The Topology of Quantum Theory and Social Choice

Based on the axioms of quantum theory we identify a class of topological singularities that separates classic from quantum probability, and explains many quantum theory’s puzzles and phenomena in simple mathematical 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 different from their counterparts in classic probability and explain mathematically 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 equivalent. We identify necessary and suffi cient conditions on how to restrict experiments to avoid these singularities and recover unicity, avoiding possible 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).


Introduction
Quantum physics is the most successful scienti…c 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 mathematical 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 that are postulated by the quantum axioms, which are quite di¤erent from their counterparts in classic physics.
Events are physical phenomena that either occur or don't occur.They are central to any probabilistic theory.In classic probability all experiments are part of one large experiment and events are described within one sample space with a single basis of coordinates or framework 1 : this is the unicity assumption of classic physics (Gri¢ ths, 2003 [10]).In quantum probability, instead, quantum events are projections maps on a Hilbert space.Quantum theory considers all possible experiments on a physical system and breaks tradition by explicitly accepting that there may be no universal experiment and no single framework to describe all observed events2 .The multiplicity of frameworks in quantum theory violates unicity: there may be no unique basis of coordinates to describe the results of all possible experiments on a physical system.When two frameworks or coordinate systems fail to be orthogonal to each other they give rise to socalled 'interaction' or 'interference' among experiments that is at the heart of quantum theory and distinguishes it from classic physics; a classic example is the two hole experiment discussed below, e.g.Gri¢ ths, 2003 [10] Gudder [12] As seen in the examples of the last section, the matter has further rami…cations as di¤erent frameworks lead to Heisenberger uncertainty and order dependence of experiments.
This article explains in simple mathematical terms the genesis of interference, where it comes from, and in particular when and when and how it can be avoided.For example, it is well known that when all experiments under consideration are part of a single larger experiment, the unicity of classic physics is recovered within quantum theory, in this case experiments do not interfere and are consistent with each other.Being part of a larger experiment is a suf-…cient condition to eliminate interference.Is it possible to …nd necessary as well as su¢ cient conditions on the range of acceptable quantum experiments to recover unicity?We show that the topological structure of the spaces of quantum events -which are also the propositions of quantum logic -explains why experiments interfere, why we typically have no common frameworks, and why quantum logic is more complex and richer than the binary logic of classic physics [12].We …nd a necessary and su¢ cient condition that, when used to restrict the domain of acceptable experiments, ensures that one can select a single framework for all experiments thus eliminating interference.This condition restricts appropriately the domain of experiments so they are consistent and do not interfere with each other.It turns out that this topological restriction by itself creates a connection between quantum theory and social choice theory, a rather unexpected connection.The necessary and su¢ cient condition that eliminates interference between experiments turns out to be the same as the restriction required to resolve Arrow's classic impossibility theorem in social choice (Arrow [1]) allowing us to aggregate individual into social preferences.It was shown in 1980 that Arrow's impossibility theorem has a topological structure, see Chichilnisky [5] [6]) and here we show that the same topological structure is at the core of the paradoxes of quantum theory.The last section illustrates the theorems and discusses simple and practical examples and new experimental predictions, examples of interference, order dependence of experiments, Heisenberg's uncertainty principle.[12] [3] and Pancharatnam-Barry geometric phases [13] all of which have a similar topological origin, and their connection with the topology of spheres.

Organization
We start by stating the axioms of quantum theory created by Von Neumann.Based on these axioms we de…ne and analyze the spaces of quantum events and of frameworks, showing that their topological structure separates classical events from the events of quantum physics.We then establish the impossibility of selecting a common framework for all experiments, based on the topological singulatrities within quantum events and frameworks.The same singularities is behind the impossibility of selecting a common social preference to di¤erent individual preferences: this is the social choice impossibility theorem.We establish that a resolution to the social choice problem is the same as a resolution to the violation of unicity, and that both cases require the same topological restriction on the domains of experiments and of preferences, respectively.Finally we illustrate the results with examples of interference, order dependence of experiments, Heisenberg's uncertainty principle,.andPancharatnam -Barry phases.

The Axioms of Quantum Theory
We start with key concepts in quantum theory and show how they di¤er in topological terms from their classical counterparts.In classic mechanics there are three important components: states, observables (dynamic variables, such as events) and dynamics.The same three components are present in quantum mechanics, but they are described by di¤erent objects.In classic mechanics the three components are described by points, functions and trajectories, while in quantum mechanics they are described by entities in a Hilbert space.This fundamental di¤erence arises in gret measure because the two theories have di¤erent goals, which can be summarized as follows: "The main goal of classic probability is the construction of a model for a single probabilistic experimentor subexperiments of a single experiment.Quantum theory is based on Hilbert space probability theory and is much more ambitious.It seeks a mathematical model for the class of all experiments that can be performed on a physical system.
Why can't we construct a classical model for each experiment and then "paste" all the models together?The problem is that we don't know how to do the pasting since we don't know how the various experiments interact of interfere with each other.The pasting is automatically done by the Hilbert space structure."S. Gudder [12] 1988, p. 68.
While classic physics attempts to explain the universe, quantum theory shares with general relativity an emphasis on the observer.For this reason "quantum events" are de…ned as maps rather than as measurable sets of objects as in classic physics.Quantum events are a key concept in this article, and they are identi…ed with projection maps (see Axiom A below), and with the subspaces of a Hilbert space onto which the projections map.Frameworks are orthonormal bases of coordinates of the Hilbert space and can be identi…ed with subspaces of the spaces of events.When two frameworks fail to be orthonormal the corresponding experiments are said to interfere with each other.In classic physics things are di¤erent: there is only one experiment -the 'Universe'-and one single framework, so quantum interference is impossible: this is the "unicity hypothesis" that is violated in quantum theory.A key di¤erence is therefore that quantum theory does not assume a single framework nor a single sample space.
In the following we consider Hilbert spaces of …nite dimension n, where n is arbitrarily large, namely euclidean spaces R n , which correspond to physical systems with n degrees of freedom.Under appropriate assumptions the theory presented here can be made applicable to in…nite dimensional Hilbert spaces.The …nite dimensional case is useful to simplify the presentation and to show that fundamental properties of quantum theory occur even within …nite dimensional real Hilbert spaces, even though full generality requires in…nite dimensional Hilbert spaces with comples coe¢ cients.
To provide a clear foundation and highlight the di¤erences, we start from basic concepts of probability theory and show the di¤erence between classic probability and quantum probability 3 .In classic theory, the set of individual outcomes of a probabilistic experiment is called a sample space and it is a non-empty set denoted X. P denotes a algebra of subsets of X, which is the collection of outcomes sets to which probabilities can be assigned, the pair (X; P ) is called a measurable space, and the sets in the algebra P are also called 'events'.By de…nition they events are all included within the common sample space X, which is the union of the outcome sets: To facilitate the comparison between classic and quantum theory, the sample space X can be assumed to be a Hilbert space with an attendant orthornomal basis of coordinates; with …nite dimensions the sample space is therefore R n .The basic postulate of unicity that divides classic from quantum physics is as follows: De…nition 1 The Unicity postulate of classic physics is the requirement that all events are included in one single sample space X.
Assume that the sample space X is a Hilbert space of dimension n; so that X = R n .

Example 2
In classic probability all events are subsets of a single sample space X = R n ; the space of all events is denoted P and is the algebra of Borelian subsets of R n , so the union of all events is the single sample space R n .In classic physics, therefore, unicity is satis…ed.
Below we show the di¤erence between classic probability and quantum problability.For a presentation of quantum probability theory, see also Gri¢ ths, 2003 [10].
The following four axioms of quantum theory were introduced by Von Neumann [17] and highlight the di¤erence between classic and quantum theories:

Von Neumann' s Axioms for Quantum Theory
(A:1) The states of a quantum system are unit vectors in a (complex) Hilbert space 4 H, (A:2) The observables are self-adjoint operators5 in H, (A:3) The probability that an observable T has a value in a Borel set A R when the system is in the state is < P T (A) ; > where P T (:) is the resolution of the identity (spectral measure) for T and (A:4) If the state at time t = 0 is ; then at time t it is t = e itH=h where H is the energy observable and h is Planck's constant.
To summarize from Axiom (A:2) above, the observables or events in a quantum theory experiment are not sets but rather self-adjoint operators T de…ned on the Hilbert space H.In further detail, by Axiom (A:3) above, the results of experiments compute the probability that the observable T has a value in the borelian set A when the system is in state and the probabiloity is < P T (A) ; > where P T (:) is the resolution of the identity (spectral measure) 6 for T .
The four axioms presented above can be greatly simpli…ed: Gudder [12] (p.50 -53) shows that these four axioms can be derived from a single axiom if we begin with a probabilitistic structure de…ned on a Hilbert space.As already mentioned, the single basic axiom of quantum theory that separates it from classic physics perteins the structure of quantum events, which are observations or physical phenomena (as de…ned above) that either occur or don't occur.The following is the single axiom from which the rest can be derived Gudder [12] (p.50 -53): Axiom (A) The events of a quantum system can be represented by (selfadjoint) projections on a Hilbert space.
The events in quantum theory are observables (as de…ned above), so quantum theory shares with relativity the emphasis on observations and the observer. 7The axioms presented above do not specify a particular Hilbert space in which the states are represented, nor which self-adjoint projection operator represents a particular physical observable or event.The next step is to show how Axiom A determines the space of quantum events.Consider a Hilbert space H of …nite dimension n; which could be very large, H = R n , and assume that H is spanned by an orthonormal set of vectors V = fV i g i=1;:::;n that forms a basis for the space 8 H. Then from Axiom A we know that a quantum event is a (self adjoint) projection 9 on a subspace of H and can be identi…ed by a subspace S H spanned by a subset V S V of basis vectors 10 ; in words the event occurs when an experimental observation lies in the subspace S H: The event S corresponds to orthogonal projections P S whose image covers S:: For each event S there is an orthonormal basis of coordinates (Gudder 1988 [12], Gri¢ ths 2003 [10]) that de…nes the subspace S. 11 If the set V has n 1 vectors, it de…nes an n 1 dimensional subspace and together with its orthonormal vector it de…nes an orthonormal base of coordinates for the entire space R n ; which is also called a framework. 12As an example if S is a subset of vectors in the basis V; and x is an observable or experiment of our physical system, corresponding to S is the event that occurs when a measurement of x results in a value in S; see e.g.Gudder [12] p. 52.Following the above description, a framework can be de…ned as an unordered unoriented orthonormal basis of coordinates of R n The space of frameworks in R n is therefore the space of all bases of coordinates (unoriented and unordered, since the order or the orientation of the coordinate vectors of S does not alter the subspace S) and is denoted F n : 13  The notions of events and frameworks just de…ned play a key role in quantum physics.The concept of interference or incompatibility between experiments is a critical new idea that distinguishes quantum probability from classic theory and is identi…ed with the 'violation of uncity'(Gri¢ th 2003 [10], Gudder 1988 [12]): "A key feature of quantum theory is that while some events may be compatible and share the same framework, or bases of coordinates consisting of vectors that are orthogonal to each other, other events may be incompatible and do not share a common basis of coordinates or framework."cf. also Busemeyer and Bruza 2012 [3].When the various bases of coordinates that appear within several quantum experiments include vectors that are not orthogonal to each other 7 Quantum probability theory is presented in Gri¢ ths [10] 8 This is an arbitrary choice and other bases can be used.The order of the vectors in the basis is not relevant. 9A self adjoint projection is similar to a a product operator, like position or momentum.1 0 The order of the vectors V i does not matter, and since we are concerned with subspaces, nor does their orientation, as is discussed further below.1 1 Events are identi…ed here with subspaces of H and are therefore given by unordered and unoriented bases of coordinates of the subspaces.Orienting the vectors does not change the main results. 1 2 The basis of coordinates need not be an ordered set and the vectors need not be oriented.1 3 No orientation is required, although similar results are obtained if the frameworks are unorderd but oriented bases of coordinates.this causes experimental 'weirdness' as shown in the illustrations of the last section.It is worth noticing that as we show here the violation of unicity can occur both in …nite or in…nite dimensional spaces and in real or complex Hilbert spaces.In all cases as seen in the last section it leads to interference between experiments, non-conmuting observations, i.e. the order of the experiments changes the observed results, the probabilistic error known as the "conjunction fallacy" by which two events are deemed to be more likely to occur together than each on its own, Busemeyer and Bruza [3], and to Heisenberg's uncertainty principle.At the heart of quantum theory is the lack of a common basis of coordinates for di¤erent observations, namely the lack of a common framework for all possible experiments on a physical system.The following sections show that this is intrinsically a topological issue.
4 Classic and Quantum Physics with n 2 degrees of freedom This section illustrates fundamental di¤erences between classic and quantum physics that emerge from the axioms, starting from the simplest possible examples.Consider initially physical systems with two degrees of freedom, n = 2: Example 3 Classic physics.The simplest possible physical system has two degreees of freedom and the Hilbert space for such systems is H = R 2 .In this case the space of events is the Boolean algebra of Borelian sets in R 2 , and the sample space is their union, namely R 2 : Therefore a classic event is a Borel set, there is a single sample space (R 2 ) and a common framework for all events, namely a single orthonormal coordinate basis for the space R 2 .Unicity is satis…ed.
Example 4 Quantum physics.From Axiom (A) above, the events of a quantum system with two degreees of freedom n = 2 are (self-adjoint) 14 projections of R 2 , and each can be identi…ed with a one-dimensional subspace (or line through the origin) L in R 2 .The space of all quantum events Q 2 in this case is the space of all one-dimensional subspaces or lines through the origin of R 2 .Observe that each projection or quantum event can also be identi…ed with an orthonormal unordered and unoriented basis of coordinates of the space R 2 , namely with a framework in R 2 , by adding a vector that is orthonormal to the line 15 L in R n In summary: 1 4 For a de…nition of self-adjoint operators see Dunford and Schwartz [11]; self adjoint operators are the closest there is to 'multiplication'operators that are used describe basic observables in physics such as position and momentum (Gudder [12]). 1 5 The orthonormal bases of coordinates in R 2 has two vectors: one is the vector spanning the line and the second is an orthonormal vector.One can choose the vectors so the space of all lines is included in the space of all orthonormal bases of vectors in R 2 and the map is one to one and onto.Observed that the bases are unordered and the vectors are unoriented.

Lemma 5
The space F2 of frameworks of a quantum system with two degrees of freedom (n = 2) in R 2 can be identi…ed with the one dimensional projective space P1 of all lines through the origin in R 2 , and the space P 1 in turn can be identi…ed with the unit circle S 1 in R 2 ; P 1 S 1 (Spanier [15], Milnor and Stashe¤ [14]) Proof.As mentioned in Example 3, each line through the origin in R 2 uniquely de…nes an unoriented, unordered system of coordinates in R 2 namely a framework.The space of all such lines is by de…nition the projective space P 1 , cf.Spanier [15], Milnor and Stashe¤ [14] who also show the identi…cation between P 1 and S 1  The following result provides a geometric characterization of the spaces of events and frameworks in quantum theory and in classic physics.It is based on the axioms stated above, and uses basic de…nitions and properties of topological spaces.A basic de…nition is De…nition 6 For n > 1; and k < n; let G(k; n) be the Grassmanian manifold of k planes of R n .(Spanier[15], Milnor and Stashe¤ [14]).

Observe that when
The following summarizes and shows the geometrical di¤erences between quantum theory and classic physics: Lemma 7 The space of classic events P in R 2 is the Boolean algebra of Borel measurable sets in R 2 :This is a convex space and is therefore topologically trivial (i.e.all its homotopy groups are zero) 17 .Unicity is satis…ed since there is a unique sample space, namely R 2 ; the space of classic frameworks has a single element, namely a (single) basis of coordinates for R 2 : In contrast, the space of quantum events Q 2 in R 2 is the space of all unoriented lines through the origin within two dimensional space R 2 also called the onedimensional projective space P 1 ; this space is the Grassmanian of 1 spaces in R 2 , denoted G(1; 2); The space G(1; 2) can be identi…ed with the unit circle S 1 ; P 1 S 1 G(1; 2): When n = 2; the space of frameworks F 2 in R 2 can be identi…ed with the space of quantum events in R 2 i.e.F 2 ' Q 2 : Both the space of quantum frameworks F 2 and the space of quantum events Q 2 can be identi…ed with the projective space P 1 G(1; 2) S 1 : 18 Neither the space of quantum events nor the space of frameworks in R 2 are contractible.
Proof.From Axiom (A) above, a quantum event in R 2 is by de…nition a projection of R 2 and therefore can be identi…ed with a (non zero) subspace of R 2 namely a line through the origin of R 2 ; in turn each line can be identi…ed with a basis of coordinates in R 2 as seen in Example 4 and in Lemma 5.The rest follows immediately from classic probability theory and the unicity postulate stated above.
De…nition 8 A singularity is a non-zero element of the homology of the space of events.Since the space of classic events is contractible it has no singularities.In quantum physics the space of events is Q 2 = S 1 and therefore has one signularity.

n 2 degrees of freedom
The next step is to characterize spaces of quantum events and frameworks in systems with n > 1 degrees of freedom, and exhibit the di¤erence with the same concepts in classic theory: The space of frameworks in R n is a manifold denoted F n , and it consists of all possible coordinate systems of R 2 : 19 Lemma 10 For n > 1; the manifold F n of all frameworks in R n is a connected subset of the manifold Q n of events in R n : Proof.Consider an event S which by Axiom (A) is a (self-adjoint) projection in R n .When the image of the projection is an n 1 dimensional subspace S of R n , de…ne an n framework by adding an orthonormal unit vector to the n 1 basis of coordinates of the subspace that represents S.This maps events, which are projections into n 1 dimensional subspaces, into frameworks of R n ; the map is continuous, one to one and onto the space of all n dimensional bases of coordinates of R n , namely the space of frameworks F n , which is a connected space.The manifold F n of frameworks is therefore contained as a connected subset of the manifold Q n of quantum events in R n .

The following summarizes:
Theorem 11 In a physical system with n > 1 degrees of freedom the space of quantum events Q n can be identi…ed with the space of all subspaces of R n and therefore can be identi…ed with the union of the Grassmanian manifolds In particular the Grassmanian G(n 1; n) consisting of all the n 1 subspaces of R n can be identi…ed with the space F n of all frameworks in R n .In particular, when n = 2; the space Q 2 of quantum events of R 2 equals the space of frameworks F 2 of R 2 and can be identi…ed with the projective space the n 1 projective space and the space F n of all frameworks of R n : Proof.This follows directly from Lemmas 7 and 9.
We analyzed the spaces of frameworks and of quantum events in R n , n > 1; and the topological di¤erence between the concept of events in classic physics.and in quantum theory.The next sections show the critical role played by the topology of spaces of frameworks and of spaces of events in separating quantum theory from classic physics.

De…nition 12
For each n a singularity is a non-zero element of the n th homology of the space of events with integer coe¢ cients.Since the space of classic events for every n > 1 is contractible, it has no singularities.In quantum physics the space of quantum events is Q n [ k<n G(k; n) and therefore it has as many signularities as generators of the homology of Q n [15] [14].: Below we explore the practical consequences of these facts with examples illustrating their connection with social choice theory.

Unicity and restricted domains of experiments
Unicity plays an important role in separating classic physics from quantum theory, and it is generally not satis…ed in quantum experiments in which more than one framework is needed to explain observations (Gudder [12], Gri¢ ths [10], Busemeyer [3] .Indeed we saw that the two theories di¤er in that quantum theory attempts to explain all possible experiments on a given physical system, which may require di¤erent frameworks, while experiments in classic physics are restricted from the outset to be part of one large experiment.havinga single framework.A natural question is whether it is possible to overcome the lack of unicity in quantum theory by restricting appropriately the domain of experiments that are performed on a physical system.There is a sinmple answer to this question and it is a¢ rmative.Restricting the domain of quantum theory experiments to all the subexperiments of a single experiment -having a single framework -has the desired e¤ect.In classic physics there is a unique experiment that contains all the rest, and from this unique experiment emerges the classic postulate of unicity.It is known that, under the same conditions, the same is true in quantum theory: if the various experiments within a restricted domain are all part of a single larger experiment, it is always possible to de…ne a common framework for all the quantum experiments, see e.g.Gudder [12].In quantum theory these are called compatible experiments, Gudder [12].Compatible observables correspond to noninterfering measurements, Gudder [12].In quantum logic there is a parallel mathematical characterization of compatible observables see Gudder [12], p. 82.Quantum theory under these restricted conditions therefore agrees with classic theory.
The question tackled in this section is whether there are more general domains of experiments where unicity can be recovered without requiring that all the experiments be subsets of a single larger experiment.
In the following we characterize restricted domains of experiments where there is a common framework for any given set of experiments within the do-main, without requiring that they are initially subexperiments of one single experiment.The ability to …nd a common framework for a set of experiments decides whether or not it is possible to reduce quantum theory to classic theory, within a restricted set of experiments.There have been indications that this may be possible: indeed it is known that in some cases it may be possible to 'prepare' appropriately the physical system before carrying out the various experiments, so that all the experiments in the domain can be observed within a common basis of coordinates of frameworks, see e.g.Cerceda [4] Gudder [12] and Busemeyer and Bruza [3] p. 158.
What follows provides a formal approach to the same problem: we identify topological conditions on a restricted domain of experiments that ensures the existence of a single or common framework for all experiments within the restricted domain.A simple example illustrates the issues: Example 13 Consider two di¤ erent bases of coordinates or frameworks in euclidean space R2 that are not orthonormal to each other. 20As an llustration consider the two orthonormal coordinate systems F 1 and F 2 in R 2 de…ned as F 1 = f(0; 1) and (1; 0)gand F 2 = f(1; 1); (1; 1)g:F 1 and F 2 are two di¤ erent orthonormal basis of coordinates or frameworks for R 2 that are not orthonormal to each other.since the vector (0; 1) is at 45 from the vector (1; 1) Having two different bases of coordinates (or frameworks) that are not orthonormal to describe the same object can create problems, since it leads to di¤ erent representations for the same object since eg the vector (x; y) in F 1 is (x y; x y) in F 2 : The problems can cause violation of unicity, interference and superposition of observations, and can lead to apparent contradictions as is illustrated in last section of this article, which provides practical examples.Nevertheless, for any two given basis of coordinates in R n such as F 1 and F 2 there is always a change of coordinates that maps one into the other, i.e. there is always a way to translate or to change one basis of coordinates into the other, denoted F 1 !F 2 .In this case the map is given by the (self adjoint) matrix M = ( 1 1 1 1 ) that maps the two vectors (1; 0); (01) into the two vectors (1; 1) and (1; 1); and more generally (x; y) into (x + y; x y):The matrix M can be thought of as a "dictionary" that translates one language or framework into another.For any two given frameworks F 1 and F 2 therefore one can de…ne a common framework by selecting one of the two framework: in this case selecting F 2 and changing the coordinates of F 1 correspondingly using the matrix M: This way , we can always de…ne a common framework for any vector v = (x; y) in R 2 : we simply consider the new vector M (v).The experiment can now be performed in the same basis of coordinates.At the end one uses the inverse matrix translate the results back into their original framework F 1 :For those two given frameworks F 1 and F 2 , therefore, unicity can be recovered.Allowing changes of coordinates seems a mild and natural way to resolve the problem for having a pair of di¤ erent frameworks that are not orthoginal and thus it resolves the problem of lack of unicity for the two given frameworks.
We will show however that this solution, while it works for any two frameworks, does not work in general.For any two given framework it is possible to select one, and translate the second into the …rst as shown above, but the question becomes whether one can always select one framework for any two given frameworks and to do it consistently, therefore resolving the lack of unicity by changes of coordinates.
In trying to do so one runs into a topological problem that identi…es the nature of lack of uncity.As we saw for any two given bases of coordinates by one can always de…ne a common basis of coordinates, but as we will see the change of coordinates that works for two given bases of coordinates does not work for all others and the selection is not be consistent: and continuous overall. 21To recover unicity one needs to be able to select a single basis of coordinates or frameworks for any number of bases of coordinates that may arise from different experiments in a way that (1) does not depend on the order of the two frameworks, (2) when the original frameworks are the same, one keeps the same framework.The map that selects a common framework for any k frameworks must be (3) continuous, so the selection of one framework among two frameworks coming from two di¤erent directions yields the same single outcome.In selecting a single framework, continuity is important in order to approximate the outcome by making increasingly accurate measurements.This is also called 'statistical su¢ ciency'and is critical for any probabiliistic theory.When continuity fails, practically identical experiments will lead to fundamentally di¤erent results, causing by itself contradictions and 'weird'observations.We need some de…nitions: De…nition 14 A map : X k !X is called symmetric if it does not depend on the order of the arguments, namely (x 1 ; :::; x k ) = (x 1 ; :::; x k ); where is any permutation of k > 1 elements.De…nition 15 A map : X k !X is called the identity on the diagonal if 8x 2 X; (x; :::; x) = x: Equivalently, is the identity on the diagonal of the product space X k ; where = f(x 1 ; :::; x k ) : 8i; j; x i = x j g when 8k > 1; the restriction map j (X k ) : (X k ) ! X is the identify map on (X k ).
Let F R n be the space of frameworks of R n ; n > 1:We can now de…ne De…nition 16 A framework selection is a way to select a single framework among any k > 2 frameworks, satisfying the conditions (1), ( 2) and (3) above.Formally, a framework selection is a sequence of continuous maps 2 1 The obvious example is when averaging the vectors in two basis of coordinates top obtain a common basis: this works in many cases but it does not work when the vectors one is trying to average are 180 degrees apart: if so, when one attempts to average both vectors one gets the zero vector.What results is therefore not a framework and the problem remains unresolved.f k g k=1;2;::: that selects one framework within any set of k > 1 frameworks,where Based on the above example, unicity can be de…ned as the possibility of selecting in a systematic way a single common framework or basis of coordinates for any set of k > 1 frameworks.

Remark 17
Observe that when a framework selection exists, the unicity of frameworks can be recovered by standard changes in coordinates as in Example 11 above.

Another example already mentionedr is as follows
Example 18 If all experiments within a restricted domain are subsets of a single larger experiment, then a common framework exists and unicity is satis…ed, see e.g.Gudder [12] p. and see also below.Observe that under these conditions the inclusion of each experiment as a subset of a larger experiment provides the framework selection required.

Is it always possible to select one common framework for any set of frameworks as de…ned above?
In general the answer is negative.We show in the next section that it is impossible to select a common framework for all the bases of coordinates of R n : As shown below the reason is topological: the ability to select one common framework among several is a property that is only satis…ed under certain topological conditions on the domain of frameworks that arises from the various experiments..

Why Unicity fails: impossibility theorems for selecting frameworks
The next step is to de…ne restricted domains of experiments within which one can recover unicity, and show why the recovery cannot be obtained in general.
.Starting with simple examples in two dimensional spaces, we extend gradually the results to provide a characterization that is valid for all dimensions. 22 First we establish that it is generally impossible to select a single common framework.Then we dentify restricted domains of experiments within which a single framework can be selected: Theorem 19 (Chichilnisky [5]) In experiments with two degrees of freedom where.H = R 2 there is no way to select a single framework for all experiments on a physical system.Formally, there exists no continuous function that selects one common framework : F 2 F 2 !F 2 that is inedpendent from the order of the framewkorks i.e. 8x; y , (x; y) = (y; x) and respects unanimity, 8x, (x; x) = x.
Proof.With two degrees of freedom the space of frameworks F 2 and the space of quantum events Q 2 coincide by Lemma 5; they are both the onedimensional projective space P 1 and this space can be identi…ed with the circle 23 S 1 , Spanier [15], and F 2 S 1 P 1 : Therefore the theorem reduces to the nonexistence of a continuous function : S 1 S 1 !S 1 that is symmetric, i.e. 8x; y (x; y) = (y; x); and respects unanimity, i.e. 8x, (x; x) = x.By de…nition, is the identity map on the diagonal D = f(x; y) 2 S 1 S 1 : x = yg namely j D = id D (x; x) = x .For a given z 2 S 1 de…ne A = f(x; z); 8x 2 S 1 g and B = f(z; x); 8x 2 S 1 g.Then A [ B can be continuously deformed into D within S 1 S 1 so by de…nition the degree mod 2 of the map on D must be the same as the degree mod 2 of the map on A [ B: Degree of j D :S 1 !S 1 is 1 since j D is the identity map, while the degreee j A[B is even, since is symmetric, which is a contradiction with (1).The contradiction arises from assuming that a map with the stated properties exists and therefore the map cannot exist.See also [5] The next result shows why the selection of a single framework is a topological problem, which can only be resolved in spaces that are contractible or topologically trivial, 24 namely in spaces of frameworks that are homotopic to a point or can be continuously deformed through themselves into a point: Theorem 20 (Chichilnisky and Heal [8]) Let X be a manifold or CW complex 25 .There exists a continuous selection map : X k !X satisfying axioms (1) (2) and (3) above, if and only if the space X is topologically trivial or contractible, i.e.X is homotopically equivalent to a point.

Proof. See Chichilnisky and Heal [8]
Theorem 21 (Chichilnisky [5]) There is no continuous function : (G(n 1; n)) k !(G(n 1; n) for any n > 1 that is symmetric and respects unanimity for all k > 1 Proof.By Theorem 13 the necessary and su¢ cient condition for the existence of a continuous map : X k !X satisfying the conditions of symmetry and respect of unanimity for all k > 1; is that the space X be contractible, see Chichilnisky and Heal [8].For every n > 1;the Grassmanian manifold The space X can be euclidean, or it can be a manifold in euclidean space.or a CW manifold cf [8] 2 5 Generally one works on CW manifolds, cf.[5] G(n 1; n) is not a contractible space (Milnor and Stashe¤ [14]).This completes the proof.The above can be summarized as follows: Theorem 22 Let H be a …nite dimensional Hilbert space, H = R n , and F n the space of its frameworks.Then F n violates unicity, i.e. there is no way to select a single framework among k frameworks because there exists is no continuous map : F k !F selecting a common framework in F n for any k frameworks 8k > 1:The space of frameworks F k can be identi…ed with the Grassmanian G(n 1; n) which is not topologically trivial as required for unicity.Violation of unicity is therefore due to the topology of the space of frameworks F k .
Proof.See Theorem above, and Chichilnisky [5] and Chichilnisky and Heal 26 : The results provided above show the topological origin of the violation of unicity.The next step is to show that by restricting the domain of experiments it is possible to recover unicity: indeed by Theorem 21 the topological condition of contractibility is necessary as well as su¢ cient for unicity.Consider now a physical system with n degress of freedom and corresponding Hilbert space H = R n Theorem 23 A necessary and su¢ cient restriction on the experiments of a quantum system with n degress of freedom to satisfy unicity, is that the corresponding space of frameworks F n is topologically trivial or contractible.
Proof.This follows from Theorem 20 above.

Quantum theory and social choice
The topological roots of the violation of unicity create an unexpected and fertile connection between quantum theory and social choice theory.
Social choice theory originated with Arrow's impossibility theorem, which de…ned reasonable axioms for the aggregation of individual into social preferences and proved that they were impossible to achieve [1].In 1980 social choice theory was rede…ned as follows: one seeks to de…ne a map that assigns a social preference to any two or more individual preferences, formally : P k !P where P representws a space of preferences [5] [6].Reasonable conditions are that map must be continuous and symmetric, depending on individual's preferences but not on the order of the individuals, and that respect unanimity so that if both indiividuals have the same preferences, the social preference is the same.Continuity means that it is possible to approximate the social preference by taking su¢ ciently accurate measurements of the individual preferences.
In 1980 the social choice problem was rewritten and given a simple geometrical form in [5] [6].Geometrically, linear preferences are vectors in a sphere S n .where n is the dimension of the space of choices.When n = 1;the problem is …nding a map that assigns a single point to every two points in the circle S 1 in a continuous way that is symmetric, so it does not depend on the order of the preferences, and respects unanimoty; Chichilnisky [5] [6].established that the problem has no solution: it is not possible to …nd such maps in the circle S 1 , or in higher dimensional spheres S n ; or even in general spaces of preferences that are co-dimension one oriented smooth foliations of R n27 [5] [6].
There is a deep connection between social choice and the topology of spheres and comes from the de…nition of preferences.Preferences are rankings or orders.A linear function f :R n !R de…nes a ranking as follows x y , f (x) > f (y): Linear preferences are de…ned by linear functions on R n .A linear function has by de…nition a constant gradient vector n R n ; and therefore a linear preference is de…ned by a single unit gradient vector in R n .The space of linear preferences P can therefore be represented by set of vectors of length one, the unit circle S 1 R 2 or more generallyu the unit sphere S n The space of all smooth preferences P is the space of all smooth co-dimension one oriented foliations of R n .[5] [6].The problem of social choice as introduced in Chichilnisky was formulated as the existence of a continuous function : P k !P satisfying two axioms ( 1) and ( 2) above.It was shown in Chichilnisky [5] that this problem has no solution, namely such a map does not exist.This non-existence result was shown to be a topological property of the space of co-dimension one foliations of R n .In the special case of linear preferences, the problem reduces to a topological property of spheres of all dimensions. 28.Chichilnisky [?] established that there are no continuous map : (S n ) k !S n that is symmetric and the identity on the diagonal S n : To understand the connection between social choice with unicity in physics we need more de…nitions: De…nition 24 A vector x in the unit circle S 1 represents a linear preference (or order) on R 2 , de…ned by the linear map f x : R 2 !R having the vector x as a gradient i.e. such that.Df x = x: The unit circle S 1 R 2 can therefore be identi…ed with the space P of all linear preferences 29 on R 2 .
De…nition 25 Let P be the space of smooth preferences or co-dimension one oriented foliations of euclidean space R n [5].for n > 1: When preferences are 2 7 This result was extended to necessary and su¢ cient conditions for the existence of such maps on manifolds of any dimension [5] [8].Formally, the problem is the non existence of a continuous function that assigns to k individual preferences a social preference, : P k !P , so that (1) is symmetric and (2) respects unanimity, as de…ned in the previous section.Here k > 1 represents the number of individuals and P is the space of preferences.One seeks to de…ne a map that assigns a common ('social') preference to any two or more individual preferences.2 8 The problem is equivalent under certain conditions to Arrow's Impossibility Theorem Arrow [1]. 2 9 Alternatively the vector is the gradient at f0g of a smooth function de…ned on R 2 that need not be linear.linear P = S n : For k > 1; a continuous function : P k !P satisfying the two axioms (1) and ( 2) is called a preference selection.ora common preference..The simplest case is n = 1 : Theorem 26 (Chichilnisky [5]) There is no continuous function : S 1 S 1 !S 1 that is symmetric, i.e. 8x; y (x; y) = (y; x); and respects unanimity, i.e. 8x, (x; x) = x, i.e.In other words: is not possible to de…ne a common preference for any two individuals with linear preferences.: Proof.See Theorem above and Chichilnisky [5] The result extends to spheres of all dimensions: Theorem 27 For n > 1; it is not possible to de…ne a common preference for any k > 1 individuals with linear preferences; there is no continuous map : (S n ) k !S n that is symmetric, and is the identity on the diagonal Proof.See Chichilnisky [5] Theorems 25 extends to general spaces P of smooth preferences consisting of oriented codimension-one smooth foliations of R n : Theorem 28 For any n > 1; it is not possible to …nd a common preference for any k > 1 smooth preferences on R n : In particular for n > 1; and 8k > 1; there is no continuous map : (P ) k !P that is symmetric, and is the identity on the diagonal P k .
Proof.See Chichilnisky [5] From the aboved resuts we can now formally establish the connection between quantun theory and social choice: Lemma 29 The space F n of frameworks in R 2 can be identi…ed with the space of linear preferences on R n: Proof.When n = 2; the result is immediate because the space of frameworks F 2 is in this case the one dimensional projective space P 1 that is the unit circle S 1 Spanier [15], and the space of linear preferences in R 2 is also the unit circle S 1 : When n > 2; by Theorem the space of frameworks can be identi…ed with the space of n 1 subspaces of R n : G(n 1; n) and each n 1 subspace A in R n de…nes an orthonormal vector v(A) in R n as shown in Theorem 13, which in turn can be identi…ed with the gradient of a linear preference in R n : This completes the proof.
We have therefore established: Theorem 30 For any restricted domain of preferences M R n n > 1; the social choice problem of aggregation of preferences in M is the existence of a map : M k !M satisfying axioms (1) (2) and (3) for all k > 1; and this problem formally coincides with the quantum theory problem of existence of unicity for all frameworks within the manifold M .
Proof.This follows from Lemma 28, from the identity between the axioms (1) and ( 2) in the two cases, in quantum theory and in social choice, and from the de…nition of unicity.
Theorem 31 The existence of common preferences is equivalent to the existence of common frameworks, or unicity.
Proof.The equivalence can be seen formally by considering the necessary and su¢ cient conditions for the existence of a selection of a single framework in restricted domains of experiments.In quantum theory, for any k 2 experiments on a given physical system, it may not be possible to de…ne in a continuous way a corresponding common framework f .In social choice theory, instead, it may not be possible to de…ne continuously a common preference in a way that respects unanimity and is anonymous, i.e. is symmetric.In this sense the general mathematical problem underlying quantum theory, which is the 'violation of unicity', can be seen as the non-existence of a continuous map : F k !F assigning a common framework to every k 2 frameworks (f 1 :::f k ) 2 F k in a way that is symmetric and respects unanimity namely 8f; (f; :::; f ) = f .9 Examples of ' weirdness'without common frameworks: conjunction fallacy, interference, Heisenberger uncertainty and order dependence this sense there is no weirdness at all.The following examples arise in experiments that have di¤erent frameworks.In each case, if the two frameworks were reduced to a common framework, of course, the so called weirdness would dissappear.While in each case one may …nd a common framework for two speci…c cases of the given experiments, the above topological results show their strength in that they demonstrate that in general this cannot be achieved: there will always exist two experiments where the common frameworks fail to exist.Quantum theory's violation of unicity has a logical, topological necessity that cannot be avoided.This is an issue that is not contemplated nor considered in the existing literature: we have shown that it is not possible to consistently reinterpret or measure all experiments -and their frameworks -to …nd always a common framework.The weirdness examples illustrated here will necesssarily emerge for some experiments, no matter how one may change the instruments and rede…ne the measurements, and therefore the frameworks, in the speci…c examples presented below.
Example 32 The conjunction fallacy Tversky & Kahneman, [16]1983 de…ned an important and common probability judgment error, called the 'conjunction fallacy', that is based on the lack of common frameworks.It is the famous 'Linda'problem.Judges are provided a brief story of a woman named Linda who used to be a philosophy student at a liberal university and was active in the anti-nuclear movement.The judges are asked to rank the likelihood of the following events: that Linda is now (a) active in the feminist movement, (b) a bank teller, (c) active in the feminist movement and a bank teller, (d) active in the feminist movement and not a bank teller, and (e) active in the feminist movement and a bank teller.The conjunction fallacy occurs when option (c) is judged to be more likely that option (b) (even though the latter contains the former).The experimental evidence shows that, surprisingly, people frequently produce conjunction fallacies for the Linda problem and for many other problems as well (Tsversky and Kahneman [16]1983).
In the following we use a geometric approach to quantum theory taken from Busemeyer and Bruza [3](2012), and explain how this relates to the results of the previous sections of this article.We refer the reader to [3] for further details and for clari…cations on the examples and on the diagram in Figure 1 below.
First we represent two answers to the feminism question by two di¤erent frameworks or basis of coordinates for euclidean space R 2 .Each framework is given by two orthogonal rays that span a two dimensional space.The answer yes to feminism is represented in Figure 1 by the ray labeled F and the answer no to the feminism question is represented by an orthogonal ray labeled F. This is the …rst framework.The person's initial belief about the feminism question which is generated from the Linda story, can be represented as a unit length vector labeled S in the …gure, within the two dimensional space spanned by these two rays.Note that the initial state vector S is close to the ray for yes to feminism, which matches the description of the Linda story.As explained geometrically by Busemeyer and Bruza [3], quantum theory computes probabilities for an event, or for a sequence of events, as follows: …rst one computes the so called 'amplitude' or inner product of two vectors denoted < F j S > for transiting from the initial state S to the ray F -this inner product equals of course the projection of the state S onto the F ray, which is the point on the F ray that intersects with the line extending up from the S state in the top panel in Figure 1 below.The quantum theory axioms postulate that the squared amplitude equals the probability of saying yes to the feminism question starting from the initial state and this is equal to j< F j S >j 2 = 0:9755 in Figure 1.Now we introduce the second framework, and rotate the axis to change from one to the other framework.In the …gure the bank teller question is represented by two orthogonal rays labeled B and B which are rotated so B is 20 below F. This de…nes the second framework, and it means that being a feminist and not being a bank teller are close in this belief space.The amplitude for transitioning from the initial state S which is close to F is also far away from the B ray (S is close to the orthogonal ray B): The amplitude < B j S > for transitioning from the initial state S to the ray B equals the projection of the state S onto the ray B ray which is illustrated by the point along the B ray that intersects with the line segment extending from S up to B in the bottom …gure.In this second framework, and according to the axioms of quantum theory, the square amplitud equals the probability of saying yes to the bank teller question starting from the initial state and this equals j< B j S >j 2 = 0:0245 in the …gure.Now consider the sequence of answers in which the person says yes to the feminism question and then says yes to the bank teller question in that order.The order that questions are processed is critical in quantum theory, and here we are assuming that the more likely event is evaluated …rst.The axioms of quantum theory imply that the amplitude for this sequence of answers equals the amplitude for the path S !F ! B and the latter equals the product of the amplitudes namely < B j F > : < F j S > : The …rst transition is from the initial state S to the ray F and the second is from the ray F to the state B: The path S !F ! B is illustrated in the top …gure.The amplitude < F j S > is the projection from S to F in the …gure which has a square magnitude equal to j< F j S >j 2 = 0:9755; and the amplitude < B j F > is the projection from the unit length basis vector aligned with F to the B ray in the …gure, which has a square magnitude equal to j< B j F >j 2 = 0:0955: By de…nition, the probability for the sequence equals the square amplitude for the path is j< B j F > : < F j S >j 2 = (0:9755):(0:0955) = 0:0932: Note that this probability exceeds the probability oif saying yes to the bank teller when starting from the initial state based on the story, j< B j S >j 2 = 0:0245:In conclusion this simple geometric model reproduces the basic facts of the conjunction fallacy.

Example 33 Order e¤ ects in observations
The same example can be used to show how quantum theory produces order e¤ects that are observed in attitude research.Note that the probability of the sequence for the order "yes to bank teller and then yes to feminism" is quite different than the probability for the opposite order.The bank teller …rst sequence has a probability equal to j< F j B > : < B j S >j 2 = (0:0955)(00245) = 0:00234 which is much smaller than the feminism …rst sequence j< B j F > : < F j S >j 2 = (0:9755)(0:0955) = 0:0932: This order e¤ect follows from the fact that the introduced a property of incompatibility between the feminism question and the bank teller question.

Example 34 Heisenberg uncertainty principle
We have assumed two frameworks, namely that the person is able to answer the feminism question using one basis of coordinates or framework fF; Fg but the person requires a di¤erent basis of coordinates or framework fB; Bg for answering the bank teller question as shown in the Figure .Observe that this implies that if the person is de…nite about the feminism question (in other words the belief state vector S is lined up with the ray F) then he or she must be inde…nite about the bank teller's question, because F and B are not orthogonal to each other and can be said to "interact or interfere" with the other.Similarly, if the person is de…nite with respect to the bank teller question then he or she must be inde…nite about the feminism question.This is essentially the Heisenberg uncertainty principle.

Example 35 Violation of unicity
Busemeyer and Bruza [3] state that, given that the two questions are treated as incompatible, we must also be violating unicity.Indeed, they say, we are assuming that the person is unable to form a single description (i.e. a single sample space) containing all the possible conjunctions fF\B; F\ B; F\B; F\ Bg: What they do not explain is why this is assumed.This article shows that, for topological reasons that are akin to those of the social choice paradox, this assumption is unavoidable.In other words, it is unavoidable that the person will be unable to form a single description for some basis of coordinates, or frameworks.The results presented here explain the violation of unicity.This implies that necessarily in some cases, the person would have never thought about conjunctions -for example those involving feminism and bank tellers -su¢ ciently to assign probabilities to all these conjunctions.Instead the person relies in such cases on two separate sample spaces: one based on elementary events fF; Fg for which they are familiar, and a second based on elementary events fB; Bg for which they are also familiar.If we did assume unicity in this example, then we could not explain the conjunction fallacy because the joint probabilities can be de…ned under unicity, and they will always be less than (or equal to) the marginal probabilities.Therefore as stated by Busemeyer and Bruza, to explain the experimental result requires the violation of unicity.The results of this article go further: they explain why the violation of unicity is a necessary logical implication when considering all possible experiments of a given physical system -as is the goal of quantum theory.And they illustrate why violation of unicity is, at its core, the same as the paradox of social choice.

The classic two-hole Experiment
The two-hole experiment is used as a famous example to show how quantum theory can explain observations that could not be explained with classic probability and physics.In the two-hole experiment illustrated in Figure 2 below S is a source of electrons all of whom have the same energy but they leave S in all directions and many impinge on a planar screen A. The screen A has two holes, 1 and 2, through which the electrons may pass.Behind the screen we have an electron detector which can be placed at distance x from the center of the screen.The detector records each passage of a single electron traveling from S through a hole in A to the point x; see Gudder [12] …g 2.1 p.58.In a classic analysis of the two hole experiment, e.g.[12] p 58-59, after performing the experiment many times with many di¤erent values of x one obtains a probability density P (x) that the electron passes from S to x as a function of x.Since an electron must pass through either hole 1 or hole 2, in classic probability theory P (x) = P (x 1 ) + P (x 2 ); where P (x i ) is the chance of arrival coming through i = 1; 2:See Figure 3 below.Figure 4 illustrates the observed distribution: the actual experimental result of the two-hole experiment is quite di¤erent, and it is shown in this …gure (see also Gudder [12] p. 59) which forces us to conclude that P 6 = P 1 + P 2 : In this sense the observations contradict classic probability.

The new two rotating hole experiment
The author proposed a variation of the classic two-hole experiment to predict new experimental observations based on the results of the article.The predicted observations are consistent with but di¤erent from the Pancharatnam -Berry Phases results that are analyzed in [13] and are illustrated below..The two rotating hole experiment (side view).isillustrated in Figure 5 below.The two planar screens A and B of the two-hole experiment are replaced by cylinders A and B: The position of each of the two holes 1 and 2 in the cylinder A can be rotated with knobs K 1 and K 2 respectively; each knob can move the respective hole around the entire cylinder A; with K 1 :rotating the hole 1 clockwise and K 2 rotaqting the hole 2 counterclockwise.On the basis of the results presented above, the author's prediction is that as hole 2 is rotated clockwise to the initial position of hole 1, and hole 1 is rotated counterclockwise to the initial position of hole 2, thus reproducing exactly the initial position of the two holes together at the end, the observations of the density distributions on the cylinder B will be di¤erent, even though in the …nal position the positions of the two holes together is undistinsgushable from the initial position of the two holes .This prediction remains to be tested experimentally, but it is close to the experimental results that have been obtained in the so called Pancharatnam-Berry phase, which is explained below, and which hsa been widely accepted, to the extent that with some good will, those can be considered experimental tests of the results of this article.
Figure 2: Two-hole experiment: S is a source of electrons all of whom have the same energy but they leave S in all directions and many impinge on a planar screen A. The screen A has two holes, 1 and 2, through which the electrons may pass.Behind the screen we have an electron detector which can be placed at distance x from the center of the screen.The detector records each passage of a single electron traveling from S through a hole in A to the point x.See Gudder [12] …g 2.1 p.58 Figure 3: Classical Analysis of the two hole experiment, S. Gudder p 58-59.After performing the experiment many times with many di¤erent values of x one obtains a probability density P (x) that the electron passes from S to x as a function of x.Since an electron must pass through either hole 1 or hole 2, P (x) = P (x 1 ) + P (x 2 ); where P (x i ) is the chance of arrival coming through i = 1; 2:   The Pancharatnam -Berry Phase can be brie ‡y summarized geometrically as follows, for a full presentation see e.g.[13].Suppose we travel on a closed path C on a sphere (Earth) while holding a vector V paralell to the surface, i e in the local tangent plane (Figure 7 below).At each point, V does not twist around the local vertical axis (the local normal vector n): This is known as paralell transport of the vector V around C: When we return to the starting point, we …nd that in general V makes an angle (C) with its initial direction: the angle , which depends only on the particular path C; (C) is known a the geometric angle, and is the classic analog of the Pancharatnam -Berry phase in quantum physics, see e.g.Ong and Lee [13] 14 Around the World in 90 days In a famous book of the same name, Jules Verne wrote a story around the concept of the "time line" about a gentleman who places a bet on being able to travel around the Earth in 90 days, and thinks he has lost by one day, arriving  2).At each point, V does not twist around the local vertical axis (the local normal vector n): This is known as paralell transport of V around C: When we return to the starting point, we …nd that in general V makes an angle (C) with its initial direction: the angle, which depends only on the particular path C is known a the geometric angle, and is the classic analog of the Pancharatnam -Berry phase in quantum physics, see e.g.Ong and Lee [13] in 91 days, only to …nd out that time went slower at the initial location so at their return they had e¤ectively won their bet..This literary piece illustrates the "time line" break in time, so that if one starts traveling around the world along a path such as C in Figure 7 at the end of the journey when one goes all around the world and reaches the initial position, the time measured by a traveling watch will be di¤erent than the time at the initial position at the moment of return, measured by a stationary watch..It can be shown that the topological problem posed by Jules Verne is the same as in the Pancharatnam Berry phases.It is well accepted that Barry phases arises from the existence of a singularity, which is the same origin that is postulated here for the basic properties of quantum theory that are described above, a topic that to be discussed in further writings.

Figure 1 :
Figure 1: The top panel shows projections for feminist and bank teller conjunction event; the bottom panel is for single bank teller event.F = yes to feminist, -F = no to feminist, B = yes to bank teller, -B = no to bank teller, S = initial state.

Figure 4 :
Figure 4: Observed Distribution: the actual experimental result of the two-hole experiment is quite di¤erent, and it is shown in this …gure (see Gudder, p 59): this forces us to conclude that P 6 = P 1 + P 2 :

Figure 6 :
Figure 6: Two rotating hole experiment (side view).The two planar screens A and B are here replaced by cylinders.The position of each of the two holes 1 and 2 can be rotated with a knob (K 1 and K 2 respectively) and each knob can move the respective hole around the entire cylinder A:

Figure 7 :
Figure 7: Pancharatnam -Berry Phase: suppose we travel on a closed path C on a sphere (Earth) while holding a vector V paralell to the surface, i e in the local tangent plane (Figure 1.2).At each point, V does not twist around the local vertical axis (the local normal vector n): This is known as paralell transport of V around C: When we return to the starting point, we …nd that in general V makes an angle (C) with its initial direction: the angle, which depends only on the particular path C is known a the geometric angle, and is the classic analog of the Pancharatnam -Berry phase in quantum physics, see e.g.Ong and Lee[13]