Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2015 / S28

In Bohmian mechanics, the corpuscles supply an actual configuration of matter, not a further branch of the wave function. Their positions and the guiding wave together specify the physical state. In a common nonrelativistic spinless formulation, the wave obeys the Schrödinger equation while the actual configuration $Q=(Q_1,\dots ,Q_N)$ obeyswhere the expression is applied away from nodes of $\Psi$. These are coupled in the sense that the wave guides the configuration; in the standard theory the wave's Schrödinger evolution does not receive a reciprocal source term from the actual particle positions. This is not a model of classical little balls carrying every ordinary observable as a pre-existing intrinsic attribute.

The point of adding a configuration is especially clear in a measurement. A wave may evolve into several macroscopically separated pointer packets. The apparatus's actual particle configuration lies in one packet, so there is a definite pointer record even though the full wave has not undergone fundamental collapse. The corpuscles thereby connect the mathematical description with the location of actual apparatus, records and observers. David Bohm's original papers provide a theory of individual processes and analyse measurement in this spirit.^[ref][ref]

For a subsystem, inserting the actual environment configuration into the universal wave gives a conditional wave function, schematically $\psi (x,t)\propto \Psi (x,Y(t),t)$. If the branches are suitably disjoint in the environment's configuration coordinates and remain dynamically isolated, the occupied branch supplies an effective collapse in Bohmian mechanics. The unoccupied branches are not simply erased by definition: before permanent separation, wave packets can recombine and affect interference. The conditional and effective wave-function constructions make this distinction precise.^[ref]

The Born rule also needs an account. The guidance current is $j_k=(\hslash /m_k)\mathrm{Im}({\Psi }^{\ast }{\nabla }_k\Psi )$, and both $|\Psi {|}^2$ and an ensemble of guided configurations satisfy the associated continuity equation. Hence an initial quantum equilibrium distribution $\rho =|\Psi {|}^2$ is preserved. Equivariance supports the ordinary statistical predictions, but the guidance law alone does not prove that an arbitrarily prepared ensemble has that distribution.

The corpuscles are therefore neither detectors that a conscious observer adds to the theory nor dispensable labels attached after the statistics have been calculated. They constitute a primitive ontology that provides definite configurations, trajectories and records. Whether this is preferable to a wave-only ontology is a further explanatory judgement; its empirical equivalence in quantum equilibrium does not make the two accounts identical in their claims about what exists.

An apparent collapse is the transition from predicting several possible recorded outcomes to describing a particular recorded result. Whether that transition is a physical discontinuity, a conditional state update or an effective description depends on what the wave function is taken to represent. The practical success of collapse-based calculation does not by itself settle that interpretive question.

The difficulty is exposed by unitary time evolution. An ideal apparatus that reliably measures a basis hasLinearity then makes a superposed input evolve asThis is an entangled state, not automatically one pointer packet or the other. A separate projection rule would instead select one recorded outcome, with Born rule weights, and assign its corresponding conditional state.

Quantum decoherence explains much of the practical appearance of a classical record. Interaction with an environment correlates the packets with nearly orthogonal environment states. The subsystem's reduced density matrix then has very small interference terms between the pointer alternatives. Local experiments on that subsystem approximately agree with a diagonal mixture. But this is an improper mixture: the total state can remain a pure superposition. Taking a partial trace has not selected one term of the total state. Suppression of observable interference and the occurrence of a unique absolute outcome are distinct claims.

There is a measurement problem if one simultaneously insists that the wave function is the complete physical description, that its evolution is universally unitary, and that a measurement has exactly one absolute outcome of the ordinary recorded kind. The ideal model shows why simply applying the same unitary law to a larger apparatus does not make that combination work. Nor does calling collapse a change in knowledge solve the problem for someone who has already taken the wave function to be a complete ontic state.

Different interpretations revise different parts of this package. Bohmian mechanics adds an actual configuration and obtains effective subsystem collapse while the total wave evolves unitarily. The many-worlds interpretation retains the total superposition and treats each record as definite relative to its decohering branch. A stochastic collapse theory instead modifies the physical dynamics. Everett's original proposal explicitly separates his relative-state account from an additional collapse postulate.^[ref]

Thus collapse is not problematic merely because conditional probabilities change when a record is learned. The problem is the status of that record and of the dynamical rule used to describe it. A defensible interpretation must explain when its updating rule is applicable, what makes a record definite in its own ontology, and why the ordinary measurement statistics follow. No appeal to human consciousness is required by the displayed unitary measurement model.

The Bell theorem gives a powerful conditional constraint, but does not by itself establish the unrestricted philosophical sentence that every interpretation contains a literal instantaneous action between distant objects. One must specify what locality means and which further assumptions are retained. In particular, Bell local causality is stronger than an inability to send a faster-than-light message.

Let $a,b$ be settings in spacelike-separated laboratories, $A,B\in \{-1,1\}$ their recorded outcomes, and $\lambda$ a sufficient specification of relevant prior physical variables. A Bell-local model factorizes asTogether with measurement independence, $\rho (\lambda \mid a,b)=\rho (\lambda )$, this constrains observable correlations. It allows stochastic response functions: predetermination of all outcomes is not a necessary separate assumption in this general formulation. Bell's 1964 paper demonstrates the incompatibility between a local completion and selected quantum correlations; the two-setting inequality below is a standard later form of the argument, not a claim about the exact notation of that paper.^[ref]

For example, write the local conditional mean outcomes as $A(a,\lambda ),B(b,\lambda )\in [-1,1]$. At each $\lambda$,Averaging over the same setting-independent distribution gives the CHSH inequality, $|S|\le 2$. A suitable spin singlet state instead predicts $|S|=2\sqrt{2}$. Consequently no model retaining the stated factorization and independence assumptions can reproduce all those quantum correlations. Merely replacing deterministic hidden variables with local random noise cannot evade the bound.

However, violation of this bound does not entail operational signalling. For an ordinary entangled state, a remote nonselective measurement leaves the local reduced density matrix unchanged. The local experimenter cannot choose the outcome needed to encode a controllable message. Quantum no-signalling and failure of Bell factorization can therefore coexist; confusing them would misstate both the prediction and the theorem.

Interpretations allocate the explanatory burden differently. In Bohmian mechanics, the guiding dynamics of an entangled configuration is explicitly nonlocal, though quantum equilibrium masks controllable signalling. A collapse interpretation must explain the status of correlations and of any proposed nonlocal state change. Everettian accounts retain the entangled global state and multiple relative records; their defenders can deny a fundamental action that selects one distant outcome. That does not produce a factorizing classical common-cause model of the observed correlations. Whether such an account deserves the word local depends on whether locality means local dynamical interactions, separability, or Bell's single-outcome causal factorization.

One can also reject setting independence or change the assumed causal organization, but such proposals must be developed as physical models, not advertised as refutations of the inequality's proof. The warranted conclusion is that the conjunction of Bell locality, independent settings and the relevant outcome framework is incompatible with the quantum correlations. Calling the failure action-at-a-distance is justified only after an interpretation specifies the physical entities, events and causal commitments to which that phrase applies.

In the specific Einstein-Podolsky-Rosen completeness criterion, completeness is not just an ability to predict all experimental probabilities. A complete theory must represent every element of physical reality. The associated sufficient reality criterion says that a quantity has such an element if its value can be predicted with certainty without disturbing the system. The EPR paper distinguishes these demands from correctness of the theory's empirical predictions.^[ref]

Their position–momentum example makes the locality assumption crucial. For two systems with ideal correlations,the two joint observables commute, although $[X_B,P_B]=i\hslash$. Measuring $X_A$ permits a certain inference of $X_B$; choosing instead to measure $P_A$ permits a certain inference of $P_B$. If a spacelike-separated choice at A cannot alter the physical reality at B, the choice should not determine which properties B possesses. Combining that independence with the reality criterion motivates assigning both quantities to B, while an ordinary individual wave function does not supply both sharp values.

This is not a procedure for experimentally measuring both noncommuting quantities on B at once. It is a counterfactual argument comparing alternative remote measurements, supported by a locality claim about the same remote system. The ideal perfectly correlated position–momentum state is also nonnormalizable; normalizable approximations approach its correlations. The philosophical inference is best assessed in the ideal limit with this qualification stated.

Niels Bohr's reply challenges the criterion's application. He distinguishes a direct mechanical disturbance of the remote system from a dependence of the conditions under which a definite prediction can be meaningfully specified. His appeal to quantum complementarity is therefore not simply the claim that the particles mechanically push one another during the final measurement. It disputes the context-independent assignment of the two elements of reality.^[ref] Whether that response answers EPR's locality challenge is itself a substantive philosophical issue; it is not resolved by restating the uncertainty relation.

A qualified assessment is preferable to an unlabelled yes or no. If EPR's locality and reality premises are retained, the wave-function description is incomplete in their sense. But the later Bell theorem shows that adding ordinary local hidden variables cannot simply recover all quantum correlations while keeping the standard independence assumptions. The natural completion EPR hoped for cannot retain that entire package.

Nonlocal Bohmian mechanics provides one kind of completion with definite configurations. Everettian wave-only accounts use another notion of completeness while rejecting the premise of one absolute set of distant outcomes. An instrumental reading can regard the quantum formalism as complete for assigning experimental probabilities without accepting EPR's representational demand at all. Thus predictive completeness, completeness of an interpretation's ontology, and completeness in EPR's particular sense are different judgements. The EPR argument exposes a conflict among commitments; it does not establish a criterion-independent verdict about every interpretation.

Born rule probabilities can be given a coherent role in an Everettian interpretation, but that role cannot simply be the chance that one branch alone will be created. In the many-worlds interpretation, unitary measurement produces all of the decohering outcome components. The question is why their squared amplitudes should govern rational expectation, typical recorded frequencies or concern for future observers.

For a state ${\sum }_i{c}_i|s_i⟩$ and a measuring apparatus, the resulting correlated state has components $c_i|s_i⟩|A_i⟩$. The squared norms $w_i=|c_i{|}^2$ supply additive branch weights. They are preserved by unitary refinements of an outcome into orthogonal subcomponents: if a branch splits internally, its descendants' weights sum to its original weight. Everett's original paper introduces the relative-state description and a measure associated with the statistical analysis; this should not be confused with an independent stochastic collapse law.^[ref]

A simple example shows why counting branches is not an alternative rule without further structure. Begin withIf an irrelevant environment interaction refines B into two orthogonal components of equal amplitude, there are now three displayed components, each of amplitude $1/\sqrt{3}$. The total weight of B is still $2/3$. Counting displayed components instead changes A's assigned share from $1/2$ to $1/3$, merely because the description has resolved additional environmental detail. Real decoherent branching is approximate and has no unique fundamental branch count. Squared norm, in contrast, respects such refinements.

There are at least two ways to understand the remaining epistemic issue. An observer may have self-locating uncertainty in Everettian quantum mechanics, especially after branching but before accessing a record: the uncertainty concerns which centred observer situation is theirs, not ignorance of the global wave function. Alternatively, a decision-theoretic account can ask how a present observer should value all future successors without assuming that exactly one successor is uniquely them. These are distinct proposals, and the second does not become incoherent simply because the first is disputed.

David Wallace's decision-theoretic programme argues that suitable rationality and measurement-equivalence assumptions lead to expected-utility weights equal to squared amplitudes. The assumptions, including the appropriate neutrality of the measurement implementation, do real work.^[ref] A critic can question whether the constraints already privilege the desired measure or whether a preference theorem establishes the empirical role of chance. The defender must answer those objections; unitarity and normalization alone do not automatically turn a deterministic global theory into ordinary single-outcome randomness.

Repeated experiments add a consistency check. With Born weights, the total weight of frequency records is binomial and concentrates near the Born frequency as the number of trials increases. Yet calling the high-weight records typical already uses a measure, so this is not by itself a noncircular derivation of the weighting rule. The balanced conclusion is that Born probabilities are not meaningless in Everettian quantum mechanics: they can organize credence, decisions and typical records. Their status requires an argued connection between branch weights and those roles, rather than a bare appeal either to all outcomes happening or to the algebraic identity ${\sum }_i|c_i{|}^2=1$.