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arxiv: 2607.00314 · v2 · pith:HEV7RIUPnew · submitted 2026-07-01 · ✦ hep-th · quant-ph

Leggett-Garg inequality in the massive scalar vacuum: No violation under spacelike-separated measurements

Yang Xiang This is my paper

Pith reviewed 2026-07-03 19:56 UTC · model grok-4.3

classification ✦ hep-th quant-ph
keywords Leggett-Garg inequalityquantum field theorymassive scalar fieldnoninvasive measurabilityvacuum correlationsspacelike separationmacrorealismtwo-time correlators
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The pith

The vacuum of a massive scalar field satisfies the Leggett-Garg inequality under spacelike-separated noninvasive measurements.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a protocol that uses three independent ensembles of the vacuum state, each measured by a different pair of observers at spacelike-separated events, to obtain the three two-time correlators needed for the Leggett-Garg test. Placing events at (0,0), ( au,L), and (2 au,2L) with L larger than au plus twice the window duration guarantees that no measurement can influence another. For the free massive scalar field in 1+1 dimensions the dichotomic observable is the sign of the smeared field, and the correlation functions are computed after absorbing time evolution into the window translation. Numerical evaluation for rectangular windows shows the Leggett-Garg parameter K3 stays below 1 for every mass value when windows overlap, while correlations decay exponentially for non-overlapping windows.

Core claim

By using independent ensembles and the spacelike configuration, the two-time correlation C( au,L) is obtained in the Heisenberg picture and the Leggett-Garg parameter is formed as K3 equals 2C( au,L) minus C(2 au,2L); this quantity remains strictly less than 1 across the entire mass range for rectangular time windows, establishing that the vacuum does not violate the inequality.

What carries the argument

The spacelike-separated independent-ensemble protocol with the dichotomic observable sign of the smeared scalar field, where time evolution is absorbed into translation of the time-window function.

If this is right

  • The vacuum respects macrorealism for temporal correlations under strict causal separation.
  • Correlations between non-overlapping windows decay exponentially with separation for massive fields.
  • The protocol supplies a benchmark for future noninvasive Leggett-Garg tests in quantum fields.
  • A clear distinction appears between spatial Bell violations and the absence of temporal Leggett-Garg violations in the same vacuum.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same protocol could be applied to other free fields or to interacting theories to test whether the no-violation result persists.
  • Analog simulators of scalar fields might allow laboratory checks of the predicted K3 values.
  • Relativistic causality appears to enforce macrorealism in the temporal sector for vacuum states.
  • Different smearing functions or window shapes could be examined to see whether violations can appear while preserving the noninvasive condition.

Load-bearing premise

The chosen spacelike event positions together with independent ensembles and the smeared sign observable guarantee zero residual influence between any pair of measurements.

What would settle it

A numerical evaluation or analytic derivation in the same protocol and rectangular window that produces K3 greater than or equal to 1 for any mass value would disprove the no-violation claim.

Figures

Figures reproduced from arXiv: 2607.00314 by Yang Xiang.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
read the original abstract

We overcome the long-standing noninvasive measurability (NIM) challenge in Leggett-Garg tests by exploiting the causal structure of quantum field theory (QFT). Our protocol uses three independent ensembles of the vacuum state, each measured by a different pair of observers at spacelike-separated events, yielding the three two-time correlators. By placing these events at positions $(0,0)$, $(\tau,L)$, and $(2\tau,2L)$ with $L>\tau+2\tau_0$, we rigorously ensure that no measurement can influence another. We investigate the vacuum state of a free massive scalar field in 1+1 dimensions, employing the dichotomic observable $Q(f)=\operatorname{sign}(\phi(f))$ where $\phi(f)$ is the smeared field. In the Heisenberg picture, the time evolution is absorbed into a translation of the time-window function, allowing us to derive the two-time correlation function $C(\tau,L)$ and the Leggett-Garg parameter $K_3=2C(\tau,L)-C(2\tau,2L)$. For non-overlapping time windows, we find that the correlation function decays exponentially with $\tau$ for a massive field. For overlapping windows, our numerical computation for a rectangular time window yields $K_3<1$ across the entire mass range, firmly establishing that the vacuum does not violate the LGI. Thus, under strict noninvasive conditions, the vacuum shows no violation of macrorealism, in stark contrast to its well-known violation of spatial Bell inequalities. Our spacelike-separated protocol provides the first LGI test in QFT with rigorously satisfied NIM, setting a methodological benchmark for future studies and highlighting the fundamental distinction between spacelike entanglement and temporal macrorealism in relativistic quantum fields.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript presents a protocol for testing the Leggett-Garg inequality (LGI) in the vacuum state of a free massive scalar field in 1+1 dimensions using spacelike-separated measurements on three independent ensembles. By positioning measurement events at (0,0), (τ,L), and (2τ,2L) with L > τ + 2τ0, and using the dichotomic observable Q(f) = sign(φ(f)) with smeared fields, the authors ensure noninvasive measurability via QFT causality. They derive the correlation function C(τ,L) in the Heisenberg picture and compute K3 = 2C(τ,L) - C(2τ,2L). Analytically, non-overlapping windows show exponential decay of correlations with τ. For overlapping rectangular windows, numerical evaluation shows K3 < 1 for all masses, leading to the conclusion that the vacuum does not violate the LGI under these strict conditions, unlike spatial Bell tests.

Significance. If the numerical results hold, this establishes the first LGI test in QFT with rigorously enforced NIM via relativistic causality, providing a methodological benchmark. The analytical exponential decay for non-overlapping windows follows directly from the standard massive scalar two-point function and is a clear strength. The work highlights a potential distinction between spatial Bell violations and temporal macrorealism in relativistic fields.

major comments (1)
  1. [Abstract and overlapping-windows results] Abstract and the section presenting results for overlapping windows: the central no-violation claim rests on the statement that 'our numerical computation for a rectangular time window yields K3<1 across the entire mass range.' No details are supplied on the quadrature method, discretization of the smeared correlators, convergence criteria, error estimates, specific ranges of m, τ, L, or the exact rectangular window function. This computation is load-bearing for the conclusion and cannot be verified from the given information.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. The single major comment concerns insufficient documentation of the numerical evaluation of K3 for overlapping windows. We address this below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract and overlapping-windows results] Abstract and the section presenting results for overlapping windows: the central no-violation claim rests on the statement that 'our numerical computation for a rectangular time window yields K3<1 across the entire mass range.' No details are supplied on the quadrature method, discretization of the smeared correlators, convergence criteria, error estimates, specific ranges of m, τ, L, or the exact rectangular window function. This computation is load-bearing for the conclusion and cannot be verified from the given information.

    Authors: We agree that the numerical implementation must be fully specified for the claim to be verifiable. In the revised manuscript we will insert a new subsection (or appendix) that explicitly states: (i) the quadrature algorithm (adaptive Gauss-Kronrod with absolute and relative tolerances of 10^{-8}); (ii) the discretization procedure for the two-point function integrals over the rectangular smearing windows; (iii) convergence criteria and a posteriori error bounds obtained by doubling the number of quadrature points; (iv) the precise parameter ranges scanned (m ∈ [0.01, 10], τ ∈ [1, 200], L > τ + 2τ0 with rectangular windows of width 2τ0 centered at the indicated times); and (v) the exact definition of the rectangular window function f(t,x). With these additions the numerical result K3 < 1 can be independently reproduced. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The two-time correlators C(τ,L) are obtained from the standard free massive scalar propagator in 1+1D; K3=2C(τ,L)−C(2τ,2L) is then evaluated directly by numerical integration over rectangular windows. Neither step reduces to a fitted parameter renamed as a prediction, nor to a self-citation chain, nor to a self-definitional relation. The NIM guarantee is asserted via the external causal structure of QFT (spacelike separation plus independent ensembles), which is not derived inside the paper. The result is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard QFT assumptions for the free massive scalar in 1+1 dimensions and the definition of the smeared sign observable; no new entities or fitted parameters are introduced in the abstract.

axioms (2)
  • domain assumption The two-point correlation functions of the free massive scalar field in 1+1D Minkowski space are given by the standard Wightman function.
    Used to obtain the time-evolved smeared-field correlators C(τ,L).
  • domain assumption The chosen event positions with L > τ + 2τ0 place all measurement pairs in spacelike separation.
    Invoked to guarantee noninvasive measurability via causal structure.

pith-pipeline@v0.9.1-grok · 5850 in / 1389 out tokens · 23603 ms · 2026-07-03T19:56:04.833202+00:00 · methodology

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