### Discussion Paper

No. 2019-20 |
March 05, 2019

A Quantum framework for economic science: new directions

## Abstract

The current paper explores the cutting-edge applications of quantum field theory and quantum information theory modelling in different areas of economic science, namely, in the behavioural modelling of agents under market uncertainty, and mathematical modelling of asset or option prices and firm theory. The paper then provides a brief discussion into a possible extension of the extant literature of quantum-like modelling based on scattering theory and statistical field theory. A statistical theory of firm based on Feynman path integral technique is also proposed very recently. The collage of new initiatives as described in the current paper will hopefully ignite still newer ideas.

## Comments and Questions

see attached file

Dear Referee,

Thank you very much for providing a critical review of the write up. I appreciate a lot the points which you mentioned. Below are my brief responses with gratitude to you.

1. I fully understand that the article was not written in a very mathematically accurate ...[more]

... way, the aim was not to present an airtight set up with accurate symbolic representations. I believe the interested reader would certainly go through the original articles as cited in the references.

2. The quantum game theory section explored in the referee report is very helping, but the discussion paper was not aimed at exploring quantum game theory, which it self can be another survey paper. Again I believe the referee might also agree that QGT is still a very nascent area, which is still not based on a suitable utility frame work. One can certainly argue that utility frame work might not be needed in a dynamic game context with adaptive behavior, but in that case it is difficult to see how the concept of Nash Equilibrium fits into the picture.

3. I agree with the looseness of the structure, but as I mentioned the main purpose was to highlight the findings in the new emerging area of quantum like modelling in economics/ social science in general.

4. I mentioned in the first few passages about the problem of building a physical theory of quantum brain, where I was alluring to the works of Roger Penrose mainly. We can recall that there have been some strong criticism of such works by noted physicists, eg, Max Tegmark from MIT. Hence while informing a larger audience I thought to be more conservative.

I welcome all the remarks of the referee, and would be keen to work on them in future.

Best Regards, Sudip Patra

The author reviews a number of well-known problems with existing financial economical theory, and then suggests that ideas from quantum mechanics might be able to solve these. He reviews the standard mathematical formalism of quantum mechanics, but does not really show how this formalism might be applied to economic modeling ...[more]

... ...[more]

... or provide the solution of any of the problems with standard economic models he has listed, leaving the reader yearning for more.

In absence of a concrete model, a critical reader may remain skeptical and unconvinced about the usefulness of QM techniques in finance or economics - in any case, this reviewer was.

In a nutshell, in QM, quantities which classically are represented by numbers are now represented by operators, e.g. matrices.

Are there any concrete examples where this might be useful in economic modeling? The author hints, but does not show. He mentions several times the example of creating/annihilation operators, which in some vague way is supposed to be related to prices going up or down, but does not really specify this more precisely and, more importantly, does not clarify what the in Quantum Mechanical modeling essential commutation relations would mean in such an economic context. (There are also classical models in which particles can be created and annihilated). All this seems very speculative, and not necessarily leading a model which is useful for economics. The author refers in his paper to a sizable body of literature on his topic, which not all readers will be familiar with. It might have been helpful if he would have gone in more detail into some of the models he mentions from that literature.

Further comments of referee:

1. Section of text:

Imagine a market at its ground state before trading starts, so when a trader buys or sells assets the price rise or falls which can be captured by operating creation (for price rise) or destruction (for price fall) operators on the market ground state.

Comment of referee:

Perhaps, but the important thing in QFT are the commutation relations between annihilation and creation operators: what form would those take for the trading example? Is there an explicit model whose details have been worked out?

2. Section of text:

However, in quantum theory randomness is inherent, it is irreducible, and hence quantumlike decision models should also contain such inherent randomness (Yukalov and Sornette, 2011 ).

Referee comment:

QM its not so much about the randomness, inherent or other, but from a classical point strange behavior of the probabilities, due to inference of wave functions

3. Section of text:

From an empirical perspective, the quantum-like modelling has been quite successful ( Khrennikov and Haven, 2013) since the results are nearly matched by the real data, whether choice under ambiguity data or asset price movement data.

Referee comment:

I am not familiar with this literature and unable to comment.

4. Section of text:

…since the days of Boole (George Boole the originator of modern Logic theory) to the modern formalization by Kolgomorov (1933 as cited in Khrennikov and Haven, 2013, 2009), Savage(1954), Knight (1921) and others has been based on the philosophy of determinism and Boolean logic so to say. In classical theory (whether physics or decision theory) randomness is a nuisance rather, it only arises in a model due to lack of information, or noise in the system4, or in other words, it is pseudo-randomness.

Referee comment:

Not sure I agree here, classical randomness can in principle be real; what distinguishes it from quantum randomness is that we can "list all possible future states" and assign a probability to each; also, classical probabilities are manipulated according to rules imposed by classical logic/set theory

5. Section of text:

Order effects P(A&B) not = P(B&A), where P(.) stands for probability and A and B are mutually exclusive events.

Referee comment:

interesting if it occurs in reality, and perhaps indicator of need for quantum proba (or at least non-classical proba) though perhaps less surprising if A&B means something like "first A and then B"?

6. Section of text:

Finite/infinite dimensional, in decision theory problems, we can work with finite dimensional H spaces.

Referee comment:

does the author mean the space of pure states? General states below are defined as density matrices (which the author then should define, for the benefit of the general readership for which this paper is intended?)

7. Description on p6. Under the superposition principle: referee comments:

Referee comment:

Sloppy writing: "(3)" comes before "(2)"? What is"(1)"? "p(e_j) " hasn't been defined

8. Section of text: P(A and B) NOT = P(B and A) (non-commutativity/incompatible variables)

Referee comment:

I don't understand this: what are the "events" A and B the author alludes to? If these correspond to linear subspaces of the (projectivized) Hilbert space, conjunction corresponds to intersection, and there obviously is commutativity. On the level of quantum probabilities the probability of the conjunction of two events is not defined: at best there is a transition probability amplitude, but although the result depends on the order, they are simply related via complex conjugation, which does not affect their absolute value for example

9. Section of text:

However, its also found that in human cognition models (?)s can also be in hyperbolic rather than standard trigonometric forms as we get in ordinary quantum mechanics (Khrennikov and Haven, 2013).

Referee comment:

??

10. Section of text: Space of belief is a finite dimensional Hilbert space H

Referee comment: Can the author give any concrete examples of when to model beliefs as a vector in Hilbert-space instead of as a set? What is the interpretation then, in the decision-making context, of an operator on such a Hilbert space

11. Section of text:

P.8: Rho=(Rho*)T (fourth line from top on page

Referee comment: Awkward as notation; the "t" should be a superscript. The author should consider using TEX or LATEX to write his manuscript (Same comment also for A-(A*)^T on that same page)

12. ‘Mathematical modeling’ – under section 3:

Referee comment: Mathematical modeling of what? Asset prices?

13. Section of text:

Entanglement state, where the composite state is pure but component states are totally mixed 14.

Referee comment: is it not the other way around?

14. Standard finance has failed to describe the asset price behaviors under uncertainty

Referee comment:

There do exist financial models with multiple probability measures, plus versions of no-arbitrage and of the fundamental theorem of asset pricing in such models

15. Atmospheric decoherence

Referee comment: is not this simply called decoherence?

16. Second paragraph at end of page 10 (just above footnote 17)

Referee comment: What would the economical interpretation of the commutation relations be then?

17. Section of text:

The idea of occupation no representation can also be extended in a market context , representation can be provided for n where a state no of traders, such that trader 1 has n1 stock, to begin with, trader 2 has n2 stocks…trader n has n k stocks, to begin with, and this state which represent the initial market state can be written in terms of creation operator:

Referee comment:

This looks rather speculative: again QFT = creation/annihilation plus (anti-)commutation relations; nothing is said about what the latter would be/mean in such an economic model

18. Text section:

QFT (for example both being continuous and non-differentiable) asset pricing or option pricing theory has been proposed based on path integral techniques

Referee comment:

"classical" or "non-quantum" mathematical finance (for want of a better designation) expresses option prices as expectations of functionals of Brownian motion, which can be already interpreted as path integrals ...

19. Text section:

provided by a modified discrete time path integral, however, if simply we need to study the propagations of prices over space-time points then a standard propagator (as widely used in QFT) based on a simple discrete time path integral is proposed.

Referee:

Isn't this just Monte Carlo simulation, which is already a standard tool in quantitative finance?

20. Section of text:

Such a representation will help us to capture many more Market configurations with similar movements of asset prices

Referee comment:

All of this subsection is very speculative: are there any concrete financial models where this is useful?

HOMEPRIVACY POLICYLEGALCONTACT & SUGGESTIONSALERTING SERVICE

Dear Referee,

Thank you very much for your quite detailed observations. I will try to respond to as many points as possible briefly, as below.

1. You have raised a very pertinent issue all over the response, what is the practical value of Quantum like modelling or Quantum Decision ...[more]

... Theory, are there any concrete models? To this I must say that over the last two decades or so, as I have referred to some seminal works, there have been many concrete models based on quantum theory frame work, which have been backed nearly perfectly by real data.

There are many such areas: asset pricing models based on path integral technique, or operator methods (creation – annihilation operators, both of Bosonic and Fermionic nature, as very rightly pointed out in the feedback. Commutation algebra does play a significant role, it depends on the constraints on the Eigenvalues realized in the specific model), explaining decision theory paradoxes at large which are non-trivial or impossible to resolve using classical measure theoretic decision making models, also quantum game theory is a budding area which allows quantum-like strategy profiles which can significantly alter cooperation / co-ordination games like prisoner’s dilemma.

Haven and Khrennikov (2013, quantum social science) have provided a brief but wide overview of such emerging models with good data support.

2. Yes, randomness is the central issue. Newtonian world view was the basis of Neoclassical economics; hence a deterministic general equilibrium theory was the ultimate goal to achieve in economic theory. However, expected utility theory, or subjective expected utility theory did change the modelling but superficially. Such models were again based on Bayesian conception of uncertainty: an epistemic problem which might resolve if full information set is available.

However, (though certainly there are many interpretations of quantum theory, with widely different views and supporters) quantum revolution opened up the possibility to think randomness or probability as ontological. Some very noted quantum information scholars (Seth Lloyd, at MIT) do believe that human decision making is ultimately truly random. The very classical world we inherit is an emergence. Certainly, emergence theory, or complexity theory can be grounded in quantum randomness, though I fully agree that we still have some very good grounds to cover.

3. Yes, order effects, failure of sure thing principle, failure of law of total probability, conjunction and disjunction fallacies have been successfully modelled by quantum like modelling, though I will never say that there are no alternatives. Recently there are many interesting studies on contextuality in human decision making, which delves into intriguing areas like entanglement in human cognition, which also have been supported by experimental data. More such robust experiments are required to demonstrate the role of quantum like modelling in human decision making.

4. There are many intriguing features generated when principles of economic theory are merged with quantum theory set up (as you have rightly pointed out the Hilbert space as state space, or even more complicated state space like Fock space): time dependent Hamiltonian, non-Hermitian Hamiltonian, time dependent state space etc, which are not regularly found in standard quantum field theory (though certainly there is a strong literature there: for eg, in quantum optics). This also shows that quantum like modelling may have a life of its own.

5. For the point 6 of the referee: Finite or infinite Dimensional Hilbert space as state space is referred to here, in decision theory context we need finite dimensional state space in many cases. Referee is true in pointing out the density matrix representation referred to here, in case of a pure state with say superposition of basis states (in information theory language the computational basis of |0> and |1>) a wave function representation will suffice. However, when we consider more general ensemble of such pure states, we need to have density matrix representation, which is a weighted average representation on many of such pure states. Certainly readers can check the important properties of density matrices, how to distinguish between pure and mixed states based on such properties, Susskind’s ‘theoretical minimum’ is a good reference to begin with.

6. For point 9, the reference here is a comprehensive text by Khrennikov and Haven (2013), ‘Quantum Social Science’, where the authors have explicitly shown that the interference terms in the modified total probability formula can be more complex in decision theory context, than standard quantum physics (for example in famous double slit experiment or alike), where the interference patterns can’t be written in terms of cosine of the phase angles, rather hyperbolic functions have to be invoked. The interpretation of such complex phase terms has not been fully found.

7. For point 12 we are referring to mathematical modelling of option pricing in general. Path integral technique has been deployed recently.

8. For point on the atmospheric decoherence, yes I agree but authors have used them in decision theory context/ cognitive modelling context.

9. For point 17: yes creation and destruction operators are referred to here, there is a strong literature now of extending the Bosonic / Fermionic operators to trading models (pioneered by Fabio Bagarello).

10. For point 19: MONTE carlo simulation techniques are used in such models.

11. For point 20: the paradigm is still unfolding, we have seen the seminal works as in the references.

Overall the article was a simple survey article without any rigorous mathematical formalism. I am indebted to the good points raised by the referees here.

Best Regards,

Sudip Patra