I was a bystander in a crash of paradigms recently -- Eric Weinstein joined a discussion prompted by Chris House when House asked the question: Why are physicists drawn to economics? House seemed to think it was out of mathematical hubris physicists felt they could jump right in. Weinstein, in comments at Orderstatistic (House's blog) and Noahpinion, pushed for an interpretation of economics in terms of gauge theory.
Now there is nothing incorrect about Weinstein's reformulation of economics in the language of fiber bundles with ordinal utility behaving like a connection (gauge field). As a physicist, I actually enjoyed the mathematics involved in reformulating gauge theory in the language of fiber bundles (and differential forms). I gave seminars on both as a grad student. Similarly there is nothing incorrect about Eric Smith and Duncan Foley's reformulation of economics as thermodynamics. Actually, the existence of both reformulations is a little unsurprising -- the partition functions found in quantum field theory and thermodynamics are closely related.
However, Steve Ellis, one of the professors on my thesis committee, asked me a question at one of those seminars that has stuck with me. What is it good for? (He would disparagingly pronounce the word JAR-GON as if it was the name of a villain from Krypton.) Paul Samuelson answered in 1960 in the case of thermodynamics with absolutely nothing ...
The formal mathematical analogy between classical thermodynamics and mathematical economic systems has now been explored. This does not warrant the commonly met attempt to find more exact analogies of physical magnitudes -- such as entropy or energy -- in the economic realm. Why should there be laws like the first or second laws of thermodynamics holding in the economic realm? Why should 'utility' be literally identified with entropy, energy, or anything else? Why should a failure to make such a successful identification lead anyone to overlook or deny the mathematical isomorphism that does exist between minimum systems that arise in different disciplines?
If he was alive today, Samuelson would probably throw in gauge theory. The thing is, thermodynamics was the big thing in physics at the time of Walras (1870s), and statistical mechanics was fully developed by the time of Fisher (late 1800s to early 1900s). Both of those economists used analogies that used the big new physics at the time. Today theoretical physics is dominated by quantum field theories, so Weinstein is in good company.
Utility as gauge field, utility as thermodynamic potential: what are they good for? If it is nothing more than a re-labeling, a translation into a new language, then I submit it's not really good for anything practical. It is analogous to translating the English "person" into Japanese -- 人 -- look at the beautiful simplicity of the representation! Of course, it doesn't help you do anything that you couldn't do with the English word besides save space or enable elegant graphic design (think Maxwell's equations on a t-shirt which uses modern mathematical notations, not Maxwell's , and definitely not the even more elegant version reformulated as differential forms). And to use it you need to learn Japanese (to use the gauge representation of utility, you need to learn gauge theory).
Maybe these reformulations help in the way most analogies help -- helping thinking and intuition?
To that end let me introduce two other (related) reformulations of economics as computer science. Cosma Shalizi wrote what is my favorite blog post ever -- ostensibly a book review, it becomes a coherent framework for thinking about markets. If running an economy is an optimization problem (allocating raw materials, goods and services), then the top-down communist model where every allocation is centrally planned is akin to trying to solve the linear programming problem directly. Shalizi demonstrates that this is beyond the capabilities of computers for thousands of years, given the size of modern economies. Instead, the market and the price mechanism are remarkably more effective for such an easy system to use.
Nick Hanauer and Eric Beinhocker seemingly independently took this idea up earlier this year, saying that capitalism is an algorithm that solves problems. Economics must therefore be the study of that algorithm. Arguing against the market as a force of nature, they argue the market as algorithm is supposed to function in the service of humanity. These top-level interpretations are the flip-side of the reformulations in mathematics. The former give purpose of the machine while the latter fiddle with the gears.
We return to Steve Ellis's question: Other than giving a general motivation for capitalism and a framework for interpreting the value of the results of capitalism, what are these reformulations as computer science good for? I mean, as far as the science of economics goes? I can see that if you believe capitalism is an algorithm, it e.g. forestalls moralizing in favor of the market distribution of resources. But it doesn't help figure out how the Phillips curve works. (I'm definitely not saying these ideas are wrong!)
What is the impact on the everyday work of particle physics if it is seen as quantum field theory as fiber bundles? I will tell you as a particle physicist: very little. Physics at least has topological solutions that are illuminated by a fiber bundle approach (instantons, Aharonov-Bohm effect, etc). I will make the bold claim that economics has no such topological solutions.
I'd venture to say that recasting economics in this or that mathematical framework or coming up with a new analogy doesn't really add new capabilities to the science of economics that didn't exist before. This pessimism may seem weird coming from a blog that's all about reformulating economics as information theory.
There are quantitative results that come from this particular reformulation -- a quantitative treatment of money as the unit of account and medium of exchange, an excellent model of the price level (including Japan and the general trend towards disinflation), and a reason for the evolution of the Phillips curve, among others.
At least, that's the answer to Steve Ellis's question .
 See this link to Einstein's special theory of relativity for a hint of how Maxwell wrote them down.
 This post can be seen as a sequel to this older post that references some of the same material.