 Welcome to Quantum Field Theory 3, photons. By way of preface, here's a quote from Richard Feynman. The theory of interaction of light with matter is called quantum electrodynamics, which was the first quantum field theory. The subject is made to appear more difficult than it actually is by the very many equivalent methods by which it may be formulated. One of the simplest is that of Fermi. This quote is almost 60 years old, and since then there have been many developments in quantum field theory. But I think the quote still has some validity. If you compare two textbooks devoted to classical mechanics, or electromagnetism, or even quantum mechanics, you'll typically see similar presentations of concepts and notation. That's not quite as true of quantum field theory. Initially, two different texts may almost seem like they're about different subjects, until you dig a bit deeper. Our approach will be to mostly follow the historical development of the field. As always, we'll strive to emphasize physical intuition, but things will get a bit mathy. Quantum field theory is among the more abstract and mathematical fields of physics. For simplicity, it will sometimes play loose with constants, defining a way or dropping some to keep the notation as simple as possible. Let's start with a review of essential material from videos one and two. In video one, we analyze the harmonic oscillator system of a unit mass on a spring. X denotes the mass's position. We keep hammering on the harmonic oscillator for good reason. It's absolutely central to quantum field theory. In classical mechanics, the total energy is one half quantity p squared plus omega squared x squared. p is the momentum mass times velocity. Since the mass is one, p is just the velocity x dot. Omega squared is the spring constant. We transition to quantum mechanics by replacing momentum and position with operators. The momentum operator, p hat, is minus i h bar slope in x, where i is the imaginary unit, the square root of minus one. The position operator, x hat, is simply multiplication by x. This gives us the Hamiltonian operator, one half quantity p hat squared plus omega squared x hat squared. We apply this to a wave function, psi of x, and look for solutions where this produces the wave function times the total energy e. This is Schrodinger's equation. The solutions have energies, e n equals n plus one half times h bar omega for n equals zero, one, two, etc. h bar is Planck's constant divided by two pi. h bar omega is the quantum of energy. The harmonic oscillator may have any number of energy quanta from zero to infinity. The one half is the zero point energy, the minimum energy allowed by the uncertainty principle. The wave functions are a constant times a polynomial times a bell curve. For simplicity, we use Dirac notation and represent the nth wave function by a ket containing the letter n, representing the state with n energy quanta. As this generally true for Hamiltonian operators, these so-called orthonormal eigenfunctions have the property that the projection of a wave function onto itself is one, while the projection of one wave function onto another is zero. In the quantum mechanic series, we saw that positioned and momentum operators do not commute. Their commutator, x hat p hat minus p hat x hat, is not zero, but i h bar. This is one way to express the uncertainty principle. It's convenient to work in units where h bar equals one. We can combine the position and momentum operators to form the operators a hat minus and a hat plus. These are the destruction and creation operators. The destruction operator applied to the state n produces square root of n times the state n minus one. The creation operator applied to the state n produces square root of n plus one times the state n plus one. The destruction operator followed by the creation operator forms the number operator, n hat. n hat applied to the state n produces n times the state n. Using the position momentum commutator, we can calculate the destruction creation commutator. The value is one. Looking at the definitions of the destruction and creation operators, it's clear that if we add them, the momentum terms cancel. If we subtract them, the position terms cancel. This gives us the expression shown for the position and momentum operators in terms of the destruction and creation operators. Photons are the quanta of the electromagnetic field. So, to understand them, we have to delve a bit into the nature of electric and magnetic fields. Electric and magnetic fields manifest as forces exerted on electric charges. Suppose we have a stationary charge Q. Electric and magnetic fields are vector fields, meaning at every point in space and time they have a magnitude and a direction. We can represent this by an arrow. The arrow length corresponds to the field magnitude and the arrow orientation indicates the field direction. We denote an electric field by the letter E. If a non-zero electric field exists at the charges position, then the charge will be subject to an electric force, Fe, equal to Q times E. We denote a magnetic field by the letter B. Only moving charges are subject to the magnetic force, which is peculiar in that it is perpendicular to both the velocity V and the magnetic field B. We write Fm equals Q times V cross B. The cross product of the velocity and magnetic field vectors is a vector with magnitude equal to magnitude of V times magnitude of B times sine of the angle between V and B, and a direction perpendicular to both V and B. Combining these two forces, we have the electromagnetic force. F equals Q times quantity E plus V cross B. It takes energy to create fields, and that energy is stored in the fields with energy density 1 half quantity magnitude E squared plus magnitude B squared. Depending on the units used, this expression can contain constants, epsilon zero and mu zero, but we'll assume units in which these are both one. Now that we know how electric and magnetic fields interact with charges, we have to consider the equations that describe how they evolve through time and space. We'll need a few concepts from vector calculus. The first is called divergence. Suppose a vector field exists at all points in space and corresponds to the velocity of water. Starting at some point, call it point one, we can move with the velocity at that point V1 for a time delta t to arrive at point two. Then we can move with the velocity V2 at that point for a time delta t to point three. And so on, tracing out a path. In the limit delta t shrinks to zero, we end up with a smoothly varying flow line. If we put a speck of glitter in the water, it would trace out a flow line. Let's call our vector field A and imagine the set of all flow lines. Now imagine a tiny closed surface, such as a square for two-dimensional flow or a cube for the three-dimensional case. At some points on the surface, the flow might be inward and at other points outward. If there is no source of water inside the surface, then any volume of water that flows into the surface must displace an equal volume of water that will flow out of the surface. We then say that the vector field has no divergence. And we symbolize this by the equation del dot A, where the del operator is represented by the triangular nabla symbol. In rectangular coordinates, the divergence of A is the X derivative of the X component plus the Y derivative of the Y component plus the Z derivative of the Z component. If there is a net outflow of water, then the divergence is positive and there must be a source in the interior. If there is a net inflow of water, then the divergence is negative and there must be a sink in the interior. Electric and magnetic fields do not represent the flow of a substance, but we can think about the net electric or magnetic flux through a closed surface as analogous to water flow. The magnetic field has no sources, so this flux and the divergence are zero. Positive and negative charges are, respectively, sources and sinks for the electric field. If there is a net flux through a closed surface, then we know it contains electric charge. In a charge-free region of space, the electric flux and divergence vanish. The next concept we need is called curl or sometimes rotation. Let's first look at the two-dimensional version. Imagine our vector field A represents the velocity of water flow on the surface of a river. On the left, near the shore, it's slow, while on the right, near the river center, it's fast. Now imagine we put a paddle wheel in the river at some point with its rotation axis pointing out of the screen. It's a mathematical wheel that doesn't change the flow of water, but does feel forces proportional to water velocity, and it's infinitesimally small. For the situation shown, the wheel would rotate counterclockwise since the flow on its right side is larger than on its left, and we say that the vector field has curl or rotation at that point. For the three-dimensional case, we put the paddle wheel at some point and we move its axis through all possible orientations until we find the one that produces the greatest rotation. Then the curl of the field at that point is represented by a vector pointing in the direction of this maximum rotation orientation and with length equal to the amount of rotation. This is symbolized by del cross A. In rectangular coordinates, this corresponds to the somewhat complicated expression shown here with EX, Y and Z being unit vectors parallel to the XY and Z axes. The final concept we need is called the Laplacian. Suppose at some point we have a scalar field with value U. A scalar is a single number. It has a magnitude but no direction in space. Suppose the neighboring field values, a tiny distance delta away in the X direction, are U of X minus delta and U of X plus delta. And similarly for the Y direction and the Z direction. Then the Laplacian of U symbolized by del squared U is a number equal to 6 over delta squared times the difference of the average of the neighboring field values and the field value of interest. It's a measure of how in equilibrium the field at a point is with neighboring field values. If the Laplacian at a point is zero, the field equals the average of its neighboring values. If the Laplacian is positive, the field is less than its average neighbor. And if it's negative, the field is greater than its average neighbor. In rectangular coordinates, the Laplacian is the sum of the second derivatives in X, Y and Z. The Laplacian of a vector field A is a vector field for which the X component is the Laplacian of the X component of A and so on. We can now tackle Maxwell's equations in empty space. These are the equations of the electromagnetic field in a region with no charges and no matter. The birth of quantum theory was the idea that the energy of this field is quantized. The quanta, called photons, can be thought of as particles of light. We assume units in which the speed of light equals one. There are four of these equations. One says that the divergence of the magnetic field is zero. The magnetic field has no sources or sinks and the magnetic flux through any closed surface is zero. A vector calculus identity is that the divergence of the curl of any vector field is zero. This suggests that we write B as the curl of another field, A. The divergence of B will then automatically vanish. A is called the vector potential. A second equation tells us how the magnetic field changes with time. The time derivative of B equals minus the curl of the electric field E. Since B is the curl of A, the time derivative of B can be written as the curl of the time derivative of A. Moving the curl of E term to the left, we have that the curl of the quantity is the time derivative of A plus E equals zero. This will be true if E equals minus the time derivative of A. We see that the electric and magnetic fields are both determined by a single field, the vector potential. This is why it makes sense to refer to the electromagnetic field as a single thing. A third equation says the divergence of E is zero. Since the divergence of E is minus the time derivative of the divergence of A, this equation is true if the divergence of A equals zero. The final equation tells us how the electric field changes with time. The time derivative of E equals the curl of B. B is the curl of A, so the curl of B is the curl of the curl of A. This messy expression can be rearranged as the sum of two terms. The first depends on the divergence of A and so is zero. The second is minus the Laplacian of A. Since E is minus the time derivative of A, the time derivative of E is minus the second derivative of A. Canceling minus signs, this leaves the second time derivative of A equals the Laplacian of A. This is a vector form of the wave equation. It consists of three scalar wave equations, one for each component of A. These plus the requirement that the divergence of A vanishes provide four equations for the components of A that we must solve to describe the electromagnetic field in empty space. If we can quantize this field, then we should obtain a rigorous quantum theory of photons, which is one of the primary goals of quantum field theory.