Quantum Theory and Social Choice Graciela Chichilnisky Columbia University and Stanford University XXVII European Workshop on General Equilibrium Theory EWGET University of Paris Sorbonne - June 28 2018 Room 114 11:00 am
Quantum theory has been called the most successful scientific theory of all time. It emerged less than a century ago from the classic axioms created by Born, Dirac and von Neumann. Based on the classic axioms of quantum theory we identify a class of topological singularities that separates classic from quantum prob- ability, by explaining quantum theory’s puzzles and phenomena in simpler mathematical terms so they are no longer ’paradoxes’. We found that the singularities emerge from one foundational ax- iom: that observations of events are projection maps or self-adjoint operators: This is due to the central role played by the observer in the classic axioms. It follows that the space of observables or quantum events is given by the projections of a Hilbert space H, and this is a union of Grass- manian manifolds and in particular it is topological complex. For example, in the simplest case, in Quantum Theory with two degrees of freedom, the Hilbert space H is a two dimensional euclidean space is R 2 , and the space of observables or quantum events is the projective space P 1 - which is also the circle S 1 .
We show that the key to the quantum paradoxes is the topology of the space of observables or quantum events and the correspond- ing ’frameworks’, as they are postulated by the classic axioms of Born, von Nemann and Dirac. We establish that the singularities in the space of observables (self-adjoint operators) explain interfer- ence, Heisenberger uncertainty, order dependence of observations and entanglement. Separatly we show a clear connection between quantum theory’s ob- servables or frameworks, and key objects of social choice theory, and of attitude and decision theory: these are preferences over choices and attitudes. In the simplest two dimensional case the space of choices is R 2 and the space of preferences over these choices is a circle S 1 Chichilnisky (1980); we show that S 1 .is also the space of quantum events or frameworks which is the projective space P 1 = S 1 and therefore also the circle. The results establish that the same topological singularities that explain interference and entanglement explain the impossibility the- orem of K. Arrow in social choice theory, as formulated and proven in Chichilnisky (1980), and imply the order effects in attitude re- search such as the so called ’conjunction fallacy’ of Tversky and Kahneman in decision theory.
1 Von Nemanns Axioms for Quantum The- ory: A1. The states of a quantum system are unit vectors in a (complex) Hilbert space H A2. The observables are self adjoint operators in H A3. The probability that an observable T has a value in a Borel set A ⊂ R when the system is in the state Ψ is � P T ( A )Ψ , Ψ � where P T ( A ) is the resolution of the identity (spectral measure) for T, and A4. If the state at time t = 0 is Ψ then at time t it is Ψ t = e − it H /h Ψ , where H is the energy observable and where h is Planck’s constant. The four axioms presented above can be greatly simplified: Gudder (1988) shows that these four axioms can be derived from a single axiom that separates quantum theory from classic physics: Axiom ( A ) : The events of a quantum sysem can be repre- sented by self-adjoint projections of a Hilbert space.
2 Quantum Theory results in the simplest case: n = 2 degrees of freedom We now show the difference between classic and quantum events or observables, and the emergence of a Singularity with quantum events. The simplest possible physical system has two degrees of freedom n = 2 , and the corresponding Hilbert space is H = R 2 . Classic events are the Boolean sets in R 2 and the sample space is their union, namely R 2 , which is contractible space with no singu- larities Instead, in Quantum Theory from axiom ( A ) above, the set of quantum events of the quantum system with two degrees of freedom n = 2 , is by definition the space of all one dimensional subspaces or lines through the origin of R 2 . By its definition therefore the space of quantum events is the projective space of dimension 1 , denoted P 1 ≈ S 1 the unit circle in R 2 . S 1 is not a contractible space. Its homology group has one generator: this generator defines
a singularity and this singularity emerges due to the role of the observer Social Theory’s results in two dimensional choice space X = R 2 : the space of preferences is now S 1 : Therefore social choice has a singularity with preference space S 1
3 Unicity and Restricted domains The difference between Quantum Theory and Classic Physics is that in the former there may exist several "frameworks" for each exper- iment, while in classic physics, there is a single framework for all observations: this is also called the "unicity" hypothesis of classic theory. The question is when is it possible to create a single frame- work for all experiments, namely a continuous framework selection map which is symmetric (it is not order dependent), it is the identity in the diagonal . This is generally impossible and it demonstrates the link between Social Choice Theory and Quantum Theory. Theorem (Chichilnisky 2016) In unrestricted experiments with two degrees of freedom there is no unicity: there is no way to select a single framework for all experiments of a physical system. Formally, there exists no continuous function Φ : F 2 × F 2 → F 2 that se- lects a single framework for all observations or experiments and is symmetric or independent of the order of the experiments .. The corresponding social choice theorem is
Theorem: (Chichilnisky 1980) In R 2 choice spaces with unrestricted preferences there exists no continuous function Φ : F 2 × F 2 → F 2 that is symmetric or independent of the order of the experiments and respects unanimity Φ( f, f ) = f. In sum: with two degrees of freedom there is no unicity: there is no way to select a single framework for all experiments of a physical system. The following example shows in decision theory that paradoxical behavior arises from multiple frameworks, and such paradoxes do not arise when one can select a single framework or preference. Example 1: The order effect and the ’conjunction fal- lacy’ in Tversky and Kahneman, aka as the "Linda prob- lem"
Example 2: The two slit experiment in physics
Example 3: The new rotating two slit experiment
Example 4: Berry Phases
The Topology of Quantum Theory and Social Choice Graciela Chichilnisky Columbia University and Stanford University August 8, 2016 Abstract Based on the axioms of quantum theory we identify a class of topo- logical singularities that separates classic from quantum probability, and explains many quantum theory’s puzzles and phenomena in simple mathe- matical terms so they are no longer ‘quantum pardoxes’. The singularities provide new experimental insights and predictions that are presented in this article and establish surprising new connections between the physical and social sciences. The key is the topology of spaces of quantum events and of the frameworks postulated by these axioms. These are quite differ- ent from their counterparts in classic probability and explain mathemat- ically the interference between quantum experiments and the existence of several frameworks or ‘violation of unicity’ that characterizes quantum physics. They also explain entanglement, the Heisenberger uncertainty principle, order dependence of observations, the conjunction fallacy and geometric phenomena such as Pancharatnam-Berry phases. Somewhat surprisingly we find that the same topological singularities explain the impossibility of selecting a social preference among different individual preferences: which is Arrow’s social choice paradox: the foundations of social choice and of quantum theory are therefore mathematically equiv- alent. We identify necessary and sufficient conditions on how to restrict experiments to avoid these singularities and recover unicity, avoiding pos- sible interference between experiments and also quantum paradoxes; the same topological restriction is shown to provide a resolution to the social choice impossibility theorem of Chichilnisky (1980). 1 Introduction Quantum physics is the most successful scientific theory of all time, having emerged less than a century ago from axioms created by Born [2] Dirac, [9] and von Neumann [17]. Based on the same axioms we identify here a class of topological singularities that separates classic from quantum probability, and explains many quantum theory’s puzzles and phenomena in simple mathemat- ical terms so they are no longer ‘quantum pardoxes’. The singularities provide 1
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