 As quantum mechanics matured during the 1920s, attention increasingly turned to quantum field theory. Here's a list of a few of the important publications from this period. In the previous video, we discussed Born Heisenberg and Jordan's 1926 paper, presenting the quantization of a vibrating string. The following year, Dirac applied these ideas to the electromagnetic field in his The Quantum Theory of the Emission and Absorption of Radiation. This paper is often cited as the birth of quantum field theory. In 1928, Jordan and Wigner's paper on the Pauli Exclusion Principle showed how to make creation and destruction operators compatible with the exclusion principle, thereby allowing the concept to be extended to electrons. We will explore these ideas in the next video. Also in 1928, Dirac published his equation of relativistic quantum mechanics for the electron. We covered this in the last video of the quantum mechanics series. The following year, Heisenberg and Pauli's paper on the quantum dynamics of wave fields presented the first general theory of quantum fields and a canonical quantization formalism that continues to play an important role in quantum field theory. Oppenheimer's 1930 paper, Note on the Theory of the Interaction of Field and Matter, highlighted a central problem in quantum field theory. The interaction of a charged particle with its own electric field produces infinite energy level shifts. Taming these infinities was a preoccupation of quantum field theorists for years to come. During the summer of 1930, Enrico Fermi visited the University of Michigan, where he delivered a series of lectures on the quantum theory of radiation. These were published as a 46-page article, which served as a mini textbook on the subject for physicists in the 1930s and beyond. As evidenced by the quote from Richard Feynman at the beginning of this video, Fermi was known for presenting difficult concepts in a simple and intuitive style. So let's look at Fermi's approach to quantizing the electromagnetic field. We start with the simplest case, the field in a one-dimensional electromagnetic cavity. We assume the field varies only along the z-axis and is confined by perfectly conducting planes to z-values between 0 and L. In this region, assume the vector potential A has only an x-component that varies sinusoidally in z and vanishes at z equals 0 and L. We write A equals ex, the unit vector in the x-direction, times sine k pi over Lz times a time-varying amplitude q of t. Here k is a positive integer. For this geometry, the wave equation requires that the second derivative in T of A equals the second derivative in z of A. Performing the derivatives and canceling common factors, we obtain q double dot equals minus omega squared q, where omega equals k pi over L. The electric field is minus the time derivative of A or minus ex sine k pi over Lz times q dot. This vanishes at the perfect conductors z equals 0 and L, as it must. The magnetic field is the curl of A, which evaluates to Ey omega cosine k pi over Lz times q. In addition to satisfying the wave equation, the vector potential must have zero divergence. Since field components are assumed to vary only with z, this will be true if the z-component of A is 0, which it is. We can repeat these steps, assuming A has only a y-component. This gives us a second solution in which all field vectors are rotated by 90 degrees from our first solution. So, for any positive integer k, we have two solutions, representing two independent field polarizations. Any possible field polarization can be represented as a linear combination of these. The most general field in our 1D cavity is then the sum over k from 1 to infinity, the sum over alpha from 1 to 2 of e alpha qk alpha of t sine k pi over Lz. Here, the unit vectors e1 and e2 represent ex and ey. For the k alpha mode, the amplitude satisfies qk alpha double dot equals minus omega k squared qk alpha, with omega k equal to k pi over L. The solution is qk alpha equals bk alpha times cosine omega k times t minus tk alpha. We can alternately express the cosine argument as omega kt plus theta k alpha, with theta k alpha equal to minus omega k tk alpha. Although the frequency omega k is fixed, the amplitude bk alpha and time offset tk alpha can be any real numbers. So, each mode can be arbitrarily scaled in amplitude and arbitrarily shifted in time. A solution of this form is called a standing wave, because the sinusoidal variation in space remains fixed and merely oscillates in amplitude. More often, we are interested in traveling waves, representing the flow of energy through unbounded space due to radiation. Let's consider a traveling wave in one-dimensional unbounded free space. Assume A is polarized in the x direction and the field travels along the z axis at the speed of light, which is 1 in our units. We write A equals ex b cosine kz minus omega t plus theta. This solves the wave equation of omega equals k. b and theta are arbitrary amplitude and phase values. To see that this is a traveling wave, we set the cosine's argument equal to a constant phi and solve for z. We find z equals t plus a constant. So for each increment of time, a given point on the wave moves a corresponding increment along the z axis. We can apply a sum of angles trig identity to write b cosine kz minus omega t plus theta as the sum of two terms. Cosine kz times b cosine omega t minus theta plus sine kz times b sine omega t minus theta. Let's denote by q the amplitude b cosine omega t minus theta. The time derivative, q dot, is minus b omega sine omega t minus theta. By analogy with a unit mass harmonic oscillator, let's consider q dot analogous to a momentum p. Then we can write the vector potential A as unit vector ex times the quantity q cosine kz minus 1 over omega p sine kz. Using this expression for A, we can calculate the electric field as minus the time derivative of A and the magnetic field is the curl of A. These operations result in the expression shown here. p equals q dot, so p dot equals q double dot, which equals minus omega squared k. Using these relations and the fact that k equals omega, we can rewrite the electric field as minus ex times the quantity p cosine kz plus omega q sine kz. The magnetic field is minus e y times the same quantity. The field energy is the field energy density, one half magnitude e squared plus magnitude b squared, averaged over space. Since e and b have the same magnitude, this is two times one half the average of p squared cosine squared kz plus omega squared q squared sine squared kz plus two omega pq cosine kz sine kz. The cosine times sine term has zero average. Cosine squared and sine squared both averaged to one half. So the field energy is one half quantity p squared plus omega squared q squared. This is formally identical to the energy of a harmonic oscillator, but in place of x, the position of a unit mass, we have q, the amplitude of an electromagnetic mode. As we progress through this series, we'll come to appreciate this statement from a quantum field theory textbook. Quantum field theory is just quantum mechanics with an infinite number of harmonic oscillators. So let's do quantum mechanics. We replace the dynamical variables q and p in the energy expression with operators q hat and p hat. q hat is multiplication by q. p hat is minus i h bar derivative with respect to q. This gives us the Hamiltonian operator h hat. Then we look for the stationary states where h hat applied to a wave function produces energy e times the wave function. But this is simply the harmonic oscillator problem that we've already solved. And we replace q and p in our field expression with q hat and p hat. So the electromagnetic field becomes an operator a hat. q hat and p hat can be expressed in terms of the creation and destruction operators a hat plus and a hat minus. With these substitutions, a hat becomes e x square root 1 over 2 omega times the quantity a hat minus plus a hat plus times cosine kz plus i a hat minus a hat plus times sine kz. Combining a hat minus and a hat plus terms, inside the square brackets we get a hat minus times quantity cosine kz plus i sine kz plus a hat plus times the same quantity with i replaced by minus i. Finally, we identify the cosine plus or minus i sine expressions as e to the i kz and its complex conjugate e to the minus i kz. So a hat equals e x square root 1 over 2 omega times the quantity a hat minus e to the i kz plus a hat plus e to the minus i kz. Now let's extend this to describe a radiation field in three dimensional free space. We take our one dimensional solution, polarized in the x direction and propagating in the z direction and rotate it in space so that it propagates in any desired direction e k. The rotation transforms e x and e y into e k1 and e k2. The propagation vector k has x, y, and z components kx, ky, and kz. Its magnitude, square root kx squared plus ky squared plus kz squared is the propagation constant and equal to the parameter k in the one dimensional wave. The propagation vector divided by its magnitude is the unit vector ek and the oscillation frequency omega k equals k. The propagation constant equals 2 pi over the wavelength lambda where the wavelength is the distance between wave peaks or valleys. In place of k times z in the one dimensional expression we will have wave vector k dot position vector x equals kxx plus kyy plus kzz. Then the field operator for this mode is a hat equals square root 1 over 2 omega k ek alpha times quantity a hat minus k alpha e to the i k dot x plus a hat plus k alpha e to the minus i k dot x. Here ek alpha can be ek1 or ek2. The most general radiation field is then the sum over all propagation vectors k and the two polarizations alpha of this mode expression. The operator a hat plus k alpha creates a photon in mode k alpha. The operator a hat minus k alpha destroys a photon in mode k alpha. The Hamiltonian operator for the entire field is simply the sum of the operators for each mode. One half quantity p hat k alpha squared plus omega k squared q hat k alpha squared which can be written as the sum over all modes of the mode number operator plus one half times h bar omega k where the number operator n hat k alpha is a hat plus k alpha times a hat minus k alpha. Finally the wave function of the radiation field is simply a list of the number of photons in each mode n k alpha called the occupation numbers. Application of the creation or destruction operator for a given mode modifies only that mode's occupation number.