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arXiv:quant-ph/9901019 · Journal of Mathematical Physics

Uncertainty principle for proper time and mass

Shoju Kudaka, Shuichi Matsumoto

Published Mar 1, 1999• 19 citations
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Abstract

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.

DOI
10.1063/1.532797

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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.
The meaning and scope of a recent type of uncertainty relation of a very general character are elucidated using the notions of time-indicating dynamical variables (clock variables) and place-indicating dynamical variables (position variables). It is shown that if the total energy (momentum) of a system is certain, all time-indicating (place-indicating) dynamical variables are completely uncertain. The quantum clock is discussed as an illustration of the energy–time uncertainty relation. The relations can be successfully applied to the thought experiments that Einstein introduced into his debate with Bohr about the uncertainty principle(s) and, in particular, to the famous photon-box experiment. It is shown that due to this general relation the photon box can never serve its purpose, independent of the details of the experiment.
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​.
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.
Abstract We present a reformulation of Bohr’s analysis of Einstein’s thought experiment with the photon box, which hopefully settles some unclear issues. The box is weighed by two position measurements, the first immediately before the emission of the photon and the second after a quarter period of oscillation of the box, or alternatively after a half period. These times are precisely defined with respect to the moving clock installed in the box, but vary statistically when measured with a stationary clock outside the box, due to gravitational time dilation in accordance with Bohr’s original idea. Moreover, we will discuss two other proposals to solve the apparent inconsistency without invoking the theory of general relativity.