In [J. Math. Phys. {\bf 40}, 1237 (1999)] Kudaka and Matsumoto derive the uncertainty relation $c^2 Δm Δτ\geq \hbar/2$ between the rest mass $m$ and the proper time $τ$, by considering the Lagrangian $M(\dot τ- c^{-1} \sqrt{-g_{μν} \dot x^μ\dot x^ν}) + e A_μ(x) \dot x^μ$. In this note we give an alternative derivation based on a special case of the time-like geodesic equation obtained using the general relativistic Lagrangian $g_{μν} \frac{dx^μ}{d τ}\frac{dx^ν}{d τ}$.
In [J. Math. Phys. {\bf 40}, 1237 (1999)] Kudaka and Matsumoto derive the uncertainty relation c2ΔmΔτ≥ℏ/2 between the rest mass m and the proper time τ, by considering the Lagrangian M(τ˙−c−1−gμνx˙μx˙ν)+eAμ(x)x˙μ. In this note we give an alternative derivation based on a special case of the time-like geodesic equation obtained using the general relativistic Lagrangian gμνdτdxμdτdxν.
Kudaka and Matsumoto derive the uncertainty relation c2ΔmΔτ≈h between the proper time τ and rest mass m of a material particle by adopting for the description of the particle the Lagrangian L=M(τ̇−c−1−gμνẋμẋν)+eAμ(x)ẋμ, defined on the enlarged space (xμ,τ,M), where M is an additional dynamical variable. A key step in the derivation of their uncertainty relation is the identification of the variable M with the rest mass m. Adopting the point of view of variational analysis, we show that such an identification goes against the relativistic requirement of the path dependence of proper time.
We review Bohr’s reasoning in the Bohr–Einstein debate on the photon box experiment. The essential point of his reasoning leads us to an uncertainty relation between the proper time and the rest mass of the clock. It is shown that this uncertainty relation can be derived if only we take the fundamental point of view that the proper time should be included as a dynamic variable in the Lagrangian describing the system of the clock. Some problems and some positive aspects of our approach are then discussed.
When we quantize a system consisting of a single particle, the proper time τ and the rest mass m are usually dealt with as parameters. In the present article, however, we introduce a new quantization rule by which these quantities are regarded as operators in addition to the position and the momentum. Applying this new rule to a scalar particle and to a particle with spin $ 1/2 $, we analyze the time evolution of the operator τ. In the former case, the evolution of the proper time perfectly matches several well-established classical formulae. In the case of the particle with spin 1/2, our new rule implies that an oscillation appears in the time evolution of the operator τ. This oscillation is similar to Zitterbewegung which is well-known in the ordinary Dirac theory. We formulate one physical effect of this oscillation by considering the interaction with a gravitational field, and estimate how small it is.
Abstract §1. In a recent paper by one of us the conclusion was reached that the proper time of a particle had a minimum value h/mc2 where m is the rest mass. This conclusion was based on a simple assumption about the metrics of space which was found to be in agreement with Schroedinger’s wave equation. The following considerations show that the same conclusion can be reached by a method which is independent of the system of metrics referred to. The expression for the momentum of a particle of rest mass m in relativity mechanics is pα = mgαβdx8/ds, where s denotes an arc of the world line. if pα (= mdxα/ds) denote the contravariant vector corresponding to pα we have pαpα = m2.
We propose that time is not a fundamental background but emerges from the quantum uncertainties of position, momentum, and energy. This is captured by the postulate Δτ=γΔEΔxΔp,where Δτ is the proper time interval, Δx,Δp,ΔE are the standard quantum uncertainties, and γ=1/1−v2/c2 is the Lorentz factor associated with the average velocity v. Using the standard wave‑packet relation ΔE=vΔp, our equation reduces exactly to the special relativistic proper time Δτ=Δx/(γv), demonstrating consistency with all existing tests. However, for non‑classical quantum states where ΔE=vΔp (e.g., a superposition of widely separated momenta), our equation predicts new measurable effects: (i) a significant modification of the decay length of unstable particles (muons), (ii) a dependence of clock rates on the spatial spread Δx even in flat spacetime, and (iii) a quantum correction to gravitational time dilation when the Lorentz factor is replaced by the general relativistic factor Γ. These predictions are falsifiable and provide a concrete experimental window into the emergent nature of time.