Valuations in the language of Topos theory

“Useful as it is under everyday circumstances to say that the world exists “out there” independent of us, that view can no longer be upheld. There is a strange sense in which this is a “participating universe” Wheeler (1983).

The above statement reveals the radical difference that exists between the view of the world given by Quantum Mechanics and the view given by Classical Physics. In fact, the existence of an “objective external world”, which is postulated by Classical Physics, seems to be rejected by Quantum Mechanics. The cause of this interpretative differences between the two theories can be traced back to the different algebras used to relate propositions0 . Precisely:

propositions in Classical Physics form a Boolean algebra, while propositions in Quantum Mechanics form a non-Boolean algebra.

This feature of Quantum Mechanics entails that properties can not be said to be possessed by a system, denying, in such a way, the existence of an independent ``outside world”.

An attempt to give Quantum Mechanics the status of a realist1 theory is given by the hidden variable theories, which postulate the following:

  • in any state $\vec{\psi}$ observables A posses an ``objectively existing value”
  • values of observables A are determined by $\vec{\psi}$ and hidden variables.

However these theories where disproved by the Kochen-Specker Theorem and Bell inequalities, both of which show that properties are not possessed by a Quantum System, therefore rejecting the first assumption of the Hidden variable theories.

In particular, the Kochen-Specker theorem asserts that it is impossible to evaluate propositions regarding values possessed by physical entities represented by projection operators, such that their truth values belong to the set $\{1,2\}$therefore depriving of meaning any statement regarding a state of affairs of a system, since, generally speaking, a statement is said to be meaningful if its validity can be assessed.

The Bell inequalities go further and show the impossibility of a local2 realist interpretation of Quantum Mechanics.

Do we then have to accept that Quantum theory is a non-realist theory and therefore regard any statement about states of affair of a system as meaningless?. Or is there a way of reformulating the Kochen-Specker theorem and the Bell inequalities so as to give Quantum theory a realist flavor?

To this question C.J.Isham of Imperial College London and J. Butterfield of All Souls’ College Oxford answer in a negative way.

In fact in a series of papers Isham, Butterfield 1998, Isham, Butterfield 1999, Isham, Butterfield, Hamilton 1999 Isham, Butterfield 1999, Isham, Butterfield 1999 they analyse the possibility to retain some realist flavor in Quantum Theory by

changing the logical structure with which propositions about the values of physical quantities are handled.

In particular they introduce a new kind of valuation for quantum quantities which is defined on all operators, so that it will be possible to assign truth values to Quantum Proposition. This new valuation is defined using Topos Theory, and it is such that truth values become multi-valued and contextual, in agreement with the mathematical formalism of Quantum Theory.

The idea behind the definition of these new valuations is given by the realization that, although the Kochen-Specker Theorem prohibits to assign truth values to propositions, nevertheless it allows the possibility of assigning truth values to generalized propositions. Therefore, by adopting generalized propositions as the domain of applicability of the valuation function, we obtain a situation in which it is meaningful to assign truth values.

The advantage of this approach is that the logic of Quantum propositions it proposes is distributive, therefore, it can be used as a deductive system of reasoning. Moreover, unlike any other multi- valued type logic, it enables to define the logical connectives in an unambiguous way, and such that the metalanguage3 /object-language4 distinction is not violated.

For anyone that is interested in the topic, apart from the references I mentioned above, a non -technical summery of the work done by Isham and Butterfield can be found at Isham 2004 .

An easier introduction can instead be found at My Master Dissertation 2005

A brief account of the subject can be found at talk which is a pdf version of a talk I gave at the Eleventh Marcel Grossmann Meeting on General Relativity in Berlin. For a more technical account see my more recent talk .

Recently it has been shown in Doering ,Isham 2011 that it is possible to regard probabilities are truth values in an appropriate topos. This definition of probabilities allows for a non instrumentalist interpretation of the latter. Details can be found in the section A Topos Representation of Probabilities

0 Propositions are defined as statements regarding properties of a given system

1 A realist theory is a theory in which the following conditions are satisfied 1) propositions are related through a Boolean algebra 2) propositions can always be assessed to be either true or false.

2 Locality means that, given a composite system, the value of a physical quantity of an individual constituent of the system is independent from what is measured on any other constituent.

3 Metalanguage is the language used to make statements about another language.

4 Object-language is the language being studied through the Metalanguage.