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I am writing my undergrad thesis on the harmonic oscillator on a lattice. So far I have implemented the Metropolis Monte Carlo algorithm to generate trajectories $x_j$ for $0 \leq j < N$, where $N$ is the number of time lattice divisions.

I get a pretty good histogram for $|\psi(x)|^2$ and it matches the expectation from the theory. Now I have to calculate the energy eigenvalues of the system. The ground energy state can be calculated with $E_0 \propto \langle x^2 \rangle$ which has worked out so far.

The first energy, $E_1$, should be computable using the exponential slope of the correlations, Creutz and Freedman (1980) wrote that this would be doable with this: $$ E_1 = \frac{-1}{\Delta \tau} \log\left( \frac{\langle x(0) x(\tau + \Delta \tau)\rangle}{\langle x(0) x(\tau) \rangle} \right)$$

With the $x_j$ that I have in my computations, I can calculate those correlations to some value of $\tau$, limited by the $N$.

My advisor has given me a paper from Blossier et al (2009) where they introduce a correlation matrix like so (2.1):

$$ C_{ij} (t) = \sum_{n=1}^\infty \langle 0 | \hat O_i | n \rangle \langle 0 | \hat O_j | n \rangle^* $$

Where $O_i$ are “some interpolating fields $O_i(x_0)$ already projected to a definite momentum and other quantum number such as parity”.

From that, they show how the eigenvalues to the generalized eigenvalue problem (GEVP) consisting of $C(t)$ and $C(t_0)$ will give the energy eigenvalues, which I am interested in.

Lüscher and Wolff (1990) then write on page 245:

In a numerical simulation, the correlation matrix $C(t)$ can be expected to be computable for some range of $t$, and the basic technical problem then is to extract the levels $W_\alpha$. from $C(t)$.

I do not see how I can get the $C_{ij}(t)$ out of my data. What are my fields $O$ for a simple harmonic oscillator to start with? Are those the ladder operators, so that $O_j$ = $(a^\dagger)^j$?

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I talked to my advisor. The fields are just the powers of $x$. In those, the ladder operators exist, and should give me the desired correlations.

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