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advanced33 min studyLesson 659 in Physics

Microcausality and local observable algebras

Distinguish spacelike vanishing commutators from nonzero vacuum correlations, and express causal compatibility through smeared local observable algebras.

Before you begin
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The question for today

What you’ll understand

  • Check the free scalar commutator at spacelike separation with fixed Fourier signs.
  • State locality for spacetime-smeared observables and nested regions.
  • Separate causal commutation from statistical independence and from locality of gauge-dependent potentials.

Relativistic causality is encoded by compatibility of physical operations in spacelike separated regions. In a free scalar field, the field commutator vanishes there even though the vacuum two-point correlation generally does not. This distinction separates causal response from preexisting correlations. Because fields are distributions, the precise statement concerns suitable spacetime smearings and the observable algebras they generate. Fermions and gauge fields require care about which operators are physical local observables before the same causal principle is applied.

The two frequency contributions cancel outside the cone

Use the same natural units, metric, and positive-energy Fourier convention as the scalar construction. Substitution of the ladder algebra into a field commutator leaves a c-number distribution containing the difference of two opposite frequency phases. The invariant mass-shell measure makes that distribution Lorentz covariant. A spacelike separation can be transformed to equal time; in that frame the two spatial Fourier contributions cancel by reversing the momentum integration variable.

The cancellation does not say that either positive-frequency term vanishes separately. A vacuum Wightman function is generally nonzero outside the light cone, reflecting correlations rather than a controllable message. A time-ordered propagator also is not itself a retarded response function. Near null or coincident separation these objects are singular distributions, so their support and boundary values must be interpreted with test functions. A finite momentum cutoff can spoil exact cancellation and should not be mistaken for a proof of fundamental acausality.

[ϕ(x),ϕ(y)]=∫d3p(2π)3 2Ep(e−ip⋅(x−y)−eip⋅(x−y))=0if (x−y)2<0[\phi(x),\phi(y)]=\int\frac{d^3p}{(2\pi)^3\,2E_{\mathbf p}}\left(e^{-ip\cdot(x-y)}-e^{ip\cdot(x-y)}\right)=0\quad\text{if }(x-y)^2<0
We use η = diag(1,−1,−1,−1), p⁰ = Ep > 0 and p·x = Ept−p·x. This is a free scalar distributional identity. The cancellation is exact after the continuum construction; neither term separately has spacelike support zero.↔ Scroll wide equations horizontally.

Regions carry algebras rather than pointwise matrices

Smear a field with a smooth real test function of compact spacetime support. Its support specifies the region in which the corresponding field probe is localized. If every point in one support is spacelike to every point in the other, their smeared free scalar commutator vanishes. Checking only the separation of the supports' centers is insufficient when their time or spatial extents overlap each other's causal cones.

A local net assigns an observable algebra to each suitable spacetime region. Enlarging the region includes its algebra in the larger one, while spacelike separated regions have commuting observable algebras. These conditions do not assert a tensor-product factorization of continuum local Hilbert spaces or independence of every vacuum probability. In a regular free-field representation, bounded Weyl operators provide a useful way to discuss local algebras without multiplying unbounded point fields as ordinary matrices.

ϕ(f)=∫d4x f(x)ϕ(x),O1⊂O2⇒A(O1)⊂A(O2),[A(O),A(O~)]=0 for spacelike regions\phi(f)=\int d^4x\,f(x)\phi(x),\qquad \mathcal O_1\subset\mathcal O_2\Rightarrow\mathcal A(\mathcal O_1)\subset\mathcal A(\mathcal O_2),\qquad [\mathcal A(\mathcal O),\mathcal A(\widetilde{\mathcal O})]=0\ \text{for spacelike regions}
The last equation means every observable in one algebra commutes with every observable in the other. Test functions carry whatever units make the smearing appropriate. Unbounded smeared fields need domains; bounded Weyl generators can be used in their regular Hilbert-space representation.↔ Scroll wide equations horizontally.

Graded fields and measurable operations are different levels

Fermionic fields anticommute, rather than commute, across spacelike separated supports. Even local combinations such as appropriately defined bilinears obey ordinary observable locality: moving an even number of odd fields past another localized even combination produces an even number of minus signs. Charged fields and gauge-fixed potentials need additional interpretation; a radiation-gauge vector potential with a transverse inverse-Laplacian kernel is not itself a local gauge-invariant observable.

If a unitary operation belongs to one local algebra, commuting observables in a spacelike separated region are unchanged by conjugation with it. Their outcome probabilities therefore do not change under that local unconditioned operation. Vacuum entanglement can still produce correlations, and conditioning on a communicated measurement outcome can change a remote inferred state. The absence of signaling is a statement about allowed operations and observable statistics, not a claim that distant regions lack quantum correlations.

UA∈A(O), B∈A(O~),[UA,B]=0 ⇒ UA†BUA=BU_A\in\mathcal A(\mathcal O),\ B\in\mathcal A(\widetilde{\mathcal O}),\quad [U_A,B]=0\ \Rightarrow\ U_A^{\dagger}BU_A=B
Uₐ is a bounded unitary local operation and B a bounded observable in a spacelike separated algebra. This elementary consequence of locality holds in every state where its expectations are defined. General measurement operations require the corresponding local, trace-preserving operational assumptions.↔ Scroll wide equations horizontally.

Work it through

Find the equal-time frame of a spacelike interval

In a one-dimensional spacetime slice with c = 1, two events differ by Δt = 3 and Δx = 5. Classify the interval and use a boost β = 3/5 to find their transformed differences. What does free scalar microcausality imply?

  1. The invariant is Δt2−Δx2=9−25=−16\Delta t^2-\Delta x^2=9-25=-16, so the separation is spacelike.

  2. The boost factor is γ=1/1−(3/5)2=5/4\gamma=1/\sqrt{1-(3/5)^2}=5/4. Thus Δt′ = (5/4)(3−3) = 0 and Δx′ = (5/4)(5−9/5) = 4.

  3. The commutator vanishes in that equal-time separated frame and, by the covariant free-field construction, in the original frame as well. This does not require the vacuum two-point function itself to vanish.

The result

The events are spacelike, transformed differences are (0,4), and the free scalar commutator vanishes at their separation in the distributional sense.

The living notebook

Explore the model

Interactive lab

Interactive lab · spacetime

Spacelike events can be simultaneous in another frame

Reset and set BOTH Relative velocity β to 0 and Event time ct to 0.3; the instrument’s fixed x = 0.6 gives interval −0.27. Then Reset and set BOTH β to 0.5 and ct to 0.3; predict ct′ = 0 and a spacelike x′ ≈ 0.520. This transforms one event only. It does not evaluate field distributions, smear finite regions, or prove locality of an observable algebra.

Spacelike events can be simultaneous in another frameA fixed x = 0.6 event is Lorentz transformed with c = 1. Playback highlights the relation between coordinate descriptions; it does not move the event. Current readings: Lorentz factor γ: 1.12; Event x′: 0.168; Event ct′: 0.817. All parameters are adjustable with the labeled controls below.−4.69−2.3502.354.69−2.2−1.101.12.2x or x′ · c = 1ct or ct′EE′ct² − x² = 0.64
Original eventTransformed eventLight cone
Lorentz factor γ
1.12
Event x′
0.168
Event ct′
0.817
t = 0 s

A fixed x = 0.6 event is Lorentz transformed with c = 1. Playback highlights the relation between coordinate descriptions; it does not move the event. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forAn equal-time frame helps the free-field argument only after the separation is established as spacelike.

Pause & reason

Check your understanding

Which statement correctly separates free-field microcausality from vacuum correlations?

Your turn

Try it yourself

In a regular free-field representation, self-adjoint smeared fields A and B have central commutator [A,B] = iπ/2 times the identity. Let U = exp(iA) and V = exp(iB). Use the central-commutator Weyl relation to find the phase relating UV to VU. What changes for spacelike separated supports with zero commutator?

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

For a central commutator, exp(X)exp(Y) = exp([X,Y])exp(Y)exp(X). Substitute X = iA and Y = iB.

Compare with the solution

Here [iA,iB]=−[A,B]=−iπ/2[iA,iB]=-[A,B]=-i\pi/2, so UV=e−iπ/2VU=−iVUUV=e^{-i\pi/2}VU=-iVU. The bounded operators fail to commute by that phase. For spacelike separated supports, the free scalar smeared commutator is zero, the phase becomes one, and UV = VU. This uses Weyl operators in a regular representation, not an unrestricted exponential manipulation of pointwise fields.

Keep the idea

Take it with you

  • Spacelike commutator cancellation is compatible with nonzero vacuum correlations.
  • Locality concerns full supports and observable algebras, not only region centers.
  • Even fermionic observables and gauge-invariant fields obey ordinary causal compatibility.

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