 Welcome to Quantum Field Theory 1, Creation and Destruction. The first video in this series on Quantum Field Theory. Let's start by reviewing a few results from our Quantum Mechanic series. These provide the context in which Quantum Field Theory was developed. The birth of Quantum Mechanics was the concept that the energy of the electromagnetic field is quantized. At frequency nu, these quanta, which we call photons, have energy and momentum, respectively, of h nu and h nu over c, which is also h over lambda. Here, h is Planck's constant, c is the speed of light, and lambda is the wavelength. Energy quantization is required to explain the Planck formula for radiation intensity versus frequency at a given temperature. This correctly predicts that intensity increases to a certain frequency, then rapidly drops off, unlike the classical prediction of an ever-increasing intensity. This wave-particle duality with the electromagnetic field having both continuous or wave and discrete or particle properties is at the heart of quantum weirdness. Because electromagnetic energy can be emitted and absorbed, it follows that photons can be created and destroyed. When this happens, their energy and momentum are transferred from, or to, electrons, typically. This occurs instantaneously in so-called quantum jumps. For example, in the Bohr model of hydrogen, the electron can only occupy a discrete set of orbits. An electron in the third orbit can suddenly transition to the second orbit with the emission of a photon. We might interpret this as three events occurring simultaneously. Destroy an electron in the third orbit, create an electron in the second orbit, and create a photon with the constraints that total energy and momentum remain constant. Eventually, the wave-particle duality idea was extended to electrons. These were found to have wave properties described by a wave function, with energy and momentum relations analogous to those of a photon. The quantum jump process now applies to a wave function, instantaneously an electron in one quantum state, transitions to a lower energy state with the emission of a photon, such that total energy and momentum are conserved. Finally, the development of relativistic quantum mechanics predicted that all space is filled with an invisible Dirac C of negative energy electrons. Pulling out an electron appears to create a visible electron and a visible hole, which behaves like an anti-electron, what came to be called a positron. In this cartoon representation, the Dirac C contains an infinite number of electrons in all possible negative energy states. This forms the background of, quote, empty space, and is assumed to be unobservable. By adding energy, we can pull an electron out of the Dirac C into a visible positive energy state. This leaves behind a hole in the Dirac C. What will be observed is the apparent creation of an electron-positron pair. The process can be reversed, with the electron falling back into the hole, so to speak. Appearing to an observer is annihilation of the electron-positron pair. As it stands, there are some loose ends and rough edges in this theory. One is that although we talk about photons being created and destroyed, we don't have a rigorous description of the process and how it is coupled with the quantum jump of an electron. Another is that the Dirac C concept is rather strange. It's hard to believe that empty space is really filled with an infinite number of invisible negative energy electrons. It would be preferable to have a theory in which electrons and positrons are actually created and destroyed, just as we suppose photons are, and corresponding more closely to what is actually observed. These and other considerations motivated the development of quantum field theory. An early contribution was Einstein's paper on the quantum theory of radiation. This was published in 1917, after development of Bohr's atomic model, but before development of the Schrodinger equation. Here we'll consider Einstein's ideas, but in a different sequence than given in his paper. Imagine a box containing a number of atoms, with each atom in either quantum state one or the higher energy quantum state two. The box also contains photons with a frequency new corresponding to the energy difference of the two quantum states. There are N1 atoms in state one and N2 atoms in state two, with respective energies E1 and E2. One possible radiation process is emission. An atom in the higher energy state can spontaneously emit a photon and transition to the lower energy state. This will decrease N2 and increase N1. Another possibility is absorption. An atom in the lower energy state can absorb a photon and transition to the higher energy state. This will decrease N1 and increase N2. Let's assume the entire system is in thermal equilibrium at temperature T. Then emission and absorption will be in balance, in the sense that there will be no net change in N1 and N2. Let's write the probability that a single atom in quantum state two will undergo emission during time dT as dW equals a constant A21 times dT. The probability is independent of the presence of photons. It depends only on the internal details of the quantum states one and two. Our picture is that an atom in state two can transition to state one with the emission of a photon and the probability of this occurring per unit time is A21. Let's write the probability that a single atom in quantum state one will undergo absorption during time dT as dW equals a constant B12 times the radiation density U of nu times dT. The probability has to depend on the radiation density because unlike emission, absorption cannot occur unless a photon is present to be absorbed. And the more photons are present, the more likely absorption becomes. Our picture is that a photon can encounter an atom in state one and be absorbed, causing the atom to transition to state two. The probability of this occurring per unit time is B12 times the radiation density. In thermal equilibrium, N1 and N2 should remain constant, so dN2, the change in N2 during a time dT, should be zero. Absorption increases N2. The number of absorptions is the number of atoms in state one, which is N1, times the probability of absorption per atom, B12 times U of nu times dT. Emission decreases N2. The number of emissions is the number of atoms in state two and two times the probability of emission per atom, A21 times dT. These two effects must cancel out. And we write the rate of change of N2 with respect to time is zero. We can solve this equation for U of nu to find that the radiation density is N2 over N1 times A21 over B12. In the first video of the quantum mechanics series, we describe the Boltzmann distribution. This tells us that at temperature T, the number of atoms with energy E is proportional to the exponential of minus E over kT, where k is Boltzmann's constant. Therefore, the ratio N2 over N1 is the ratio of the corresponding exponentials. We can write this as the exponential of minus the quantity E2 minus E1 over kT, which equals exponential of minus H nu over kT. Therefore, the radiation density is A21 over B12 times E to the minus H nu over kT, or equivalently, 1 over E to the H nu over kT. However, this doesn't agree with the observed radiation density in thermal equilibrium given by Planck's law, which we also described in the first quantum mechanics video. This has a denominator E to the H nu over kT minus 1, while the expression we just obtained lacks the minus 1. Let's try starting with Planck's law and work backward to see what we may have missed. We replace E to the H nu over kT by N1 over N2. We also identify the leading factor in Planck's law with A21 over B12. After multiplying numerator and denominator by N2, we obtain A21 over B12 times N2 over N1 minus N2. This has an extra N2 term relative to our previous analysis, implying that there is some additional emission process that we've overlooked. Multiplying through by B12 times N1 minus N2, we get N1 B12 U of nu minus N2 B12 U of nu equals N2 A21. Moving all terms to the left-hand side and combining the two N2 terms, we identify the expression for the equilibrium of absorption and emission. The absorption term is the same as before, but now the emission term has N2 multiplied by a sum of two terms. For consistency of notation, let's define B21 equal to B12. Then the terms in brackets imply that the probability that an atom in state 2 will undergo emission during a time dt has two contributions. The first term we call spontaneous emission. It's independent of the radiation density, that is, it doesn't require the presence of photons. It's what we originally assumed was the only emission process. The second term we call stimulated emission. It's proportional to the radiation density, so like absorption, it only occurs in the presence of photons. Photons are required to stimulate this type of emission. We see that the form of Planck's law requires that such a process must exist. Stimulated emission implies that the following process might be possible. An atom in state 2 could generate a photon through spontaneous emission. That photon could stimulate emission of a second photon. Those two photons could stimulate the emission of a third photon and so on. As the number of photons increases, the rate of stimulated emission will also increase. Thus, there should be an exponential increase in the radiation density. In this manner, we might be able to build a device that uses stimulated emission to amplify light. Such a device could be called a laser for light amplification by stimulated emission of radiation. Thus, Einstein's 1917 paper predicted the possibility of building lasers, although due to technical challenges, this wasn't actually accomplished until 1960. Furthermore, any quantum field theory we develop must be able to explain the existence and relative strengths of these two emission processes.