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Symmetry Conditions on Dirac Observables
Proceedings of Institute of Mathematics of NAS of Ukraine 2004, Vol. 50, Part 2, 671–676 Symmetry Conditions on Dirac Observables Heinz Otto CORDES Department of Mathematics, University of California, Berkeley, CA 94720 USA E-mail: [email protected] Using our Dirac invariant algebra we attempt a mathematically and philosophically clean theory of Dirac observables, within the environment of the 4 books [10,9,11,4]. All classical objections to one-particle Dirac theory seem removed, while also some principal objections to von Neumann’s observable theory may be cured. Heisenberg’s uncertainty principle appears improved. There is a clean and mathematically precise pseudodifferential Foldy–Wouthuysen transform, not only for the supersymmeytric case, but also for general (C∞-) potentials. 1 Introduction The free Dirac Hamiltonian H0 = αD+β is a self-adjoint square root of 1−∆ , with the Laplace HS − 1 operator ∆. The free Schr¨odinger√ Hamiltonian 0 =1 2 ∆ (where “1” is the “rest energy” 2 1 mc ) seems related to H0 like 1+x its approximation 1+ 2 x – second partial sum of its Taylor expansion. From that aspect the early discoverers of Quantum Mechanics were lucky that the energies of the bound states of hydrogen (a few eV) are small, compared to the mass energy of an electron (approx. 500 000 eV), because this should make “Schr¨odinger” a good approximation of “Dirac”. But what about the continuous spectrum of both operators – governing scattering theory. Note that today big machines scatter with energies of about 10,000 – compared to the above 1 = mc2. So, from that aspect the precise scattering theory of both Hamiltonians (with potentials added) should give completely different results, and one may have to put ones trust into one of them, because at most one of them can be applicable. -
Degruyter Opphil Opphil-2020-0010 147..160 ++
Open Philosophy 2020; 3: 147–160 Object Oriented Ontology and Its Critics Simon Weir* Living and Nonliving Occasionalism https://doi.org/10.1515/opphil-2020-0010 received November 05, 2019; accepted February 14, 2020 Abstract: Graham Harman’s Object-Oriented Ontology has employed a variant of occasionalist causation since 2002, with sensual objects acting as the mediators of causation between real objects. While the mechanism for living beings creating sensual objects is clear, how nonliving objects generate sensual objects is not. This essay sets out an interpretation of occasionalism where the mediating agency of nonliving contact is the virtual particles of nominally empty space. Since living, conscious, real objects need to hold sensual objects as sub-components, but nonliving objects do not, this leads to an explanation of why consciousness, in Object-Oriented Ontology, might be described as doubly withdrawn: a sensual sub-component of a withdrawn real object. Keywords: Graham Harman, ontology, objects, Timothy Morton, vicarious, screening, virtual particle, consciousness 1 Introduction When approaching Graham Harman’s fourfold ontology, it is relatively easy to understand the first steps if you begin from an anthropocentric position of naive realism: there are real objects that have their real qualities. Then apart from real objects are the objects of our perception, which Harman calls sensual objects, which are reduced distortions or caricatures of the real objects we perceive; and these sensual objects have their own sensual qualities. It is common sense that when we perceive a steaming espresso, for example, that we, as the perceivers, create the image of that espresso in our minds – this image being what Harman calls a sensual object – and that we supply the energy to produce this sensual object. -
Relational Quantum Mechanics and Probability M
Relational Quantum Mechanics and Probability M. Trassinelli To cite this version: M. Trassinelli. Relational Quantum Mechanics and Probability. Foundations of Physics, Springer Verlag, 2018, 10.1007/s10701-018-0207-7. hal-01723999v3 HAL Id: hal-01723999 https://hal.archives-ouvertes.fr/hal-01723999v3 Submitted on 29 Aug 2018 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. Noname manuscript No. (will be inserted by the editor) Relational Quantum Mechanics and Probability M. Trassinelli the date of receipt and acceptance should be inserted later Abstract We present a derivation of the third postulate of Relational Quan- tum Mechanics (RQM) from the properties of conditional probabilities. The first two RQM postulates are based on the information that can be extracted from interaction of different systems, and the third postulate defines the prop- erties of the probability function. Here we demonstrate that from a rigorous definition of the conditional probability for the possible outcomes of different measurements, the third postulate is unnecessary and the Born's rule naturally emerges from the first two postulates by applying the Gleason's theorem. We demonstrate in addition that the probability function is uniquely defined for classical and quantum phenomena. -
Identical Particles
8.06 Spring 2016 Lecture Notes 4. Identical particles Aram Harrow Last updated: May 19, 2016 Contents 1 Fermions and Bosons 1 1.1 Introduction and two-particle systems .......................... 1 1.2 N particles ......................................... 3 1.3 Non-interacting particles .................................. 5 1.4 Non-zero temperature ................................... 7 1.5 Composite particles .................................... 7 1.6 Emergence of distinguishability .............................. 9 2 Degenerate Fermi gas 10 2.1 Electrons in a box ..................................... 10 2.2 White dwarves ....................................... 12 2.3 Electrons in a periodic potential ............................. 16 3 Charged particles in a magnetic field 21 3.1 The Pauli Hamiltonian ................................... 21 3.2 Landau levels ........................................ 23 3.3 The de Haas-van Alphen effect .............................. 24 3.4 Integer Quantum Hall Effect ............................... 27 3.5 Aharonov-Bohm Effect ................................... 33 1 Fermions and Bosons 1.1 Introduction and two-particle systems Previously we have discussed multiple-particle systems using the tensor-product formalism (cf. Section 1.2 of Chapter 3 of these notes). But this applies only to distinguishable particles. In reality, all known particles are indistinguishable. In the coming lectures, we will explore the mathematical and physical consequences of this. First, consider classical many-particle systems. If a single particle has state described by position and momentum (~r; p~), then the state of N distinguishable particles can be written as (~r1; p~1; ~r2; p~2;:::; ~rN ; p~N ). The notation (·; ·;:::; ·) denotes an ordered list, in which different posi tions have different meanings; e.g. in general (~r1; p~1; ~r2; p~2)6 = (~r2; p~2; ~r1; p~1). 1 To describe indistinguishable particles, we can use set notation. -
A Simple Proof That the Many-Worlds Interpretation of Quantum Mechanics Is Inconsistent
A simple proof that the many-worlds interpretation of quantum mechanics is inconsistent Shan Gao Research Center for Philosophy of Science and Technology, Shanxi University, Taiyuan 030006, P. R. China E-mail: [email protected]. December 25, 2018 Abstract The many-worlds interpretation of quantum mechanics is based on three key assumptions: (1) the completeness of the physical description by means of the wave function, (2) the linearity of the dynamics for the wave function, and (3) multiplicity. In this paper, I argue that the combination of these assumptions may lead to a contradiction. In order to avoid the contradiction, we must drop one of these key assumptions. The many-worlds interpretation of quantum mechanics (MWI) assumes that the wave function of a physical system is a complete description of the system, and the wave function always evolves in accord with the linear Schr¨odingerequation. In order to solve the measurement problem, MWI further assumes that after a measurement with many possible results there appear many equally real worlds, in each of which there is a measuring device which obtains a definite result (Everett, 1957; DeWitt and Graham, 1973; Barrett, 1999; Wallace, 2012; Vaidman, 2014). In this paper, I will argue that MWI may give contradictory predictions for certain unitary time evolution. Consider a simple measurement situation, in which a measuring device M interacts with a measured system S. When the state of S is j0iS, the state of M does not change after the interaction: j0iS jreadyiM ! j0iS jreadyiM : (1) When the state of S is j1iS, the state of M changes and it obtains a mea- surement result: j1iS jreadyiM ! j1iS j1iM : (2) 1 The interaction can be represented by a unitary time evolution operator, U. -
Two-State Systems
1 TWO-STATE SYSTEMS Introduction. Relative to some/any discretely indexed orthonormal basis |n) | ∂ | the abstract Schr¨odinger equation H ψ)=i ∂t ψ) can be represented | | | ∂ | (m H n)(n ψ)=i ∂t(m ψ) n ∂ which can be notated Hmnψn = i ∂tψm n H | ∂ | or again ψ = i ∂t ψ We found it to be the fundamental commutation relation [x, p]=i I which forced the matrices/vectors thus encountered to be ∞-dimensional. If we are willing • to live without continuous spectra (therefore without x) • to live without analogs/implications of the fundamental commutator then it becomes possible to contemplate “toy quantum theories” in which all matrices/vectors are finite-dimensional. One loses some physics, it need hardly be said, but surprisingly much of genuine physical interest does survive. And one gains the advantage of sharpened analytical power: “finite-dimensional quantum mechanics” provides a methodological laboratory in which, not infrequently, the essentials of complicated computational procedures can be exposed with closed-form transparency. Finally, the toy theory serves to identify some unanticipated formal links—permitting ideas to flow back and forth— between quantum mechanics and other branches of physics. Here we will carry the technique to the limit: we will look to “2-dimensional quantum mechanics.” The theory preserves the linearity that dominates the full-blown theory, and is of the least-possible size in which it is possible for the effects of non-commutivity to become manifest. 2 Quantum theory of 2-state systems We have seen that quantum mechanics can be portrayed as a theory in which • states are represented by self-adjoint linear operators ρ ; • motion is generated by self-adjoint linear operators H; • measurement devices are represented by self-adjoint linear operators A. -
Chapter 6. Time Evolution in Quantum Mechanics
6. Time Evolution in Quantum Mechanics 6.1 Time-dependent Schrodinger¨ equation 6.1.1 Solutions to the Schr¨odinger equation 6.1.2 Unitary Evolution 6.2 Evolution of wave-packets 6.3 Evolution of operators and expectation values 6.3.1 Heisenberg Equation 6.3.2 Ehrenfest’s theorem 6.4 Fermi’s Golden Rule Until now we used quantum mechanics to predict properties of atoms and nuclei. Since we were interested mostly in the equilibrium states of nuclei and in their energies, we only needed to look at a time-independent description of quantum-mechanical systems. To describe dynamical processes, such as radiation decays, scattering and nuclear reactions, we need to study how quantum mechanical systems evolve in time. 6.1 Time-dependent Schro¨dinger equation When we first introduced quantum mechanics, we saw that the fourth postulate of QM states that: The evolution of a closed system is unitary (reversible). The evolution is given by the time-dependent Schrodinger¨ equation ∂ ψ iI | ) = ψ ∂t H| ) where is the Hamiltonian of the system (the energy operator) and I is the reduced Planck constant (I = h/H2π with h the Planck constant, allowing conversion from energy to frequency units). We will focus mainly on the Schr¨odinger equation to describe the evolution of a quantum-mechanical system. The statement that the evolution of a closed quantum system is unitary is however more general. It means that the state of a system at a later time t is given by ψ(t) = U(t) ψ(0) , where U(t) is a unitary operator. -
Abstract for Non Ideal Measurements by David Albert (Philosophy, Columbia) and Barry Loewer (Philosophy, Rutgers)
Abstract for Non Ideal Measurements by David Albert (Philosophy, Columbia) and Barry Loewer (Philosophy, Rutgers) We have previously objected to "modal interpretations" of quantum theory by showing that these interpretations do not solve the measurement problem since they do not assign outcomes to non_ideal measurements. Bub and Healey replied to our objection by offering alternative accounts of non_ideal measurements. In this paper we argue first, that our account of non_ideal measurements is correct and second, that even if it is not it is very likely that interactions which we called non_ideal measurements actually occur and that such interactions possess outcomes. A defense of of the modal interpretation would either have to assign outcomes to these interactions, show that they don't have outcomes, or show that in fact they do not occur in nature. Bub and Healey show none of these. Key words: Foundations of Quantum Theory, The Measurement Problem, The Modal Interpretation, Non_ideal measurements. Non_Ideal Measurements Some time ago, we raised a number of rather serious objections to certain so_called "modal" interpretations of quantum theory (Albert and Loewer, 1990, 1991).1 Andrew Elby (1993) recently developed one of these objections (and added some of his own), and Richard Healey (1993) and Jeffrey Bub (1993) have recently published responses to us and Elby. It is the purpose of this note to explain why we think that their responses miss the point of our original objection. Since Elby's, Bub's, and Healey's papers contain excellent descriptions both of modal theories and of our objection to them, only the briefest review of these matters will be necessary here. -
Hamilton Equations, Commutator, and Energy Conservation †
quantum reports Article Hamilton Equations, Commutator, and Energy Conservation † Weng Cho Chew 1,* , Aiyin Y. Liu 2 , Carlos Salazar-Lazaro 3 , Dong-Yeop Na 1 and Wei E. I. Sha 4 1 College of Engineering, Purdue University, West Lafayette, IN 47907, USA; [email protected] 2 College of Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61820, USA; [email protected] 3 Physics Department, University of Illinois at Urbana-Champaign, Urbana, IL 61820, USA; [email protected] 4 College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou 310058, China; [email protected] * Correspondence: [email protected] † Based on the talk presented at the 40th Progress In Electromagnetics Research Symposium (PIERS, Toyama, Japan, 1–4 August 2018). Received: 12 September 2019; Accepted: 3 December 2019; Published: 9 December 2019 Abstract: We show that the classical Hamilton equations of motion can be derived from the energy conservation condition. A similar argument is shown to carry to the quantum formulation of Hamiltonian dynamics. Hence, showing a striking similarity between the quantum formulation and the classical formulation. Furthermore, it is shown that the fundamental commutator can be derived from the Heisenberg equations of motion and the quantum Hamilton equations of motion. Also, that the Heisenberg equations of motion can be derived from the Schrödinger equation for the quantum state, which is the fundamental postulate. These results are shown to have important bearing for deriving the quantum Maxwell’s equations. Keywords: quantum mechanics; commutator relations; Heisenberg picture 1. Introduction In quantum theory, a classical observable, which is modeled by a real scalar variable, is replaced by a quantum operator, which is analogous to an infinite-dimensional matrix operator. -
On Relational Quantum Mechanics Oscar Acosta University of Texas at El Paso, [email protected]
University of Texas at El Paso DigitalCommons@UTEP Open Access Theses & Dissertations 2010-01-01 On Relational Quantum Mechanics Oscar Acosta University of Texas at El Paso, [email protected] Follow this and additional works at: https://digitalcommons.utep.edu/open_etd Part of the Philosophy of Science Commons, and the Quantum Physics Commons Recommended Citation Acosta, Oscar, "On Relational Quantum Mechanics" (2010). Open Access Theses & Dissertations. 2621. https://digitalcommons.utep.edu/open_etd/2621 This is brought to you for free and open access by DigitalCommons@UTEP. It has been accepted for inclusion in Open Access Theses & Dissertations by an authorized administrator of DigitalCommons@UTEP. For more information, please contact [email protected]. ON RELATIONAL QUANTUM MECHANICS OSCAR ACOSTA Department of Philosophy Approved: ____________________ Juan Ferret, Ph.D., Chair ____________________ Vladik Kreinovich, Ph.D. ___________________ John McClure, Ph.D. _________________________ Patricia D. Witherspoon Ph. D Dean of the Graduate School Copyright © by Oscar Acosta 2010 ON RELATIONAL QUANTUM MECHANICS by Oscar Acosta THESIS Presented to the Faculty of the Graduate School of The University of Texas at El Paso in Partial Fulfillment of the Requirements for the Degree of MASTER OF ARTS Department of Philosophy THE UNIVERSITY OF TEXAS AT EL PASO MAY 2010 Acknowledgments I would like to express my deep felt gratitude to my advisor and mentor Dr. Ferret for his never-ending patience, his constant motivation and for not giving up on me. I would also like to thank him for introducing me to the subject of philosophy of science and hiring me as his teaching assistant. -
1 Does Consciousness Really Collapse the Wave Function?
Does consciousness really collapse the wave function?: A possible objective biophysical resolution of the measurement problem Fred H. Thaheld* 99 Cable Circle #20 Folsom, Calif. 95630 USA Abstract An analysis has been performed of the theories and postulates advanced by von Neumann, London and Bauer, and Wigner, concerning the role that consciousness might play in the collapse of the wave function, which has become known as the measurement problem. This reveals that an error may have been made by them in the area of biology and its interface with quantum mechanics when they called for the reduction of any superposition states in the brain through the mind or consciousness. Many years later Wigner changed his mind to reflect a simpler and more realistic objective position, expanded upon by Shimony, which appears to offer a way to resolve this issue. The argument is therefore made that the wave function of any superposed photon state or states is always objectively changed within the complex architecture of the eye in a continuous linear process initially for most of the superposed photons, followed by a discontinuous nonlinear collapse process later for any remaining superposed photons, thereby guaranteeing that only final, measured information is presented to the brain, mind or consciousness. An experiment to be conducted in the near future may enable us to simultaneously resolve the measurement problem and also determine if the linear nature of quantum mechanics is violated by the perceptual process. Keywords: Consciousness; Euglena; Linear; Measurement problem; Nonlinear; Objective; Retina; Rhodopsin molecule; Subjective; Wave function collapse. * e-mail address: [email protected] 1 1. -
Qualification Exam: Quantum Mechanics
Qualification Exam: Quantum Mechanics Name: , QEID#43228029: July, 2019 Qualification Exam QEID#43228029 2 1 Undergraduate level Problem 1. 1983-Fall-QM-U-1 ID:QM-U-2 Consider two spin 1=2 particles interacting with one another and with an external uniform magnetic field B~ directed along the z-axis. The Hamiltonian is given by ~ ~ ~ ~ ~ H = −AS1 · S2 − µB(g1S1 + g2S2) · B where µB is the Bohr magneton, g1 and g2 are the g-factors, and A is a constant. 1. In the large field limit, what are the eigenvectors and eigenvalues of H in the "spin-space" { i.e. in the basis of eigenstates of S1z and S2z? 2. In the limit when jB~ j ! 0, what are the eigenvectors and eigenvalues of H in the same basis? 3. In the Intermediate regime, what are the eigenvectors and eigenvalues of H in the spin space? Show that you obtain the results of the previous two parts in the appropriate limits. Problem 2. 1983-Fall-QM-U-2 ID:QM-U-20 1. Show that, for an arbitrary normalized function j i, h jHj i > E0, where E0 is the lowest eigenvalue of H. 2. A particle of mass m moves in a potential 1 kx2; x ≤ 0 V (x) = 2 (1) +1; x < 0 Find the trial state of the lowest energy among those parameterized by σ 2 − x (x) = Axe 2σ2 : What does the first part tell you about E0? (Give your answers in terms of k, m, and ! = pk=m). Problem 3. 1983-Fall-QM-U-3 ID:QM-U-44 Consider two identical particles of spin zero, each having a mass m, that are con- strained to rotate in a plane with separation r.