A unified relativistic path integral origin for noise-activated collapse and decoherence

Abstract

Relativity and quantum mechanics are two cornerstones of modern physics, yet their unification within a single-particle path integral and a dynamic explanation of quantum measurement remain unresolved. Historically, these two problems have been treated as separate, but we in this work show they are intimately linked. We construct a relativistic path integral that recovers the Dirac, Klein-Gordon, and Schrödinger equations, while also exposing a latent nonlocal term in the propagator. This term dormant in differentiable potentials but is activated by non-differentiable noise, driving outcome probabilities through bounded-martingale stochastic process. In this regime, the pointer basis emerges as absorbing boundaries, Born's rule arises from first-passage statistics, and collapse occurs in finite, parameter-dependent time, thereby reducing measurement axioms to dynamical consequences. Crucially, our work recovers the standard GKSL master equation by taking the ensemble average over the noise, and thus provides a first-principles foundation for decoherence. Because the trigger is the noise spectrum, our work shows that engineering ``colored'' noise can expedite or steer collapse, suggesting practical routes to fast qubit reset, coherence preservation, and quantum sensing beyond the standard quantum limit.

Figures (4)

• Figure 1: Relativistic pushforward -- pullback mechanism and nonlocality.A) Schematic of the covariant evolution. The laboratory equal–time slice (rest frame) and the particle’s moving slice (moving frame, i.e. the particle’s proper frame) are different hyperplanes in spacetime, so viewing the lab state on the moving slice introduces (i) a change of spatial volume element, (ii) a boosted spin frame, and (iii) an amplitude renormalization that preserves probability flux (detailed in SI. Sec. S4). Accordingly, the state is pushed forward to the moving frame by $L_n(\mathbf v)=L_B F_V^{ n}F_\Psi$, evolved by $K$, and then pulled back by $\hat{L}_n^{-1}(\hat{\mathbf v})$ to yield the observable result in the rest frame: $\psi(\mathbf r_1,t_1)= \int \hat{L}_n^{-1} K L_n \psi(\mathbf r,t) \mathrm dV$(detailed in SI Sec. S4). B) differentiable environments. Although individual moving–frame histories may include superluminal segments (black curve outside the cone joining $P_1$ and $P_2$), the pullback mapping exactly cancels this intrinsic nonlocality, and the image on the rest-frame space is a local connection between $P_1$ and $P_2$. C) Non-differentiable (noisy) environments. Random, non-differentiable noise (blue path) disrupts the smooth geometry of the mapping. Geometrically, the rough path prevents the pushforward and pullback transformations from perfectly cancelling. This resulting geometric mismatch is precisely the residual holonomy, a finite nonlocal correction that becomes the physical kernel for quantum collapse.
• Figure 2: Noise-induced wave-function collapse and Born’s rule. The collapse equation derived from our single-particle relativistic path integral, Eq. (\ref{['eq:Collapse_SDE_final_concise']}), is a bounded-martingale SDE that naturally yields stochastic collapse and Born’s rule. In this figure, we simulate a spin-$1/2$ two-level system at $T=300~\mathrm{K}$ with magnetic-field noise $\sigma_B=1.25~\mu\mathrm{T}/\sqrt{\mathrm{Hz}}$. A) Time traces for the initial state $|\psi(0)\rangle=\tfrac{\sqrt{3}}{2}|\uparrow\rangle+\tfrac{1}{2}|\downarrow\rangle$. For each tick of $1~\mathrm{ns}$ on the horizontal axis we generate $10^{4}$ micro-samples. The shaded bands depict the oscillation envelope spanned by these micro-samples and the dots in bands are block averages over 500 consecutive micro-samples; the red (blue) series corresponds to $p_\uparrow(t)$ ($p_\downarrow(t)=1-p_\uparrow(t)$). Because Eq. (\ref{['eq:Collapse_SDE_final_concise']}) is a bounded martingale, trajectories are absorbed at $p_\uparrow=1$ or $0$ (collapse to $|\uparrow\rangle$ or $|\downarrow\rangle$); the inset shows the opposite outcome under a different noise realization. B) Monte-Carlo convergence of collapse's results with the number of trials $N$. The running estimates $P_\uparrow(N)$ (red) and $P_\downarrow(N)$ (blue) approach the initial weights $p_\uparrow(0)$ and $p_\downarrow(0)$, respectively. C) Monte-Carlo validation of Born’s rule. For many choices of $|\psi(0)\rangle$, the fraction of collapses to $|\uparrow\rangle$ (points; colors denote independent batches) plotted against $|\langle\uparrow|\psi(0)\rangle|^{2}$ follows the Born-rule prediction (black line), $P_\uparrow(\infty)=p_\uparrow(0)=|\langle\uparrow|\psi(0)\rangle|^{2}$ and $P_\downarrow=1-P_\uparrow$. Together, panels A–C visualize boundary absorption and show that absorption probabilities are fixed by the initial state—i.e., Born’s rule emerges from Eq. (\ref{['eq:Collapse_SDE_final_concise']}).
• Figure 3: Crossover from unitary evolution to measurement-like collapse.A) Schematic Bloch–sphere snapshots at three noise levels (a1 $\to$ a3). The thick red curve is a visual guide to the transition between the unitary and measurement-dominated regimes. In a1 (negligible noise or low temperature) the state follows a smooth, unitary equatorial trajectory (blue); under a $\pi$-pulse of duration $\tau_\pi = {2 m \pi}/{(g q B)}$ it reaches the target state $|\uparrow\rangle-|\downarrow\rangle$. In a2 (moderate noise or moderate temperature) the stochastic path on the sphere (red) deviates from the unitary arc but, within one $\pi$-pulse, still ends near the same target—this is the critical region: increasing the temperature $T$ or the noise amplitude pushes the dynamics across the boundary. In a3 (strong noise or high temperature) the trajectory becomes erratic (red) and, within a $\pi$-pulse, the state randomly collapses to $|\uparrow\rangle$ or $|\downarrow\rangle$. B) Monte-Carlo phase map of the collapse weight $W_M$ (see Methods) as a function of magnetic-noise intensity $\sigma_B$ ($\mathrm{nT}/\sqrt{\mathrm{Hz}}$) and temperature $T (\mathrm{K})$. The map is computed on a $300\times300$ grid in $(\sigma_B,T)$; at each grid point $W_M$ is estimated by averaging over 2000 stochastic trajectories of the collapse SDE. Colors run from blue ($W_M \approx 0$, near-unitary) to red ($W_M \approx 1$, measurement limit), with black iso-contours. The dashed red line overlays the same transition guide shown in panel A and closely follows the $W_M \approx 0.5$ contour. Together, panels A and B show that unitary evolution and quantum measurement arise as two limits of a single mechanism, with environmental noise and temperature setting where the system lies along this continuum.
• Figure 4: Noise shaping steers quantum collapse and advances decoherence-reduction technologies.A) Collapse time under white and colored noise. The effective collapse time $\tau_{\mathrm{clps}}$ is plotted versus the observation–bandwidth cutoff $\Omega_c$ for several spectra. White noise (blue) provides the baseline and shortens roughly linearly with $\Omega_c$. Ornstein–Uhlenbeck noise (orange/green; correlation times $t_{\mathrm{cor}}=10 \mu\mathrm{s}$ and $100 \mu\mathrm{s}$) yields longer $\tau_{\mathrm{clps}}$ owing to spectral narrowing. High-pass–shaped noise (magenta) reallocates spectral weight toward the measurement coupling and can therefore accelerate collapse relative to the white-noise case. B) Deviation from the Born line under varying 1/$f$ noise fraction. Monte Carlo estimates of the absorption probability $P(\uparrow)$ versus the initial weight $p_0=|\langle\uparrow|\psi(0)\rangle|^2$. The black diagonal is the Born rule ($P(\uparrow)=p_0$). Colored dashed curves correspond to different values of $r$, which quantifies the relative weight of 1/$f$ noise in the total spectrum, $r=S_{\mathrm{1/f}}(0)/[S_{\mathrm{1/f}}(0)+S_{\mathrm{white}}(0)]$, with $S(0)$ the zero-frequency power spectral density. Larger $r$ produces stronger memory-induced drifts, biasing absorption away from the Born prediction and revealing the controllable impact of long-time correlations on collapse (this bias provides a decisive test of our framework, see Methods). C) A resource-effectiveness map for benchmarking decoherence-suppression strategies. We introduce this platform-agnostic map to chart technical effectiveness ($C_*$, y-axis; the larger the value, the stronger the suppression of decoherence) against resource cost ($\mathcal{R}_*$, x-axis; the larger the value, the more qubits, control, or overhead a method requires), where the ideal technology would occupy the top-left corner. Formal definitions of the indices are provided in the Methods. Bubble area is proportional to the resource cost, and error bars show the reported performance ranges for established methods (numbered) and our directed-collapse approach (red). The map highlights that our partial-collapse technique is already competitive with existing mid-resource methods, while the complete-collapse variant promises access to a new, highly effective, low-cost regime.