# Dirac matrix algebra from bosonic creation/anihilation operators?

Using some generic fermionic creation/anihilation operators $$a_i$$ and $$a_i^{\dagger}$$ ($$i, j = 1, 2, 3, \dots, N$$) such that \begin{align} \{ a_i, \, a_j^{\dagger} \} &= \delta_{ij}, \tag{1} \\[1ex] \{ a_i, \, a_j \} = \{ a_i^{\dagger}, \, a_j^{\dagger} \} &= 0, \tag{2} \end{align} it is possible to construct new operators $$\gamma_a$$ (where $$a, b = 1, 2, 3, \dots, 2 N$$) that produces a Clifford algebra on an even dimensional vector space: $$$$\tag{3} \{ \gamma_a, \, \gamma_b \} = 2 \, \delta_{ab}.$$$$ Of course $$\{ A, \, B \} \equiv A B + B A$$. For example: \begin{align} \gamma_i &= a_i + a_i^{\dagger}, \tag{4} \\[1ex] \gamma_{n+i} &=-\, i (a_i - a_i^{\dagger}). \tag{5} \end{align} For an odd dimensional vector space, we could introduce a new $$\gamma_{2n + 1}$$ by multiplication of all the $$\gamma_a$$ together: $$$$\tag{6} \gamma_{2n + 1} = -\, i^n \, \gamma_1 \, \gamma_2 \, \gamma_3 \cdots \gamma_{2 n},$$$$ such that \begin{align} \{ \gamma_{2 n + 1}, \, \gamma_a \} &= 0, &\{ \gamma_{2 n + 1}, \, \gamma_{2 n + 1} \} = 2. \end{align} It is also well known that using a pair of bosonic creation/anihilation operators $$b_1$$, $$b_1^{\dagger}$$, $$b_2$$, $$b_2^{\dagger}$$ combined with the three Pauli matrices, we could reproduce the whole angular momentum theory of quantum mechanics.

Now, I'm wondering if it's possible to built the same Clifford algebra (3) with a generic set of bosonic operators $$b_i$$, $$b_i^{\dagger}$$ (where $$i, j = 1, 2, 3, \dots, N$$) such that \begin{align} [ b_i, \, b_j^{\dagger} ] &= \delta_{ij}, \tag{7} \\[1ex] [ b_i, \, b_j ] = [ b_i^{\dagger}, \, b_j^{\dagger} ] &= 0. \tag{8} \end{align} So is there some special combination of the $$b_i$$, $$b_i^{\dagger}$$ (analogous to (4), (5), maybe non-linear) that could reproduce (3)?

• Mar 8 '20 at 15:06