Music of the Spheres: Teaching Quantum Field Theory at the Introductory Level

TL;DR

This paper proposes an accessible approach to teaching quantum field theory at the introductory level, making its fundamental concepts understandable using familiar classical and quantum mechanics principles.

Contribution

It introduces a novel pedagogical method that simplifies quantum field theory for beginners, bridging the gap in current physics education.

Findings

Provides a step-by-step teaching framework

Demonstrates how core concepts can be grasped early

Suggests integration into standard curricula

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.

Equations

$$E=h\nu =\hslash \omega ,$$

$$p=\frac{h}{\lambda }=\hslash k,$$

$$\lambda =\frac{h}{p}i.e.,k=\frac{p}{\hslash }$$

$$\nu =\frac{E}{h}i.e.,\omega =\frac{E}{\hslash },$$

$$\frac{p^2}{2m}+U=E,$$

$$-\frac{{\hslash }^2}{2m}{\nabla }^2\psi +U\psi =i\hslash \frac{\partial \psi }{\partial t},$$

$$-\frac{{\hslash }^2}{2m}{\nabla }^2\psi +U\psi =E\psi ,$$

$$-\frac{{\hslash }^2}{2m}\frac{d^{\psi }}{d{x}^2}+\frac{1}{2}m{\omega }^2\psi =E\psi .$$

$${\psi }_n(x)={\left(\frac{m\omega }{\hslash }\right)}^{1/4}(2^{n}n!\sqrt{\pi }{)}^{-1/2}{H}_n\left(\sqrt{\frac{m\omega }{\hslash }}x\right)e^{-\frac{1}{2}\frac{m\omega }{\hslash }{x}^2},$$

$$E_n=(n+\frac{1}{2})\hslash \omega ,$$

$$E_0=\frac{1}{2}\hslash \omega$$

$$y_1(t)$$

$$y_2(t)$$

$$\varphi =\frac{1+\sqrt{5}}{2}=1.618\dots$$

$$\frac{E^2}{c^2}-p^2=m^2{c}^2,$$

$${\hslash }^2\left(\frac{1}{c^2}\frac{{\partial }^2}{\partial t^2}-{\nabla }^2\right)\phi +m^2{c}^2\phi =0,$$

$$({\gamma }^{\mu }{\hat{p}}_{\mu }-mc)\Psi =0,$$

$${\mu }^{-}\to e^{-}+{\bar{\nu }}_e+{\nu }_{\mu }$$

Full text

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 motivation for searches for a “quantum theory of gravity” or “Theory of Everything,” from Schrödinger in the 1940’s through today’s string theory, M-theory, and loop quantum gravity. 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.111The usual distinction is between “relativistic” and “nonrelativistic” theories, but this is incorrect: Newtonian mechanics and Schrödinger quantum mechanics are just as relativistic as their Einsteinian and Dirac counterparts, only with a different group of transformations between reference frames. I shall therefore attempt quixotically to rationalize the terminology here. 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 introductory 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 Hobson (2010) and Redmount Redmount (2016) 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 Hobson (2005, 2007, 2011, 2013) and Huggins Huggins (2007) 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.

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 momentum $p$, are related to the properties of the corresponding waves, viz., frequency $\nu$, angular frequency $\omega=2\pi\nu$, wavelength $\lambda$, and wave number $k=2\pi/\lambda$, via the familiar relations

[TABLE]

where $h=2\pi\hbar=6.626\ldots\times 10^{-34}~{}\hbox{J s}$ is the “fundamental quantum of action” introduced 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, e.g., in the experiments of Max von Laue and Davisson and Germer, displayed interference patterns. Schrödinger 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:

[TABLE]

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 $x_{0}$ and $p_{0}$. A wave description does not allow this—that is the content of the Heisenberg Uncertainty Principle. In quantum mechanics a basic problem is the determination of a particle’s wave function $\psi(x,t)$, given its initial configuration $\psi(x,0)$. 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 interpretations 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 Schrödinger wave equation, is obtained from the Newtonian energy-momentum relation

[TABLE]

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” $k=m\omega^{2}$, the time-independent Schrödinger equation (3c) for the energy levels of the system takes the form

[TABLE]

in its lowest-energy or ground state. And the excitation energies of all the levels above this come in equal increments $\Delta E=\hbar\omega$. 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 described 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 modes.) And each normal mode behaves as an individual harmonic oscillator, uncoupled from the others.

Consider, for example, the system illustrated in Fig. 1: one simple

harmonic oscillator hung from another, one of the simplest coupled systems imaginable. Even if the oscillators have equal masses $m$ and identical spring constants $k$, the springs are essentially massless, and friction and drag are negligible, the motions of the masses about their equilibrium positions are complicated and (ideally) nonperiodic. Yet they can always be represented as a superposition of simple harmonic motions with two different frequencies:

[TABLE]

the Golden Ratio.222That the Golden Ratio of the ancient Greeks should appear in this simple but nontrivial mechanics problem is one of nature’s undeserved gifts to humanity. Each of the two terms on the right-hand sides of Eqs. (5a) represents the contribution of one of the system’s normal modes to the motion of each mass. For any system, the number of independent normal modes is equal to the number of dynamical degrees of freedom the system possesses.

IV QUANTUM FIELD THEORY

The emergence of quantum field theory from quantum mechanics was impelled, first, by the need to apply quantum-mechanical principles to electromagnetic fields, and second, by the need to shift quantum mechanics from its foundation in Newtonian mechanics to a foundation in Einsteinian mechanics, i.e., in the dynamics associated with the Special Theory of Relativity formulated by Albert Einstein in 1905. In fact these impeti are the same, as classical electromagnetic fields are inherently Einsteinian. The bad news is that this shift is extremely difficult: The conceptual frameworks of quantum mechanics and Einsteinian mechanics are almost inconsistent with one another. The good news is that if one finds a way to do this, it’s probably correct: There cannot be too many competing options.

IV.1 Free fields and particles

IV.1.1 Einsteinian Quantum Mechanics

It is straightforward to construct a quantum-mechanical wave equation by applying the de Broglie relations (2) to the Einsteinian energy-momentum relation for a free particle of mass $m$:

[TABLE]

where I follow convention in using $\varphi$ for the wave function in place of $\psi$. This should serve as the Einsteinian version of the Schrödinger equation (3b). The physics it describes, however, is distinguished by certain curious features: Unlike the Schrödinger equation, the Klein-Gordon equation does not define a conserved, positive-definite probability density (for which the total probability of finding a particle is constant in time). And the solutions of the Klein-Gordon equation are equally divided between positive-energy and negative-energy states. These would appear to be fundamental flaws in the theory.

In part to circumvent these flaws, Paul Adrien Maurice Dirac introduced in 1927 the equation which bears his name:

[TABLE]

where $\hat{p}_{\mu}$ is a differential operator in accord with the de Broglie relations, and the first term on the left is summed over the four dimensions of spacetime. The price paid for the apparent simplicity of this equation is that this is a matrix equation: The $\gamma^{\mu}$ are $4\times 4$ matrices. The wave function $\Psi$ is not a single function, but is represented by a $4\times 1$ column matrix; it is an object called a Dirac spinor.

The Dirac equation—with electromagnetic forces incorporated—is chemists’ go-to equation for calculating the energy levels of electrons in atoms and molecules correctly. It reproduces the successes of Schrödinger quantum mechanics, but includes corrections associated with Einsteinian mechanics. Moreover, it describes a spinning electron, with intrinsic angular momentum $\hbar/2$ but corresponding magnetic dipole moment twice that which would be expected from a classical spinning, charged particle. It is possible to describe a spinning particle in the context of Newtonian, i.e., Schrödinger quantum mechanics, but the spin and its interactions must be put in “by hand.” In Dirac’s formulation, these features are built in—there is no non-spinning Dirac particle. And that factor of 2 in the magnetic dipole moment—called the “reduced gyromagnetic ratio” of the electron—is almost right. Getting it exactly right is one of the triumphs of the quantum-field-theoretic approach.

Dirac’s approach does not dispense with the positive and negative energies of the Klein-Gordon treatment. To reconcile this with observation—the electrons around us do not fall through an endless sequence of negative-energy states—Dirac posited a “sea” of electrons which fill all the negative-energy states. These are not ordinarily observed, because they fill space uniformly. But if sufficient energy is supplied, an electron can be promoted from the negative-energy sea into a positive-energy state, where it can be observed. The vacancy or “hole” created in the sea behaves exactly like a positive-energy electron with opposite electric charge—an “antielectron” or “positron.” This is not science fiction: The positron was discovered by Carl Anderson in 1932. The antiproton and antineutron were observed later. All known particles have corresponding antiparticles.

The Dirac formulation of Einsteinian quantum mechanics thus yields some spectacularly successful predictions. But it actually abandons the notion of a single particle, introducing a multi-particle description even of an isolated electron. This calls for the more general framework of quantum field theory.

IV.1.2 “Second quantization”

The desired multi-particle, Einsteinian quantum theory is obtained by applying the principles of quantum mechanics to fields, entities defined throughout some region of space: electromagnetic fields, or the quantum-wave fields which describe other particles. The application of quantum dynamics to wave fields from quantum mechanics is described by the historic term “second quantization,” although no quantity is “quantized” twice.333Ordinary quantum mechanics is “first quantization.” There is such a thing as “third quantization”; it’s an attempt to incorporate gravity and spacetime geometry into the picture. The states of the theory are characterized by different numbers of universes. The ground state of the theory is the “no-universe state,” though the physical meaning of that is not entirely clear. The values of the field throughout space at one time become the dynamical variables of the theory, like the coordinates of a particle in ordinary quantum mechanics. The fundamental object of the (second-) quantized theory is the “wave functional” $\Psi[\psi(x),t]$, which assigns a probability amplitude to the field configuration $\psi(x)$ at time $t$. The squared magnitude of $\Psi$ is the probability density for finding the field in that configuration. The wave functional (so called because its argument is an entire function, not simply a number) is the complete description of the quantum state of the field. For technical reasons, most calculations in quantum field theory do not use the wave functional, but an operator formalism like that of Heisenberg. There are problems, however, for which the “functional Schrödinger” formalism is useful.

This is where the basic principles of the theory come together: The field $\psi$ can be decomposed into its harmonics or normal modes, each of which behaves as an independent harmonic oscillator. The quantum field theory, then, describes a collection of quantum harmonic oscillators, one for each normal mode. Each has a zero-point energy and excitations in uniform energy increments, as in Eqs. (4c) and (4d). These energy increments are the particles of the theory; the numbers of excitations in all the normal modes define the state of the field.

The description of the particles of matter not as “lumps of stuff,” but as excitations of field modes—more like musical notes than classical particles—is a sea change in our basic description of existence itself. We do not hear the ancient Greeks’ “Music of the Spheres,” nor do we make it: We are the Music of the Spheres.

The payoffs, in terms of understanding, of this approach are huge. The states of the theory are defined by numbers of particles in each mode. The quantum transitions of the theory, then, can involve both changes of state (mode) of a particle, or changes in the number of particles. In any other formulation of physics—Newtonian, Einsteinian (special- or general-relativistic), ordinary quantum-mechnical—particle number is a conserved quantity. But nature does display processes in which particles are created or destroyed: An electron and a positron can annihilate, leaving two or three photons; a collision of protons at Fermilab or the Large Hadron Collider can produce thousands of new particles. Only quantum field theory provides a framework into which such processes fit naturally.

The Dirac equation describes spinning electrons. In Einsteinian quantum field theory, all particles are characterized by “intrinsic spin” as a matter of course. This feature appears as the answer to the question, “How many different states can describe a single particle?” Since the theory is relativistic, the answer is “an infinite number”: Any state for a particle, transformed into a different reference frame, must also be a valid state for that particle. That is, the particle can appear with all possible values for its momentum. Aside from that, then: “How many different states can describe a single, stationary particle?” (This assumes a massive particle. For massless particles, like the photon, which are never at rest, the argument differs slightly in detail.) Since the theory is quantum-mechanical, the states must form a vector space. The dimension of the space is tabulated by the spin quantum number of the particle. If the space is one-dimensional—there is only one state—the particle is spinless: $s=0$. A two-dimensional space—the states are combinations of “spin up” and “spin down”—corresponds to $s=\tfrac{1}{2}$, as for the electrons, the neutrinos, and the quarks. Three dimensions mean $s=1$; four, $s=\tfrac{3}{2}$; five, $s=2$; in general the dimension of the space of stationary-particle states is $2s+1$. Because the states can be transformed into one another by all the changes of reference frame which do not change the velocity of the particle, i.e., rotations, this tabulator of the number of states takes on the characteristics of quantum angular momentum. The intrinsic spin of particles and all its consequences, such as magnetism, arise in quantum field theory as a simple bookkeeping device.

An even simpler bookkeeping device has even more profound implications. Consider a state $\Psi_{kk^{\prime}}$ of two particles, the first in mode $k$ and the second in mode $k^{\prime}$. How is this related to state $\Psi_{k^{\prime}k}$, with the first particle in mode $k^{\prime}$ and the second in mode $k$? We observe in nature that elementary particles of the same species are absolutely identical; no individual distinguishing features have ever been observed. Classically, this is remarkable, but in quantum field theory, it is entirely natural, as the particles are excitations of the same oscillator. So exchanging the roles of the two particles in states $\Psi_{kk^{\prime}}$ and $\Psi_{k^{\prime}k}$ can have no observable consequences—all the probabilities determined by the two wave functionals must be the same. What can you do to a quantum wave functional which 1) doesn’t change its squared magnitude (probability), and 2) if you do it twice, doesn’t change anything at all, since exchanging the particles twice undoes the exchange? There are exactly two possibilities: Multiply the wave functional by $+1$, or multiply it by $-1$. Particles for which the former applies are called “bosons,” after Satyendra Nath Bose, and particles for which the latter applies are called “fermions,” after Enrico Fermi. This classification is termed the “statistics” of the particles or field. (More precisely, bosons obey “Bose-Einstein statistics,” fermions “Fermi-Dirac statistics.”) Consider now a state $\Psi_{kk}$ with two particles in the same mode. Exchanging the particles here changes nothing. But there is only one value which is unchanged upon multiplication by $-1$: Necessarily, for fermions $\Psi_{kk}=0$. That is, there is no probability of finding two fermions in the same mode, or quantum state. This is the Pauli Exclusion Principle, introduced by Wolfgang Pauli in 1925. Like spin, it can be introduced by hand into ordinary quantum mechanics. But in quantum field theory it is built in, a necessity to keep the particle-exchange rules consistent. The Pauli Exclusion Principle gives ordinary matter its rigidity and impenetrability, and undergirds all of chemistry: Because electrons in atoms must be stacked at most two to an orbital (one spin-up, one spin-down), atoms with different atomic numbers have different electron configurations, hence different chemical properties. All these aspects of matter, then, are quantum-field-theoretic effects.

It is a remarkable result of Einsteinian quantum field theory that the spin and statistics characteristics of particles are not independent. Bosons have integral spin quantum numbers, fermions half-integral values, and conversely. This result is known as the Spin-Statistics Theorem. A proof is beyond the scope of the present discussion.

IV.2 Interacting fields and renormalization

As important as they are, these features of the quantum theory of free fields are only the kinematics of the theory. The dynamics of quantum field theory emerges in the description of interacting fields. Since a field entails an infinite number of dynamical variables or “degrees of freedom”—e.g., the value of the field at each point in space—such a description is inherently problematic. The challenges presented by such theories have shaped the evolution of physics for the past seventy years.

It is typical to envision an elementary-particle reaction, such as the decay

[TABLE]

of a muon ($\mu^{-}$) into an electron ($e^{-}$), an electron antineutrino (${\bar{\nu}}_{e}$), and a muon neutrino ($\nu_{\mu}$), as an explosion. But while this event is far more violent—in terms of the fraction of particle mass converted into kinetic energy—than any chemical or even nuclear explosion, such a description is not apt. The features we associate with explosions are gas-dynamic phenomena, involving septillions of particles. The muon decay is closer akin the the production of a note from the strings of a piano by singing into the piano: In that case energy is transferred from the vibrating vocal cords of the singer to the air, then to the sound board and strings of the piano, then back to the air, then to the ears of the listener. In the muon decay, the excitation energy of the muon field (which constitutes the muon particle) is transfered to the electron, electron-neutrino, and muon-neutrino fields, appearing as those particles. Another example is illustrated in Fig. 2, which shows a NaI-crystal

scintillation detector for gamma rays. Incoming gamma-ray photons produce electrons in the crystal via Compton scattering, photoelectric effect, or pair production. The electrons interact with atoms in the crystal to produce flashes of visible light, or scintillations. These interact with the plates of a photomultiplier tube to produce a cascade of electrons; the resulting current pulse is detected and analyzed by suitable circuitry. From a field-theoretic standpoint, the incoming energy is transferred between the electromagnetic (photon) and electron fields three times.

Because of the complexities involved with infinities of interacting degrees of freedom, there are no exact analytic solutions of interacting quantum field theories, except perhaps in a few contrived examples. Instead, field interactions are treated as perturbations of free-field theories. Physical quantities are calculated via infinite series of “correction” terms. This approach is represented graphically via the figures known as Feynman diagrams, introduced by Richard Feynman. For example, for two electrons scattering via electrostatic repulsion, the simplest diagram is shown in Fig. 3.

It is tempting to interpret this classically: The electron ($e^{-}$) on the left emits the virtual photon ($\gamma$) and recoils; the electron on the right absorbs the photon and is deflected. But how could the attraction of oppositely charged particles be described in this way? The first time I heard this question asked, the answer given was, “One mustn’t take these things too literally.” As unsatisfying as that seems, it is in fact the correct answer. Feynman diagrams are not literal depictions of events in physical spacetime; rather, they are mnemonic devices for constructing the terms in an infinite series for, say, a scattering amplitude or reaction probability. The same diagram represents attraction or repulsion; only the algebraic signs of the corresponding expressions are different.

The perturbation approach has one glaring drawback. The result of any calculation of a physical quantity is: infinity. The expressions corresponding to any diagram more complex than Fig. 3 are divergent integrals. That is, the individual terms of the perturbation series for any quantity are infinite, not even to speak of the convergence of the series. Only after Richard Feynman, Julian Schwinger, and Tomonaga Shin-ichiro discovered (independently) in 1948 how to evade this quandary did it become possible to extract physical predictions from quantum field theory. The three men shared the Nobel Prize in Physics in 1965 for their discovery. It is one of the most fundamental, and at the same time most obscure, ideas in science. It is called “renormalization.”

Renormalization can be described in a variety of ways, not obviously connected with one another: It corrects the parameters of the theory for the effects of the interactions between fields. It absorbs the inaccessible high-energy behavior of the theory into its parameters. It compensates for the fact that the quantum states of the interacting-field theory do not exist in the same abstract vector space as the states of the free-field theory. Perhaps the simplest way to understand it is to note that any quantum field theory contains certain parameters, such as the masses and charges of the particles described, and the scale of the fields. Given finite values of these parameters, the theory gives infinite results for any physical quantity. But by allowing the parameters to become infinite in the opposite direction, the infinities can be canceled, yielding finite answers to physical questions.

The parameters allowed to become infinite are called the “bare parameters” of the theory. They represent, e.g., the masses and charges of particles, and the scales of fields, stripped of their interactions. But no particle or field is ever observed stripped of its interactions, so there is no conflict with observation.

The difference between two infinite quantities, i.e., two quantities diverging to infinity in some suitable limit, can be any finite value. The finite values of renormalized quantities are pinned down by requiring them to match specific meaurements: For example, the renormalized mass of the electron at zero interaction energy must match the mass of the electron measured in low-energy experiments. This has the consequence that the parameters of the theory have values which vary with interaction energy; the theory is said to have “running coupling constants.” That looks like a contradiction, except that nature actually does this: For example, the strength of the electric interaction between electrons is measurably greater at the energies attained in high-energy accelerators than in low-energy events. This is one of the signal successes of renormalized quantum field theory.

The infinities or divergences which arise in field-theory calculations cannot be dismissed as mere artifacts of the calculation. They have observable, and observed, consequences. The probabilities for some processes consist of a factor which is formally zero, multiplied by a diverging factor, giving a finite, nonzero result. Such processes, called “anomalies,” actually occur.

The results of renormalization are spectacularly vindicated by experiment. Dirac’s treatment of the electron predicts that certain states of the hydrogen atom should be degenerate, i.e., have the same energy. The energies actually differ very slightly; the difference can be measured via microwave spectroscopy. Renormalized quantum electrodynamics correctly predicts this difference, known as the “Lamb shift.” Dirac’s treatment also predicts that the ratio of the magnetic dipole moment of the electron to its spin angular momentum—its “gyromagnetic ratio”—should be exactly twice that obtained from a classical treatment of the electron as a spinning ball of electric charge. The actual factor is about one-tenth of one percent greater than two. The discrepancy can be calculated using renormalized quantum electrodynamics. At present, calculation and measurement agree to some eighteen significant figures. Nowhere else in science does a prediction agree with observation to such precision and accuracy.444As the precision of the calculation is increased, the number of perturbation terms which must be included grows rapidly, into the hundreds of millions. This led in the 1970’s and 1980’s to an unprecedented situation: It was more expensive to calculate the next digit in the gyromagnetic ratio than to measure it! The precision of this result is now so high that calculating further digits is beyond the scope of quantum electrodynamics alone. The contributions of nuclear interactions must now be included.

As more and more perturbation terms are included in any calculation, more and more divergences must be absorbed into the same set of parameters. The form of the field theory must be rather special for this to be possible. Such a theory is called “renormalizable.” This property, “renormalizability,” seems extraordinarily arcane, but it has been regarded as a sine qua non of physical theory for the last seventy years. Feynman, Schwinger, and Tomonaga established that quantum electrodynamics, the theory of electromagnetic fields interacting with charged particles, is renormalizable. Gerard t‘Hooft and Martin Veltman were awarded the Nobel Prize in Physics in 1999 for their proof that Yang-Mills theories—a class of generalizations of quantum electrodynamics—are also renormalizable. Our current theories of both strong and weak nuclear interactions are Yang-Mills theories. But Einstein’s General Theory of Relativity, our best available description of gravitation, is not a renormalizable field theory. This has motivated decades of searching for a quantum theory of gravitation, a so-called “Theory of Everything.”

Since the 1990’s it has been suggested that the ultimate physical theory might not need to be perturbatively renormalizable. The hope is that more sophisticated, nonperturbative calculations—large-scale numerical calculations, perhaps—might be able to extract the physical content of the theory, including the effects of its interactions, without generating and then absorbing infinite terms. The search is in its early stages, and has not yet produced definitive results.

IV.3 Newtonian quantum field theory

The conceptual framework of quantum field theory—treating the dynamics of an extended system via the quantum excitations of its normal modes—is not restricted to Einsteinian quantum theory and elementary-particle physics. Quantized fields in the Newtonian limit are employed in other areas. In solid-state physics the excitations of the vibrations of an atomic lattice are called “phonons,” quanta of sound or thermal vibration. They interact with the electron field in the material. In nuclear physics, excitations of the collective motions of protons and/or neutrons in nuclei are called “quasiparticles.” Different numbers and states of quasiparticles characterize different excited states of the nucleus.

IV.4 The Standard Model and beyond

Quantum field theory might be termed a “meta-theory,” i.e., a conceptual framework, within which specific physical theories are formulated by specifying the fields in play and their interactons. The current “Standard Model” of particle physics, our best description to date of the fundamental nature of matter, consists of three Yang-Mills theories combined, treating the quark and lepton fields which make up the particles we observe, and their interactions: the “color” interaction between quarks, which engenders the strong nuclear force (and has nothing to do with optical color); the weak nuclear forces, and the electromagnetic forces described by the original quantum electrodynamics of Feynman, Schwinger, and Tomonaga. This model is consistent with all available data on the physics of elementary particles.555Strictly, the Standard Model is a theory, i.e., a well-tested hypothesis and conceptual framework, while individual Yang-Mills constructions are models or hypotheses. The reverse usage, however, is traditional.

A great deal of recent excitement has centered on the 2012 discovery of the “Higgs boson” at the Large Hadron Collider in Geneva, Switzerland. Higgs bosons are the quantum excitations of the Higgs field. This field was proposed in 1964—by Peter Higgs, by François Englert and Robert Brout, and by Tom Kibble, Carl R. Hagen, and Gerald Guralnik—in hopes of accounting for the particle/field masses in the Standard Model as interactions between fields, rather than as arbitrary parameters of the model. The identification of the Higgs particle in data from the Large Hadron Collider has apparently vindicated these theorists’ insights.

Many attempts have been made to merge the three theories making up the Standard Model into a single Yang-Mills theory, allowing processes in which quarks, e.g., in nucleons might transform into leptons. Such constructs are called Grand Unified Theories (GUTs). The simplest models, proposed in the 1970’s, made predictions in conflict with later observations and are thus ruled out. But efforts to construct a successful Grand Unified Theory continue today.

Neither the Standard Model nor the Grand Unified Theories incorporate gravitational interactions between particles. Our most accurate and precise understanding of gravitation to date is contained in Einstein’s 1915 General Theory of Relativity, which describes gravitation in terms of the geometry of spacetime. But this is a classical theory. Attempts to construct a quantum theory of gravitation go at least as far back as Schrödinger in the 1940’s. Such attempts are hampered by the fact that Einstein’s theory, expressed as a quantum field theory, is not renormalizable: Physical predictions cannot be extracted from a quantum version by setting the values of a finite number of parameters. Approaches to an eventual quantum theory of gravitation are varied. As a first step, ordinary quantum field theories can be formulated in the curved spacetime of Einstein’s theory—the normal modes of the fields reflect the spacetime geometry. Stephen Hawking’s 1974 discovery that black holes might radiate via quantum processes emerged from this approach. Or Einstein’s theory might be recast as a quantum theory without relying on the usual perturbative calculations—various attempts in this direction are ongoing.

In the other direction, quantum field theories of the usual form can be constructed incorporating a gravitational interaction, e.g., via a “graviton” field. Such a model is called a “super-unified” theory or, with some hyperbole, a “Theory of Everything.” One class of such candidate models links particles/fields of different spins (and statistics). This feature is called “supersymmetry” (SUSY) or, when a graviton field is involved, “supergravity.” These have been studied extensively since the 1970’s, but the goal of a satisfactory super-unified theory has not yet been reached.

Perhaps the most extensively explored extensions of standard quantum field theory are string theories and a generalization of them, M-theories. These are quantum field theories in which the fields “live” not on spacetime points but on extended structures, “strings” or “branes”—short for “membranes.” (The M in M-theories stands for “matrix”; the field values in these theories are matrices, rather than single real or complex numbers.) Originating in models for strong nuclear forces proposed in the 1960’s, the current versions of these theories were launched by the work of Michael Green and John H. Schwartz in the 1980’s. These theories or, more properly, hypotheses appear to offer the promise of, first, a finite theory without infinities; second, a quantum theory incorporating a gravitational interaction; third, a unique theory—a single, mathematically consistent theory which must be correct, there being no consistent alternative. To date, however, these theories have yet to fulfill any of these promises.

V SUMMARY

Quantum field theory has formed the conceptual framework of much of contemporary physics since the middle of the 20th century. It treats the dynamics of extended systems—from atomic lattices in solids to the fields which constitute the fundamental structure of matter—in terms of the quantum excitations of the systems’ normal modes. It can thus describe dynamical processes in which excitations or particles are created or destroyed. It allows a consistent melding of quantum and Einsteinian mechanics, incorporating key features of elementary particles, such as spin and statistics, in a natural and inevitable way. It relies on the renormalization procedure to absorb the infinities or divergences in a theory into its parameters, but in so doing yields agreements between calculation and measurement unrivaled anywhere else in science. Much of the current frontier of theoretical physics lies within the scope of quantum field theory.

Yet as fundamental and transformative as it is, quantum field theory is rarely even mentioned in introductory physics, from secondary-school through upper-division undergraduate courses. As illustrated here, this omission—comparable to omitting DNA from introductory biology—is not necessary. The theory can be described using no more advanced concepts than the harmonics of extended systems and the energy levels of the quantum harmonic oscillator. It is to be hoped that with such inclusion, the introductory treatment of physics can be brought into greater harmony with the current state of the field.