What does it mean to say that something "is a gauge theory"?

[questioner1]

Hello,

Brief context to this question:
I'm an economics student and I've recently seen a lecture by phycisist-turned-economist Eric Weinstein, who says that "neoclassical economics is a naturally occurring gauge theory". In response to this, I tried to find out about group theory and gauge theory. My understanding is very rudimentary: My impression is that a group is something along the lines of a matrix of numbers combined with a rule of operation on that matrix.

So I am planning to develop a rigorous understanding of gauge theory, for its application in economics.My question is: could someone possibly explain what it means to say that "neoclassical economics is a gauge theory"? (to someone who does not have an understanding of gauge theory from a physics background)

Also, what exactly is the difference between gauge theory and group theory?

I hope my question is to the point, and for context, I am actually planning to study the mathematics of this in great detail once I have developed a general understanding.

Thank you :)

 

[MisterX]

Also, what exactly is the difference between gauge theory and group theory? Would it mean the same thing to say "neoclassical economics is a group theory"? if not, why?

Perhaps someone else can give a more complete explanation, but the answer is no, it would not mean the same thing. In fact, it would not make any sense at all. A group is a set alongside a binary operation which together satisfy the group axioms. I encourage you to look these up. We don't generally classify economic or physical theories as being group theories. They may involve groups. But theories of physics or economics that involve groups are not called group theories (that name is reserved for group theory itself, which is purely mathematical).

In contrast, gauge theory is a label by which we may classify physical or economic theories. This is a distinct concept from group theory. Someone else may be able to explain what a gauge theory is better than I.

 

[questioner1]

Perhaps someone else can give a more complete explanation, but the answer is no, it would not mean the same thing. In fact, it would not make any sense at all. A group is a set alongside a binary operation which together satisfy the group axioms. I encourage you to look these up. We don't generally classify economic or physical theories as being group theories. They may involve groups. But theories of physics or economics that involve groups are not called group theories (that name is reserved for group theory itself, which is purely mathematical).

In contrast, gauge theory is a label by which we may classify physical or economic theories. This is a distinct concept from group theory. Someone else may be able to explain what a gauge theory is better than I.

Thank you! This clarifies that part of my question :).
Im still wondering what the exact relation between gauge theory and group theory is, then, and what it means to say that something "is a gauge theory".

Thank you. :)

 

[WannabeNewton]

Im still wondering what the exact relation between gauge theory and group theory is, then, and what it means to say that something "is a gauge theory".

A gauge theory is just a field theory (classical or quantum) that possesses a gauge symmetry. The relation to group theory arises from the observation that the set of all gauge transformations of a given field theory allow codification through some Lie group.

Electromagnetic theory is the simplest and most familiar example of a gauge theory, with gauge group $U(1)$, but any further elucidation would require a proper physics background.

 

[Matterwave]

This is an interesting prospect. In order to turn neoclassical economics into a gauge theory, you must first define the class of symmetries that act on the variables in the theory.

A gauge theory is a theory that exhibits a gauge symmetry. It's not always obvious, even to physicists, exactly the nature of the gauge symmetry and what differentiates a gauge symmetry from other kinds of symmetries. But roughly speaking, a gauge symmetry is an internal symmetry of the field that is independent of the space-time symmetries of the field. Gauge symmetries arise when the physical objects of interest are field strengths which are (roughly speaking) special derivatives of potentials. Because of the special construction of the special derivatives, the field strengths are unchanged by "gauge transformations" of the underlying potential.

Unless you want to dive into the deep water and start learning about principle bundles of manifolds and principle connections and all that jazz, it's hard to give a more precise definition of a gauge theory.

As such, it's hard to classify a non-physics theory into such a category of a "gauge theory" rigorously. However, we might be able to draw analogies to "gauge theory" and this is perhaps what your professor meant.

In neoclassical economics, we assume consumers maximize utility and firms maximize profits in a rational manner. In this assumption, there is an ambiguity of the definition of utility. We cannot say with any specificity that util=1 actually means anything. For example, the statement "that pizza gives me 10 utils" is meaningless because a util is undefined in any absolute sense. The only statement that matters are statements such as "x gives me more utils than y".

The central problem of neoclassical economics is given products $x_1, x_2, ...,x_n$ maximize the utility function $U(x_1,x_2,...,x_n)$ given some budget constraint $B=p_1 x_1+p_2 x_2+...+p_n x_n$. There is a "gauge symmetry" in the utility function $U(x_1,x_2,...,x_n)$ because absolute utilities don't matter, only relative utilities do. For example, take just two products x and y and an extremely simple utility function $U(x,y)=x+3y$ a different utility function $U(x,y)=2x+6y$ will give exactly the same results (in this extremely simple case, one only has to see what the utility to price ratio is of x and y and whichever is higher, only purchase that product). The usefulness of this "gauge symmetry" is yet to be determined. I have also not given a rigorous definition of the gauge symmetry because I haven't specified the gauge group of the symmetry. As this is the first time I have heard about this concept, I haven't the time right now to develop this any further. But the basic idea would be to ask yourself "what utility functions $U'(x_1,...,x_n)$ will give me the same behavior as the utility function $U(x_1,...,x_n)$ and what transformations take me from one to the other?".

I am also not sure that this is the "gauge theory" that your professor was talking about. But it seems to me to be the most obvious one.