One particularly fascinating example of this I have found is the following. The delta function potential has no effect in nonrelativistic quantum mechanics in spatial dimensions greater than or equal to 4. This was first proven here: https://www.sciencedirect.com/science/article/pii/002212367290033X.

The QFT analog is the result that the $\phi^4$ theory is trivial in spacetime dimensions $d>4$. This was proven here: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.47.1.

The reason I call this analogous to the nonrelativistic result is that if you construct a (nonrelativistic) many body quantum mechanics system with a delta function potential interaction, this will give you an interaction term equivalent to the $\phi^4$ in the QFT.

My question is, is it reasonable for the nonrelativistic result to be, at least suggestive of an analogous QFT result?

Now, the stability of the hydrogen atom in spatial dimensions higher than 3 is, as far as I am aware, an open question. I guess the reason for this question is my wanting for the answer to the stability of the hydrogen atom to have any suggestive power to nonperturbative results of QED in high spacetime dimensions.

  • $\begingroup$ I am only at the beginning of learning field theory, but what I suspect is that it is happening because it is a feature of field theory and not particularly of quantum field theory. Since many-body quantum mechanics is essentially quantum mechanics of non-relativistic fields, it is supposed to uphold general characteristics of field theory. And the triviality of $\phi^4$ theory in $d>4$ seems to me to be a generic field-theoretic feature, for example, the $m^4$ Landau-Ginzburg Hamiltonian in the statistical physics of fields also becomes trivial in $d>4$. $\endgroup$ – Dvij Mankad Feb 24 at 1:09
  • $\begingroup$ @Dvij Mankad Perhaps I should emphasize a bit more. The result cited in the first paragraph of my question holds for one particle quantum mechanics. Since this feature of triviality at spacetime dimensions $d>4$ occurs in all of the examples we have cited, I believe it is more a feature of the type of interaction than a generic feature of field theory. The Landau-Ginzburg model has the same type of interaction as the $\phi^4$ theory. It has more to due I believe with how "local" the interaction is. $\endgroup$ – LucashWindowWasher Feb 24 at 21:26

I think the two theories (non-relativistic fermions in 4+1 and a relativistic scalar in 3+1) are quite different, even though the Feynman diagrams look superficially similar (there is one important difference: in the non-relativistic theory the propagators are not invariant under $p_0\to-p_0$, and the propagators have a direction).

What happens in $(d=4)+1$ non-relativistic field theory is that the particle-particle loop diagram is (see, for example, equ.(83) of https://arxiv.org/abs/nucl-th/0609075 ) $$ {\cal A} \sim \left[\Gamma(1-d/2)\right]^{-1} (-p_0+E_p/2)^{1-d/2}. $$ The Gamma function reflects power divergences in $d>2$. Since the Gamma function has poles at $d=2,4,\ldots$ the scattering amplitude vanishes in those cases. This means that the interacting theory defined in $d=3$ has an uper and lower critical dimension $d=2$ and $d=4$. This can be used to set up and $\epsilon$ expansion around these two cases.

In $\phi^4$ theory the loop function is logarithmically divergent in $(d=3)+1$ dimensions, and the loop amplitude is proportional to $\Gamma(3/2-d/2)$ (note that the Gamma function is in the numerator, not the denominator). This gives the usual logarithmic running, and a Landau pole. The Landau pole implies that for an interacting theory the cutoff cannot be taken to infinity. This seems quite different from what happens in the non-relativistic theory.

There is a similarity, however. I can take the euclidean $\phi^4$ theory in $4-\epsilon$ dimensions and set up an $\epsilon$ expansion to study the interacting theory in 3 dimensions (the Wilson-Fisher fixed point).


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