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Physica Scripta

Why the mass and proper time must be considered as dynamical variables

Daniel M. Greenberger

Published Jan 25, 2019• 0 citations
Publisher

Abstract

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.

DOI
10.1088/1402-4896/ab01ed

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Mass and Proper Time as Conjugated Observables

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This work discloses some inconsistencies of quantum mechanics (QM) in two different contexts. On the one hand the equivalence principle in his two formulations is the starting point of all gravitational theories; yet there is no way to make it consistent with the non-relativistic formulation of QM. Moreover, the evaluation of the Schrödinger equation for a particle in an accelerated reference frame sheds light on a trajectory-dependent phase term subject to proper time. On the other hand, the Bargmann theorem unmistakably proves that the Schrödinger equation does not admit any superposition of different masses if the symmetry chosen for the system is the Galilean one. However, this is incompatible with relativity, since mass and energy are equivalent and one can certainly superimpose different energies. Both inconsistencies support the hypothesis that mass and proper time could be treated as variables. In the third chapter we will describe a theory in which the Galilean symmetry is extended by assuming particle mass and proper time as canonically conjugated classical variables, or conjugated observables in the quantum theory. The first three chapters discuss subjects illustrated in several articles of the professor Daniel M. Greenberger, which are listed in the bibliography. Whereas in the last chapter the results of a recent theory on QW, say a discrete model for quantum particle evolution, will be examined. In particular, the striking result is that, in the most natural physical interpretation of QW dynamics, it is not possible to have a constant and invariant mass in any manner. The long term aim of this work is indeed to study the reasons and consequences of a theory of particles with variable mass in the context of wider studies still in progress on QW.

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

R. A. Krikorian

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.

Journal of Mathematical Physics•Aug 1, 2001•0 citations

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The concept of proper time cannot be just taken over from classical theory and applied to quantum theory. There are a number of serious ambiguities related to it. Similarly, the concept of mass has some inconsistencies attached to it. We argue that both of these concepts only make sense if they are included in the theory as dynamical variables, and subsequently as conjugate operators. We discuss the necessary uncertainty principle between them.

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Abstract It is shown that the generally accepted statement that one cannot superpose states of different mass in non-relativistic quantum mechanics is inconsistent. It is pointed out that the extra phase induced in a moving system, which was previously thought to be unphysical, is merely the non-relativistic residue of the "twin-paradox" effect. In general, there are phase effects due to proper time differences between moving frames that do not vanish non-relativistically. There are also effects due to the equivalence of mass and energy in this limit. The remedy is to include both proper time and rest energy non-relativis-tically. This means generalizing the meaning of proper time beyond its classical meaning, and introduc­ ing the mass as its conjugate momentum. The result is an uncertainty principle between proper time and mass that is very general, and an integral role for both concepts as operators in non-relativistic physics.

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Fluctuations and criticality (dedicated to Reinhard Folk on his 60th birthday)

Harald Iro, Yurij V. Holovatch, I. M. Mryglod +1 more

Fluctuations and criticality (dedicated to Reinhard Folk on his 60th birthday)It is our pleasure and honour to present a collection of papers dedicated to Reinhard Folk on the occasion of his 60th birthday.With this Festschrift, colleagues and friends of the jubilee pay tribute to his long lasting and fruitful work in the field of condensed matter physics.His research includes ferroelectric and ferromagnetic phase transitions in regular and disordered systems, neural networks, superconductivity, critical dynamics -this list can be continued.Those, who work in the above mentioned fields certainly know and often use results obtained by R.F. which are currently turning into scientific Folk-lore.However, those who know him personally, recognizing his important contribution to physics, enjoy marvelous features of his extraordinary personality.Openness, readiness for discussion, erudition, interest in people and concern in their problems, sense of humour -that's what makes communication with him so desirable and pleasant.Moreover, his vivid interest in science on a larger scale and a deep concern in its future lead R.F. to activities in science organizations in Austria and the European Union, to his personal participation in different projects aiming at strengthening collaboration with Central and Eastern European countries.His work in this respect is widely recognized and acknowledged by the scientific community.Dear Reinhard, may your work in future lead you to new discoveries and solutions of puzzles put to us almost on every step of research, may it give you satisfaction of the achieved results, and may we, your friends, enjoy for many more years the wonderful intellectual atmosphere created by your work and your spirit.

Condensed Matter Physics•Jan 1, 2005•0 citations
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.
This work discloses some inconsistencies of quantum mechanics (QM) in two different contexts. On the one hand the equivalence principle in his two formulations is the starting point of all gravitational theories; yet there is no way to make it consistent with the non-relativistic formulation of QM. Moreover, the evaluation of the Schrödinger equation for a particle in an accelerated reference frame sheds light on a trajectory-dependent phase term subject to proper time. On the other hand, the Bargmann theorem unmistakably proves that the Schrödinger equation does not admit any superposition of different masses if the symmetry chosen for the system is the Galilean one. However, this is incompatible with relativity, since mass and energy are equivalent and one can certainly superimpose different energies. Both inconsistencies support the hypothesis that mass and proper time could be treated as variables. In the third chapter we will describe a theory in which the Galilean symmetry is extended by assuming particle mass and proper time as canonically conjugated classical variables, or conjugated observables in the quantum theory. The first three chapters discuss subjects illustrated in several articles of the professor Daniel M. Greenberger, which are listed in the bibliography. Whereas in the last chapter the results of a recent theory on QW, say a discrete model for quantum particle evolution, will be examined. In particular, the striking result is that, in the most natural physical interpretation of QW dynamics, it is not possible to have a constant and invariant mass in any manner. The long term aim of this work is indeed to study the reasons and consequences of a theory of particles with variable mass in the context of wider studies still in progress on QW.
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.
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.
Abstract It is shown that the generally accepted statement that one cannot superpose states of different mass in non-relativistic quantum mechanics is inconsistent. It is pointed out that the extra phase induced in a moving system, which was previously thought to be unphysical, is merely the non-relativistic residue of the "twin-paradox" effect. In general, there are phase effects due to proper time differences between moving frames that do not vanish non-relativistically. There are also effects due to the equivalence of mass and energy in this limit. The remedy is to include both proper time and rest energy non-relativis-tically. This means generalizing the meaning of proper time beyond its classical meaning, and introduc­ ing the mass as its conjugate momentum. The result is an uncertainty principle between proper time and mass that is very general, and an integral role for both concepts as operators in non-relativistic physics.
Fluctuations and criticality (dedicated to Reinhard Folk on his 60th birthday)It is our pleasure and honour to present a collection of papers dedicated to Reinhard Folk on the occasion of his 60th birthday.With this Festschrift, colleagues and friends of the jubilee pay tribute to his long lasting and fruitful work in the field of condensed matter physics.His research includes ferroelectric and ferromagnetic phase transitions in regular and disordered systems, neural networks, superconductivity, critical dynamics -this list can be continued.Those, who work in the above mentioned fields certainly know and often use results obtained by R.F. which are currently turning into scientific Folk-lore.However, those who know him personally, recognizing his important contribution to physics, enjoy marvelous features of his extraordinary personality.Openness, readiness for discussion, erudition, interest in people and concern in their problems, sense of humour -that's what makes communication with him so desirable and pleasant.Moreover, his vivid interest in science on a larger scale and a deep concern in its future lead R.F. to activities in science organizations in Austria and the European Union, to his personal participation in different projects aiming at strengthening collaboration with Central and Eastern European countries.His work in this respect is widely recognized and acknowledged by the scientific community.Dear Reinhard, may your work in future lead you to new discoveries and solutions of puzzles put to us almost on every step of research, may it give you satisfaction of the achieved results, and may we, your friends, enjoy for many more years the wonderful intellectual atmosphere created by your work and your spirit.