Intro to Quantum Field Theory

Field Theories are usually always a good step up from discrete theories in complexity. In the case of quantum physics, this is also a relatively intuitive expansion on the theories we know, since in quantum physics our solution space is already a linear space. It thus can be abstracted to using fields instead of waves, which is nicer for looking at dynamics beyond propagation. In general we adopt the notation of classical field theory, meaning that our calculations will be largely done using tensor calculus. The transition of the known quantities, like the Lagrangian then is somewhat confined to their definitions.

Free Scalar Theory

Replacing a wave function with some scalar field is the naive approach to establish a new theory. Technically this doesn't yet change much, other than notation. We get some other handy equations from the Lagrangian. Keep in mind that the Lagrangian describes the energy within the system.

The last equation here is the Klein-Gordon Equation. It will take the place of the energy conservation. If energy is conserved, we will assume the Klein-Gordon equation is true and we are operating on-shell. From here, we can derive the Noether theorem and their currents. The next departure from operations comes when we try to define the Hamiltonian.

We will also take note of the commutators as we go along, as they will later aid in solving the integrals. Here, the definition of the mode Π lets it interact with Φ, so that

[Φ, Π] = iδ(x-y) and [Φ, Φ] = [Π, Π] = 0

Where there is modes, there is mode expansions, but these are not generally different to that of regular quantum physics, though each dimension in the integral divides out a factor of 2π. The definitions of the ladder operators are as they come. The Hamiltonian has an expression utilizing those operators, which also has an additional summand that remains even if the ladder operators annihilate. That then means that some amount of energy remains even in the least energetic state. We call that remainder the vacuum energy.

IR/UV divergence

Splitting the scalar fields into a real and imaginary parts gets us a complex field, each with its own set of ladder operators and all Fourier transform. This new multitude of ladder operators gives us enough tools to define propagators via commutators.

Using this commutator, one can define the behavior of the fields as they spread through space. We are interested in the probability of a particle moving from some point y to x. In the Fock-space we already know how to express this, and for our new theory, we only need to apply definitions.

For two points in space that is totally fine as a definition, as we need to define a starting point and a destination for the problem to even make sense. At the same time, we notice that this problem is very much dependent on which way the particle is traveling. In more complex paths, we then also need to pay attention to the order - causality is retained. For such scenarios, the propagator needs to be time-ordered. We call it a Feynman Propagator.

There is the time-ordering operator, which orders the following fields according to their 0-component (remember, in 4-vectors, the 0-component denotes the time). The name Feynman Propagator is maybe best illustrated through its integral form, but as we would need to be familiar with another set of concepts, I will postpone talking much about that definition until we can draw direct comparison.

Interacting Scalar Fields

So far we have assumed no interactions between the particles, which has the nice bonus of our theory being exactly solvable. Interactions between scalar fields are expressed as a linear combinations of higher-order terms, and as they only affect the potential of the fields, the potential of the full Lagrangian can be expressed as a power series. This however changes the way the vacuum energy is defined, as those potential terms contribute to all the states. Where we have written |0> for the free vacuum energy, we write |Ω> for the interacting one. These are not equal. A spectral representation by means of the energy eigenstates can be found through the usage of Lorentz invariance and the commutative properties of 3-momentum and the Hamiltonian operator. First though, it should be noted that the equations of motions aren't free anymore, due to the interaction terms. The Klein-Gordon equation then evaluates to some current j. Familiar concepts like the completeness theorem are augmented by the usual normalization factors we are familiar with from every integral we've seen so far. From the Lorentz-invariance (which applies to a whole group of unitary operators) every eigenstate can be related to the null-eigenstate.

We can use this definition then to define the Feynman propagator in interacting scalar theory. We discretely sum over each lambda with the associated mass and for each, the integral is normalized by the absolute square of the null energy. Using this notion then, we can construct a framework for the particles to travel in between states as well.

Just like in quantum physics, the way for transfers in between states can be achieved by some operator with some incoming state leading to some outgoing state. This however isn't as simple as ladder operators, though we condense it down to a S-Matrix. The way to get there explicitly is the LSZ Reduction Formula. We want to remember the definitions of the ladder operators for this. All S-matrix elements are of the form

Where α are some improper terms that we tend not to think too much about. The derivation of the formula for this is long-winded and honestly very hard to explain without a whole chapter to this. We just want to have seen this once, because evaluating parts of it can be beneficial later. The last part of the product, for example is the (n+r)-correlation function of the full interaction theory with the Hamiltonian. We work in the Dirac-Picture (since it keeps the interactions flexible) and apply it to the fields. It is handy, since we regain the free Klein-Gordon equation. With this we also regain the classical commutator relations of the unperturbated Hamiltonian with the ladder operators of the Dirac picture. From this, we gain an expression for the time-evolution operator and with that for the field operator in the interaction picture (the full Hamiltonian and the unperturbated one don't commute, so the definition makes sense and doesn't just resolve to 1).

Of course this relation can be reversed to relate interacting fields back to the free ones. This case is somewhat rare, and considering the not quite simple nature of this relation it doesn't really make sense to dwell on it. It is most prevalent in the formulation of Wick's Theorem, which in essence gives a short-hand method for computation of time-ordered products through convenient contractions. Central to this is the Normal-order operator which pushes all creation operators to the left.

Feynman Diagrams

For the last segment this week, we want to apply all this stuff. A product of Feynman propagators can be translated into a graphical sketch. Each Feynman propagator defines a start and destination point, which are connected by a vertex. Orientation isn't interesting to us in this case. Instead we take a look at the symmetry group of the diagram to introduce a symmetry factor, which divides the product in the integral. We understand the integral to travel along a vertex and through the points in between those vertices. Points that are connected, but not integrated over are considered external and don't factor into the integral explicitly. The specific factors and the form of the propagators of course depend on the space the calculation is done in, so position space and momentum space have separate Feynman rules, which at least work similarly.

We make a distinction between disconnected subdiagrams and partially connected subdiagrams. Since we don't require that transitions between all points in the diagram be possible, some points might not be connected to any other points other than itself by vertices. Such are disconnected subdiagrams. All other are partially connected. One finds that the full correlator divides the partition function from the full time-ordered product. That means that the full correlator is equal to the sum of all partially connected diagrams with as many external points as are fields in the product.

Analogously, the 1-particle-irreducible diagrams are such, which can't be disassembled into two separate non-trivial diagrams by cutting one vertex. They are handy in that they can be summed up into a propagator.

α is some term regular at the squared mass, or the pole mass, which happens to be the first analytical pole of the propagator and reflects the physical 1-particle mass. In sum, these can be used to check how many particles occupy identical states through scattering. In essence, scattering falls off inversely to the squared sum of incoming/outgoing momentum.