Particles, Individuals, and Discernibility

Two electrons in a helium atom. Are they two things — individuals, each with its own identity — or is that question ill-formed?

Classical physics gives an easy answer: two particles are two, distinguished at least by their trajectories. Quantum mechanics does not, and the reason is not interpretive delicacy but a structural feature of the theory. This is the case where Black's two spheres stops being a thought experiment.

Permutation invariance

For indistinguishable particles the state must be symmetric (bosons) or antisymmetric (fermions) under exchange, so a permutation yields the same physical state. The consequence shows up directly in counting.

Take two particles and two available single-particle states, $|1⟩$ and $|2⟩$. How many distinct two-particle arrangements are there?

The classical count treats "$a$ in 1, $b$ in 2" and "$b$ in 1, $a$ in 2" as different arrangements — which presupposes that there is a fact about which particle is where. The quantum counts do not, and the quantum counts are the ones borne out by experiment. The world does not recognise the distinction that individuality would underwrite. The formal apparatus is in Identical Particles, and the field-theoretic version — where indistinguishability is automatic because all electrons are excitations of one field — in Particles as Excitations.

French and Redhead

The 1988 argument draws the ontological moral. In a symmetric or antisymmetric state, the two particles share all monadic properties and all relational properties: any property one has, the other has, because the state is invariant under their exchange.

By the Principle of the Identity of Indiscernibles, they should therefore be one. They are two. So PII is false — not merely questionable in an imagined symmetric universe, but false of the actual world, at its most basic level, in a way established by physics rather than by intuition about a story.

Two options follow, and both are uncomfortable:

• Particles are not individuals. They lack identity in the ordinary sense: they can be counted but not named, aggregated but not distinguished. This was Schrödinger's own view — that the concept of an individual particle does not survive the transition to quantum mechanics.
• Particles are individuals with haecceities. Each has a primitive thisness that makes it the one it is. This preserves individuality at the cost of positing something that makes no difference to anything — and worse, the symmetrisation postulate must then be imposed as a brute restriction on which states are physically possible, to prevent the extra states that individuality would allow.

The second option's difficulty is precisely that of the hole argument: a metaphysics of primitive identity generates surplus structure that the physics must then be constrained to ignore.

Weak discernibility

Saunders' intervention, using Quine's notion, is the most important recent move in a debate that had been static since 1952.

Distinguish three grades:

• Absolute discernibility: some monadic formula is satisfied by one and not the other.
• Relative discernibility: some asymmetric relation holds one way — $Rab$ and not $Rba$.
• Weak discernibility: some irreflexive relation holds between them.

Two fermions in the spin-singlet state

$$\frac{1}{\sqrt{2}}(|\uparrow {⟩}_a|\downarrow {⟩}_b-|\downarrow {⟩}_a|\uparrow {⟩}_b)$$

are not absolutely or relatively discernible. But they stand in the relation has opposite spin to, which is irreflexive — nothing has opposite spin to itself. So they are weakly discernible, and therefore two.

What this delivers is objecthood without qualitative difference and without primitive thisness. Their being two is an objective structural fact — the obtaining of an irreflexive relation — requiring no further individuator. PII survives in the weak form, and the dilemma above is dissolved.

Muller and Seevinck extended the treatment to bosons, using suitable observables to construct the required irreflexive relations, which addresses the natural objection that the singlet case is special.

The remaining objection is that weak discernibility may be too cheap: it tells us that there are two without saying what makes this one this one, and one may hold that genuine individuality requires more than a relation that presupposes plurality in order to obtain. The reply is that this demand is precisely what the physics gives no purchase on, and that a notion of individuality with no physical consequences is not obviously a notion of anything.

Quasi-set theory

If particles are non-individuals, standard set theory is the wrong formal tool, since its objects have determinate identity — for any $x$ and $y$, either $x=y$ or not.

Quasi-set theory (French and Krause) is a formal system in which identity is not universally defined. Some objects — m-atoms, representing quanta — admit of cardinality without ordinality: one can say there are three of them without there being a first, second, and third, and without any of them being available to be named. Aggregation without individuation is made formally respectable.

Whether the formalism is needed, or whether weak discernibility makes it unnecessary, is the live question. Its interest for this section is that it is a case of the logical apparatus being modified to fit an ontological thesis, rather than the ontology being read off standard logic — a reversal of the usual direction, and a reminder that the choice of formalism is itself an ontological choice.

Where this sits

This is the individuation debate conducted with an actual case rather than an imagined one, and the two should be read together: the general PII argument belongs there, the physics here.

Its wider significance is methodological. Black's spheres was an intuition pump — one could always deny the possibility of the scenario. The quantum case cannot be denied that way, because the statistics are measured. This is the clearest instance in the section of physics doing genuine ontological work: not settling a question, but removing an escape route that armchair reflection had left open.