Ineffective Theory

Harmonic Oscillator Weirdness

The phrase “low-energy Hilbert space” is massively deceptive.

Some background: working with infinite-dimensional vector spaces is hard. We like to truncate them, to obtain a finite-dimensional space, so that operators can be treated as matrices on a computer. The standard truncation is to take some well-understood Hamiltonian, diagonalize it, and consider only the (vector space spanned by the) lowest $N$ eigenstates. As $N$ is increased, the approximation is improved, and the properties of the true (infinite-dimensional) system are recovered in the limit.

Physically, the idea is that high-energy physics can be ignored when you know that the energy of your system is, well, not that high. Since we took only the low-energy states, it seems natural to call the resulting vector space the “low-energy Hilbert space”.

So far, seems fine. Now consider the (not normalized) state of the harmonic oscillator (hamiltonian $H = x^2 + p^2 - 1$): $$ \Psi(x) = \frac{1}{1 + x^2} \text. $$ This is, of course, just the glorious Cauchy-Lorentz distribution. The expectation value $\langle x^2\rangle$ is some finite number. Same for $\langle p^2\rangle$. Therefore, the energy of this state is finite! The expectation value $\langle x^4 \rangle$, though, is infinite: $$ \langle x^4\rangle = \int dx\; \Psi(x)^* x^4 \Psi(x) = \int dx\; \frac{x^4}{x^4 + O(x^2)} “=” \infty \text. $$

But wait! The low-energy Hilbert space (for any finite cutoff) is spanned by states with finite $\langle x^4\rangle$ (and indeed finite expectations of every polynomial in $x$ and $p$). And here we have a low-energy state with an infinite expectation value. What gives?

Well, the state $\Psi$ is not in any low-energy Hilbert space. It has amplitudes (decaying algebraically) for every excited state of the harmonic oscillator. For certain observables (like $x^4$), its physics is dominated by those high-lying states. It can be approximated arbitrarily well within the cutoff, but some qualitative features can never be recovered.

This isn’t just about divergent integrals, either. If $|E\rangle$ is the harmonic oscillator eigenstate with energy $E$, then by considering the state $|0\rangle + \epsilon^2|\frac 1 \epsilon\rangle$, we see that there are arbitrarily low-energy states that don’t lie within any low-energy Hilbert space.

In short: the “low-energy Hilbert space” does not contain all (or most) low-energy states. Conversely, the space of low-energy states is not a vector space, since it fails to be closed under linear combination. Moreover, if you ignore the fact that it’s not a vector space, it appears to be infinite-dimensional; more evidence that it really looks nothing like the space constructed from a truncation of the eigenbasis.

Of course, this is not actually about the harmonic oscillator. The same logic should apply to every quantum system, including field theories.