Gauge Theory of Economics

Gauge theory of economics is the application of differential geometric methods to economic problems. This was first developed by Pia Malaney and Eric Weinstein in Malaney's 1996 doctoral thesis The Index Number Problem: A Differential Geometric Approach.

Examples

Gauge theory is all you need to break out of the economics flatland. The following is an equation that Eric Weinstein talked about. We are going to break it down together and picture the meaning of each part in a geometrical and intuitive way. The results of this work will be interesting for the present economics community.

[math]\displaystyle{ q = \frac{ {p_0}\cdot{q} }{ {p_0}\cdot{q_0} } q_0 + (q - \frac{ {p_0}\cdot{q} }{ {p_0}\cdot{q_0} } q_0) }[/math]

where we label the first term as Reference Basket and the second one as Barter.

Suppose that we live in a world where there are only 3 different types of items for sale: apples, berries and cherries (A, B and C respectively.) Say today we pick up our basket and go to the market. At the market, the price of each item is posted up as a number on the wall where we can see. So, we represent the prices by a [math]\displaystyle{ {1}\times{3} }[/math] row vector [math]\displaystyle{ p }[/math]. On the other hand, we buy different quantities of each item and so a [math]\displaystyle{ {3}\times{1} }[/math] column vector [math]\displaystyle{ q }[/math] denotes the list of 3 quantities for items A, B and C.

The next day, we go back to the market and now we are interested in measuring price changes.

Inflation

(Video and text from Pull that up, Jamie)

Here is shown a red curve/function, possibly representing the inflation of a currency as time progresses to the right along with a blue function representing the notion of “constant wage.” The concept at play is that the definition of constant depends on the faded horizontal ticks in the background space, and in the example of a wage, it makes conceptual sense to set such a notion of constant against the inflation curve rather than the axis. This represents one spirit of what is done in the “calculus of gauges,” and in reality we tend not to see god-given coordinate grids sprawling over space thus encouraging us to adopt a different language of geometry/calculus.

Reference Material

Papers

• The Index Number Problem: A Differential Geometric Approach by Pia Malaney.
• An Extension of Intertemporal Ordinal Welfare to Changing Tastes: Economics as Gauge Theory (Draft for University of Chicago Money and Banking Workshop. In preparation.) by Pia Malaney and Eric Weinstein
• Tracking Changes in Bilateral Trading Patterns in the Absence of PPP by Eric Weinstein
• Time and symmetry in models of economic markets by Lee Smolin.
• Gauge Invariance, Geometry and Arbitrage by Samuel Vázquez and Simone Farinelli

Books

• The Physics of Wall Street by James Weatherall.

Lectures

Lectures, presentations, and panels by Pia Malaney and Eric Weinstein on the topic.

Outlet	Title	Link	Air Date
University of Chicago Money and Banking Workshop	Gauge Theory, Inflation and Geometric Marginalism: Are our Inflation, Productivity and Trade Indicators All Off Because The Marginal Revolution Was Based on the Wrong Version of The differential Calculus?	Abstract	2021-11-10
Fields Institute	Towards a Mathematics of New Economic Thinking for Reflexive Markets	Watch	2013-11-02
Institute for New Economic Thinking	What Math and Physics Can Do for New Economic Thinking	Watch	2013-10-30
Stanford University	Systems Architecture, Kabuki Capitalism, and the Economic Manhattan Project	Watch	2013-06-21
Perimeter Institute for Theoretical Physics	A Science Less Dismal: Welcome to the Economic Manhattan Project	Watch	2009-01-05
Perimeter Institute for Theoretical Physics	Sheldon Glashow Owes me a Dollar (and 17 years of interest)	Watch	2008-09-11
Perimeter Institute for Theoretical Physics	Gauge Theory and Inflation: Enlarging the Wu-Yang Dictionary	Watch	2006-05-24

Interviews

Outlet	Title	Link	Air Date
Institute for New Economic Thinking	Economic Thinking In A Fallible World	Watch	2014-06-22
Institute for New Economic Thinking	What Math and Physics Can Do for New Economic Thinking	Watch	2013-10-30

Other Sources

• Op-Ed by mathematician Edward Frenkel
• Eric talking about this on Quora