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W. Heisenberg's Measurement Uncertainty Principle and A. Einstein's General Theory of Relativity

Thierry Periat

Published Nov 1, 2020• 0 citations
OpenAlex

Abstract

In exposing his uncertainty principle, W. Heisenberg has pointed out the fact that positions and speeds form two separate and independent sets of observables. He also suggested that infinitesimal variations of energies and lapses of times were following a similar logic. Unfortunately, there is up to now no operator representing the time and, at least at a technical level, this absence is blocking the progression of any development for the second pair. Whatever the interpretation of that principle is, a fact is that it introduces a finite limit for the relative variations of these pairs. The situations realizing the minimum value of a product between both elements of a given pair are labeled as “quantum limit”. And this is the information on which I build my theoretical proposition. Following a well-known and accepted principle affirming that the laws of physics must be the same everywhere, I look for technical circumstances allowing to go further on that road. Concretely, that quantum limit must be the same for two observers, wherever they stay in the world. I consider the simplest transposition of Heisenberg’ suggested interdependence between the elements of the pair (delta energy, delta lapse of time) in a simple one-dimensional world. And I ask me where, in the arsenal of formulas and laws of physics, I can find a plausible manner to introduce these ingredients softly.

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In exposing his uncertainty principle, W. Heisenberg has pointed out the fact that positions and speeds form two separate and independent sets of observables. He also suggested that infinitesimal variations of energies and lapses of times were following a similar logic. Unfortunately, there is up to now no operator representing the time and, at least at a technical level, this absence is blocking the progression of any development for the second pair. Whatever the interpretation of that principle is, a fact is that it introduces a finite limit for the relative variations of these pairs. The situations realizing the minimum value of a product between both elements of a given pair are labeled as “quantum limit”. And this is the information on which I build my theoretical proposition. Following a well-known and accepted principle affirming that the laws of physics must be the same everywhere, I look for technical circumstances allowing to go further on that road. Concretely, that quantum limit must be the same for two observers, wherever they stay in the world. I consider the simplest transposition of Heisenberg’ suggested interdependence between the elements of the pair (delta energy, delta lapse of time) in a simple one-dimensional world. And I ask me where, in the arsenal of formulas and laws of physics, I can find a plausible manner to introduce these ingredients softly.
One of the most widely known building blocks of modern physics is Heisenberg's indeterminacy principle. Among the different statements of this fundamental property of the full quantum mechanical nature of physical reality, the uncertainty relation for energy and time has a special place. Its interpretation and its consequences have inspired continued research efforts for almost a century. In its modern formulation, the uncertainty relation is understood as setting a fundamental bound on how fast any quantum system can evolve. In this topical review we describe important milestones, such as the Mandelstam–Tamm and the Margolus–Levitin bounds on the quantum speed limit, and summarise recent applications in a variety of current research fields—including quantum information theory, quantum computing, and quantum thermodynamics amongst several others. To bring order and to provide an access point into the many different notions and concepts, we have grouped the various approaches into the minimal time approach and the geometric approach, where the former relies on quantum control theory, and the latter arises from measuring the distinguishability of quantum states. Due to the volume of the literature, this topical review can only present a snapshot of the current state-of-the-art and can never be fully comprehensive. Therefore, we highlight but a few works hoping that our selection can serve as a representative starting point for the interested reader.
Are General Relativity and Quantum Mechanics incompatible? Each in their world, that of the infinitely large and that of the infinitely small, they did not seem to interfere as long as they avoided each other. However, it is their fundamental oppositions that prevent the scientific community from achieving a unification of physics. The proposal of this paper is to provide a mathematical proof of incompatibility, beyond the fact that they have fundamentally different principles, between the foundations of General Relativity and Quantum Mechanics, namely the deformation of the space-time geometry and the Uncertainty Principle. It will thus be possible to provide an absolute limitation in establishing a unifying theory of physics, if any. Moreover, while respecting the conditions fixed by the Uncertainty Principle, it will be tempted to determine with accuracy and simultaneity, the position and the speed of a non-relativistic particle, by application of relativistic principles and bypassing the problems raised by such an operation. The Uncertainty Principle as stated by Werner Heisenberg will be then, in the light of observations made on the measurement of the time dilatation and in accordance with its own terms, refuted by the present. - Physics Essays, Volume 31, Issue 3 (September 2018), Article 12 - https://physicsessays.org/browse-journal-2/product/1667-12-alexandre-georges-incompatibility-between-einstein-s-general-relativity-and-heisenberg-s-uncertainty-principle.html
Abstract One of the most widely known building blocks of modern physics is Heisenberg’s indeterminacy principle. Among the different statements of this fundamental property of the full quantum mechanical nature of physical reality, the uncertainty relation for energy and time has a special place. Its interpretation and its consequences have inspired continued research efforts for almost a century. In its modern formulation, the uncertainty relation is understood as setting a fundamental bound on how fast any quantum system can evolve. In this topical review we describe important milestones, such as the Mandelstam–Tamm and the Margolus–Levitin bounds on the quantum speed limit , and summarise recent applications in a variety of current research fields—including quantum information theory, quantum computing, and quantum thermodynamics amongst several others. To bring order and to provide an access point into the many different notions and concepts, we have grouped the various approaches into the minimal time approach and the geometric approach , where the former relies on quantum control theory, and the latter arises from measuring the distinguishability of quantum states. Due to the volume of the literature, this topical review can only present a snapshot of the current state-of-the-art and can never be fully comprehensive. Therefore, we highlight but a few works hoping that our selection can serve as a representative starting point for the interested reader.
One of the most striking deviations of quantum mechanics from classical mechanics is that any measurement of a microscopic system involves the interaction with the measuring instruments. The interaction cannot be neglected and in fact leads to uncontrollable disturbance on the system. In 1927, Werner Karl Heisenberg in his celebrated paper established the quantitative expression for unavoidable momentum disturbance caused by any position measurement. The crucial word here is simultaneously. The uncertainty relation gives the lower bound on the product of the uncertainties of two pairs of variables. But it says nothing about the accuracy with which individual observables can be measured. The physical meaning of energy-time uncertainty is different from that of x-px uncertainty. Position and momentum are observables to which Hermitian operators are assigned and which are measured at a particular time. An important implication of the uncertainty principle is that quantum systems do follow classical trajectories which conflict with our everyday experience.
This paper presents a conceptual reinterpretation of the Heisenberg Uncertainty Principle within the relational framework developed in the preceding papers of this series. In conventional quantum mechanics, the uncertainty relation between position and momentum is typically understood as a limitation on the simultaneous precision with which these quantities can be measured. In this work, we explore an alternative interpretation in which the uncertainty principle reflects a structural constraint on the types of physical states that can exist within spacetime. Within the relational lens framework, physical systems exist between two fundamental limiting conditions: gravitational singularities, representing perfect spatial localization, and the speed of light, representing maximal propagation through spacetime. From this perspective, position and momentum can be interpreted as expressions of two complementary tendencies—localization and propagation—that define the allowable configurations of physical systems. Under this interpretation, the uncertainty relation expresses the minimum balance that must exist between these tendencies for a physical system to remain within spacetime. Planck’s constant (ħ) therefore represents the smallest relational scale separating these two boundary conditions. Rather than reflecting a limitation of measurement alone, the uncertainty principle may be understood as a structural feature of relational spacetime itself. This paper is conceptual in nature and does not introduce new empirical claims. Instead, it proposes a relational interpretation of the uncertainty principle that links quantum behavior with the broader boundary structure explored in earlier papers in this series.
A vast number of internal “time” quantities, intrinsically associated with some particular phenomena, is all that can be consistently defined in physics. This leads us to interpret the notion of “uncertainty in time” specifically, for each of the considered phenomena. However, a universal conception of “time” can be proposed from the unifying point of view of the Observer, considering only the information process underlying any phenomenon. A change in the available information relative to that phenomenon can be calculated with the use of a universal “time”-information operator. This operator computes the algorithmic information content of the predictive tool associated with any experimental situation. Its standard deviation is then used to interpret the notion of “uncertainty in time” as a limit of predictability for any evolution. Heisenberg’s principle of “time”-energy uncertainty can then be rigorously regarded as a cognitive principle governing the evolution of any physical system.