Logic, Geometry And Probability Theory Federico Holik 1 1-Center Leo Apostel for Interdisciplinary Studies and, Department of Mathematics, Brussels Free University Krijgskundestraat 33, 1160 Brussels, Belgium Whither Quantum Structures In The 21th Century? Brussels - 2013 Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com
Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Outline Abstract General Framework Boolean algebra Quantum probabilities Kolmogorov R. T. Cox Quantum Probabilities The von Neumann program General Scheme General Method Compound quantum systems Conclusions Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com
Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Outline Abstract General Framework Boolean algebra Quantum probabilities Kolmogorov R. T. Cox Quantum Probabilities The von Neumann program General Scheme General Method Compound quantum systems Conclusions Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com
Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Abstract ◮ We study the origin of quantum probabilities as arising from non-boolean propositional-operational structures and deduce non-kolmogorvian probability measures of quantum mechanics. ◮ We discuss some problems posed by von Neumann regarding the development of a quantum probability theory. ◮ We present an alternative perspective to the problem of compound quantum systems. Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com
Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Abstract ◮ We study the origin of quantum probabilities as arising from non-boolean propositional-operational structures and deduce non-kolmogorvian probability measures of quantum mechanics. ◮ We discuss some problems posed by von Neumann regarding the development of a quantum probability theory. ◮ We present an alternative perspective to the problem of compound quantum systems. Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com
Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Abstract ◮ We study the origin of quantum probabilities as arising from non-boolean propositional-operational structures and deduce non-kolmogorvian probability measures of quantum mechanics. ◮ We discuss some problems posed by von Neumann regarding the development of a quantum probability theory. ◮ We present an alternative perspective to the problem of compound quantum systems. Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com
Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Outline Abstract General Framework Boolean algebra Quantum probabilities Kolmogorov R. T. Cox Quantum Probabilities The von Neumann program General Scheme General Method Compound quantum systems Conclusions Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com
Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Regularities Given a physical theory: ◮ We concentrate on events or states of affairs . Given a certain state of affairs, we ask for how likely is that another state of affairs occurs. How can we measure this? ◮ How are events of a given theory structured? Are they structured in some way? Is there a link between the event structure and probability theory? ◮ The existence of well defined events is a necessary condition in order to have a scientific theory. This is independent of the position that we adopt, realism, empiricism, etc. Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com
Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Regularities Given a physical theory: ◮ We concentrate on events or states of affairs . Given a certain state of affairs, we ask for how likely is that another state of affairs occurs. How can we measure this? ◮ How are events of a given theory structured? Are they structured in some way? Is there a link between the event structure and probability theory? ◮ The existence of well defined events is a necessary condition in order to have a scientific theory. This is independent of the position that we adopt, realism, empiricism, etc. Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com
Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Regularities Given a physical theory: ◮ We concentrate on events or states of affairs . Given a certain state of affairs, we ask for how likely is that another state of affairs occurs. How can we measure this? ◮ How are events of a given theory structured? Are they structured in some way? Is there a link between the event structure and probability theory? ◮ The existence of well defined events is a necessary condition in order to have a scientific theory. This is independent of the position that we adopt, realism, empiricism, etc. Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com
Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Different kinds of regularities Several possibilities: ◮ We can predict with absolute certainty (example: classical mechanics, determinism). Absolute regularity . ◮ We can only compute probabilities: given two equivalent preparations, we may have different effects, but with probabilities cogently defined (example: quantum mechanics). Statistical regularity . ◮ There is no regularity at all. Several reasons: ontological, impossibility of defining equivalent preparations, ect. Even in this case, we can (sometimes) define probabilities (but we will not discuss this here). Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com
Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Different kinds of regularities Several possibilities: ◮ We can predict with absolute certainty (example: classical mechanics, determinism). Absolute regularity . ◮ We can only compute probabilities: given two equivalent preparations, we may have different effects, but with probabilities cogently defined (example: quantum mechanics). Statistical regularity . ◮ There is no regularity at all. Several reasons: ontological, impossibility of defining equivalent preparations, ect. Even in this case, we can (sometimes) define probabilities (but we will not discuss this here). Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com
Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Different kinds of regularities Several possibilities: ◮ We can predict with absolute certainty (example: classical mechanics, determinism). Absolute regularity . ◮ We can only compute probabilities: given two equivalent preparations, we may have different effects, but with probabilities cogently defined (example: quantum mechanics). Statistical regularity . ◮ There is no regularity at all. Several reasons: ontological, impossibility of defining equivalent preparations, ect. Even in this case, we can (sometimes) define probabilities (but we will not discuss this here). Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com
Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Events in the QL approach von Neumann: “And one also has the parallelism that logics corresponds to set theory and probability theory corresponds to measure theory and that a given system of logics, so given a system of sets, if all is right, you can introduce measures, you can introduce probability and you can always do it in very many different ways.”(vN-Amsterdam Talk) Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com
Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions In this way, the connection between Logic, Set Theory, and Probability Theory is clear. What does this means? The definition of Cantor of a set reads: “A set is a gathering together into a whole of definite, distinct objects of our perception [Anschauung] or of our thought —which are called elements of the set.” (Cantor) Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com
Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Events in the QL approach Propositional systems: ◮ Operational questions in CM correspond to subsets of phase space . ◮ Operational questions in QM correspond to subspaces (projections) in a Hilbert space . Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com
Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Events in the QL approach Propositional systems: ◮ Operational questions in CM correspond to subsets of phase space . ◮ Operational questions in QM correspond to subspaces (projections) in a Hilbert space . Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com
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