The Future of Everything

December 14, 2021

Quantum economics – the story so far

This piece gives a brief summary of my work to date (2016-2021) in quantum economics.

The idea that the financial system could best be represented as a quantum system came to me (dawned on me? evolved?) while working on The Evolution of Money (Columbia University Press, 2016). “Money objects bind the virtual to the real, and abstract number to the fuzzy idea of value, in a way similar to the particle/wave duality in quantum physics,” I offered. “Money serves as a means to quantify value, in the sense of reducing it to a mathematical quantity – but as in quantum measurement, the process is approximate.” Price is best seen as an emergent feature of the financial system. I summarised this theory in two papers for the journal Economic Thought: “A Quantum Theory of Money and Value” and “A Quantum Theory of Money and Value, Part 2: The Uncertainty Principle“.

While I had some background in quantum physics – I studied the topic in undergraduate university, taught a course on mathematical physics one year at UCL, and encountered quantum phenomena first-hand while working on the design of particle accelerators in my early career – my aim in the book (co-authored with Roman Chlupaty) was not to impose quantum ideas onto the economy. My primary research interest was in computational biology and forecasting and I had not touched quantum mechanics in many years. The dual real/virtual nature of money just had an obvious similarity to the dual nature of quantum entities, and in fact I was surprised that I appeared to have been the first to make this connection in a serious way and come up with a quantum theory of money.

I was aware that a number of researchers were working in applying quantum models to cognition and psychology, but it was only after finishing the book that I learned about the area of quantum finance (I also discovered a separate paper on “Quantum economics” by the physicist Asghar Qadir from 1978, which argued that the quantum formalism was well suited to modelling things like economic preferences). The reason I hadn’t come across these works in my research about money was because just like in neoclassical economics there was no discussion of that topic. Nor was there much discussion of quantum phenomena such as entanglement or interference. Instead the emphasis in quantum finance (as this paper notes) was on using quantum techniques to solve classical problems such as the Black-Scholes option-pricing algorithm, or portfolio optimisation.

My motivation was completely different. In books such as Economyths, and The Money Formula (with Paul Wilmott), I had investigated the drawbacks and limitations of these traditional models – so rather than invent more efficient ways of solving them, I wanted to replace them with something more realistic. Money was the the thing which linked finance and psychology, so a quantum theory of money could be a first step in developing a new approach to economics.

I sketched out the basic idea as an Economic Thought paper “Quantum economics” which served as a blueprint for my 2018 book of the same name. It tied together the quantum theory of money, with ideas from quantum finance, quantum cognition, quantum game theory, and the broader field of quantum social science. The ideas were also summarised in a piece for Aeon magazine – which was when I found out why no one had probably bothered to develop a quantum theory of money. The article was not well received, by economists but especially it seemed by physicists, some of whom went out of their way to trash the idea.

I was not new to having my work come under criticism. Indeed, much of my career has focused on pointing out the drawbacks and limitations of mathematical models, which has frequently brought me into conflict with people who don’t see it that way, starting with my D.Phil. thesis on model error in weather forecasting (see Apollo’s Arrow). My book Economyths also drew howls of outrage from some economists. However quantum economics felt different, and seemed to touch on a range of taboos, in particular from physicists who have long resisted the adoption of quantum ideas by other fields. But quantum mathematics is not owned by physicists, it is simply an alternative version of probability which was first used to model subatomic particles, but also can be used to describe phenomena such as uncertainty, entanglement, and interference which affect mental systems including the economy.

While writing the book I developed in parallel an online mathematical appendix which presented some key results from quantum cognition, finance, and game theory (an early version was translated into Russian). Because my aim was to develop a theory of quantum economics, I also started applying quantum methods to some key economic problems, including supply and demand, option pricing, stock market behaviour, and the debt relationship which underlies the creation of money. This online appendix later grew into my technical book Quantum Economics and Finance: An Applied Mathematics Introduction, first published in 2020 and now in its second edition.

For supply and demand, my idea was to model the buyer and seller in terms of a propensity function, which describes a probabilistic propensity to transact as a function of price. A simple choice is to describe the propensity function as a normal distribution. The joint propensity function is the product of the buyer and seller functions. The next step is to use the concept of entropic force to derive an expression for the forces which describes the tendency for each party to move the price closer to their preferred price point. The joint force is just the sum of the forces for the buyer and seller. However there is a contradiction because the probability distribution does not match that produced by an oscillation. To resolve this, we quantize the force to obtain a quantum harmonic oscillator whose ground state matches the joint propensity function. This model, which sounds elaborate but is actually quite minimal in terms of parameters, applies to economic transactions in general, so has numerous applications, including the stock market. The paper “A quantum model of supply and demand” was published in the journal Physica A in 2020.

Typical propensity functions for buyer (to the left), seller (right), and joint (shaded).

The question of how to price options is one of the oldest problems in finance. The modern method dates back to a 1900 thesis by Bachelier and is based on the concept of a random walk. For the quantum version, the logical place to start was with the quantum version of this which is a quantum walk. Instead of assuming that the log price will follow a normal distribution with a standard deviation that grows with the square-root of time, the model has two peaks which speed away from each other linearly in time. It therefore captures the psychological stance of an investor who has a bullish or bearish view on the asset (e.g. price might grow by 10 percent each year), but balances that with the possibility that the opposite might happen in order to obtain a fair price for the option. When coupled with the quantum model of supply and demand, the algorithm can be used to predict option price and volume. “A quantum walk model of financial options” was published in Wilmott magazine in 2021, and the theory was reported on the same year by the Economist in an article “A quantum walk down Wall Street“.

Probability distribution for a quantum walk (solid) versus random walk (dashed).

Finally a main question in quantum economics is the interaction between mind and money which underlies the debt relationship, and also the creation of money objects in the first place. Both of these topics are traditionally neglected in mainstream economics. In quantum economics it is easy to show that the debt relationship can be modelled as a simple circuit with two qubits, representing the debtor and creditor, entangled by a C-NOT gate which represents the loan contract. Interestingly, it turns out that the same circuit can be used to represent the decision-making process within the mind of a single person, where there is an interplay between a subjective context and the final decision. In quantum cognition, this is usually modelled as a two-stage process; however it can also be modelled using two entangled qubits, in which the context and the decision are separated out, as in the debt model. This result was published in a 2021 Frontiers in Artificial Intelligence paper, co-authored with Monireh Houshmand, called “Quantum propensity in economics“. A related paper published in Quantum Reports, that discusses applications including mortgage default, is “The color of money: threshold effects in quantum economics“. 

Two-qubit entanglement circuit for debt contract (A is debtor, B is creditor), or quantum cognition (A is context, B is decision).

For a full list of my research in quantum economics and finance, including links to these and other papers, please see the page Quantum Economics Resources. These findings and others are also presented in my technical book Quantum Economics and Finance: An Applied Mathematics Introduction, and for a general audience in Money, Magic, and How to Dismantle a Financial Bomb: Quantum Economics for the Real World (available 02/2022). The work continues! – if readers are interested in getting involved, please drop me a line here or through LinkedIn.

March 1, 2021

QEF11 – The money bomb

In quantum economics money plays a role rather like energy – so what happens when we get an awful lot of energy together in the same place?

In 1945 the Trinity Test convincingly demonstrated that quantum forces are not limited to the subatomic domain, but can scale up to affect our lives. Since then we’ve learned to exploit the energy inside the atom in different ways, from peaceful nuclear reactors to more recently quantum computing.

Money is a quantum social technology with quantum properties that also scale up to affect the economy as a whole.

Perhaps the most important concept in physics is that of energy. The Trinity Test released some 92 terajoules of energy which is about the same as 20 tank cars full of oil. There is also energy in finance. As an example an empty house close to where I live recently sold for 2.24 million Canadian dollars. One way to equate that with energy is to do a thermodynamic analysis of the world economy and figure out how much energy is needed to maintain a dollar bill. Another way is to simply figure out how much oil it would buy, and actually either method gives roughly similar results which is about 40 tank cars of oil or two Trinity Tests. That’s a lot of energy, so where does this energy come from?

Most money is created not by central banks but by private banks lending money for things like mortgages on houses. As the deputy governor of the Bank of Norway explained in a 2017 speech, when you borrow from a bank the bank just credits your bank account, the bank does not transfer the money from someone else or from a vault, it’s simply created by the bank itself out of nothing. We have already seen this process for a Medieval tally stick, so now instead of a sovereign backed by the divine right of kings, we have your local bank making a loan to someone who wants to buy a house. Instead of a tally stick the output is going to be new money which as we have seen is associated with energy through the equation \Delta E \approx \frac{\hbar a x_0}{2} \log \frac{1}{p} . Here the \frac{1}{p} term represents the degree of coercion, so this is ultimately where the energy is coming from.

Now in physics Bohr’s principle of correspondence states that at large enough scales quantum mechanics should converge to classical, but things like nuclear devices don’t wash out. Nuclear reactors are based on self-sustaining nuclear chain reactions, and we get the same kind of self-sustaining exponential growth in the economy. Banks make loans on real estate, this adds to the money supply, it’s used to buy more real estate, and so on, so you find that the growth in house prices in Canada at least matches quite closely the growth in the money supply. And the system needs to grow continuously in order to pay the interest on the debt. It’s ironic that Aristotle thought that money was sterile and should not “breed” as he put it by handing out interest but we ended up with what amounts to a financial breeder reactor. And the product of all this is asset price inflation and inequality.

On top of all this debt is a system of derivatives which has an estimated notional value of some one quadrillion dollars. This quadrillion dollars is really a kind of a magical number because it’s larger than the actual economy. So how do macroeconomists model this financial sector which is so huge? Well really they don’t – one reason the banking crisis of 2007/8 wasn’t predicted was because banks weren’t in the models.

Since then there have been efforts to incorporate financial “frictions” into these models, but when you think about it this is odd because finance is really the the opposite of friction, and a crisis is what happens when friction breaks down completely, which is why plots of price changes during the crisis resemble plots of tremors during earthquakes.

These concerns about the financial system are not exactly new. One person to raise them was Frederick Soddy who was awarded the Nobel Prize in chemistry in 1921 but switched to economics because of his fear that a financial crash would lead to nuclear conflict. Soddy described money as virtual wealth that gives the illusion of being unbound by earthly constraints, and believed that we had to align human law and convention with the needs of thermodynamics.

His solution to all these problems was basically to stop paying tax in order to make “a clean sweep of all the webs woven to entangle humanity by the magicians who have discovered how to get something out of nothing and moreover to make it bear perennial interest.” I’m not sure that strategy would really work today but the split between the virtual and the real which he highlighted was very evident in the spring of 2020 when we had two headlines at the same time: US stocks have their best month since 1987; US now has 22 million unemployed.

This real-virtual split in money is perhaps most evident in our attitude towards the environment. Mainstream economics treats the planet as an inert object, and environmental damage as a market failure (even though markets are optimizing the numbers by growing as fast as they can). One of the main contributions of the quantum approach is to draw our attention to this dual real-virtual nature of money and help us to align our economy and our use of energy with the thermodynamic realities of the planet.

Further reading:

Orrell D (2018) Quantum Economics: The New Science of Money. London: Icon Books.

Soddy F (2003). The Role of Money: What It Should Be, Contrasted with What It Has Become. London: Routledge.

Previous: QEF10 – A Quantum Option Pricing Model

Quantum Economics and Finance playlist

Playlist: Quantum Economics and Finance

February 14, 2018

Of minds and money

Filed under: Economics — Tags: — David @ 3:04 pm

Post written for the Rebuilding Macroeconomics blog of the UK’s Economic and Social Research Council (ESRC), 23 February 2018

In 2017, when the ESRC announced that it was setting up a network of experts from different disciplines to “revolutionise” the field of economics, I thought it sounded like a very good idea. But in the spirit of that disruptive mandate, what if the revolution has already happened – and economics just needs to recognise it? I am talking about the quantum revolution which overturned physics at the start of the last century.

Of course, the thought that economics has anything to do with quantum physics will sound to many economists (or physicists) like a particularly severe case of physics envy. But my point is not that economics can be reduced to quantum physics (in my book Economyths I argue against such reductionist approaches). Nor is to assert that we can use quantum physics as a fuzzy metaphor to understand the economy. Instead, it is to say that the economy is a complex quantum system in its own right and should be treated as such. This recognition has the potential to disrupt all the pillars of mainstream economics, including its models of human behaviour, markets, and the macroeconomy.

To see why (and starting with the obvious), recall that quantum physics grew out of the empirical discovery by scientists that energy is transmitted, not continuously, but in discrete chunks they called quanta, from the Latin for “how much”. This violated the basic principle that Natura non facit saltum (nature makes no sudden leaps), as the epitaph of Alfred Marshall’s Principles of Economics read at the time (and until the final 1920 edition). But of course the same is true of money. When you pay for an item with a card, the money doesn’t drain out in a continuous flow, it goes as a discrete transmission.

Quantum objects can magically appear out of the void, and disappear back into it. In the economy, we have the creation of money by private banks issuing loans, and its destruction when loans are repaid. Again, these are discrete processes that take place “instantaneously and discontinuously” as a 2015 Bank of England working paper observed, so don’t fit easily with the continuous nature of conventional models – one reason they are typically excluded.

Another discovery of quantum theory is that matter has two complementary aspects, a “real” particle aspect and a “virtual” wave aspect. Money too combines the virtual properties of a number, with the real properties of an owned object – a bitcoin seems pretty virtual, unless you lose the computer it’s on – and it is this dualistic nature which gives money its confounding properties.

In quantum mechanics, a particle doesn’t have a well-defined location or momentum. In the economy, prices of something like a stock, or a house, are similarly indeterminate, and there is no unique intrinsic value (or eigenvalue in the quantum jargon). This is why some traders, along with researchers in the field of quantum finance, model the markets as a quantum system, with its own version of an uncertainty principle.

One of the more mysterious features of quantum reality is that subatomic particles can become entangled so that a measurement on one instantaneously affects the state of the other. Suppose I take a mortgage out with a bank. If I choose to default, then from that moment on the state of the loan has changed. Of course it will take time for the bank to learn what happened and settle the matter, but it takes time to measure particles too, during which other effects may come into play. It is a financial version of the social entanglement studied in quantum social science (see for example the 2015 book Quantum Mind and Social Science by political scientist Alexander Wendt).

Finally, another key property of quantum systems is interference: as in the double-slit experiment, where shining a light beam through two slits produces an interference pattern on the other side, even if only one photon passes at a time. Of course, money objects don’t interfere with each other, which is good because otherwise they might cancel out in your pocket. But the purpose of money is to put a number on the idea of value, and dissonance between these two things – objective number and subjective value – does produce interference patterns in our minds, of the sort modelled by researchers working in the area of quantum cognition.

In fact, instead of following classical logic when making decisions, it turns out that we are better described as following a version of quantum logic. As physicists Vyacheslav Yukalov and Didier Sornette note, “It is the appearance of interference terms that makes the structure of quantum expressions richer than the related classical ones and that allows one to explain those psychological phenomena that, otherwise, are inexplicable in classical decision making.”

As an example of a quantum economics phenomenon, suppose that you come under financial stress and are deciding whether to pay the latest installment on your mortgage, or default. Your mental state – and that of the loan – can then be treated, following the methods of quantum cognition, as being in an indeterminate superposition of two states. The emergent consequences in the entangled economy can be modelled at a societal level using complexity methods, including quantum agent-based models, though as always the system eludes reduction or exact computation.

Until now, money has played no more role in the quantum social sciences than in neoclassical economics, where it has traditionally been treated as an inert medium of exchange. But as seen above, quantum ideas are perfectly suited to things like the creation of money, entanglement through loans and other contracts, and credit default – all of which were at the heart of the 2007/8 crisis, but nowhere in the macroeconomic models.

As I argue in a discussion paper and in more detail in a forthcoming book, the quantum approach has implications not just for finance, or macroeconomic modelling, but for all the basic assumptions and conclusions of mainstream economics; and points the way to a new economics which is to its neoclassical version what quantum physics was to its classical version. Economics isn’t physics, but can learn from it; and if the intention is to rebuild macroeconomics, the natural place to start is with the quantum, dualistic nature of both mind and money.

October 20, 2017

A Quantum Theory of Money and Value, Part 2: The Uncertainty Principle

Filed under: Economics, Forecasting — Tags: — David @ 4:53 pm

New paper in Economic Thought

Abstract: Economic forecasting is famously unreliable. While this problem has traditionally been blamed on theories such as the efficient market hypothesis or even the butterfly effect, an alternative explanation is the role of money – something which is typically downplayed or excluded altogether from economic models. Instead, models tend to treat the economy as a kind of barter system in which money’s only role is as an inert medium of exchange. Prices are assumed to almost perfectly reflect the ‘intrinsic value’ of an asset. This paper argues, however, that money is better seen as an inherently dualistic phenomenon, which merges precise number with the fuzzy concept of value. Prices are not the optimal result of a mechanical, Newtonian process, but are an emergent property of the money system. And just as quantum physics has its uncertainty principle, so the economy is an uncertain process which can only be approximated by mathematical models. Acknowledging the dynamic and paradoxical qualities of money changes our ontological framework for economic modelling, and for making decisions under uncertainty. Applications to areas of risk analysis, forecasting and modelling are discussed, and it is proposed that a greater appreciation of the fundamental causes of uncertainty will help to make the economy a less uncertain place.

Published in Economic Thought Vol 6, No 2, 2017. Read the full paper here.

October 10, 2016

More quantum money

Filed under: Economics — Tags: , , — David @ 2:37 pm

New discussion paper at Economic Thought is called A Quantum Theory of Money and Value, Part 2: The Uncertainty Principle.

Here is the abstract:

Economic forecasting is famously unreliable. While this problem has traditionally been blamed on theories such as the efficient market hypothesis or even the butterfly effect, an alternative explanation is the role of money – something which is typically downplayed or excluded altogether from economic models. Instead, models tend to treat the economy as a kind of barter system in which money’s only role is as an inert medium of exchange. Prices are assumed to almost perfectly reflect the ‘intrinsic value’ of an asset. This paper argues, however, that money is better seen as an inherently dualistic phenomenon, which merges precise number with the fuzzy concept of value. Prices are not the optimal result of a mechanical, Newtonian process, but are an emergent property of the money system. And just as quantum physics has its uncertainty principle, so the economy is an uncertain process which can only be approximated by mathematical models. Acknowledging the dynamic and paradoxical qualities of money changes our ontological framework for economic modelling, and for making decisions under uncertainty. Applications to areas of risk analysis and economic forecasting are discussed, and it is proposed that a greater appreciation of the fundamental causes of uncertainty will help to make the economy a less uncertain place.

Download the paper here.

October 8, 2016

Notes on the quantum theory of money and value

Filed under: Economics — Tags: , — David @ 12:38 am

Following the publication in Economic Thought of my paper “A Quantum Theory of Money and Value” I have received a number of interesting comments and questions from readers, and this post is an attempt to clarify some of the points which came up. For a description of the theory, please see the paper, or (for the book version) The Evolution of Money.

What is a money object?

These are objects – either real or virtual – which have a fixed numerical value in currency units. Just as quantum objects have dual real/virtual properties, so do money objects (bitcoins don’t seem like objects, until you lose the hard drive they are located on). Money objects are unique in that they have a fixed numerical price. Other objects or services attain their price by being traded for money objects in markets.

Is money an emergent phenomenon?

Money objects are designed (e.g. by the state) to have a set price. The prices of other things emerge as the by-product of money-based markets, which themselves emerge into being as money objects become commonly used. Therefore prices and markets can be viewed as emergent phenomena, but money itself is better seen as a carefully designed technology. (Of course the way that e.g. cybercurrencies emerge into actual use, as markets develop around them, can also be described as an emergent phenomenon.)

What does money measure?

Nothing. Because prices emerge from the use of money objects, one consequence is that price should not be viewed as an accurate measure of “labor”, “utility”, “economic value”, or any other quantity. Money is better viewed as a fundamental quantity, like electrical charge. Money objects, as used in markets, are a way of attaching numbers to things, but that is not the same as measuring them in some way. Of course market forces tend to align prices with some vague idea of value, but the process is far from exact, and money has its own dynamics (which is one reason CEOs in the US earn over 300 times the median wage of their employees). Note this contradicts the Aristotelian idea, later expressed by Aquinas, that money was “the one thing by which everything should be measured.”

Why quantum?

The comparison with quantum theory comes about because money is treated as a fundamental quantity (from the Latin quantum); and money objects are a way of combining the notions of number and value, which are as different from one another as the dual wave/particle properties of matter. For example, number is stable, while value varies with time. Money objects are therefore fundamentally dualistic.

As mentioned in The Evolution of Money, other authors and economists (and many others) have used the term “quantum” in different ways. One example is Charles Eisenstein’s Sacred Economics, where in an appendix called “Quantum Money and the Reserve Question” he notes “the similarity between fractional-reserve money and the superposition of states of a quantum particle,” in the sense that money can seem to exist in more than one place at the same time. The quantum macroeconomics school, also known as the theory of money emissions, which dates to the 1950s, gained its name from the idea that production is an instantaneous event that quantizes time into discrete units. A completely different concept is quantum money, which exploits quantum physics in an encryption technique.

What inspired the approach?

One thing is the history of money. The most concrete example of a money object is a coin, which consists of a number pressed into a piece of metal. These date to the time when Greek philosophers were developing the first theories of mathematics. Pythagoras believed that the universe was based on number, and money can be seen as a way of making that true by impressing numbers onto the real world. However mixing the properties of number and things produces a strange kind of alchemy. See this presentation for the 2015 Marshall McLuhan lecture at transmediale in Berlin for a discussion.

How does this differ from the usual understanding of the role of money?

One consequence of the theory is that it inverts the usual narrative of mainstream economics. Since the time at least of Adam Smith, economists have downplayed the importance of money, seeing it as a kind of neutral chip that emerged as a way of facilitating barter. But instead of money emerging from markets, it is more accurate to say that the use of money (jumpstarted by the state) prompted the emergence of markets. And far from being an inert chip, money is an active, dualistic substance with powerful and contradictory properties. Putting numbers on things changes the way they behave.

What is the mathematical map or connection between price and value?

In general there is no such map. Price is an emergent property, which means it need not be computable at all. Of course it is possible to come up with some rules of thumb, but there are no fundamental laws as in physics.



The quantum properties of money and the economy as a whole are explored in the new book Quantum Economics: The New Science of Money. For a discussion of the mathematical background, please see: Introduction to the mathematics of quantum economics.

September 6, 2015

The true value of money: What would a revolution of economics look like?

Filed under: Economics — Tags: , — David @ 11:50 am

Economics needs a revolution.

This sentiment has been expressed by people from the physicist turned hedge-fund manager Jean-Philippe Bouchaud (in a 2008 paper), to the Bank of England’s Andrew Haldane (in a 2014 foreword for Manchester’s student-run Post-Crash Economics), to activist groups such as Kick It Over. So what would such a revolution look like?

Perhaps the archetypal model for a scientific revolution is the quantum revolution that shocked the world at the turn of the last century. In the space of a few short years, almost everything that was known about the nature of matter was overturned. The Newtonian view of the world as a predictable machine crumbled with it.

Except, that is, in economics – which continues to base its models on quasi-Newtonian economic laws.

Read the full article at Adbusters.

August 27, 2015

99 Bitcoins interview

Filed under: Economics — Tags: , , — David @ 1:00 pm

Chatted about cryptocurrencies with Maria Santos from 99 Bitcoins, full interview is here.

August 19, 2015

Bitcoin needs a new theory of money

Filed under: Economics — Tags: , — David @ 4:06 pm

Is bitcoin money? To its users the answer will be a resounding yes, but to many people, including former Federal Reserve Chairman Alan Greenspan, the answer is less clear. Indeed, one of the things holding back the adoption of cybercurrencies such as bitcoin is that they do not conform with traditional ideas about money.

Read the rest of the article at Bitcoin Magazine.

July 19, 2015

A Quantum Theory of Money and Value

Filed under: Economics — Tags: , — David @ 9:09 pm

Abstract of a new paper posted to SSRN:

The answer to the question “what is money?” has changed throughout history. During the gold standard era, money was seen as gold or silver (the theory known as bullionism). In the early 20th century, the alternative theory known as chartalism proposed that money was a token chosen by the state for payment of taxes. Today, many economists take an agnostic line, and argue that money is best defined in terms of its function, e.g. as a neutral medium of exchange. This paper argues that none of these approaches adequately describe the nature of money, and proposes a new theory, inspired by non-Newtonian physics, which takes into account the dualistic real/virtual properties and complex nature of money. The theory is applied to the example of the emergence of cybercurrencies.


Read the full article here.

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