The Loophole

Now that it’s clear what’s at stake, it doesn’t take many words to state what’s wrong with the previous argument. It’s simply that we don’t know for sure the equations of effective field theories (RGEs) have solutions which can be analytically continued from high resolution to all lower resolutions.

Landau poles are typical examples. A Landau pole is a divergence in a coupling constant that determines the strength of an interaction. Such a divergence happens, for example in QCD at around 100MeV or in QED at energies far beyond the Planck energy. These poles are clearly non-physical and must mean that the extrapolation for the running of the coupling breaks down because the theories become strongly coupled. And QED of course is believed to be absorbed in a grand unified symmetry long before the Landau pole, which may or may not actually happen.^[XIX]

So, a theory can’t be extended beyond its Landau pole which would mean strong emergence is viable, but also Landau poles shouldn’t be there to begin with because they are not physical. Landau poles, thus, don’t help. But note that just because a function can’t be continued doesn’t necessarily mean it diverges and therefore can be discarded as non-physical. A function can be perfectly regular, indeed be differentiable up to all orders, and still can’t be continued.

A good example for a non-divergent function that can’t be continued is the function f (x) := exp(-1 /x²) for x > 0, which cannot be Taylor-expanded around zero and hence can’t be continued to x < 0. If you haven’t come across this function before, I encourage you to do the Taylor-expansion at zero. You will find it’s just identical to zero at all orders.

Because of this you can complete the function f (x) beyond zero with any other function that has a similar behavior, say, g(x) := a exp(-1 /x²) for any value of a. The combination of both functions (f (x) for x > 0 and g(x) for x < 0) will then be well-defined and differentiable at all orders.

To translate the mathematical example to the physical case, f (x) corresponds to some coupling constant of the effective theory, x correspond to the scale of resolution, and of course the transition would not be at zero, but should be shifted to some finite value, say a distance of a nanometer. But the central conclusion remains: There isn’t a priori any reason why it must be possible to continue the constants of the theory at high resolution to any lower resolution. If you run into a point where the coupling can’t be continued, you will need new initial values that have to be determined by measurement. Hence, strong emergence is viable.

I will admit that this example would be more convincing if I could come up with a system that has a beta-function which actually displays such a feature. I don’t have any such example, and if I had I’d have written a proper paper and not an essay with many pictures and few equations. But I also do not know of any reason why it should not happen.

With this, the ball is back in the court of physicists. The argument that effective field theory proves reductionism even though no one is able to at least derive the properties of an atomic nucleus from QCD undeniably has an air of physicists’ hubris to it. It is thus only fair on those philosophers who like to believe that strong emergence exists that physicists first show that the coupling constants of a quantum field theory can always be continued to low energies for physically realistic systems.

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