# Gauge Theory Does Not Fix This

a gauge theory is a type of field theory in which the Lagrangian does not change (is invariant) under local transformations from certain Lie groups.

“We imagine we have some countries. Each country has its own currency. Let us imagine that the countries are arranged on a regular grid on a flat world. Each country is connected with its neighbors with a bridge. At the bridge there is a bank. There you are required to change the money you are carrying into the new currency, the currency of the country you are crossing into. There is an independent bank at each bridge. There is no central authority coordinating all the exchange rates between the various countries. Each bank is autonomous and sets the exchange rate in an arbitrary way. The bank charges no commission. For example, assume that the currency in your original country is dollars and the one in the new country is euros. Suppose that the exchange rate posted by the bank at the bridge between two countries is 1.5 dollar = 1 euro. Then if you have 15 dollars the bank converts it to 10 euros as you cross the border. If you decide to come back your 10 euros will be converted to 15 dollars. Therefore, if you go to a neighboring country and you come right back, you end up with your original amount of money. Another rule is that you can only go from one country to the neighboring country. From there you can continue to any of its neighbors and so on. However, you cannot fly from one country to a distant country without passing through the intermediate ones. You can only walk from one to the next, crossing the various bridges and changing your money to the various currencies of the intermediate countries. The final assumption is that the only thing you can carry from one country to the next is money. You cannot carry gold, silver, or any other good.

Where is the symmetry? The gauge symmetry is the following. Imagine that one of the countries has accumulated too many zeros in its currency and wants to drop them. This is fairly common in the real world in countries with high inflation. What happens is that one day the local government decides that they will change their currency units. For example, instead of using Pesos now everybody needs to use “Australes”. The government declares 1,000 Pesos will now be worth 1 Austral, or 1,000 Pesos = 1 Austral. So everybody changes all prices and exchange rates accordingly. If you needed to pay 5,000 Pesos for a banana, now you will need to pay 5 Australes. If your salary was 1 million Pesos, it will now be 1 thousand Australes. Suppose the neighboring country is the USA. If the exchange rate was 3,000 Pesos = 1 Dollar, it will now be 3 Australes = 1 Dollar. See figure 5. We call this a “symmetry” because after this change nothing really changes, nobody is richer or poorer and the change offers no new economic opportunities. It is done purely for convenience. You can see this gauge symmetry in action in some Argentinean banknotes in figure 6. It is called a “gauge” symmetry because it is a symmetry of the units we use to measure or “gauge” the value of various quantities.

This symmetry is “local”, in the sense that each country can locally decide to perform this change, independently of what the neighboring countries decide to do. Some countries might like to do it more frequently than others. In the real world, Argentina has eliminated thirteen zeros through various actions of this “gauge symmetry” since the 1960s, so that 1 Peso of today = 10^13 Pesos of the 1960s.

Now, in physics the countries are analogous to points, or small regions, in space. The whole set of exchange rates is a configuration of the magnetic potentials throughout space. A situation like the one in figure 7, where you can earn money, is called a magnetic field. The amount of gain is related to the magnetic field. The speculators are called electrons or charged particles. In the presence of magnetic fields, they simply move in circles in order to earn money. In fact, the total gain along the circuit is the flux of the magnetic field through the area enclosed by the circle. Now imagine that you are a speculator that has debt instead of having money. In that case you would go around these countries in the opposite direction! Then your debts would be reduced in the same proportion. In the example of figure 7, your debts would be reduced by a factor of 1/1.5 by circulating in the direction opposite to the green arrow. In physics, we have positrons, which are particles like the electron but with the opposite charge. In fact, in a magnetic field positrons circulate in the opposite direction as compared to electrons.

In physics, we imagine that this story about countries and exchange rates is happening at very, very short distances, much shorter than the ones we can measure today. When we look at any physical system, even empty space, we are looking at all these countries from very far away, so that they look like a continuum … When an electron is moving in the vacuum, it is seamlessly moving from a point in spacetime to the next. In the very microscopic description, it would be constantly changing between the different countries, changing the money it is carrying, and becoming “richer” in the process. In physics we do not know whether there is an underlying discrete structure like the countries we have described. However, when we do computations in gauge theories we often assume a discrete structure like this one and then take the continuum limit when all the countries are very close to each other.

Electromagnetism is based on a similar gauge symmetry. In fact, at each point in spacetime the symmetry corresponds to the symmetry of rotations of a circle. One way to picture it is to imagine that at each point in spacetime we have an extra circle, an extra dimension. See figure 9(a). The “country” that is located at each point in spacetime chooses a way to define angles on this extra circle in an independent way. More precisely, each “country” chooses a point on the circle that they call “zero angle” and then describe the position of any other point in terms of the angle relative to this point. This is like choosing the currency in the economic example. Now, in physics, we do not know whether this circle is real. We do not know if indeed there is an extra dimension. All we know is that the symmetry is similar to the symmetry we would have if there was an extra dimension. In physics we like to make as few assumptions as possible. An extra dimension is not a necessary assumption, only the symmetry is. Also the only relevant quantities are the magnetic potentials which tell us how the position of a particle in the extra circle changes as we go from one point in spacetime to its neighbor.

“keep in mind that our goal is not to explain the real economy. Our goal is to explain the real physical world. The good news is that the model is much simpler than the real economy. This is why physics is simpler than economics!”

“The proposal of Malaney and Weinstein is that to construct models of economies that have real dynamics and time dependence in them- so that for example, preferences of households can change in time-it is necessary to hypothesize that the dynamics is con- strained by much larger groups of gauge invariances.

As we have seen in the discussion above, the need for gauge invariance stems from a fundamental fact about prices, which is that they appear to be at least in part arbitrary. It seems that each agent in an economic system is free to put any value they like on any object or commodity subject to trade. How do we describe dynamics of a market given all this freedom? To get started we recall that in the Arrow-Debreu description of economic equilibrium, there is a gauge symmetry corresponding to scaling all prices. This may suffice for equilibrium, but it is insufficient for describing the dynamics out of equilibrium, because away from equilibrium there may be no agreement as to what the prices are. There is then not one price, but many views as to what prices should be. Each agent should then be free to value and measure currency and goods in any units they like- and this should still not change the dynamics of the market. It should not even matter if two agents trading with each other use different units. Thus we require an extension of the gauge symmetry in which the freedom is given to each agent, so they may each scale their units of prices as they wish, independently of the others.

There is a further difficulty with price which is that even after issues of measurement units are accounted for different agents will value different currencies or goods differently. Different agents have different views of the economy or market they are in, they have diverse experiences, strategies and goals, and consequently have different views of the values of currencies, goods and financial instruments. Consequently, in a given economy or market it is often possible to participate in a cycle of trading of currencies, goods or instruments and make a profit or a loss, without anything actually having been produced or manufactured. This is called arbitrage.

In equilibrium, all the inconsistencies in pricing are hypothesized to vanish. This is what is called the no arbitrage assumption. But out of equilibrium there will exist generically inconsistencies in pricing. In fact, we are very interested in the dynamics of these inconsistencies because we want to understand how market forces act out of equilibrium on inconsistencies and differences in prices to force them to vanish. This is essential to answer the questions the static notion of equilibrium in the Arrow-Debreu model does not address.

However, in analyzing the dynamics that results from the inconsistencies, we need to be careful to untangle meaningful differences and inconsistencies in prices from the freedom each agent has to rescale the units and currencies in which those prices are expressed. This is precisely what the technology of gauge theories does for economics.

How does gauge theory accomplish this? As in applications of gauge theory to particle physics and gravitation, the key is to ask what quantities are meaningful and observable, once the freedom to rescale and redefine units of measure are taken into account. The answer is that out of equilibrium the meaningful observables are not defined at a single event, trade or agent. Because of the freedom each agent has to rescale units and choose different currencies, the ratios of pairs of numerical prices held by two agents in a single trade are not directly meaningful.

To define a meaningful observable for an economic system one must compare ratios of prices of several goods of one agent, or consider the return, relative to doing nothing, to an agent of participating in a cycle of trading. This might be a cycle of trades that starts in one currency, goes through several currencies or goods and ends up back in the initial currency. Because the starting and ending currencies are the same, their ratio is meaningful and invariant under rescalings of the currency’s value. This is true whether one agent or several are involved in the cycle of trades. We say that these kinds of quantities are gauge invariant.

Such quantities, defined by cycles of trades such that they end up taking the ratio of two prices held by the same agent in the same currency have a name: they are called curvatures. The ratios of prices given to a good by different agents also have a name: they are called connections. The latter do depend on units and hence are gauge dependent, the former are invariant under arbitrary scalings of units by each agent and are gauge invariant or gauge covariant (this means they transform in simple fashion under the gauge transformations.)

It is interesting that the quantities that are invariant under the gauge transformations include arbitrages, which should vanish in equilibrium. This does not mean that they are irrelevant, indeed, they may be precisely the quantities one needs to understand how the non-equilibrium dynamics drives the system to equilibrium. That is, it is natural to frame the non-equilibrium dynamics in terms of quantities that vanish in equilibrium. These are the quantities that the law of supply and demand acts on, in order to diminish them.

There is a precise analogy to how gauge invariance works in physics. In gauge theories in physics, local observables are not defined because of the freedom to redefine units of measure from place to place and time to time. Instead, observables are defined by carrying some object around a closed path and comparing it with a copy of its configuration left at the starting point. These observables are called curvatures. The result of carrying something on a segment of open path is called a connection and is dependent on local units of measure. But when one closes the path, one makes comparison to the starting point possible, so one gets a meaningful observable, which is a curvature.

In general relativity exactly the same thing is true. Here curvature corresponds to inconsistencies in measurements, for example, you can carry a ruler around a closed path and it comes back pointing in a different direction from its start. The dynamics is then given by the Einstein equations, which are expressed as equations in the curvature. But in the ground state-which is roughly analogous to equilibrium in an economic model-the curvature vanishes. The state with no curvature, called flat spacetime, is the geometry of spacetime in the absence of matter or gravitational forces. It is the state where all observers agree on measurements, such as which rulers are parallel to which.

But while the curvatures vanish in the ground state, the physics of that state is best understood in terms of the curvatures. For example, suppose one perturbs flat spacetime a little bit. The result are small ripples of curvature that propagate at the speed of light. These are gravitational waves. The stability of flat spacetime is explained by the fact that these ripples in curvature require energy.

Similarly, it may be that the stability of economic equilibrium can be studied by modeling the dynamics of small departures from equilibrium. These are states where prices are inconsistent, ie where arbitrage or curvature is possible. By postulating that the dynamics is governed by a law that says that inconsistencies evolve, one gets a completely different understanding of the underlying dynamics than in a theory that simply says that inconsistencies or curvatures vanish.

If economics follows the model from physics, then the next step, after one has an- swered the question of what are the observables, is to ask what are the forms of the laws that govern the dynamics of those observables. Given what we have said, the choice of possible laws is governed by a simple principle: since only gauge invariant quantities are meaningful, the dynamics must be constructed in terms of them alone.