Abstract
The concept of infinity took centuries to achieve recognized status in the field of mathematics, despite the fact that it was implicitly present in nearly all mathematical endeavors. Here I explore the idea that a similar development might be warranted in physics . Several threads will be speculatively examined, including some involving nonstandard analysis . While there are intriguing possibilities, there also are noteworthy difficulties.
My thanks for helpful comments and clarifications from: Paulo Bedaque, Juston Brodie, Jeff Bub, Jean Dickason, Sam Gralla, Dan Lathrop, Carlo Rovelli, Ray Sarraga, and two anonymous reviewers—none of whom however is to be blamed for any errors or outrageousnesses that remain.
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Notes
- 1.
In [8] Martin Davis includes a discussion of infinity in mathematics in terms of imaginative powers of our minds (my words, not his), and (partly) justifies this by analogy with physics—somewhat the reverse of my point here, but one I am equally sympathetic to.
- 2.
One prominent example that will not be discussed at any length here are the divergent Feynman integrals (among others) of quantum field theory (QFT). See for instance the excellent Wikipedia entry for Renormalization [28].
- 3.
I can’t resist noting that in roughly 1968-9 Martin Davis mentioned to me that in his estimation a huge unclarity underlay foundational issues in mathematics and in particular set theory: what counts as a thing?
- 4.
That the topic is appropriate to a volume dedicated to Martin Davis, I justify with the observations that (i) Martin helped instill in me a general love for ideas on topics far and wide; and (ii) at least two of Martin’s writings bear on related themes: nonstandard analysis [7] and quantum physics [6]. I note that Rovelli [21] entertains an idea already present in [6], namely that of observer-dependent reference frames in quantum mechanics; and (personal note from Rovelli) this also apparently has come up in writings of Kochen and Isham as well, all after Martin’s contribution appeared. See also [24] for more on this theme.
- 5.
E.g., when associated to the spectral lines found by Balmer in 1886.
- 6.
A very recent result [9] even derives the famous centuries-old Wallis formula for \(\pi \) from the very same infinite sequence of hydrogen’s energy levels, something no one had the faintest idea could happen, suggesting that the infinitude has yet further significance—although just what that may be is unclear.
- 7.
For instance, had Schrödinger’s calculation led instead to a sequence of values for \(E_{n}\) that stopped after \(n=20\), surely there would have been a frenzied attempt by experimentalists to find twenty-one energy levels to test the theoretical result.
- 8.
The chapter by Blass and Gurevich in this volume similarly comments on “infinitely many possible values, for example of position or momentum” and the corresponding infinite-dimensional Hilbert space 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] is not suspect on the basis of the infinity involved.
- 9.
More so some decades ago; it seems now a minority view.
- 10.
This is reminiscent of the early uses of imaginary numbers: they were clearly useful, but it was far less clear that such a number could be a thing in any sense available back then. Eventually two developments 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 all that is needed in order for it to be an object of mathematical study.
- 11.
But see for instance [23].
- 12.
See [27] for an interesting discussion of electrons as black holes. A related set of issues involve the self-force and self-energy of an electron (or any point charge): the field created by a charge affects not only space surrounding the charge but also at the charge location(s) as well. Thus an electron’s field influences it’s own behavior. Similar considerations apply to any particle with non-zero mass: the associated gravitational field should affect the particle itself; see [25].
- 13.
Namely: \(\int _{-\infty }^{+\infty }f_1(x)g(x) = \int _{-\infty }^{+\infty }f_2(x)g(x)\) for all “test” functions g.
- 14.
This is not to say that successful application to non-linear differential equations is an automatic benefit; as noted, it is not the product per se but rather integration properties of products that is at issue.
- 15.
I apologize for introducing the term coat for this; already in use are: monad, haze, cloud, halo. My excuse is that a coat of paint is thin, hugs close to its target, and is not to be touched by other entities (at least while wet).
- 16.
Details can get a bit complicated; see [7].
- 17.
- 18.
See [12] for a rare exceptional—but alas all too preliminary—treatment of NSA’s nonstandard universe itself as having physical significance, in this case to QFT.
- 19.
Further investigation (I am unaware of any work on this topic) may reveal advantages to particular “natural” choices for a delta function in particular applications. For now I simply point out one from Robinson’s book (p. 138): \(\frac{1}{\sqrt{\varepsilon \pi }} \exp (- \frac{x^2}{\varepsilon })\). For real values of \(\varepsilon \) this is just an ordinary Gaussian, which arises quite naturally in many situations, and has very nice mathematical properties. Possibly in the nonstandard realm it will also play a helpful role. Note that this is not claimed to resolve issues about non-linear applications where integration properties of products arise.
- 20.
Note that this means the ball will be a proper subset of the coat, since coats have no boundary; if they did, then for instance \(2\varepsilon _e\) would be outside the coat, which makes no sense for it too is infinitesimal .
- 21.
- 22.
For many purposes; but in QFT for instance this is not quite right.
- 23.
It is no good trying to wriggle out of this by supposing T is an NSA sort of infinity ; that would correspond to v being “almost” the same as c (in the same coat, so that v / c is in the coat of 1). For in fact we need—for the Goudsmit/Uhlenbeck model—that v be even greater than c. And then \(\gamma \) actually has an imaginary value! This leads into the even stranger physics of tachyons.
- 24.
This can actually be given a positive spin (pun intended). The Higgs field endows particles with mass according to whether they are retarded by it—retarded from traveling at light-speed, that is. Particles that are not so retarded are by definition massless!
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Perlis, D. (2016). Taking Physical Infinity Seriously. In: Omodeo, E., Policriti, A. (eds) Martin Davis on Computability, Computational Logic, and Mathematical Foundations. Outstanding Contributions to Logic, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-319-41842-1_9
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