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Quantum field theory in $d$ spatial dimensions is equivalent to statistical field theory in $d+1$ spatial dimensions by Wick rotating the path integral $\int \mathcal{D} \phi \, e^{iS}$ to the partition function $\int \mathcal{D} \phi \, e^{-\beta H}$. I'm wondering how the description of gapless modes works on both sides of the correspondence.

It seems to be taken for granted that the existence of gapless modes (which I take to mean a sequence of states whose energy approaches zero) on both sides of this correspondence is equivalent. For example, in these lecture notes a heuristic proof of the Mermin-Wagner theorem is given in $d = 1$ as:

In $d = 1$, the physics is straightforward: there are no gapless modes [in the statistical field theory, because] in the language of quantum mechanics, the spectrum of a particle moving on $S^{N-1}$ is discrete and gapped.

How is this reasoning done, in more detail?

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The $d$ in the beginning of your question is not the same as that of your quote. The quote is comparing ordinary quantum mechanics to statistical field theory in one spatial dimension (it is true Mermin-Wagner also holds in 2 Euclidean dimensions but the quote is explaining it is especially easy to see in 1). So I'll write this answer discussing 1 Euclidean dimension to avoid unnecessary complications.

The partition function on the quantum mechanics side is $$Z=Tr\left(e^{-H\tau}\right)$$ where this corresponds to the statistical field theory in finite volume $\tau$, but in the end we can send $\tau\rightarrow \infty$. This can be expanded in the usual way to write it as a path integral but it is useful to also take the trace with respect to energy eigenstates.

For instance a correlation function for an operator $x$ is something like $$\frac{Tr\left(e^{-H(\tau-\tau_f)}xe^{-H(\tau_f-\tau_i)}xe^{-H\tau_i}\right)}{Tr(e^{-H\tau})}=\frac{\sum_ie^{-E_i(\tau-\tau_f)}\langle i|xe^{-H(\tau_f-\tau_i)}x|i\rangle e^{-E_i\tau_i}}{\sum_i e^{-E_i\tau}}$$$$=\frac{\sum_{i,j}e^{-E_i\tau}e^{-(E_j-E_i)(\tau_f-\tau_i)}|\langle j|x|i\rangle|^2 }{\sum_i e^{-E_i\tau}}\rightarrow \sum_j e^{-(E_j-E_0)(\tau_f-\tau_i)}|\langle j|x|0\rangle|^2$$ where in the last line we are taking the limit $\tau\rightarrow \infty$.

The slowest decaying term in the correlation function (assuming $x$ has zero expectation value) is that of the lowest energy state $j$ for which the matrix element coefficient doesn't vanish. So a gapped quantum mechanics where there is a nonzero $E_j-E_0$ for some minimum $E_j$ corresponds to finite correlation length in the statistical model. If there is no gap, for instance in the quantum mechanics of a free particle, this argument breaks down and there are 'long range correlations' (e.g. power law or logarithmic).

In a quantum field theory where there is a mass gap the exact same argument holds. A field $x$ creates a particle at rest with mass $m$ (and various higher energy states which affect the short range behavior), and this mass gap becomes the correlation length on the statistical side.

The quote is explaining that since the quantum mechanics of a particle on a sphere is gapped (the eigenstates are spherical harmonics), there must be finite correlation length for any field. There are no 'gapless' modes with long range correlations.

This idea of 'gapless' in terms of correlation length is not the same as the definition you proposed. There are configurations of classical Goldstone modes with arbitrarily low free energy that you are summing over in the path integral, but they don't correspond to states on the quantum mechanics side, and correlation functions have a finite correlation length.

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