DOCSLIB.ORG

Hydrodynamics of the Dark Superfluid: I. Genesis of Fundamental Particles Marco Fedi

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Hydrodynamics of the Dark Superfluid: I. Genesis of Fundamental Particles Marco Fedi Hydrodynamics of the dark superfluid: I. genesis of fundamental particles Marco Fedi To cite this version: Marco Fedi. Hydrodynamics of the dark superfluid: I. genesis of fundamental particles. 2017. hal- 01549082v1 HAL Id: hal-01549082 https://hal.archives-ouvertes.fr/hal-01549082v1 Preprint submitted on 28 Jun 2017 (v1), last revised 19 Jul 2017 (v2) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. manuscript No. (will be inserted by the editor) Hydrodynamics of the dark superfluid: I. genesis of fundamental particles. Marco Fedi Received: date / Accepted: date Abstract Here we consider the existence of an ubiquitous 1 Introduction: a superfluid universe? dark superfluid which fills the universe corresponding to dark energy (∼ 70% of the universe mass-energy, also expressed K.Huang has dedicated a book [9] to the possibility that our in the cosmological constant) and dark matter. As in oth- universe possess superfluid features. Here we want to con- er superfluids, quantum vortices would originate, whose ge- tinue the theoretical investigation, also taking into account ometry can describe the spin of fundamental particles, sug- recent experiments and observations and in this first paper gesting the validity of a quantum hydrodynamic approach concerning the hydrodynamics of the dark superfluid we fo- to particle physics. This seething dark superfluid would also cus on fundamental particles described as topological de- justify the continuous formation and annihilation of virtu- fects (quantum vortices) in this dark medium. Data from the al particles and antiparticles in quantum vacuum as vortex- ESA Planck Space Observatory say that dark energy con- antivortex pairs, letting us reinterpret quantum vacuum as stitutes 69:1% of the universe mass-energy. The only de- hydrodynamic spontaneous fluctuations of this dark super- tectable effect currently attributed to dark energy is its re- fluid. In this case, it would be correct to say that dark en- pulsive force, which opposes gravity and causes the expan- ergy (or rather the dark superfluid) does not interact with sion of the universe. Since it is otherwise undetectable and our baryon world unless it is hydrodynamically perturbed, its density does not affect the motion of the celestial bodies, in which case it could even correspond to the particles of it is believed that it does not interact with ordinary matter. the Standard Model. Antimatter and its absence in the uni- On the contrary, we speculate that it may be a fundamental verse, as well as the decay of particles may also have place superfluid, whose internal pressure opposes gravity [1] –– in a superfluid model, completing the picture, while funda- indeed the energy of vacuum per unit volume (whose val- mental forces exerted by particles, such as gravity or elec- ue is currently believed to be ∼ 5:7 · 10−10J=m3)1 has units tromagnetism, have their hydrodynamic equivalents in the of pressure –– and whose hydrodynamic perturbation may Bernoulli force which may cause attraction or repulsion be- even correspond to the known massive fundamental parti- tween quantum vortices. As regards the repulsive force at- cles, as quantized vortices, a picture thanks to which we can tributed to dark energy in cosmology, it would depend on also understand quantum vacuum and the infinite pairs of the internal pressure of the dark superfluid, represented by virtual particles and antiparticles which continuously form its energy density, while the cosmic microwave background and annihilate in it, probably as vortex-antivortex pairs, as at ∼ 2:72K would be its superfluidity temperature, consis- indeed observed in other superfluids and in polariton con- tent with those of other superfluids as 4He. densates [35,36]. Quantum vacuum or here the cosmic and ubiquitous dark superfluid2 (DS), would be a cosmic Bose- Keywords superfluid vacuum · dark energy · dark matter · particle physics · spin 1 resorting to the upper value of the cosmological constant, other- PACS 95.36.+x · 03.75.Kk · 03.65.-w · 03.75.Nt · 95.35.+d wise a much higher value according to quantum and stochastic electro- dynamics. M. Fedi 2 we believe that here the term dark energy may be confusing as it Ministero dell’Istruzione, dell’Universita` e della Ricerca (MIUR), does not underline its mass density and hypothesized superfluidity. The Rome, Italy definition dark superfluid also helps to understand dark matter as a hy- E-mail: [email protected] drodynamic manifestation of superfluid vacuum, as we argue (Fig.1). 2 Marco Fedi Einstein condensate (BEC) [1–10] of dark energy’s quanta, we call them dark superfluid quanta (DSQ). The existence of a false vacuum with non-zero energy content is definitively accepted and proven in a lot of physical phenomena as the Lamb shift, the Casimir effect, the Unruh effect, the anoma- lous magnetic moment, vacuum birefringence [29] etc. All these phenomena could be rooted in the ubiquitous DS. Its quantum granular, foamy nature seems to have been con- firmed by G. Amelino-Camelia in a recent statistical analy- sis on the observation of neutrinos and photons using to data from the IceCube and the Fermi GLAST [37]. Let us reflect on vacuum fluctuations 4E4t ≥ h¯: (1) H The circulation in a quantized vortex, C v · dl= nh=m, mul- tiplied by mass becomes the Bohr-Sommerfeld quantization condition, expressing mass circulation in a quantized vortex I p · dx = nh: (2) C Fig. 1 Left: (a) Metal atoms trapped in superfluid helium vortices high- For n = 1, Eq. (2) tells us that the quantum of action, h; light a structure of vortex filaments [25]; (b) galactic filaments of dark actually refers to a complete turn along a circular path of matter which galaxies aggregate on. Here the relationship between dark energy and dark matter is the same existing between superfluid helium a quantum whose momentum is p. In (1) 2p (h¯ = h=2p) and the vortex filaments which manifest in it, i.e. we suppose dark mat- also refers to a complete turn and we can interpret vacuum ter is a hydrodynamic manifestation of dark energy [4,10,38]. fluctuations as the circulation of DSQ in quantized vortices during a time 4t (1), before vortex-antivortex annihilation occurs. structure of a vortex web arising in a familiar superfluid such 4 Quantum vortices are known to manifest in superfluids, as He. Within this analogy all the space among the fila- as in superfluid 4He [8,11], so this is a first clue to consid- ments is occupied by liquid helium in (a) and by the DS in er quantum vacuum as a superfluid. Furthermore, fermion- (b). s spin-½ may be described in hydrodynamic terms as the Moreover, the equation of state of cosmology for a single- circulation of DSQ in a torus vortex (see Sect. 3). Thus, if fluid model can be referred to superfluid dark energy, where vacuum fluctuations are superfluid vortices, which is the un- Pd and rd are respectively the pressure and the density of derlying superfluid in which they arise? dark energy [31] Also according to Huang [9], the possible answers are Pd either the Higgs field or dark energy, both observed as “dark” w = (3) rd scalar fields. Being the Higgs boson the fundamental excita- tion of the Higgs field and very massive, it is probably a vor- Treating the universe as a fluid medium is therefore already tex itself, so we opt for dark energy, as a cosmic fundamen- a standard praxis in cosmology. The analogy shown in Fig. 1 tal scalar field with superfluid features. After all, we know it let us also suppose that the universe was (or still is) spinning. constitutes ∼ 69% of the mass-energy of the universe, also But to be precise, it would be the DS filling the universe that expressed in the cosmological constant, L = kr0, of Einstein would be spinning. Indeed, we know that rotatory motion 00 field equation, where r0 (T ; as regards the stress-energy produces vortex structures in laboratory superfluids. The ob- tensor) indicates the density of dark energy. Along with dark served asymmetry in spiral galaxies chirality [26] suggests matter, which can be interpreted as “condensed” dark ener- the rotation of the cosmic superfluid, and this fact would also gy [4,10] and whose existence is evident in the dark halos explain the left-handed bias in the world of fundamental par- of spiral galaxies which the flat profiles of orbital velocities ticles and even of organic molecules. Lathrop [9,22] has hy- are believed to be due to, we arrive at ∼ 95%. Thus, we are pothesized the genesis of spiral galaxies from giant vortices probably facing a superfluid universe [9,10]. of dark matter on which matter particles accrete and gravi- The temperature of the cosmic microwaves background tate, up to creating a black hole at the center of any galaxy. radiation (CMB), ∼ 2:72 K, would be in agreement with the Moreover, if particles have formed as quantized vortices in temperature of other superfluids such as 4He. Fig. 1 shows the spinning DS, this would also be the reason for the non- dark matter distribution in the universe in analogy with the detection of antimatter in the universe (vortex-particles have Hydrodynamics of the dark superfluid: I. genesis of fundamental particles. 3 arisen as either left- or right-handed) and would be a further gle sustained by vortex reconnection) in the early universe clue for a superfluid origin of fundamental particles.
Load more

Recommended publications
  • Quantum Trajectories: Real Or Surreal?
    entropy Article Quantum Trajectories: Real or Surreal? Basil J. Hiley * and Peter Van Reeth * Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK * Correspondence: [email protected] (B.J.H.); [email protected] (P.V.R.) Received: 8 April 2018; Accepted: 2 May 2018; Published: 8 May 2018 Abstract: The claim of Kocsis et al. to have experimentally determined “photon trajectories” calls for a re-examination of the meaning of “quantum trajectories”. We will review the arguments that have been assumed to have established that a trajectory has no meaning in the context of quantum mechanics. We show that the conclusion that the Bohm trajectories should be called “surreal” because they are at “variance with the actual observed track” of a particle is wrong as it is based on a false argument. We also present the results of a numerical investigation of a double Stern-Gerlach experiment which shows clearly the role of the spin within the Bohm formalism and discuss situations where the appearance of the quantum potential is open to direct experimental exploration. Keywords: Stern-Gerlach; trajectories; spin 1. Introduction The recent claims to have observed “photon trajectories” [1–3] calls for a re-examination of what we precisely mean by a “particle trajectory” in the quantum domain. Mahler et al. [2] applied the Bohm approach [4] based on the non-relativistic Schrödinger equation to interpret their results, claiming their empirical evidence supported this approach producing “trajectories” remarkably similar to those presented in Philippidis, Dewdney and Hiley [5]. However, the Schrödinger equation cannot be applied to photons because photons have zero rest mass and are relativistic “particles” which must be treated differently.
    [Show full text]
  • The Stueckelberg Wave Equation and the Anomalous Magnetic Moment of the Electron
    The Stueckelberg wave equation and the anomalous magnetic moment of the electron A. F. Bennett College of Earth, Ocean and Atmospheric Sciences Oregon State University 104 CEOAS Administration Building Corvallis, OR 97331-5503, USA E-mail: [email protected] Abstract. The parametrized relativistic quantum mechanics of Stueckelberg [Helv. Phys. Acta 15, 23 (1942)] represents time as an operator, and has been shown elsewhere to yield the recently observed phenomena of quantum interference in time, quantum diffraction in time and quantum entanglement in time. The Stueckelberg wave equation as extended to a spin–1/2 particle by Horwitz and Arshansky [J. Phys. A: Math. Gen. 15, L659 (1982)] is shown here to yield the electron g–factor g = 2(1+ α/2π), to leading order in the renormalized fine structure constant α, in agreement with the quantum electrodynamics of Schwinger [Phys. Rev., 73, 416L (1948)]. PACS numbers: 03.65.Nk, 03.65.Pm, 03.65.Sq Keywords: relativistic quantum mechanics, quantum coherence in time, anomalous magnetic moment 1. Introduction The relativistic quantum mechanics of Dirac [1, 2] represents position as an operator and time as a parameter. The Dirac wave functions can be normalized over space with respect to a Lorentz–invariant measure of volume [2, Ch 3], but cannot be meaningfully normalized over time. Thus the Dirac formalism offers no precise meaning for the expectation of time [3, 9.5], and offers no representations for the recently– § observed phenomena of quantum interference in time [4, 5], quantum diffraction in time [6, 7] and quantum entanglement in time [8]. Quantum interference patterns and diffraction patterns [9, Chs 1–3] are multi–lobed probability distribution functions for the eigenvalues of an Hermitian operator, which is typically position.
    [Show full text]
  • Bohmian Mechanics Versus Madelung Quantum Hydrodynamics
    Ann. Univ. Sofia, Fac. Phys. Special Edition (2012) 112-119 [arXiv 0904.0723] Bohmian mechanics versus Madelung quantum hydrodynamics Roumen Tsekov Department of Physical Chemistry, University of Sofia, 1164 Sofia, Bulgaria It is shown that the Bohmian mechanics and the Madelung quantum hy- drodynamics are different theories and the latter is a better ontological interpre- tation of quantum mechanics. A new stochastic interpretation of quantum me- chanics is proposed, which is the background of the Madelung quantum hydro- dynamics. Its relation to the complex mechanics is also explored. A new complex hydrodynamics is proposed, which eliminates completely the Bohm quantum po- tential. It describes the quantum evolution of the probability density by a con- vective diffusion with imaginary transport coefficients. The Copenhagen interpretation of quantum mechanics is guilty for the quantum mys- tery and many strange phenomena such as the Schrödinger cat, parallel quantum and classical worlds, wave-particle duality, decoherence, etc. Many scientists have tried, however, to put the quantum mechanics back on ontological foundations. For instance, Bohm [1] proposed an al- ternative interpretation of quantum mechanics, which is able to overcome some puzzles of the Copenhagen interpretation. He developed further the de Broglie pilot-wave theory and, for this reason, the Bohmian mechanics is also known as the de Broglie-Bohm theory. At the time of inception of quantum mechanics Madelung [2] has demonstrated that the Schrödinger equa- tion can be transformed in hydrodynamic form. This so-called Madelung quantum hydrodynam- ics is a less elaborated theory and usually considered as a precursor of the Bohmian mechanics. The scope of the present paper is to show that these two theories are different and the Made- lung hydrodynamics is a better interpretation of quantum mechanics than the Bohmian me- chanics.
    [Show full text]
  • The Quantum Potential And''causal''trajectories For
    REFERENCE IC/88/161 \ ^' I 6 AUG INTERNATIONAL CENTRE FOR / Co THEORETICAL PHYSICS THE QUANTUM POTENTIAL AND "CAUSAL" TRAJECTORIES FOR STATIONARY STATES AND FOR COHERENT STATES A.O. Barut and M. Bo2ic INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION IC/88/161 I. Introduction International Atomic Energy Agency During the last decade Bohm's causal trajectories based on the and concept of the so called "quantum potential" /I/ have been evaluated United Nations Educational Scientific and Cultural Organization in a number of cases, the double slit /2,3/, tunnelling through the INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS barrier /4,5/, the Stern-Gerlach magnet /6/, neutron interferometer /8,9/, EPRB experiment /10/. In all these works the trajectories have been evaluated numerically for the time dependent wave packet of the Schrodinger equation. THE QUANTUM POTENTIAL AND "CAUSAL" TRAJECTORIES FOR STATIONARY STATES AND FOR COHERENT STATES * In the present paper we study explicitly the quantum potential and the associated trajectories for stationary states and for coherent states. We show how the quantum potential arises from the classical A.O. Barut ** and M. totii *** action. The classical action is a sum of two parts S ,= S^ + Sz> one International Centre for Theoretical Physics, Trieste, Italy. part Sj becomes the quantum action, the other part S2 is shifted to the quantum potential. We also show that the two definitions of causal ABSTRACT trajectories do not coincide for stationary states. We show for stationary states in a central potential that the quantum The content of our paper is as follows. Section II is devoted to action S is only a part of the classical action W and derive an expression for the "quantum potential" IL in terms of the other part.
    [Show full text]
  • Photon Wave Mechanics: a De Broglie - Bohm Approach
    PHOTON WAVE MECHANICS: A DE BROGLIE - BOHM APPROACH S. ESPOSITO Dipartimento di Scienze Fisiche, Universit`adi Napoli “Federico II” and Istituto Nazionale di Fisica Nucleare, Sezione di Napoli Mostra d’Oltremare Pad. 20, I-80125 Napoli Italy e-mail: [email protected] Abstract. We compare the de Broglie - Bohm theory for non-relativistic, scalar matter particles with the Majorana-R¨omer theory of electrodynamics, pointing out the impressive common pecu- liarities and the role of the spin in both theories. A novel insight into photon wave mechanics is envisaged. 1. Introduction Modern Quantum Mechanics was born with the observation of Heisenberg [1] that in atomic (and subatomic) systems there are directly observable quantities, such as emission frequencies, intensities and so on, as well as non directly observable quantities such as, for example, the position coordinates of an electron in an atom at a given time instant. The later fruitful developments of the quantum formalism was then devoted to connect observable quantities between them without the use of a model, differently to what happened in the framework of old quantum mechanics where specific geometrical and mechanical models were investigated to deduce the values of the observable quantities from a substantially non observable underlying structure. We now know that quantum phenomena are completely described by a complex- valued state function ψ satisfying the Schr¨odinger equation. The probabilistic in- terpretation of it was first suggested by Born [2] and, in the light of Heisenberg uncertainty principle, is a pillar of quantum mechanics itself. All the known experiments show that the probabilistic interpretation of the wave function is indeed the correct one (see any textbook on quantum mechanics, for 2 S.
    [Show full text]
  • Relativistic Quantum Field Theory - Implications for Gestalt Therapy (Brian O’Neil)
    Relativistic Quantum Field Theory - Implications for Gestalt Therapy (Brian O’Neil) It is timely to address how the terms ‘field’ and ‘field theory’ have developed in physics and Gestalt therapy and the inter-relation between the two. Gestalt therapy writers such as Parlett have noted advances in physics which have moved beyond the initial formulation of Maxwell’s electromagnetic field. This includes the quantum field, the relativitistic quantum field, and now post relativistic quantum fields through the work of Bohm and others. These advanced notions of field are beyond the original field theory taken up by Smuts, Lewin, Wertheimer and Gestalt therapy. The possible implication of this expanded field perception in physics for Gestalt therapy is outlined theoretically and clinically. Arjuna Said What are Nature and Self? What are the field and its Knower, knowledge and the object of knowledge? Teach me about them Krishna. Krisna: I am the Knower of the field in everybody Arjuna. genuine knowledge means knowing both the field and the Knower The Bhagavad Gita This paper is offered as a heuristic device to better understand the impact and application of field theory in Gestalt therapy and to refresh and enrich the dialogue between Gestalt therapy and other domains, specifically physics and biology. In particular, to what extent are the terms ‘field’ and field theory’ used as an epistemology (ie as a method of obtaining and validating knowledge) and as an ontology (ie expressing the nature of being), with the understanding that each term is not exclusive of the other. In the present state of physics there is an acceptance of ‘field’ as an ontological reality, while many Gestalt therapy writers relate to ‘field’ epistemologically as a metaphor or method.
    [Show full text]
  • Multi-Field and Bohm's Theory
    Multi-field and Bohm’s theory Davide Romano∗ June 02, 2020 forthcoming in Synthese Abstract In the recent literature, it has been shown that the wave function in the de Broglie–Bohm theory can be regarded as a new kind of field, i.e., a “multi-field”, in three-dimensional space. In this paper, I argue that the natural framework for the multi-field is the original second- order Bohm’s theory. In this context, it is possible: i) to construe the multi-field as a real scalar field; ii) to explain the physical interaction between the multi-field and the Bohmian particles; and iii) to clarify the status of the energy-momentum conservation and the dynamics of the theory. Contents 1 Introduction2 2 Is the wave function nomological?5 2.1 The nomological view: a brief introduction...........5 2.2 The analogy between the wave function and the Hamiltonian function..............................6 2.3 Top-down ( ) versus bottom-up (H) functions.........6 2.4 Time-dependence of the wave function.............9 2.5 Interaction between V and ................... 10 3 Bohm’s theory and Bohmian mechanics 12 3.1 Bohmian mechanics........................ 12 3.2 Bohm’s theory........................... 13 ∗University of Salzburg, philosophy department. E-mail: [email protected] 1 4 Why Bohm’s theory 15 5 Multi-field(s) from Bohm’s theory 17 6 Further considerations 19 6.1 Why not first order?....................... 19 6.2 Energy and momentum conservation.............. 22 6.3 A possible objection: the no-back reaction problem...... 23 7 Conclusions 27 1 Introduction Quantum mechanics describes the behavior of microscopic systems via a mathematical function defined on the system’s configuration space: the wave function.
    [Show full text]
  • Quantum Mechanics of the Electron Particle-Clock
    Quantum Mechanics of the electron particle-clock David Hestenes Department of Physics, Arizona State University, Tempe, Arizona 85287-1504∗ Understanding the electron clock and the role of complex numbers in quantum mechanics is grounded in the geometry of spacetime, and best expressed with Spacetime Algebra (STA). The efficiency of STA is demonstrated with coordinate-free applications to both relativistic and non- relativistic QM. Insight into the structure of Dirac theory is provided by a new comprehensive analysis of local observables and conservation laws. This provides the foundation for a comparative analysis of wave and particle models for the hydrogen atom, the workshop where Quantum Mechanics was designed, built and tested. PACS numbers: 10,03.65.-w Keywords: pilot waves, zitterbewegung, spacetime algebra I. INTRODUCTION The final Section argues for a realist interpretation of the Dirac wave function describing possible particle De Broglie always insisted that Quantum Mechanics is paths. a relativistic theory. Indeed, it began with de Broglie`s relativistic model for an electron clock. Ironically, when Schr¨odingerintroduced de Broglie`s idea into his wave II. SPACETIME ALGEBRA equation, he dispensed with both the clock and the rela- tivity! Spacetime Algebra (STA) plays an essential role in the This paper aims to revitalize de Broglie's idea of an formulation and analysis of electron theory in this paper. electron clock by giving it a central role in physical inter- Since thorough expositions of STA are available in many pretation of the Dirac wave function. In particular, we places [2{4], a brief description will suffice here, mainly aim for insight into structure of the wave function and to establish notations and define terms.
    [Show full text]
  • The Quantum Theory of Motion Firstly Introduced by Louis De Broglie, and Later and Independently by David Bohm, Has Several Attractive Features
    INERTIAL TRAJECTORIES IN DE BROGLIE-BOHM QUANTUM THEORY: AN UNEXPECTED PROBLEM1 PABLO ACUÑA INSTITUTO DE FILOSOFÍA PONTIFICIA UNIVERSIDAD CATÓLICA DE VALPARAÍSO AVENIDA EL BOSQUE 1290, 2530388, VIÑA DEL MAR, CHILE [email protected] A salient feature of de Broglie-Bohm quantum theory is that particles have de- terminate positions at all times and in all physical contexts. Hence, the trajec- tory of a particle is a well-defined concept. One then may expect that the closely related notion of inertial trajectory is also unproblematically defined. I show that this expectation is not met. I provide a framework that deploys six different ways in which de Broglie-Bohm theory can be interpreted, and I state that only in the canonical interpretation the concept of inertial trajectory is the customary one. However, in this interpretation the description of the dynamical interac- tion between the pilot-wave and the particles, which is crucial to distinguish inertial from non-inertial trajectories, is affected by serious difficulties, so other readings of the theory intend to avoid them. I show that in the alternative in- terpretations the concept at issue gets either drastically altered, or plainly un- defined. I also spell out further conceptual difficulties that are associated to the redefinitions of inertial trajectories, or to the absence of the concept. 1. INTRODUCTION The quantum theory of motion firstly introduced by Louis de Broglie, and later and independently by David Bohm, has several attractive features. Most notably: it is not affected by the measurement problem, it provides an intuitive and visualisable explanation of quantum phenomena, it offers a clear account of the classical limit, particles are always distinguishable, and they have definite positions at all times and in all contexts.
    [Show full text]
  • Quantum Mathematics
    Quantum Mathematics by Peter J. Olver University of Minnesota Table of Contents 1. Introduction .................... 3 2. Hamiltonian Mechanics ............... 3 Hamiltonian Systems ................ 3 Poisson Brackets ................. 7 Symmetries and First Integrals ........... 9 Hamilton-Jacobi Theory .............. 11 3. Geometric Optics and Wave Mechanics ....... 25 The Wave Equation ................ 25 Maxwell’s Equations ................ 27 High Frequency Limit and Quantization ........ 29 4. Basics of Quantum Mechanics ........... 36 Phase space ....................36 Observables ....................37 Schr¨odinger’s Equation ............... 38 Spectrum .....................39 The Uncertainty Principle ............. 43 5. Linear Operators and Their Spectrum ....... 44 Hilbert Space and Linear Operators ......... 44 The Spectral Theorem ............... 46 Spectrum of Operators ............... 48 Dispersion and Stationary Phase ........... 55 6. One–Dimensional Problems ............. 59 The Harmonic Oscillator .............. 63 Scattering and Inverse Scattering .......... 65 Isopectral Flows ..................75 9/9/21 1 c 2021 Peter J. Olver 7. Symmetry in Quantum Mechanics ......... 77 Why is Symmetry Important? ............ 77 Group Representations ............... 79 Lie Algebra Representations ............. 81 Representations of the Orthogonal Group ....... 83 The Hydrogen Atom ................ 91 The Zeeman Effect ................. 96 Addition of Angular Momenta ............ 97 Intensities and Selection Rules ............104
    [Show full text]
  • Reality of the Wave Function and Quantum Entanglement Mani Bhaumik1 Department of Physics and Astronomy, University of California, Los Angeles, USA.90095
    Reality of the wave function and quantum entanglement Mani Bhaumik1 Department of Physics and Astronomy, University of California, Los Angeles, USA.90095 Abstract The intrinsic fluctuations of the underlying, immutable quantum fields that fill all space and time can support the element of reality of a wave function in quantum mechanics. The mysterious non-locality of quantum entanglement may also be understood in terms of these inherent quantum fluctuations, ever-present at the most fundamental level of the universe. 1. Introduction The reality of the wave function in quantum mechanics has been controversial ever since Schrödinger introduced it in his equation of motion. Nevertheless, it would be reasonable to embrace at least a tentative conception of the quantum reality since quantum formalism invariably underpins the reality of our entire objective universe. 1 e-mail: [email protected] 1 In a recent article [1], it was pointed out that the wave function plausibly represents a structural reality of the very foundation of our universe. In brief, contemporary experimental observations supported by the quantum field theory demonstrate the universal presence and immutability of the abiding quantum fields at the ultimate level of reality as well as some definitive confirmation of the existence of inherent ceaseless fluctuations of these fields. Let us now explore how an elementary particle like electron can have its inescapable associated wave as a result of the incessant fluctuations of the primary quantum field. According to QFT, an electron represents a propagating discrete quantum of the underlying electron field. In other words, an electron is a quantized wave (or a ripple) of the electron quantum field, which acts as a particle because of its well-defined energy, momentum, and mass, which are conserved fundamentals of the electron.
    [Show full text]
  • THE PARTICLE ENIGMA-Restructured
    THE PARTICLE ENIGMA Gerald E. Marsh ABSTRACT The idea that particles are the basic constituents of all matter dates back to ancient times and formed the basis of physical thought well into modern times. The debate about whether light was a wave or a stream of particles also lasted until relatively recently. It was the advent of de Broglie’s work and its implications that revolutionized the concept of an elementary particle—but unfortunately did not banish the idea of a point particle despite its difficulties in both classical and quantum physics. Some of these problems are discussed in this essay, which covers chiral oscillations, Penrose’s “zigzag” picture of particles satisfying the Dirac equation, and some ideas derived from string theory. 1 What is a particle? We all know that the concept of a particle comes from Democritus’ idea of atoms. His conception, and what today we would call Brownian motion, was related by Lucretius to the origin of all motion in his poem On the Nature of Things (50 B.C.E.): Whence Nature all creates, and multiplies And fosters all, and whither she resolves Each in the end when each is overthrown. This ultimate stock we have devised to name Procreant atoms, matter, seeds of things, Or primal bodies, as primal to the world. ! ! ! For thou wilt mark here many a speck, impelled By viewless blows, to change its little course, And beaten backwards to return again, Hither and thither in all directions round. Lo, all their shifting movement is of old, From the primeval atoms; for the same Primordial seeds of things first move of self, And then those bodies built of unions small And nearest, as it were, unto the powers Of the primeval atoms, are stirred up By impulse of those atoms' unseen blows, And these thereafter goad the next in size; Thus motion ascends from the primevals on, And stage by stage emerges to our sense, Until those objects also move which we Can mark in sunbeams, though it not appears What blows do urge them.
    [Show full text]