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

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 $(0,0)$, $(τ,L)$, and $(2τ,2L)$ with $L>τ+2τ_0$, 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 $Q(f)=\operatorname{sign}(ϕ(f))$ where $ϕ(f)$ 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 $C(τ,L)$ and the Leggett-Garg parameter $K_3=2C(τ,L)-C(2τ,2L)$. For non-overlapping time windows, we find that the correlation function decays exponentially with $τ$ 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.