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Quantization of Time, Newton's Second Law of Motion, Maximal Proper

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Quantization of Time, Newton's Second Law of Motion, Maximal Proper International Journal of Theoretical and Mathematical Physics 2020, 10(3): 51-59 DOI: 10.5923/j.ijtmp.20201003.01 Relativistic Dynamics in the Quantum Limit of Maximal Proper Acceleration Maciej Rybicki Sas-Zubrzyckiego, Kraków, Poland Abstract Quantum extension of the Newton’s second law of motion in the relativistic form (referred to as Einstein’s second law) is proposed in order to adjust it to the realm of elementary particles subjected to extreme accelerations. An underlying idea is the quantization of proper time, set in a close connection with the concept of maximal proper acceleration. The relativistic concept of proper time 휏 is identified with the quantum notion of evolution parameter. The postulated −1 quantum of proper time 휏푞 is thought to depend on the particle mass: 휏푞 ~푚푝푎푟 ; consequently relates to the Compton −1 3 wavelength as 휏푞 ~휆퐶 and to the Caianiello’s maximal proper acceleration as 휏푞 ~퐴푚푎푥 , 퐴푚푎푥 = 2푚푐 ℏ. Quantization of proper time makes the relativistic increase of particle mass/energy discrete, which impacts on the general shape of Newton’s second law, now including both velocity and acceleration limits. Introducing the acceleration-dependent term results in a gradual neutralization of the mass increase as determined by the mass-velocity relation, together with the increasing proper acceleration. The new formula satisfies the correspondence principle with respect to the classical (Newtonian and relativistic) cases, and to the relevant formula connecting Planck units of force, mass and acceleration. The obtained results are juxtaposed with the quantization of spacetime proposed by the Causal Sets approach to quantum gravity. Keywords Quantization of time, Newton’s second law of motion, Maximal proper acceleration, Causal sets spontaneous creation of virtual electron-positron pairs. The 1. Introduction resultant maximal limit for the proper acceleration of electron proves to be: The idea of maximal acceleration has been the topic of 3 퐸푆 푞푒 푚푒푐 29 −2 lively debate during last decades. Different investigations 푎푆 = = ≈ 2.33 × 10 푚푠 (2) 푚푒 ℏ have brought different results, all of them rooted in Another example is the Planck acceleration, defined as various quantum theories [1], [2], [3], [4]. The limits put on acceleration from zero speed to the speed of light within the proper acceleration (defined as acceleration measured by Planck time. It is sometimes interpreted as the proper an onboard accelerometer) appeared to depend on the acceleration at the event horizon of a Planck black hole, i.e. mass of an accelerated point-like particle. A remarkable black hole with the Schwarzschild radius equal to the early example involving the concept of maximal proper Compton wavelength. Planck acceleration binds together the acceleration refers to electron. Developing the earlier works fundamental constants of physics, yielding an inconceivably by Sauter [5], Heisenberg and Euler [6] in the framework of great value: Quantum Electrodynamics (QED), Schwinger derived the 7 1 2 limit [7]: 푐 51 −2 푎푃 = ≈ 5.56 × 10 푚푠 (3) 2 3 ℏ퐺 푚푒 푐 퐸푆 = (1) 푞푒 ℏ It is not clear whether Planck acceleration is a purely theoretical quantity, refers solely to the early universe, or (푚푒 - electron mass, 푞푒 - elementary charge, 푐 - speed is too present in definite quantum phenomena currently of light, ℏ - reduced Planck constant, 퐸푆 - Schwinger limit of the dimension 푇푐 , where 푇 /Tesla/ is the magnetic observed. Special Theory of Relativity (STR) does not allow induction). Beyond this limit electromagnetic field for massive body to achieve the speed of light; however, it does the quantum vacuum becomes nonlinear making the not basically prevent from changing mass for pure energy, i.e. photon-photon scattering inelastic, in consequence causing from replacing massive particle by the massless photon. Like any other acceleration, Planck acceleration is expressible by * Corresponding author: different combinations of kinematical units, here specified as [email protected] (Maciej Rybicki) Planck length, Planck time and speed of light: Published online at http://journal.sapub.org/ijtmp 푐 ℓ 푐2 Copyright © 2020 The Author(s). Published by Scientific & Academic Publishing 푃 푎푃 = = 2 = (4) This work is licensed under the Creative Commons Attribution International 푡푃 푡푃 ℓ푃 License (CC BY). http://creativecommons.org/licenses/by/4.0/ 52 Maciej Rybicki: Relativistic Dynamics in the Quantum Limit of Maximal Proper Acceleration 3 All the basic Planck units consist of fundamental constants: 퐅 = 훾 퐯 푚0퐚∥ + 훾 퐯 푚0퐚⊥ (8) ℏ, 푐, 퐺; hence, the due conversion of any from the above where 퐚 and 퐚 are the parallel and perpendicular expressions gives the Planck acceleration related to Planck ∥ ⊥ components of acceleration, respectively. Hence, in the case mass: of net force acting upon an object in the direction parallel to 푚 푐3 the instantaneous velocity vector (actually considered), the 푎 = 푃 (5) 푃 ℏ Einstein’s second law of motion is: 3 (푡푃 - Planck time, ℓ푃 - Planck length, 푚푃 - Planck mass). 퐅 = 훾 퐯 푚0풂∥ (9) Both Schwinger and Planck accelerations, regardless of (hereinafter, we shall denote invariant/rest mass simply possible difference in their physical meaning have identical by 푚 , referred to as “mass”). In practice, energy and structure. Therefore, they both can be considered as specific momentum of a particle are obtainable from the independent manifestations of the general concept known as Caianiello’s measurements, whereas mass becomes deduced from these maximal proper acceleration, derived from quantum measurements according to the relationship: mechanics (QM), in particular from the Heisenberg 2 2 2 2 4 uncertainty principle [8], [9], [10], [11]: 퐸 = 푃 푐 + 푚 푐 (10) 2 2푚푐3 where 푃 is the dot product of 3-momentum 퐏 , i.e. 퐴 = (6) 푚푎푥 ℏ 푃 = 퐏 ∙ 퐏 . This formula applies both to massive and It is clearly evident from Eqns. (2), (5) and (6) that massless particles. In the latter case (of photon) one has maximal proper acceleration is basically a quantum always: phenomenon. Special relativity does not apply itself a limit 퐸2 − 푃2푐2 = 0 (11) on acceleration, which results in the absence of respective yielding zero mass. term in the relativistic formula for the Newton’s second law Time in QM is usually treated not as an observable (thus of motion, hereinafter called the Einstein’s second law of as a quantity defined by respective time-operator) but as the motion. Meanwhile, it is reasonable to expect that the wave function parameter. This conforms with the classical mass-dependent limit of the proper acceleration, assuming it dynamics and, in fact, with physics in general, considering really exist, should affect an ultimate shape of the second law. the latter conceived as a description of how the inanimate Using simple methods, we shall try to fulfil this expectation. objects behave in time. In accord with that, the wave function Our goal is to implement the Caianiello’s limit into the in the Schrodinger equation is basically time-dependent Einstein’s second law of motion, so as to make it applicable (except solutions with stationary waves/states). On the other to the extreme accelerations likely present in the quantum hand, to comply with STR, time should be treated on equal realm of elementary particles. This may provide us with a footing with space. Hence, temporal and spatial coordinates deeper insight into the atomic physics of “quantum jump”, (taken together) should be treated either as parameters or as usually interpreted as an abrupt (basically timeless) operators. It is admittedly possible to treat both time and transition from one quantum state to another, e.g. in the pair space as parameters (as it is in Quantum Field Theory). production/annihilation or photon emission/absorption. Consequently however, it should be also possible to treat These questions are the subject matter of Secs. 2 and 3. all constituents of the position 4-vector as operators. Instead, in Sec. 4 the obtained results are briefly compared Considering the correctness of this premise, another question with the idea of quantization of spacetime proposed by the arises: how the STR notions of coordinate time and proper Causal Sets Theory. time relate to the QM notions of time-as-parameter and time-as-operator? The answer here proposed is the following. 2. Dynamics Including Maximal Proper Proper time is an observable (operator) and plays the role of evolution parameter. Instead, coordinate time is the function Acceleration. Quantization of Proper parameter. The reason for such distinction comes out from Time the fact that proper time is an invariant of Lorentz transformation; besides, in the context of acceleration, it STR applies a universal limit on velocity identified with refers to mass, specific for any (type of) elementary particle. the speed of light. Together with the principle of relativity, This links the proper time with the concept of maximal this determines both relativistic kinematics and dynamics, proper acceleration, unique for each particle. Consequently, thereby redefines the Newtonian concepts of force, we postulate the proper time to be a discrete quantity. momentum, energy and mass. Mass, an invariant in the Expressed in terms of operators, energy and momentum Newton’s second law: are respectively: 퐸 = 푖ℏ 휕 휕푡 , 푝 = −푖ℏ 휕 휕푥 (along 푑퐏 푑 푚퐯 퐅 = = (7) 푥 dimension). In connection with Eq. (10) this makes the 푑푡 푑푡 starting point for any theory of relativistic quantum is subject in STR to the mass-velocity relation. The mechanics (RQM), i.e. QM formulated in the Poincare momentum is defined as 퐏 = 훾 퐯 푚0퐯 , where 훾 퐯 = covariant form. However, RQM does not say what exactly 1 − 푣2 푐2 −1 2 is the Lorentz factor. Differentiating gives happens in the time range during which the particles convert the force formula: into one another, e.g. in the fermions-bosons exchange or in International Journal of Theoretical and Mathematical Physics 2020, 10(3): 51-59 53 deep inelastic scattering.
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