## ABSTRACT

I’m not going to attempt anything along the lines of a summing up of what Hilary has meant, and what he continues to mean, to my own particular corner of philosophy here. But it needs at least to be mentioned – on an occasion like this – that Hilary has thought about the foundations of physics harder, and longer, and more deeply, and with more openness, and with more wonder, and with more determination, and with more courage, than anyone now living. Sidney Morgenbesser used to say that Hilary was the most quantum-mechanical philosopher in the world, because he and his philosophical position could not simultaneously be identiﬁed. And that (as with all of Sidney’s jokes) is exactly right. Nobody, to this day, is as young, and as curious, and as willing to be surprised, and as ready to turn his back on everything he has ever believed, and as full of the exuberant expectation of the impossible, as Hilary. And this is how to learn about the world. And I want to do something in that spirit here. There is a topic

that Hilary and I have been talking and talking and talking to one another about for decades, and about which we have both lately been surprised, about which we have both lately come to understand that we never really knew anything at all. And that’s what I want to talk about today. I’m going to start oﬀ with some mildly technical remarks about the beha-

viors of quantum-mechanical wave-functions under Lorentz-transformations. But bear with me just a bit – I think something of what’s at stake in these considerations, something of what’s philosophically interesting, will come through well enough later on, whether you’re able to follow the quantum mechanics or not. Consider a system of four distinguishable quantum-mechanical spin-1/2

particles. Call it S. And suppose that the complete history of the motions of those particles in position-space – as viewed from the perspective of some particular Lorentz-frame K – is as follows: Particle 1 is permanently located in the vicinity of some particular spatial point, and particle 2 is permanently located in the vicinity of some other spatial point, and particles 3 and 4 both move with uniform velocity along parallel trajectories in space-time.1 The trajectory of particle 3 intersects the trajectory of particle 1 at space-time

point P – as in Figure 7.1 – and the trajectory of particle 4 intersects the trajectory of particle 2 at space-time point Q. And P and Q are simultaneous, from the perspective of K. And suppose that the state of the spin degrees of freedom of S, at t = −1,

is [φ >12 [φ >34, where

[φ >AB = 1/√2[">A[#>B – 1/√2[#>A[">B. (1)

I want to compare the eﬀects of two diﬀerent possible Hamiltonians on this system. In one, S evolves freely throughout the interval from t =−1 to t = +1. The other includes an impulsive contact interaction term that exchanges spins – a term (that is) which is zero except when two of the particles occupy the same point, and which (when it isn’t zero) generates precisely the following unitary evolution:

[#>A[">B![">A[#>B (2)

[#>A[#>B![#>A[#>B A minute’s reﬂection will show that the entire history of the quantum state of this system, from the perspective of K – the complete temporal sequence (that is) of the instantaneous quantum-mechanical wave-functions of this system, even down to the overall phase, from the perspective of K – will be identical on these two scenarios. On both scenarios (that is) the state of S, from the perspective of K, throughout the interval from t = −1 to t = +1, will be precisely [φ >12 [φ >34.