Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 59 / 5

Locality in quantum theory is not one interchangeable condition. Spatial concentration of a wave packet concerns a state description. Quantum no-signalling restricts operational changes of remote statistics. Microcausality concerns commutation of spacelike observable algebras. The spectrum condition constrains energy-momentum and stability, while primitive causality concerns determination of future-domain expectations by initial-region data. The distinctions are clearest when all five are discussed; none should be replaced silently by the Bell factorization examined elsewhere.

Wave-function tails and entangled correlations are not by themselves controllable signals. Conversely, absence of signals does not by itself supply a local initial-value dynamics or positive-energy spectrum. Each claim must specify the observable algebras, allowed operations and relevant causal regions.

A free nonrelativistic Gaussian wave packet with initial position variance ${\sigma }_0^2$ has wave functionThe free dispersion relation $E=p^2/(2m)$ makes its different momentum components acquire different phases. Fourier evolution gives a position varianceThis spreading is ordinary unitary dynamics. A Gaussian already has nonzero tails initially, so its width increase alone is not an example of information suddenly appearing outside an initially compact support.

For genuinely compactly supported nonzero initial data, the nonrelativistic free kernel yieldsAt $t\ne 0$, the integral is an entire Fourier transform of a compactly supported function. It cannot vanish on an open exterior interval unless the whole transform, and hence the initial state, vanishes. Thus this theory has no strict finite propagation cone for nonzero compact initial wave functions. It is a nonrelativistic theory, so that result is not a contradiction in a fundamental relativistic field theory.

Positive-energy relativistic particle descriptions raise a subtler issue. The Hegerfeldt theorem concerns a Hamiltonian bounded below and a positive localization effect $A$. The function $p_A(t)=\Vert A^{1/2}{e}^{-iHt}\psi {\Vert }^2$ either vanishes identically or is nonzero for almost every time. Vanishing for an open interval makes the analytic continuation of the vector matrix elements vanish identically. A sharp exterior localization probability which remains zero for a nonzero time interval but becomes positive later is therefore incompatible with those assumptions.

The conclusion restricts simultaneous assumptions about positive energy and sharp particle localization. It does not establish an admissible procedure for controllable superluminal messaging. A particle-position projection need not belong to the algebra of a physically local relativistic measurement, and strictly localized one-particle preparation can fail the required assumptions. In field theory the appropriate locality test concerns local operations and observables, not merely pointwise values of a one-particle wave function. Localization, propagation of field disturbances and transmission of information must be distinguished.

For a bipartite state ${\rho }_{AB}$, let a local operation on A have Kraus operators $M_{\alpha }$ satisfying ${\sum }_{\alpha }{M}_{\alpha }^{†}{M}_{\alpha }=I$. Without selecting its outcome, the state becomesFor every remote observable $B$, cyclicity of the trace givesTherefore the remote partial trace is unchanged:This proves quantum no-signalling for local trace-preserving quantum channels, including choices between different such operations. It makes no assumption that ${\rho }_{AB}$ is separable.

Conditioning on one outcome instead uses an unnormalized state with only that outcome's Kraus operators and then divides by its probability. Its remote conditional state can change, yielding quantum steering. However, the outcome cannot be selected deterministically by choosing the measurement setting, and the remote observer needs an ordinary message identifying the selected ensemble. Ignoring the outcome restores the unchanged marginal. No-signalling thus concerns operationally accessible unconditional statistics, not equality of every conditional probability.

The proof presupposes that the intervention is genuinely confined to A and is trace-preserving when outcomes are ignored. A nonlocal operation or setting-dependent postselection does not meet those premises. In field theory local algebras need not come with a simple finite-dimensional tensor factorization; the corresponding statement is expressed using commuting local algebras and local operations, as below.

In algebraic quantum field theory, regions carry observable algebras $\mathcal{A}(O)$. Microcausality requiresField operators are usually distributions, so the corresponding formula uses fields smeared with test functions of spacelike-separated support. Fermionic fields obey graded locality, while their physical even observables commute. This separates the physical observable condition from an unqualified demand that all fields commute.

For a free Klein–Gordon field, the commutator is proportional to the difference of retarded and advanced fundamental solutions. Their causal support makes this commutator vanish outside the light cone. Vacuum two-point correlations can nevertheless be nonzero at spacelike separation: the condition concerns the commutator, not the expectation of a product or statistical independence of outcomes.

To see its operational relevance, take local Kraus operators $M_{\alpha }\in \mathcal{A}(O_1)$ with ${\sum }_{\alpha }{M}_{\alpha }^{†}{M}_{\alpha }=I$. For a remote $B\in \mathcal{A}(O_2)$ the dual operation satisfiesThus such a local unselected operation changes no remote expectation in any state, proving an algebraic no-signalling result without assuming a tensor-product decomposition of the local algebras. The allowed measurement operations must actually be local; calling an arbitrary ideal projector a measurement is not enough.

Spacelike commutation allows entanglement and Bell-inequality violation while preventing signalling by these local operations. It is a kinematic algebraic condition. An additional dynamical condition is needed to say that initial-region data determine fields in the domain of dependence.

For translations represented by $U(a)=e^{i{a}_{\mu }{P}^{\mu }}$, the energy-momentum generators have a joint spectral measure. The spectrum condition isusing signature $(+,-,-,-)$. It excludes negative-energy and spacelike momentum states in the vacuum representation, apart from the allowed zero momentum of the vacuum. It is stronger than bounding the Hamiltonian below in just one chosen frame: energy is nonnegative in every future-directed inertial frame. Thermal infinite-system representations need not satisfy the same vacuum spectrum condition, so the representation matters.

The condition supports stability and controls analyticity. For a translation-invariant vacuum, insertion of the spectral measure into a two-point function gives a formReplacing $x$ by $x-iy$, with $y$ in the future timelike cone, inserts the decaying factor $e^{-p\cdot y}$ and gives analytic continuation into the appropriate tube, subject to the usual distributional bounds. This is a consequence of spectral support, not an assertion of compact spatial support of $W$.

Positive energy is not a commutator condition or a guarantee of finite-speed particle-position propagation. It supplies none of the region-algebra inclusions used in primitive causality by itself. Its tension with sharp localization is precisely why the particle-localization discussion needs separate hypotheses. Spectrum, spacelike commutation and operational no-signalling constrain different structures and cannot simply substitute for one another.

Primitive causality is an initial-region completeness requirement. A state's full restriction to a suitable region $O$ determines expectations of observables in its future domain of dependence. The domain of dependence is not the whole causal future: it contains points whose past-inextendible causal curves all meet the specified initial data region. Merely being reachable by one causal curve from $O$ is not enough.

A standard local time-slice axiom makes this precise when $O$ contains a neighbourhood of a Cauchy slice of its causal development:with the regions and algebra identifications understood in the specified local net. Definitions applying primitive causality to all bounded open regions can be stronger than a time-slice property stated only for suitable causally convex Cauchy neighbourhoods; these formulations should not be conflated without extra hypotheses.

For a local hyperbolic Klein–Gordon equation, a later field is determined by the field and its normal derivative, or canonical momentum, on a Cauchy slice. Its value can be written using the causal fundamental solution, with integration supported where the past light cone meets the initial slice. If that support lies in the initial patch, the later smeared field is generated by the patch's initial field and momentum operators. Their algebra therefore generates the observables of the patch's domain of dependence. This is the local differential-equation mechanism behind the time-slice property.

The required information is the entire restricted quantum state: it assigns expectations to every observable and product in the initial algebra. Mean values of the initial field and momentum alone do not determine their fluctuations or higher correlations and thus are insufficient. Knowledge of the restricted state fixes later probability distributions and correlations; it does not fix which individual outcome a future measurement will produce.

Finally, primitive causality concerns propagation of information already encoded in initial data, while microcausality controls compatibility of spacelike measurements and quantum no-signalling controls possible interventions. The spectrum condition adds a separate positivity restriction. A causal initial-value evolution, positive energy and no-signalling are complementary requirements, not equivalent definitions of quantum locality.