## Tuesday, March 03, 2009

### Gauge invariance in economics

Consider a victimist alternative physicist who has literally made a career out of his nonsensical proposed alternatives to key theories in physics - such as quantum mechanics, relativity, or string theory.

His alternative theories make no sense whatsoever and the author doesn't have the sufficient knowledge, creativity, concentration, and luck to do valuable original research. But there are always many people who find it "unjust" that some ideas are important while others are not. And there are always many people who can't distinguish crap from gold. So they will always pay for anything that carries the "alternative" label.

So without having anything valuable to offer, the author becomes a symbol of rebellion, fight for the rights of the suppressed, and even independence. By nonsensical mixtures of jargons from different disciplines, he can also become a symbol of interdisciplinary research and the unity of knowledge, in the eyes of the many people who have no idea about either discipline.

Important sciences such as theoretical physics are critical enough so that they can apparently afford to feed a few 100% horseleeches.

Now, imagine that the very same person decides to apply the very same survival strategy to economics. What would the details look like? Well, he would probably choose classical or neoclassical economics as his target because it is the most successful and robust paradigm in economics.

He would claim that it has some particular bugs - which it certainly doesn't have - and he would propose concepts from very different disciplines as justifications of his conclusions that differ from neoclassical economics at random critical places, even though no implication of this sort can ever be proven rationally.

What I have told you was not just a thought experiment. Here is the paper (hat tip: Dan):
Time and symmetry in models of economic markets (PDF)
So Lee Smolin chooses neoclassical economics, as formulated in the Bourbaki style of mathematics in the 1950s by Arrow, Debreu, and McKenzie (ADM model of general equilibrium theory), as his target. The next thing you should expect is a followup to his book The Trouble With Physics. A female collaborator - let's call her Leslie Winkle - could recommend him an original title: the new work could be called The Trouble With Economics. The subtitle is "The Rise of the Free Markets, the Fall of the Human Race, and What Comes Next." :-)

But back to the ADM theory.

In this neoclassical setup, prices are assigned not only to particular products but to particular products at arbitrary locations and different moments in the future. The price space is highly multi-dimensional.

Lee Smolin begins with some universal promotion of alternative theories in general that he essentially copies from his victimist physics papers. A discussion of the ADM theory follows; it makes less sense than the text on Wikipedia. Very soon, he offers you absurd claims, e.g. that neoclassical economics knows no mechanisms that drive the system towards the equilibrium (Smolin, middle of page 4/41).

Well, the classical mechanism doing this particular thing is very simple. If the current price is lower than the (future) equilibrium price, demand exceeds the supply and the prices go up. As they go up, the demand is decreasing and the production is increasing, helping to reach the equilibrium point.

Later in the paper, on page 8, Smolin seems to be aware of this mechanism. On that page, he repeats these well-known things with some colorful words (like "making people happier"), suggesting that he was perhaps the father of classical economics.

Landscape in economics

Smolin tries to import two physics buzzwords, "landscape" and "gauge invariance", to economics. In doing so, he reveals that he completely misunderstands these basic notions in physics, too.

Concerning the "landscape problem", he claims that much like string theory, economics has a lot of "equilibrium points". That's simply a wrong analogy. What we have in economics is a lot of possible configurations. But none of them is static.

These configurations of prices are analogous to points in the configuration space or phase space in physics (depending on the degrees of freedom you include). But the landscape in string theory is composed out of vacua which are perfectly stationary states.

In any useful enough application of economics, one always works in some vicinity of an equilibrium. That would be analogous to knowing the right compactification of string theory. There may exist "very different equilibria" but they're "very far on the configuration space" which also implies that they can't become relevant for the dynamics in a near future. So the degeneracy of the equilibrium points - certainly the large degeneracy - is irrelevant for any realistic enough homework problem in economics. After all, configurations in economics can be viewed as special examples of those in physics, and we know that there is no "additional" exponentially degeneracy of the vacua coming from non-fundamental laws of physics.

What Smolin misunderstands even more is that we almost always have to work with effective field theories that only concentrate on some dynamical degrees of freedom, corresponding to an interval of time scales.

In this approximation, all faster degrees of freedom (than this interval) are assumed to take on the equilibrium values instantly (like the relative electron-proton wave functions in a Born-Oppenheimer approximation: they sit into the ground state "instantly"). And all slower degrees of freedom (than this interval) play the role of external parameters (that influence the internal dynamics of the system but are unaffected by it, and can be either constant or otherwise functionally given).

Whenever a proper truncation to an effective theory is made in this way, the markets will have a small number of equilibrium points (for each configuration of the external parameters). If the space of possible out-of-equilibrium metastable states grows arbitrarily, it is always possible to demonstrate this diversity as a consequence of effects at ever slower time scales.

If we tried to incorporate arbitrarily long time scales into economics, e.g. billions of years, we would need to know all processes that are faster than this time scale and economics would become geology. It's very clear that geological degrees of freedom must be treated as external parameters in a sensible model of economics.

Whenever we avoid this trap, it may be seen that a sensible and useful model of economics will only have a small number of equilibrium points. Smolin's exponentially large "landscape" is an artifact of his sloppy treatment of time scales.

Gauge invariance

Smolin's misunderstanding of gauge invariance is comparable to his misunderstanding of effective theories and the landscape. On page 16, he claims that the neoclassical Arrow-Debreu model has the following "gauge invariances":
1. prices get rescaled by a constant, Lambda
2. utility functions are rescaled by different constants, lambda_a, that can be chosen to differ for different households because their evaluations of utility are independent
He seems to claim that these "gauge invariances" are extremely deep. However, they're not deep and they're not gauge invariances, either.

Any symmetry that should be called "gauge invariance" must have time-dependent parameters describing the transformations; otherwise it is not a "gauge symmetry" but just a "global symmetry". Clearly, if prices are rescaled by a function of time, we deal with inflation which has a profound impact on expectations in particular and the economy in general. This time-dependent rescaling is surely not a symmetry.

Concerning the utility functions, the overall rescaling can be time-dependent - only relative utilities matter for each households at each moment. However, this observation is an inconsequential tautology. The utility functions determine how the money will be spent by the household so the natural normalization of the utility functions makes it proportional to the overall wealth of the household, anyway. That will tell you the household's weight - how much they will spend for XY depending on their preferences. It would be very uneconomic to assign two comparable households with two parametrically different overall utility functions.

You can rescale the utility functions by a new function of time, lambda_a(t), but such a new description of the equations with a gauge symmetry will be completely useless because the new unphysical degrees of freedom can be easily and "algebraically" eliminated.

Only gauge symmetries in which some degrees of freedom transform by "lambda" itself while others transform according to time-derivatives of "lambda" (e.g. Yang-Mills gauge symmetry) are useful. It never makes sense to ask "what gauge symmetries objectively exist in a given situation".

The identity of gauge symmetries is only associated with a description (i.e. a formalism), not with actual dynamics, and there exist many descriptions with different gauge symmetries that describe the same dynamics. Gauge-fixing is an old example; dualities are even more powerful modern examples . For example, in AdS/CFT, one equivalent description has a U(infinity) Yang-Mills symmetry in 3+1 dimensions while the other naturally includes diffeomorphisms in 9+1 dimensions, among others.

So gauge symmetries are only redundancies of a description that are worth talking about if they give us a simpler description of a physical system and if they cannot be easily eliminated.

Noether's currents and cosmic inflation in economics

At some points, it is not clear whether Lee Smolin is just joking or whether he is serious. For example, the rescaling of all prices is a symmetry. It is only a global symmetry, not a gauge symmetry as Smolin incorrectly thinks, but a global symmetry should be enough to yield a Noether current. According to Emmy Noether's theorem, continuous symmetries are in one-to-one correspondence with conserved quantities.

On page 16, Smolin asks what is the Noether current associated with the rescaling of all the prices. Now, this is a question that an A-student of a physics course should be able to answer. Smolin is clearly not among them.

The numerical value of prices clearly depends on the choice of units. In physics, we have many examples of this sort. For example, you can change the units of mass in your physical theory. It is very clear that such a trivial operation cannot ever give you any useful, nontrivial conserved charge.

Nevertheless, it is also very easy to see what the charge corresponding to the noncompact U(1) symmetry actually is. If all prices "p" are rescaled by "Lambda", it is analogous to all charge-one fields in QED being rescaled by "exp(i lambda)". So it is the very property that "p" is a price that assigns it with the charge "Q=1" which guaranteed that it gets multiplied by the 1st power of "Lambda" under the gauge transformations.

So prices carry charge one, inverse prices carry charge minus one, non-prices and price ratios carry a vanishing charge, squared prices carry charge two, and so on. It's as simple as that.

It is also easy to see that unlike the electric charge, the Smolin-Noether price charge is a completely useless stupidity. Why? Because in QED, the fields such as the Dirac fields "Psi" are able to create (discrete) particles and the vacuum can be defined to carry no particles, i.e. to have "Q=0".

In economics, prices or inverse prices don't create any excitations. More importantly, there is no "vacuum state without particles" that would have a well-defined value of "Q". The analogy between economics and QED seemed promising but one key aspect made them very different: QED is a quantum theory while economics is a classical theory, especially classical economics. ;-)

This means that in the exponents "exp(i lambda)" added to the charged fields in QED should really be written as "exp(i lambda / hbar)" where "hbar" is Planck's constant and "lambda" is approximately classically-behaving, as far as the scales and gradients go. It means that in the classical limit, the phase (e.g. the phase of a macroscopically charged object) is changing extremely quickly with time and space.

On the other hand, the scaling factor "Lambda" has no "hbar" in it. Even though the Smolin-Nother charge of all sensible quantities is integer-valued, you can't find any "quanta" that discretely change the charge of a configuration. So whatever you define to be the charge of the "vacuum" or "equilibrium", and this question has no good answer either (because it is inherently quantum-mechanical), you won't obtain any states with different values of the charge. The only "application" of the new charge is to remember whether a quantity has "USD" in the numerator or the denominator - i.e. dimensional analysis.

If Smolin had elementary knowledge about Noether's theorem and if he had spent 30 seconds to think about the problem, he would have to know that this idea can lead to no interesting results.

An even more explosive joke appears in the footnote on page 16. While Smolin was not able to find the Noether's charge for the price-scaling symmetry, he already wants to know that there is a Goldstone boson which is an inflaton! ;-) Inflation in cosmology and inflation in economics, you get it?

Now, can he possibly be serious about this comment? It's clear that the only thing that cosmic inflation and price inflation share is the word "inflation" that describes an exponential increase of "something" in time. Linguistics plays no role in the scheme of things, so there is no reason for any details to agree. And of course, no other details agree.

Of course, price inflation is linked to the increase of overall prices in time, so whenever you define something, e.g. a "Goldstone mode", to be linked to the increase of prices, it will be linked to price inflation, too. ;-)

But one can also see that all the reasons that make Goldstone bosons useful concepts in physics will be absent in this trivial example of overall price rescaling. Most importantly, there will be no "bosons" because economics is not a quantum theory so its excitations or "waves" are not quantized. As I will explain below, the Goldstone theorem is not applicable at all.

Second, Goldstone fields are useful in physics because they're massless (or very light, in the case of pseudo-Goldstone bosons) fields but these fields are otherwise analogous to other dynamical fields. Again, this nontrivial conclusion about the low mass will be absent in economics.

The "mass" or "energy" (related by "E=mc^2") can't be defined for evolution in economics (except as the physical mass/energy!) because economics is a classical theory and classical physics has no "E=hf" link between frequencies and masses. And if you thought that there is a statement about low frequencies of the Goldstone modes, you would be wrong, too. It is very clear that inflation influences economic dynamics at all time scales.

But the detail about the Goldstone theorem that makes an even more devastating impact on the analogy is that the Goldstone theorem only works in 2+1 dimensions and higher. There are no Goldstone bosons in 2 (or lower) dimensions, as Coleman proved. Economics is formally a 0+1-dimensional system so the Goldstone theorem is not applicable (and the concept of a fixed mass would mean a completely fixed frequency of a periodic process linked to inflation which is clearly wrong in economics).

The only way to make Goldstone's theorem relevant would be to define a field theory and the masslessness (or lightness) of the Goldstone modes would mean that certain signals propagate (almost) by a preferred speed, a counterpart of the speed of light, along the surface of Earth. Clearly, such patterns haven't yet been detected in economics.

Summary

To summarize, Smolin's ideas make no sense in either discipline but he relies on the hypothesis that people in one discipline will always think that Smolin might have some meaningful expertise about a different discipline which could be helpful in the first discipline. Smolin has no expertise in any discipline and everything he writes is bullshit but he can get away with it because when it comes to some politically correctly sounding clichés, people are amazingly uncritical.

And that's the memo.