# Projection of bosonic and fermionic creation and annihilation operators

For the creation and annihilation operator $$a(\varphi)$$ and $$a^\dagger(\varphi)$$ and the orthogonal projections i would like to understand why the following holds $$$$P_\pm a(\varphi) P_\pm = a(\varphi) P_\pm \,\neq \,P_\pm a(\varphi)$$$$ and

$$$$P_\pm a^\dagger(\varphi) P_\pm = P_\pm a^\dagger(\varphi) \neq \,a^\dagger(\varphi) P_\pm$$$$

I know how $$a(\varphi)$$ and $$a^\dagger(\varphi)$$ are defined, but I am not sure how to the projection operators act on them.

Edit: We defined the projection operators in the following way: $$$$P_+ := \frac{1}{N!} \sum\limits_{\sigma \in S_N} U_\sigma \quad\text{and}\quad P_- := \frac{1}{N!} \sum\limits_{\sigma \in S_N} \text{sgn}(\sigma) U_\sigma$$$$ and annihilation and creation operator as $$$$(a^\dagger(\varphi)\Psi)_N := \sqrt{N}\varphi\otimes \psi_{N-1}$$$$

$$$$(a(\varphi)\Psi)_N := \sqrt{N+1}\langle \varphi, \psi_{N+1}\rangle_1$$$$

• I think you need to define your $P_\pm$. There are many projection operators out there! May 11 '21 at 12:29
• you are absolutely right, i added it May 11 '21 at 13:15
• You should maybe also add the definition of creation/annihilation that you are using May 11 '21 at 14:04
• i did, thanks!! May 11 '21 at 16:04

First of all, the range of $$P_\pm a^\dagger(\varphi)$$ is a subspace of the (anti)-symmetric states (who are stable under $$P_\pm$$), while this is not true of $$a^\dagger(\varphi)P_\pm$$. For example, if $$\psi\neq \varphi$$ is another $$1$$-particle state, then : $$a^\dagger(\varphi)P_\pm \psi =a^\dagger(\varphi) \psi = \sqrt{2} \varphi \otimes \psi$$

is not symmetric.

To prove the other part, let $$\psi_0,\psi_1,\ldots, \psi_n$$ be $$1$$-particle states. Then :

\begin{align} P_\pm a^\dagger(\psi_0) P_\pm \psi_1 \otimes \ldots \psi_n &= \frac{\sqrt{n+1}}{(n+1)!n!}\sum_{\sigma \in \mathfrak S_{n+1}} \sum_{\sigma' \in \mathfrak S_n}\varepsilon^\pm(\sigma)\varepsilon^\pm(\sigma')U_\sigma (\psi_0 \otimes \psi_{\sigma'(1)} \otimes \ldots \psi_{\sigma'(n)}) \\ &= \frac{\sqrt{n+1}}{(n+1)!n!}\sum_{\sigma \in \mathfrak S_{n+1}} \varepsilon^\pm(\sigma\circ\sigma') (\psi_{\sigma(0)} \otimes \psi_{\sigma\circ\sigma'(1)} \otimes \ldots \psi_{\sigma\circ\sigma'(n)}) \\ &= \frac{\sqrt{n+1}}{(n+1)!}\sum_{\sigma \in \mathfrak S_{n+1}} \sum_{\sigma' \in \mathfrak S_n} \varepsilon^\pm(\sigma) (\psi_{\sigma(0)} \otimes \psi_{\sigma(1)} \otimes \ldots \psi_{\sigma(n)}) \\ &= P_\pm a^\dagger(\psi_0) \psi_1 \otimes \ldots \psi_n \end{align}

where we consider $$\sigma' \in \mathfrak S_n \subset \mathfrak S_{n+1}$$ by setting $$\sigma'(0)=0$$.

The results for $$a(\varphi)$$ are just the adjoint of the ones for $$a^\dagger(\varphi)$$.

• thank you very much! May 12 '21 at 12:15