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On a minimum uncertainty of space-time geometry

Shuichi Matsumoto, 修一 松本

Published Dec 27, 1997• 0 citations
OpenAlex

Abstract

We formulate some fluctuations of space-time which do not generate any fluctuations in the curvature, but which nevertheless do give rise to a spontaneous breakdown of the quantum superposition principle for macroscopic objects.

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The complete quantum metric of a parametrized quantum system has a real part (usually known as the Provost-Vallee metric) and a symplectic imaginary part (known as the Berry curvature). In this paper, we first investigate the relation between the Riemann curvature tensor of the space described by the metric, and the Berry curvature, by explicit parallel transport of a vector in Hilbert space. Subsequently, we write a generating function from which the complex metric, as well as higher-order geometric tensors (affine connection, Riemann curvature tensor), can be obtained in terms of gauge-invariant cumulants. The generating function explicitly relates the quantities which characterize the geometry of the parameter space to quantum fluctuations. We also show that for a mixed quantum-classical system both real and imaginary parts of the quantum metric contribute to the dynamics, if the mass tensor is Hermitian. A many-operator generalization of the uncertainty principle results from taking the determinant of the complex quantum metric. We also calculate the quantum metric for a number of Lie group coherent states, including several representations of the SU(1,1)\text{SU}(1,1)SU(1,1) group. In our examples nontrivial complex geometry results for generalized coherent states. A pair of oscillator states corresponding to the SU(1,1)\text{SU}(1,1)SU(1,1) group gives a double series for its spectrum. The two minimal uncertainty coherent states show trivial geometry, but, again, for generalized coherent states nontrivial geometry results.
In this paper, we extend the framework introduced in [1], which proposed a two-dimensional model of time as a means to reconcile relativistic invariance with quantum discreteness. Building on this foundation, we derive key principles of quantum theory such as the uncertainty and exclusion principles and also introduce a fourth postulate: “The probability of an event is the same in all inertial frames of reference (IFR), independent of the observer's position in space.” This postulate enables us also to derive the probability amplitude interference from a purely relativistic time-phase geometry. We show that trajectories in the two-dimensional time plane, governed by relativistic constraints and discrete frequency-phase dynamics, naturally give rise to quantized measurements and path-based superposition. Using the Mach-Zehnder interferometer as an illustrative case, we demonstrate that the square of the proper time trajectory corresponds to quantum probability and interference phenomena emerge from the structure of phase evolution in time. Our formulation also aligns with Feynman’s path integral framework, showing that quantum mechanics can be interpreted as a direct geometric consequence of extended relativistic principles.