\left|u_{i}(1),u_{j}(2)\right\rangle =\left|u_{i}(2),u_{j}(1)\right\rangle \end{aligned}}$$

. This operator satisfies the following properties:

• it is hermitian: $P_{21}=P_{21}^{\dagger }$$P_{21} = P_{21}^{\dagger}$
• $P_{21}^{2}=\mathbb {1}$$P_{21}^2 = \mathbb{1}$
• from the previous properties it follows that $P_{21}$$P_{21}$ is unitary: $P_{21}^{\dagger }P_{21}=P_{21}P_{21}^{\dagger }=\mathbb {1} \rightarrow P_{21}^{\dagger }=P_{21}^{-1}$$P_{21}^{\dagger}P_{21}=P_{21}P_{21}^{\dagger}=\mathbb{1} \rightarrow P_{21}^{\dagger}=P_{21}^{-1}$

For a number $N>2$$N>2$ of identical particles the situation is more complex. There are indeed $N!$$N!$ permutation operators, which we'll call $P_{\alpha }$$P_{\alpha}$, with $\alpha$$\alpha$ referring to an arbitrary permutation. It is simple to prove that the permutation operators do not commute with each other. Furthermore, the properties which we have seen to be valid for $P_{21}$$P_{21}$ are not necessary respected by $P_{\alpha }$$P_{\alpha}$. However we can define transposition operators as linear operators which exchange the roles of two particles without changing the others. Of course the pecularity of $N=2$$N=2$ is that the only permutation operator is also a transposition operator. This is why it has the properties previously written. Since any permutation operators can be broken down into the product of transposition operators, which are unitary, $P_{\alpha }$$P_{\alpha}$ are also unitary. However they are not necessary hermitian and the decomposition is not unique, even if it can be shown that the parity of transposition operators that constitues a permutation operator $P_{\alpha }$$P_{\alpha}$ is always the same. This allows us to define a parity for permutation operators,according to the eveness or oddness of the number of transposition operators in their decomposition. Since permutation operators do not commute with each other it is not possibile to form a basis of common eigenvectors. Nevertheless there exist special kets which are simultaneously eigenvectors of all permutation operators. These kets are the ones completely symmetric and completely antisymmetric with respect to all permutation operators:

$${\begin{aligned}{\mbox{completely symmetric ket}}\quad P_{\alpha }\left|\phi _{S}\right\rangle =\left|\phi _{S}\right\rangle \quad \forall {\mbox{permutation }}\alpha \\{\mbox{completely antisymmetric ket}}\quad P_{\alpha }\left|\phi _{A}\right\rangle =\epsilon _{\alpha }\left|\phi _{A}\right\rangle \quad \epsilon _{\alpha }={\begin{cases}+1\quad \forall {\mbox{even permutation}}\\-1\quad \forall {\mbox{odd permutation}}\end{cases}}\\\end{aligned}}$$

$${\begin{aligned}{\mbox{completely symmetric ket}}\quad P_{\alpha }\left|\phi _{S}\right\rangle =\left|\phi _{S}\right\rangle \quad \forall {\mbox{permutation }}\alpha \\{\mbox{completely antisymmetric ket}}\quad P_{\alpha }\left|\phi _{A}\right\rangle =\epsilon _{\alpha }\left|\phi _{A}\right\rangle \quad \epsilon _{\alpha }={\begin{cases}+1\quad \forall {\mbox{even permutation}}\\-1\quad \forall {\mbox{odd permutation}}\end{cases}}\\\end{aligned}}$$

The symmetrization postulate

The set of $\left|\phi _{S}\right\rangle$$\left|\phi_S\right\rangle$ constitutes a vector subspace $\left(\xi _{S}\right)$$\left(\xi_S\right)$ of the state space, as well as the set of $\left|\phi _{A}\right\rangle$$\left|\phi_A\right\rangle$ $\left(\xi _{A}\right)$$\left(\xi_A\right)$. However the state space $\xi$$\xi$ is not the direct sum of $\xi _{S}$$\xi_S$ and $\xi _{A}$$\xi_A$. There exist indeed kets which aren't neither completely symmetric nor antisymmetric. However the symmetrization postulate saves us, since it states that in system composed of identical particles physical kets are, depending on the nature of the identical particles,either completely symmetric or completely antisymmetric with respect to permutation of these particles. Those particles whose physical kets are completely symmetric are called bosons and those for which physical kets are completely antisymmetric are called fermions. This postulate limits the state space: it is no longer the tensor product of the state spaces of each identical particle, but just a subspace, $\left(\xi _{S}\right)$$\left(\xi_S\right)$ for bosons and $\left(\xi _{A}\right)$$\left(\xi_A\right)$ for fermions. A never contradicted empirical rule enable us to identify fermions with particles of half-integral spin (elecrtons,positrons,neutrons,protons,muons,etc) and bosons with particles of integral spin (photons,mesons,etc). This is called spin-statistics theorem and it is proved in quantum field theory.

Permutation operators and exchange degeneracy

Let $\left|u\right\rangle$$\left|u\right\rangle$ be a ket which can mathematically describe the state of a physical system of N identical particles. Then for all $\alpha$$\alpha$ $P_{\alpha }\left|u\right\rangle$$P_{\alpha}\left|u\right\rangle$ is also a proper ket for the description of the system state, as well as any ket belonging to the subspace $\xi _{u}$$\xi_u$ spanned by $\left|u\right\rangle$$\left|u\right\rangle$ and every $P_{\alpha }\left|u\right\rangle$$P_{\alpha}\left|u\right\rangle$. Depending on the ket $\left|u\right\rangle$$\left|u\right\rangle$ the dimension of $\xi _{u}$$\xi_u$ can vary between 1 and $N!$$N!$. If this dimension is $>1$$>1$ there is an exchange degeneracy since several mathematical kets correspond to the same physical state. The new postulate restricts the class of mathematical kets which are proper description of a physical state: they have to belong to $\xi _{S}$$\xi_S$ or $\xi _{A}$$\xi_A$ according to the nature of identical particles. In order to prove the uniqueness of the mathematical ket able to describe a physical system (we can call it physical ket) we define the projectors on $\xi _{S}$$\xi_S$ and $\xi _{A}$$\xi_A$ as:

$${\begin{aligned}S={\frac {1}{N!}}\sum _{\alpha }P_{\alpha }\\A={\frac {1}{N!}}\sum _{\alpha }\epsilon _{\alpha }P_{\alpha }\end{aligned}}$$

$${\begin{aligned}S={\frac {1}{N!}}\sum _{\alpha }P_{\alpha }\\A={\frac {1}{N!}}\sum _{\alpha }\epsilon _{\alpha }P_{\alpha }\end{aligned}}$$

They are also called symmetrizer and antisimmetrizer since they act on an arbitrary ket belonging to $\xi$$\xi$ and give respectively a completely symmetric and completely antisymmetric ket, belonging to $\xi _{S}$$\xi_S$ and $\xi _{A}$$\xi_A$. These property can be written as:

$${\begin{aligned}P_{\alpha }S\left|\phi \right\rangle =S\left|\phi \right\rangle \quad S\left|\phi \right\rangle {\mbox{ is completely symmetric}}\\P_{\alpha }A\left|\phi \right\rangle =\epsilon _{\alpha }A\left|\phi \right\rangle \quad A\left|\phi \right\rangle {\mbox{ is completely antisymmetric}}\\\end{aligned}}$$

$${\begin{aligned}P_{\alpha }S\left|\phi \right\rangle =S\left|\phi \right\rangle \quad S\left|\phi \right\rangle {\mbox{ is completely symmetric}}\\P_{\alpha }A\left|\phi \right\rangle =\epsilon _{\alpha }A\left|\phi \right\rangle \quad A\left|\phi \right\rangle {\mbox{ is completely antisymmetric}}\\\end{aligned}}$$

They commute with each $P_{\alpha }$$P_{\alpha}$ and the following properties hold:

$${\begin{aligned}P_{\alpha }S=SP_{\alpha }=S\\P_{\alpha }A=AP_{\alpha }=\epsilon _{\alpha }A\end{aligned}}$$

$${\begin{aligned}P_{\alpha }S=SP_{\alpha }=S\\P_{\alpha }A=AP_{\alpha }=\epsilon _{\alpha }A\end{aligned}}$$

Then

$${\begin{aligned}S\left|u\right\rangle =SP_{\alpha }\left|u\right\rangle \\A\left|u\right\rangle =\epsilon _{\alpha }AP_{\alpha }\left|u\right\rangle \end{aligned}}$$

$${\begin{aligned}S\left|u\right\rangle =SP_{\alpha }\left|u\right\rangle \\A\left|u\right\rangle =\epsilon _{\alpha }AP_{\alpha }\left|u\right\rangle \end{aligned}}$$

We have proved that the action of $S$$S$ and $A$$A$ on kets spanning $\xi _{u}$$\xi_u$ give collinear kets, which correspond to the same state. Finally we can state that there is a unique (within a constant factor) ket belonging to $\xi _{u}$$\xi_u$, called physical ket, which is able to describe the physical state of the system: $S\left|u\right\rangle$$S\left|u\right\rangle$ for bosons and $A\left|u\right\rangle$$A\left|u\right\rangle$ for fermions.