Download Topological ordered spaces as a foundation for a quantum

Topological ordered spaces as a foundation
for a quantum spacetime theory
Ettore Minguzzi
Università Degli Studi Di Firenze
Castiglioncello, September 19, 2012
Talk based on
• Normally preordered spaces and utilities, Order, to appear.
• Topological conditions for the representation of preorders by continuous
utilities, Appl. Gen. Topol. 13, 81-89 (2012).
• Quasi-pseudo-metrization of topological preordered spaces, Topol. Appl. 159,
2888-2898 (2012).
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Ideas for a quantum spacetime theory
Physical theories rely on a hierarchy of specialized mathematical
structures
• (pseudo)-Riemannian structure (causal order)
• Differential structure (smoothness)
• Topological structure (closeness and continuity)
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Ideas for a quantum spacetime theory
Physical theories rely on a hierarchy of specialized mathematical
structures
• (pseudo)-Riemannian structure (causal order)
• Differential structure (smoothness)
• Topological structure (closeness and continuity)
History of science suggests: let them be dynamical
Some of these levels have become dynamical in the evolution of science: e.g. the
rigid Euclidean space metric of classical mechanics has been replaced by the the
dynamical Lorentzian metric of general relativity.
We expect that the same process should continue. Thus we must start removing
structure from our theory trying to preserve physical content.
Castiglioncello, September 19, 2012
Topological ordered spaces as a foundation. . .
2/22
Ideas for a quantum spacetime theory
Physical theories rely on a hierarchy of specialized mathematical
structures
• (pseudo)-Riemannian structure (causal order)
• Differential structure (smoothness)
• Topological structure (closeness and continuity)
History of science suggests: let them be dynamical
Some of these levels have become dynamical in the evolution of science: e.g. the
rigid Euclidean space metric of classical mechanics has been replaced by the the
dynamical Lorentzian metric of general relativity.
We expect that the same process should continue. Thus we must start removing
structure from our theory trying to preserve physical content.
Quantum regime
Differentiability does not hold at the fundamental level, we expect it to be an
emerging phenomenon. Thus we are left with
• Topology and Order
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Topology, Order and their unification
Main ingredients
• Topology: gives as a notion of continuity to work with.
• Order: Will be interpreted in a causal fashion, thus causality is still
meaningfull at a quantum regime.
• (Measure): The σ-algebra will follow from the topology.
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Topology, Order and their unification
Main ingredients
• Topology: gives as a notion of continuity to work with.
• Order: Will be interpreted in a causal fashion, thus causality is still
meaningfull at a quantum regime.
• (Measure): The σ-algebra will follow from the topology.
Difference with Causal Set Theory
We do not infer topology from order, rather we impose compatibility conditions
between topology and order. Causal set theory is less general because from the
order you can define only few topologies: Alexandrov, Scott, etc. Any causal set is
a topological ordered space, but not conversely.
Castiglioncello, September 19, 2012
Topological ordered spaces as a foundation. . .
3/22
Topology, Order and their unification
Main ingredients
• Topology: gives as a notion of continuity to work with.
• Order: Will be interpreted in a causal fashion, thus causality is still
meaningfull at a quantum regime.
• (Measure): The σ-algebra will follow from the topology.
Difference with Causal Set Theory
We do not infer topology from order, rather we impose compatibility conditions
between topology and order. Causal set theory is less general because from the
order you can define only few topologies: Alexandrov, Scott, etc. Any causal set is
a topological ordered space, but not conversely.
Role of quasi-uniformities
Topology and Order are two faces of this mathematical entity.
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Classical spacetime physics motivation:
Relationship between time and closed relations
Stable Causality
(M, g) is stably causal if there is g 0 > g with (M, g 0 ) causal.
Here g 0 > g if the light cones of g are everywhere strictly larger than those of g.
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Classical spacetime physics motivation:
Relationship between time and closed relations
Stable Causality
(M, g) is stably causal if there is g 0 > g with (M, g 0 ) causal.
Here g 0 > g if the light cones of g are everywhere strictly larger than those of g.
The causal relation
J + = {(x, y) : there is a causal curve connecting x and y}
is not both closed and transitive
Seifert’s relation JS+ =
T
g 0 >g
Jg+ (1971)
JS+ is closed, transitive and contains J + .
Castiglioncello, September 19, 2012
Topological ordered spaces as a foundation. . .
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Classical spacetime physics motivation:
Relationship between time and closed relations
Stable Causality
(M, g) is stably causal if there is g 0 > g with (M, g 0 ) causal.
Here g 0 > g if the light cones of g are everywhere strictly larger than those of g.
The causal relation
J + = {(x, y) : there is a causal curve connecting x and y}
is not both closed and transitive
Seifert’s relation JS+ =
T
g 0 >g
Jg+ (1971)
JS+ is closed, transitive and contains J + .
Theorem
The spacetime is stably causal if and only if JS+ is antisymmetric.
Castiglioncello, September 19, 2012
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Classical spacetime physics motivation:
Relationship between time and closed relations
Stable Causality
(M, g) is stably causal if there is g 0 > g with (M, g 0 ) causal.
Here g 0 > g if the light cones of g are everywhere strictly larger than those of g.
The causal relation
J + = {(x, y) : there is a causal curve connecting x and y}
is not both closed and transitive
Seifert’s relation JS+ =
T
g 0 >g
Jg+ (1971)
JS+ is closed, transitive and contains J + .
Theorem
The spacetime is stably causal if and only if JS+ is antisymmetric.
Theorem
In a stably causal spacetime the Seifert relation coincides with the smallest closed
and transitive relation containing J + (also denoted K + by Sorkin and Woolgar).
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Powerfulness of topological ordered space theory
Time function
A continuous real function such that p < q ⇒ t(p) < t(q).
Hawking proved: Stable causality ⇒ time function.
Lorentzian geometry is actually not needed in the previous problem on the
existence of time functions. We shall see that
Levin’s theorem
Every second countable locally compact space endowed with a closed order (e.g. a
stably causal spacetime endowed with the relation JS+ ) admits a strictly increasing
continuous function.
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Powerfulness of topological ordered space theory
Time function
A continuous real function such that p < q ⇒ t(p) < t(q).
Hawking proved: Stable causality ⇒ time function.
Lorentzian geometry is actually not needed in the previous problem on the
existence of time functions. We shall see that
Levin’s theorem
Every second countable locally compact space endowed with a closed order (e.g. a
stably causal spacetime endowed with the relation JS+ ) admits a strictly increasing
continuous function.
Thus we can infer an important theorem on the existence of time functions
without using the differentiable structure. It seems that we are going in the right
direction. . .
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Preorders
Relation
A relation on a set E is a subset R ⊂ E × E.
A special relation is the diagonal
∆ = {(x, x) : x ∈ E}.
It is an identity for the composition ◦ of relations.
We define:
R−1 = {(x, y) ∈ E : (y, x) ∈ R}.
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Preorders
Relation
A relation on a set E is a subset R ⊂ E × E.
A special relation is the diagonal
∆ = {(x, x) : x ∈ E}.
It is an identity for the composition ◦ of relations.
We define:
R−1 = {(x, y) ∈ E : (y, x) ∈ R}.
Preorder and order
• A preorder on a set E is a reflexive and transitive relation on E.
• A preorder which is antisymmetric “x ≤ y and y ≤ x ⇒ x = y” is an (partial)
order (R ∩ R−1 = ∆).
We shall write x ≤ y for (x, y) ∈ R. Then R is called the graph of ≤.
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Preorders II
Increasing/decreasing hulls
Let S ⊂ E, the increasing and decreasing hulls are
i(S) = {y ∈ E : x ≤ y for some x ∈ S},
d(S) = {y ∈ E : y ≤ x for some x ∈ S}.
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Preorders II
Increasing/decreasing hulls
Let S ⊂ E, the increasing and decreasing hulls are
i(S) = {y ∈ E : x ≤ y for some x ∈ S},
d(S) = {y ∈ E : y ≤ x for some x ∈ S}.
Increasing/decreasing, monotone sets
Subsets S for which i(S) = S are called increasing, while subsets for which
d(S) = S are called decreasing. Increasing and decreasing sets are monotone.
The complement of an increasing set is decreasing and the other way around.
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Preorders III
Convex set
A set S is convex if S = i(S) ∩ d(S). As a consequence a set S is convex if and
only if it is the intersection of an increasing set B and a decreasing set A.
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Preorders III
Convex set
A set S is convex if S = i(S) ∩ d(S). As a consequence a set S is convex if and
only if it is the intersection of an increasing set B and a decreasing set A.
Discrete order
• If R = ∆ the order is discrete. This means i(x) = d(x) = {x}.
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Preorders III
Convex set
A set S is convex if S = i(S) ∩ d(S). As a consequence a set S is convex if and
only if it is the intersection of an increasing set B and a decreasing set A.
Discrete order
• If R = ∆ the order is discrete. This means i(x) = d(x) = {x}.
Isotone functions
A function between two preordered spaces f : E → E 0 is isotone if
x ≤ y ⇒ f (x) ≤ f (y) and anti-isotone if x ≤ y ⇒ f (y) ≤ f (x).
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Topological preordered space
Topological preordered space
A topological preordered space is triple (E, T , ≤) given by a topological space
(E, T ) endowed with a preorder ≤.
If R = ∆ we want to recover the usual topology.
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Topological preordered space
Topological preordered space
A topological preordered space is triple (E, T , ≤) given by a topological space
(E, T ) endowed with a preorder ≤.
If R = ∆ we want to recover the usual topology.
Upper and lower topologies
• U = T ] is the topology generated by the open increasing sets,
• L = T [ is the topology generated by the open decreasing sets.
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Topological preordered space
Topological preordered space
A topological preordered space is triple (E, T , ≤) given by a topological space
(E, T ) endowed with a preorder ≤.
If R = ∆ we want to recover the usual topology.
Upper and lower topologies
• U = T ] is the topology generated by the open increasing sets,
• L = T [ is the topology generated by the open decreasing sets.
Bitolopogy
The theory of topological preordered spaces is very much connected with the
theory of bitopological spaces.
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Compatibility conditions between topology and preorder I
Local convexity
The topology T admits a base of convex neighborhoods.
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Compatibility conditions between topology and preorder I
Local convexity
The topology T admits a base of convex neighborhoods.
Not all open and convex sets are the intersection of and increasing open set and a
decreasing open set.
Convexity
The topology T is generated by U and L :
T = sup(U , L ).
Convexity is much more important than local convexity!
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Compatibility conditions between topology and preorder II
1 Quasi-pseudo-metrizable preordered space
2 Normally preordered space
3 Completely regularly preordered space (Tychonoff-preordered space)
(Quasi-uniformizable space)
4 Closed preordered space
There are some gaps in the expected relative strengths, for instance 1 does not
imply 2, and convexity is needed to infer that 2 implies 3.
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Closed preordered spaces
Closed preordered space
E is a closed preordered space if the graph of the preorder ≤ is closed in the
product topology of E × E.
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Closed preordered spaces
Closed preordered space
E is a closed preordered space if the graph of the preorder ≤ is closed in the
product topology of E × E.
The Hausdorff condition is equivalent to the statement: “∆ is closed in the
product topology”.
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Closed preordered spaces
Closed preordered space
E is a closed preordered space if the graph of the preorder ≤ is closed in the
product topology of E × E.
The Hausdorff condition is equivalent to the statement: “∆ is closed in the
product topology”.
Proposition
• Every closed preordered space in which R = ∆ is Hausdorff (T2 ).
• Every closed ordered space is Hausdorff. (because R ∩ R−1 = ∆)
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Closed preordered spaces
Closed preordered space
E is a closed preordered space if the graph of the preorder ≤ is closed in the
product topology of E × E.
The Hausdorff condition is equivalent to the statement: “∆ is closed in the
product topology”.
Proposition
• Every closed preordered space in which R = ∆ is Hausdorff (T2 ).
• Every closed ordered space is Hausdorff. (because R ∩ R−1 = ∆)
General philosophy
Do not assume Hausdorffness and any other usual separability conditions! We
avoid to double the preorder-separability conditions.
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Normally preordered spaces
Normally preordered space
E is a normally preordered space if it is a closed preordered space and for every
pair of closed increasing set B, and closed decreasing set A such that A ∩ B = ∅,
there are an open increasing set V and an open decreasing set U such that,
A ⊂ U , B ⊂ V , U ∩ V = ∅.
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Normally preordered spaces
Normally preordered space
E is a normally preordered space if it is a closed preordered space and for every
pair of closed increasing set B, and closed decreasing set A such that A ∩ B = ∅,
there are an open increasing set V and an open decreasing set U such that,
A ⊂ U , B ⊂ V , U ∩ V = ∅.
Theorem (Nachbin’s extension of Urysohn’s lemma)
The topological preordered space (E, T , ≤) is normally preordered if and only if
for any two disjoint closed subsets A, B ⊂ E, A is decreasing and B is increasing,
there exist on E a continuous isotone real-valued function f such that f (x) = 0
(x ∈ A), f (x) = 1 (x ∈ B), and 0 ≤ f (x) ≤ 1 (x ∈ E).
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Completely regularly preordered spaces
Completely regularly preordered space (Tychonoff-preordered space)
E is a completely regularly preordered space if
(i) The topology is the weak topology of the family of continuous isotone
functions.
(ii) x ≤ y ⇔ ∀f continuous and isotone, f (x) ≤ f (y)
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Completely regularly preordered spaces
Completely regularly preordered space (Tychonoff-preordered space)
E is a completely regularly preordered space if
(i) The topology is the weak topology of the family of continuous isotone
functions.
(ii) x ≤ y ⇔ ∀f continuous and isotone, f (x) ≤ f (y)
Theorem
Every completely regularly preordered space is convex
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Completely regularly preordered spaces
Completely regularly preordered space (Tychonoff-preordered space)
E is a completely regularly preordered space if
(i) The topology is the weak topology of the family of continuous isotone
functions.
(ii) x ≤ y ⇔ ∀f continuous and isotone, f (x) ≤ f (y)
Theorem
Every completely regularly preordered space is convex
Theorem
Every completely regularly preordered space can be compactified.
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How do we improve separability properties?
In usual (discrete order) topology the following results are essential
Theorem
Every locally compact Hausdorff space is completely regular (Tychonoff).
This is usually proved using the Alexandrov one-point compactification.
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How do we improve separability properties?
In usual (discrete order) topology the following results are essential
Theorem
Every locally compact Hausdorff space is completely regular (Tychonoff).
This is usually proved using the Alexandrov one-point compactification.
Using the fact that completely regular implies regular, we make the next jump
with
Urysohn’s metrization theorem
Every second countable regular space is metrizable.
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How do we improve separability properties?
In usual (discrete order) topology the following results are essential
Theorem
Every locally compact Hausdorff space is completely regular (Tychonoff).
This is usually proved using the Alexandrov one-point compactification.
Using the fact that completely regular implies regular, we make the next jump
with
Urysohn’s metrization theorem
Every second countable regular space is metrizable.
None of these results holds in the preordered case. I used a different path proving
Theorem
Every locally compact σ-compact space equipped with a closed preorder is
normally preordered.
It is an essential generalization for the manifold case.
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Quasi-pseudo-metrizability
Quasi-pseudo-metric
A quasi-pseudo-metric on a set X is a function p : X × X → [0, +∞) such that for
x, y, z ∈ X
(i) p(x, x) = 0,
(ii) p(x, z) ≤ p(x, y) + p(y, z).
The function q(x, y) := p(y, x) is the conjugate quasi-pseudo-metric.
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Quasi-pseudo-metrizability
Quasi-pseudo-metric
A quasi-pseudo-metric on a set X is a function p : X × X → [0, +∞) such that for
x, y, z ∈ X
(i) p(x, x) = 0,
(ii) p(x, z) ≤ p(x, y) + p(y, z).
The function q(x, y) := p(y, x) is the conjugate quasi-pseudo-metric.
Quasi-pseudo-metrizable space
A topological preordered space (E, T , ≤) is quasi-pseudo-metrizable if there is a
pair of conjugate quasi-pseudo-metrics p, q, said admissible, such that T is the
topology generated by the pseudo-metric p + q, and the graph of the preorder is
given by
G(≤) = {(x, y) : p(x, y) = 0}.
Every quasi-pseudo-metrizable preordered space is a completely regularly
preordered space.
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Quasi-pseudo-metrizability and ordered Hilbert cube
Theorem
The following conditions are equivalent for a topological preordered space
(E, T , ≤)
(a) (E, T , ≤) is a second countable completely regularly preordered space,
(b) (E, T , ≤) is separable and quasi-pseudo-metrizable.
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Quasi-pseudo-metrizability and ordered Hilbert cube
Theorem
The following conditions are equivalent for a topological preordered space
(E, T , ≤)
(a) (E, T , ≤) is a second countable completely regularly preordered space,
(b) (E, T , ≤) is separable and quasi-pseudo-metrizable.
Ordered Hilbert cube
The Hilbert cube H = [0, 1]N once endowed with the product topology and the
product order is a strict quasi-pseudo-metric
ordered I-space with
P
n
quasi-pseudo-metric p(x, y) = ∞
n=1 max(xn − yn , 0)/2 .
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Quasi-pseudo-metrizability and ordered Hilbert cube
Theorem
The following conditions are equivalent for a topological preordered space
(E, T , ≤)
(a) (E, T , ≤) is a second countable completely regularly preordered space,
(b) (E, T , ≤) is separable and quasi-pseudo-metrizable.
Ordered Hilbert cube
The Hilbert cube H = [0, 1]N once endowed with the product topology and the
product order is a strict quasi-pseudo-metric
ordered I-space with
P
n
quasi-pseudo-metric p(x, y) = ∞
n=1 max(xn − yn , 0)/2 .
Order embedding into the ordered Hilbert cube
The following conditions are equivalent for a topological ordered space (E, T , ≤)
(a) (E, T , ≤) is a second countable completely regularly ordered space,
(b) (E, T , ≤) is order embeddable in the ordered Hilbert cube H.
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Convexity?
From local convexity to convexity
A locally convex closed preordered space which is topologically a locally compact
σ-compact space is necessarily convex.
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Convexity?
From local convexity to convexity
A locally convex closed preordered space which is topologically a locally compact
σ-compact space is necessarily convex.
Justifying local convexity
Every compactly generated closed ordered space is locally convex.
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Conclusions
• We have given an introduction to the theory of topological preordered spaces.
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Conclusions
• We have given an introduction to the theory of topological preordered spaces.
• This theory has many applications and is mathematically quite natural
(quasi-uniformities).
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Conclusions
• We have given an introduction to the theory of topological preordered spaces.
• This theory has many applications and is mathematically quite natural
(quasi-uniformities).
• The main message seems to be:
Order and topology are two aspects of the same object
and should be studied jointly.
We expect that it will be central for a formulation of a quantum spacetime
theory.
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