Causality, Support, and Interpretation in Nonrelativistic QM

The nonrelativistic free-particle kernel on $\mathbb R^d$ is

K_0^{(d)}(\mathbf x,T;\mathbf x',0) = \left( \frac{m}{2\pi i\hbar T} \right)^{d/2} \exp\left[ \frac{im\lvert\mathbf x-\mathbf x'\rvert^2} {2\hbar T} \right]

for $T\gt0$ , with the usual real-time convergence prescription. It is nonzero at every finite separation $\lvert\mathbf x-\mathbf x'\rvert$ , however large, for every positive time, however small.

That fact is sometimes described as instantaneous spreading or infinite propagation speed. It is a real mathematical property of Schrödinger evolution. It is not a hidden relativistic light cone, and it should not be disguised by calling the tails exactly zero. But several further claims do not follow from the kernel alone:

$K$ is not a normalized arrival-position wavefunction;
a nonzero amplitude is not by itself an operational signaling protocol;
relativistic QFT does not require every two-point function to vanish at spacelike separation;
a time-retarded Green function need not have finite spatial propagation speed.

Keeping those distinctions separate prevents most apparent paradoxes.

Nonrelativistic Propagation Has Different Causal Structure

The free Schrödinger equation is

i\hbar\frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2m}\nabla^2\psi.

Its plane-wave dispersion relation is

E(\mathbf p)=\frac{\mathbf p^2}{2m}, \qquad \omega(\mathbf k)=\frac{\hbar\mathbf k^2}{2m},

so the group velocity is

\mathbf v_g = \nabla_{\mathbf k}\omega = \frac{\hbar\mathbf k}{m} = \frac{\mathbf p}{m}.

The model permits arbitrarily large momentum and therefore has no finite upper bound on group velocity. Its spacetime symmetry is Galilean, not Lorentzian. There is no invariant speed $c$ in the Schrödinger equation and no division of separations into timelike, null, and spacelike classes.

This differs from a relativistic hyperbolic equation such as the Klein–Gordon equation,

\left( \frac{1}{c^2}\frac{\partial^2}{\partial t^2} -\nabla^2 +\frac{m^2c^2}{\hbar^2} \right) \phi=0,

whose retarded fundamental solution has support only on and inside the future light cone. The contrast is about the differential equation and the object being solved for, not merely about replacing $p^2/(2m)$ by a relativistic energy formula.

Support of the Free Kernel

The support of a function or distribution is the closure of the region where it does not vanish. For fixed $T\gt0$ ,

\operatorname{supp}_{\mathbf x} K_0^{(d)}(\mathbf x,T;\mathbf x',0) = \mathbb R^d.

Its magnitude is independent of separation:

\left\lvert K_0^{(d)}(\mathbf x,T;\mathbf x',0) \right\rvert = \left( \frac{m}{2\pi\hbar T} \right)^{d/2}.

This does not say that every distant detector has the same probability to click. The ideal position ket $\lvert\mathbf x'\rangle$ is not normalizable, and

\left\lvert K_0^{(d)}\right\rvert^2

is not a normalized transition-probability density for a particle prepared at an exact point. Physical predictions require a normalizable initial wave packet, a detector model, and a specified measurement.

The kernel’s oscillatory phase is essential. When it is integrated against a wavefunction, distant contributions can be extremely small through destructive interference even though they are not exactly zero.

Instantaneous Tails from Compact Support

Suppose $\psi_0(\mathbf x')$ is nonzero, integrable, and supported in a bounded region $\Omega$ . Free evolution gives

\psi(\mathbf x,T) = \int_\Omega d^dx'\, K_0^{(d)}(\mathbf x,T;\mathbf x',0) \psi_0(\mathbf x').

Expanding the square in the kernel phase yields

\begin{aligned} \psi(\mathbf x,T) &= C_T e^{im\lvert\mathbf x\rvert^2/(2\hbar T)} F_T\left( \frac{m\mathbf x}{\hbar T} \right), \end{aligned}

where

C_T = \left( \frac{m}{2\pi i\hbar T} \right)^{d/2},

and

\begin{aligned} F_T(\mathbf k) &= \int_\Omega d^dx'\, e^{-i\mathbf k\cdot\mathbf x'} e^{im\lvert\mathbf x'\rvert^2/(2\hbar T)} \psi_0(\mathbf x'). \end{aligned}

Because the integrand has bounded spatial support, $F_T$ extends to an entire function of complex $\mathbf k$ . If $\psi(\mathbf x,T)$ vanished on a nonempty open spatial region, analytic uniqueness would force $F_T$ to vanish identically, which would force $\psi_0=0$ .

Therefore a nonzero compactly supported initial wavefunction cannot remain zero on any open spatial region after free evolution through a nonzero time. It may have isolated nodes or nodal surfaces, but its support becomes all of $\mathbb R^d$ .

This argument is stronger than observing that the kernel itself is nonzero. It shows how strict localization is lost after convolution with a physical, normalizable state.

Momentum-Space Interpretation

A compactly supported wavefunction cannot also have compact momentum support unless it is identically zero. Its Fourier transform contains arbitrarily large momenta, although those components may be strongly suppressed for a smooth initial state.

In the nonrelativistic dispersion relation,

\mathbf v_g=\frac{\mathbf p}{m},

so those high-momentum components have arbitrarily large formal group velocities. This offers useful intuition for rapid tails, but it is not a classical decomposition into tiny particles traveling on definite trajectories. Instantaneous spreading is a property of coherent unitary evolution and Fourier analyticity.

Imposing a strict momentum cutoff would bound $\lvert\mathbf v_g\rvert$ , but then the wavefunction would be bandlimited and could not have strict compact spatial support. Exact localization and an exact momentum cutoff are incompatible.

Beyond the Explicit Free Kernel

The free-kernel calculation is the simplest example. A broader result, commonly associated with Hegerfeldt, uses a Hamiltonian bounded below together with a positive localization operator. Under the theorem’s assumptions, a state that is strictly localized cannot remain confined to a causally expanding bounded region for a finite time interval: the probability outside becomes nonzero immediately, except in the alternative where it vanishes for all times.

The precise theorem is conditional. Its interpretation depends on:

the localization observable being used;
positivity and regularity assumptions;
what counts as strict localization;
whether a single-particle description is physically adequate.

The result is especially important when one tries to build relativistic positive-energy single-particle localization. It helps explain why relativistic locality is formulated in terms of local fields and observable algebras rather than a naive relativistic position wavefunction alone.

Why This Is Not an Internal Contradiction

Nonrelativistic quantum mechanics does not postulate a universal finite signal speed. Instantaneous tails therefore do not contradict one of its own axioms. They instead mark a limitation when the theory is compared with special relativity.

The nonrelativistic approximation is normally used when

\lvert\mathbf p\rvert\ll mc,

and kinetic and interaction energies are small compared with $mc^2$ . Arbitrarily sharp localization requires high-momentum components that eventually leave this regime. Extrapolating the Schrödinger model to arbitrarily large distance divided by arbitrarily small time asks it to answer outside the scale hierarchy that justifies it.

This does not make the tails mathematically fictitious. It means the low-energy theory should not be promoted to an exact fundamental account of relativistic communication.

Tail amplitude is not yet a signal

A nonzero kernel element specifies an amplitude. To demonstrate signaling one must specify at least:

a controllable operation at a source region;
a detector observable in a destination region;
two source choices that change detector statistics;
timing and localization assumptions;
the dynamical theory connecting those operations.

The kernel alone supplies none of that operational structure. Conversely, one should not conclude that nonrelativistic theory secretly enforces exact relativistic no-signaling. It does not contain microcausality. The correct statement is narrower: instantaneous support is not itself a complete signaling argument.

Entanglement no-signaling is a different statement

In a tensor-product model, local trace-preserving operations on subsystem $A$ cannot change subsystem $B$ ‘s reduced state without communication. That quantum-information no-signaling result does not imply finite-speed spatial propagation. It concerns subsystem operations and marginals, while instantaneous spreading concerns the support generated by a spatial Hamiltonian.

Conflating the two uses of “no-signaling” obscures both.

Temporally Retarded Does Not Mean Light-Cone Supported

For the Schrödinger operator, a retarded Green function can be written as

G_{mathrm R}^{\mathrm{NR}} (\mathbf x,t;\mathbf x',t') = -\frac{i}{\hbar} \Theta(t-t') K(\mathbf x,t;\mathbf x',t').

It vanishes for $t\lt t'$ , so the response does not precede the source in the chosen absolute time coordinate. For a free particle, however, it is nonzero at every spatial separation when $t\gt t'$ .

Thus “retarded” can mean only future in time, while relativistic causal support means inside the future light cone. The second condition is stronger.

The distinction is developed at the inverse-kernel level in Retarded and Advanced Green Functions and at the many-body commutator-response level in Retarded and Advanced Response.

Relativistic Locality Preview

In local relativistic QFT, a standard bosonic microcausality condition is

[\phi(x),\phi(y)]=0

when

(x-y)^2\lt0.

Spacelike-separated local observables are therefore compatible. Fermionic fields use spacelike anticommutation relations, while observable quantities built from them obey the appropriate locality condition.

For a scalar field, define the commutator distribution schematically by

C(x-y) = \langle0\rvert [\phi(x),\phi(y)] \lvert0\rangle.

Microcausality gives

C(x-y)=0 \qquad \text{for spacelike }x-y.

A retarded response function is proportional to

\Theta(x^0-y^0)C(x-y),

so its support is restricted to the future light cone in a local relativistic theory.

This is the QFT object most directly tied to causal influence. It is not the same object as a one-particle position kernel.

QFT Propagator Comparison

The word “propagator” covers several inequivalent objects. Their support properties must be compared definition by definition.

Object	Schematic definition	Spacelike behavior	Main role
nonrelativistic kernel	$\langle\mathbf x\rvert e^{-iHT/\hbar}\lvert\mathbf x'\rangle$	generally nonzero at every separation for $T\gt0$	evolve one-particle wavefunctions
nonrelativistic retarded Green function	$\Theta(T)K$	time-retarded but spatially noncompact	solve a driven Schrödinger equation
Pauli–Jordan or commutator distribution	$\langle[\phi(x),\phi(y)]\rangle$	zero for spacelike separation in local QFT	express microcausality
relativistic retarded Green function	$\Theta(x^0-y^0)\langle[\phi(x),\phi(y)]\rangle$	supported in the future light cone	causal response
Feynman propagator	$\langle0\rvert\mathcal T\phi(x)\phi(y)\lvert0\rangle$	generally nonzero at spacelike separation	time-ordered perturbation theory
Wightman function	$\langle0\rvert\phi(x)\phi(y)\lvert0\rangle$	generally nonzero at spacelike separation	vacuum correlation and spectral structure

The nonvanishing of a Feynman or Wightman two-point function outside the light cone is not a violation of microcausality. Vacuum correlations need not vanish at spacelike separation. Causal response is controlled by commutators of local observables, not by demanding that every correlator be zero.

Likewise, “virtual particles travel faster than light” is not a sound interpretation of a spacelike Feynman propagator. The Feynman propagator is a time-ordered field correlation used inside amplitudes, not a measured particle trajectory or a retarded signal.

The broader dictionary is From QM Propagators to QFT Propagators.

Boundaries, Lattices, and Other Qualifications

Hard boundaries can restrict propagation to an allowed domain, but they do not create a relativistic light cone within a connected Schrödinger region. The appropriate image or spectral kernel generally develops nonlocal support across the domain rather than a sharp finite-speed front; symmetries, nodes, and special revival times can still produce exact zeros. Disconnected components remain dynamically isolated only if the Hamiltonian contains no coupling between them. See Propagators and Boundary Conditions.

On a quantum lattice with finite-range interactions, a Lieb–Robinson bound produces an effective light cone: commutators outside it are exponentially suppressed, not generally exactly zero. That is a different mechanism from Lorentzian microcausality and a different setting from the continuum free-particle kernel. See Lieb–Robinson Bound.

Common Mistakes

Saying the free Schrödinger kernel vanishes beyond a finite propagation radius.
Squaring $K(\mathbf x,T;\mathbf x',0)$ and treating the result as a normalized detector probability for an exact position state.
Claiming that a small tail is exactly zero because it is experimentally negligible.
Treating the group-velocity explanation as a classical mixture of definite particle paths.
Assuming a strict momentum cutoff and strict spatial localization can hold simultaneously.
Calling a temporally retarded Schrödinger Green function light-cone supported.
Treating instantaneous spreading as the same phenomenon as entanglement or measurement collapse.
Claiming that a nonzero amplitude alone proves an operational faster-than-light signal.
Claiming that nonrelativistic mechanics satisfies exact relativistic microcausality.
Expecting the QFT Feynman propagator or Wightman function to vanish at spacelike separation.
Interpreting a spacelike Feynman correlator as a virtual particle trajectory.
Applying a positive-energy single-particle localization argument without stating the localization observable and theorem assumptions.

Cross-Links

References

G. C. Hegerfeldt, “Instantaneous spreading and Einstein causality in quantum theory,” Annalen der Physik 7, 716–725 (1998), arXiv:quant-ph/9809030.
G. C. Hegerfeldt, “Causality, particle localization and positivity of the energy,” in Irreversibility and Causality, Lecture Notes in Physics 504, 238–245, Springer, 1998, arXiv:quant-ph/9806036.
M. Reed and B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness, Academic Press, 1975.
B. Thaller, The Dirac Equation, Springer, 1992.
R. Haag, Local Quantum Physics, 2nd ed., Springer, 1996.
R. F. Streater and A. S. Wightman, PCT, Spin and Statistics, and All That, Princeton University Press, 2000.
S. Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press, 1995.

Exercises

Show that the free kernel has full spatial support but that its modulus squared is not a normalized probability density.

Solution

For $T\gt0$ ,

\left\lvert K_0^{(d)}(\mathbf x,T;\mathbf x',0) \right\rvert^2 = \left( \frac{m}{2\pi\hbar T} \right)^d,

which is positive and independent of $\mathbf x-\mathbf x'$ . Thus the kernel is nonzero at every finite separation.

However,

\int_{\mathbb R^d}d^dx\, \left\lvert K_0^{(d)}\right\rvert^2 = \infty.

The initial position ket is delta-normalized rather than normalizable, so this square is not a physical position probability density. One must first convolve $K_0^{(d)}$ with a normalizable initial wavefunction.

Complete the compact-support argument showing that a nonzero free wavefunction cannot vanish on an open region at $T\ne0$ .

Solution

For bounded initial support $\Omega$ ,

\begin{aligned} F_T(\mathbf k) &= \int_\Omega d^dx'\, e^{-i\mathbf k\cdot\mathbf x'} e^{im\lvert\mathbf x'\rvert^2/(2\hbar T)} \psi_0(\mathbf x') \end{aligned}

is the Fourier transform of a compactly supported integrable function. It extends to an entire function of complex $\mathbf k$ .

The evolved state is a nonzero phase and normalization factor times

F_T\left( \frac{m\mathbf x}{\hbar T} \right).

If $\psi(\mathbf x,T)$ vanished on an open set, then $F_T$ would vanish on an open set. Analytic uniqueness would imply $F_T=0$ everywhere. Fourier-transform uniqueness would then imply $\psi_0=0$ , contrary to the assumption. Hence a nonzero state cannot retain an open region of exact zero amplitude.

Explain why bounding the nonrelativistic group velocity with a strict momentum cutoff does not preserve strict spatial localization.

Solution

\widetilde\psi(\mathbf p)=0 \qquad \text{for }\lvert\mathbf p\rvert\gt p_{\max},

then

\lvert\mathbf v_g\rvert \le \frac{p_{\max}}{m}.

But the inverse Fourier transform of a compactly supported momentum function is analytic in position. A nonzero analytic function cannot vanish on an open exterior region, so it cannot have compact spatial support. The cutoff trades strict localization for band limitation; it does not provide both simultaneously.

Compare the support of a nonrelativistic retarded Green function with a relativistic retarded Green function.

Solution

For Schrödinger evolution,

G_{\mathrm R}^{\mathrm{NR}} = -\frac{i}{\hbar}\Theta(t-t')K.

It is zero for $t\lt t'$ , but the free kernel $K$ is nonzero at every spatial separation for $t\gt t'$ .

For a local relativistic scalar theory, the retarded function is proportional to

\Theta(x^0-y^0) \langle0\rvert[\phi(x),\phi(y)]\lvert0\rangle.

Microcausality makes the commutator vanish at spacelike separation. The relativistic retarded function is therefore supported only on and inside the future light cone. Temporal retardation and light-cone support are distinct conditions.

Why can a Feynman propagator be nonzero at spacelike separation without enabling a spacelike signal?

Solution

The Feynman propagator is a time-ordered vacuum correlation function:

D_F(x-y) = \langle0\rvert \mathcal T\phi(x)\phi(y) \lvert0\rangle.

Correlations need not vanish at spacelike separation. A controllable response to a local source is governed instead by a retarded commutator. In a local relativistic theory,

[\phi(x),\phi(y)]=0

for spacelike-separated points, so local operations cannot use the Feynman correlation as a faster-than-light response channel. Nonzero correlation and causal influence are different statements.