Multiple Uses of Infinity in physics

Quantum mechanics provides us with many intriguing examples of our subject; I give three here. First, Schrodinger’s solution of his wave equation for the energy levels of the hydrogen atom involves an argument in which infinity plays the role of a kind of reductio, or proof by contradiction, leading to the rejection of the infinity. Second, that same solution results in an infinite set of energy levels, which are pointedly not rejected. Third, Dirac introduced the (infinite-valued on an infinitesimal interval) delta function because it provided a highly simplifying and intuitively satisfying notation for his vastly influential treatment of quantum mechanics. I briefly summarize each of these uses of infinity below.

In a 1926 paper, Schrodinger solved his famous wave equation for the special case of the hydrogen atom. Along the way he had to set to zero certain series terms, since otherwise they would lead to variables with infinite values. (The remaining terms provide solutions for energy levels of the hydrogen atom that are the familiar Bohr ones that closely match experiment[1]—but not quite close enough; later refinements were needed, including spin and relativistic effects.) So in this case, a variable taking on an infinite value is used as a reason to reject it and instead consider only alternative lines of argument. This of course is not new to Schrodinger but in fact is a common form of argument, applicable whenever the variable in question is something one has reason to think should be finite. I provide this particular example of such a reductio use of infinity here (as opposed to any number of others) simply because it is curious that it arises in the same setting in which the next example occurs. We may refer to this first as a dense physical infinity: a physical variable (that in principle might be measured by means of instruments within certain physical confines) taking on (but perhaps should not do so) an infinite value. This is employed via a reductio to eliminate the infinity (sometimes easily as above, sometimes with enormous effort and controversy as in QFT).

Yet a result of Schrodinger’s argument is that the distinct possible energy levels of the hydrogen atom alluded to above are infinite in number, and in fact a specific formula is derived for the possible energies, En where n = 1, 2,.... This infinitude is not shrugged off as unphysical; each and every En is taken as representing an in-principle possible physical energy for the atom.[2] Indeed, it is the excellent matchup with experiment that makes the Schrodinger result so convincing.[3] Of course, it is similar in kind to the infinitude of possible heights (or potential energies) of a projectile above ground level, which is also not seen as unusual. These perhaps amount, in the end, to little more than the fact that the infinite (unbounded) set of real numbers, R, is taken as the possible range of values for many physical variables (with some limitations as dictated by a given situation—but the infinitude is not in general ruled out). This is a range-of-values physical infinity: a mere listing of possible values, of which there may be infinitely many. Yet it is a possibility that, in some sense, describes (a working picture of) the universe: the universe has in it an unbounded range of allowable values for certain variables.[4]

One way to make these two standard physical uses of infinity more intuitive may be this: if a variable represents a measurable quantity, something that one might detect in an experiment, then the measured value must be finite: we have no means to measure an actual infinity; whereas any—even an infinite—number of finite values might be measured (given enough time). Or: there may be an infinite amount of space, matter, or energy, in the universe; but not right where the measuring instruments are located. Note that we are not taking a stand on such a view; in fact, we are exploring alternative possibilities!

Indeed, one can reason: there may be things physically present that we cannot measure. One such that comes to mind is the wavefunction itself; this is sometimes[5] characterized as the fundamental “reality” of which our measurements ferret out (and even modify) some features but never reveal the full thing in itself. If the wavefunction is really there, yet never fully revealed, why not also infinite energies and other quantities? Or consider space and time (or spacetime) themselves: we never measure all of space or time, by any means. Yet in measuring bits and pieces, we convince ourselves that there is a great deal more, and in the case of some theories even that the universe has an infinitude of such pieces, either extended (range-of-values) or densely packed.

Our third example is Dirac’s delta function. This is in wide use by physicists (and not only in quantum mechanics). Yet the delta function is routinely viewed as a useful fiction, not something to take seriously except as a convenient shorthand for a much more cumbersome and less intuitive set of tools. This mode we then call the useful fiction infinity: we use it but we don’t believe it corresponds to anything physical.[6] Nonetheless, it seems to fall also into the dense mode of infinity.

Thus we have cases where a dense infinity is outlawed (by reductio), and others where it is accepted as a useful fiction; and there are also cases (range-of-values) where infinity is accepted as quite physically sensible. Much of what we are considering here is whether some of the “fiction” cases should perhaps be considered as less fiction and more real physics. Delta functions are one case in point (we shall return to them below) but not the only one.

• [1] E.g., when associated to the spectral lines found by Balmer in 1886.
• [2] A very recent result [9] even derives the famous centuries-old Wallis formula for n from the verysame infinite sequence of hydrogen’s energy levels, something no one had the faintest idea couldhappen, suggesting that the infinitude has yet further significance—although just what that may beis unclear.
• [3] For instance, had Schrodinger’s calculation led instead to a sequence of values for En that stoppedafter n = 20, surely there would have been a frenzied attempt by experimentalists to find twenty-oneenergy levels to test the theoretical result.
• [4] The chapter by Blass and Gurevich in this volume similarly comments on “infinitely many possiblevalues, for example of position or momentum” and the corresponding infinite-dimensional Hilbertspace of such a system’s states. This is closely realted to the idea of an infinite extent of space,which may or may not be the case—but such is not seen as a reason to reject a model outright.Similarly, the infinitely-many possible reference frames in quantum mechanics suggested in [6] isnot suspect on the basis of the infinity involved.
• [5] More so some decades ago; it seems now a minority view.
• [6] This is reminiscent of the early uses of imaginary numbers: they were clearly useful, but it wasfar less clear that such a number could be a thing in any sense available back then. Eventually twodevelopments helped: (i) the observation that imaginary numbers can be interpreted as rotations,and (ii) formal/abstract methodology: if something has a consistent mathematical use, that is allthat is needed in order for it to be an object of mathematical study.