On the universal length appearing in the theory of elementary particles - in 1938

Special relativity and quantum mechanics are characterized by two universal constants, the speed of light, c, and Planck's constant, ℏ. Yet, from these constants one cannot construct a constant of dimension length (or mass respectively as a length can be converted to a mass by use of ℏ and c). In 1899, Max Planck pointed out that adding Newton's constant G to the universal constants c and ℏ allows one to construct units of mass, length and time. Today these are known as Planck-time, Planck-length and Planck-mass respectively. As we have seen in this earlier post, they mark the scale at which quantum gravitational effects are expected to become important. But back in Planck's days their relevance was in their universality, since they are constructed entirely from fundamental constants.

In the early 20th century, with the advent of quantum field theory, it was widely believed that a fundamental length was necessary to cure troublesome divergences. The most commonly used regularization was a cut-off or some other dimensionful quantity to render integrals finite. It seemed natural to think of this pragmantic cut-off as having fundamental significance, though the problems it caused with Lorentz-invariance. In 1938, Heisenberg wrote "Über die in der Theorie der Elemtarteilchen auftretende universelle Länge" (On the universal length appearing in the theory of elementary particles), in which he argued that this fundamental length, which he denoted r₀, should appear somewhere not too far beyond the classical electron radius (of the order some fm).

This idea seems curious today, and has to be put into perspective. Heisenberg was very worried about the non-renormalizability of Fermi's theory of β-decay. He had previously shown that applying Fermi's theory to the high center of mass energies of some hundred GeV lead to an "explosion," by which he referred to events of very high multiplicity. Heisenberg argued this would explain the observed cosmic ray showers, whose large number of secondary particles we know today are created by cascades (a possibility that was discussed at the time of Heisenberg's writing already, but not agreed upon). We also know today that what Heisenberg actually discovered is that Fermi's theory breaks down at such high energies, and the four-fermion coupling has to be replaced by the exchange of a gauge boson in the electroweak interaction. But in the 1930s neither the strong nor the electroweak force was known. Heisenberg then connected the problem of regularization with the breakdown of the perturbation expansion of Fermi's theory, and argued that the presence of the alleged explosions would prohibit the resolution of finer structures:

"Wenn die Explosionen tatsächlich existieren und die für die Konstante r₀ eigentlich charakeristischen Prozesse darstellen, so vermitteln sie vielleicht ein erstes, noch unklares Verständnis der unanschaulichen Züge, die mit der Konstanten r₀ verbunden sind. Diese sollten sich ja wohl zunächst darin äußern, daß die Messung einer den Wert r₀ unterschreitenden Genauigkeit zu Schwierigkeiten führt... [D]ie Explosionen [würden] dafür sorgen..., daß Ortsmessungen mit einer r₀ unterschreitenden Genauigkeit unmöglich sind."

("If the explosions actually exist and represent the processes characteristic for the constant r₀, then they maybe convey a first, still unclear, understanding of the obscure properties connected with the constant r₀. These should, one may expect, express themselves in difficulties of measurements with a precision better than r₀... The explosions would have the effect... that measurements of positions are not possible to a precision better than r₀.")

In hindsight we know that Heisenberg was, correctly, arguing that the theory of elementary particles known in the 1930s was incomplete. The strong interaction was missing and Fermi's theory indeed non-renormalizable, but not fundamental. Today we also know that the standard model of particle physics is perturbatively renormalizable and know techniques to deal with divergent integrals that do not necessitate cut-offs, such as dimensional regularization. But lacking that knowledge, it is understandable that Heisenberg argued gravity had no role to play for the appearance of a fundamental length:

"Der Umstand, daß [die Plancklänge] wesentlich kleiner ist als r₀, gibt uns das Recht, von den durch die Gravitation bedingen unanschaulichen Zügen der Naturbeschreibung zunächst abzusehen, da sie - wenigstens in der Atomphysik - völlig untergehen in den viel gröberen unanschaulichen Zügen, die von der universellen Konstanten r₀ herrühren. Es dürfte aus diesen Gründen wohl kaum möglich sein, die elektrischen und die Gravitationserscheinungen in die übrige Physik einzuordnen, bevor die mit der Länge r₀ zusammenhängenden Probleme gelöst sind."

("The fact that [the Planck length] is much smaller than r₀ gives us the right to leave aside the obscure properties of the description of nature due to gravity, since they - at least in atomic physics - are totally negligible relative to the much coarser obscure properties that go back to the universal constant r₀. For this reason, it seems hardly possible to integrate electric and gravitational phenomena into the rest of physics until the problems connected to the length r₀ are solved.")

Today, one of the big outstanding questions in theoretical physics is how to resolve the apparent disagreements between the quantum field theories of the standard model and general relativity. It is not that we cannot quantize gravity, but that the attempt to do so leads to a non-renormalizable and thus fundamentally nonsensical theory. The reason is that the coupling constant of gravity, Newton's constant, is dimensionful. This leads to the necessity to introduce an infinite number of counter-terms, eventually rendering the theory incapable of prediction.

But the same is true for Fermi's theory that Heisenberg was so worried about that he argued for a finite resolution where the theory breaks down - and mistakenly so since he was merely pushing an effective theory beyond its limits. So we have to ask then if we are we making the same mistake as Heisenberg, in that we falsely interpret the failure of general relativity to extend beyond the Planck scale as the occurence of a fundamentally finite resolution of structures, rather than just the limit beyond which we have to look for a new theory that will allow us to resolve smaller distances still?

If it was only the extension of classical gravity, laid out in many thought experiments (see eg. Garay 1994), that made us believe the Planck length is of fundamental importance, then the above historical lesson should caution us we might be on the wrong track. Yet, the situation today is different from that which Heisenberg faced. Rather than pushing a quantum theory beyond its limits, we are pushing a classical theory and conclude that its short-distance behavior is troublesome, which we hope to resolve with quantizing the theory. And several attempts at a UV-completion of gravity (string theory, loop quantum gravity, asymptotically safe gravity) suggest that the role of the Planck length as a minimal length carries over into the quantum regime as a dimensionful regulator, though in very different ways. This feeds our hopes that we might be working on unraveling another layer of natures secrets and that this time it might actually be the fundamental one.

---

Aside: This text is part of the introduction to an article I am working on. Is the English translation of the German extracts from Heisenberg's paper understandable? It sounds funny to me, but then Heisenberg's German is also funny for 21st century ears. Feedback would be appreciated!