Solace of Quantum

Early on in Leonard Susskind and Art Friedman’s Quantum Mechanics: The Theoretical Minimum, the authors illustrate a difference between the logic of predicates and sets, and the logic of observations at quantum scale. One way to look at it is in terms of modelling.

If the state space of a classical system is the set of all the states the system can be in, then a proposition about the system corresponds to a predicate P that picks out certain states as possible states of the system if that proposition is true. Susskind’s example is a single roll of a six-sided die. Let the “state” of the die be the number that is facing upwards after the roll – the state space S is then the set {1, 2, 3, 4, 5, 6}. To the proposition “the value of the die is even” corresponds the predicate P that picks out the subset of the state space S' = {2, 4, 6}. We have S, the state space, P, the predicate, and S' = {x ε s & P(x)}, the subset of the state space which satisfies the predicate.

The state space is thus a model for propositions about the system, which means that we can translate statements about propositions into statements about predicates and subsets. If two propositions about the system are true at once, then the states the system can be in are those which are picked out by both of their corresponding predicates. For example, take the propositions “the value of the die is even”, and “the value of the die is greater than three”. Considered separately, these correspond to two predicates, P1 and P2, which pick out two different subsets of S, S1 = {2, 4, 6} and S2 = {4, 5, 6}. The state subset corresponding to the proposition “the value of the die is even, AND is greater than three” is the intersection of the subset S1 picked out by P1 and the subset S2 picked out by P2: {4, 6}. The subset corresponding to the proposition “the value of the die is even, AND/OR is greater than three” is the union of S1 and S2: {2, 4, 5, 6}. There is thus a “primitive” or “direct” set-theoretic interpretation of propositions concerning the state of the system, based on Boolean algebra.

An important characteristic of classical systems is that they “hold still” while being observed: the state space is a static image of possible measurement results within a given reference frame, and we can combine measurements freely without that image shifting beneath our feet. You can measure both the position and the velocity of a billiard ball, in whichever order you like, since there exists a state of the billiard-ball-system that captures both of these properties simultaneously.

The way we model the state of a quantum system is necessarily different, because the state of such a system is not independent of observation: to every measurement of the truth of a proposition about a quantum system corresponds a new distribution of possible states of the system. We can lose information by measuring (for example, if we observe the velocity of a particle, we lose information about its position). The order in which observations are made is therefore significant: O1 followed by O2 may well get you a different result to O2 followed by O1, which means that the algebra of quantum observation – unlike the Boolean algebra of classical systems – is noncommutative.

Susskind makes the following eyebrow-raising statement – “the space of states of a quantum system is not a mathematical set; it is a vector space” – and then qualifies it in a footnote: “To be a little more precise, we will not focus on the set-theoretic properties of state spaces, even though they may of course be regarded as sets” (italics mine). This is a subtle distinction. A vector space is a set, or at least is not not a set – working in vector spaces doesn’t in any sense take you out of the set-theoretic mathematical universe. But the way states “fit together” in a quantum system – their way of being compossible – is not readily definable in terms of primitive set-theoretic operators like union and intersection: you need the language of linear algebra, of orthonormal bases and inner products, to make sense of it.

In the light of this, it becomes a little clearer why Laruelle mobilises a metaphorics of “quantumness” – of vectors and matrices, superpositions and complex conjugates – as a way of undermining or circumventing the static philosophical architecture of “Being”: the claim he’s trying to establish is that non-standard philosophy is in some sense to standard philosophy as quantum mechanics is to classical mechanics…