PIAF Workshop, Sydney, February 2008 Quantum measurement and quantum reality from a topos theoretical perspective. John V Corbett Department of Mathematics, Macquarie Uni- versity, N.S.W. 2109, Australia, Centre for Time, Philosophy Department, Syd- ney University, N.S.W. 2006, Australia. 1
Quantum measurement and quantum reality from a topos theoretical perspective. ABSTRACT: Most approaches to understand- ing quantum mechanics use mathematical con- cepts founded on set theory. Topos theory gives an alternative foundation for mathemat- ics. I argue that re-interpreting quantum for- malism in terms of topos theory will help to solve some of the central conceptual problems in quantum theory. In this talk I firstly review the quantum measurement problem to obtain a concept of reality and then reformulate the problem using quantum real numbers which are the topos theoretical numerical values of the physical qualities of the quantum systems. If I have time a particular example of a position measurement will be given. The first part of the talk is based on work with Dipankar Home, the latter parts were de- veloped in work with Thomas Durt and with Frank Valckenborgh. 2
Outline of quantum theory in a topos. 1. Topoi - variable sets with intuitionistic logic, prototype Shv ( X ), with X a topological space. (a) Internal logic is intuitionistic, truth values Ω = O ( X ). (b) Ring of real numbers; Dedekind reals R D ( X ). (c) Collation and restriction of data. 2. Application to quantum systems. Adelman and Corbett (1995-2001), Butterfield and Isham(1998- 2000), Corbett and Durt (2002 - 2007), Heunen and Spitters (2007), Isham and D¨ oring (2007), Landsman (2007), Trifonov (2008). 3. The quantum measurement problem. QT predicts only probability distribution, but probabili- ties can’t be determined as outcomes indistinguishable. 4. Quantum real numbers interpretation. O ∗ - algebra of observables M , X = E S ( M ) state space. Value of ˆ A in condition W ∈ O ( X ) is a ( W ) ∈ R D ( W ). Prepared in W , final conditions V j ∈ O ( W ), values a ( V j ). 3
1. In 1970, Lawvere showed that toposes can be viewed as a “variable” set theory whose in- ternal logic is intuitionistic. (a) A topos is a category which generalizes the category S et. It has a subobject classifier Ω of truth values. The in- ternal logic is intuitionistic; only constructive arguments are acceptable; neither excluded middle nor the axiom of choice can be used. A spatial topos, Shv ( X ), is category of sheaves on a topological space X with Ω = O ( X ) so that ⊤ = X and ⊥ = ∅ . Heyting algebra. Propositional Calculus. If V, W ∈ O ( X ), negation: ¬ V = int( X \ V ), or: V ∨ W = V ∪ W , and: V ∧ W = V ∩ W , implication: V ⇒ W is the largest open set U such that U ∩ V ⊂ W . Intuitionistic as V ∪ ¬ V = X \ ∂V � = X . (b) It has a ring of real numbers, R D ( X ), the Dedekind reals, which is the sheaf C ( X ) of con- tinuous real-valued functions on X . R D ( X ) is a complete metric space , a residue field , has an interval topology in which Q D ( X ) is dense. 4
(c) Infinitesimal, local and global data. Locally true everywhere does not necessarily imply glob- ally true. A sheaf encodes the passage from local to global by collating. A sheaf F on X is a variable set F ( U ) indexed by U ∈ O ( X ) in which compatible local ele- ments { f i ∈ F ( U i ) } I collate to a unique global element f ∈ F ( ∪ I U i ). (1) If a property holds globally for a sheaf A then it holds for all subsheaves A ( U ), U ∈ O ( X ), and (2) if it holds for each subsheaf A ( U α ), where { U α } α ∈ J form an open cover of X , then it holds globally. Restriction map R V maps A ( U ) → A ( U ∩ V ). A local problem P on X is a problem that makes sense in every V ∈ O ( X ). P has a so- lution in U ∈ O ( X ) when ∀ x ∈ U, ∃ U x ⊆ U , an open neighbourhood of x such that P has a solution in U x and in every open V x ⊆ U x . 5
A section of the sheaf C ( X ) over U ∈ O ( X ) is a continuous function s from U to R such that ∀ x ∈ U the projection of s ( x ) onto X is x . A section f defined on the open set U can be restricted to sections f | V on open sets V ⊂ U and, conversely, the section f on U can be recovered by patching together the sections f | W α where { W α } α ∈ J is an open cover of U . To say that there exists a local section f | W for which a property is true means that there is an open cover { W α } α ∈ J of W such that for each α ∈ J there is a local section f | W α for which the property holds. There is a parallelism between the vocabularies of toposes and sets. Sheaves in a topos cor- respond to sets, sub-sheaves to subsets and local sections to elements. As long as a proof in set theory does not use the law of excluded middle or the Axiom of Choice then it can be translated into a proof in topos theory. 6
2. The different topos theoretical models of quantum systems start from different (gener- alized) topological base spaces X . A common aim is to formalize a notion of contextuality, motivated by Bohr’s requirements As regards the specification of the conditions for any well-defined application of the formal- ism, it is moreover essential that the whole ex- perimental arrangement be taken into account. and however far the phenomena transcend the scope of classical physical explanation, the account of all evidence must be expressed in classical terms. The base space X models the contexts. 7
Quantum Real Numbers Model Mathematical structure is built upon the stan- dard quantum mechanical formalism. Every physical quality of a microscopic entity possesses definite quantum real number (qrum- ber) values even in the absence of a specific experimental arrangements. The qrumber val- ues exist to extents given by open subsets of the state space. The theory is realist in the sense that it postu- lates the existence of entities possessing prop- erties corresponding to qualities such as the position, momentum, spin and mass but does not identify the ontological quantitative values of these qualities with their observed numerical values. ∗ ∗ c.f. Bohr’s severing of the “direct connection between observation properties and properties possessed by the independently existing object”. 8
Assume that any quantum system has a Hilbert space H , the carrier space of a unitary irre- ducible projective representation U of a Lie group G , the symmetry group of the system. The physical properties (qualities) of the system are rep- resented by self-adjoint operators in the O ∗ -algebra M , the representation dU of the enveloping algebra of G . The state space E S ( M ) is the space of normalized linear functionals on M . For each self-adjoint operator ˆ A ∈ M define the function a : E S ( M ) → R given by a ( ρ ) = Tr ˆ Aρ, ∀ ρ ∈ E S ( M ). X = E S ( M ) has the weakest topology that makes all the functions like a continuous. The system has its own real numbers, R D ( E S ( M )), called its quantum real numbers. When restricted to an open set U , a ( U ) = a | U , a local section of C ( X ), is the quantum real number value of the quality ˆ A when the system exists to extent U . The expectation values of the standard quan- tum mechanical formalism are order theoreti- cal infinitesimal quantum real numbers . Each infinitesimal quantum real number is an intuitionistic nil- square infinitesimal in the sense that it is not the case that it is not a nilsquare infinitesimal. 9
The ontological and epistemological condition of a quantum system. If a system is experimentally prepared in an open set W of state space, e.g.by passage through a slit I with W the largest open set such that a ( W ) ∈ I , then the epistemological condition of the sys- tem is the sieve S ( W ) generated by W . ∗ S ( W ) = O ( W ) \ ∅ because for any non-empty open set V ⊂ W then a ( V ) ∈ I , that is, the system in the condi- tion V will pass through the slit. The ontological condition of the system is an open set V ∈ E S ( M ) which is postulated to provide a complete description of the state of an individual system. Each system has an on- tological condition always. A system behaves classically when its ontological con- dition V is such that ∀ n ∈ N , a n ( V ) = a ( V ) n for all its qualities ˆ A n . ∗ A sieve S ( W ) on an open set W is a family of open subsets of W with the property if U ∈ S ( W ) and V ⊂ U then V ∈ S ( W ). 10
The quantum measurement process Each experiment has a measure of precision, ǫ > 0, that defines the level of accuracy re- quired to verify the predictions. It is usual to divide a quantum mechanical measurement process into three sub-processes; (i) In the first, the system S and the measure- ment apparatus A are considered separately. S is prepared in such a way that it can inter- act with A . At the same time, A has been prepared so that it can realize standard real number values of certain physical qualities of S to the required accuracy ǫ . (ii) In the second, S and A interact. During this interaction the state of A is changed. (iii) In the third, the change in A gives rise to a classical output which is registered. If from this output we can deduce a value for some quantity belonging to S , this is the measured value. 11
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