*Laurie M Brown. New Dictionary of the History of Ideas. Editor: Maryanne Cline Horowitz. Volume 2. Detroit: Charles Scribner’s Sons, 2005.*

The German physicist Max Planck (1858-1947) introduced the quantum of action *h* (Planck’s constant) into physics in his successful theory of blackbody radiation in 1900 and thus started quantum theory, one of several revolutionary discoveries that occurred in physics at the turn of the twentieth century. Others were Albert Einstein’s (1879-1955) special theory of relativity, his theory of the light quantum, and the experimental proof of the existence of atoms and molecules, based on Einstein’s theory of Brownian motion, all published in the single year 1905. These theories and the general theory of relativity distinguish modern physics from the classical physics that was dominant in 1900.

The desire to formulate physics in terms of observable quantities motivated Einstein’s special theory of relativity, which eliminated the unobservable ether assumed by James Clerk Maxwell (1831-1879) and his followers. The “new quantum theory” of the 1920s was also supposed to be a theory of observables, as it eliminated the visualizable orbits of the “old quantum theory” that preceded it, but the new version introduced concepts such as Erwin Schrödinger’s (1887-1961) wave function (psi-function) that involves the imaginary number *i* 1. Quantum mechanics and quantum field theory employ other mathematical concepts that are far from the observable realm.

While other developments of modern physics, such as relativity, can be seen as generalizing classical physics, quantum mechanics makes a much stronger break with the past. Even the basic notion of measurement is problematic in quantum theory. One of its consequences is a fundamental indeterminacy that prevents, for example, the simultaneous determination of the position and momentum of a particle. That brings into question the entire epistemology on which classical physics is based. The psi-function of an atom, for example, is the solution of a differential equation that is linear in the time variable, and once given, psi is determined for all time. The psi-function thus provides a deterministic description of the system, but from this complete description one can predict with certainty the outcome of only special “compatible” measurements; for most measurements, only the probability of a given result can be calculated.

The implications of quantum theory are so profound that even its creators, such as Planck and Einstein, found it difficult to accept and wrestled with its concepts all their lives. Skeptical physicists devised ways to avoid the apparent contradictions, and these proposals have led to experimental tests. All of the tests performed so far have confirmed the predictions of quantum theory, although its laws are subject to differing physical interpretations.

## Planck’s Paper of 1900

As the nineteenth century drew to a close, many physicists felt that the fundamental laws of physics were well understood, and would be a permanent part of humanity’s worldview. Although new experimental results, such as the recent discoveries of X-rays, radioactivity, and the electron, could modify the view of the microworld, it was felt that classical mechanics, electromagnetism, thermodynamics, and statistical mechanics, based on classical ideas, would sooner or later provide a satisfactory account of the new phenomena. When William Thomson (Lord Kelvin; 1824-1907) gave a talk at the Royal Institution in London in April 1900 concerning two clouds over the theories of heat and light, he referred not to the recent discoveries but to problems with the ether and the failure of the classical theorem that heat energy should be distributed uniformly among the various possible motions of molecules.

Einstein’s relativity theory disposed of the ether (at least for the next century), while the more obscure problems of molecular motion would find their solution in a new mechanics that was born in 1900 from the revolutionary reasoning of a conservative middle-aged physicist, Max Planck. In 1860 Gustav Robert Kirchhoff (1824-1887) had shown that the relative amounts of energy emitted at different wavelengths from any surface was the same for any material and dependent only on the temperature. He also argued that the ratio of the emissive power to the absorptive power of any given material was constant. Kirchhoff conceived of an ideal blackbody that absorbed all the radiation that fell upon it, and he saw that this situation could be realized by an enclosure at constant temperature (such as the interior of a furnace). The radiation from a small hole in such an isothermal enclosure would not depend upon the size or shape of the enclosure or upon the material of the walls. Therefore it had a universal character that called for a fundamental explanation, one that was independent of theories regarding the structure of matter (which were essentially unknown). The spectrum, and hence the color, of the radiation depends only upon the temperature, whether the source be a furnace, the filament of an incandescent lamp, or a glowing star in the sky.

Planck found that explaining the spectrum of blackbody radiation was especially challenging, and hoped to understand its intensity as a function of wavelength or frequency (the two being related by λν = *c, c* being the velocity of light in vacuum). Born in 1858 in the Baltic city of Kiel to an academic family, he studied in Munich and Berlin, where one of his professors was Kirchhoff, to whose chair he succeeded in 1887 after receiving his Ph.D. in Munich in 1879. Planck lived a long and productive life, undergoing terrible personal losses in both world wars and also during the Nazi period. He died in 1947, after suffering the execution of his younger son for participating in a plot to kill Adolf Hitler.

Planck’s Ph.D. thesis dealt with the second law of thermodynamics, which says that heat does not flow from a cooler body to a hotter body (one of several formulations). The first law of thermodynamics expresses the conservation of energy. Planck was attracted to the laws of thermodynamics because they were held to have a universal significance. From about 1897 on he devoted himself to understand the laws of black-body radiation from the standpoint of thermodynamics. Another motivation for him was that experimental scientists at the *Physikalisch-Technische Reichsanstalt* (the German Bureau of Standards) in Berlin were making an accurate determination of the form of the blackbody spectrum. Using reasoning based first on thermodynamics and then on the statistical mechanics of the Austrian physicist Ludwig Boltzmann (1844-1906), Planck derived a formula that gave precise agreement with the latest experimental results, although even its author, who spent more than a decade afterward trying to improve it, later questioned its original derivation.

Planck’s formula for u, the energy per unit volume of radiation in a cavity at frequency, reads:

*u*_{ν} = (8π*h*ν^{3} / *c*^{3}) (1 / exp(*h*ν / *kT*) − 1)

where T is the absolute temperature and *k* is called Boltzmann’s constant. Plotted against, the curve for *u* rises from zero at 0 to form a curved peak with its maximum near *h* 3 *kT,* and then falls exponentially to zero for large, where the “1” in the denominator becomes negligible. The peak position that determines the color of the radiation thus increases linearly with the absolute temperature.

The important things to notice about Planck’s formula, which he published in 1900, are that it gives the correct spectrum of blackbody radiation and that, aside from the variable quantities frequency and temperature, it contains only the fundamental physical constants *h, c,* and *k.* Boltzmann’s constant *k,* which was first introduced by Planck, is the so-called gas constant per mole divided by the number of molecules in a mole (Avogadro’s number) and its significance is that *kT* is a molecular energy. (A monatomic gas molecule has energy (3/2) *kT.*) The universal constant *c* is the velocity of light in vacuum. Planck’s constant is *h* 6.626 × 10^{34} Joule-seconds. Its dimension of energy times time is called “action,” and *h* is called the quantum of action. Evidently Planck’s *h* is a very small quantity and, like *k,* its most important role is in atomic and molecular physics. The laws of motion of baseballs and planets do not depend upon *h,* but without it one cannot understand anything about the microworld; its size sets the scale for the elementary particles of which the world is built.

## Einstein’s Light Quantum

In deriving his blackbody formula Planck assumed the complete validity of classical electromagnetism and, not having any theory of atomic structure, used the fact that the radiation was independent of the material of the radiator to assume that it would be sufficient to consider the cavity walls as modeled by a collection of simple harmonic oscillators, each capable of absorbing and emitting a particular frequency. He then calculated the average energy of such an oscillator in equilibrium with the cavity radiation, using a method previously developed by Boltzmann for the statistical treatment of an ideal gas. As an aid to calculating the various possible “partitions” of the total energy among the gas molecules, Boltzmann had assumed each to have one of a discrete set of energy values, but afterward had let the separation of the values tend to zero, so that the molecules could have a continuous range of energy. However, Planck found good agreement with the experimental frequency distribution only when the oscillator energy values, separated by *h* in his theory, remained separated by that amount, with *h* retaining the finite value that we have given above.

Several physicists criticized Planck’s derivation, claiming that he had misapplied Boltzmann’s method to his problem, and pointing out that Boltzmann had treated his molecules as distinguishable from one another. Planck’s treatment implied that oscillators of a given frequency were indistinguishable. Einstein’s view was that the derivation was questionable, but the formula was undoubtedly correct and this brought into question certain aspects of Maxwell’s electromagnetic theory. Einstein also had other grounds for questioning Maxwell’s theory. In the photoelectric effect, radiation (e.g., ultraviolet light) falls on a metal plate and ejects electrons. Below a certain threshold frequency, dependent on the metal, no electrons are ejected. Above that threshold, electrons are ejected with kinetic energies that increase linearly with the difference of the light frequency from the threshold. At a fixed frequency, increasing the light intensity causes more electrons to be emitted, without increasing the energy per electron. This phenomenon is best explained by assuming that the light consists of concentrations of energy that can be transferred to individual electrons, as in a collision of particles. However, that picture is totally inconsistent with the smooth continuous wave picture of Maxwell’s electromagnetic theory of light.

Einstein conjectured that the cavity radiation treated by Planck was like a gas consisting of “particles” called light quanta, each having energy *hν.* (In 1926 the American chemist Gilbert N. Lewis named these particles “photons.”) Arguing this way, Einstein obtained Planck’s formula without having to make any arbitrary assumption concerning the cavity walls, such as their being modeled by simple harmonic oscillators. Einstein also regarded that this particulate aspect belonged to radiation in general, whether in an enclosure or in free space. That amounted to a revolutionary modification of the existing theory of optics and electromagnetism. Planck and others found Einstein’s proposal very difficult to accept. Although it explained phenomena that were otherwise mysterious, it was hard to see how such optical phenomena as refraction or diffraction (the spreading of light around a sharp edge) that since the early nineteenth century had been treated with a wave theory of light, could be reconciled with almost Newtonian particles of light. For at least two decades, the quantum achievements of Planck and Einstein were regarded with suspicion. The Nobel Prize Committee recognized Planck’s great discovery only in 1918; Einstein’s in 1921 (but awarded the prize in 1922).

## Neils Bohr and the “Old Quantum Theory”

In the same year (1922) in which Einstein was awarded the Nobel Prize, largely for his explanation of the photoelectric effect, the Danish physicist Niels Bohr (1885-1962) received the same recognition for his quantum theory of the structure of atoms and their radiations, first put forward in 1913. Before Bohr, the quantum of action had been associated with heat radiation and light. It had also dissipated (in the hands of Einstein and others) one of Kelvin’s clouds hanging over physics, that of the specific heats. Bohr seized upon the quantum to make a durable model of the atom itself.

In 1909 the New Zealander Ernest Rutherford (1871-1937), a professor in Manchester, England who was famous for his work in radioactivity, asked one of his students to study the absorption of alpha particles in a thin foil of gold. As an atom was supposed to be a sphere of diffuse positive charge with electrons stuck in it, able to vibrate in response to light or to emit light when set into vibration, Rutherford expected the alpha particle, a high speed helium nucleus, about eight thousand times heavier than the electron, to pass through with only a small deviation. Instead, many alpha particles suffered deflections by large angles, and some were even reflected in the backward direction. After verifying this amazing result and analyzing it mathematically, in 1911 Rutherford announced that the atom, about 10^{8} cm in radius, was almost entirely empty, that the positive charge was concentrated in a region no larger than 10^{12} cm (the nucleus), and the electrons (79 in the case of gold) circled the nucleus as the planets circle the sun.

Niels Bohr came from a well-known academic family in Copenhagen, his father a professor of physiology and his mother belonging to a rich and cultivated Jewish family. Niels’s brother Harald (1887-1951) became a famous mathematician and his son Aage (b. 1922) also won a Nobel Prize in physics. Born in 1885, Niels Bohr received his Ph.D. from the University of Copenhagen in 1911 and traveled for further study to England, first to Cambridge to work with Joseph John Thomson (1856-1940), who had discovered the electron in 1897. Thomson was responsible for the atomic model that was replaced by Rutherford’s planetary picture. After a short time, Bohr moved to Manchester to work with Rutherford and began to think about atoms.

Comparing Rutherford’s model of the simplest atom, hydrogen—one light electron circling a heavy positive nucleus—with a simplified solar system containing only the earth and the sun, the mechanical picture would seem to be nearly identical, with the inverse square electrostatic force in the atom replacing the inverse square gravitational force in the solar system. However, atoms that continuously interact and collide with other atoms (e.g., in a gas) remain remarkably stable, retaining their size and shape, which is not true of solar systems. Moreover, according to classical electromagnetic theory even an isolated atom would be unstable, as the electron in its orbit accelerates toward the nucleus, it should radiate energy like a small antenna. Within a small fraction of a second, the electron should spiral in toward the nucleus and be absorbed. Yet atoms appear to be almost indestructible under ordinary conditions.

In 1913 Bohr published an atomic theory that solved these difficulties, but it required the acceptance of two very new principles. The electrons revolve around the nucleus, but only in certain well-separated orbits called “stationary states.” In these allowed states, the electron does not radiate, but in passing from state of energy *E* to another of lower energy *E*′ it emits a light quantum of energy *hν = E − E′*. Similarly it absorbs such a quantum (if present) in passing from the state of lower energy to the higher. In the case of hydrogen, the allowed energy states are circular orbits that are restricted by the condition *mvr = nh/2π*, where *m* and *v* are, respectively, the mass and speed of the electron and *r* is the radius of the circular orbit. The “quantum number” *n* is a positive integer. One can then show easily that the energies of the stationary states are given by *E*_{n} = −(1/*n*^{2}) me^{4}/2(*h/2*π)^{2} = −13.6/*n*^{2} electron volts. (The negative energy means that the electron is in a bound state. An electron with positive energy is outside of an atom.)

In 1860, Kirchhoff and Robert Bunsen (1811-1899) at the University of Heidelberg started systematic investigations of atomic spectra and showed that the frequencies of the spectral lines were characteristic of elements, even when those atoms formed part of a chemical compound. Any good theory of atomic structure would have to explain those frequencies. For the case of hydrogen, combining Bohr’s two principles gave the result *hv* = *E − E* = −13.6 (1/n^{2} − 1/*n*′^{2}) electron volts. With *n*′ = 2, Bohr’s expression gave the Balmer formula for the hydrogen lines in the visible part of the spectrum, known since 1885. Other hydrogen lines predicted to be in the ultraviolet were found, as well as those of another “one-electron” atom, namely once-ionized helium. Another success was the prediction of X-ray lines arising from heavier elements, which gave an accurate determination of the element’s atomic number and hence its correct place in the periodic table of the chemical elements. This showed that there was a “missing” element, with atomic number 43, still to be discovered.

The analysis of the spectra of heavier elements was not so simple, and even the observed hydrogen spectrum showed more structure than Bohr’s first model allowed. Bohr himself attacked this problem, as did others, especially Arnold Sommerfeld (1868-1951) of the University of Munich. Besides Bohr’s *n,* the “principle quantum number,” Sommerfeld introduced an “angular momentum quantum number” *l*, which could take on the integer values from 0 to *n* − 1, and a “magnetic quantum number” with integer values from − *l* to + *l.* With their help, Sommerfeld could account for many spectral lines, including the “splitting” into several components, an effect known as “fine structure.” Even the hydrogen lines showed fine structure, and Sommerfeld was able to account for small relativistic effects.

Because the Bohr-Sommerfeld model had electrons following definite visualizable orbits, however restricted, including ellipses as well as circles, the model could be described as semi-classical. In the calculation of the probabilities of transitions between stationary states, which give the relative brightness of the various spectral lines, Bohr used another semiclassical idea, to which in 1923 he gave the name “correspondence principle.” According to this, the classical radiation theory is valid whenever the quantum numbers have large values, so that both the frequency and the intensity of light emission is that which would arise classically from the electron’s acceleration in its orbit.

## The New Quantum Mechanics of Heisenberg, Schrödinger, and Dirac

From 1925 to 1927 three equivalent new versions of quantum mechanics were proposed that extended the Bohr-Sommerfeld theory, cured its main difficulties, and produced an entirely novel view of the microworld. These new theories were the matrix mechanics of Werner Heisenberg (1901-1976), the wave mechanics of Erwin Schrödinger (1887-1951), and the transformation theory of Paul A. M. Dirac (1902-1984), the last being a more general version that includes both of the other versions. The three physicists were awarded Nobel prizes in physics at a single award ceremony in Stockholm in 1933, Heisenberg receiving the prize for 1932 and Dirac and Schrödinger sharing the prize for 1933.

In his Nobel address, Heisenberg stated:

*“**Quantum** mechanics … arose, in its formal content, from the endeavor to expand Bohr’s principle of correspondence to a complete mathematical scheme by refining his assertions. The physically new viewpoints that distinguish **quantum** mechanics from classical physics were prepared … in analyzing the difficulties posed in Bohr’s theory of atomic structure and in the radiation theory of light.” (*Nobel Lectures in Physics,* p. 290)*

The difficulties were many, including that of calculating the intensities of spectral lines and their frequencies in most cases (hydrogen being an exception) and in deducing the speed of light in various materials. In classical theory an atomic electron would emit radiation of the same frequency with which it orbited a nucleus, but the frequencies of spectral lines are practically unrelated to that orbital frequency and depend equally upon the final and initial atomic states. Heisenberg argued that the electron orbiting the nucleus was not only “un-observable,” but also “unvisualizable,” and that perhaps such an orbit did not really exist! He resolved to reformulate the theory following Einstein’s procedure in formulating the theory of relativity, using only observable quantities. Heisenberg started, therefore, with the spectral lines themselves, not an atomic model, and introduced a transition “amplitude” A_{if,} depending on an initial state *i* and a final state *f,* such that the line intensity would be given by the square of A_{if}. That is analogous to the fact that light intensity in Maxwell’s optical theory is given by the square of the field intensity. (Actually, A_{if} is a complex number, involving the square root of −1, so in quantum mechanics we use the absolute square, which is a positive real number.)

In calculating the square to produce the light intensities, Heisenberg found it necessary to multiply amplitudes together, and he discovered that they did not behave like ordinary numbers, in that they failed to commute, meaning that in multiplying two different A’s the result depended on the order in which they were multiplied. Heisenberg was working in Göttingen under the direction of Max Born, and when he communicated his new result, Born realized that the mathematics involving arrays of numbers such as A_{if} was a well-known subject known as matrix algebra. Born, together with his student Pascual Jordan (1902-1980) and Heisenberg, then worked out a complete theory of atoms and their transitions, known as matrix mechanics. Born also realized that the matrix A_{if} was a probability amplitude, whose absolute square was a transition probability. This meant that the law for combining probabilities in quantum mechanics was entirely different from that of classical probability theory.

Heisenberg was only twenty-four years old when he made his major discovery. The son of a professor of classics at the University of Munich, he received his Ph.D. under Sommerfeld at Munich in 1923, after which he went to work with Born and later with Bohr. He had an illustrious career, not unmarked by controversy. The author of wave mechanics, Erwin Schrödinger, on the other hand, was already in 1926 an established professor at the University of Zurich, holder of a chair once held by Einstein, and an expert on thermodynamics and statistical mechanics. He was born in Vienna to a wealthy and cultured family and received his Ph.D. at the University of Vienna in 1910.

Schrödinger’s theory was based on an idea that the French physicist Louis de Broglie (1875-1960) put forward in his Ph.D. thesis at the University of Paris in 1924. Einstein had advanced as further evidence of the particle character of the photon that it had not only an energy *hν*, but also a momentum *p* = *hν/c*. The American experimentalist Arthur Compton (1892-1962) showed in 1923 that X-rays striking electrons recoiled as if struck by particles of that momentum. Photons thus have wavelength λ = c/*ν* = *h/p*. De Broglie conjectured that all particles have wave properties, their wavelengths being given by the same formula as photons, with *p* being given by the usual particle expressions (*p = m*v, nonrelativistically, or *p = m*vβ in relativity, with β(1 − v^{2}/c^{2})^{−1/2}). C. J. Davidson and L. H. Germer in America and G. P. Thomson in England in 1927 showed the existence of electron waves in experiments. An important application is the electron microscope.

Schrödinger’s theory merged the particle and wave aspects of electrons, stressing the wave property by introducing a “wave function,” often denoted by the Greek letter psi (ψ), which is a function of space and time and obeys a differential equation called Schrödinger’s equation. Psi has the property that its absolute square at a certain time and place represents the probability per unit volume of finding the electron there at that time. The stationary atomic states of Bohr are those whose probability density is independent of time; the electron in such a state is spread out in a wavelike manner, and does not follow an orbit. The real and imaginary parts of the one-particle wave function are separately visualizable, but not for *n* particles, as it exists in a 3 *n*-dimensional space.

Although the pictures of Heisenberg and Schrödinger are totally different, Schrödinger (and others) proved that their experimental consequences were identical. In 1926 the English physicist Paul Dirac showed that both pictures could be obtained from a more general version of quantum mechanics, called transformation theory, based on a generalization of classical mechanics. When he did this work, the basis of his doctoral thesis in 1926, he was the same age as Heisenberg. Dirac was born in Bristol, the son of an English mother and a Swiss father. The latter taught French language in the Merchant Venturers’ School, where Dirac received his early education, going on to earn his Ph.D. at Cambridge. The influence of Dirac’s treatise *The Principles of **Quantum** Mechanics,* published in 1930, has been compared by Helge Kragh to that of Newton’s *Principia Mathematica.*

## Uncertainty, Exclusion Principle, Spin, and Statistics

One of the striking consequences of the quantum theory is the noncommutation of pairs of “operators” representing physical quantities, such as position and momentum (whether regarded as matrices, derivatives acting upon the psi-function or, as Dirac would put it, abstract algebraic quantities). If *p* and *q* are two such operators, their commutation relations read: *p q − q p = h*/2π*i*, where *i* is the square root of −1. From this condition Heisenberg proved that *q* and *p* cannot be simultaneously measured with arbitrary accuracy, but that each must have an error, the product of the errors exceeding *h* /2. Since an exact knowledge of the initial conditions is necessary in order to make predictions of the future behavior of the system in question, and since such exactness is in principle impossible, physics is no longer a deterministic science. Even in classical physics, exact knowledge is not usually possible, but the important difference is that quantum mechanics forbids exactness “in principle.” Heisenberg’s uncertainty relations and other “acausal” predictions of quantum mechanics have given rise to an enormous amount of philosophical debate, and Einstein and other prominent older physicists never accepted the so-called Copenhagen interpretation, developed by the school of Bohr and Heisenberg. Bohr’s principle of complementarity states that the world (or the part being studied) reveals itself to experimental probing in many different guises, but no one of these pictures is complete. Only the assembly of all possible pictures can reveal the truth.

One of the great aims of the Bohr-Sommerfeld quantum theory was to explain the periodic table of the chemical elements in terms of atomic structure, especially the periods 2, 8, 18, 32, and so on, having the general form 2 *n*^{2}, where *n* is an integer. Bohr and others made great progress in this direction, based on the quantum numbers of the allowed stationary states, except for the “2” in the formula. Wolfgang Pauli (1900-1958), who had worked with Born and Bohr and was studying the classification of atomic states in the presence of a strong magnetic field, suggested in 1925 that the electron must possess a new nonclassical property, or quantum number, that could take on two values. With this assumption he could describe the atomic structures in terms of shells of electrons by requiring that no two electrons in the atom could have identical quantum numbers. This became known as Pauli’s “exclusion principle.”

A short time after this, two young Dutch physicists, Samuel Goudsmit and George Uhlenbeck, found Pauli’s new property in the form of spin angular momentum. This is the analog for an elementary particle of the rotation of the earth on its axis as it revolves around the sun, except that elementary particles are pointlike and there is no axis. However, they do possess a property that behaves as and combines with orbital angular momentum and that can take on values of the form s(*h/2*π), where *s* can be zero, integer, or half-integer. That was a totally unexpected development, especially the half-integer quantum numbers. The electron is one of several elementary particles that has ½ unit of spin, while the photon has spin of one unit.

The subject of quantum statistics, as opposed to classical statistical mechanics, is an important field. (Recall that *h* entered physics through Planck’s analysis of the statistics of radiation oscillators.) It is found that identical stable elementary particles of half-integer spin (electron, protons) form shells, obey the exclusion principle, and follow a kind of statistics called Fermi-Dirac, worked out by Dirac and the Italian-American physicist Enrico Fermi (1901-1954). Other Fermi-Dirac systems are the conduction electrons in a metal and the neutrons in a neutron star. On the other hand, identical elementary particles of integer spin, such as photons of the same frequency, tend to occupy the same state when possible, and obey Bose-Einstein statistics, worked out by the Indian physicist Satyendranath Bose (1894-1974) and by Einstein. This results in phenomena that characterize lasers, superfluid liquid helium at low temperature, and other Bose-Einstein condensates that have been studied recently.

## Relativistic Quantum Theory and Antimatter

The effects of relativity are large when the speed of a particle approaches that of light or, equivalently, when its energy is an appreciable fraction of its rest energy *mc*^{2}*.* This occurs in atomic physics only in the inner shells of heavier elements and plays a relatively minor role even in the physics of atomic nuclei. However, in treating such problems as the scattering of X-rays and gamma rays (the Compton effect), relativity and quantum theory are both important. In 1928 Dirac used his transformation theory to deduce for the electron a relativistic analog of the Schrödinger equation with remarkable properties. He found that it required four associated wave functions, where Schrödinger used only one. Dirac’s equation automatically endowed the electron with its observed spin of ½ (h/2π), and gave it its observed magnetic moment and the relativistic fine structure of spectral lines. That accounted for one doubling of the number of wave functions. The second doubling resulted from the formalism of relativity and was very troubling, as it allowed electrons to have negative energy, which in relativity implies meaningless negative mass. After several years of pondering the problem, Dirac suggested an interpretation of the theory that predicted the existence of a positive electron, capable of annihilating with a negative electron, the result being gamma rays, an example of the transmutation of mass into energy. A year later, in 1932, the positive electron (anti-electron or positron) was found in the cosmic rays and then was produced in the laboratory. All particles have been found to have antiparticles (some are their own antiparticles) and so there exists a world of antimatter.

## Quantum Field Theory and Renormalization

Although the quantum theory began with the study of radiation, it took more than two decades before the electromagnetic field itself was quantized. The results obtained before 1928 used semiclassical approaches such as Bohr’s correspondence principle. Dirac first made a fully quantum theory of electrons interacting with photons in 1927. Heisenberg, Pauli, and later Fermi, extended the theory, but although it produced valuable results not otherwise obtainable, it was found to be mathematically unsound at very high energy, and predicted infinite results for the obviously finite charge and mass of the electron. The problem was solved only in the late 1940s by the renormalization theories of the American theorists Richard Feynman, Julian Schwinger, Freeman Dyson, and the Japanese theorist Sin-itiro Tomonaga. Elementary particle physics in the twenty-first century is dominated by a so-called Standard Model, which is based entirely on the use of renormalized quantum fields.