 Welcome to Quantum Field Theory 7, Oppenheimer and Beta. In this video, we consider results from two papers, one from each of these authors, that describe the problem of infinities in quantum field theory. Oppenheimer wrote his paper in 1929, while a physics professor at University of California. Beta wrote his paper nearly 20 years later in 1947, while a physics professor at Cornell University. In between, both played central roles in the Manhattan Project. Oppenheimer, his director of Los Alamos Laboratory, oversaw all scientific aspects of the project, while Beta led the theoretical physics division. Oppenheimer's paper, published in 1930 during the early years of quantum field theory development, was titled, Note on the Theory of the Interaction of Field and Matter. His paper develops a method for the systematic integration of the relativistic wave equations for the coupling of electrons and protons with each other and with the electromagnetic field. We have, to some extent, also been doing this in this video series, although using the non-relativistic equation for the electron. Oppenheimer concluded that, it is impossible on the present theory to eliminate the interaction of a charge with its own field, and the theory leads to false predictions when it is applied to compute the energy levels and the frequency of the absorption and emission lines of an atom. Oppenheimer's work was, in part, responsible for Fermi, concluding his seminal 1932 review of the then-current state of the quantum theory of radiation. With the statement, practically all the problems in radiation theory which do not involve the structure of the electron have their satisfactory explanation, while the problems connected with the internal properties of the electron are still very far from their solution. During the 1930s, researchers struggled with these perplexing problems which manifested as infinities in various quantum field theory calculations. Frustration grew to the point that some feared the field was a dead end. Others took these problems as an indication that something was wrong with the basic concepts and methods of physics itself. In 1937, Dirac proclaimed, because of its extreme complexity, most physicists will be glad to see the end of quantum electrodynamics. In 1936, Polly's opinion was, it seems to me that our present methods are not fundamental enough and there are two possibilities for overcoming the difficulties. The first is to change our concept of space and time in small regions. The second, to change the concept of state for systems with an infinite number of degrees of freedom. With the onset of World War II, quantum field theory research effectively came to an end as most research activity was redirected toward developing technologies for the war effort. Beta's paper was written in 1947, shortly after the end of World War II, when attention was returning to peacetime research topics, and when some of the technological advances made during the war could be applied to new experimental investigations. One of these was the application of microwave technology, originally developed for radar, to study very small energy differences in atomic energy levels. Lam and Rutherford used this to conduct, in Beta's words, some very beautiful experiments. These show that the 2S level of the hydrogen atom, which should have the same energy as the 2P level in both the Schrodinger and Dirac theories, is actually higher by about 1,000 megacycles or 1 GHz. This is on the order of only about 1 millionth the energy of a typical hydrogen transition. Beta references the suggestion that a possible explanation might be the shift of energy levels by the interaction of the electron with a radiation field. However, he notes that this shift comes out infinite in all existing theories, and has therefore always been ignored. He continues, However, it is possible to identify the most strongly divergent term in the level shift with an electromagnetic mass effect. This is the quantum version of the classical electromagnetic mass effect we described in video 5. The solution Beta suggested in this paper was the same we considered in that video, renormalization of the electron mass. Let's look at the calculation of the electron self-energy in quantum field theory. Oppenheimer and Beta treated the problem in the context of the electron in an atom, and Oppenheimer used relativistic theory. We are going to deal with the much simpler problem of a free electron using non-relativistic theory, but the main conclusions will be the same. In the previous video, we studied photon-electron scattering via the H-hat-i double prime part of the interaction Hamiltonian. This allowed us to describe the destruction of an electron and a photon, followed by the creation of a different electron and a different photon. We interpreted this as electron-photon scattering, and we found the following transition rate. Due to the factor nL Beta, if there are no photons present initially, there is no scattering. So we would expect the electron to continue on in its pure momentum state forever. However, it turns out that even if there are no photons present initially via the H-hat-i double prime term, an electron can emit a photon and immediately reabsorb it. In this way, it, quote, interacts with its own field. This is the type of process Oppenheimer was referring to. Consider the scattering process where the same electron and photon are destroyed and created. H-hat-i double prime contains operators that destroy any electron and create any other electron. A special case is the destruction and recreation of the same electron. Likewise, the destruction and recreation of the same photon. Since the destruction operator is to the right of the creation operator, this suggests that the photon is first absorbed, then emitted. Of course, that would require the photon to already exist. But there is also a factor with the creation operator on the right. This suggests the photon is first emitted and then absorbed. Now keep in mind our previous caution to not take Feynman diagrams literally. The Feynman diagram is really just a visual representation of a term in the interaction Hamiltonian. Even though the diagram seems to suggest the photon would fly off to infinity and so could never come back around from the other side to be absorbed, we cannot literally represent this process on a space-time diagram. An alternate illustration is to show the photon being emitted and looping back around to be absorbed. Again, we should not take this literally. The key point is that the electron is emitting and then absorbing a photon instantaneously. It is interacting with its own field. This is the quantum version of the classical electron self-force we described in video 5. Assume our quantum state is a single electron with momentum P in vacuum. The radiation state with zero photons. We want to calculate the energy of this state. There are three terms in this expression. One for the electron Hamiltonian, one for the radiation Hamiltonian, and one for the interaction Hamiltonian. The electron Hamiltonian is the sum over all momentum states Q of the kinetic energy Q squared over 2 Me times the number operator B hat Q plus B hat Q minus. This term counts the single electron in the P momentum state and extracts the electron kinetic energy P squared over 2 Me. The radiation Hamiltonian is the sum over all photon modes K alpha of omega K times the number operator A hat K alpha plus A hat K alpha minus plus one half. There are no photons, so this leaves one half omega K for each mode. There are two polarizations alpha for each K vector which cancels the one half and we obtain the sum over all K of omega K. Since there are infinite number of photon modes and omega K increases with K, the sum equals infinity. This infinity is the zero point energy we discussed in video 2. Previously we have ignored this infinity arguing that it is unobservable and will cancel when we calculate energy differences. As we have seen before, the non-relativistic interaction Hamiltonian has two terms H hat I prime and double prime. The first contains a product between the electromagnetic field operator A hat and the momentum operator P hat. While the second has the product of A hat with itself. A hat contains a sum over all photon states of destruction and creation operator terms. Therefore H hat I prime can create or destroy a single photon. A hat squared contains products of a destruction operator for a photon in the state K alpha with a creation operator for state L beta or a creation operator for a photon in the state K alpha with a destruction operator for state L beta. If K alpha and L beta are the same state then H hat I double prime can create and destroy the same photon. Let's call these terms in A hat squared A hat self squared. This is the sum over all photon states K alpha of 1 over 2 omega K times creation operator destruction operator plus destruction operator creation operator for the K alpha photon. In this case the only factors in H hat I double prime with spatial dependence are E to the minus I P dot X and E to the I Q dot X. The integral of this product over space vanishes unless P equals Q in which case the product is one. This simply tells us that if we create and destroy the same photon we must create and destroy the same electron to conserve momentum and energy. Let's call the result the H hat I double prime self operator. When this operates on our quantum state product of B hat operators destroys and recreates the electron in momentum state P. The first product of A hat operators tries to destroy a photon but the vacuum state contains no photons so this produces nothing. Our problem is that the second product of A hat operators creates and then destroys a photon in the K alpha state which takes us back to the vacuum state. So we end up with a sum over all photon states of 1 over 2 omega K times the vacuum state. Projecting this on to our quantum state we obtain E squared over 2 Me times the sum over all photon states of 1 over 2 omega K. In our units omega K equals the magnitude of K. For a given omega K we sum all states over a sphere of area 4 pi K squared. The sum over the two polarizations for each state gives us a factor of 2 and the sum over K values becomes 1 over 2 pi cubed times the integral over all K of 4 pi K squared. The result is that the self energy of the electron becomes a constant times an integral that diverges to infinity. Therefore the electron has infinite self energy due to its interaction with the vacuum state of the electromagnetic field. What are we supposed to do with this result? As beta noted physicists generally ignored this. The argument being that all electrons will have this additional infinite energy so it will cancel when energy differences are calculated and we can only observe energy differences, not absolute energies. This is the same argument we used previously to ignore the infinite zero point energy of the photon field. Another argument was to note that the infinity is due to the infinite upper limit of the integral. If we assume instead a finite upper frequency limit K-limb then the integral will be finite. Since wavelength is 2 pi over K this is equivalent to setting a smallest wavelength limit. If the physical predictions this leads to are independent of this limit then the problem goes away. As it stands approach number two is as much of a hand-waving exercise as approach number one. That is unless a physical argument can be made for a specific cutoff. Heisenberg investigated if just as the uncertainty principle sets a lower limit on the product of position and momentum uncertainties and Planck's constant sets a lower limit on action nature might also set a lower limit on distance. But unlike the other two examples no experimental evidence for a lower distance limit could be observed. Others as alluded to by Pauli's quote earlier in this video took these infinities as evidence that something in the way we conceive of space and time might break down at very small scales. Niels Bohr in particular argued that a revolutionary new type of physics was called for. However no such revolutionary new physics was successfully developed. Instead quantum field theory was molded into a form that produced amazingly precise and experimentally verified physical predictions. This was done by developing arguably mathematically and philosophically jarring workarounds for the infinities that played it. The infinities we have encountered so far can be dealt with by a trick called normal ordering. Both the infinite zero point energy of the electromagnetic field and the infinite self energy of the electron come from the operator combination A hat minus A hat plus for a given photon mode acting on the vacuum state because the expectation value of this is one and this is then summed over the infinite number of photon modes. If these operators were in the opposite order there would be no problem since the destruction operator operating on the vacuum state vanishes. Now operators for different photon states commute but operators for the same state do not. Instead as we saw in the first video in this series the commutator of photon destruction and creation operators is one. And from that it follows that A hat minus A hat plus equals A hat plus A hat minus plus one. That substitution would result in the same expectation value as before. But how about this? Let's just ignore the commutation relation and make the replacement as if the operators did commute. That is what we call normal ordering. Here's the self interaction part of the H hat I double prime operator. Applying normal ordering, the second problematic A hat operator term becomes identical to the first term. This gives us two times the first term which cancels a factor of two in the denominator. The resulting operator applied to the photon vacuum state vanishes and along with that are infinite electron self energy vanishes also. We can apply the same trick to the problematic one half term in the radiation Hamiltonian. We multiply it by the A hat minus A hat plus commutator which has a value of one. The first and third terms combine to give one half times the first term. Applying our trick to the second term we end up with the sum over all K alpha of omega K A hat plus K alpha A hat minus K alpha. Just as in the previous case this will vanish when applied to the photon vacuum state. Normal ordering is the convention that we put all creation operators to the left of destruction operators without regard to commutation relations. This is a trick that avoids the embarrassing infinities we have encountered so far. Heisenberg called this the Klein-Jordan trick after the authors who first applied it. It is important to note that this is no more mathematically rigorous than simply saying let's just ignore the one half term in the radiation Hamiltonian. And in the nearly 90 years since no mathematically consistent way to deal with these infinities has been developed. As a recently published textbook says when you normal order something you just pick up the operators and move them. Just manhandle them over without any commuting. So even though we have a catchy notation for normal ordering placing an expression between colons this process ignores commutation relations. And what are commutation relations? As discussed in video one of this series they are a manifestation of the uncertainty principle of quantum mechanics. And as such they are essential to the inner workings of quantum field theory. Yet in a special case we ignore them ignoring the foundations of quantum mechanics in order to hide embarrassing infinities. This is a compromise we make in order to move onward with the theory. However as we do so we will find other infinities that cannot be hidden with this trick.