Sheaves of Structures in Physics: From Relativistic Causality to the Foundations of Quantum Mechanics

Transcription

1 Sheaves of Structures in Physics: From Relativistic Causality to the Foundations of Quantum Mechanics J. Benavides

2 The Stability of the Global Hyperbolic Property and Cosmic Censor Whether black holes are fundamental phenomena related to the gravitational collapse of massive objects or simply a particular feature of certain kind of gravitational collapse models, is one of the more important open problems in general relativity.

3 The Stability of the Global Hyperbolic Property and Cosmic Censor Whether black holes are fundamental phenomena related to the gravitational collapse of massive objects or simply a particular feature of certain kind of gravitational collapse models, is one of the more important open problems in general relativity. Cosmic Censorship Hypothesis (CCH) The complete gravitational collapse of a body results in the formation of a black hole rather than a naked singularity.

4 The Stability of the Global Hyperbolic Property and Cosmic Censor Whether black holes are fundamental phenomena related to the gravitational collapse of massive objects or simply a particular feature of certain kind of gravitational collapse models, is one of the more important open problems in general relativity. Cosmic Censorship Hypothesis (CCH) The complete gravitational collapse of a body results in the formation of a black hole rather than a naked singularity. The fundamental principle acting on the formation of black holes is normally associated to a process acting to conserve classical determinism to some degree.

5 Weak Cosmic Censorship (WCC) In physical reasonable spacetimes the black hole exterior region is globally hyperbolic.

6 Strong Cosmic Censorship (SCC) Every physical reasonable spacetime is globally hyperbolic. Weak Cosmic Censorship (WCC) In physical reasonable spacetimes the black hole exterior region is globally hyperbolic.

7 Deficiencies of WCC and SCC The definition of the black hole exterior region makes use of curves that escape to spacetime infinity= no local description. The SCC does not give a definition of black holes. The SCC discards as physical reasonable any spacetime where causality is violated.

8 Deficiencies of WCC and SCC The definition of the black hole exterior region makes use of curves that escape to spacetime infinity= no local description. The SCC does not give a definition of black holes. The SCC discards as physical reasonable any spacetime where causality is violated. Can we obtain a satisfactory formulation of the CCH within the classical relativistic formalism? do WCC or SCC capture the fundamental property related to the formation of event horizons?

9 Deficiencies of WCC and SCC The definition of the black hole exterior region makes use of curves that escape to spacetime infinity= no local description. The SCC does not give a definition of black holes. The SCC discards as physical reasonable any spacetime where causality is violated. Can we obtain a satisfactory formulation of the CCH within the classical relativistic formalism? do WCC or SCC capture the fundamental property related to the formation of event horizons? We can try to answer these questions studying stability properties of the global hyperbolic property.

10 The Stability of the Global Hyperbolic Property The Stability of the Global Hyperbolic Property The global hyperbolic property A global hyperbolic spacetime (M, g) is a causal spacetime such that for every p, q M, J + (p) J (q) is a compact set.

11 The Stability of the Global Hyperbolic Property The Stability of the Global Hyperbolic Property The global hyperbolic property A global hyperbolic spacetime (M, g) is a causal spacetime such that for every p, q M, J + (p) J (q) is a compact set. Interval Topology Let Lor(M) be the space of Lorentzian metrics over M, given g, g Lor(M), we say that g < g if for every p M, the causal cone of g in T pm is contained in the timelike cone of g, we say that g g if for every p M, the causal cone of g in T pm is contained in the causal cone of g. The intervals (g, g ) = {h Lor(M) : g < h < g }, with g, g Lor(M), generate a topology in Lor(M)

12 The Stability of the Global Hyperbolic Property A property of the spacetime (M, g) is said to be stable with respect to the interval topology if there exists an interval (h, h ) of the interval topology such that g (h, h ) and for all g (h, h ) the property holds for the spacetime (M, g ).

13 The Stability of the Global Hyperbolic Property A property of the spacetime (M, g) is said to be stable with respect to the interval topology if there exists an interval (h, h ) of the interval topology such that g (h, h ) and for all g (h, h ) the property holds for the spacetime (M, g ). Theorem Global hyperbolicity is stable in the interval topology.

14 The Stability of the Global Hyperbolic Property A property of the spacetime (M, g) is said to be stable with respect to the interval topology if there exists an interval (h, h ) of the interval topology such that g (h, h ) and for all g (h, h ) the property holds for the spacetime (M, g ). proof Let h be a complete Riemannian metric over M. Fix m M, let K 0 = m and define inductively K n+1 = J + (B h (m, r n )) J (B h (m, r n )), with r n > r n such that K n IntB h (m, r n ). Theorem Global hyperbolicity is stable in the interval topology.

15 The Stability of the Global Hyperbolic Property K n is compact for all n. n=0 K n = M, For every compact set C there exists n such that C K n, For n 0, J + (K n ) J (K n ) intk n+1,

16 The Stability of the Global Hyperbolic Property K n is compact for all n. n=0 K n = M, For every compact set C there exists n such that C K n, For n 0, J + (K n ) J (K n ) intk n+1, Then we construct a family of metrics {g n } n s.t. g n > g, g n+1 < g n, g n+1 = g n on K n and J g + n (K n ) Jg n (K n ) intk n+1.

17 The Stability of the Global Hyperbolic Property K n is compact for all n. n=0 K n = M, For every compact set C there exists n such that C K n, For n 0, J + (K n ) J (K n ) intk n+1, Then we construct a family of metrics {g n } n s.t. g n > g, g n+1 < g n, g n+1 = g n on K n and J g + n (K n ) Jg n (K n ) intk n+1. Let g 0 > g be sucht that (M, g 0 ) is stably causal. Suppose we have constructed g n, since = J + (K n+1 ) J (K n+1 ) K n+2 = g<g <g n (J + g (K n+1 ) J g (K n+1 ) K n+2 ), g < g n+1 < g n such that J + (K g n+1 n+1 ) J g n+1(k n+1 ) K n+2 = and J + (K g n+1 n+1 ) J g n+1(k n+1 ) IntK n+2. Consider χ : M R continuous s.t. 0 χ 1, χ = 0 on K n and χ = 1 on M \ intk n+1. Let g n+1 = χg n+1 + (1 χ)g n.

18 The Stability of the Global Hyperbolic Property K n is compact for all n. n=0 K n = M, For every compact set C there exists n such that C K n, For n 0, J + (K n ) J (K n ) intk n+1, Then we construct a family of metrics {g n } n s.t. g n > g, g n+1 < g n, g n+1 = g n on K n and J g + n (K n ) Jg n (K n ) intk n+1. Let g 0 > g be sucht that (M, g 0 ) is stably causal. Suppose we have constructed g n, since = J + (K n+1 ) J (K n+1 ) K n+2 = g<g <g n (J + g (K n+1 ) J g (K n+1 ) K n+2 ), g < g n+1 < g n such that J + (K g n+1 n+1 ) J g n+1(k n+1 ) K n+2 = and J + (K g n+1 n+1 ) J g n+1(k n+1 ) IntK n+2. Consider χ : M R continuous s.t. 0 χ 1, χ = 0 on K n and χ = 1 on M \ intk n+1. Let g n+1 = χg n+1 + (1 χ)g n. Define h > g on M such that h (p) = g n(p) on K n, h g 0 is stably causal, J + h (p) J h (q) is closed p, q and J + h (p) J h (q) J + h (K n) J h (K n) K n+1 Then (M, h ) is globally hyperbolic.

19 The Stability of the Global Hyperbolic Property The stability of other familiar properties as the energy conditions, or the existence of Hawking-Penrose singularities follow straightforward from the regularity of the differential structure. On the other hand, the stability of global hyperbolicity is highly non-trivial.

20 The Stability of the Global Hyperbolic Property The stability of other familiar properties as the energy conditions, or the existence of Hawking-Penrose singularities follow straightforward from the regularity of the differential structure. On the other hand, the stability of global hyperbolicity is highly non-trivial. Can we expect the existence of generic properties within the relativistic formalism able to capture collapsing gravitational scenarios and the formation of event horizons,and that imply the validity of the global hyperbolic property?

21 The Stability of the Global Hyperbolic Property The stability of other familiar properties as the energy conditions, or the existence of Hawking-Penrose singularities follow straightforward from the regularity of the differential structure. On the other hand, the stability of global hyperbolicity is highly non-trivial. Can we expect the existence of generic properties within the relativistic formalism able to capture collapsing gravitational scenarios and the formation of event horizons,and that imply the validity of the global hyperbolic property? Or should we take seriously counterexamples of the SCC, particularly those which contain causality violations? Denying the physical importance of these kind of examples we are taking a position that is very reminiscent to the position assumed during the first years of general relativity in confront to solutions of the Einstein equations which predicted the formation of singularities.

22 Causality Violations and Quantum Geometry Causality violations as black holes are one of the most intriguing predictions of classical relativistic formalism. These phenomena have been normally considered as unphysical mainly because, within the classical formalism, causality violations seem to impose more initial data constraints than those normally required in causality respecting spacetimes.

23 Causality Violations and Quantum Geometry Causality violations as black holes are one of the most intriguing predictions of classical relativistic formalism. These phenomena have been normally considered as unphysical mainly because, within the classical formalism, causality violations seem to impose more initial data constraints than those normally required in causality respecting spacetimes. Even if these kind of initial constraints have not been observed so far, these do not provide a conclusive answer respect to the unphysical character of causality violations.

24 Causality Violations and Quantum Geometry Causality violations as black holes are one of the most intriguing predictions of classical relativistic formalism. These phenomena have been normally considered as unphysical mainly because, within the classical formalism, causality violations seem to impose more initial data constraints than those normally required in causality respecting spacetimes. Even if these kind of initial constraints have not been observed so far, these do not provide a conclusive answer respect to the unphysical character of causality violations. Then it results important to establish whether violations of causality are indeed in real conflict with some fundamental physical principles or otherwise causality violating regions as those predicted by Kerr s solution have to be considered seriously, possibly as a fundamental hint towards a quantum gravity theory.

25 Computational Networks and The Classical Time Travel Paradoxes Computational Networks and The Classical Time Travel Paradoxes The study of physical effects of closed timelike curves via computational networks allows to abstract away most of the geometry of such situations avoiding the technicalities associated to strange objects linked to these violations (e.g singularities or wormholes) and at the same time it allows to include quantum mechanics, something not possible yet in the geometrical approach due to the lack of a satisfactory quantum gravity theory.

26 Computational Networks and The Classical Time Travel Paradoxes Computational Networks and The Classical Time Travel Paradoxes The study of physical effects of closed timelike curves via computational networks allows to abstract away most of the geometry of such situations avoiding the technicalities associated to strange objects linked to these violations (e.g singularities or wormholes) and at the same time it allows to include quantum mechanics, something not possible yet in the geometrical approach due to the lack of a satisfactory quantum gravity theory. The models obtained, even if constituted just of finite particles travelling along fixed trajectories and interacting only at short range, do not result restrictive because such models include the class of classical and quantum computational networks

27 Computational Networks and The Classical Time Travel Paradoxes Computational Networks and The Classical Time Travel Paradoxes The study of physical effects of closed timelike curves via computational networks allows to abstract away most of the geometry of such situations avoiding the technicalities associated to strange objects linked to these violations (e.g singularities or wormholes) and at the same time it allows to include quantum mechanics, something not possible yet in the geometrical approach due to the lack of a satisfactory quantum gravity theory. The models obtained, even if constituted just of finite particles travelling along fixed trajectories and interacting only at short range, do not result restrictive because such models include the class of classical and quantum computational networks The Turing Principle (Deutsch 1985) Every finitely realizable physical system can be perfectly simulated* by a universal model computing machine operating by finite means

28 Computational Networks and The Classical Time Travel Paradoxes Computational Networks and The Classical Time Travel Paradoxes The study of physical effects of closed timelike curves via computational networks allows to abstract away most of the geometry of such situations avoiding the technicalities associated to strange objects linked to these violations (e.g singularities or wormholes) and at the same time it allows to include quantum mechanics, something not possible yet in the geometrical approach due to the lack of a satisfactory quantum gravity theory. The models obtained, even if constituted just of finite particles travelling along fixed trajectories and interacting only at short range, do not result restrictive because such models include the class of classical and quantum computational networks The Turing Principle (Deutsch 1985) Every finitely realizable physical system can be perfectly simulated* by a universal model computing machine operating by finite means * A computing machine C is capable of perfect simulating a physical system S, if there exists a program π(s) for C that renders C computationally equivalent to S.

29 Computational Networks and The Classical Time Travel Paradoxes A spacetime bounded network can be simplified without altering its computational character, or in other words obtaining an alternative network where the outputs are the same function of their inputs, in the following way:

30 Computational Networks and The Classical Time Travel Paradoxes Localize all the Interactions in Gates. Closed temporal trajectories on carriers are replaced by trajectories that goes first in the ambiguous future of all gates, the with a single negative delay in the ambiguous past and then forward to the future (S 2 ). A spacetime bounded network can be simplified without altering its computational character, or in other words obtaining an alternative network where the outputs are the same function of their inputs, in the following way: Replace the particles travelling in the network by sets of particles which carry a 2-state internal degree of freedom (the carriers of the bits)

31 Computational Networks and The Classical Time Travel Paradoxes Localize all the Interactions in Gates. Closed temporal trajectories on carriers are replaced by trajectories that goes first in the ambiguous future of all gates, the with a single negative delay in the ambiguous past and then forward to the future (S 2 ). A spacetime bounded network can be simplified without altering its computational character, or in other words obtaining an alternative network where the outputs are the same function of their inputs, in the following way: Replace the particles travelling in the network by sets of particles which carry a 2-state internal degree of freedom (the carriers of the bits) Figure: Chronology violating network.

32 Computational Networks and The Classical Time Travel Paradoxes Grandfather Paradox (Classical Perspective) We can understand the grandfather paradox as a particle that travels around a closed timelike curve arriving back before its departure preventing itself from setting out.

33 Computational Networks and The Classical Time Travel Paradoxes Grandfather Paradox (Classical Perspective) We can understand the grandfather paradox as a particle that travels around a closed timelike curve arriving back before its departure preventing itself from setting out. The Values 1 or 0 represent whether the simulated particle is on that carrier or not respectively. The first two bits represent two possible trajectories that the particle can follow. Before going on their separate ways the particles carrying the bits find an older version of the right one which acts on them in the next way, x x 1 y x y x y 1 y i.e if they meet the older version of the particle they change the value of their bit changing the trajectory of the simulated particle, otherwise they do not change their value.

34 Computational Networks and The Classical Time Travel Paradoxes Grandfather Paradox (Classical Perspective) We can understand the grandfather paradox as a particle that travels around a closed timelike curve arriving back before its departure preventing itself from setting out. The Values 1 or 0 represent whether the simulated particle is on that carrier or not respectively. The first two bits represent two possible trajectories that the particle can follow. Before going on their separate ways the particles carrying the bits find an older version of the right one which acts on them in the next way, x x 1 y x y x y 1 y i.e if they meet the older version of the particle they change the value of their bit changing the trajectory of the simulated particle, otherwise they do not change their value. Consistency Condition: x y 1 = y, then x = 1. The particle has to take the left

35 Local Causality Violations and Singularities Local Causality Violations and Singularities The above analysis arise from the false premise that classical physics is approximately true near causality violating regions.

36 Local Causality Violations and Singularities Local Causality Violations and Singularities The above analysis arise from the false premise that classical physics is approximately true near causality violating regions. Causal Tube N in (M, g) N = S 1 S 2 T, is compact The S i are disjoint spacelike manifolds, S i = S2, T = S i [0, 1], timelike S 2 I + ( S 1 ), every future causal vector field V in M tangent to T points inside N in S 1 and outside N in S 2, I N (S 1) = = I + N (S 2)

37 Local Causality Violations and Singularities Theorem Let (M, g) be a spacetime where Einstein s equation and the weak energy condition hold. Let N be a causal tube in M which contains a chronology violating region and such that on every null geodesic σ past inextendible contained in N, there exists a point where R(σ, σ ) 0. Then N is non compact.

38 Local Causality Violations and Singularities Theorem Let (M, g) be a spacetime where Einstein s equation and the weak energy condition hold. Let N be a causal tube in M which contains a chronology violating region and such that on every null geodesic σ past inextendible contained in N, there exists a point where R(σ, σ ) 0. Then N is non compact. This result can be interpreted as evidence of the formation of singularities in the region N or as the incapacity of the classical formalism to give a local description of the formation of chronology violating regions. From this theorem follows that classical (non-quantum) physics cannot give an accurate description close to causality violating regions.

39 Local Causality Violations and Singularities Sketch of the Proof Let D + N (S 1) be the set of points p such that every inextendible past causal curve in N from p intersects S 1 and H + N (S 1 ) = D + N (S 1) I N (D+ N (S 1 ))

40 Local Causality Violations and Singularities Sketch of the Proof Since N contains a chronology violating region, there exists a closed timelike curve contained in N, then N D + N (S 1) and H + N (S 1) is not empty. Let D + N (S 1) be the set of points p such that every inextendible past causal curve in N from p intersects S 1 and H + N (S 1 ) = D + N (S 1) I N (D+ N (S 1 ))

41 Local Causality Violations and Singularities Sketch of the Proof Let D + N (S 1) be the set of points p such that every inextendible past causal curve in N from p intersects S 1 and Since N contains a chronology violating region, there exists a closed timelike curve contained in N, then N D + N (S 1) and H + N (S 1) is not empty. The weak energy condition and the fact that Ric(σ, σ ) 0 at some point on every past null geodesic σ contained in N imply that for every null geodesic generator γ of H + N (S 1), Ric(γ, γ ) 0 on every point and is not zero at some point. H + N (S 1 ) = D + N (S 1) I N (D+ N (S 1 ))

42 Local Causality Violations and Singularities Sketch of the Proof Let D + N (S 1) be the set of points p such that every inextendible past causal curve in N from p intersects S 1 and H + N (S 1 ) = D + N (S 1) I N (D+ N (S 1 )) Since N contains a chronology violating region, there exists a closed timelike curve contained in N, then N D + N (S 1) and H + N (S 1) is not empty. The weak energy condition and the fact that Ric(σ, σ ) 0 at some point on every past null geodesic σ contained in N imply that for every null geodesic generator γ of H + N (S 1), Ric(γ, γ ) 0 on every point and is not zero at some point. From this follows that H + N (S 1) is a closed non-compact subset of N

43 Local Causality Violations and Singularities Sketch of the Proof Let D + N (S 1) be the set of points p such that every inextendible past causal curve in N from p intersects S 1 and H + N (S 1 ) = D + N (S 1) I N (D+ N (S 1 )) Since N contains a chronology violating region, there exists a closed timelike curve contained in N, then N D + N (S 1) and H + N (S 1) is not empty. The weak energy condition and the fact that Ric(σ, σ ) 0 at some point on every past null geodesic σ contained in N imply that for every null geodesic generator γ of H + N (S 1), Ric(γ, γ ) 0 on every point and is not zero at some point. From this follows that H + N (S 1) is a closed non-compact subset of N Then N is non-compact.

44 Causality Violations: The quantum Approach Causality Violations: The Quantum Approach Since classical physics cannot give an appropriate description near causality violating regions, we require quantum computational networks to simulate the behaviour of particles close to such regions. This means:

45 Causality Violations: The quantum Approach Causality Violations: The Quantum Approach Since classical physics cannot give an appropriate description near causality violating regions, we require quantum computational networks to simulate the behaviour of particles close to such regions. This means: 1 To consider the carrier of the bits (qubits) as a quantum physical systems which non trivial observables are Boolean (i.e. observables which have just two eigenvalues). 2 The inputs/outputs of quantum gates are not necessarily elements of the computational basis (i.e. they are not necessarily eigenstates of the tensor product of the observables). 3 Computational Gates are unitary transformations over composed n-qubit systems.

46 Causality Violations: The quantum Approach A general Chronology Violating Network The initial joint state of the m + 2n bits is given by ρ 12 ρ 3,. The final joint state is given by U(ρ 12 ρ 3 )U, where U is an arbitrary Unitary interaction. Let ρ 12 = ρ 1 ρ 2 H 1 H 2 be a density operator representing the initial state of the non time travelling bits and of the younger versions of the time travelling bits. Let ρ 3 be the density operator representing the initial state of the older versions of the time travelling bits.

47 Causality Violations: The quantum Approach A general Chronology Violating Network The initial joint state of the m + 2n bits is given by ρ 12 ρ 3,. The final joint state is given by U(ρ 12 ρ 3 )U, where U is an arbitrary Unitary interaction. Consistency Condition: Tr 1,3 (U(ρ 12 ρ 3 )U ) = ρ 3, (1) i.e. the n older time travelling bits enter the gate in the same state as the younger bits leave it Let ρ 12 = ρ 1 ρ 2 H 1 H 2 be a density operator representing the initial state of the non time travelling bits and of the younger versions of the time travelling bits. Let ρ 3 be the density operator representing the initial state of the older versions of the time travelling bits.

48 Causality Violations: The quantum Approach A general Chronology Violating Network The initial joint state of the m + 2n bits is given by ρ 12 ρ 3,. The final joint state is given by U(ρ 12 ρ 3 )U, where U is an arbitrary Unitary interaction. Consistency Condition: Tr 1,3 (U(ρ 12 ρ 3 )U ) = ρ 3, (1) i.e. the n older time travelling bits enter the gate in the same state as the younger bits leave it Let ρ 12 = ρ 1 ρ 2 H 1 H 2 be a density operator representing the initial state of the non time travelling bits and of the younger versions of the time travelling bits. Let ρ 3 be the density operator representing the initial state of the older versions of the time travelling bits. Theorem (Deutsch) The equation 1 admits a solution ρ 3 for every initial state ρ 12 and every interaction U.

49 Causality Violations: The quantum Approach A general Chronology Violating Network The initial joint state of the m + 2n bits is given by ρ 12 ρ 3,. The final joint state is given by U(ρ 12 ρ 3 )U, where U is an arbitrary Unitary interaction. Consistency Condition: Tr 1,3 (U(ρ 12 ρ 3 )U ) = ρ 3, (1) i.e. the n older time travelling bits enter the gate in the same state as the younger bits leave it Let ρ 12 = ρ 1 ρ 2 H 1 H 2 be a density operator representing the initial state of the non time travelling bits and of the younger versions of the time travelling bits. Let ρ 3 be the density operator representing the initial state of the older versions of the time travelling bits. Theorem (Deutsch) The equation 1 admits a solution ρ 3 for every initial state ρ 12 and every interaction U. Quantum theory does not impose initial data constrains

50 Causality Violations: The quantum Approach The Grandfather Paradox Revisited If the input state is the one which classically leads to a paradoxical situation, i.e. ρ 12 = 0 1, we obtain that equation 1 admits as unique solution ρ 3 = 1 2 I Thus, the density operator of the two output bits is Tr 3 (U(ρ 12 ρ 3 )U ) = 1 ( ). 2 Therefore, the probability that there will be two particles in the output, or zero particles in the output is 1/2 in both cases, and the probability to obtain exactly one single particle, as we will expect based on our classical intuition, is zero! The quantum state history of the GF paradox 1 The original input state of the particle is that one that models the particle taking the trajectory that will take it back in time if nothing intervened, 2 The particle will meet an older version of itself in a mixed state ρ 3 = 1 I of being 2 present and absent, 3 The younger version enters the gate in a mixed state of being prevented or not from going back in time, 4 Finally in the unambiguous future we find the mixed state which is a state of there being two particles present and none particles present.

51 Causality Violations: The quantum Approach The Grandfather Paradox Revisited If the input state is the one which classically leads to a paradoxical situation, i.e. ρ 12 = 0 1, we obtain that equation 1 admits as unique solution ρ 3 = 1 2 I Thus, the density operator of the two output bits is Tr 3 (U(ρ 12 ρ 3 )U ) = 1 ( ). 2 Therefore, the probability that there will be two particles in the output, or zero particles in the output is 1/2 in both cases, and the probability to obtain exactly one single particle, as we will expect based on our classical intuition, is zero! The quantum state history of the GF paradox 1 The original input state of the particle is that one that models the particle taking the trajectory that will take it back in time if nothing intervened, 2 The particle will meet an older version of itself in a mixed state ρ 3 = 1 I of being 2 present and absent, 3 The younger version enters the gate in a mixed state of being prevented or not from going back in time, 4 Finally in the unambiguous future we find the mixed state which is a state of there being two particles present and none particles present. What happened to the modelled particle?

52 Causality Violations: The quantum Approach The GF paradox in the statistical interpretation of QM When the older and younger version encounter each other, the older is in a mixed state of being absent and being present. Under the statistical interpretation this means that in one half of the elements of the ensemble, that the quantum state represents, there was not encounter and in the other half there was an encounter. In those elements of the statical ensemble where there was not an encounter, the younger version travels back in time and experiences the encounter in the same event of the ensemble where there was not encounter leading to the same classical paradox, By consequence the statistical interpretation require some constrains on the original states as in the classical case, but we saw that the quantum formalism does not impose such constraints.

53 Causality Violations: The quantum Approach The GF paradox in the statistical interpretation of QM When the older and younger version encounter each other, the older is in a mixed state of being absent and being present. Under the statistical interpretation this means that in one half of the elements of the ensemble, that the quantum state represents, there was not encounter and in the other half there was an encounter. In those elements of the statical ensemble where there was not an encounter, the younger version travels back in time and experiences the encounter in the same event of the ensemble where there was not encounter leading to the same classical paradox, By consequence the statistical interpretation require some constrains on the original states as in the classical case, but we saw that the quantum formalism does not impose such constraints. The GF paradox in the Deutsch-Everett interpretation In all the universes the particle approaches the chronology violating region in a state which would take it back in time (i.e. ρ 12 = 01 ), but just in half of the universes the particle follows the time back travelling trajectory, because in half of the universes there is an encounter with a particle that looks like an older version of the particle (this follows from the fact that ρ 3 = 1 I), which changes the trajectory of 2 the younger version to avoid the time travel trajectory. After that, both versions live together in the unambiguous future. In the other half of the universes there is no encounter and the particle takes the time travelling trajectory, but travelling back to the past of one of the other half universes where the encounter happens. This is consistent with the the density operator of the output bits Tr 3 (U(ρ 12 ρ 3 )U ) = 1 ( ). 2

54 Towards a Quantum Geometry Towards a Quantum Geometry: The Deutsch-Everett Interpretation If given an observable ˇB with eigenvectors b 1,..., b n, the state vector ψ(t) is expressed in the basis formed by the eigenvectors of ˇB as: ψ(t) = α 1 b α m b m. Then a measurement of B at time t will give as a result one of the eigenvalues b i with probability α i 2 respectively, and none other result that is not one of the eigenvalues is obtained.

55 Towards a Quantum Geometry Towards a Quantum Geometry: The Deutsch-Everett Interpretation If given an observable ˇB with eigenvectors b 1,..., b n, the state vector ψ(t) is expressed in the basis formed by the eigenvectors of ˇB as: ψ(t) = α 1 b α m b m. Then a measurement of B at time t will give as a result one of the eigenvalues b i with probability α i 2 respectively, and none other result that is not one of the eigenvalues is obtained. Measurement Problem The quantum formalism is not able to distinguish the actual result of measurement from all the possible results.

56 Towards a Quantum Geometry Towards a Quantum Geometry: The Deutsch-Everett Interpretation If given an observable ˇB with eigenvectors b 1,..., b n, the state vector ψ(t) is expressed in the basis formed by the eigenvectors of ˇB as: ψ(t) = α 1 b α m b m. Then a measurement of B at time t will give as a result one of the eigenvalues b i with probability α i 2 respectively, and none other result that is not one of the eigenvalues is obtained. Measurement Problem The quantum formalism is not able to distinguish the actual result of measurement from all the possible results. Deutsch-Everett Everett 1957: QM is consistent with the idea that all possible results of the measurement actually happen, being the single universe where we perceive one single outcome part of a larger structure of many-worlds where the different outcomes happen. The multiverse is a set with a measure, whose elements are maximal sets of observables with definite values that correspond to different universes or different histories. In this context the terms α i 2 associated to the expression ψ(t) = α 1 b α m b m represent the values of the measure of the sets of universes where the observable ˇB assume the value b i respectively.

57 Towards a Quantum Geometry Different sets of compatible observables determine different expressions of the state vector ψ(t), these different forms to express the state vector correspond to different foliations of the multiverse in the same sense that a region of spacetime can be foliated by spacelike surfaces in different ways.

58 Towards a Quantum Geometry Different sets of compatible observables determine different expressions of the state vector ψ(t), these different forms to express the state vector correspond to different foliations of the multiverse in the same sense that a region of spacetime can be foliated by spacelike surfaces in different ways. Each universe in any foliation is associated to a classical system, which corresponds to the classical physical world where we see the measuring apparatus taking one unique value.

59 Towards a Quantum Geometry Different sets of compatible observables determine different expressions of the state vector ψ(t), these different forms to express the state vector correspond to different foliations of the multiverse in the same sense that a region of spacetime can be foliated by spacelike surfaces in different ways. Each universe in any foliation is associated to a classical system, which corresponds to the classical physical world where we see the measuring apparatus taking one unique value. The universes interact via interference phenomena, but such interactions are suppressed at the classical level described by classical physics.

60 Towards a Quantum Geometry The Geometry of The Grand Father Paradox Classical Evolution Zone Interference Zone

61 Towards a Quantum Geometry The Geometry of The Grand Father Paradox Classical Evolution Zone Interference Zone There is not yet a sound mathematical model of the quantum multiverse, but it is clear that if such model exists, it is probably the key towards a quantum gravity theory.

62 Sheaves of Structures Sheaves of structures (Motivation) Figure: Galilean Spacetime Galilean Spacetime A topological space X =temporal line For each x X there exists a structure A x constituted by: -A universe E x formed by snapshots of extended objects in time. - Functions f x 1, f x 2,... and relations R x 1, R x 2,..., that give the instantaneous properties of extended objects. The different worlds E x attach in an extended universe E, in such a way that the different functions and relations attach in a continuous way. A x = (E x, R x 1, R x 2..., f x 1, f x 2...)

63 Sheaves of Structures Sheaves of Structures (Definition) Definition Given a fix type of structures τ = (R 1,..., f 1,..., c 1,...) a sheaf of τ-structures A over a topological space X is given by:

64 Sheaves of Structures Sheaves of Structures (Definition) Definition Given a fix type of structures τ = (R 1,..., f 1,..., c 1,...) a sheaf of τ-structures A over a topological space X is given by: a-) A sheaf (E, p) over X( i.e a local homeomorphism).

65 Sheaves of Structures Sheaves of Structures (Definition) Definition Given a fix type of structures τ = (R 1,..., f 1,..., c 1,...) a sheaf of τ-structures A over a topological space X is given by: a-) A sheaf (E, p) over X( i.e a local homeomorphism). b-) For each x X, a τ-structure A x = (E x, R x 1, Rx 2..., f x 1,..., cx 1,...), where E x = p 1 (x) (the fiber that could be empty) is the universe of the τ-structure A x, and the following conditions are satisfied:

66 Sheaves of Structures Sheaves of Structures (Definition) Definition Given a fix type of structures τ = (R 1,..., f 1,..., c 1,...) a sheaf of τ-structures A over a topological space X is given by: a-) A sheaf (E, p) over X( i.e a local homeomorphism). b-) For each x X, a τ-structure A x = (E x, R1 x, Rx 2..., f 1 x,..., cx 1,...), where E x = p 1 (x) (the fiber that could be empty) is the universe of the τ-structure A x, and the following conditions are satisfied: i. R A = x R x is open in x E x n seeing as subspace of E n, where R is an n-ary relation symbol. ii. f A = x f x : x E x m x E x is a continuous function, where f is an m-parameter function symbol. iii. h : X E such that h(x) = c x, where c is a constant symbol, is continuous.

67 Sheaves of Structures Sheaf Logic (Motivation) A sheaf of structures is a space extended over the base space X of the sheaf as Galilean spacetime extends over time. The elements of this space are not the points of E but the sections of the sheaf conceived as extended objects. The single values of these sections represent just point-wise descriptions of the extended object. As the objects of a sheaf of structures are the sections of the sheaf, the logic which governs them should define when a property for an extended object holds in a point of its domain of definition.

68 Sheaves of Structures Sheaf Logic (Motivation) A sheaf of structures is a space extended over the base space X of the sheaf as Galilean spacetime extends over time. The elements of this space are not the points of E but the sections of the sheaf conceived as extended objects. The single values of these sections represent just point-wise descriptions of the extended object. As the objects of a sheaf of structures are the sections of the sheaf, the logic which governs them should define when a property for an extended object holds in a point of its domain of definition. Contextual Truth Paradigm If a property for an extended object holds in some point of its domain then it has to hold in a neighbourhood of that point.

69 Sheaf Logic Sheaf Logic, Point-Wise Semantics Sheaf Logic (Definition) A x ϕ[σ 1,..., σ n ] exists U neighbourhood of x such that for all y U, A y ϕ[σ 1 (y),..., σ n (y)]. A x (ϕ ψ)[σ 1,..., σ n ] Exists U neighbourhood of x such that for all y U if A y ϕ[σ 1 (y),..., σ n (y)] then A y ψ[σ 1 (y),..., σ n (y)]. A x vϕ(v, σ 1,..., σ n ) exists U neighbourhood of x such that for all y U and all σ defined in y, A y ϕ[σ(y), σ 1 (y),..., σ n (y)].

70 Sheaf Logic Sheaf Logic, Local Semantics Given an open subset U X and sections defined over U, we say that a proposition about these sections holds in U if it holds at each point in U or in other words: A U ϕ[σ 1,..., σ n] x U, A x ϕ[σ 1,..., σ n]

71 Sheaf Logic Sheaf Logic, Local Semantics Given an open subset U X and sections defined over U, we say that a proposition about these sections holds in U if it holds at each point in U or in other words: A U ϕ[σ 1,..., σ n] x U, A x ϕ[σ 1,..., σ n] Kripke-Joyal Semantics A U (ϕ ψ)[σ 1,..., σ n] there exist open sets V, W such that U = V W, A V ϕ[σ 1,..., σ n] and A W ψ[σ 1,..., σ n]. A U vϕ(v, σ 1,..., σ n) there exists {U i } i an open cover of U and µ i sections defined on U i such that A Ui ϕ[µ i, σ 1,..., σ n] for all i. A U vϕ(v, σ 1,..., σ n) for any open set W U and µ defined on W, A W ϕ(µ, σ 1,..., σ n).

72 Sheaf Logic The logic just defined can be seen as a multivalued logic with truth values that variate over the Heyting algebra of the open sets of the base space X.

73 Sheaf Logic The logic just defined can be seen as a multivalued logic with truth values that variate over the Heyting algebra of the open sets of the base space X. Let σ 1,..., σ n be sections of a sheaf A defined over an open set U, we define the truth value of a proposition ϕ in U as: [[ϕ(σ 1,..., σ n )]] U := {x U : A x ϕ[σ 1,..., σ n ]} (2) [[ϕ(σ 1,..., σ n )]] U is an open set, thus we can define a valuation as a topological valuation on formulas: T U : ϕ [[ϕ(σ 1,..., σ n )]] U.

74 Sheaf Logic The logic just defined can be seen as a multivalued logic with truth values that variate over the Heyting algebra of the open sets of the base space X. Let σ 1,..., σ n be sections of a sheaf A defined over an open set U, we define the truth value of a proposition ϕ in U as: [[ϕ(σ 1,..., σ n )]] U := {x U : A x ϕ[σ 1,..., σ n ]} (2) [[ϕ(σ 1,..., σ n )]] U is an open set, thus we can define a valuation as a topological valuation on formulas: T U : ϕ [[ϕ(σ 1,..., σ n )]] U. The definition of the logic allows to define the value of the logic operators in terms of the operations of the algebra of open sets.

75 Quantum Set Theory Variable Sets Using the comprehension axiom in classical set theory, given a proposition ϕ(x) and a set A, we can construct a set B such that x B if and only if x A and ϕ(x) is truth for x, or in other words, B = {x A : ϕ(x)}.

76 Quantum Set Theory Variable Sets Using the comprehension axiom in classical set theory, given a proposition ϕ(x) and a set A, we can construct a set B such that x B if and only if x A and ϕ(x) is truth for x, or in other words, B = {x A : ϕ(x)}. Consider the following proposition: ϕ(x) x is an even number greater or equal than 4 and x can be written as the sum of two prime numbers.

77 Quantum Set Theory Variable Sets Using the comprehension axiom in classical set theory, given a proposition ϕ(x) and a set A, we can construct a set B such that x B if and only if x A and ϕ(x) is truth for x, or in other words, B = {x A : ϕ(x)}. Consider the following proposition: ϕ(x) x is an even number greater or equal than 4 and x can be written as the sum of two prime numbers. "Now and here" the following is valid in our temporal sheaf Now and Here ({x N : ϕ(x)} = {x N : x 4 (x is even )}), because now and here we do not know if the Goldbach conjecture is valid.

78 Quantum Set Theory Variable Sets Instead of conceiving sets as absolute entities, we can conceive them as variable structures which variate over our Library of the states of knowledge.

79 Quantum Set Theory Variable Sets Instead of conceiving sets as absolute entities, we can conceive them as variable structures which variate over our Library of the states of knowledge. It is natural then to conceive the set of nodes where our states of Knowledge variates as nodes in a partial order or points in a topological space, that can represent, for instance, the causal structure of spacetime.

80 Quantum Set Theory Variable Sets Instead of conceiving sets as absolute entities, we can conceive them as variable structures which variate over our Library of the states of knowledge. It is natural then to conceive the set of nodes where our states of Knowledge variates as nodes in a partial order or points in a topological space, that can represent, for instance, the causal structure of spacetime. Our states of Knowledge will be then structures that represent the sets as we see them in our nodes. Therefore from each node we will see arise a cumulative Hierarchy of variable sets, which structure will be conditioned by the perception of the variable structures in the other nodes that relate to it. Or more precisely.

81 The valuation V over the open sets constitute an exact presheaf of structures, the respective sheaf of germs V X constitute the cumulative hierarchy of variable sets. Quantum Set Theory Definition (The Hierarchy of Variable Sets) Let X be an arbitrary topological space. Given U Op(X) we define inductively: V 0 (U) = V α+1 (U) ={f : Op(U) P(V α(w )) : 1. If W U then f (W ) V α(w ), W U 2. If V W U, then for all g f (W ), g Op(V ) f (V ), 3. Given {U i } i an open cover of U and g i f (U i ) such that g i op(ui U j )= g j op(ui U j ) for any i, j, there exists g f (U) such that g op(ui )= g i for all i} V λ (U) = α<λ V α(u) if λ is a limit ordinal, V (U) = α On V α(u).

82 Quantum Set Theory For each U Op(X) the set V (U) is a set of functions defined over Op(U) which values for W Op(U) are functions over Op(W ) which values for V Op(W ) are functions over Op(V ) and so on.

83 Quantum Set Theory For each U Op(X) the set V (U) is a set of functions defined over Op(U) which values for W Op(U) are functions over Op(W ) which values for V Op(W ) are functions over Op(V ) and so on. the relation f U g (i.e. U f g) f g(u), i.e. that respect to the context U, f belongs to g if and only if f g(u) as classical sets.

84 Quantum Set Theory For each U Op(X) the set V (U) is a set of functions defined over Op(U) which values for W Op(U) are functions over Op(W ) which values for V Op(W ) are functions over Op(V ) and so on. the relation f U g (i.e. U f g) f g(u), i.e. that respect to the context U, f belongs to g if and only if f g(u) as classical sets. Theorem For any topological space X V X ZF

85 a â(u) defines an embedding of V in V (U) for any open set U X. Quantum Set Theory For each U Op(X) the set V (U) is a set of functions defined over Op(U) which values for W Op(U) are functions over Op(W ) which values for V Op(W ) are functions over Op(V ) and so on. the relation f U g (i.e. U f g) f g(u), i.e. that respect to the context U, f belongs to g if and only if f g(u) as classical sets. Theorem For any topological space X V X ZF V V (U) To each classical set a we can associate a constant set â(u) : Op(U) V (W ) â(u)(w ) = { b(u) Op(W ) : b a}. W U

86 Quantum Set Theory Using this embedding, it can be proved that: N(U) = N(U), Z(U) = Ẑ(U), Q(U) = Q(U) for any open set U X

87 Quantum Set Theory Using this embedding, it can be proved that: N(U) = N(U), Z(U) = Ẑ(U), Q(U) = Q(U) for any open set U X These tools provide a mechanism to construct new mathematical universes over arbitrary topological spaces. If we find a topological space able to capture the essence of quantum logic this will provide a mathematical quantum universe that will probably improves our understanding of quantum mechanics.

88 Quantum Set Theory Quantum Variable Sets Foliations Let U be an abelian Von Neumann subalgebra of the algebra of operators of the Hilbert space of a quantum system. Each self-adjoint operator Ǎ U admits a spectral decomposition in U, i.e a family of projections {ˇP r } r R U such that Ǎ = rd ˇP r

89 Quantum Set Theory Quantum Variable Sets Foliations Let U be an abelian Von Neumann subalgebra of the algebra of operators of the Hilbert space of a quantum system. Each self-adjoint operator Ǎ U admits a spectral decomposition in U, i.e a family of projections {ˇP r } r R U such that Ǎ = rd ˇP r We perceive the quantum system through an abelian Von Neumann frame of observables in analogous way as we perceive classical spacetime through an inertial frame which determines a particular foliation of a spacetime region.

90 Quantum Set Theory Quantum Variable Sets The base space=the space of Histories The Gelfand spectrum S A of U is the space of positive linear functions σ : U C of norm 1 such that σ(ab) = σ(a)σ(b) for all A, B U. These are the histories because when restricted to the self-adjoint operators of U they become valuations. A valuation is a function λ from the set of self-adjoint operators B sa (U) on U to the real numbers, λ : B sa (H) R, which satisfies: 1.λ(Ǎ) belongs to the spectrum of Ǎ, for all Ǎ B sa(u) 2.λ(ˇB) = f (λ(ǎ)) whenever ˇB = f (Ǎ) with f : R R a continuous function.

91 Quantum Set Theory Quantum Variable Sets The Topology: Similar Histories Interfere Consider a self-adjoint operator Ǎ U such that Ǎ = N a n ˇPn is the spectral representation of Ǎ in U. Fix m such that 1 m N; and λ S U such that λ(ǎ) = am, then λ(ˇp m) = 1. n=1

92 Quantum Set Theory Quantum Variable Sets The Topology: Similar Histories Interfere Consider a self-adjoint operator Ǎ U such that Ǎ = N a n ˇPn n=1 is the spectral representation of Ǎ in U. Fix m such that 1 m N; and λ S U such that λ(ǎ) = am, then λ(ˇp m) = 1. Thus if Ǎ represents a physical observable, we have that in all the histories λ such that λ(ˇp m) = 1, the physical observable A assumes the value a m. Therefore, given a projection ˇP P(U) the set P = {λ S U : λ(ˇp) = 1} (3) is a context of histories which are similar in the sense that some physical observables assume the same values or the values satisfy the same inequalities in each history.

93 Quantum Set Theory Quantum Variable Sets The Topology: Similar Histories Interfere Consider a self-adjoint operator Ǎ U such that Ǎ = N a n ˇPn n=1 is the spectral representation of Ǎ in U. Fix m such that 1 m N; and λ S U such that λ(ǎ) = am, then λ(ˇp m) = 1. Thus if Ǎ represents a physical observable, we have that in all the histories λ such that λ(ˇp m) = 1, the physical observable A assumes the value a m. Therefore, given a projection ˇP P(U) the set P = {λ S U : λ(ˇp) = 1} (3) is a context of histories which are similar in the sense that some physical observables assume the same values or the values satisfy the same inequalities in each history. {P}ˇP P(U) defines a topology in S U.