Brief introduction to

Doi-Peliti Field Theory

Doi-Peliti Field Theory is a method to describe stochastic processes which can be cast in a Master Equation. It is an equivalent representation to the Master Equation and does inprinciple not involve simplifications or approximations which can occur when deriving Fokker-Planck Equations or Langevin Equations. The reason why I present some introductory information about Doi-Peliti Field Theory here is that there are not many textbook-level notes avilable and easily accessible. Here are my lecture notes on the topic from the course that I created and taught in Part III at Cambridge in 2020 (online) and 2021 (in-person). What follows below is a super-quick overview. The derivation of Doi-Peliti Field theory follows three steps:

Set-up of Master Equation
Using Second Quantization to derive an equivalent Equation of the Probability Generating Function
Deriving the Action of the Field Theory

A useful introductory example is the system in which particles undergo an extinction process and a spontaneous creation process. We use \(\epsilon\) as extinction rate and \(\gamma\) as spontaneous creation rate. Then the three derivation steps work out as follows:

Master Equation

Let’s denote the probability that the system contains \(n\) particles at time \(t\) by \(P(N=n,t)\). Then the Master Equation is

\[\frac{\partial}{\partial t}P(N=n,t)=\epsilon\Bigl((n+1)P(N=n+1,t)-nP(N=n,t)\Bigr)+\] \[\qquad\quad+\gamma\Bigl(P(N=n-1,t)-P(N=n,t)\Bigr)\]

Although we only wrote one equation, the Master Equation is a system of infinitely many linear coupled ordinary differntial equations - one equation for every \(n\in\mathbb{N}_0\).

The Master Equation could be solved directly, but here we want to illustrate how to derive a Doi-Peliti field theory in principle.

Second Quantization

Second Quantization is a language that hides some of the complicated functions we are dealing with behind nice notation. If the system contains \(n\) particles, we write \(\vert n\rangle\). We introduce operators \(a\) and \(a^\dagger\) which have the following properties:

\[[a,a^\dagger]=aa^\dagger-a^\dagger a=1\] \[a|n\rangle=n|n-1\rangle\] \[a^\dagger|n\rangle=|n+1\rangle\]

The operators allow us to increase or decrease the number of particles in the system one particle at a time, much like climbing up or down a ladder. The operators are therefore called ladder operators. Given their effect on the particle number, \(a\) is called annihilation operator and \(a^\dagger\) is called creation operator. We can write the probability generating function of the system as follows

\[|\mathcal{M}(t)\rangle=\sum\limits_{n=0}^\infty P(N=n,t)|n\rangle.\]

Taking its time-derivative and using the Master Equation to replace the occuring time-derivatives of \(P\), we find

\[\frac{\partial}{\partial t}|\mathcal{M}(t)\rangle=\underbrace{\Bigl(\epsilon(1-a^\dagger)a+\gamma(a^\dagger-1)\Bigr)}_{=:\mathcal{H}[a^\dagger,a]}|\mathcal{M}(t)\rangle\]

The Action of the Field Theory

Once the equation for \(\partial\vert\mathcal{M}(t)\rangle/\partial t\) is derived the action of the field theory can be readily written down as follows

\[\mathcal{A}[\widetilde\phi,\phi]=\int\widetilde\phi(-\partial_t)\phi+\mathcal{H}[\widetilde\phi+1,\phi]\,\text{d}t\]

\(n\)-point correlation functions can then be calculated from the Path Integral

\[\langle\bullet\rangle=\int\mathcal{D}[\widetilde\phi,\phi]\bullet e^{\mathcal{A}[\widetilde\phi,\phi]}\]

For example, in order to calculate the mean number of particles in the steady state, we calculate

\[\mathbb{E}[N]=\langle\phi(t)\rangle=\int\underbrace{\frac{\delta(\omega+\omega')e^{-i\omega t}}{-i\omega+\epsilon}}_{\text{propagator}}\underbrace{\gamma\delta(\omega')}_{\text{source}}\text{d}\omega'\text{d}\omega=\frac{\gamma}{\epsilon},\]

which would be represented by this admittedly simple Feynman diagram

In particular, the Feynman diagrams of a Doi-Peliti Field theory are read from right to left.