Localization Problems

Relativistic one-particle localization is a tradeoff among positive energy, sharp position probabilities, Lorentz transformation properties, and causal propagation. A position operator can be constructed, and a wavefunction can be sharply localized on one time slice. The difficulty is retaining all the additional properties expected of a local relativistic measurement. This page treats a free massive spin-zero particle; it does not assert one unqualified localization theorem for every spin, detector, or observable.

Required background. Relativistic Phase Space supplies the invariant momentum norm, the Square-Root Hamiltonian supplies positive-energy evolution, and Causality and Light Cones distinguishes propagation support from operational signaling.

Position in an invariant momentum Hilbert space

Take $E_{\mathbf p}=\sqrt{c^2\mathbf p^2+m^2c^4}$ with $m>0$. Absorb constant Fourier normalization factors into $f$ and use the norm

$$\|f\|^2=\int\frac{d^3p}{2E_{\mathbf p}}|f(\mathbf p)|^2.$$

Multiplication by $p_i$ defines momentum. The naive expression $i\hbar\partial_{p_i}$ is not symmetric in this weighted measure: integration by parts also differentiates $1/(2E_{\mathbf p})$. Introduce a unitary map to flat momentum measure,

$$g(\mathbf p)=\frac{f(\mathbf p)}{\sqrt{2E_{\mathbf p}}}, \qquad \|g\|_{L^2(d^3p)}=\|f\|.$$

Transporting ordinary position through this map gives

$$\begin{aligned} X_{\mathrm{NW},i}f &=\sqrt{2E_{\mathbf p}}\,i\hbar\partial_{p_i} \left(\frac{f}{\sqrt{2E_{\mathbf p}}}\right)\\ &=i\hbar\left(\partial_{p_i} -\frac{c^2p_i}{2E_{\mathbf p}^2}\right)f. \end{aligned}$$

This is the massive spin-zero Newton–Wigner position operator in this normalization. Its self-adjoint realization is obtained by transporting the usual self-adjoint derivative domain in flat momentum space. On a common smooth core,

$$[X_{\mathrm{NW},i},P_j]=i\hbar\delta_{ij}, \qquad [X_{\mathrm{NW},i},X_{\mathrm{NW},j}]=0.$$

Thus the issue is not the absence of any canonical position operator. The choice of what its spectral projectors mean physically is essential.

A positive density on a chosen time slice

Fourier transforming $g$ defines

$$\psi_{\mathrm{NW}}(\mathbf x,t) =\int\frac{d^3p}{(2\pi\hbar)^{3/2}} g(\mathbf p)e^{i\mathbf p\cdot\mathbf x/\hbar-iE_{\mathbf p}t/\hbar}.$$

Its squared modulus integrates to one for a normalized state and defines the Newton–Wigner position distribution on that simultaneity surface. One can choose smooth compactly supported $\psi_{\mathrm{NW}}(\mathbf x,0)$. It is then a positive-energy state evolving under the square-root Hamiltonian.

The square-root kernel is nonlocal, so that compact support generally does not propagate within a finite light cone. This does not contradict the local KG equation: the corresponding scalar covariant amplitude includes the energy-dependent Fourier weight, and its two initial data are not arbitrary compactly supported functions.

Nor does $|\psi_{\mathrm{NW}}|^2$ transform as a local scalar or as the time component of a four-current. Boosts change momenta, energy weights, and the simultaneity surface. One can organize localization observables into Poincaré-covariant families indexed by spacelike hyperplanes, as discussed by Moretti. It is therefore too strong to say that Newton–Wigner localization has no covariant formulation at all. The sharper limitation is the combination of sharp localization and causal propagation of its probabilities.

What a positive-energy spreading theorem says

A useful form of Hegerfeldt’s result has explicit hypotheses. Let $H$ be self-adjoint and bounded below, let $A$ be a bounded positive operator, and let $\psi_t=e^{-iHt/\hbar}\psi$. Define

$$p_A(t)=\langle\psi_t,A\psi_t\rangle\geq0.$$

Then either $p_A(t)=0$ for all times, or it is positive for almost every time and on an open dense set. In particular, it cannot vanish throughout an open time interval and then become nonzero later. The result does not say it must be positive at every individual time.

The spectral lower bound makes matrix elements of $A^{1/2}e^{-iHz/\hbar}\psi$ analytic in a half-plane after shifting the lower energy bound. Vanishing boundary values on a time interval, together with the positivity identity $p_A(t)=\|A^{1/2}\psi_t\|^2$, force the relevant analytic functions to vanish identically. This is the mechanism behind the dichotomy; the cited paper gives the precise analytic argument.

If $A$ measures localization outside a region, the theorem constrains claims that a state remains exactly confined for a finite time and later escapes. To conclude that a particular distant region gains probability, one must also exclude the alternative that its probability is identically zero. The theorem alone supplies neither a detector model nor a local preparation operation. Interpreting it as a protocol for faster-than-light signaling would add assumptions that have not been established.

The reduced Compton scale is a crossover, not a universal bound

The uncertainty estimate $\Delta p\gtrsim\hbar/(2\Delta x)$ implies that confinement on scales comparable to $\hbar/(mc)$ probes momenta comparable to $mc$. The nonrelativistic expansion then ceases to be controlled. Interactions used to prepare or resolve such a packet can also open additional particle channels, depending on their kinematics and strength.

This reasoning does not prove $\Delta x\geq\hbar/(mc)$ for every state. The sharp one-particle localization construction above is a mathematical counterexample to that blanket statement. What breaks down is the simple identification of arbitrarily sharp one-particle position with an arbitrary local measurement in a relativistic interacting theory. An actual pair-production threshold depends on the complete process and momentum balance, not just on a proposed position uncertainty.

Exercises

1. Derive the correction to $i\hbar\partial_{p_i}$ by integrating by parts in the weight $w(\mathbf p)=1/(2E_{\mathbf p})$.

Solution

For compactly supported smooth functions, the adjoint of $\partial_i$ is $-\partial_i-\partial_i\log w$. Therefore $i\hbar(\partial_i+\tfrac12\partial_i\log w)$ is symmetric. Since $\partial_i\log w=-c^2p_i/E^2$, it is precisely the displayed Newton–Wigner operator. Self-adjointness follows from the unitary map with the specified domain, not from formal symmetry alone.

2. Why does Hegerfeldt’s result not by itself prove instantaneous influence on every distant detector?

Solution

It concerns expectation values of a chosen positive operator under a semibounded Hamiltonian. A detector may not implement a sharp one-particle position projector, the identically-zero alternative must be excluded, and the preparation/intervention must be shown to be local. None of these follows merely from the theorem’s spectral hypothesis.

3. Find the low-momentum size of the correction in $X_{\mathrm{NW},i}$. Does it shift the canonical commutator?

Solution

For $|\mathbf p|\ll mc$, the correction is $-i\hbar p_i/(2m^2c^2)$ with corrections of order $\hbar|\mathbf p|^3/(m^4c^4)$. It is a multiplication operator and commutes with every $P_j$, so $[X_i,P_j]=i\hbar\delta_{ij}$ remains exact.

References

• G. C. Hegerfeldt, “Causality, Particle Localization and Positivity of the Energy,” in Irreversibility and Causality in Quantum Theory—Semigroups and Rigged Hilbert Spaces, A. Bohm, H.-D. Doebner, and P. Kielanowski, eds., Lecture Notes in Physics 504, Springer, 1998, pp. 238–245, arXiv:quant-ph/9806036 — the positive-operator spreading dichotomy and its interpretation.
• V. Moretti, “On the Relativistic Spatial Localization for Massive Real Scalar Klein–Gordon Quantum Particles,” Letters in Mathematical Physics 113, 66, 2023, doi:10.1007/s11005-023-01689-5 — hyperplane-indexed localization and causal qualifications.
• T. D. Newton and E. P. Wigner, “Localized States for Elementary Systems,” Reviews of Modern Physics 21, 400–406, 1949, doi:10.1103/RevModPhys.21.400 — the original sharp-localization construction.