Interpretations (Evidence-Anchored)

The formalism of The Postulates of Quantum Mechanics predicts, with no known exception, the outcome statistics of every experiment in this book. What it does not do is say what happens in a single run. The unitary evolution is linear and deterministic; the measurement postulate is nonlinear and probabilistic; and the theory as usually stated does not say which of the two applies when, nor where the boundary between system and apparatus lies. That is the measurement problem, and it is the only reason this chapter exists. It is placed after entanglement (Entanglement and Bell Tests) because the sharpest constraints on the question are Bell-type theorems, and before decoherence (Open Quantum Systems and Decoherence) because decoherence is the piece of the answer that is not interpretation at all but ordinary quantum dynamics of an open system.

This chapter is written under a rule that the rest of the treatise applies silently and that is worth stating aloud here. What experiment constrains is physics; what it does not constrain is interpretation, and this book will label it as such. A great deal of the subject is genuinely empirical — Gleason's theorem [Gleason:1957], Kochen–Specker contextuality [Kochen:1967], Bell's analysis of von Neumann's impossibility proof [Bell:1966], the PBR theorem [Pusey:2012], Leggett–Garg tests [Leggett:1985], the whole decoherence programme [Zeh:1970] [Zurek:2003], and above all dynamical collapse models [Ghirardi:1986], which make numerically different predictions and are being ruled out region by region [Vinante:2020] [Donadi:2021]. The remainder — whether the wavefunction is a thing, whether the other branches exist, whether probabilities are credences — has, at present, no experiment attached. Both kinds of statement appear below, and the ledger in Section 85.6 says which is which. Standard references are [Bell:1987] [Peres:1993] [Wheeler:1983]; for decoherence, [Schlosshauer:2007]; for collapse models, [Bassi:2013].

The measurement problem

The Born rule

[Reserved: Born's probabilistic reading of the wavefunction, introduced in a footnote of the first collision paper [Born:1926a] and developed in the second [Born:1926b]; the statement \(p=\abs{\braket{a}{\psi}}^{2}\) and its generalization to positive operator-valued measures; the fact that this is the only postulate of The Postulates of Quantum Mechanics that mentions probability, and the question — taken up in Section 85.3 — of whether it can be derived from the others rather than assumed; the experimental tests of the exponent, via three-slit interference bounding higher-order terms in the Born rule, listed with their bounds.]

The von Neumann chain

[Reserved: von Neumann's model of measurement as a unitary system–apparatus coupling that produces an entangled state, and the separate “process 1” that reduces it [vonNeumann:1932]; the chain that follows, in which including the next apparatus only moves the cut; the demonstration that the cut may be placed anywhere without changing predictions, which is the reason the problem is hard rather than the reason it is solved; von Neumann's impossibility proof for hidden variables and Bell's identification of the assumption that makes it inapplicable [Bell:1966], a case study in Epistemology and the Scientific Method of a theorem being widely believed to say more than it says.]

Three problems, stated separately

[Reserved: the useful decomposition into the problem of outcomes (why one result), the problem of the preferred basis (why this observable is definite), and the problem of statistics (why these probabilities), following Maudlin's separation [Maudlin:1995]; the observation that decoherence solves the second, is silent on the first, and gives at best a circular account of the third; that each proposal in Section 85.3 is best classified by which of the three it addresses and at what cost.]

What experiment has already settled

Gleason's theorem

[Reserved: that in a Hilbert space of dimension at least three the only probability assignment on the lattice of projections that is additive on orthogonal families is \(\tr(\rho P)\) [Gleason:1957]; the consequence that the Born rule is not an independent choice once non-contextual additivity is granted, and the equally important consequence that dimension two is exempt; the modern Busch–Caves–Fuchs–Schack strengthening to POVMs, which removes the dimension restriction; the status of this as a mathematical theorem constraining any interpretation, not as an interpretation.]

Contextuality

[Reserved: the Kochen–Specker theorem, that no assignment of definite values to all observables can reproduce the functional relations among commuting sets [Kochen:1967]; the original \(117\)-vector construction and the small modern sets; the distinction from Bell's theorem — contextuality needs no spatial separation and no locality assumption, only compatibility — and the resulting statement that value-definiteness, not merely locality, is what fails; the experimental tests of state-independent contextuality inequalities, and the loopholes (finite precision, compatibility) that keep them weaker than Bell tests.]

Bell and Leggett–Garg constraints

[Reserved: what the measured Bell violations of Entanglement and Bell Tests and Experiment: Bell Tests exclude, restated as a constraint on interpretations rather than on theories: any account must be either non-local, or non-value-definite, or must abandon measurement independence. The temporal analogue of Leggett and Garg [Leggett:1985], whose assumptions are macroscopic realism and non-invasive measurability, and its violation in superconducting circuits and nuclear spins; the honest caveat that the non-invasiveness assumption is far harder to enforce than spatial separation, so a Leggett–Garg violation is weaker evidence than a Bell violation.]

PBR and the status of the quantum state

[Reserved: the Pusey–Barrett–Rudolph theorem [Pusey:2012], that if a system has a real physical state determining measurement statistics, and if independently prepared systems have independent physical states, then distinct quantum states cannot correspond to overlapping distributions — so the wavefunction is ontic, not merely a state of knowledge, within that framework; the preparation-independence assumption, which is exactly where epistemic accounts resist; the ontological-models framework of Harrigan and Spekkens in which the statement is formulated; the experimental tests, which bound the degree of overlap rather than settle the question.]

Decoherence

[Reserved: environmental monitoring destroys local coherence on timescales far shorter than any dissipative timescale, as first argued by Zeh [Zeh:1970], computed for a mass coupled to scattering particles by Joos and Zeh [Joos:1985], and developed into einselection and quantum Darwinism by Zurek [Zurek:2003] [Zurek:2009]; the derivation belongs to Open Quantum Systems and Decoherence and is referenced, not repeated, here. The observations: the controlled decay of a mesoscopic cavity superposition [Brune:1996], interference of C\(_{60}\) molecules [Arndt:1999], and its controlled destruction by gas collisions [Hornberger:2003]. The crucial negative statement, which this chapter will make plainly: decoherence explains why the interference terms are unobservable and which basis is stable, and it does not explain why one outcome occurs — the state after decoherence is still an improper mixture.]

The interpretive programmes

Copenhagen and complementarity

[Reserved: Bohr's complementarity as announced at Como [Bohr:1928] — that wave and particle descriptions are mutually exclusive and jointly exhaustive, and that the apparatus must be described classically; Heisenberg's uncertainty paper and its microscope argument [Heisenberg:1927], with the essential correction that the modern inequality is a property of states, proved by Kennard [Kennard:1927] and in general form by Robertson [Robertson:1929], and not a statement about measurement disturbance — a misattribution worth correcting explicitly, since the distinction is measurable and has been measured in weak-measurement experiments; the quantitative wave–particle duality relations that replaced qualitative complementarity; what “Copenhagen” does and does not commit one to, given that Bohr and Heisenberg disagreed.]

Pilot-wave theory

[Reserved: de Broglie's pilot wave as presented at Solvay [deBroglie:1927] and Bohm's two-part revival [Bohm:1952a] [Bohm:1952b]; particles with definite positions guided by the wavefunction through the velocity field of the probability current, with the quantum potential of the polar decomposition already written in Hamilton–Jacobi Theory and the Optical–Mechanical Analogy; equivariance of the \(\abs{\psi}^{2}\) distribution, so that the theory reproduces every prediction of The Postulates of Quantum Mechanics exactly and is therefore not distinguishable by experiment; the explicit non-locality of the guidance equation for many particles, which is what Bell's theorem says any such theory must have; the genuine costs — a preferred foliation, and the unsettled relativistic and field-theoretic extension.]

Everett's relative states

[Reserved: Everett's proposal that the unitary evolution is all there is and that measurement outcomes are relative to a branch [Everett:1957]; the modern form in which decoherence supplies the branch structure, so that the preferred-basis problem is answered by Section 85.2.5; the remaining and hardest problem, the status of probability in a theory where every outcome occurs, and the proposed answers — Deutsch's decision-theoretic derivation [Deutsch:1999] and Zurek's envariance argument [Zurek:2005]; the statement that no experiment currently distinguishes this from any other interpretation, and the fair note that its advocates regard that as its virtue.]

Consistent histories

[Reserved: the reformulation of quantum mechanics in terms of sequences of projections with a decoherence functional whose off-diagonal terms vanish, giving families of histories to which probabilities may consistently be assigned; Griffiths' formulation [Griffiths:1984], Omnès' development [Omnes:1992], and the Gell-Mann–Hartle decoherent-histories version aimed at closed systems and cosmology [GellMann:1993]; the set-selection problem — many mutually incompatible consistent families exist and the formalism does not choose — which is where its critics locate the measurement problem in new clothes.]

Epistemic and relational accounts

[Reserved: QBism, in which the quantum state is an agent's personal degree of belief and the Born rule a normative coherence condition, as set out by Fuchs, Mermin and Schack [Fuchs:2014]; Rovelli's relational account, in which state assignments are always relative to a physical system [Rovelli:1996]; the Frauchiger–Renner argument that a set of natural-looking assumptions about agents reasoning about agents is inconsistent [Frauchiger:2018], and the several ways different programmes escape it; the treatise's assessment that these are positions on the meaning of probability, on which no experiment discussed in this book bears.]

Dynamical collapse: interpretation made testable

Collapse models are the reason this chapter is not merely a survey of opinions. They modify the Schrödinger equation itself, so they predict different numbers, and the numbers are being measured.

GRW and continuous spontaneous localization

[Reserved: the Ghirardi–Rimini–Weber model [Ghirardi:1986], in which each constituent undergoes spontaneous localization at a rate \(\lambda\) with width \(r_{\text{C}}\), so that an \(N\)-particle centre-of-mass superposition collapses \(N\) times faster, so that the transition from micro to macro becomes a computed crossover rather than a postulated cut; the parameters originally proposed, a rate \(\lambda\) of \(10^{-16}\,/\mathrm{s}\) at a localization length \(r_{\text{C}}\) of \(10^{-7}\,\mathrm{m}\); the continuous stochastic reformulation of Pearle [Pearle:1989] and its mass-proportional identical-particle version [Ghirardi:1990], which is the model actually confronted with data; the unavoidable side effect, that the noise heats matter and makes charged particles radiate, which is what the experiments of Section 85.4.3 exploit; the unresolved problems — energy non-conservation and no relativistic version with a white-noise field.]

Gravitationally induced collapse

[Reserved: Diósi's proposal that the noise is gravitational and the collapse rate therefore fixed by Newton's constant, leaving only a short-distance regulator free [Diosi:1989], and Penrose's independent argument from the ill-definedness of the time-translation Killing field for a superposition of spacetimes, giving a decay time set by the gravitational self-energy difference [Penrose:1996]; the resulting model is parameter-free up to the regulator, which makes it the most falsifiable proposal in the chapter; the relation to Quantum Gravity: The Honest Status, where the question of whether gravity must be quantized at all is examined against evidence.]

The experimental bounds

[Reserved: the section that makes the chapter empirical. The searches use the predicted spontaneous X-ray emission from germanium and the predicted heating of mechanical oscillators. A cantilever-based force sensor at millikelvin temperatures excludes the mass-proportional CSL rate over a wide band of localization lengths [Vinante:2020]; an underground germanium experiment excludes the parameter-free Diósi–Penrose model outright, requiring a regulator larger than about \(5\times 10^{-11}\,\mathrm{m}\), which is bigger than the size of a nucleus and therefore not a natural choice [Donadi:2021]; the review [Bassi:2013] for the full parameter plane. The section will tabulate, in SI, the excluded region of the \((\lambda,r_{\text{C}})\) plane with its uncertainties, and state plainly what remains open — which is the point: this is the one part of the subject where a treatise on evidence-based physics has data to report.]

Adjacent experiments that constrain the discussion

Delayed choice and quantum erasure

[Reserved: Wheeler's delayed-choice gedanken experiment [Wheeler:1978] and its realization with a single-photon interferometer whose output beam splitter is inserted after the photon has entered [Jacques:2007]; the correct reading — that which-path information and interference are complementary properties of the whole experimental arrangement — and the incorrect reading, that the past is retroactively altered, which the no-signalling theorem of Entanglement and Bell Tests already excludes; the quantum eraser as the same statement about post-selected subensembles.]

Weak measurement

[Reserved: the weak-value formalism of Aharonov, Albert and Vaidman [Aharonov:1988], in which a weakly coupled pointer, post-selected on a final state, registers a complex quantity outside the eigenvalue spectrum; the reconstruction of average photon trajectories in a two-slit interferometer [Kocsis:2011], which agree with the Bohmian guidance field of Section 85.3.2 — and the careful statement that this confirms a calculation both formalisms share, and selects neither; the use of weak measurements to separate measurement disturbance from the state-intrinsic uncertainty relation of Section 85.3.1.]

The ledger

[Reserved: the closing section, a two-column table rather than an argument. One column lists the statements this chapter regards as physics because an experiment bears on them — the Born rule and its tested exponent, Gleason, Kochen–Specker, Bell and Leggett–Garg violations, PBR under its stated assumptions, decoherence rates measured against computed ones, and the excluded regions of collapse-model parameter space. The other lists the statements no experiment in this treatise distinguishes: whether an unobserved branch exists, whether the wavefunction is a thing or a credence, whether a particle has a trajectory between measurements. The chapter closes by naming what would move an entry from the second column to the first — an observed deviation from linear evolution, a measured Born-rule violation, or a macroscopic interference experiment beyond the collapse bounds of Section 85.4.3 — and by noting that these are experiments now under construction, not thought experiments.]