Locality and Causality Warnings

A relativistic kernel can be nonzero outside the light cone without carrying a signal there. The missing information is what the kernel represents: a positive-frequency correlation, a time-ordered correlation, or a response to a local source. This worked free-scalar example applies the causal principles owned by Causality and Light Cones. It assumes the standard free-field vacuum correlations as a bridge input; it does not derive field quantization from a one-particle wave equation.

Required background. Causality and Light Cones distinguishes signals from correlations, Relativistic Phase Space supplies the invariant measure, and the Klein–Gordon Equation supplies the wave operator.

A positive-frequency kernel at spacelike separation

Use $c=\hbar=1$ throughout this page. Define

$$W(x)=\int\frac{d^3p}{(2\pi)^3\,2E_{\mathbf p}} e^{-iE_{\mathbf p}t+i\mathbf p\cdot\mathbf x}, \qquad E_{\mathbf p}=\sqrt{\mathbf p^2+m^2}.$$

In the standard real free scalar theory this is $\langle0|\widehat\phi(x)\widehat\phi(0)|0\rangle$. As an on-shell distribution it solves $(\Box+m^2)W=0$. It is not a fundamental solution with a delta-function source.

For a spacelike separation, a proper orthochronous transformation takes $x$ to an equal-time separation of length $r=\sqrt{-x^2}>0$. At equal time, angular integration gives

$$W(0,r)=\frac{1}{4\pi^2r} \int_0^\infty dp\,\frac{p\sin(pr)}{\sqrt{p^2+m^2}} =\frac{mK_1(mr)}{4\pi^2r}.$$

The oscillatory integral is understood as a regulated distributional limit. It follows by differentiating the cosine-transform identity for $K_0$; the NIST DLMF gives the relevant modified-Bessel representations. For $r>0$ it is nonzero. At large $mr$ it is exponentially suppressed, whereas the massless limit is

$$W(0,r)\longrightarrow\frac{1}{4\pi^2r^2}.$$

Neither behavior is compact support. A nonzero vacuum correlation at spacelike separation is therefore an explicit calculation, not a hypothetical exception to be ignored.

The commutator cancels outside the cone

For this free real scalar field, the commutator is a c-number distribution:

$$[\widehat\phi(x),\widehat\phi(0)]=C(x)I, \qquad C(x)=W(x)-W(-x).$$

At equal time, changing $\mathbf p$ to $-\mathbf p$ shows $W(0,\mathbf x)=W(0,-\mathbf x)$, so $C(0,\mathbf x)=0$. Lorentz invariance extends this cancellation to every spacelike separation. It is a difference of two nonzero terms, not a claim that each term vanishes.

This calculation is the free-field realization of microcausality described by Tong. The general causal principle is imposed on the algebra of local observables. For fermions the field-level algebra uses anticommutators, while even local observables commute at spacelike separation.

A retarded inverse and its source normalization

Define a scalar Green function by the explicit convention

$$G_R(x)=i\theta(t)C(x).$$

It vanishes in the past and at spacelike separation. Its source normalization is fixed by the equal-time data

$$C(0,\mathbf x)=0, \qquad \partial_tC(0,\mathbf x)=-i\delta^{(3)}(\mathbf x).$$

Differentiating $\theta(t)C(x)$ distributionally therefore gives

$$(\Box+m^2)G_R(x)=\delta^{(4)}(x).$$

For a compact source $J$, the retarded solution of $(\Box+m^2)\phi=J$ is

$$\phi(x)=\int d^4y\,G_R(x-y)J(y),$$

up to independently specified homogeneous initial data. The source-generated term has support only in the causal future of the source support. This is the appropriate source-response statement; replacing $G_R$ by $W$ would not even solve the same inhomogeneous equation.

Time ordering has a different purpose

The time-ordered vacuum correlation is

$$D_F(x)=\theta(t)W(x)+\theta(-t)W(-x).$$

With Fourier convention $e^{-ip\cdot x}$, its momentum expression is $i/(p^2-m^2+i0)$, and it satisfies $(\Box+m^2)D_F=-i\delta^{(4)}$. The extra factor differs from the convention chosen for $G_R$ above. One must compare both the boundary prescription and the delta-source normalization before using a table of propagators.

At spacelike separation $W(x)=W(-x)$, so $D_F(x)$ equals the same nonzero correlation. Time ordering is useful in transition amplitudes and perturbative field theory; it is not the instruction to evolve a classical source with retarded boundary conditions.

Exercises

1. Derive $\partial_tC(0,\mathbf x)=-i\delta^{(3)}(\mathbf x)$ directly from the two on-shell integrals.

Solution

Each time derivative cancels its $2E$ denominator up to a factor $1/2$. At equal time the result is $-\tfrac{i}{2}\int d^3p\,[e^{i\mathbf p\cdot\mathbf x} +e^{-i\mathbf p\cdot\mathbf x}]/(2\pi)^3=-i\delta^{(3)}(\mathbf x)$.

2. Use $K_1(z)\sim1/z$ at small positive $z$ to verify the massless spacelike correlation. Does its nonzero value change the commutator?

Solution

$mK_1(mr)/(4\pi^2r)\to1/(4\pi^2r^2)$. It is still even under $x\mapsto-x$ at spacelike separation, so the two terms in $C$ cancel.

3. A calculation reports a nonzero kernel at spacelike separation. State the minimum additional information needed before calling it an acausal response.

Solution

Identify the kernel’s operator or source definition, its boundary condition, and which local intervention and measured observable it connects. A correlator alone does not answer those questions. For the explicit retarded source problem above, causal support can be tested directly.

References

• R. Haag, Local Quantum Physics: Fields, Particles, Algebras, 2nd ed., Springer, 1996 — local observable algebras and causality.
• National Institute of Standards and Technology, Digital Library of Mathematical Functions, sections 10.30 and 10.32 — modified-Bessel limits and integral representations.
• D. Tong, Lectures on Quantum Field Theory, University of Cambridge, 2006, sections 2.6.1 and 2.7 — free-field causality and propagators.