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arXiv:2607.00314

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

Yang Xiang

hep-thquant-phSubmitted Jul 1, 2026• Announced Jul 2, 2026• 0 citations
v1 · Jul 1, 2026v2 · Jul 2, 2026
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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.

Comments
10 pages, 1 figure

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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)(0,0)(0,0), (τ,L)(τ,L)(τ,L), and (2τ,2L)(2τ,2L)(2τ,2L) with L>τ+2τ0L>τ+2τ_0L>τ+2τ0​, 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)=sign⁡(φ(f))Q(f)=\operatorname{sign}(φ(f))Q(f)=sign(φ(f)) where φ(f)φ(f)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)C(τ,L)C(τ,L) and the Leggett-Garg parameter K3=2C(τ,L)−C(2τ,2L)K_3=2C(τ,L)-C(2τ,2L)K3​=2C(τ,L)−C(2τ,2L). For non-overlapping time windows, we find that the correlation function decays exponentially with τττ for a massive field. For overlapping windows, our numerical computation for a rectangular time window yields K3<1K_3<1K3​<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.
In the tests for macrorealism proposed by Leggett and Garg, the temporal correlation functions of a dichotomic variable QQQ must be measured in a noninvasive way to rule out alternative classical explanations of Leggett-Garg inequality violations. Ideal negative measurements in which a null result is argued to be a noninvasive determination of the system's state are often used. From a quantum-mechanical perspective, such a measurement collapses the wave function and will therefore typically be found to be invasive under any experimental check. Here, a simple modified ideal negative measurement protocol is described for measuring the correlation functions, which is argued to be noninvasive from both classical and quantum perspectives and hence the noninvasiveness can then be checked experimentally, thereby permitting a quantitative measure of the degree of clumsiness of the measurement. It is also shown how this procedure may be extended to measure higher-order correlation functions, and a number of higher-order conditions for macrorealism are derived.
Wigner's form of the local realist inequality is used to derive its temporal version for an oscillating two-level system involving two-time joint probabilities. Such an inequality may be regarded as a form of the Leggett-Garg inequality (LGI) constituting a necessary condition for macrorealism. The robustness of its quantum mechanical (QM) violation against unsharpness of measurement is investigated by using a suitable model of unsharp measurements. It is found that there exists a range of values of the sharpness parameter (characterizing precision of the relevant measurements) for which the usual LGI is satisfied by QM, but Wigner's form of the LGI (WLGI) is violated. This implies that for such unsharp measurements, the QM violation of macrorealism cannot be tested using the usual LGI, but can be tested using WLGI. In showing this, we take into account the general form of the usual LGI involving an arbitrary number of pairs of two-time correlation functions. Another recently proposed necessary condition for macrorealism, called ``no-signaling in time,'' is also probed, showing that its QM violation persists for arbitrarily unsharp measurements.
Quantum mechanics predicts that measurement is inherently invasive, challenging the macrorealist view that physical properties exist independently of observation. The Leggett–Garg inequality (LGI) provides a test of this worldview, but its violation in systems in collective systems is often challenging due to technical measurement disturbance. By embedding an Elitzur–Vaidman interaction-free measurement (“quantum bomb-testing”) protocol into a microwave-driven Rydberg superatom (a mesoscopically coherent collective system) interferometer—we realize a high-fidelity, sequential measurement test. We observe a clear LGI violation that exceeds the classical bound by nearly 50 standard deviations in such a mesoscopic collective system. This result demonstrates the failure of noninvasive measurability for a collective quantum observable, establishing a scalable platform for probing temporal quantum correlations beyond the single-particle limit.
We consider two different types of conditions which were proposed to test macrorealism in the context of a system described by a single dichotomic variable QQQ. This is the view that a macroscopic system evolving in time possesses definite properties which can be determined without disturbing the future or past state. The Leggett-Garg (LG) inequalities, the most commonly studied test, are only necessary conditions for macrorealism, but, building on earlier work J. J. Halliwell, Phys. Rev. A 93, 022123 (2016), it is shown that when the four three-time LG inequalities are augmented with a certain set of two-time inequalities also of the LG form, Fine's theorem applies and these augmented conditions are then both necessary and sufficient. A comparison is carried out with a very different set of necessary and sufficient conditions for macrorealism, namely the no-signaling in time (NSIT) conditions proposed by J. Kofler and C. Brukner, Phys. Rev. A 87, 052115 (2013) and L. Clemente and J. Kofler, Phys. Rev. A 91, 062103 (2015), which ensure that all probabilities for QQQ at one and two times are independent of whether earlier or intermediate measurements are made in a given run, and do not require (but imply) the LG inequalities. We argue that tests based on the LG inequalities have the form of very weak classicality conditions and can be satisfied in the face of moderate interference effects, but those based on NSIT conditions have the form of much stronger coherence witness conditions, satisfied only for zero interference. The two tests differ in their implementation of noninvasive measurability and so are testing different notions of macrorealism: the augmented LG tests are indirect, entailing a combination of the results of different experiments with only compatible quantities measured in each experimental run, in close analogy with Bell tests, and are primarily tests for macrorealism per se; in contrast, the NSIT tests entail sequential measurements of incompatible quantities and are primarily tests for noninvasiveness.
The Leggett--Garg inequality (LGI) constrains temporal correlations under the assumptions of macrorealism and noninvasive measurability. We study the four-time quantity K4K_4K4​ for two-flavor neutrino oscillations using a pseudo-density matrix (PDM) under static conditions, instead of non-invasive measurement. With this approach, it becomes possible to examine a Horodecki-type criterion for the temporal Pauli-correlation tensor. The Horodecki criterion had been proven for CHSH inequality (R. Horodecki et al. Phys. Lett. \textbf{A200}, 340), For any unitary qubit evolution, the corresponding scalar is MR=2M_R=2MR​=2. Although, this result characterizes the correlation tensor and the optimized temporal construction, it does not imply that a fixed flavor observable violates the LGI at every neutrino energy or baseline. We therefore evaluate K4K_4K4​ explicitly for flavor measurements in matter of constant density. The numerical examples illustrate how matter effects modify the temporal correlations, with a pronounced matter--vacuum separation near the MSW resonance for the chosen parameters and generally smaller differences in the selected off-resonance cases. This finding motivates further investigation of matter-induced temporal correlations in atmospheric neutrinos and their potential relevance to future studies with long-baseline and atmospheric-neutrino experiments, including DUNE and Hyper-Kamiokande.
In 1985, Leggett and Garg formulated a class of inequalities for testing the compatibility between macrorealism and quantum mechanics. In this paper, we point out that, based on the same assumptions of macrorealism that are used to derive Leggett-Garg inequalities (LGIs), there is a scope of formulating another class of inequalities different from the standard LGIs. By considering the three-time measurement scenario in a dichotomic system, we first propose an interesting variant of the standard LGIs and show that its quantum violation is larger than the standard LGIs. By extending this formulation to the nnn-time measurement scenario, we found that the quantum violations of variants of the LGIs for a qubit system increase with nnn, and, for a sufficiently large nnn, an algebraic maximum can be reached. We then compare the violations of the standard and variants of the LGIs in an unsharp measurement scenario and show that, for any arbitrary nnn, the violation of the latter is more robust to unsharpness than the former. Furthermore, we examine the relation between the quantum violations of the variants of the LGIs and an another formulation macrorealism, known as no-signaling in time conditions.