Tuesday, September 15, 2015

Physics!

There were a couple of posts in my feed recently that make some references to physics, and as the econoblogosphere's resident Phd physicist (Noah Smith was an undergrad major, so were John Cochrane and Paul Romer ... sort of) I thought I'd weigh in.

First is this one ('You're Not Irrational, You're Just Quantum Probabilistic') linked to by Mark Thoma (and brought up by Tom Brown in comments). I might be missing something, but it seems that the "quantum" is unnecessary. They're really just using properties of non-orthogonal state vectors and non-commuting operators -- that is, the basic mathematics of vector spaces. The Hilbert space of quantum mechanics (and its non-commuting operators of position and momentum) is an example of such a model, but it's not the only one. Therefore using that specific model is probably not entirely accurate. It's like saying your model is a Ferrari, when what you're actually presenting is a model of car. Sure, "X is like a Ferrari" is a better headline than "X is like a car". That makes me think that the only reason they use the word quantum is because it's cool. You're Not Irrational, Your Brain Just Uses Noncommuting Operators isn't as exciting of a title. Nor is What Is the General Linear Space Model of Cognition, and How Is It Applied to Psychology? That, or the authors don't really understand what it is they've come up with.

The second one -- which is far more nit-picking on my part in what is a great post -- is this quote from Robert Waldmann:
Physicists are quite sure general relativity is not the truth (because it is inconsistent with quantum mechanics and therefore a lot of data).
I'm not sure "truth" is a useful concept here. Physicists don't think of general relativity as a thing that is true or false in a way that you could choose "false". They think of it as an effective theory: a theory that captures the phenomenology (describes the empirical data) and a theory that you could derive from whatever the "truth" is. String theory may be the "truth" -- and you can derive general relativity from it. Entropic gravity based on states on a horizon may be the truth (i.e. gravity isn't 'real', it's just thermodynamics) -- and you can derive general relativity from it. So general relativity is as much "truth" as the theories that contain it. As a mathematical consequence of truth, it must be truth, too.

General relativity is not inconsistent with quantum mechanics (you can't use canonical quantization on it without the time dimension disappearing, but that is a theoretical consideration, not empirical). The two theories have non-overlapping regions where they have been empirically tested. All empirical tests of quantum mechanics are at short distances and all tests of general relativity are at long distances. You actually can't test them at the same time because the effects of general relativity beyond Newtonian gravity on the quantum scale are basically zero.

Actually, we could say quantum mechanics is inconsistent with general relativity -- it predicts a value of the cosmological constant that is off by 100+ orders of magnitude!

PS

I am under the impression that there is as big of a schism between undergraduate vs graduate economics as there is in undergraduate vs graduate physics. It's not that anything taught is wrong (or wrong-headed), it's just that it seems there are different ways of going about things -- so different that in other pursuits, they might even be considered different fields.

When someone makes the claim of being an undergraduate physics major, they're usually well versed in Newtonian mechanics, Lagrangian/Hamiltonian approaches, thermodynamics and quantum physics (and some other stuff like math). Unfortunately, this tends to leave out two major organizing principles and one major perspective shift:

Physics is the study of symmetry. You learn energy and momentum conservation as an undergraduate. As a graduate, you learn time and space translation invariance (symmetry). Newton's laws are basically a consequence of Poincare symmetry at speeds less than the speed of light.

All theories are effective theories. This is referenced in my discussion of Waldmann's post above. Physicists don't think general relativity is "wrong" because it isn't quantum. General relativity is an effective field theory of physics at large length scales. Actually, general relativity is the effective field theory of physics at large length scales. Whatever the fundamental theory of everything is, it must reduce to general relativity in the proper limit. So whatever the fundamental theory of everything is, it contains general relativity. That's why people like string theory: it contains general relativity. Entropic gravity also contains general relativity. All empirically valid theories are viewed sort of like Taylor expansions, analytic continuations or ensemble averages of a more fundamental theory. Newton's laws are derivable from quantum mechanics and you can show < F > = m < a >. It's quite fun to look at the Earth-Moon gravitational system as an extremely high energy level of the Hydrogen atom solution to the Schrodinger equation with a different coupling constant.

Mathematical objects have physical reality. There are some objects called D-branes and NS-branes in string theory. Those names come from Dirichlet (function value) and Neumann (function derivative value) boundary conditions from your undergraduate differential equations class (they determine how you treat the ends of strings). The assumptions and theoretical constructs you make (or derive) in your theory have a physical reality. That's why Dirac proposed the positron (a consequence of making the Schrodinger equation relativistic for spin 1/2 particles). That's why physicists were convinced the Higgs existed and why people are out there looking for supersymmetric particles.

PPS

This last bit is why I tend to look at utility, expectation operators (those E's) and even equilibrium conditions in DSGE models as posing a lot of questions. If your model has these things, they must represent an economic reality. An equilibrium condition must be measurable economic construct that has dynamics just like a Dirichlet boundary condition is a D-brane with an energy density flying through an 11-dimensional universe. A good example (in my head) is that your Taylor rule in your New-Keynesian DSGE model has a physical reality as the central bank. The Taylor rule is the central bank. At least, in that specific model.

3 comments:

  1. Jason, you are right that "quantum" is unnecessary. However, IIUC, quantum logic arose as a result of trying to understand quantum mechanics. Some years ago I noted, as a curiosity, that you could apply it to games, and possibly vice versa. Their use of the term, "quantum", is not just because it is cool.

    I have long held that, contra many economists with the notable exception of Keynes, rationality does not depend upon reduction to real numbers.

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    1. Hi Bill,

      My point above is that quantum mechanics is a specific instance of a Hilbert space with operators in a general linear space GL(n, ℝ) or GL(n, ℂ). You get something like quantum behavior in a lot of different systems. Quantum mechanics is an instance of a more general mathematical framework; it's not the only way to get non-commuting operators and non-orthogonal states. Saying it's "quantum" is limiting.

      Additionally, the word quantum has to deal with quantization -- e.g. discrete energies of electrons or energy packets in photons. The psychologists don't think there is a quantum of aggression or that someone can make a transition from the aggression = 4 state to the aggression = 5 state. They're using quantum to refer to the vector space properties that don't have to be (and actually aren't) quantized.

      As an example, the groups SO(n) (classical rotations in n dimensions) don't have to be quantum at all but have all of the properties the psychologists are after. Operators don't commute (you can rotate something around the x-axis and then the z-axis that doesn't end up in the same place doing the same rotations but around the z-axis and then the x-axis) and you can have non-orthogonal states. But:

      What Is Rotational Cognition, and How Is It Applied to Psychology?

      ... isn't as cool of a title.

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    2. Hi, Jason.

      As we know several years ago there were a lot of popular texts relating quantum mechanics and cognition in a loosey goosey way. I only skimmed this paper, but I do not think that it falls into that category. It is based, at least in part, on what is called quantum logic. The authors have found experimental results that could be the result of humans using quantum logic instead of Aristotelian logic or standard predicate logic. I think that that justifies what they call their theory.

      Sure, there are other things they could call it. As I said, I think the key is that certain entities are matrices instead of real numbers. But there are historical reasons for using the term, "quantum", even without quanta. ;)

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