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each associated with a possible position of the electron. How does this relate to Chapter 2 in which we discussed there being finitely many distinct configurations for particles? To avoid constant qualifications that would be of minimal relevance to the major points I am explaining in this chapter, I have not emphasized the fact, encountered in Chapter 2, that to pinpoint the electron’s location with ever-greater accuracy your device would need to exert ever-greater energy. As physically realistic situations have access to finite energy, resolution is thus imperfect. For the spiked quantum waves, this means that in any finite energy context, the spikes have nonzero width. In turn, this implies that in any bounded domain (such as a cosmic horizon) there are finitely many measurably distinct electron locations. Moreover, the thinner the spikes are (the more refined the resolution of the particle’s position) the wider are the quantum waves describing the particle’s energy, illustrating the trade-off necessitated by the uncertainty principle.

6. For the philosophically inclined reader, I’ll note that the two-tiered story for scientific explanation which I’ve outlined has been the subject of philosophical discussion and debate. For related ideas and discussions see Frederick Suppe, The Semantic Conception of Theories and Scientific Realism (Chicago: University of Illinois Press, 1989); James Ladyman, Don Ross, David Spurrett, and John Collier, Every Thing Must Go (Oxford: Oxford University Press, 2007).

7. Physicists often speak loosely of there being infinitely many universes associated with the Many Worlds approach to quantum mechanics. Certainly, there are infinitely many possible probability wave shapes. Even at a single location in space you can continuously vary the value of a probability wave, and so there are infinitely many different values it can have. However, probability waves are not the physical attribute of a system to which we have direct access. Instead, probability waves contain information about the possible distinct outcomes in a given situation, and these need not have infinite variety. Specifically, the mathematically inclined reader will note that a quantum wave (a wavefunction) lies in a Hilbert space. If that Hilbert space is finite-dimensional, then there are finitely many distinct possible outcomes for measurements on the physical system described by that wavefunction (that is, any Hermitian operator has finitely many distinct eigenvalues). This would entail finitely many worlds for a finite number of observations or measurements. It is believed that the Hilbert space associated with physics taking place within any finite volume of space, and limited to having a finite amount of energy, is necessarily finite dimensional (a point we will take up more generally in Chapter 9), which suggests that the number of worlds would similarly be finite.

8. See Peter Byrne, The Many Worlds of Hugh Everett III (New York: Oxford University Press, 2010), p. 177.

9. Over the years, a number of researchers including Neill Graham; Bryce DeWitt; James Hartle; Edward Farhi, Jeffrey Goldstone, and Sam Gutmann; David Deutsch; Sidney Coleman; David Albert; and others, including me, have independently come upon a striking mathematical fact that seems central to understanding the nature of probability in quantum mechanics. For the mathematically inclined reader, here’s what it says: Let be the wavefunction for a quantum mechanical system, a vector that’s an element of the Hilbert space H. The wavefunction for n-identical copies of the system is thus . Let A be any Hermitian operator with eigenvalues αk, and eigenfunctions. Let Fk(A) be the “frequency” operator that counts the number of times appears in a given state lying in . The mathematical result is that lim. That is, as the number of identical copies of the system grows without bound, the wavefunction of the composite system approaches an eigenfunction of the frequency operator, with eigenvalue . This is a remarkable result. Being an eigenfunction of the frequency operator