Let's have real massless scalar field theory:

$$

L = \frac{1}{2}(\partial_{\mu}\varphi )^{2}

$$

Corresponding EOM reads

$$

\partial^{2}\varphi = 0

$$

Let quantize it: for $\pi_{\varphi} = \partial_{0}\varphi$

$$

[\hat{\varphi}(\mathbf x), \hat{\pi}_{\varphi}(\mathbf y)] = i\delta (\mathbf x - \mathbf y), \quad [\hat{\varphi}(\mathbf x), \hat{\varphi}(\mathbf y)] = [\hat{\pi}_{\varphi}(\mathbf x), \hat{\varphi}(\mathbf y)] = 0,

$$

or in terms of creation-destruction operators,

$$

[\hat{a}_{\mathbf p}, \hat{a}_{\mathbf k}^{\dagger}] = \delta (\mathbf p - \mathbf k), \quad [\hat{a}_{\mathbf p}, \hat{a}_{\mathbf k}] = [\hat{a}_{\mathbf p}^{\dagger}, \hat{a}_{\mathbf k}^{\dagger}] = 0

$$

Suppose we turn on mass. Then we have EOM

$$

(\partial^{2} +m^{2})\varphi = 0

$$

with relations

$$

[\hat{b}_{\mathbf p}, \hat{b}^{\dagger}_{\mathbf k}] = \delta (\mathbf p - \mathbf k), \quad [\hat{b}^{\dagger}_{\mathbf p}, \hat{b}^{\dagger}_{\mathbf k}] = [\hat{b}_{\mathbf p}, \hat{b}_{\mathbf k}] = 0

$$

The question: how to derive (at least for zero mode) commutation relations between $a_{\mathbf p}, b_{\mathbf k}$,

$$

[\hat{a}_{\mathbf p}, \hat{b}_{\mathbf k}^{\dagger}] = ?

$$

**My attemption**

My idea is that to rewrite creation operator in terms of $\hat{\varphi}, \hat{\pi}_{\varphi}$:

$$

\hat{a}_{\mathbf p} = \int d^{3}\mathbf x (i\hat{\pi}^{0}_{\varphi}(x) + p^{0}_{0}\hat{\varphi}^{0}(x))e^{-ip^{0}x},

$$

$$

\hat{b}_{\mathbf p} = \int d^{3}\mathbf x (i\hat{\pi}^{M}_{\varphi}(x) + p_{0}^{M}\hat{\varphi}^{M}(x))e^{-ip^{M}x},

$$

where superscripts $M, 0$ denote massive and massless theory correspondingly.

Then

$$

[\hat{a}_{\mathbf k}, \hat{b}^{\dagger}_{\mathbf p}] = \int d^{3}\mathbf x d^{3}\mathbf y e^{ip^{M}x - ik^{0}y}[i\pi^{0}_{\varphi}(x) + k^{0}_{0}\hat{\varphi}^{0}(x), -i\hat{\pi}^{M}_{\varphi}(y) + p^{M}_{0}\hat{\varphi}^{M}(y)]

$$

So the task is "reduced" to definition of canonical commutation relations

$$

[\hat{\varphi}^{0}(\mathbf x), \hat{\pi}^{M}_{\varphi}(\mathbf y)], ...

$$

Is it true that

$$

[\hat{\varphi}^{0}(\mathbf x), \hat{\pi}^{M}_{\varphi}(\mathbf y)] = ie^{-ip^{M}_{0}t + ip^{0}_{0}t}\delta (\mathbf x - \mathbf y)

$$

If yes, how to argue this statement?