Simple QFT simulation - how to do it

I would like to write a simple QFT simulation for a free scalar field with a cubic interaction term. However, I got stuck a bit. I will try to describe what I think I understand.

I want to have a look at a field with Lagrangian density

$$\mathcal{L} = \frac{1}{2} \dot{\phi}^2 - \frac{1}{2} \phi'^2 - K \phi^3.$$

Then, for a classical field, the equation of motion (from Euler-Lagrange equation) would be

$$\ddot{\phi} - \phi'' = -3K \phi^2.$$

However, I would like to work with a quantum field. Using the standard procedure of second quantisation, I should construct the Hamiltonian

$$H = \int \mathrm{d} x \left [ \frac{1}{2} \Pi^2 + \frac{1}{2} \phi'^2 + K \phi^3 \right ]$$

where $\Pi = \partial \mathcal{L} / \partial \dot{\phi}$.

Now I should make a transformation $\Pi(x,t) \to \hat{\Pi}(x,t)$ and $\phi(x,t) \to \hat{\phi}(x,t)$. So far so good. Defining commutation relation $[\hat{\phi}(x,t),\hat{\Pi}(x',t')] = i \delta(x-x',t-t')$, I should somehow make contact with the ladder operators $a$ and $a^{\dagger}$. However, I don't really see how. For a free field, I find the free classical wave-like solutions, and then promote the weights of different modes to operators. For a field with interaction, I don't really know what to do. This is my first problem.

Secondly, once I make contact with ladder operators, I can then express any state of the system as a concatenation of ladder operators on the vacuum state $|0\rangle$ such as $a^{\dagger}_{p_2} a^{\dagger}_{p_1} |0\rangle$. A general state should then be described as a superposition of all possible states of all particle numbers

$$|\psi\rangle = \left( \lambda_0 + \sum_{\mathrm{p_1}} \lambda_1(p_1) a^{\dagger}(p_1) + \sum_{\mathrm{p_1,p_2}} \lambda_2(p_1,p_2) a^{\dagger}(p_2) a^{\dagger}(p_1) + \begin{bmatrix}\mbox{higher particle}\\\mbox{number states}\end{bmatrix} \right) |0\rangle.$$

Such state has parameters functions $\lambda_0$, $\lambda_1(p_1)$, $\lambda_2(p_1,p_2)$ etc. giving amplitudes of different particular particle number states with different momenta.

Now, I am confused as to whether the evolution of the system should be entirely contained in the $\lambda$s as functions of time, or whether the ladder operators should evolve in time. In any case, how can I get the equation of motion of the QFT system? This is my second problem.

I would be grateful if anyone could help me with this. In particular, my goal is to see the functions $\lambda$ evolving in time on a computer.

Thanks a lot.

SSF

• Your first commutation relations are wrong. They should be at equal time, i.e $[{\hat \phi} (t,x) , {\hat \Pi}(t,x') ] = i \delta(x-x')$. – Prahar Nov 3 '15 at 17:40
• I get the $i$ bit - that was a mistake. But why should I write them manifestly at the same time. After all, the objects $\phi$ and $\Pi$ in classical field theory were functions of both space and time. – SSF Nov 3 '15 at 18:19
• In canonical quantization, one quantizes the theory on an arbitrary time slice $t$. One then defines fields and their conjugate momenta on this time slice and finally, imposes "equal-time commutation relations" on this time slice. Everything is done on a single time slice. To move between slices, you can then use the time-evolution operator. – Prahar Nov 3 '15 at 18:33
• Indeed, one of the issues people have with the Hamiltonian formulation is that it breaks manifest Lorentz invariance by picking out a special time as described above. – Okazaki Nov 3 '15 at 18:58
• Note that the system you are simulating is unstable. It has no vacuum. You're going to get a mess if you try to simulate it on a computer. Try starting instead with the simple harmonic oscillator, considered as a 1d QFT. – user1504 Nov 3 '15 at 20:26

• I see. I thought that there might be a general way - without the need for a perturbative series solution. The same way, when in ordinary QM, I can just use $e^{-i \hat{H} t} \approx 1 - i\hat{H}t$ and evolve the system step by step, even without knowing the actual solution to $\hat{H}$. – SSF Nov 10 '15 at 9:16