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arXiv:1802.03911 · Physical review. D/Physical review. D.

Detecting discrete spacetime via matter interferometry

Todd A. Brun, Leonard D. Mlodinow

Published Jan 7, 2019• 15 citations
arXivHTMLPDF

Abstract

If the structure of spacetime is discrete, then Lorentz symmetry should only be an approximation, valid at long length scales. At finite lattice spacings there will be small corrections to the Dirac evolution. In particular, the lattice structure will be reflected in a modification of the free-particle dispersion relation. We show that these can produce a surprisingly large phase shift between the two arms of a nonparallel interferometer. This method could be employed to test any model that predicts a direction-dependent dispersion relation. Here, we calculate the size of this phase shift for a particular model, the three-dimensional quantum walk on the body-centered cubic lattice, which has been shown to give rise to the Dirac equation in the continuum limit. Though the details of this model will affect the size of the shift, its magnitude is set largely by dimensional analysis, so there is reason to believe that other models would yield similar results. We find that, with current technology, a modest-sized neutron interferometer could detect lattice spacings as small as ${10}^{\ensuremath{-}31}\text{ }\text{ }\mathrm{m}$, 13 orders of magnitude smaller than the scale probed by the LHC. A suitably scaled-up experiment could possibly detect lattice spacings even at the Planck scale.

DOI
10.1103/physrevd.99.015012

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If the structure of spacetime is discrete, then Lorentz symmetry should only be an approximation, valid at long length scales. At finite lattice spacings there will be small corrections to the Dirac evolution. In particular, the lattice structure will be reflected in a modification of the free-particle dispersion relation. We show that these can produce a surprisingly large phase shift between the two arms of a nonparallel interferometer. This method could be employed to test any model that predicts a direction-dependent dispersion relation. Here, we calculate the size of this phase shift for a particular model, the three-dimensional quantum walk on the body-centered cubic lattice, which has been shown to give rise to the Dirac equation in the continuum limit. Though the details of this model will affect the size of the shift, its magnitude is set largely by dimensional analysis, so there is reason to believe that other models would yield similar results. We find that, with current technology, a modest-sized neutron interferometer could detect lattice spacings as small as 10\ensuremath−31  m{10}^{\ensuremath{-}31}\text{ }\text{ }\mathrm{m}10\ensuremath−31  m, 13 orders of magnitude smaller than the scale probed by the LHC. A suitably scaled-up experiment could possibly detect lattice spacings even at the Planck scale.
It has been observed that quantum walks on regular lattices can give rise to wave equations for relativistic particles in the continuum limit. In this paper, we define the three-dimensional discrete-time walk as a product of three coined one-dimensional walks. The factor corresponding to each one-dimensional walk involves two projection operators that act on an internal coin space; each projector is associated with either the ``forward'' or ``backward'' direction in that physical dimension. We show that the simple requirement that there is no preferred axis or direction along an axis---that is, that the walk be symmetric under parity transformations and steps along different axes of the cubic lattice be uncorrelated---leads, in the case of the simplest solution, to the requirement that the continuum limit of the walk is fully Lorentz-invariant. We show further that, in the case of a massive particle, this symmetry requirement necessitates the use of a four-dimensional internal space (as in the Dirac equation). The ``coin flip'' operation is generated by the parity transformation on the internal coin space, while the differences of the projection operators associated with each dimension must all anticommute. Finally, we discuss the leading correction to the continuum limit, and the possibility of distinguishing through experiment between the discrete random walk and the continuum-based Dirac equation as a description of fermion dynamics.
The hypothesis that spacetime possesses a discrete structure at the Planck scale - potentially composed of quantized tetrahedral or simplicial building blocks - arises naturally in background-independent approaches to quantum gravity such as Loop Quantum Gravity (LQG) and Causal Dynamical Triangulations (CDT). In these frameworks, spatial geometry emerges from spin networks of quantum tetrahedra with discrete area and volume spectra, while spacetime evolution proceeds via spin foams or summed triangulations. Roger Penrose’s Objective Reduction (OR) mechanism further posits that quantum superpositions of massive states generate incompatible spacetime curvatures whose gravitational self-energy difference triggers an objective collapse when the associated energy uncertainty reaches ħ/τ, implicitly requiring a non-continuous spacetime geometry. This article presents an exhaustive theoretical foundation and proposes a hybrid tabletop experiment - the Planck Geometry Interferometer - designed to simultaneously measure gravitational phase shifts arising from proper-time differences and spontaneous collapse lifetimes. By placing mesoscopic masses in spatial superpositions and monitoring quantized deviations in phase or collapse time, the experiment can distinguish discrete tetrahedral geometries (LQG-style) from continuous spacetime or pure Penrose foam. Building directly on existing proposals, the setup is shown to be within reach of current optomechanical, cryogenic, and interferometric technologies. Detailed protocols, expected signatures of discreteness, feasibility assessments, and implications for falsifying or confirming Planck-scale discreteness are provided. A negative result would constrain or refute these models; a positive detection of quantization would constitute direct empirical evidence for discrete spacetime.
Abstract Quantum gravity theories generically predict a discrete structure of spacetime at the Planck scale, yet laboratory‑scale testable signals have remained elusive for a long time. We propose a tabletop experiment based on a squeezed‑light interferometer that exploits the exponential amplification property of squeezed states to probe the intrinsic phase noise arising from microscopic spacetime structures. Two complementary measurements are designed: (i) a Mach–Zehnder interferometer to measure the exponential enhancement of the residual visibility with the squeezing parameter ; (ii) intensity correlation (Hanbury Brown–Twiss) to measure the third‑ and fourth‑order correlation functions of the optical field, from which the non‑Gaussian characteristic ratio is extracted. This ratio is independent of the squeezing parameter and is determined solely by the recursive structure of a discrete spacetime, thus serving as a statistical fingerprint that distinguishes continuous from discrete microscopic structures. We give a clear null hypothesis: a continuous spacetime corresponds to Gaussian phase fluctuations ( with no enhancement), while a discrete spacetime predicts either (binary branch) or (ternary branch) together with the exponential scaling. The two criteria are independent and must be satisfied simultaneously, forming a double falsifiability of the model. Based on existing quantum‑optics technology (squeezing parameter , detection efficiency ), the predicted signal magnitude and the ratios or lie within a detectable range. The proposed scheme does not rely on any specific ultraviolet theory and can be implemented on existing platforms. It can also be combined with cosmological observations of primordial non‑Gaussianity and high‑energy Lorentz violation tests to form a cross‑scale consistency check. 摘要 量子引力理论普遍预言时空在普朗克尺度下呈现离散结构,但实验室尺度的可检验信号长期缺失。本文提出一种基于压缩态光干涉仪的桌面实验方案,利用压缩态的指数放大特性探测时空微观结构导致的内禀相位噪声。我们设计两套互补的测量:(i)马赫‑曾德干涉仪测量压缩光残余可见度随压缩参数 的指数增强行为 ;(ii)强度关联(Hanbury Brown‑Twiss)测量光场三阶与四阶关联函数,提取非高斯特征比值 。该比值与压缩参数无关,仅由离散时空的递归结构决定,构成区分连续/离散微观结构的统计指纹。 本文给出明确的零假设判据:连续时空对应高斯相位涨落( 且无 增强),离散时空则预言 (二元分支)或 (三元分支)且满足指数标度。两条判据独立且必须同时满足,构成模型的双重可证伪性。基于现有量子光学技术(压缩度 、探测效率 ),预言信号量级 、比值 或 ,处于可探测范围。本方案不依赖具体紫外理论,可直接在现有平台上实现,并与宇宙学非高斯性观测、高能宇宙线洛伦兹破缺检验形成跨尺度自洽性检验。
We introduce a new theoretical framework based on Feynman diagrams to compute phase shifts in matter wave interferometry. The method allows for analytic computation of higher order quantum corrections, beyond the traditional semi-classical approximation. These additional terms depend on the finite size of the initial matter wavefunction and/or have higher order dependence on ℏ. We apply the method to compute the response of matter wave interferometers to power law potentials and potentials with an arbitrary spatial dependence. The analytic expressions are validated by comparing to numerical simulations, and estimates are provided for the scale of the quantum corrections to the phase shift response to the gravitational field of the earth, anharmonic trapping potentials, and gravitational fields from local proof masses. We also find that for certain experimentally feasible parameters, these corrections are large enough to be measured and could lead to systematic errors if they are not mitigated. We find that to first order in a spatially dependent potential, quantum corrections vanish when the initial matter wavepacket has spherical symmetry and the potential satisfies Laplace's equation. We anticipate these quantum corrections will be especially important for trapped matter wave interferometers and for free-space matter wave interferometers in the presence of proof masses. These interferometers are becoming increasingly sensitive tools for mobile inertial sensing, gravity surveying, tests of gravity and its interplay with quantum mechanics, and searches for dark energy.
Very Long Baseline Atom Interferometry (VLBAI) corresponds to ground-based atomic matter-wave interferometry on large scales in space and time, letting the atomic wave functions interfere after free evolution times of several seconds or wave packet separation at the scale of meters. As inertial sensors, e.g., accelerometers, these devices take advantage of the quadratic scaling of the leading order phase shift with the free evolution time to enhance their sensitivity, giving rise to compelling experiments. With shot noise-limited instabilities better than 10−910^{-9}10−9 m/s2^22 at 1 s at the horizon, VLBAI may compete with state-of-the-art superconducting gravimeters, while providing absolute instead of relative measurements. When operated with several atomic states, isotopes, or species simultaneously, tests of the universality of free fall at a level of parts in 101310^{13}1013 and beyond are in reach. Finally, the large spatial extent of the interferometer allows one to probe the limits of coherence at macroscopic scales as well as the interplay of quantum mechanics and gravity. We report on the status of the VLBAI facility, its key features, and future prospects in fundamental science.
We study a single quantum particle in discrete spacetime evolving in a causal way. We see that in the continuum limit, any massless particle with a two-dimensional internal degree of freedom obeys the Weyl equation, provided that we perform a simple relabeling of the coordinate axes or demand rotational symmetry in the continuum limit. It is surprising that this occurs regardless of the specific details of the evolution: it would be natural to assume that discrete evolutions giving rise to relativistic dynamics in the continuum limit would be very special cases. We also see that the same is not true for particles with larger internal degrees of freedom, by looking at an example with a three-dimensional internal degree of freedom that is not relativistic in the continuum limit. In the process, we give a formula for the Hamiltonian arising from the continuum limit of massless and massive particles in discrete spacetime.