# Second quantisation for dynamical systems

The paper "Perturbative approach to an $$A + B \rightarrow C$$ reaction-diffusion system", (Z. Phys. B 96, 137-144 (1994)), by Conrad and Trimper, applies the Fock Space formalism for the master equation developed by Doi and applies it to a $$A+B \rightarrow C$$ system.

The probability distribution $$P (n,t)$$ can be related to a state vector $$|F(t)\rangle$$ in a Fock space according to $$P (n,t) = \langle n|F(t)\rangle$$ with the basis vectors $$|n \rangle = | {n_1 n_2 \dots n_i \dots}\rangle$$. As a consequence, the master equation is is re-written as an equation in Fock space $$\begin{equation} \partial_t |F(t)\rangle = \hat{L} |F(t)\rangle \end{equation}$$.

Some other useful relations,

$$|F(t)\rangle = \sum_{n_i} P(n,t) |n\rangle$$

$$\langle \hat{O} (t) \rangle = \langle s|O|F(t)\rangle$$

$$\partial_t\langle \hat{O}(t) \rangle = \langle s|[\hat{O},\hat{L}]|F(t)\rangle$$

$$\langle s| = \sum\limits_{n_i} \langle n|$$

$$\langle s|F\rangle = 1$$

$$\langle s|\hat{L} = 0$$

Now as far as I understand in the paper linked above they define bose annihilation operators $$a_i$$, $$b_i$$, $$c_i$$ at each lattice site, and similarly introduce bose creation operators $$a^\dagger_i$$, $$b^\dagger_i$$, $$c^\dagger_i$$. And number operators $$A_i = a^\dagger_i a_i$$ etc.

The reaction Liouvillean also makes sense to me derived as $$L_R = \eta \sum_i c^\dagger_ia_ib_i - a^\dagger_ia_ib^\dagger_i b_i$$

What I am having a hard time deriving is the time derivative of the expectation values, although the quoted result makes intuitive sense to me.

$$\partial_t \langle C_i \rangle = \eta \langle A_i B_i\rangle$$

The issue I am having is the time derivative given above should be the expectation value of the commutator, $$[c^\dagger_i c_i,c^\dagger_ia_ib_i - a^\dagger_ia_ib^\dagger_i b_i] = c^\dagger_i a_i b_i \neq a^\dagger_ia_ib^\dagger_ib_i$$ according to my computations. Any help would be appreciated

• This formalism is called Doi-Peliti approach. The definition of creation/annihilation operators is different in this method, this might be the reason for discrepancy (for calculating species c, you might have to use operator c). – Sunyam Jun 6 at 21:31
• Hi!@Sunyam the authors seem to have defined $C_i = c^\dagger_i c_i$ in the standard way (text above equation 8 in the linked paper). – jcp Jun 6 at 21:35
• Is $\hat L = L_{R} + L_{\text{Diffusion}}$. If so is $\langle s|L_{\text{Diffusion}}=0$, then $\langle s|L_{R}^{}=0 \Rightarrow \langle s| \eta \sum_i c_i^\dagger a_i b_i = \langle s| \eta \sum_i a_i^\dagger a_i b_i^\dagger b_i$? – Sunyam Jun 6 at 22:09
• Minor comment to the post (v1): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot. – Qmechanic Jun 6 at 22:54
• @Sunyam. Yes that is true for $\hat{L}$. I am not sure about the other relations. I have only just begun studying this formalism. But I would assume in a process without diffusion, these relations should still hold? (I am not sure). – jcp Jun 6 at 23:35