DOCSLIB.ORG

Symmetries of the Point Particle

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Symmetries of the Point Particle UPPSALA UNIVERSITY BACHELOR THESIS 15 HP Symmetries of the Point Particle Author: Alexander Söderberg Supervisors: Ulf Lindström Subject Reader: Maxim Zabzine June 23, 2014 Abstract: We study point particles to illustrate the various symmetries such as the Poincaré group and its non-relativistic version. In order to find the Noether charges and the Noether currents, which are conserved under physical symme- tries, we study Noether’s theorem. We describe the Pauli-Lubanski spin vector, which is invariant under the Poincaré group and describes the spin of a particle in field theory. By promoting the Pauli-Lubanski spin vector to an operator in the quantized theory we will see that it describes the spin of a particle. Moreover, we find an action for a smooth spinning bosonic particle by compactifying one string dimension together with one embedding dimension. As with the Pauli- Lubanski spin vector, we need to quantize this action to confirm that it is the action for a smooth spinning particle. Sammanfattning: Vi studerar punktpartiklar för att illustrera olika symemtrier som t.ex. Poincaré gruppen och dess icke-relativistiska version. För att hitta de Noether laddningar och Noether strömmar, vilka är bevarade under symmetrier, studerar vi Noether’s sats. Vi beskriver Pauli-Lubanksi spin vektorn, vilken har en invarians under Poincaré gruppen och beskriver spin hos en partikel i fältteori. Genom att låta Pauli-Lubanski spin vektorn agera på ett tillstånd i kvantfältteori ser vi att den beskriver spin hos en partikel. Dessutom finner vi en verkan för en spinnande partikel genom att kompaktifiera en bosonisk sträng dimension till- sammans med en inbäddad dimension. Som med Pauli-Lubanski spin vektorn, kvantiserar vi denna verkan för att bekräfta att det är en verkan för en spinnande partikel. BACHELOR PROGRAM IN PHYSICS DEPARTMENT OF PHYSICSAND ASTRONOMY DIVISIONOF THEORETICAL PHYSICS 1 ACKNOWLEDGEMENT I would like to give many thanks to my supervisor, Ulf Lindström, for the intro- duction to this subject, and for many interesting discussions. Moreover, would I like to thank Susanne Mirbt and my friends for good advices, and my family for the support they have given me. For given me something to look forward to, I would like to thank Shigeru Miyamoto. 2 CONTENTS 1. Introduction 4 2. Method 5 2.1. Problem Formulation . 6 3. The Point Particle 7 3.1. The Non-Relativistic Action for a Free Massive Particle . 7 3.2. The Galilean Transformations . 7 3.3. The Relativistic Action for a Free Massive Particle . 9 3.4. Equations of Motion for a Free Particle . 10 4. The Lorentz Group 11 4.1. Boosts . 11 4.2. Rotations Around a Fixed Axis . 13 5. The Poincaré Group 14 5.1. Translations . 14 6. Noether’s Theorem 15 6.1. Noether’s Theorem . 16 6.2. Noether’s Theorem in Field Theory . 16 7. Pauli-Lubanski Spin 17 7.1. Pauli-Lubanski Spin . 17 7.2. Quantization of the Pauli-Lubanski Spin . 19 8. Smooth Spinning Particles 20 8.1. The Action of a Bosonic String . 20 8.2. The Action of a Rigid Particle . 21 8.3. Classical Dynamics of a Rigid Particle . 23 8.4. Quantization of a Rigid Particle’s Momenta and Spin . 24 9. Conclusion 26 A. Derivations 30 3 1. INTRODUCTION When we study the total angular momentum of a particle in quantum mechanics we need to study not only the orbital angular momentum (as we do in the classical case) but also the internal angular momentum of the particle. This internal angular momentum is what we call spin, and it determines in which state the particle is in. Spin is a conserved quantity and therefore it is important to study this quantity, e.g. in particle colliders is spin of big impor- tance. In this thesis we will try to get a better understanding of the quantum mechanical property spin by studying its classical counterpart. This will we do by using symmetries. We shall take the classical particle as our starting point for a discussion of symmetries. If a quantity has a symmetry under a transformation, then this quantity is invariant under this transformation, i.e. it is the same before and after this transformation. We have a practical understanding of what a symmetry operation does, e.g. when a sphere is rotated an angle around an axis through its center. In physics we are interested in symmetries of the equa- tions of motion. There are two main categories of symmetries. Space-time symmetries, such as boosts, rotations and space-time translations, and internal symmetries such as the gauge transformations in electromagnetism. As an example, consider that a physical system has a time translational symmetry. This would mean that performing an experiment before or after a time translation for instance yields the same result. Mathematically this leads to invariance of the equations of motion, or equivalently of the action for the system, under the mathemat- ical transformation. From the action of a system one can find the equations of motion associated with this sys- tem. Therefore, we study the action of systems. We will proceed from the action of a system and look at transformations under which the action is invariant. Through symmetries we will describe a quantity which describes spin in classical field theory, and an action for a smooth spinning particle. A smooth spinning particle has spin once it is quantized. We will find an ac- tion for a smooth spinning particle by compactifying one string dimension together with one embedding dimension. We will only consider strings with integer spin, i.e. bosonic strings, and therefore we will only find an action for smooth spinning particles with integer spin, i.e. bosonic smoth spinning particles. In general, to see if a quantity describes spin, one need to study whether its quantized version introduce spin or not. Not all systems in physics can be described by classical models. When one is considering sys- tems with smaller distances and higher energies, one needs to use quantum mechanics. On even smaller distances and higher energies one needs to apply string theory. However, both quantum mechanics and string theory have their mathematical basis in symmetries, many of which are relevant also classically. Therefore it is important to study these symmetries in the classical case. Symmetries of classical theories correspond to conserved quantities that are essential in de- scribing e.g. the motion of a system. It is often these conserved quantities we study in ex- 4 periments, e.g. the energy and the momentum. Symmetries in quantum mechanics have a similar purpose, and also label the quantum numbers of a system. Classical symmetries have their quantum mechanical counterpart, but the opposite is not always true. Spin is one ex- ample of a quantum mechanical property which is not fully covered in the classical theory. The classical counterpart is best described by the Pauli-Lubanski vector. In this thesis we will always study massive particles. This means we study particles having a mass, e.g. we study the Pauli-Lubanski spin for a particle with mass and not for a photon. The total angular momentum in classical physics is identical with orbital angular momen- tum. However, in quantum mechanics one also has to consider the spin angular momentum. The spin angular momentum is what describes spin in quantum mechanics. It has no mean- ing in classical physics, but since some classical quantities introduce spin when quantized we would like to say that these quantities have spin in classical physics. It is important to study such quantities since they give us a better understanding of quantum mechanical properties. 2. METHOD In this thesis we will gather important information about symmetries and spin in classical physics in one place. Often is information about symmetries and spin in classical physics written down in different kind of books. So in this thesis we will get a better understanding how one can with symmetries describe spin in classical physics. Moreover will most of the calculations be done so one can follow the mathematical steps. This project is a literature study. The references in this thesis are mostly to books. They have been chosen based on their importance in the various areas which have been studied in this thesis. We will use all of the references as guidelines and do most of the calculations. We can in principal divide this thesis into three parts. In the first part we will illustrate the im- portance of symmetries by proving that the action for a point particle is invariant under the transformations in the Poincaré group (the transformations used in special relativity) and its non-relativistic version. In the second part we will describe a quantity in classical field the- ory which describes the spin of a particle. Finally in the third part, we will find an action for a smooth spinning particle. The first part consist of section 3,4 and 5. Here we will follow [1]-[4]. Reference [1] and [3] are about special relativity and reference [2] and [4] are about quantum field theory. Since the Poincaré group are important in quantum field theory, [2] and [4] are two good references to follow when we want to prove the invariance of the relativistic action for a point particle under the transformations in the Poincaré group. The second part consist of section 6 and 7. Here we will follow [4]-[6]. Reference [5] is a book about some mathematics which exists in physics, and reference [6] is another book of 5 quantum field theory. We follow reference [5] since we want to know the mathematical ver- sion of Noether’s theorem and why we can apply it to physical systems.
Load more

Recommended publications
  • Elementary Particles: an Introduction
    Elementary Particles: An Introduction By Dr. Mahendra Singh Deptt. of Physics Brahmanand College, Kanpur What is Particle Physics? • Study the fundamental interactions and constituents of matter? • The Big Questions: – Where does mass come from? – Why is the universe made mostly of matter? – What is the missing mass in the Universe? – How did the Universe begin? Fundamental building blocks of which all matter is composed: Elementary Particles *Pre-1930s it was thought there were just four elementary particles electron proton neutron photon 1932 positron or anti-electron discovered, followed by many other particles (muon, pion etc) We will discover that the electron and photon are indeed fundamental, elementary particles, but protons and neutrons are made of even smaller elementary particles called quarks Four Fundamental Interactions Gravitational Electromagnetic Strong Weak Infinite Range Forces Finite Range Forces Exchange theory of forces suggests that to every force there will be a mediating particle(or exchange particle) Force Exchange Particle Gravitational Graviton Not detected so for EM Photon Strong Pi mesons Weak Intermediate vector bosons Range of a Force R = c Δt c: velocity of light Δt: life time of mediating particle Uncertainity relation: ΔE Δt=h/2p mc2 Δt=h/2p R=h/2pmc So R α 1/m If m=0, R→∞ As masses of graviton and photon are zero, range is infinite for gravitational and EM interactions Since pions and vector bosons have finite mass, strong and weak forces have finite range. Properties of Fundamental Interactions Interaction
    [Show full text]
  • On the Reality of Quantum Collapse and the Emergence of Space-Time
    Article On the Reality of Quantum Collapse and the Emergence of Space-Time Andreas Schlatter Burghaldeweg 2F, 5024 Küttigen, Switzerland; [email protected]; Tel.: +41-(0)-79-870-77-75 Received: 22 February 2019; Accepted: 22 March 2019; Published: 25 March 2019 Abstract: We present a model, in which quantum-collapse is supposed to be real as a result of breaking unitary symmetry, and in which we can define a notion of “becoming”. We show how empirical space-time can emerge in this model, if duration is measured by light-clocks. The model opens a possible bridge between Quantum Physics and Relativity Theory and offers a new perspective on some long-standing open questions, both within and between the two theories. Keywords: quantum measurement; quantum collapse; thermal time; Minkowski space; Einstein equations 1. Introduction For a hundred years or more, the two main theories in physics, Relativity Theory and Quantum Mechanics, have had tremendous success. Yet there are tensions within the respective theories and in their relationship to each other, which have escaped a satisfactory explanation until today. These tensions have also prevented a unified view of the two theories by a more fundamental theory. There are, of course, candidate-theories, but none has found universal acceptance so far [1]. There is a much older debate, which concerns the question of the true nature of time. Is reality a place, where time, as we experience it, is a mere fiction and where past, present and future all coexist? Is this the reason why so many laws of nature are symmetric in time? Or is there really some kind of “becoming”, where the present is real, the past irrevocably gone and the future not yet here? Admittedly, the latter view only finds a minority of adherents among today’s physicists, whereas philosophers are more balanced.
    [Show full text]
  • Chapter 5 the Relativistic Point Particle
    Chapter 5 The Relativistic Point Particle To formulate the dynamics of a system we can write either the equations of motion, or alternatively, an action. In the case of the relativistic point par- ticle, it is rather easy to write the equations of motion. But the action is so physical and geometrical that it is worth pursuing in its own right. More importantly, while it is difficult to guess the equations of motion for the rela- tivistic string, the action is a natural generalization of the relativistic particle action that we will study in this chapter. We conclude with a discussion of the charged relativistic particle. 5.1 Action for a relativistic point particle How can we find the action S that governs the dynamics of a free relativis- tic particle? To get started we first think about units. The action is the Lagrangian integrated over time, so the units of action are just the units of the Lagrangian multiplied by the units of time. The Lagrangian has units of energy, so the units of action are L2 ML2 [S]=M T = . (5.1.1) T 2 T Recall that the action Snr for a free non-relativistic particle is given by the time integral of the kinetic energy: 1 dx S = mv2(t) dt , v2 ≡ v · v, v = . (5.1.2) nr 2 dt 105 106 CHAPTER 5. THE RELATIVISTIC POINT PARTICLE The equation of motion following by Hamilton’s principle is dv =0. (5.1.3) dt The free particle moves with constant velocity and that is the end of the story.
    [Show full text]
  • Theory of More Than Everything1
    Universally of Marineland Alimentary Gastronomy Universe of Murray Gell-Mann Elementary My Dear Watson Unified Theory of My Elementary Participles .. ^ n THEORY OF MORE THAN EVERYTHING1 V. Gates, Empty Kangaroo, M. Roachcock, and W.C. Gall 2 Compartment of Physiques and Astrology Universally of Marineland, Alleged kraP, MD ABSTRACT We derive a theory which, after spontaneous, dynamical, and ad hoc symmetry breaking, and after elimination of all fields except a set of zero measure, produces 10-dimensional superstring theory. Since the latter is a theory of only everything, our theory describes much more than everything, and includes also anything, something, and nothing. (More text should go here. So sue me.) 1Work supported by little or no evidence. 2Address after September 1, 1988: ITP, SHIITE, Roc(e)ky Brook, NY Uniformity of Modern Elementary Particle Physics Unintelligibility of Many Elementary Particle Physicists Universal City of Movieland Alimony Parties You Truth is funnier than fiction | A no-name moose1] Publish or parish | J.C. Polkinghorne Gimme that old minimal supergravity. Gimme that old minimal supergravity. It was good enough for superstrings. 1 It's good enough for me. ||||| Christian Physicist hymn1 2 ] 2. CONCLUSIONS The standard model has by now become almost standard. However, there are at least 42 constants which it doesn't explain. As is well known, this requires a 42 theory with at least @0 times more particles in its spectrum. Unfortunately, so far not all of these 42 new levels of complexity have been discovered; those now known are: (1) grandiose unification | SU(5), SO(10), E6,E7,E8,B12, and Niacin; (2) supersummitry2]; (3) supergravy2]; (4) supursestrings3−5].
    [Show full text]
  • Gravitational Field of Massive Point Particle in General Relativity
    Gravitational Field of Massive Point Particle in General Relativity P. P. Fiziev∗ Department of Theoretical Physics, Faculty of Physics, Sofia University, 5 James Bourchier Boulevard, Sofia 1164, Bulgaria. and The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34014 Trieste, Italy. Utilizing various gauges of the radial coordinate we give a description of static spherically sym- metric space-times with point singularity at the center and vacuum outside the singularity. We show that in general relativity (GR) there exist a two-parameters family of such solutions to the Einstein equations which are physically distinguishable but only some of them describe the gravitational field of a single massive point particle with nonzero bare mass M0. In particular, we show that the widespread Hilbert’s form of Schwarzschild solution, which depends only on the Keplerian mass M < M0, does not solve the Einstein equations with a massive point particle’s stress-energy tensor as a source. Novel normal coordinates for the field and a new physical class of gauges are proposed, in this way achieving a correct description of a point mass source in GR. We also introduce a gravi- tational mass defect of a point particle and determine the dependence of the solutions on this mass − defect. The result can be described as a change of the Newton potential ϕN = GN M/r to a modi- M − 2 0 fied one: ϕG = GN M/ r + GN M/c ln M and a corresponding modification of the four-interval. In addition we give invariant characteristics of the physically and geometrically different classes of spherically symmetric static space-times created by one point mass.
    [Show full text]
  • Fall 2009 PHYS 172: Modern Mechanics
    PHYS 172: Modern Mechanics Fall 2009 Lecture 16 - Multiparticle Systems; Friction in Depth Read 8.6 Exam #2: Multiple Choice: average was 56/70 Handwritten: average was 15/30 Total average was 71/100 Remember, we are using an absolute grading scale where the values for each grade are listed on the syllabus Typically, exam grades (500/820 points) are lower than homework, recitation, clicker, & labs grades Clicker Question #1 You get in a parked car and start driving. Assume that you travel a distance D in 10 seconds, with a constant acceleration a. Define the system to be the car plus the driver (you), whose total mass is M and acceleration a. Choose the correct statement for this system: A. The final total kinetic energy is Ktotal= MaD. B. The final translational kinetic energy is Ktrans= MaD. C. The total energy of the system changes by +MaD. D. The total energy of the system changes by -MaD.. E. The total energy of the system does not change. Clicker Question Discussion You get in a parked car and start driving. Assume that you travel a distance D in 10 seconds, with a constant acceleration a. Define the system to be the car plus the driver (you), whose total mass is M and acceleration a. Choose the correct statement for this system: A. The final total kinetic energy is Ktotal= MaD.No, this is just the translational K. There is also relative K (wheels, etc.) B. The final translational kinetic energy is Ktrans= MaD. Correct, use point particle system; force is friction applied by the road on the tires.
    [Show full text]
  • Relativistic Quantum Mechanics 1
    Relativistic Quantum Mechanics 1 The aim of this chapter is to introduce a relativistic formalism which can be used to describe particles and their interactions. The emphasis 1.1 SpecialRelativity 1 is given to those elements of the formalism which can be carried on 1.2 One-particle states 7 to Relativistic Quantum Fields (RQF), which underpins the theoretical 1.3 The Klein–Gordon equation 9 framework of high energy particle physics. We begin with a brief summary of special relativity, concentrating on 1.4 The Diracequation 14 4-vectors and spinors. One-particle states and their Lorentz transforma- 1.5 Gaugesymmetry 30 tions follow, leading to the Klein–Gordon and the Dirac equations for Chaptersummary 36 probability amplitudes; i.e. Relativistic Quantum Mechanics (RQM). Readers who want to get to RQM quickly, without studying its foun- dation in special relativity can skip the first sections and start reading from the section 1.3. Intrinsic problems of RQM are discussed and a region of applicability of RQM is defined. Free particle wave functions are constructed and particle interactions are described using their probability currents. A gauge symmetry is introduced to derive a particle interaction with a classical gauge field. 1.1 Special Relativity Einstein’s special relativity is a necessary and fundamental part of any Albert Einstein 1879 - 1955 formalism of particle physics. We begin with its brief summary. For a full account, refer to specialized books, for example (1) or (2). The- ory oriented students with good mathematical background might want to consult books on groups and their representations, for example (3), followed by introductory books on RQM/RQF, for example (4).
    [Show full text]
  • “WHAT IS an ELECTRON?” Contents 1. Quantum Mechanics in A
    \WHAT IS AN ELECTRON?" OR PARTICLES AS REPRESENTATIONS ROK GREGORIC Abstract. In these informal notes, we attempt to offer some justification for the math- ematical physicist's answer: \A certain kind of representation.". This requires going through some reasonably basic physical ideas, such as the barest basics of quantum mechanics and special relativity. Contents 1. Quantum Mechanics in a Nutshell2 2. Symmetry7 3. Toward Particles 11 4. Special Relativity in a Nutshell 14 5. Spin 21 6. Particles in Relativistic Quantum Mechaniscs 25 0.1. What these notes are. This are higly informal notes on basic physics. Being a mathematician-in-training myself, I am afraid I am incapable of writing for any other audience. Thus these notes must fall into the unfortunate genre of \physics for math- ematicians", an idiosyncracy on the level of \music for the deaf" or \visual art for the blind" - surely possible in some fashion, but only after overcoming substantial technical obstacles, and even then likely to be accussed by the cognoscenti of \missing the point" of the original. And yet we beat on, boats against the current, . 0.2. How they came about. As usual, this write-up began its life in the form of a series of emails to my good friend Tom Gannon. Allow me to recount, only a tinge apocryphally, one major impetus for its existence: the 2020 New Year's party. Hosted at Tom's apartment, it featured a decent contingent of UT math department's grad student compartment. Among others in attendence were Ivan Tulli, a senior UT grad student working on mathematics inperceptably close to theoretical physics, and his wife Anabel, a genuine experimental physicist from another department, in town to visit her husband.
    [Show full text]
  • New Varying Speed of Light Theories
    New varying speed of light theories Jo˜ao Magueijo The Blackett Laboratory,Imperial College of Science, Technology and Medicine South Kensington, London SW7 2BZ, UK ABSTRACT We review recent work on the possibility of a varying speed of light (VSL). We start by discussing the physical meaning of a varying c, dispelling the myth that the constancy of c is a matter of logical consistency. We then summarize the main VSL mechanisms proposed so far: hard breaking of Lorentz invariance; bimetric theories (where the speeds of gravity and light are not the same); locally Lorentz invariant VSL theories; theories exhibiting a color dependent speed of light; varying c induced by extra dimensions (e.g. in the brane-world scenario); and field theories where VSL results from vacuum polarization or CPT violation. We show how VSL scenarios may solve the cosmological problems usually tackled by inflation, and also how they may produce a scale-invariant spectrum of Gaussian fluctuations, capable of explaining the WMAP data. We then review the connection between VSL and theories of quantum gravity, showing how “doubly special” relativity has emerged as a VSL effective model of quantum space-time, with observational implications for ultra high energy cosmic rays and gamma ray bursts. Some recent work on the physics of “black” holes and other compact objects in VSL theories is also described, highlighting phenomena associated with spatial (as opposed to temporal) variations in c. Finally we describe the observational status of the theory. The evidence is slim – redshift dependence in alpha, ultra high energy cosmic rays, and (to a much lesser extent) the acceleration of the universe and the WMAP data.
    [Show full text]
  • MITOCW | L0.8 Introduction to Nuclear and Particle Physics: Relativistic Kinematics
    MITOCW | L0.8 Introduction to Nuclear and Particle Physics: Relativistic Kinematics MARKUS Now back to 8.701. So in this section, we'll talk about relativistic kinematics. Let me start by saying that one of KLUTE: my favorite classes here at MIT is a class called 8.20, special relativity, where we teach students about special relativity, of course, but Einstein and paradoxes. And it's one of my favorite classes. And in their class, there's a component on particle physics, which has to do with just using relativistic kinematics in order to understand how to create antimatter, how to collide beams, how we can analyze decays. And in this introductory section, we're going to do a very similar thing. I trust that you all had some sort of class introduction of special relativity, some of you maybe general relativity. What we want to do here is review this content very briefly, but then more use it in a number of examples. So in particle physics, nuclear physics, we often deal particles who will travel close to the speed of light. The photon travels at the speed of light. We typically define the velocity as v/c in natural units. Beta is the velocity, gamma is defined by 1 over square root 1 minus the velocity squared. Beta is always smaller or equal to 1, smaller for massive particle. And gamma is always equal or greater than 1. The total energy of a particle with 0 mass-- sorry, with non-zero mass, is then given by gamma times m c squared.
    [Show full text]
  • Chapter 11 the Relativistic Quantum Point Particle
    Chapter 11 The Relativistic Quantum Point Particle To prepare ourselves for quantizing the string, we study the light-cone gauge quantization of the relativistic point particle. We set up the quantum theory by requiring that the Heisenberg operators satisfy the classical equations of motion. We show that the quantum states of the relativistic point particle coincide with the one-particle states of the quantum scalar field. Moreover, the Schr¨odinger equation for the particle wavefunctions coincides with the classical scalar field equations. Finally, we set up light-cone gauge Lorentz generators. 11.1 Light-cone point particle In this section we study the classical relativistic point particle using the light-cone gauge. This is, in fact, a much easier task than the one we faced in Chapter 9, where we examined the classical relativistic string in the light- cone gauge. Our present discussion will allow us to face the complications of quantization in the simpler context of the particle. Many of the ideas needed to quantize the string are also needed to quantize the point particle. The action for the relativistic point particle was studied in Chapter 5. Let’s begin our analysis with the expression given in equation (5.2.4), where an arbitrary parameter τ is used to parameterize the motion of the particle: τf dxµ dxν S = −m −ηµν dτ . (11.1.1) τi dτ dτ 249 250 CHAPTER 11. RELATIVISTIC QUANTUM PARTICLE In writing the above action, we have set c =1.Wewill also set =1when appropriate. Finally, the time parameter τ will be dimensionless, just as it was for the relativistic string.
    [Show full text]
  • Fully Symmetric Relativistic Quantum Mechanics and Its Physical Implications
    mathematics Article Fully Symmetric Relativistic Quantum Mechanics and Its Physical Implications Bao D. Tran and Zdzislaw E. Musielak * Departmemt of Physics, University of Texas at Arlington, Arlington, TX 76019, USA; [email protected] * Correspondence: [email protected] Abstract: A new formulation of relativistic quantum mechanics is presented and applied to a free, massive, and spin-zero elementary particle in the Minkowski spacetime. The reformulation requires that time and space, as well as the timelike and spacelike intervals, are treated equally, which makes the new theory fully symmetric and consistent with the special theory of relativity. The theory correctly reproduces the classical action of a relativistic particle in the path integral formalism, and allows for the introduction of a new quantity called vector-mass, whose physical implications for nonlocality, the uncertainty principle, and quantum vacuum are described and discussed. Keywords: relativistic quantum mechanics; generalized Klein–Gordon equation; path integral formulation; nonlocality; quantum measurement; uncertainty principle; quantum vacuum 1. Introduction Relativistic quantum mechanics (RQM) primarily concerns free relativistic fields [1,2] Citation: Tranl, B.D.; Musielak, Z.E. described by the Klein–Gordon [3,4], Dirac [5], Proca [6], and Rarita–Schwinger [7] wave Fully Symmetric Relativistic equations, whereas the fundamental interactions and their unification are considered by Quantum Mechanics and Its Physical the gauge invariant quantum field theory (QFT) [8,9]. The above equations of RQM are Implications. Mathematics 2021, 9, P = SO( ) ⊗ 1213. https://doi.org/10.3390/ invariant with respect to all transformations that form the Poincaré group 3, 1 s ( + ) ( ) math9111213 T 3 1 , where SO 3, 1 is a non-invariant Lorentz group of rotations and boosts and T(3 + 1) an invariant subgroup of spacetime translations, and this structure includes Academic Editor: Rami Ahmad reversal of parity and time [10].
    [Show full text]