## Abstract

This paper tallies the links between fluid mechanics and quantum mechanics, and attempts to show whether those links can aid in beginning to build a formal template which is usable in economics models where time is (a)symmetric and memory is absent or present. An objective of this paper is to contemplate whether those formalisms can allow us to model information in economics in a novel way.

## 1. Introduction

The social sciences, especially economics and finance, have for many years been interacting, to varying degrees of intensity, with the exact sciences. We should really distinguish two important movements. First, the ‘Econophysics’ movement which applies formalisms from statistical mechanics to the social sciences, and championed by Eugene Stanley and others (for instance [1–6]). Second, the movement which applies the mathematical apparatus from quantum information to the cognitive and social sciences, and championed by Andrei khrennikov and others (see [7–12]). In both of the above research domains, the *Philosophical Transactions of the Royal Society A* had recent special issues: see Burger *et al.* [13] and Haven & Khrennikov [14]. There is an intense amount of work which is underway in applying quantum physical concepts to other areas of (social) science, especially in psychology and now also in biology. The recent work by Asano *et al.* [15] sheds light on how quantum information theory can explain information processing in biological systems.

In the rest of the paper we will consider the topic of formalisms which may allow us to model ‘information’ in an economics environment. We will consider several approaches. In §2, we consider a model, which is essentially a ‘no memory’ model with an asymmetry between past and future times. We also briefly discuss a ‘no memory’ model but which assumes symmetry between past and future times. Section 3 of the paper considers another model which is a memory-based model. We conclude in §4 of the paper.

## 2. A no memory model with asymmetry between past and future time

We commonly use in asset pricing a version of Brownian motion which allows for the normal density of price returns and the lognormal density of prices. The so-called Black–Scholes PDE [16] which, when solved, yields the prices of financial option contracts, assumes exactly this process for the time evolution description of the asset which underlies the financial option. Such a Brownian motion has white noise, i.e. it does not allow for memory between price occurrences. However, Commissariat [17] reports that the usual assumption that ‘kicks’ made to a descending particle, in say a glass of water, are random, may not always be true. When the densities of the particle and fluid are similar the kicks are not random, and ‘persistent correlations’ are predicted between the motions of the fluid and the particle. This gives rise to the idea of so-called coloured noise. We will briefly come back to density differences between liquids in §3 of our paper.

Brownian motions with a so-called Hurst exponent allow for the introduction of memory. Stochastic differential equations with such exponent were popular some years ago (but less so now). The paper by Rodgers [18] is an excellent example of fruitful applications in the area of arbitrage.

In some recent work (Haven [19,20]), we have tried to make the case that there may be scope to use elements of the Nelson approach [21,22], which essentially looks at quantum mechanics from a Newtonian perspective. For a discussion on how a particle model can agree with quantum mechanics, see Sutherland [23].

Nelson's approach shows the emergence of the so-called quantum potential, which we will discuss below.

We provide some important elements of the theory. We follow Paul & Baschnagel [24]. The theory assumes a Brownian motion: d*x*(*t*)=*b*_{+}(*x*,*t*) d*t*+*σ* d*W*(*t*), where d*W*(*t*) is a Wiener process, *σ* is the diffusion coefficient and *b*_{+}(*x*,*t*) is the drift function. It is assumed that *E*(d*W*(*t*))=0, and one defines
2.1

Similarly, a Brownian motion of the type: d*x*(*t*)=*b*_{−}(*x*,*t*) d*t*+*σ* d*W*(*t*) is then proposed, and one defines then
2.2

Two velocity fields are then defined: 2.3and 2.4

In Newtonian mechanics *b*_{+}(*x*,*t*)=*b*_{−}(*x*,*t*), and hence *u*(*x*,*t*)=0. We note that and if *u*(*x*,*t*)≠0, there is asymmetry between past and future time. Thus, the model proposed here has no memory (using the Wiener process construction) but there is explicit asymmetry between past and future time. This asymmetry between future and past time is important. One can show that
2.5

where *p*(*x*,*t*) is a probability density function. Say if, , with *R* some amplitude function, then *u*(*x*,*t*)=*σ*^{2}∇*R*(*x*,*t*). We could make the argument that *u*(*x*,*t*) is narrowly related to Fisher information, which we already defined above and which we can also write as
2.6

The quantum potential, which we mentioned before, can be defined as (omitting mass and Planck constant): (1/*R*)∇^{2}*R*. It seems thus quite clear that without this asymmetry between past and future we cannot make this link to Fisher information. When Fisher information is being optimized (and that even within an economics-based information setting), it has been shown by Hawkins & Frieden [8,25] to be closely linked to a Schrödinger-like PDE, for details, see the discussion on Fisher information in Heifetz & Cohen [26].

What we thus see here is an emergent structure. The asymmetry of past and future time allows us to use Fisher information which itself is closely allied to a quantum-like construction, i.e. the Schrödinger-like PDE. We can also take this in another (but related) direction. The quantum potential which emerges out of the Nelson approach is also of use in economics as an information device. An extended Newton law: *m*.*a*=−∇(*V* +*Q*), where *V* and *Q* are, respectively, the real and quantum potentials can be estimated from real data. *Q* can be estimated (from the probability density function on return data for instance). One tears out *R* from the pdf, , and substitutes this *R* into *Q*=(1/*R*)∇^{2}*R*. In a recent paper, Tahmasebi *et al.* [27] derive the quantum potential for the Standards & Poor financial index and they find a quantum potential with infinite walls for short-time scales (small price variations). They find the quantum potential does change when the time scales are lengthened. This is a very intuitive finding. The real potential has now also been shown to be of great practical use in recent work by Baaquie [28] where he shows that this potential, when minimized, yields an equilibrium price formulation which ends up to be richer than the traditional basic microeconomics equilibrium formulation. Furthermore, real potentials have now been estimated for real-world commodities by the same author.

The above development involving the quantum potential is also known under a different name: Bohmian mechanics (see Bohm & Hiley [29]).^{1} We have invoked in previous work, the use of Bohmian mechanics in finance, especially from the promise such approach brings in terms of being able to formalize information in the setting of the pricing of assets (see, for instance, [8,19,25,30,31]. Bohmian mechanics brings with itself the issue of non-locality which means that the outcome of a measurement on one particle is immediately notable on another particle, irrespective of the distance between the particles. One may argue that non-locality is closely connected to the appearance of the quantum potential [31], pp. 106–107 and [32], p. 282. The link between non-locality and hidden variables is well known and important recent experimental evidence indicates that non-locality may exist *without* invoking hidden variables [33].

What can we do with an approximation of non-locality at a macroscopic level, especially in the area of asset pricing? Consider a simple ‘Alice and Bob’ example, where each of the protagonists are rocketed away to planets which are very far apart from each other. Assume Bob is going to planet ‘Tito’ and Alice is going to planet ‘Nito’.

Consider three simple scenarios.

—

*Scenario*1. Alice and Bob each have an envelope with either a white or black square inside. They know there are two squares in all—one black and one white. Bob opens the envelope when he is on planet Tito and let us assume he has a black square. Hence, he instantaneously knows that Alice, on planet Nito, has the white square. This is not precisely non-locality. Clearly, no physical message is sent from Alice to Bob (as this would imply there is a physical transmission speed in excess of the speed of the light and so on). Note also there is no uncertainty here at all.—

*Scenario*2. Assume from the start there is 1 black square and 5 white squares. Bob and Alice know this distribution of squares from the start. If Bob opens the envelope when he is on planet Tito and he finds 1 black square then he instantaneously knows Alice must have a white square. However, if he opens the envelope and finds a white square instead, then he is unsure what Alice may have. This scenario clearly exemplifies the existence of uncertainty.—

*Scenario*3. Assume from the start there are 3 white and 3 black squares. For either Bob or Alice when they open their envelopes, there is uncertainty as to what the other person has as square colour. This scenario again exemplifies the presence of uncertainty.

Now let us consider the above scenarios again, but now with a twist. Imagine that when both the spaceships of Alice and Bob pass some red line in the universe, the colours of their squares instantaneously reverse. Alice and Bob do not know about this colour reversion. Let us re-consider the scenarios.

—

*Scenario*1′. Bob had before the ship crossed the red line a black square. Bob has now a white square when he opens the envelope. He knows instantaneously that Alice must have black. There was no uncertainty in scenario 1 and the ‘red line event’ has not created any less or more uncertainty.—

*Scenario*2′. Having crossed the red line, there are now 5 black squares and 1 white one. Bob's white square now makes him think (using the scenario 2) he is not sure about Alice's colour. Clearly, the ‘red line’ event has augmented the uncertainty in this scenario.—

*Scenario*3′. After the red line event, the 3 white squares have become 3 black squares and the 3 black squares have become 3 white squares. The uncertainty, which we already had before the red line event, has not been changed.

After considering these simple scenarios, we could argue that the presence of information reduces uncertainty. If an information event were to augment uncertainty, then in fact this information adds to dis-information, i.e. the opposite of information. A formal definition of information can be given by the so-called Fisher information, *I*, defined as , where *P*(.) is the pdf on noise ‘*x*’ derived from *x*_{obs}:*x*_{obs}=*x*_{0}+*x*. Hence, in scenario 2′, the red line event creates more noise, and thus augments uncertainty. Hence, the red line event reduces (Fisher) information. The red line event does not create (or reduce) information in either scenario 1′ or scenario 3′.

Since the red line event does not change uncertainty in scenarios 1, and 1′; 3 and 3′, if we were to look for an approximation of instantaneous knowledge (approximating the notion of non-locality), we could find it in scenarios 1 and 1′, and (with uncertainty) in scenarios 3 and 3′. But in scenarios 2 and 2′: the event space has changed unbeknownst to Alice and Bob and this change has diminished information. We can almost argue for an approximate level of ‘non-local’ dis-information.

Translating the Alice and Bob example in the so-called risk-neutral pricing approach, which we use in modern asset pricing, would lead to some conundrums. Risk-neutral pricing essentially uses non-event-based probabilities to force expected returns of risky assets to be risk free. The importance of using a risk-free rate of return on risky assets provides for a superior advantage in modelling the pricing of assets: such a rate is *objectively* determinable.^{2} The essential problem with using risky rates of return is that they reflect so-called preferences for risk by decision-makers, and those preferences are of course not unique and hence they are notoriously difficult to model. In fact, the risk-neutral pricing setting would require the red line event to be much more sophisticated. Instead of simply swapping colours, one would need to create a red line event where black and white squares would now be converted into black and white squares with varying degrees of intensity. In other words, the level of ‘non-local’ dis-information would be much higher than in scenario 2′. If we want to approximate instantaneous transmission of information, then this would be even more difficult in this setting.

Here are a couple of points we may want to keep in mind while progressing through this paper:

— In a risk-neutral price setting, an approximation to non-locality when it relates to instantaneous transmission of information seems to be difficult to fathom.

— Non-locality can be shown to be tightly connected to the existence of the quantum potential.

— The quantum potential emerges in the Nelson approach.

— The Nelson approach implicitly shows us that asymmetry of past and future time seems to be an important condition for Fisher information to exist.

— Optimizing Fisher information (in an economics setting) leads to Schrödinger-like PDE's.

— The quantum and real potentials can be estimated from real data and their interpretation can be shown to be quite intuitive.

— Finally, we discussed so far a no-memory model: the stochastic differential equations used here employ white noise.

An important issue which we should worry about is whether asymmetry between past and future information is a palatable condition in social science. Modern academic finance has debated such asymmetry, while some finance academics will continue to assume that only current information in the price of an asset can be used to predict a future price, behavioural finance will think otherwise [34,35].

An interesting model, which firmly resides in modern finance and which clearly assumes symmetry between past and future time, is developed by Detemple & Rindisbacher [36], who show that one can provide for a formalism which models, next to public information, so-called private information. The authors define the private information price of risk (PIPR) as representing the changed price of risk when private information becomes available [37]. They argue that the information gain (public information with the addition of private information) can be measured with an appropriate entropy measure. The full model occurs within a memoryless (white noise) environment.

## 3. Laying the seeds for a ‘new’ memory model?

The former section of this paper attempted to argue that the Nelson approach and the ensuing quantum potential (which is also part of Bohmian mechanics) can be used in a financial setting. We also discussed some issues which are much harder to resolve such as the asymmetry of time which is seemingly needed to be able to define Fisher information. We also tried to highlight, in a very heuristic way, some possible problems which may surface when considering an approximation to non-locality (which itself is very much tied to the existence of the quantum potential).

All the arguments so far have been tied to a framework where we essentially claim that we use concepts and techniques from quantum mechanics, without in any way pretending, that the macroscopic world is quantum mechanical. In this section, we actually consider a macroscopic version of a model which is steeped in quantum mechanics. The model which will be discussed is also supported by experimental evidence (see [38–42]).

The proposed model looks at a so-called walking droplet and the explicit ‘context’ here is definable as follows:

— The preparation is necessary

*before*time*t*=0: the droplet bounces on a surface which needs to vibrate in a certain way (droplets will otherwise coalesce with the liquid they drop on (the droplet is made out of the same liquid)).— The preparation affects the trajectory on

*t*=0 and following: the damping time of the wave formed by the impact of the droplet on the vibrating surface, is dependent on the vibration frequency distance from a certain threshold.

Given the above conditions, we may wonder whether we have here ‘context conditioning’ versus ‘event conditioning’? We do not want to imply that, even if there were to be scope for so-called context conditioning, one could allocate a quantum mechanical flavour to this experiment. We also have not come forward with any critical condition(s) which could help us in deciding this.

What is interesting in the conducted experiment is that the so-called walker droplet exhibits wave particle duality. In a double slit experiment, the walker droplet chooses one slit over the other, but as Bush ([38], p. 274) indicates, ‘the guiding wave passes through both slits and the walker droplet ‘feels’ the second slit by virtue of its pilot wave.’

According to Fort *et al.* ([40], p. 17515) ‘The motion of a droplet is…driven by its interaction with a superposition of waves emitted by the points it has visited in the recent past’. This is the argument by which one can claim that there is a memory property embedded in the walker droplet process ([40], p. 17516).

Bush [38] and Fort *et al.* [40] discuss how close this macroscopic experiment is to Bohmian mechanics, which is not explained in detail in this paper. What we do believe is that this macroscopic model is holding great potential as an explicit formalism to model information in an economics/finance setting. Note that in the following examples there is a formalism established (see, for instance, [38,40]) to describe the events contained in the examples. Thus, the promise that analogous arguments, which are formulated below, hold is that this formalism could be used in an economics or finance setting. We did not explain the formalism in detail in this paper.

### Example 3.1

(i) The droplet bounces at time *t* at a height *h* (in the bath) and (ii) an orbital wave is generated upon impact. Two coordinates can be determined: time and position (height, *h*) ([40], p. 17518). The formalism used in the above example can be used in analogy within a financial setting: (i) a price level is generated at time *t* and (ii) it has an information impact. Two coordinates are determined: price level occurring at time *t*.

### Example 3.2

The time duration of the information impact (damping time, *τ*) is dependent on the context. The preparation comes into play: the fluid frequency in the bath is close or far from the so-called Faraday threshold ([40], p. 17518). In a financial setting, by analogy, we could for instance wonder why initial public offerings and their ensuing price level (height) have such differing impact before or after an election?

### Example 3.3

Past bounces of the droplet affect the current trajectory and hence memory is important. Here again, by analogy, in mathematical finance this is resisted: past information is not informative of the future but in behavioural finance, memory effects can be accepted.

### Example 3.4

The gradient of the height, ∇*h*, determines the surface slope (and will determine the way the droplet bounces) ([40], p. 17518). By analogy, in an financial setting, in so-called technical analysis (see §2 of this paper), a detailed analysis of past price increments can be used to chart future price behaviour. The way a price increases or decreases in a time interval may say something about future price.

It has been remarked by Fort *et al.* ([40], p. 17520) that the path memory we mentioned earlier can be related to non-locality.

## 4. Conclusion

What have we tried to claim in this paper? There is relevance in applying concepts like Fisher information and the quantum potential to finance. There is evidence that those concepts have applicability. We refer to works by Khrennikov [30]; Hawkins & Frieden [25]; Tahmasebi *et al.* [27] and Haven & Khrennikov [31]. But consider again the macroscopic, experimentally tested, version of Bohmian mechanics, as briefly described in §3. This is macroscopic (!!) and the richness of the formalism (not shown here in this paper) especially in terms of available parameters is promising. We believe that this model seems to provide for a primer: a Bohmian mechanics model adapted to a macroscopic environment. We are aware that analogies need to be taken seriously and this can only be done with the use of hard data (for instance, estimate real and quantum potentials from data [27,28]). In the analogies presented in the above section, the mission should be similar: what are hard data-driven analogies on say damping time (*τ*); height (*h*(*τ*)); ∇*h* and context?

## Competing interests

The author declares that there are no competing interests.

## Funding

I received no funding for this study.

## Acknowledgements

The author thanks Andrei Khrennikov for the opportunity given to him to present this work in a special session at the QDTF conference in June 2015.

## Footnotes

One contribution of 14 to a theme issue ‘Quantum foundations: information approach’.

↵1 We note that the Bohmian mechanics approach arrives to the quantum potential in a different way as compared to the Nelson approach.

↵2 One can peruse the financial newspapers to find the risk free rate. Typically, it will be the rate of return of some very safe investment product such as a government bond from a country which has an outstanding financial position.

- Accepted February 22, 2016.

- © 2016 The Author(s)