On the basis of three physical axioms, we prove that if the choice of a particular type of spin 1 experiment is not a function of the information accessible to the experimenters, then its outcome is equally not a function of the information accessible to the particles. We show that this result is robust, and deduce that neither hidden variable theories nor mechanisms of the GRW type for wave function collapse can be made relativistic. We also establish the consistency of our axioms and discuss the philosophical implications.
Introduction The two theories that revolutionized physics in the twentieth century, relativity and quantum mechanics, are full of predictions that defy common sense. Recently, we used three such paradoxical ideas to prove “The Free Will Theorem” (strengthened here), which is the culmination of a series of theorems about quantum mechanics that began in the 1960s. It asserts, roughly, that if indeed we humans have free will, then elementary particles already have their own small share of this valuable commodity. More precisely, if the experimenter can freely choose the directions in which to orient his apparatus in a certain measurement, then the particle's response (to be pedantic—the universe's response near the particle) is not determined by the entire previous history of the universe. Our argument combines the well-known consequence of relativity theory, that the time order of spacelike separated events is not absolute, with the EPR paradox discovered by Einstein, Podolsky, and Rosen in 1935 and the Kochen–Specker Paradox of 1967 (see [5]). We follow Bohm in using a spin version of EPR and Peres in using his set of thirty-three directions, rather than the original configuration used by Kochen and Specker. More contentiously, the argument also involves the notion of free will, but we postpone further discussion of this to the last section of the chapter.
A finite relativistic model for free particles, which describes the collapse of the state vector, is presented. The interaction of particles with a classical fluctuating field
w(x) causes collapse, as in the (so far) physically satisfactory nonrelativistic continuous spontaneous localization (CSL) model. In previous relativistic models, where
w(x) has a white-noise spectrum, collapse is accompanied by spontaneous creation of particles with all energies out of the vacuum [amounting to infinite energy/(sec-volume)]. In this paper we explore one way of eliminating such vacuum excitation. We present a formalism which allows an arbitrary spectrum for
w(x). We point out, in lowest order of perturbation theory, that it is timelike components of this spectrum which are responsible for vacuum excitation but (appropriate, it seems, to a fundamentally nonlocal phenomenon) it is spacelike components which are responsible for collapse. We restrict the spectrum of
w(x) to that of a tachyon of mass $\ensuremath{\mu}=\ensuremath{\Elzxh}/ac\ensuremath{\approx}1 \mathrm{eV},$ where $a\ensuremath{\approx}{10}^{\ensuremath{-}5} \mathrm{cm}\mathrm{}$ is the Ghirardi-Rimini-Weber (GRW) collapse model scale parameter used in CSL. However, in higher orders than the first, there is still vacuum excitation due to the energy-momentum supplied by particle propagators. We opt to explore a simple means of eliminating this vacuum excitation: by removing the time-ordering operation from the state vector evolution so that particle propagators are on the mass shell. The result is a finite theory: there is no vacuum excitation (and no need for renormalization) as each vertex describes a finite physical process, the spontaneous emission or absorption of a real tachyon by a particle (pair creation or annihilation is not permitted by energy-momentum conservation). This process may be thought of as analogous to the GRW ``hitting'' description of collapse. The cost of no time ordering is that at each vertex a mass M particle's wave function can spread by as much as $\ensuremath{\approx}cT(\ensuremath{\mu}/M)$ in time T. This has the result that, beginning at perturbation order $2M/\ensuremath{\mu},$ there is a nonvanishing probability for the particle to be found outside its light cone: for electrons and nucleons, the probability of this spacelike transport is negligibly small. Apart from this effect, the density matrix of particles in one region evolves independently of particles in another spacelike separated region. We thus obtain a finite, reasonably sensible, relativistic collapse model for free particles.
A logical physical ontology is code theory, wherein reality is neither deterministic nor random. In light of Conway and Kochen’s free will theorem [The free will theorem, Found. Phys. 36(10) (2006) 1441–1473] and strong free will theorem [The strong free will theorem, Not. Am. Math. Soc. 56(2) (2009) 226–232], we discuss the plausibility of a third axiomatic option — geometric language; the code-theoretic axiom. We suggest that freewill choices at the syntactically-free steps of a geometric language of spacetime form the code-theoretic substrate upon which particle and gravitational physics emerge.