DOCSLIB.ORG

Confronting the Faster-Than-Light Neutrinos?

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Confronting the Faster-Than-Light Neutrinos? Confronting the Faster-Than-Light Neutrinos? Utpal Sarkar Physical Research Laboratory, Ahmedabad 380 009, INDIA [email protected] http://www.prl.res.in/ utpal Fundamental Theories: Newtonian Mechanics Classical Mechanics is not applicable to fast moving particles: Special Theory of Relativity Uncertainty Principle at short distances: Quantum Mechanics Interactions of Relativistic particles at short distances: Quantum Field Theory Newton’s Law of gravitation fails for relativistic particles: General Theory of Relativity How well do we know Special Theory of Relativity, Quantum Mechanics, General Theory of Relativity (and hence, the Equivalence Principle), CPT theorem, Spin-Statistics theorem ??? Structure of the talk ² Recent experiment on faster-than-light neutrino ² It’s implication on Special Theory of Relativity and Lorentz invariance ² Confronting the experiment using phenomenological constraints ² Testing the violation of Lorentz invariance from Cosmic Ray data ² Constraining equivalence principle violation ² The Standard Model and the Higgs scalar ² Cosmology and beyond the standard model Recent Experiment on faster-than-light neutrino: A muon neutrino º¹beam was produced at CERN, Geneva: A 400 GeV/c proton beam ! charged pions and kaons ! ¼ ! ¹ +º ¹¹ K ! ¹ +º ¹¹ OPERA Collaboration at the Gran Sasso Underground Laboratory (LNGS), detected the º¹from CNGS at a distance of 730 kms away. The arrival time reported by the OPERA collaboration is earlier than expected from luminal speed by a time interval ±t = (60:7 § 6:9stat § 7:4syst) ns: This implies a superluminal propagation velocity for neutrinos: (vº ¡ c) ¡5 ±cº = = (2:48 § 0:28stat § 0:30syst) £ 10 c for an average energy of 17 GeV. Special Theory of Relativity: 1 The principle of relativity: The laws of physics are not affected by the motion of any inertial reference frames, i.e. the frames that are moving with constant velocity with respect to each other. 2 The principle of invariant light speed: Speed of light in empty space (c) is constant and same in all inertial frames. O P B P A B O A (a) (b) Both the observers at O and P will find that light travelled by a distance c in all directions to the circle in 1 sec. (a) Observer O will find himself at the center of the circle, (b) while observer at P will also find himself at the center of the circle. Particles with v > c are the Tachyons. Tachyon: Consider energy-momentum relationship: E2 = ~p 2c2 + m2c4 m2c4 For m ¿ ~p, we can express: E = ~pc + 2~pc dE m2c4 From the Hamilton-Jacobi equation: v = = c ¡ 2 d~p 2~p c Ordinary massive particles : m2 > 0 ) v < c Light and other massless particles m2 = 0 ) v = c T achyons m2 < 0 ) v > c Tachyons will find light travelling with velocity ¡c ) t !¡t ¡5 If º¹ s are tachyons, for the OPERA value of cº¹ = 2:5 £ 10 SN1987A neutrinos would have reached us four years earlier. Apparently observed faster-than-light neutrinos cannot be Tachyons. ¡9 Observed supernovae SN1987A neutrinos implies: j±cºj < 2 £ 10 Lorentz Invariance and OPERA: Assume energy-momentum is conserved in a particular reference frame A Lorentz transformation for an observer moving at velocity ~v yields ~p = γ m ~v ; and E = γ m c2 ; p where γ = 1= 1 ¡ v2=c2 is the Lorentz factor. ¡ ¢ 2 ¹ E 2 2 2 2 The invariant momentum is defined as p = p¹p = c ¡ j~pj = m c ; so that E2 = j~pj2c2 + m2c4 For any massive particle we thus require v < c, which contradicts OPERA result. For a muon neutrino with mass around m » 1 eV and energy E » 17 GeV (c ¡ v)=c » 10¡20. Violation of Lorentz Invariance: Gauge invariant Lorentz non-invariant interaction (Coleman & Glashow) h i y i u D0 ¡ i(1 ¡ ²=2)~σ ¢ D~ u : The parameter ² is a measure of violation of Lorentz invariance. This implies a modified energy-momentum dispersion relation: p2c2 = E2 ¡ j~pj2c2 = m2c4 + ²j~pj2c2 : In other words, writing the dispersion relation as E2 = j~pj2c2 + m2 c4 : º¹ º¹ º¹ the maximum attainable velocity, and the mass are now modified: c2 = (1 + ²)c2 = (1 + ²); m = m=(1 + ²) º¹ º¹ q With the new Lorentz factor γ = 1= 1 ¡ v2=c2 , we have º¹ º¹ 2 2 (cº¹ ¡ v) m E = γº mº c = γº m with ¼ ; for very large v ¹ ¹ º¹ ¹ 2 cº¹ 2E . The present result from OPERA gives (v ¡ c) = v ¡ 1 = (2:48 § 0:28 § 0:3) £ 10¡5 : c When Lorentz invariance is violated, the Lorentz factor contains (c ¡ v) m2 º¹ ¼ ¼ 10¡20 : 2 cº¹ 2E Since c < v < cº¹ , the present measurement of the º¹ velocity v becomes consistent with the modified Lorentz factor and the º¹ remains time-like. Different particles can have different limiting velocities (ci), if Lorentz invariance is violated. If the limiting velocities of the three neutrinos are different, that would lead to neutrino ¡22 oscillations. Present experimental constraints imply jcºi ¡ cºj j < 10 for i 6= j; i; j = e; ¹; ¿. Simplest solution is to assume, cºe = cº¹ = cº¿ = cº. OPERA then requires ¡5 cº = 2:5 £ 10 . Phenomenological Constraints: Consider a decay process: A ! B + C This is allowed if MA > MB + MC. 2 In case of VLI, one may write effective mass of the particles as mX = mX + ®~p . Thus if A is superluminal, the decay condition becomes 2 2 2 MA + ®A~p A > MB + ®B ~p B + MC + ®C ~p C Consider pion decay: ¼ ! ¹ + º¹ If º¹ is superluminal (®¼ = ®¹ = 0 and ®º¹ = ®), at some high energies 2 2 (m¼ ¡ m¹)=® < mº¹ pions cannot decay. 2 On the other hand, at some even higher energies, when mº¹ + ®~p , º¹ can decay º¹ ! ¼ + ¹ These processes imply stringent constraints. But they are approximate and not correct. The decay constraint is obtained using energy-momentum conservation in the rest frame of pions, not in the laboratory frame. In VLI models they are not equivalent. Pion Decay Kinematics D E @E Defining, in general (for massless particles and LIV), @p = 1 + ® as the effective average from pmin up to the typical momenta of » 17 GeV of neutrinos, we write the energy-momentum relation Eº = pº(1 + ®) ; where ® corresponds to VLI. To avoid other constraints, assume all particles (except the neutrinos) have terminal 2 2 1=2 velocity c = 1, and satisfy, Ei = (pi + mi ) ; The four vector of the decaying pion can then be written as (E¼; p¼; 0; 0) and those of the final neutrino and muon can be given by ¡ ¢ (Eº; pº`; pºt; 0) and E¹; p¹`; p¹t; 0 : The longitudinal components of momenta are pº` = ´p¼ and p¹` = (1 ¡ ´)p¼ and the transverse components are pºt = ¡p¹t = pt. 2 The energy conservation E¼ = Eº + E¹ ; then gives (for large p¼ and positive pt : · ¸ 1 m2 ®´ · m2 ¡ ¹ 2 ¼ 2p¼ (1 ¡ ´) In OPERA average neutrino energy is » 17 GeV that arise from the decay of pions with a mean energy of » 60 GeV, so that the typical value of h´i ¼ 0:3. We thus obtain a bound that rules out all models of VLI: · ¸ 1 m2 ® · m2 ¡ ¹ » 4 £ 10¡6 : OP ERA 2 ¼ 0:6p¼ 0:7 Cosmic ray muon spectra at higher energies give stronger constraint on VLI The observed differential energy spectrum is well represented by · ½ ¾ » ¯¡1 h1 ¡ ´i¼E¼ f¹(E) = A¼h1 ¡ ´i ¼ E + h1 ¡ ´i E ½ ¼ ¼ ¾¸ ¯¡1 h1 ¡ ´iKEK ¡¯ AKh1 ¡ ´iK E ; E + h1 ¡ ´iKEK where ¯ = spectral index of the cosmic ray spectrum » 2:65; E¼ = h0(θ)m¼=c¿¼; EK = h0(θ)mK=c; ¿K; A¼=K = Constants. 5 h0(θ) = 7 £ 10 sec θ cm, the scale height of the Earth’s atmosphere at a zenith angle θ ¿¼=K = decay lifetimes of pions/kaons at rest h1 ¡ ´i¼=K = the fractional momenta carried away by the muons in pion/kaon decay averaged over the spectrum of cosmic rays, around the energy band of interest. Considering the pion and kaon contributions and ´ distributions, we can arrive at the following bounds: Muon spectra at Baksan fits very well with this theoretical estimate up to » 4 £ 104 GeV. Thus the muon data imply ® < 10¡11 : Similarly the atmospheric neutrino spectrum up to » 400 TeV by the Ice-Cube Collaboration imply a constraint ® < 10¡13 : To explain OPERA we need ® » 10¡5, which is not allowed by cosmic ray data. Violation of Equivalence Principle In weak field limit with Newtonian potential Á = GM=r ¿ 1 consider effective metric: g44 = 1 ¡ 2®Á gij = ¡±ij(1 + 2γÁ) : In GTR, ® = γ = 1. Assuming γ = 1 and different ®i (VEP), the dispersion relation becomes: m2 E0 = p + ¡ ®Áp ; 2p µ ¶ ®1Áp If two VEP states jºGi = differ from the flavor eigenstates ®2Áp jºF i = U(θG)jºGi, there can be neutrino oscillations. In presence of both mass oscillation and VEP oscillation (with same mixing angles θM = θG), the oscillation length becomes £ ¤¡1 L = 2¼ ¢m2=2p + pÁ¢® Cosmic rays data constraints VEP, because the dispersion relation is modified. The Standard Model and the Higgs Scalar: Strong Interaction: Force between the quarks – constituents of nucleons and mesons (hadrons) Electromagnetic Interaction: Force between charged particles Weak Interaction: Neutron decay n ! p + e¡ +º ¹. Only left-handed particles take part in this interaction (charged current). At energies above 250 GeV, they combine into Strong and Electroweak Interaction: The Standard Model Symmetry group SU(3)c £ SU(2)L £ U(1)Y breaks into SU(3)c £ U(1)Q At higher energies before the symmetry breaking SU(3)c: 8 massless Gluons mediating strong interaction among quarks §;3 SU(2)L: 3 massless gauge bosons W , that interact with only the left-handed particles.
Load more

Recommended publications
  • 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]
  • 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]
  • 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]
  • 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]
  • Theoretical Investigation of Subluminal Particles Endowed with Imaginary Mass
    Article Theoretical Investigation of Subluminal Particles Endowed with Imaginary Mass Luca Nanni Faculty of Natural Science, University of Ferrara, 44122 Ferrara, Italy; [email protected] Abstract: In this article, the general solution of the tachyonic Klein–Gordon equation is obtained as a Fourier integral performed on a suitable path in the complex w-plane. In particular, it is proved that this solution does not contain any superluminal components under the given boundary conditions. On the basis of this result, we infer that all possible spacelike wave equations describe the dynamics of subluminal particles endowed with imaginary mass. This result is validated for the Chodos equation, used to describe the hypothetical superluminal behaviour of the neutrino. In this specific framework, it is proved that the wave packet propagates in spacetime with subluminal group velocities and that it behaves as a localized wave for sufficiently small energies. Keywords: tachyons; Klein–Gordon equation; tachyon-like Dirac equation; relativistic wave packet PACS: 03.65.Pm; 03.65.Ge; 03.65.Db 1. Introduction The physics of tachyons is an intriguing and fascinating subject that has attracted the attention of physicists in both the pre-relativistic [1–3] and post-relativistic epochs [4–10]. Citation: Nanni, L. Theoretical However, the majority of the scientific community still considers it a speculative theory, and Investigation of Subluminal Particles Endowed with Imaginary Mass. there are only a few efforts aimed at finding the experimental evidence that could make it a Particles 2021, 4, 325–332. https:// candidate for extending the current Standard Model [11]. A comprehensive overview of the doi.org/10.3390/particles4020027 most relevant experiments dealing with the superluminal behaviour of waves and particles is detailed in [9,12–15].
    [Show full text]
  • What Might the Matter Wave Be Telling Us of the Nature of Matter?
    What might the matter wave be telling us of the nature of matter? Daniel Shanahan PO Box 301, Cleveland, Queensland, 4163, Australia August 21, 2019 Abstract Various attempts at a thoroughly wave-theoretic explanation of mat- ter have taken as their fundamental ingredient the de Broglie or matter wave. But that wave is superluminal whereas it is implicit in the Lorentz transformation that influences propagate ultimately at the velocity c of light. It is shown that if the de Broglie wave is understood, not as a wave in its own right, but as the relativistically induced modulation of an underlying standing wave comprising counter-propagating influences of velocity c, the energy, momentum, mass and inertia of a massive particle can be explained from the manner in which the modulated wave struc- ture must adapt to a change of inertial frame. With those properties of the particle explained entirely from wave structure, nothing remains to be apportioned to anything discrete or “solid”within the wave. Considera- tion may thus be given to the possibility of wave-theoretic explanations of particle trajectories, and to a deeper understanding of the Klein-Gordon, Schrödinger and Dirac equations, all of which were conceived as equations for the de Broglie wave. Keywords de Broglie wave Planck-Einstein relation wave-particle duality inertia pilot wave theory· Dirac bispinor Lorentz· transforma- tion · · · · 1 Introduction It might be thought that the de Broglie wave can say very little regarding the nature of solid matter. As this “matter wave”is usually understood, it seems to make no sense at all.
    [Show full text]
  • Particle Mass Oscillation Through Tachyon Interaction
    Particle Mass Oscillation through Tachyon Interaction L. Nanni† University of Ferrara 44100-Ferrara, Italy †E-mail: [email protected] Abstract: In this study, a novel theory to investigate the mass oscillation of particles is proposed. It has been proven that, at high-energy conditions, the fermion field described by Dirac’s Lagrangian interacts with the half-integer spin tachyon field with negative energy, causing the formation of composite particles whose mass depends on the total angular momentum. The proposed theory is based on a new interpretation of the Majorana equation for particles with arbitrary spin and shows that mass oscillation is a phenomenon in which the component of particle decay prevails over that of mixing mass states. Using the kinematic of Lemke for spacelike particle decay, we propose a mechanism able to explain the neutrino flavour change. The proposed mechanism is also investigated concerning the shape of its spectrum. Finally, the Lagrangian field of composite particles is formulated. Keywords: Mass oscillation; bradyon-tachyon interaction; infinite-component spinor; Standard Model. PACS numbers: 03.65.Pm, 03.70.+k 1. Introduction The mass oscillation of particles is one of the unsolved problems of particle physics and, together with those concerning the origin of masses, flavours and quantum numbers, make the current Standard Model an unsatisfactory and incomplete theory [1-2]. To date, the only case of mass oscillation experimentally observed is the flavour change of the neutrino during free flight [3-4], a phenomenon for which there is not yet a robust theory capable of explaining it. To overcome this impasse, the route of supersymmetry theory has been undertaken, which is expected to be the best candidate to definitively resolve all the current shortcomings of the Standard Model [5-7].
    [Show full text]
  • Symmetries of Massive and Massless Neutrinos
    Symmetries of massive and massless Neutrinos Y. S. Kim University of Maryland since 1962 College Park, Maryland nr. Washington, DC, USA Gatchina, Russia, June 2016 Einstein invented Lorentz transformations for point particles, but never worried about the orbit of the hydrogen atom, even though he talked with Bohr before and after 1927. Newton New Mathematics People seem to have this picture of the Lorentz-boosted orbit. Included in the symmetries of particles are Massless particles. 2. Are they Lorentz-boosted massive particles? 3. Why are the spins parallel to momentum? 4. Why are neutrinos polarized? Among the massless particles Photon can be both right-handed and left-handed, while neutrinos are polarized. Photons have a gauge degree of freedom, while neutrinos do not. Neutrino polarization as a consequence of gauge invariance? Internal Space-time symmetries Eugene Paul Wigner On his little groups 1939 Wigner got his Nobel in 1963, but not for his 1939 paper addressing issues on internal space-time symmetries. It is generally agreed that he deserved the prize for his 1939 paper on the little groups. What is wrong with the Wigner did not explain the 1939 paper? physics of this matrix. This is why he did not get the prize for his This paper contains the matrix 1939 paper. This matrix remained as the ugliest matrix in physics. Lorentz group consists of three rotational and thee boost degrees of freedom. The rotation group O(3) is a subgroup of the Lorentz group. Wigner’s Little Groups: Subgroups of the Lorentz group whose transformations keep the given momentum of a particle invariant.
    [Show 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.
    [Show full text]
  • Overcoming the Bradyon–Tachyon Barrier
    Overcoming the Bradyon–Tachyon Barrier *Luca Nanni *corresponding author e-mail: [email protected] Pacs: 03.65Pm Keywords: particles with arbitrary spin, light barrier, tachyon. Abstract In this study, the problem of overcoming the infinite energy barrier separating the bradyonic and tachyonic realms is investigated. Making use of the Majorana equation for particles with arbitrary spin and the Heisenberg uncertainty principle, it is proved that, under certain conditions of spatial confinement, quantum fluctuations allow particles with very small mass and velocity close to the speed of light to pass in the tachyonic realm, avoiding the problem of the infinite barrier (bradyon–tachyon tunnelling). This theoretical approach allows an avoidance of the difficulties encountered in quantum field theory when it is extended to particles with imaginary rest mass. 1 Introduction Faster than light particles (tachyons) have been a topic that the most eminent theoretical physicists of the last century, such as Sommerfeld [1], Pauli [2] and Feynman [3], have dealt with in the course of their research. Although there is still no experimental evidence for the existence of tachyons (or even experimental evidence that definitively proves their false physical reality), several researchers have been working to introduce this hypothetical particle into the modern quantum theory without compromising the foundations [4–6]. In particular, Surdashan, Binaliuk and Recami [7–9] have proved that, by introducing the reinterpretation principle and extending the theory of relativity to superluminal frames and objects, it is possible to solve the problem concerning the violation of the principle of causality that since the beginning has led physicists to consider the tachyon as a particle that is incompatible with physical reality [11–12].
    [Show full text]
  • Notes on Nuclear and Elementary Particle Physics
    Notes on Nuclear and Elementary Particle Physics Daniel F. Styer; Schiffer Professor of Physics; Oberlin College Copyright c 2 June 2021 Abstract: Even in intricate situations where closed-form solutions are unavailable, relativity and quantum mechanics combine to provide qualitative explanations for otherwise inexplicable phenomena. The copyright holder grants the freedom to copy, modify, convey, adapt, and/or redistribute this work under the terms of the Creative Commons Attribution Share Alike 4.0 International License. A copy of that license is available at http://creativecommons.org/licenses/by-sa/4.0/legalcode. Contents 1 Characteristics of Nuclei 3 2 Radioactivity 9 3 \Something Is Missing" 13 4 Nuclear Stability 19 5 Balance of Masses 26 6 Elementary Particles 27 7 Epilogue 38 2 Chapter 1 Characteristics of Nuclei Contents. A nucleus is made up of protons and neutrons. (The word \nucleon" 1 means \either a proton or a neutron". Nucleons are spin- 2 fermions.) For example, the most common isotope of oxygen has 8 protons and 8 neutrons. This nucleus is denoted by 16 8 O: The number of protons, in this case 8, is called the \atomic number" Z. (The symbol Z originates from the German \Zahl", meaning \number".) The total number of nucleons, in this case 16, is called the \mass number" A. It is called \mass number" because the mass of the nucleus is nearly A times the mass of a proton. This relation is not exactly true because (1) a neutron is slightly more massive (about 0.14% more) than a proton and (2) as we learned in relativity, the mass of the nucleus is slightly less than the sum of the masses of its constituents.
    [Show full text]