Here is a ramifying extract from the Wharton essay
@Soupie referenced first:
". . . Even if we were in possession of a unified physical theory that explained all known experimental and observational data, we would still not be able to definitively answer the question of whether reality was fundamentally continuous or discrete. If our unified theory was based on a continuous ontology, there would always be the possibility that this was simply an approximation for an underlying discrete reality, too fine for our experiments to detect. Conversely, if the unified theory were based on discrete elements, future experiments might one day reveal that those elements emerged from an underlying continuous structure. Examples in this category include solitons (particle-like solutions of non-linear wave equations), or the discrete modes in continuous boundary-value problems, such as laser cavities.
Of course, we are not in possession of a unified theory. Of the two pillars of modern physics, general relativity (GR) is clearly in the continuum camp, but GR fundamentally conflicts with quantum theory (QT), a formalism with both continuous elements (quantum fields) and discrete elements (atomic energy levels, etc.). Naıvely, one might give more weight to GR’s principle-based framework than to the wishy-washy position of QT. But this view is not currently popular. Many physicists think that QT implies a discrete substructure for GR’s continuous spacetime, and the vast majority of the effort in reconciling these pillars lies in “quantizing gravity”, as opposed to finding a GR-compatible revision of QT.
But this conclusion is by no means definitive, especially when one considers that one of the biggest unresolved issues of QT – the “Measurement Problem” – is directly related to this discrete/continuous division. In a simplified nutshell, the QT formalism distinguishes between observations made from outside a system (which yield discrete results) and interactions within a system (which are treated much more continuously).
This isn’t tenable because there’s no objective definition of a “system”; it can always be expanded to include any “outside” observations. The measurement problem is how objectively different mathematical procedures could possibly be associated with an apparently subjective parsing of joint vs. separate systems.
In this essay, I argue that the details of this unresolved tension (between the continuous and discrete aspects of QT) provide us with important clues as to whether nature is fundamentally digital or analog. Far from implying that fundamentally nature is somehow both (a meaningless position that many otherwiserational people have tried to adopt), QT is more likely describing a parameter regime where one of these descriptions is emerging from the other. After all, there are many known examples of discreteness arising from an underlying continuity, and vice-versa. This essay argues that the best analogies to QT are found in the discrete modes of continuous boundary value problems. If the discreteness of QT does indeed arise in this manner, then (together with GR) all of modern physics would be built upon a continuous foundation. 1 1 Whether this foundation would be fundamental, or emergent from a deeper discreteness, would remain to be seen.
Although discreteness only enters QT when one takes a measurement, many popular descriptions give the contrary impression that QT describes discrete entities, regardless of whether or not a “measurement” occurs. This is a fault that lies partially with poor word choices (including the very word “quantum”), but also arises from the natural assumption that measurements merely reveal an underlying reality. Here is a typical quote from Wikipedia [1]:
'A quantum mechanical system or particle that is bound – that is, confined spatially – can only take on certain discrete values of energy. This contrasts with classical particles, which can have any energy.'
This quote implies that a quantum system must always exist in one of several discrete modes, but such a statement is contradicted by the mathematics of QT. The fact is that a quantum system can be in any “superposition” of these different energy states, corresponding to a smooth continuum. An unmeasured single electron in an atom is therefore completely unconstrained by anything discrete, a stubborn fact that is difficult for many students to wrap their heads around after learning about atomic shells and quantum numbers.
Of course, when an energy measurement is made on an atomic electron, discreteness finally rears its head; the electron is never found in a superposition of different energy states, but always has one of several discrete energy values. One could argue that the mathematical structure of quantum mechanics is built around these discrete energies, but the way that these values come about is by solving a continuous boundary value problem (the wavefunction of the electron has to fall off to zero in all directions, sufficiently fast), making this an example of how quantized values naturally emerge when solving such problems in a bounded continuum. (More examples of “boundary-induced quantization” will be covered in Section III.)
Retreating to something that doesn’t have a spacetime representation – like quantum spin – doesn’t help the case for a fundamental discreteness. Inaccurate comments such as: ‘The spin of the electron can only be up or down’ are simply false, unless one is talking about the result of an actual measurement. In QT an electron spin lives on a continuous mathematical surface known as the Bloch sphere. Indeed, there exists a well-known, one-to-one correspondence between the mathematics of an unmeasured (quantum) electron spin and the (classical) polarization of a plane electromagnetic wave.[2, 3] This connection, combined with the clear continuum nature of the latter, makes it untenable to claim that former is built upon a discrete foundation.
The final line of retreat would be that quantum mechanics talks about discrete particles, which is true to a point. Quantum mechanics was designed to discuss a well-defined number of particles; in that sense, a discreteness has been put in by hand. But a more general theory which can handle particle creation and destruction – quantum field theory (QFT) – allows superpositions of different particle number states. (This should be no surprise, as QFT is built upon continuous fields!) Even when it comes to something as basic as particle number, nature is only discrete when we actually take a measurement.
But if things are discrete whenever we look, why wouldn’t they also be discrete when we don’t look? While this classical logic is tempting, it simply doesn’t stand up to the most basic quantum experiments. Take the double-slit experiment, where single particles pass through two closely-spaced slits (one particle at a time), and form a two-slit interference pattern as if the particles were continuous waves. Whenever one looks to see which slit the particle passes through, the interference pattern disappears. It only reappears when we don’t look. Any theory which demanded the same behavior, whether there was a measurement or not, would be unable to explain a vast array of similar observations.
Further insights can be gleaned from imprecise, or “weak” measurements. True, all measurements are imprecise to some extent. But there’s evidence that this non-exactness is not merely due to measurement 3 imprecision. For example, every observed transition between two atomic energy states has a “linewidth”, or range of possible energies, a non-discrete phenomenon that significantly impacts the real-world behavior of lasers. In another example, if one sends particles through a slit, the resulting behavior is only explicable in terms a wave that passes through the entire span of the slit. As one changes the size of the slit, the discreteness somehow automatically compensates. Dialing down the measurement sensitivity, as in recent “weak measurement” experiments, has also been observed to gradually destroy the discreteness predicted by quantum mechanics.[4, 5] It is as if particles somehow know how precisely they are being measured, and conspire to only be as discrete as the measurement precision demands.
This behavior is consistent with the known limit: If one takes away all measurements, QT turns out to be not quantized at all. This is in itself evidence of a continuous foundation for QT – with the enormous elephant in the room being the measurement problem, and the apparent discreteness that this process somehow generates. In the next sections, I’ll talk more about how the act of ‘looking’ could have such dramatic consequences, and how this might be our biggest clue as to what is really going on.
III. BOUNDARY-INDUCED QUANTIZATION
. . . . .
http://fqxi.org/data/essay-contest-files/Wharton_FQX3.pdf
