# Modeling non-quantum objects (in finance, sociology etc) using fermionic fields?

Please provide (if any) applications of fermionic field theory in non-physics macro contexts (finance, sociology etc). I see only bosonic fields being used mostly. The only (minor) application of fermionic fields that I have come across is in section 7.9.1 in Baaquie's Quantum Finance book where nonlinear forward interest rates are modeled using fermionic fields.

Is there an intuitive reason for such asymmetry between bosonic and fermionic field usage at such scales?

Edit (6 Jan 2015): After posting this question, I came to know that bosons and fermions (along with bosonic and fermionic exclusion principles) are just two special cases of a more general statistic called Haldane's exclusion statistics. While this may sound like the familiar anyon statistics in condensed matter physics, they are not the same (for example, while anyons exist only in 2+1 dimensions, Haldane's statistics can be formulated for any dimension). For more on Haldane statistics:

So, the question now is, does Haldane's generalized statistics be used to describe non-quantum objects (in finance, sociology etc)?

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Determinantal Point Processes (1)(2) is a more formal theory for using fermionic models in stats and machine learning. – crackjack Mar 30 at 4:32
Maybe I should also highlight that the name 'Determinantal' in DPP comes about because matrix determinant is a way to encode fermionic properties in physics. For more, see pgs 4 & 5 of ref (2) in above comment. – crackjack Mar 30 at 17:54

Actually a paper recently came out, and highlighted in Popular Science, discussing using fermionic field concepts to model crowd avoidance at Netflix. You can imagine that the same concept could be used to consider in any situation where there are large numbers of people competing for limited preferred items.

## Update

Now that we have a few minutes, rather than give an expose of linked paper, I find it more appropriate to discuss some theory. First off, fermionic behavior is actually very relevant and widely used outside physics when it comes to economic problems. To illustrate this, first consider the graph below.

$\hskip1in$

People should recognize this as the standard textbook supply and demand diagram. There are different variations available, but I have made a few simplifications to aid in the explanation. Here both price and quantity are discrete (e.g. all quantities and prices are rational numbers), and are always positive.

To review some features:

• The demand curve is downward sloping and represents how price responds to the decreasing utility of increasing quantities. This means that as an individual acquires more of some product, the product has less and less value to the individual. This is reflected in the price a person is willing to pay for an additional item of the same product after they already have several.

• The supply curve represents the marginal cost a supplier incurs for each additional unit of a product they have. This can be a complicated concept to convey, because one usually thinks on the demand side of the equation (e.g. price goes down as quantity goes up...e.g. learning curves etc). One way to think about the supply curve is to realize that as a seller increases their quantity of goods, it becomes increasingly difficult to sell the good at a profitable price, so overhead and carrying costs begin to dominate the sellers cost, and so carrying large quantities of goods increases the sellers marginal cost.

• The point where the supply curve and the demand curve meet is the equilibrium point for the particular product. The equilibrium point has an associated equilibrium price and quantity. It is the point of where maximum efficiency and profit is achieved for a particular product.

• For our example, I am assuming that our seller is a retailer, and therefore does not produce its own products. So here the retailer buys a product on the wholesale market for a fixed price, and then adds some minimum markup to each item in order to reflect their basic fixed operating expenses (and in some cases their minimum profit margin). The dynamics of the wholesale market are not directly represented here, but we include a max quantity line where one would have to adjust the wholesale price if that quantity were exceeded. For our retailer, this represents an upper bound on total quantity of any one product that the retailer could possibly sell (this is one type of "fermionic" attribute...if it were "bosonic", we would assume that the wholesale price would be completely independent of quantity, and the quantity could be infinite).

Now the seller/retailer (or producer in our graph) wants the price to be as high as possible, and the individual buyer wants the price to be as low as possible. If the price is the equilibrium price, and the supply and demand curves hold true, then seller and buyer will maximize their benefits at the equilibrium price. When the price is lower, the buyer thinks they are "getting a bargain", and when the price is higher then the seller will make more profit off an individual sale. However, if the curves hold true, then the changes in price change the quantities that are sold, and a higher price will result in fewer sales, and a lower price will begin to reduce profits, and the utility of the product to the buyer.

The equilibrium price has an equilibrium quantity associated with it. This is another "fermionic" attribute, it is a natural limit to the quantities of a product that is being sold. It does not benefit the seller/retailer to carry more than the equilibrium quantity of any good within the context of that good. The statement "within the context of that good" is important when one starts considering the case of multiple products, where the function to be minimized is now a function of many goods.

Getting to some equations:

For an individual product, if we say $S(Q)$ is the supply price as a function of quantity, and $D(Q)$ is the demand price as a function of quantity, and $Q$ is quantity sold, then we could say the equilibrium is reached when:

$$\left( S(Q) - D(Q) \right) Q = 0$$

This just tells us that when the difference of the price functions is zero for a particular quantity, then you are at the equilibrium quantity, and the measured inefficiency is zero. For our particular graph we could write:

$$\left( K_SQ - K_{D_1}Q + K_{D_2}\right) Q = K_SQ^2 - K_{D_1}Q^2 + K_{D_2}Q = \left[ \left( K_S - K_{D_1} \right)Q^2 + K_{D_2}Q \right]= 0$$

Where the $K$'s are constants controlling the slopes of the curves and the demand curve has two constant values, one controlling the downward slope, the other controlling the individual's "ground state" price, or rather the price they would pay when the possess none of the commodity (it should be pointed out that this is example specific).

We can expand the minimizing function to account for individual buyers $i$ and individual commodities $j$ as follows:

$$\sum_{i,j} \left[ \left( S(Q_i) - D_i(Q_i) \right) Q_i \right]_j = 0$$ $$\sum_{i,j} Q_{ij} = Q_{total}$$

it is assumed that the supply curve $S(Q)$ for a commodity $j$ is not dependent on the buyer $i$ (e.g. the seller's curve is not tailored to an individual buyer...which is not always the case) but is determined by the total quantity $Q_j$ for some commodity $j$, and that $Q_{total}$ is the total number of items sold across the entire set of commodities $j$.

In our specific example we could write this equation as:

$$\sum_{i,j} \left[ \left( K_{S} - K_{{D_1}_i} \right)Q_i^2 + K_{{D_2}_i}Q_i \right]_j =0$$

and defining: $$\left( K_{S} - K_{{D_1}_i} \right) = M_i$$ we can write: $$\sum_{i,j} \left[ M_iQ_i^2 + K_{{D_2}_i}Q_i \right]_j =0$$ $$\sum_{i,j} Q_{ij} = Q_{total}$$

It should be recognized that in our simple example, the equation to be minimized $$\sum_{i,j} \left[ M_iQ_i^2 + K_{{D_2}_i}Q_i \right]_j =0$$ can be thought of as a series of hypersurfaces where inefficiency is present when the sum $\neq 0$.

In real systems, it is not likely that sum will $=0$. However, although there are squared quantities, and are not linear individually, the quantities are linear in our summation making it possible to consider linear optimization as a problem solving technique. In addition, the first derivatives of the individual equations should also sum to zero, taking away any particular concern about the squared quantities.

$$\sum_{i,j} \partial_{Q_i}\left[ M_iQ_i^2 + K_{{D_2}_i}Q_i\right]_j =\sum_{i,j} \left[ 2M_iQ_i + K_{{D_2}_i} \right]_j = 0$$

Since the $K_{{D_1}_i}$'s are constants they can be summed and moved to the right hand side of the equation:

$$\sum_{i,j} \left[ K_{{D_2}_i} \right]_j = K_{total}$$

$$\sum_{i,j} \left[ 2M_iQ_i \right]_j = -K_{total}$$

## Discussion

It should be pointed out that there is only one optimal equilibrium point, however, if in real systems it can be shown that the optimal point is unobtainable and is instead replaced by some other minimum value, then there may be multiple solutions to a given problem. This means there may be multiple allocation schemes that are deemed optimal.

It should also be pointed out that individuals who understand this simple theory are able to manipulate a given situation by temporarily moving a system from its equilibrium point. This is often a focus of wall street trading, but can make itself apparent in other areas of life. One example that comes to mind is the current use of qualifications in order to limit job opportunities to a select set of individuals.

Although job qualifications can be good by maintaining a minimum standard for employment, they can also be manipulated to limit job allocation to a smaller population. The inefficiency appears when people who are otherwise capable of performing a job are excluded due to some arbitrary set of constraints unrelated to job performance. Recent evidence indicates that some of the current economic malaise is directly associated with a market inefficiency trend in the form of qualification standards. This and many other examples can be shown to represent "fermionic" behavior in financial and sociological settings.

To get to the specific point as to whether "fermionic" models are better than "bosonic" models, the simple answer is that they already have been used and are regularly used, in particular, even the linked paper above discusses how many of these problems simplify to standard type linear resource allocation problems.

UPDATE 2

As discussed in the comments below and in the linked paper, occupancy constraints by restricting quantities or by reducing utility with increased quantities reflect "crowd avoidance" and thus represent "fermionic" behavior.

Edit Adjusted so that the supply and demand curves were combined in a more intuitive system based on signs of their slope. Further corrected to reflect the need for an additional constant to represent the downward sloping demand curve.

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I'd +1 if you could summarize the essential point. Links do go bad. In particular, what's the advantage of fermionic fields over a bosonic field that avoids itself due to charge? Say, a box full of $\pi^+$ particles. – DarenW Feb 14 '13 at 4:22
@DarenW incentives are always good, I will have to come back to this later today or tomorrow – user11547 Feb 14 '13 at 11:36
Hal, I've explained fermions as particles that need "personal space", and that get very upset (require more high-cost energy) if they don't get enough of it. For real particles this "space" has addresses with three parts (location, momentum, and spin), but many types of address spaces could be used without changing the idea. So, I found your answer fascinating... and alas, utterly incomprehensible! So, um... could you maybe point out to me (and likely others) exactly where the "fermionic" part of all that was? Again, I'm interested but baffled. – Terry Bollinger Feb 17 '13 at 5:44
@HalSwyers, we're in sync then. From my first very quick scan, I just didn't pick up that the critical anti-crowding component, which is indeed what makes a fermion a fermion. We see it everyday -- two objects don't like to be in the same space, because ordinary matter is fermionic. In the past I've had some brief but interesting email exchanges with Ginestra Bianconi, whose work was pointed out to me by Albert-László Barabási, on both bosonic and fermionic networking models. While my conclusion was that one must be careful in defining the entities involved, both concepts can be useful. – Terry Bollinger Feb 18 '13 at 18:52
@TerryBollinger Concur with your statements. What is interesting though is that I think we are coming full circle since economics has, as you point out, been fermionic in nature since its beginning. What would be interesting is how economic insights might translate into research into objects like Marjorana fermions – user11547 Feb 18 '13 at 19:24

According to this book (p. 61) "[this] paper discusses in detail how a Fermion and Boson field can be used to express the state of the human psyche."

I cannot expand.

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Apologies, I guess this doesn't qualify as a "macro context". – Keep these mind Feb 17 '13 at 13:18
@TerryBollinger Thanks? I'm not quite sure what you did to my answer. The line in the book is "The paper discusses in detail how a Fermion and Boson field can be used to express the state of the human psyche." I replaced 'The' with '[this]', as per convention. You removed '"[' and ']'. What did you "fix"? – Keep these mind Feb 22 '13 at 8:49
My bad; I just rolled back to your original correct version. I saw the square brackets and must have interpreted it as a slightly mangled link attempt. – Terry Bollinger Feb 22 '13 at 13:34
@TerryBollinger Thanks again. :) – Keep these mind Feb 22 '13 at 13:40