One of the main objectives of quantum-gravity research is to establish how the Heisenberg position-momentum uncertainty principle is modified by Planck-scale effects. We here focus on some residual gray areas of our understanding of the interplay between the uncertainty principle and special relativity, which might have been an obstacle for the further generalization needed in order to include gravitational effects. In its original formulation the uncertainty principle is inconsistent with special relativity, and in nearly a century of investigation not much progress has been made toward a satisfactory reformulation. Some partial insight has been gained in the ultra-high-velocity regime, but a full description is still missing and in particular we have no clue about the intermediate regime of particles whose speeds are much smaller than the speed of light but still high enough for tangible departures from the Heisenberg formulation. As we stress here, that intermediate regime is also our best chance for testing experimentally our understanding of the implications of special relativity for the uncertainty principle. We introduce a new approach toward facing these challenges, based mainly on the observation that the only operative notion of position of a particle at a given time involves the crossing of the worldline of that particle with the worldline of a test particle. We find that the worldline-crossing perspective opens a path toward a special-relativistic version of the uncertainty principle, which indeed could be tested experimentally.
One of the main objectives of quantum-gravity research is to establish how the Heisenberg position-momentum uncertainty principle is modified by Planck-scale effects. We here focus on some residual gray areas of our understanding of the interplay between the uncertainty principle and special relativity, which might have been an obstacle for the further generalization needed in order to include gravitational effects. In its original formulation the uncertainty principle is inconsistent with special relativity, and in nearly a century of investigation not much progress has been made toward a satisfactory reformulation. Some partial insight has been gained in the ultra-high-velocity regime, but a full description is still missing and in particular we have no clue about the intermediate regime of particles whose speeds are much smaller than the speed of light but still high enough for tangible departures from the Heisenberg formulation. As we stress here, that intermediate regime is also our best chance for testing experimentally our understanding of the implications of special relativity for the uncertainty principle. We introduce a new approach toward facing these challenges, based mainly on the observation that the only operative notion of position of a particle at a given time involves the crossing of the worldline of that particle with the worldline of a test particle. We find that the worldline-crossing perspective opens a path toward a special-relativistic version of the uncertainty principle, which indeed could be tested experimentally.
In this paper we suggest that through working with the Planck mass and its link to other particles in a simple way, it possible to “convert” the Heisenberg uncertainty principle into a very simple quantum probabilistic model. We further combine this with key elements from special relativity theory and get an interesting quantum relativistic probability theory. Some of the key points presented here could help to eliminate negative and above unity (pseudo) probabilities that often are used in standard quantum mechanics. These fake probabilities may be rooted in a failure to understand the Heisenberg principle fully in relation to the Planck mass. When properly understood, the Heisenberg principle seems to give a probabilistic range of quantum probabilities that is sound. There are no instantaneous probabilities and the maximum probability is always unity. In our formulation, the Planck mass particle is always related to a probability of one. Thus, we have certainty at the Planck scale for the Planck mass particle, or for particles accelerated to reach Planck energy. We are also presenting a relativistic extension of the McCulloch Heisenberg-derived Newton equivalent gravity theory. Our relativistic version requires much less mass than the Newtonian theory to explain gravitational phenomena, and initial investigation indicates it is consistent with perihelion of Mercury.
In this note, we consider the implications of the Heisenberg uncertainty principle (HUP) when computing uncertainties that affect the main dynamical quantities, from the perspective of special relativity. Using the well-known formula for propagating statistical errors, we prove that the uncertainty relations between the moduli of conjugate observables are not relativistically invariant. The new relationships show that, in experiments involving relativistic particles, limitations of the precision of a quantity obtained by indirect calculations may affect the final result.
In this paper, we will revisit the derivation of Heisenberg's uncertainty principle. We will see how the Heisenberg principle collapses at the Planck scale by introducing a minor modification. The beauty of our suggested modification is that it does not change the main equations in quantum mechanics; it only gives them a Planck scale limit where uncertainty collapses. We suspect that Einstein could have been right after all, when he stated, ``God does not throw dice." His now-famous saying was an expression of his skepticism towards the concept that quantum randomness could be the ruling force, even at the deepest levels of reality. Here we will explore the quantum realm with a fresh perspective, by re-deriving the Heisenberg principle in relation to the Planck scale. We will show how this idea also leads to an upper boundary on uncertainty, in addition to the lower boundary. These upper and lower boundaries are identical for the Planck mass particle; in fact, they are zero, and this highlights the truly unique nature of the Planck mass particle. Further, there may be a close connection between light and the Planck mass particle: In our model, the standard relativistic energy momentum relation also seems to apply to light, while in modern physics light generally stands outside the standard relativistic momentum energy relation. We will also suggest a new way to look at elementary particles, where mass and time are closely related, consistent with some of the recent work in experimental physics. Our model leads to a new time operator that does not appear to be in conflict with the Pauli objection. This indicates that both mass and momentum come in quanta, which are perfectly correlated to an internal Compton `clock' frequency in elementary particles.
Several theories of quantum gravity propose the existence of a minimal measurable length and maximum measurable momentum near the Planck scale. When integrated into the framework of quantum mechanics, such restrictions lead to the generalization of the Heisenberg uncertainty principle, which is commonly referred to as the generalized uncertainty principle (GUP). The GUP has been applied to a plethora of physical models in order to provide further insights on our understanding of the Universe, both in the macroscopic and the microscopic regime. Nevertheless, the corrections related to GUP are expected to be extremely small, which thus renders experimental verification a difficult task. Therefore, phenomenological approaches on a qualitative level are typically addressed with the aim of enhancing our possibility of detecting these tiny effects. Motivated by a recent work on a universal decoherence mechanism stemming from the standard formulation of GUP, we extend the ideas contained therein by considering a more general form of GUP which includes linear and quadratic momentum terms to examine its implications. To achieve such a picture, we equip the deformation parameter appearing in front of all momentum terms with a stochastic nature, so that the effective decoherence mechanism emerges by taking the average over fluctuations. We find that, despite the apparently small differences among the two generalizations of the Heisenberg uncertainty principle, the consequences at the level of the universal decoherence mechanism they entail is significant, as they predict decoherence times that are completely uncorrelated and distinct.
COGNITIVE ASPECTS OF THE HEISENBERG PRINCIPLE I T HE Heisenberg uncertainty principle, like its rival theory of relativity/ has been one of the truly crucial phases in the physics of our day~ It did not, like the discovery of a new element or the Maxwell equations, simply indent the story of classical physics for a new paragraph. It was not only a new chapter but a new book. Detected, in the words of Heisenberg himself, when physics became self-critical, the principle is a statement of what is observable-like its contemporary, the theory of relativity. It was anew advance in method, so revolutionary indeed that physics has virtually lost meaning as content and become synonymous , as logical empiricism amply indicates, with mere rules of procedure. That science is developing into a system of logic, combining the Cartesian stress on clearness with the Kantian regulative idea, actually confirms a contention of traditional philosophy. It reveals that once we deny the colors and sounds, the life, lights, and purpose in our universe-as pure scientific method actually does-we are fated to deny much more than quality. Quality is in the order of form, substance, and being. The moment quality is banished from the real world, by the same 1 That all is not as peaceful and unified in physics as the layman sometimes believes can be seen in the deep cleavage between quantum and relativity mechanics. For the first, the exact specification of space (position) and time (velocity, momentum) has no meaning, while in relativity theory such exact description is the only meaning that can be had. Despite the efforts 'of leading physicists, including Dirac, to unite the two theories, the contradiction is so profound that one of these systems must eventually be abandoned or, more probably, both of them will be superseded. The recent work .of Schroedinger and apparently also of Einstei~ himself are directed to extend the general theory of relativity, making the current theory a special case just as Newtonian physics is a special case of present relativity mechanics. 474 COGNITIVE ASPECTS OF THE HEISENBERG PRINCIPLE 475 decree being itself-because it is an immediate, non-inertial (formal) thing like quality-is driven into its only possible exile, nothingness. It is not surprising then that, after modern physics· has finished its task, the universe should not only be emptied of its lights, its fragrance, its sweetness, and its music but should also be without being and without existence. The uncertainty principle has been much debated in modern thought. Its impact has brought a sad end to the pride of 19th century science since it prevents a complete knowledge of anything at all by shading into the darkness at least one of the two major factors that specify it. Shortly after the principle was born it was hailed by various authors as a proof of free will, of the possibility of religion, and of non-mechanistic concepts in biology. In a more consistent and challenging sense, it has often been taken to refute what the physicist calls causality and what the philosopher calls the order of nature. The fact of the matter is that if the senses alone supply knowledge, truly only one thing at a time can be apprehended. If Hume was right in philosophy, so is Heisenberg. But when sensism yields to an intellectual insight into reality, it is easy to refute the scientist's claim to dethrone the doctrine of nature's order which he could never detect, even in classical physics, by pointerreadings .2 But whatever the fate of the Heisenberg principle in cosmology , it has had a most fruitful career in the field of physics. To Heisenberg himself, the principle was a point of departure for an original development of quantum mechanics based on the " uncertainty " of position and momentum,.: 3 If, for example, measurements are made, under a microscope, of the position of an electron, the photon (light-particle), which impinges on the eye and thus conveys the required knowledge to the physicist, has had to collide with the electron, thus imparting an impact known as the Compton recoil. When • For ·a discussion of the order of nature, cf. J. Marling: The...
Relativity of simultaneity is shown to limit the accuracy in simultaneous measurement of position and momentum of a moving body by an inertial observer. The inaccuracy or uncertainty product derived from special relativity theory is found to be similar to the much known quantum mechanical uncertainty relation. However, whether the above two uncertainty relations indicate the same physics is debatable and any conclusion is not attempted here. A modification of this relativistic uncertainty formula under gravitational field is achieved using general relativity principles, and a small gravitational correction term has to be introduced. This gravitational term being small remains negligible in ordinary conditions but becomes appreciable at small time scale.