PIAF Workshop, Sydney, February 2008 Quantum measurement and quantum - - PDF document

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.

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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.

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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 RD(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¨

• ring (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 = ES(M) state space. Value of ˆ A in condition W ∈ O(X) is a(W) ∈ RD(W). Prepared in W, final conditions Vj ∈ O(W), values a(Vj).

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• 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 Set.

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

• f choice can be used.

A spatial topos, Shv(X), is category of sheaves

• n 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, RD(X), the

Dedekind reals, which is the sheaf C(X) of con- tinuous real-valued functions on X. RD(X) is a complete metric space, a residue field, has an interval topology in which QD(X) is dense.

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(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 {fi ∈ F(Ui)}I collate to a unique global element f ∈ F(∪IUi).

(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(Uα), where {Uα}α∈J form an open cover of X, then it holds globally.

Restriction map RV 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, ∃Ux ⊆ U, an

• pen neighbourhood of x such that P has a

solution in Ux and in every open Vx ⊆ Ux.

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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|Wα where {Wα}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 {Wα}α∈J of W such that for each α ∈ J there is a local section f|Wα for which the property holds.

There is a parallelism between the vocabularies

• f 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.

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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

• f classical physical explanation, the account
• f all evidence must be expressed in classical

terms.

The base space X models the contexts.

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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

• f these qualities with their observed numerical
• values. ∗

∗c.f. Bohr’s severing of the “direct connection between

• bservation properties and properties possessed by the

independently existing object”.

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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 ES(M) is the space of normalized linear functionals on M. For each self-adjoint operator ˆ A ∈ M define the function a : ES(M) → R given by a(ρ) = Tr ˆ Aρ, ∀ρ ∈ ES(M).

X = ES(M) has the weakest topology that makes all the functions like a continuous. The system has its own real numbers, RD(ES(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.

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The ontological and epistemological condition

• f a quantum system.

If a system is experimentally prepared in an

• pen 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

• pen set V ∈ ES(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, an(V ) = a(V )n for all its qualities ˆ An.

∗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).

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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.

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The quantum measurement problem The standard quantum theory predicts only a probability distribution of values that would be

• btained by measuring a physical quality on an

ensemble of identically prepared systems. The probabilistic predictions can only be ver- ified by measurements if each outcome is ob- servationally distinct. Therefore, after the interaction the ensemble (of system and apparatus) must yield a series

• f distinguishable apparatus states with each

distinct state corresponding to a distinct out-

• come. This does not happen in the standard

QM formulation. The question of how to resolve this inconsis- tency is the quantum measurement problem. Until it is resolved it is difficult to give a logi- cally consistent meaning to the probabilities of the outcomes.

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Quantum Real Numbers Model 1. Initial preparation determines initial open sets W0(S) ⊂ ES(S) and W0(A) ⊂ ES(A) the initial epistemic conditions of S and A. The pointer of A has a qrumber value xA(W0(A)) which is classical.

S is prepared so that if the values of particular qualities were measured they would yield standard real number values with an accuracy ǫ. This is done by the ǫ sharp collimation. The single slit experiment is the prototype of ǫ sharp collimation process in a slit I. Given ǫ > 0, ∃U, such that the qrumber zQ(U) is well approximated by a standard real number when I =]z1, z2[ is sufficiently narrow.

• 2. S interacts with A changing A’s condition.

The interaction between S and A is described by qrumber equations of motion. 3. In each run of the experiment a distinct approximately classical value of A’s pointer is registered with accuracy ǫ. From this output we can deduce a classical value for the requi- site quality belonging to S, this is its measured value.

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A simple example.

S and A are massive particles, called A and B, with Hilbert spaces HA and HB and HA,B = HA ⊗ HB for the combined system. For j = A, B, there are physical qualities represented by ∗-algebras Mj and state spaces ESj. ESA,B is the combined system state space. Wj ∈ O(ESj), have superset WA,B, the smallest open set in ESA,B such the partial traces TrHAˆ ρ ∈ WB and TrHBˆ ρ ∈ WA for all ˆ ρ ∈ WA,B. If ˆ A acting on HA, then aQ(WA) = aQ(WA,B) because ˆ A ⊗ IB represents the same physical quality as ˆ

• A. Similarly for particle B.

The initial epistemic condition of particle j is Wj(0) = Λ(ρj, δj) ⊂ ESj, an open neighbourhood of the pure states ˆ ρA = ˆ PΨ or ˆ ρB = ˆ PΦ. Λ(ρj, δj) = {ρ ∈ ESj ; Tr|ρ−ρj| < δj}. The value xB

Q(WB(0)) of the pointer ˆ

XB is approximately classical when Φ is an approximate eigenvector of ˆ XB. The initial epistemic condition of the combined system is the superset WA,B(0) of WA(0) and WB(0). For η > 0 sufficiently small, Λ(ρA ⊗ ρB, η) ⊂ WA,B(0). Any open set Uj(0) ⊂ Wj(0) defines an ontic condition which is consistent with the initial epistemic condition

• n particle j. Any open set UA,B(0) ⊂ WA,B(0) will sat-

isfy all the initial conditions.These variations account for the possibility of variation in the outcomes when the experimental controls are satisfied.

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(ii) The interaction is given by a von Neumann Hamilto- nian H(A, B)Q(WA,B) = [gxApB]Q(WA,B). The coupling gQ(WA,B) is a constant g1Q(WA,B) of dimension T −1. The qrumber value of the pointer changes by an amount ξB

Q(WA,B)(∆t) = xB Q(WA,B)(∆t) − xB Q(WA,B)(0) propor-

tional to xA

Q(WA,B)(0).

ξB

Q(WA,B)(∆t) = (g∆t)xA Q(WA,B)(0)

(1) Interpret this to say that the condition WA,B(0) changed to the condition WA,B(∆t), so that qualities IA⊗XB and (g∆t)(XA ⊗ IB) have equal values on WA,B(∆t). If (g∆t)αj = βj , j = 1, 2 then for η > 0 sufficiently small, Λ(ρA,B(∆t), η) ⊂ WA,B(∆t) where ρA,B(∆t) = ˆ PΘ(∆t) and Θ(∆t) = a1ψ1 ⊗ φ1 + a2ψ2 ⊗ φ2. (iii). When the qrumber xB

Q(WA,B)(∆t) is approximately

classical, a standard real number outcome can be recorded. There are a pair of slits, Is =]αs − δs, αs + δs[, s = 1, 2, where αs is the eigenvalue of ˆ XA with eigenvector ψs and δs > 0. On passing through Is the qrumber value xA

Q(UA,B) is

reduced to xA

Q(UA,B ∩ Us A,B). If Us A,B ∩ UA,B = ∅ and δ is

small enough, then xA

Q(Us A,B∩UA,B) is approximately αs so

that ((xA)2)Q(Us

A) ≈ (xA Q(Us A))2, where Us A is the partial

trace of Us

A,B∩UA,B over HB. Then xB Q(Us A,B∩UA,B)(∆t) =

g∆t.xA

Q(Us A,B ∩ UA,B)(0) is approximately (g∆t)αj = βj.

Taking the partial trace over HA the value of xB

Q(Us B) is

approximately βj, which can be recorded.

The “collapse of the wavefunction” rule gives an approximation to the qrumber result.

If just before entering the slit s the system (A, B) is in VA,B = Λ(ρ0; δ) ⊂ WA,B(∆t) and ˆ XA is measured on Us

A

so that its value xA(Us

A ∩ VA) is ǫ-realized in an interval

• Is. Let ˆ

Ps be the projection operator of Is and assume that ˆ Psρ0 = ˆ 0. Then ˆ XA has the value xA

Q(Us ∩ VA,B)

given to within ǫ by (Trρ′

0 ˆ

XA ⊗ IB) · 1(Us ∩ VA,B), where ρ′

0 = ˆ Psρ0 ˆ Ps Tr(ρ0Ps) which is a state. ρ′ 0 is the reduction of ρ0.

Realism: If an ensemble of systems is pre- pared identically in an ontic condition, a par- ticular outcome of a measured variable will be

• btained with certainty.

Hence the wave- function is not ontic (not objective) and the

• ntic condition of a system gives a more com-

plete description of an individual system than a QM wavefunction does. But the wave function plays an important role at the infinitesimal level. For example, the Schr¨

• dinger time evolution applies infinitesi-

mally.

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