Can Quantum Physics Select a Here and Now?

Relativity does not abolish the local present. It abolishes the assumption that a local present automatically determines a global present throughout space.

An ordinary quantum measurement is often described as selecting one outcome, or one branch, from a superposition. But could relativistic quantum physics contain a more ambitious operation—one that selects not only what happens, but also a spacetime “here and now” in which it happens?

The short answer is that relativistic quantum theory contains several structures that come close, but no universally accepted sharp projection operator that performs both tasks. The closest operational object is a spacetime-indexed quantum instrument: a family of quantum operations whose outcomes may include both a detector result and a spacetime event. Objective-collapse theories can go further and treat such events as elements of physical reality, but they add structure beyond standard quantum field theory.

1. What an ordinary projection selects

Suppose that an observable has mutually orthogonal spectral projections \(P_a\). If the system is in the state \(\rho\), the probability of outcome \(a\) is

\[ p(a\mid\rho)=\operatorname{Tr}(\rho P_a). \]

Conditioned on obtaining that outcome, the Lüders update is

\[ \rho\longmapsto \rho_a= \frac{P_a\rho P_a} {\operatorname{Tr}(\rho P_a)}. \]

This selects an eigenspace—or, in phenomenological language, a branch associated with the recorded outcome. But the projector does not normally determine where or when the measurement occurs. Its spacetime setting is supplied separately by the detector, its motion, its switching function, and its coupling to the measured system.

Thus an ordinary projection answers:

Which outcome occurred, given that this measurement was performed?

It does not by itself answer:

At which spacetime event did an outcome become actual?

2. Relativity permits a local present, but not a universal one

Special relativity does not assign an invariant global present to the universe. Two spatially separated events that are simultaneous in one inertial frame need not be simultaneous in another. A projection onto “the state of everything now” would therefore require a chosen spacelike hypersurface or foliation.

A local event \(x\), however, is perfectly well defined. Observers may assign different coordinates to it, but they agree that it occurred, and they agree on its causal relations to other events. Likewise, an observer travelling on a worldline

\[ x=\gamma(\tau) \]

has an invariant proper time \(\tau\). The event \(\gamma(\tau_0)\) can provide a completely relativistic local meaning of “here and now.”

The tension is therefore not between relativity and a local present. It is between relativity and the promotion of one local present into a global simultaneity relation.

3. Local observables in algebraic quantum field theory

Algebraic quantum field theory begins not with a global position operator, but with a net of local observable algebras,

\[ \mathcal O\longmapsto\mathcal A(\mathcal O), \]

where \(\mathcal O\) is a bounded spacetime region and \(\mathcal A(\mathcal O)\) contains observables localizable in that region.

The net satisfies isotony:

\[ \mathcal O_1\subseteq\mathcal O_2 \quad\Longrightarrow\quad \mathcal A(\mathcal O_1)\subseteq\mathcal A(\mathcal O_2), \]

and, under the usual microcausality condition, observables in spacelike separated regions commute:

\[ [A,B]=0, \qquad A\in\mathcal A(\mathcal O_1),\; B\in\mathcal A(\mathcal O_2), \]

whenever \(\mathcal O_1\) and \(\mathcal O_2\) are spacelike separated.

A local measurement outcome may be represented by an effect

\[ 0\leq E_{a,\mathcal O}\leq\mathbf 1, \qquad E_{a,\mathcal O}\in\mathcal A(\mathcal O), \]

or, in an idealized sharp case, by a projection \(P_{a,\mathcal O}\). For a measurement performed in a fixed region \(\mathcal O\),

\[ p_{\mathcal O}(a\mid\rho) = \operatorname{Tr}(\rho E_{a,\mathcal O}). \]

The notation \(p_{\mathcal O}(a\mid\rho)\) is important. The region \(\mathcal O\) is part of the experimental arrangement, not an outcome selected by the operation. The pair

\[ (\mathcal O,P_{a,\mathcal O}) \]

says: “outcome \(a\), in a measurement whose coupling region was \(\mathcal O\).” It does not say that the theory randomly selected \(\mathcal O\) as the location of the event.

4. From projections to quantum instruments

A projection records less information than a physical measurement process. A more general description uses completely positive quantum operations,

\[ \mathcal I_{a,\mathcal O}:\rho\longmapsto \mathcal I_{a,\mathcal O}(\rho). \]

For a fixed coupling region \(\mathcal O\), the outcome-indexed family

\[ \{\mathcal I_{a,\mathcal O}\}_a \]

is the instrument; each \(\mathcal I_{a,\mathcal O}\) is an individual operation. It supplies both the probability

\[ p_{\mathcal O}(a\mid\rho) = \operatorname{Tr} \bigl[\mathcal I_{a,\mathcal O}(\rho)\bigr] \]

and the conditional state

\[ \rho_{a,\mathcal O} = \frac{\mathcal I_{a,\mathcal O}(\rho)} {p_{\mathcal O}(a\mid\rho)}. \]

The associated effect is

\[ E_{a,\mathcal O} = \mathcal I_{a,\mathcal O}^{*}(\mathbf 1). \]

In the abstract algebraic formulation, the trace notation can be replaced by the action \(\omega(E_{a,\mathcal O})\) of a state functional \(\omega\).

This generalization is not merely a concession to imperfect detectors. It is also required by relativistic causality.

5. Why a local projection is not automatically a local measurement

If \(P\in\mathcal A(\mathcal O)\) and \(C\) belongs to a spacelike separated algebra, microcausality gives \([P,C]=0\). A single non-selective Lüders operation then leaves \(C\) unchanged:

\[ PCP+(\mathbf 1-P)C(\mathbf 1-P)=C. \]

Therefore it would be incorrect to say that every local projection directly produces superluminal signalling.

Microcausality does not, however, guarantee that arbitrary sequences of textbook state reductions remain causal. In a suitable three-region arrangement, an intervention in one region, followed by an ideal measurement in an intermediate extended region, can alter later statistics in a third region even when the first and third regions are spacelike separated. This is the problem raised by Rafael Sorkin’s Impossible Measurements on Quantum Fields.

Sorkin’s explicit field projector was not itself an operation implementable by apparatus confined to a bounded local laboratory. That qualification matters: membership in \(\mathcal A(\mathcal O)\) identifies where an observable is localizable, but does not by itself prove that its Lüders update can be physically realized by an apparatus confined there. Under explicit assumptions, Albertini and Jubb have obtained a stronger causality obstruction for ideal measurements of compactly smeared real scalar fields; see Are Ideal Measurements of Real Scalar Fields Causal?.

The Fewster–Verch framework approaches the problem differently. A system field is coupled to a probe field inside a bounded spacetime region, and the resulting operation is derived from that local dynamics. Causal factorization then ensures that operations in causally disjoint regions compose consistently. Bostelmann, Fewster and Ruep show that the Sorkin-type “impossible measurement” would require correspondingly non-local apparatus; see Impossible Measurements Require Impossible Apparatus.

The move from an abstract projector to a locally generated instrument is therefore principled: an algebraically local observable and a physically local measurement operation are not the same thing.

6. Letting the spacetime event become an outcome

If the event’s position is not fixed in advance, its spacetime coordinate can itself be treated as a random measurement outcome. Let

\[ E_a(\Delta) \]

be the effect associated with outcome \(a\) occurring in a spacetime region \(\Delta\). If the measure admits a density, one may write

\[ E_a(\Delta) = \int_{\Delta}E_a(d^4x). \]

The joint probability is then

\[ p(a,x\in\Delta\mid\rho) = \operatorname{Tr}\!\left[\rho E_a(\Delta)\right]. \]

Schematically, Poincaré covariance requires

\[ U(g)E_a(\Delta)U(g)^\dagger = E_{g\cdot a}(g\Delta), \]

where \(g\cdot a\) denotes the induced action of the Poincaré transformation \(g\) on the outcome space. For a scalar detector outcome this action is trivial, so the right-hand side reduces to \(E_a(g\Delta)\).

This is a spacetime POVM rather than, in general, a four-dimensional projection-valued position measure. Toller studied Poincaré-covariant POVMs for quantum events in Localization of Events in Space-Time. Anastopoulos and Savvidou likewise construct relativistic measurement probabilities while treating the spacetime coordinates of detection events as random variables in Measurements on Relativistic Quantum Fields: I. Probability Assignment.

The no-detection outcome

A detector need not fire. For a spacetime observation window \(W\), the detected-event effects should therefore be completed by a no-detection effect:

\[ \sum_a\int_W E_a(d^4x)+E_{\varnothing} = \mathbf 1, \qquad E_{\varnothing}\geq0. \]

Equivalently,

\[ \sum_a\int_W E_a(d^4x)\leq\mathbf 1. \]

The detected outcomes fail to sum to the identity precisely when \(E_{\varnothing}\neq0\). If detection is guaranteed—or if no-click has already been included among the outcome labels—the relevant outcomes may sum to \(\mathbf 1\).

Thus a spacetime event can be contingent: the detector may register an event at \(x\), or it may register no event at all. The no-click effect does not, by itself, prove that the observable cannot be projection-valued; a PVM could also contain a no-click projection. The deeper reasons for using POVMs arise from covariance, causal localization, positive-energy conditions, and the finite resolution of physically realizable detectors.

7. The closest single object: a spacetime instrument

A spacetime POVM supplies probabilities but not the conditional state change. The two structures can be combined into an operation-valued measure

\[ \Delta\longmapsto\mathcal I_a(\Delta). \]

It assigns a completely positive operation to the outcome “\(a\) occurred somewhere in \(\Delta\).” It determines both

\[ p(a,x\in\Delta\mid\rho) = \operatorname{Tr} \bigl[\mathcal I_a(\Delta)(\rho)\bigr] \]

and the corresponding conditional state

\[ \rho_{a,\Delta} = \frac{\mathcal I_a(\Delta)(\rho)} {p(a,x\in\Delta\mid\rho)}. \]

Its associated spacetime POVM is

\[ E_a(\Delta) = \mathcal I_a(\Delta)^*(\mathbf 1). \]

This is arguably the closest operational answer to the original question. It can select both an outcome and a spacetime region. But it is generally an instrument, not a sharp projector, and the spacetime coordinate refers to a detector event rather than to a universal metaphysical present.

8. An observer’s local here and now

For a detector travelling along \(x=\gamma(\tau)\), the measurement may instead be indexed by the detector’s proper time:

\[ p(a,\tau\in J\mid\rho) = \operatorname{Tr}\!\left[\rho F_a(J)\right], \]

where \(F_a(J)\) is the effect for outcome \(a\) during a proper-time interval \(J\). A real detector occupies a narrow world-tube rather than a mathematical worldline, so both temporal and spatial resolution remain finite.

A record at \(\tau_0\) identifies the invariant local event

\[ x_0=\gamma(\tau_0). \]

This is a relativistically legitimate “here and now.” It does not generate an observer-independent plane of simultaneous events across the universe. Different observers can agree on \(x_0\) while associating it with different distant events under their respective simultaneity conventions.

9. Why not an exact spacetime projection?

Several distinct obstructions stand between the intuitive idea and a universal sharp spacetime projector.

Fields are distributions

Quantum fields such as \(\phi(x)\) are operator-valued distributions, not ordinary operators defined at mathematical points. Well-defined observables are obtained by smearing:

\[ \phi(f) = \int d^4x\,f(x)\phi(x), \]

where \(f\) has finite spacetime support. This already replaces an exact point by a region and a detector profile.

Newton–Wigner localization is hypersurface-relative

The Newton–Wigner construction assigns a sharp spatial PVM

\[ Q_{n,t}(\Delta) \]

to a spatial region \(\Delta\) on a rest-space hypersurface \(\Sigma_{n,t}\). The full hypersurface-indexed family can be Poincaré covariant when the state, the region, and the hypersurface are transformed together. It is therefore too crude simply to say that Newton–Wigner localization “is not covariant.”

This covariance does not turn \(Q_{n,t}\) into a single four-dimensional event observable, nor does it remove the causal problem. Under positive-energy assumptions, Hegerfeldt’s theorem implies instantaneous spreading: a state sharply localized in a bounded region develops non-zero tails arbitrarily far away at later times. See Instantaneous Spreading and Einstein Causality in Quantum Theory.

Malament’s theorem reaches a related no-go result. Under assumptions combining sharp localizability, translation covariance, energy bounded below, and a relativistic locality condition, all bounded sharp localization projections become trivial. A useful discussion is given by Halvorson and Clifton in No Place for Particles in Relativistic Quantum Theories?.

Unsharp localization can avoid some of these conclusions. Moretti, for example, studies a Poincaré-covariant family of localization POVMs that approximates Newton–Wigner localization without admitting perfectly sharply localized states: On the Relativistic Spatial Localization for Massive Real Scalar Klein–Gordon Quantum Particles. Busch has likewise shown that relativistic locality can force localization observables to be strongly unsharp under suitable assumptions; see Unsharp Localization and Causality in Relativistic Quantum Theory.

Local QFT algebras are not ordinary subsystem algebras

In standard algebraic quantum field theories, local von Neumann algebras are typically of type III. They contain many projections, but no minimal projections corresponding to one-dimensional local pure states. Nor does the global Hilbert space generally factorize in the naive form

\[ \mathcal H = \mathcal H_{\mathcal O} \otimes \mathcal H_{\mathcal O'}, \]

with a separate finite subsystem for every sharply bounded region.

Consequently, one should not imagine a local projector as a rank-one operator that isolates a complete microscopic state of a spacetime cell. Local effects and operations exist, but their structure is subtler than the subsystem picture inherited from non-relativistic laboratory quantum mechanics. For an accessible review of the role and physical consequences of type III factors in relativistic quantum field theory, see Jakob Yngvason, The Role of Type III Factors in Quantum Field Theory.

10. Does the event become objectively actual?

An instrument tells us how to calculate probabilities and conditional states. It does not, by itself, decide whether the update represents a physical collapse, a change in information, a relative state, or the formation of a decohered record.

In standard relativistic quantum field theory, the statement that outcome \(a\) occurred at \(x\) normally refers to a detector record. Whether that record selects one objectively unique branch is an interpretive question not settled by the instrument formalism alone.

Objective-collapse theories add a stochastic process intended to settle this question dynamically. The closest example to a literal ontology of selected spacetime points is Tumulka’s relativistic GRW flash model. In its flash ontology, matter is represented by a discrete and comparatively sparse set of spacetime events—the centres of spontaneous collapses—rather than by continuous particle trajectories. Macroscopic objects correspond to dense patterns, or “galaxies,” of flashes.

The original model treated \(N\) non-interacting distinguishable particles; see A Relativistic Version of the Ghirardi–Rimini–Weber Model. A later construction allows interactions between \(N\) distinguishable particles, assuming a given interaction-local Tomonaga–Schwinger evolution: A Relativistic GRW Flash Process With Interaction.

The latter model still does not cover indistinguishable particles or variable particle number, and is therefore not a general relativistic quantum field theory of known matter. Its dynamics can also be paired with a matter-density ontology, so the sparse event ontology is a feature of the specifically “flash” interpretation rather than of every possible presentation of the model.

The flashes are nevertheless conceptually striking: they are not merely places where an observer happens to look. They are postulated physical events. This makes rGRWf much closer to an actual “selection of a here and now” than standard operational quantum field theory, at the price of modifying the usual dynamics.

11. Can local descriptions be glued into a global one?

The net

\[ \mathcal O\longmapsto\mathcal A(\mathcal O) \]

suggests a further local-to-global question. Calling the net “presheaf-like” requires some care. The observable algebras map covariantly under inclusions,

\[ \mathcal O_1\subseteq\mathcal O_2 \quad\Longrightarrow\quad \mathcal A(\mathcal O_1)\longrightarrow \mathcal A(\mathcal O_2), \]

whereas states on the larger algebra restrict contravariantly:

\[ \omega_{\mathcal O_2} \longmapsto \omega_{\mathcal O_2}\!\mid_{\mathcal A(\mathcal O_1)}. \]

In this limited sense, observables assemble outward while state descriptions restrict inward. But isotony alone does not make the observable net a cosheaf, and ordinary additivity is weaker than a full descent theorem.

Brunetti, Fredenhagen and Verch extend this local perspective across different spacetimes by formulating a locally covariant quantum field theory as a covariant functor from a category of globally hyperbolic spacetimes to a category of algebras; see The Generally Covariant Locality Principle. This functorial formulation does not by itself supply a descent theorem, but it provides a natural bridge from locality on a fixed spacetime to the question of how theories and observables behave under restriction, embedding and reconstruction.

Suppose that several regions \(\mathcal O_i\) carry local states \(\omega_i\), and that the states agree wherever the regions overlap. Must there be a global state on

\[ \mathcal A\!\left(\bigcup_i\mathcal O_i\right) \]

whose restrictions reproduce all the \(\omega_i\)?

Not in general. Pairwise agreement on overlaps may fail to supply all the compatibility conditions required for a global extension. Even when an extension exists, it need not be unique, because local restrictions do not necessarily determine correlations between separated regions.

This gives a precise form to a broader question:

Under what additional compatibility and correlation conditions can many local quantum descriptions belong to one coherent global description?

A related—but distinct—modern programme formulates descent conditions for entire Haag–Kastler theories on open covers of spacetime. See Benini, Grant-Stuart and Schenkel, Haag–Kastler Stacks. The distinction is important: gluing local states is not identical to reconstructing a complete quantum field theory from local theories.

Conclusion

Relativistic quantum physics does not contain one canonical projection operator that simultaneously chooses a wavefunction branch and creates a universal spacetime present.

What it offers instead is a hierarchy of increasingly rich structures:

A local projection or effect selects an outcome for a measurement whose spacetime region has already been specified.
A local quantum instrument describes both outcome probabilities and conditional state changes.
A spacetime POVM allows the detector event’s position and time to become random outcomes.
A spacetime-indexed instrument combines the event distribution with the corresponding conditional state update.
An observer’s worldline and proper time provide an invariant local meaning of “here and now.”
Objective-collapse theories may add actual spacetime events, but this goes beyond standard relativistic quantum field theory.

The nearest operational analogue of the proposed “spacetime projection” is therefore not a projector at all. It is a causally realizable, spacetime-indexed quantum instrument.

Such an instrument can say:

This outcome was recorded in this spacetime region.

It cannot, without further interpretive or dynamical assumptions, say:

This event selected the one globally real present for the entire universe.

The deeper problem may therefore not be how to project the global state onto one universal “now,” but how local events, local states, and their correlations can form a coherent global description without introducing a preferred simultaneity structure.

The relativistic lesson is not that reality has no here and now. It is that every operationally identifiable here and now is local—and that constructing a coherent whole from many such events is a further physical and mathematical problem.

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Gustaf Ullman — Perspectives on Physics and Philosophy