Quantum of Solace

atomic orbitalsWhat does quantum mean? With a new Hollywood blockbuster just out, and containing the word “quantum” in the title, it seems we’d better get to the bottom of this.

The word “quantum” was substantively introduced into atomic physics by Einstein, in his 1905 paper (PDF) on the thermodynamics of radiation. (Einstein won the 1921 Nobel Prize in physics for this work.) The meaning of “quantum” in this paper is clear: Einstein describes heat radiation as behaving thermodynamically as if it were made up of “energy quanta” — discrete chunks of energy of exceedingly small size size hf (where f is frequency and h is Planck’s constant, the latter being equal to about 0.0000000000000000000000000000000006). In other words:

    Meaning #1: quantum = very small chunk.

However, when someone refers to a “quantum theory” today, it almost always means much more than Meaning #1. As I’ve mentioned before, we can point to three categories of phenomena that can be tested in a laboratory, which in large part form the empirical basis of quantum theory. They are:

(1) particle diffraction;
(2) superposition; and
(3) discrete energy spectra.

Einstein’s original use of the word “quantum” was an instance of item (3). But today, when one often refers to a “quantum particle,” “quantum tunneling,” “quantum teleportation,” and the like, what is meant (broadly speaking) is:

    Meaning #2: quantum = exhibiting properties (1), (2) and (3).

Of course, this characterization is perhaps heavy on the side of the experimentalist. The theoretician might prefer to think of “quantum” as referring to the structure of a typical quantum theory. Unfortunately, it’s not that easy to say precisely what that structure must be like. For example, it’s certainly not a simple matter of casting one’s theory in terms of bounded operators on Hilbert space, since Koopman and von Neumann showed that this is also possible for classical theories. But I think it’s fair to ask that, in order for a theory to be “quantum,” it must admit:

(5) a unitary representation of the Canonical Commutation Relations; or
(6) a unitary representation of the Canonical Anticommutation Relations.

Since the theories typically described as “quantum” all tend to admit at least one of these two properties, we now have available:

    Meaning #3: quantum = characterized by (5) or (6).

There’s one last meaning for “quantum” that I should mention. It’s a unique (and, for the moment, inescapable) feature of all our current quantum theories, but perhaps an undesirable one. Namely, quantum theories describe the world in terms of the following two kinds of processes. The first, called unitary (Schrodinger) evolution, is continuous through time. The second, called state reduction (or for von Neumann (1932), “wave function collapse”), is discontinuous. So, according to quantum theory, the typical lifetime of an entity in a particle physics lab consists in a sequence of successive evolutions and “jumps.” A system evolves continuously, until a measurement interaction occurs. It then undergoes a discontinuous transition to a new point, and then begins evolving continuously again. (The nature of these jumps is a matter of debate among interpreters of quantum theory, but let’s bracket that.) We thus have:

    Meaning #4: quantum = discontinuous transition.

This meaning seems most closely related to the use of the word in popular jargon. Indeed, my dear friend and fellow blogger Justin over at My Mind is Made Up thinks that the main way people use the word is in phrases like “a quantum leap.” If this simply means a severe, discontinuous jump, then I’d say “the folk” have picked up on Meaning 4. Interestingly, Justin tells me that Hollywood has managed to pick up on something more like Meaning #1, which would be more historically accurate. But of course, I’m no expert on Hollywood, or the folk — so check out what Justin has to say!

Soul Physics is authored by Bryan W. Roberts. Thanks for subscribing.
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