Music of the Spheres: Teaching Quantum Field Theory at the Introductory Level Ian H. Redmount∗ Department of Physics Saint Louis University St. Louis, MO 63103–2010 USA (Dated: August 6, 2019) Abstract Quantum field theory has formed the conceptual framework of most of physics for more than sixty years. It incorporates a complete revision of our conception of the nature of matter and existence itself. Yet it is rarely taught, or even mentioned, in introductory physics—from high school, college, and university survey courses through upper-division “modern physics” courses. This omission is not necessary: This paper describes an approach through which the fundamental concepts and consequent insights of quantum field theory can be grasped, building upon familiar notions from classical and quantum mechanics. arXiv:1908.01579v1 [physics.hist-ph] 27 Jun 2019 ∗[email protected] 1 I. INTRODUCTION Quantum field theory, or the theory of quantized fields, forms the current conceptual framework of almost all of physics, and has ever since Richard Feynman, Julian Schwinger, and Tomonaga Shin-ichiro discovered in 1948 how to make the theory give finite answers to physical questions. Only gravitation lies outside the theory’s purview; this is the mo- tivation for searches for a “quantum theory of gravity” or “Theory of Everything,” from Schr¨odinger in the 1940’s through today’s string theory, M-theory, and loop quantum grav- ity. Einsteinian quantum field theory is the foundation of elementary-particle physics, while the Newtonian limit undergirds our understanding of solid-state physics, superfluidity, and superconductivity.[8] The theory affords a sweeping syntheses of many features of physics, and a profound new understanding of the nature of existence itself. Given the centrality and scope of the theory, it is remarkably difficult to find an introduc- tory physics text—whether a high-school text, an algebra/trig-based college physics text, a calculus-based university physics text, or even a text for a higher-level “modern physics” course—which even mentions quantum field theory, much less explains it. (Recent texts by Hobson [1] and Redmount [2] are notable exceptions). There are semi-popular books which describe it, but its absence from introductory physics texts and courses is striking. As a result, for example, intrductory treatments of elementary particles—which do appear in all the texts—read like classical botany, all terminology and taxonomy. It has been proposed to teach string theory at the high-school level, but this is surely putting the cart before the horse, as all the motivations for string theory are rooted in issues which arise in quantum field theory; indeed, string theory is a quantum field theory. Recently Hobson [3–6] and Huggins [7] have argued forcefully for a more central role for quantum field theory in the teaching of introductory physics, in particular to clarify many aspects of quantum mechanics. Of course it would be difficult even to set up the quantum-field-theoretic calculation of anything at the introductory level. But the fundamental ideas and breathtaking insights of the theory are readily understandable. They can be explained by building on concepts from classical and quantum physics which are accessible in elementary treatments. My purpose in this paper is to sketch out one such approach. The necessary foundations are described in Secs. II and III. The emergence of quantum field theory from these roots, and the features and implications of the theory, are detailed in Sec. IV. 2 II. QUANTUM MECHANICS Quantum field theory is the quantum mechanics of fields—extended systems—e.g., fluids, or electromagnetic fields, or the quantum waves of elementary particles. Quantum mechanics is built on two principles, both inferred from observation in the early decades of the 20th century: Electromagnetic waves, described classically as continuous fields, actually come in discrete “packets,” “lumps,” “quanta,” or “photons.” And conversely, classical particles of matter, such as electrons, protons, and neutrons, must be described via continuous wave fields. The particulate nature—quantization—of electromagnetic waves was introduced by Max Planck in 1900, in order to get the spectrum of radiation from hot bodies right. (This is usually considered the opening shot of the quantum revolution; classical electromagnetic theory fails miserably here.) Albert Einstein invoked it in 1905 to explain the photoelectric effect, and Arthur Holly Compton demonstrated it with his x-ray scattering experiments in 1923. The mechanical properties of the particles or photons, viz., energy E and momen- tum p, are related to the properties of the corresponding waves, viz., frequency ν, angular frequency ω =2πν, wavelength λ, and wave number k =2π/λ, via the familiar relations E = hν = ~ω , (1a) and h p = = ~k , (1b) λ where h = 2π~ = 6.626 ... 10−34 J s is the “fundamental quantum of action” intro- × duced by Planck. These equations are not as radical as they might appear: They accord with the classical relations for the energies and momenta of electromagnetic wave packets which follow from the Maxwell equations. They do not, however, represent a throwback to Newton’s corpuscular theory of light. The phenomena which reveal the wave nature of light—interference and diffraction—remain, even when the experiments are performed with one photon at a time. Wave and particle natures coexist, the famous wave-particle duality of quantum mechanics. The converse principle, that matter particles exhibit behaviors which must be described with waves, was revealed during the same period by experiments akin to those that originally revealed the wave nature of light. The scattering of electrons and neutrons from crystals, 3 e.g., in the experiments of Max von Laue and Davisson and Germer, displayed interference patterns. Schr¨odinger tunneling of particles through potential barriers they do not have the energy to surmount is simply the counterpart of a phenomenon in wave optics, where it is known as Frustrated Total Internal Reflection. The relationships connecting matter-wave properties with particle properties were formulated by Louis de Broglie in 1921: h p λ = i.e., k = (2a) p ~ and E E ν = i.e., ω = , (2b) h ~ analogous to the photon relations above. Incorporating the wave nature of particles into mechanics fundamentally alters the nature of the science; the questions which are asked and answered are changed. A basic problem in classical mechanics is to determine the trajectory x(t) of a particle, given its initial position and momentum x0 and p0. A wave description does not allow this—that is the content of the Heisenberg Uncertainty Principle. In quantum mechanics a basic problem is the deter- mination of a particle’s wave function ψ(x, t), given its initial configuration ψ(x, 0). The squared magnitude of the wave function is associated with the probability density for finding the particle, in the standard or Copenhagen interpretation of the theory. Alternative inter- pretations have been and continue to be proposed and debated, but all are interpretations. They change the outcome of no calculation or prediction of the theory. The basic “equation of motion” in ordinary quantum mechanics, the Schr¨odinger wave equation, is obtained from the Newtonian energy-momentum relation p2 + U = E, (3a) 2m for a particle of mass m with potential energy U, by imposing the de Broglie relations (2) in the form of differential operators applied to the wave function. Calculus speakers will recognize the resulting form: ~2 ∂ψ 2ψ + U ψ = i~ , (3b) − 2m ∇ ∂t with i = √ 1 the imaginary unit, and the time-independent form − ~2 2ψ + U ψ = E ψ , (3c) − 2m ∇ 4 for wave functions describing particle states of definite energy E. Despite the formidable appearance of these equations, their physical content is simply that of the energy-momentum relation (3a). There are only five exactly, analytically soluble problems in quantum mechanics: the free particle, the “particle in a box,” the free rotor, the hydrogen atom, and the harmonic oscillator. (If this seems rather limited, it may be recalled that there are not that many more exactly, analytically soluble problems in classical mechanics.) The last of these is the key to understanding quantum field theory. For an oscillator in one dimension, with mass m and “spring constant” k = mω2, the time-independent Schr¨odinger equation (3c) for the energy levels of the system takes the form ~2 dψ + 1 mω2 ψ = E ψ . (4a) − 2m dx2 2 The resulting wave functions are 1 mω mω 1/4 mω − x2 n √ −1/2 2 ~ ψn(x)= ~ (2 n! π) Hn ~ x e , (4b) r with Hn the Hermite polynomials discovered in the 19th century. The corresponding energy levels are 1 ~ En =(n + 2 ) ω , (4c) where n is a non-negative integer. The oscillator has zero-point energy 1 ~ E0 = 2 ω (4d) in its lowest-energy or ground state. And the excitation energies of all the levels above this come in equal increments ∆E = ~ω. These features will prove crucial in what is to follow. III. DYNAMICS OF SYSTEMS: NORMAL MODES The other physical principle underlying quantum field theory is a result from classical mechanics: The dynamics of any system, however complex, obeying a linear equation of motion—including almost any system slightly perturbed from any equilibrium—can be de- scribed via a collection of independent harmonic oscillators. The most general motion of the system can be represented as a combination of patterns called normal modes or harmonics. (The harmonics of the vibrating strings or air columns of musical instruments are normal 5 k m k m FIG. 1: The not-quite-simplest system of two coupled harmonic oscillators. modes.) And each normal mode behaves as an individual harmonic oscillator, uncoupled from the others. Consider, for example, the system illustrated in Fig.