Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2017 / S28

Bell's theorem makes precise an obstruction to reproducing quantum correlations with a local causal completion. Let measurement settings be $a,b$, outcomes $A,B=\pm 1$ and a more complete state $\lambda$. Bell locality factorizes $P(A,B|a,b,\lambda )=P(A|a,\lambda )P(B|b,\lambda )$; setting independence takes the distribution of $\lambda$ to be independent of the freely chosen settings. These assumptions imply a Bell inequality. For example deterministic assignments satisfy $|A(B+B^{\prime })+A^{\prime }(B-B^{\prime })|\le 2$, and averaging gives the CHSH bound $|S|\le 2$. The stochastic version follows by averaging local response probabilities too.

An entangled spin singlet predicts $E(a,b)=-a\cdot b$ and suitable four directions give $|S|=2\sqrt{2}$. Thus the quantum predictions are incompatible with the stated local model, not merely with an unnecessarily specific mechanical hidden-variable proposal. Perfect anticorrelation in the EPR argument had motivated inferring pre-existing properties from remote predictions under locality and a completeness criterion. Bell shows that retaining such a local completion cannot recover the broader quantum correlation pattern.

It would overstate the result to say that it proves the impossibility of every hidden-variable theory. Bohmian mechanics supplies a nonlocal example, and rejecting setting independence or changing the causal model changes the theorem's premises. It would also be wrong to equate Bell locality with operational no-signalling. Quantum outcome probabilities at either wing can remain independent of the distant setting after averaging, even though the joint correlations have no local factorized explanation.

The theorem therefore sharpens the EPR dilemma: one cannot preserve all of the local causal assumptions and the quantum predictions. Empirical Bell violations bear on that conjunction. They do not alone choose one interpretation, make controllable superluminal communication available, or settle every philosophical sense of “realism.”^[ref][ref]

In the many-worlds interpretation, the universal state evolves unitarily. A measurement transforms $(\alpha |0⟩+\beta |1⟩)|M_{\ast }⟩$ into $\alpha |0,M_0⟩+\beta |1,M_1⟩$. Environmental decoherence suppresses interference between macroscopically distinct records, so each branch contains a consistent observer record of one outcome. There is no stochastic selection of a unique global result in that dynamics. Thus fundamental indeterministic collapse is not recovered; it is deliberately absent.

The appearance of indeterminism can nevertheless be described branch-relatively. Before examining a record, an observer lacks information about which recorded result will be encountered in a future branch-relative history. After measurement, each descendant has one stable record and experiences a definite result. This is more than saying that a classical observer does not know a predetermined hidden bit, because the global state includes multiple continuing record structures rather than one unknown selected trajectory.

The hard question is why uncertainty or rational expectation should follow the Born weights $|\alpha {|}^2,|\beta {|}^2$. Decoherence explains approximately autonomous records and a preferred coarse-grained basis, but does not by itself turn all branch counts into a unique measure. Literal counting is ill-defined because branches do not have a uniquely fixed number. Decision-theoretic, symmetry or typicality arguments try to justify squared-amplitude weights and their role in observed frequencies, but their auxiliary assumptions must be assessed rather than silently called consequences of unitary dynamics alone.

The interpretation can therefore account structurally for the appearance of definite unpredictable outcomes without adding dynamical randomness. Whether that completely accounts for probability depends on a successful connection from branch weights to expectation and confirmation. Its strength is a uniform dynamical law for system and observer; its cost is an ontology and probability account that go beyond the experience of one apparently unique outcome.^[ref]

Bohmian mechanics retains definite particle positions and deterministic trajectories, which makes it resemble classical mechanics at one level. But the guiding dynamics depends on a wavefunction obeying the Schrödinger equation, with velocity schematically ${\dot{Q}}_i=(\hslash /m_i)\mathrm{Im}({\nabla }_i\psi /\psi )(Q)$. For many particles this wavefunction is defined on configuration space and can be entangled. The velocity of one particle can consequently depend on the configuration of distant particles. Classical-looking trajectories therefore do not entail classical locality or separability.

Writing $\psi =R{e}^{iS/\hslash }$ also yields a Hamilton-Jacobi-like equation with quantum potential $Q=-{\sum }_i{\hslash }^2{\nabla }_i^{2}R/(2m_{i}R)$. The quantum potential is not generally a sum of ordinary pair potentials in three-space; its configuration dependence carries the nonclassical correlations. A stationary real bound-state wavefunction can give a stationary particle configuration, unlike the orbit one might infer from a classical Coulomb picture. Measurement outcomes are contextual: the actual apparatus interaction helps determine which position record is produced.

With the quantum-equilibrium distribution $\rho =|\psi {|}^2$, the guidance and wave equations preserve that distribution and reproduce the usual quantum statistics. Determinism alone does not establish that distribution; equilibrium or typicality is an additional explanatory component. The theory also leaves philosophical choices about whether the wavefunction is a physical field, a law-like object or part of a broader ontology.

The significant departure is therefore not simply replacing chance by hidden classical motion. It is the entangled guiding structure, nonlocal dependence and contextual relation between measurements and properties. The retained particles make the account continuous with some classical intuitions, while its actual law and ontology depart substantially from classical many-body mechanics.^[ref][ref]

Quantum mechanics has a highly successful predictive calculus, but its standard informal presentation does not transparently supply one consistent physical description of all processes. An isolated system evolves unitarily, while a measurement is often said to select one outcome stochastically. If both the apparatus and the system are physical and governed by the same unitary dynamics, their interaction generally produces an entangled superposition of records. Explaining how this relates to an experienced definite result is the measurement problem.

For a limited laboratory task, an operational rule can assign probabilities to recorded outcomes without first resolving every ontological question. In that pragmatic sense, a calculation need not await agreement on interpretation. But specifying what counts as an intervention, a record and an outcome still supplies a relation between the mathematics and practice; calling this “no interpretation” may merely leave it implicit.

The need becomes sharper in cosmology or a universal theory, where an external classical observer cannot simply be assumed. Distinct responses change what is physically asserted: collapse theories alter dynamics, Bohmian mechanics adds configurations and guidance, and many-worlds accounts keep unitary dynamics and branch-relative records. Some proposals are empirically equivalent in standard regimes, whereas modified collapse models can have additional testable noise or heating. Interpretation is therefore not always an idle choice among verbal paraphrases of one unchanged theory.

The appropriate conclusion distinguishes applications from completeness. One can make reliable conditional predictions using a modest operational account, but a claim to describe the physical world requires saying what the formal state and its alternatives represent and how definite evidence arises. That need does not prove that any particular realist interpretation is correct, or that every philosophical dispute must be settled before doing useful physics.

A dynamical collapse law specifies stochastic changes of the wavefunction, including a rate, localization scale and probabilities. Such changes can suppress macroscopic superpositions, but a law for a mathematical state does not automatically identify the stuff composing a macroscopic table. If the wavefunction is on configuration space, one must still explain how it represents a three-dimensional distribution of matter and records.

One option defines mass density in ordinary space by $m(\mathbf{x},t)={\sum }_i{m}_i\int dq|\psi (q,t){|}^2\delta (\mathbf{x}-{\mathbf{q}}_i)$. A localization then changes this density so that a single dominant macroscopic pattern remains. The formula is additional interpretive structure, even if it introduces no independent evolving variable beyond the wavefunction. Another option takes discrete spacetime localization events as the primitive ontology. These two proposals can accompany closely related wavefunction dynamics while telling different stories about what matter is.

This is why mass density in particular is not logically mandatory, but some account of the relation between collapse and ordinary physical objects is required if the theory is offered as a complete ontology. A wavefunction-only view may argue that no further primitive stuff is necessary, yet it owes an explanation of spatial localization and records on that basis. Merely asserting that interference terms become small is not that explanation.

Additional posits can also have costs: small wavefunction tails survive finite localization, and the ontology must say what their tiny matter components or corresponding events mean. Choosing primitive ontology and handling such tails are distinct from specifying the stochastic dynamical law. A defensible collapse interpretation therefore makes both commitments explicit rather than suggesting that adding randomness alone solves every part of the measurement problem.^[ref][ref]