The "Many Worlds" of the Financial Markets

 

“’The Garden of Forking Paths’” is an incomplete, but not false, image of the universe as conceived by [the garden’s designer].  Unlike Newton and Schopenhauer, [the designer] did not believe in a uniform and absolute time; he believed in an infinite series of times, a growing, dizzying web of divergent, convergent, and parallel times.  That fabric of times that approach one another, fork, are snipped off, or are simply unknown for centuries, contains ALL possibilities….Time forks, perpetually, into countless futures.”

     Jorge Luis Borges, “The Garden of Forking Paths”

 

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     May I propose a new paradigm for comprehending the mutable nature of the financial markets, alternative to the ever-changing cycles of Robert Bacon, and Newton’s law of cooling as adapted by Martin Knight?  Although both of these concepts have their merits, I find deficiencies in each.

 

     Bacon’s ever-changing cycles concept, while predicting the inevitability of change, in no way accounts for making predictions as to what the prevailing cycle will change TO.  Its utility is thus very limited.  Although Knight’s adaptation of Newton’s law of cooling “provides a quantitative estimate of the rate at which cycles will change,” I believe those quantitative estimates will often be misleading, because they do not account for the interactive, self-referencing aspects of investors’ behavior, as described by Sorosian reflexivity.  Market bubbles (very extended upmoves out of standard deviation bounds), for example, which can be the most profitable times for speculation, are not adequately accounted for -- indeed, they are contradicted by -- Knight’s entropic model.  To invert Spinoza’s criticism of Cartesian interactionist dualism regarding human conduct, Knight’s model does not:

 

     “…conceive man in Nature as a kingdom within a kingdom.  [It does not recognize] that he disturbs rather than follows Nature’s order…[Knight’s model erroneously presumes that] Nature is always the same, and its force and power of acting is everywhere one and the same.” [Spinoza, “The Ethics,” Part III, “Concerning the Origin and Nature of the Emotions,” (Samuel Shirley trans.)]

 

     It’s the essential problem with applying the determinism of classical physics to systems in which their laws are of limited applicability:

 

“In classical physics, the complete specification of the state of a closed physical system at any time (t0, say) serves to determine (via the equations of motion) the result of ANY other measurement on the system, carried out at ANY time, either before or after t0.  Measurements carried out on the system at times other than t0 are therefore in a certain sense redundant:  since their results can be deduced from the result of the t0 experiment, they produce, in principle, no additional information about the system (either about its past or about its future).” [1][Aharonov, Albert and D’Amato, “Multiple-Time Properties of Quantum-Mechanical Systems,” 32 Physical Review D p. 1975 (1985)]

 

As Laplace so famously wrote:

 

“If an intelligence, at a given instant, knew all the forces that animate nature and the position of each constituent being; if, moreover, this intelligence were sufficiently great to submit these data to analysis, it could embrace in the same formula the movements of the greatest bodies of the universe and those of the smallest atoms:  to this intelligence nothing would be uncertain, and the future, as the past, would be present to its eyes.”

 

     The determinism of classical physics is inadequate to explain movements in the financial markets for two reasons.  First, the mutable nature of those markets changes the underlying dynamics -- not merely the state -- of the system being observed.  Second, the influence of the observer’s (trader’s) actions on those systems changes the systems’ character.

 

     What model can account for markets’ movements by incorporating those two   distinctive attributes?

 

      I submit the answer lies in quantum mechanics (“QM”), which incorporates those attributes, while at the same time permitting classical determinism to operate to the extent to which it is pertinent.  It’s analogous to the Theory of Relativity, which has general applicability, while incorporating Newtonian physics to the extent to which the latter is apposite.

 

     First, QM explicitly accounts for the mutable nature of the system being described, because it recognizes the necessity for continually-updated observation:

 

“An essential difference (perhaps THE essential one) between quantum and classical theories lies precisely here; because every new complete measurement on a quantum-mechanical system will, in general, augment our information about that system.”   Aharonov et al [1 above].

 

     Second, QM recognizes the “observer effect” on the system, as instantiated by the double-slit experiment, in which observation of electrons passing through two slits changes the distribution of their target destination.  (For an explanation, see “The Feynman Lectures on Physics,” (Addison-Wesley 1963) chapter 37)  “[W]hen we look at the electrons the distribution of them on the screen is different than when we do not look.”  Id. at 37-8.

 

     The question then arises, which interpretation of QM is most useful for the trader?

 

     I find the conventional (“Copenhagen”) interpretation of QM to be unsatisfactory.  First, its essence is to make observations external to the system in question.  [2][Wheeler, “Assessment of Everett’s ‘Relative State’ Formulation of Quantum Theory,” 29 Reviews of Modern Physics 463 (July 1957).]  Traders are part of the pertinent system, and hence need to make observations of a system to which they are integrally connected.  Second, the Copenhagen interpretation ignores what it cannot measure or explain.  It makes assertions about reality only after the collapse of the pertinent state vector, as instantiated by the Paradox of Schrödinger’s Cat.  [See, e.g., DeWitt, “Quantum Mechanics and Reality,” Physics Today, Sept 1970, pp 30-35; Wolf, “Parallel Universes” (Simon & Schuster 1988), pp 51-52.]  For the trader, who must make predictions about future states of the market before the state vector collapse occurs, this is not helpful.

 

      I submit that the most useful approach to QM for traders is the “Many Worlds Interpretation” (“MW”) proposed by Hugh Everett, III, later expanded upon by Everett, John Wheeler and Neill Graham.  (An excellent exposition of MW is found in [3] [DeWitt and Graham, eds. “The Many-Worlds Interpretation of Quantum Mechanics,” (Princeton 1973)]).  Under MW, every physical system in the world is a QM system, all such systems evolve in accordance with the linear QM equations of motion, and every self-adjoint local operator of such systems can, at least in principle, be measured.  [4] [Albert, “How to Take a Photograph of Another Everett World,” 480 Annals of the NY Academy of Sciences, Dec. 30, 1986, p 498.]

 

     The distinguishing characteristic of MW is self-consistency (a quality which traders seeking a rational explanation for markets’ movements should find highly desirable); it was formulated with that specific intent.  Everett, “The Theory of the Universal Wave Function,” in DeWitt and Graham[3] at 8-9;  [5]Everett, “’Relative State’ Formulation of Quantum Mechanics,” 29 Reviews of Modern Physics 454, 455 (July 1957).  The seeming paradox of the interference of two mutually exclusive alternatives in the double-slit experiment is resolved by the simple expediency of concluding that both alternatives exist simultaneously.  The putative Paradox of Schrödinger’s Cat is solved by postulating the contemporaneous existence of BOTH state vectors.  Accepting the simultaneous existence of many state vectors – many worlds – avoids all the contradictions caused by postulating discontinuous change brought about by separating observation from the state vector.  Everett in [3] at 3-9.

 

     Most importantly for the trader, MW has a place for observations only insofar as they take place within an isolated system, Wheeler [2] at 463, and “postulates that a wave function that obeys a linear wave equation everywhere and at all times supplies a complete mathematical model for every isolated physical system without exception.”  Everett [5] at 455.  For traders, who must make observations within isolated systems, a consistent formulation for determining the resolution of those observations provides the answer to predicting markets’ movements. 

 

     The law of causality in QM applies:  The initial eigenstate of each isolated system can at least in theory be determined (by observation) to a level of precision limited only by the Heisenberg uncertainty relation.  Everett in [3] at 117.  The trader makes observations about the current state of the markets to the best of his ability, given the resources at his disposal.  He then attempts to discern the eigenfunction (the investment concept) which will act on that eigenstate.  This process is easier said than done!  In Everett’s words:

 

     “There are…fundamental restrictions to the knowledge than an observer can obtain about the state of the universe.  It is impossible for any observer to discover the total state function of any physical system, since the process of observation itself leaves no independent state for the system or the observer, but only a composite state in which the object-system states are inextricably bound up with the observer states [hence incorporating the observer effect - AC].  As soon as the observation is performed, the composite state is split into a superposition for which each element describes a different object-system state and an observer with (different) knowledge of it.  Only the totality of these observer states, with their diverse knowledge, contains complete information about the original object-system state – but there is no possible communication between the observers described by these separate states.  Any single observer can therefore possess knowledge only of the relative state function (relative to his state) of any system….”  Everett in [3] at 98-99.

 

     If the eigenfunction – the wave function psi(x,y,z,t0) whose (absolute) square is interpreted as a probability distribution of finding the system at time t0 at position x,y,z -- acting on that totality of states can be correctly identified, then by the dynamics of the Schrödinger wave equation the probability distribution of the future eigenstate psi(x,y,z,t) will be completely determined.  If the trader can accurately discern the investment concept which will act on that totality of states which constitutes the market(s) under consideration -- no easy task, to be sure -- then the probability distribution of the future state of those markets can be predicted.

 

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     Lemma:  The phrase “Many Worlds” has caused skepticism on the part of many, because it makes the MW Interpretation seem more “science fiction” than science.  See, e.g., Albert[4] at 499.  MW seems to be contradicted by experience, because we are unaware of any splitting or superposition of observer states.  Everett’s response to this skepticism was to cite the criticism which greeted the Copernican theory of a heliocentric solar system.  Because we feel no motion by the Earth, the Earth could not be moving around the Sun:

 

“In both cases the argument fails when it is shown that the theory itself predicts that our experience will be what it in fact is.  (In the Copernican case the addition of Newtonian physics was required to be able to show that the earth’s inhabitants would be unaware of any motion of the earth.)”  Everett[5] at 459 fn.

“[The] total lack of effect of one branch on another also implies that no observer will ever be aware of any  ‘splitting’ process.”  Id.

 

     Personally, I have no problem accepting the seemingly bizarre MW process, because I view it as being akin to “sense data” from the Theory of Perception.  This latter concept may be illustrated by the following simple experiment:  Gather together with a few others to observe an everyday object, e.g., a bowl of fruit.  Although all participants in the experiment will agree that they are observing the same physical object, the objective (not merely subjective) reality is that the sensory data received by the retinas of each observer differ, in that each observer will see different parts of the bowl, different pieces of fruit, the same pieces of fruit from different angles, etc.  It is the totality of all possible sensory data pertaining to the fruit bowl which constitutes the reality that rises to the level of our common perception.  Similarly, it is the superposition of all isolated MW systems which rises to the level of everyday experience.

 

 

     The trader makes observations of the markets to the best of his ability, but should recognize that he is operating within the context of an isolated system – no one individual can have complete knowledge.  He then attempts, in light of his observations, to determine which investment concept(s) are most likely to be operative in changing the markets’ states.  This done, it becomes a (comparatively simple) mechanistic exercise to determine how the markets will be affected.

 

     In his classic work “The Sleepwalkers,” Arthur Koestler described Johannes Kepler’s discovery of the modern model of the solar system as “The Watershed” development in the history of Western scientific thought.  Kepler’s debate with other scientists of his time regarding the nature of scientific proof is an exact parallel with the dispute between technical analysts satisfied with visual charting techniques, and those who put “statistics on the table.”  Kepler maintained that “only quantitative mathematical proof is a characteristic of objective science.”  Beer, “Kepler’s Astrology and Mysticism,” p 399 at 416, in Beer & Beer eds, “Kepler:  Four Hundred Years,” (Pergamon Press 1967).  Objecting to those who believed that the key to scientific understanding lay in the interpretation of diagrams, Kepler declared:  “Without mathematical demonstration I am like a blind man….” Id.

 

     Yet Kepler recognized that such mathematical demonstration was in itself of secondary importance.  “[F]or him it is not the number, size and motions of the planets…but instead the causes of these mathematical entities that are the really essential thing…. [H]e is looking for the plan that underlies these values…”  Haase, “Kepler’s Harmonies, between Pansophia and Mathesis Universalis,” in Beer & Beer at p 519, 523.  Kepler knew that the essential question was to determine the underlying dynamics of the system.  It was only after that question had been answered that mathematics should be applied to work out the resolution of the underlying plan.

    

     A modern example of this principle applied to the field of human events (which may be more pertinent for those concerned with human conduct in the financial markets) is given by George Lindsay’s “The Other History” (Vantage Press 1969).  Lindsay applied analytical techniques he had used in the stock market in order to determine the underlying connection between historical events, in the tradition of non-Herodotean historiography.  Lindsay declared it “certain that an analysis of details will not predict the future.”  Id. at 17-18.  Instead, he sought the one underlying principle – the eigenfunction – which held true under dissimilar circumstances, in order to enable him to make probabilistic predictions about the future.  Id.  It is the search for that eigenfunction which should be the trader’s primary concern.

 

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     In conclusion, the many worlds interpretation of quantum mechanics provides an apt paradigm for the trader.  The trader makes observations of the markets in order to determine their states, recognizing that:  (1) his knowledge cannot be complete, (2) his actions will affect those states, and (3) the underlying characteristics of those states will change over time.  In a dynamic process, the trader attempts to discern those influences which are most pertinent to the markets at each point in time.  He then works out the probabilistic effects those influences will have, and trades accordingly.  It is the superposition of all such observations, actions and trades by all market participants which constitutes those market movements that rise to the level of common experience. 

 

“Actualities seem to float in a wider sea of possibilities from out of which they were chosen; and somewhere, indeterminism says, such possibilities exist, and form part of the truth.”             William James, “The Dilemma of Determinism”