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Journal of Mathematical Physics

Comment on “Uncertainty principle for proper time and mass” [J. Math. Phys 40, 1237 (1999)]

R. A. Krikorian

Published Aug 1, 2001• 0 citations
Publisher

Abstract

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.

DOI
10.1063/1.1367329

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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.
In [J. Math. Phys. {\bf 40}, 1237 (1999)] Kudaka and Matsumoto derive the uncertainty relation c2ΔmΔτ≥ℏ/2c^2 Δm Δτ\geq \hbar/2c2ΔmΔτ≥ℏ/2 between the rest mass mmm and the proper time τττ, by considering the Lagrangian M(τ˙−c−1−gμνx˙μx˙ν)+eAμ(x)x˙μM(\dot τ- c^{-1} \sqrt{-g_{μν} \dot x^μ\dot x^ν}) + e A_μ(x) \dot x^μM(τ˙−c−1−gμν​x˙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 gμνdxμdτdxνdτg_{μν} \frac{dx^μ}{d τ}\frac{dx^ν}{d τ}gμν​dτdxμ​dτdxν​.
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.
Abstract We present a ‘proof’ that the standard treatment of the mass and proper time as parameters, rather than dynamical variables, is actually inconsistent, and we show how they can be introduced as a generalization of the standard Hamiltonian Dynamics. We discuss various topics such as the equivalence principle and particle decay and the uncertainty principle from this point of view. I am honored to be able to give this talk as part of the 60th birthday festival for professor. Wolfgang Schleich, a long-term colleague, a very good friend, and a beloved and highly respected member of the quantum community, and I hope our collaboration can continue for many years to come. I extend my best wishes to him and his wonderful wife Kathy on this occasion.
We study the quantum properties of an oscillator in proper time. This proper time oscillator is a particle model with mass that is on shell. Its internal time can be treated as a self-adjoint operator. The displaced time and displaced time rate of the oscillator obey an uncertainty relation resembling the one between position and momentum, which is different from the usual energy-time uncertainty relation. In addition, we demonstrate that a matter field with proper time oscillators satisfies the Klein-Gordon equation. It has properties of a zero-spin quantum field. The formulations adopted permits a more symmetrical treatment between time and space in a matter field.
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.