Field Theory

Effective field theory.

A field is tracking information at every point in space (and time). Effective means addressing relevant questions.

In effective field theory, there are 4 categories of organizing our ignorance: known knowns, known unknowns, unknown knowns, and unknown unknowns.

All forces are considered fields.

Known knowns: a model and its predictions (Galilean gravity constant, fluid mechanics, etc.).
Known unknowns: consistent extensions, corrections requiring measurements (Coriolis effect, viscosity, radial dependence of q, etc.)
Unknown knowns: unexpected consequences, questions we haven't thought to ask yet (orbits, turbulence, laminar flow, etc.).
Unknown unknowns: new physics (Newtonian gravity, general relativity superfluids, non-Newtonian fluids, phase transitions, etc.).

Quantum field theory.

Divergences arise when calculating Feynman diagrams due to integrals over loop momenta that lead to infinite results.

3 types of divergence in quantum field theory.

UV divergence occurs due to the behavior of Feynman integrals at high momenta (large energy scales), so short distances.

IR divergence occurs due to the behavior of Feynman integrals at low momenta (small energy scales), so long distances.

In quantum electrodynamics, the electron self-energy diagram exhibits UV divergence when integrating over large momenta, and the soft photon emissions lead to IR divergences in loop and real emission diagrams.

Collinear divergence arise when massless particles (such as photons and gluons) are emitted or absorbed in a way that they become parallel (collinear) to another massles or light particle.

In quantum chromodynamics, when a gluon is emitted by a quark, the probability of emission becomes singular if the emitted gluon is collinear with the quark. Collinear divergence is related to IR divergence but specific to massless particle emissions.

The simplest expression in quantum field theory is ⟨0∣Tφ(x)φ(y)∣0⟩

It is called the Feynman propagator, or 2-point correlation function.

-The ⟨0∣ and ∣0⟩ represent the vacuum state, which is the lowest-energy state in the quantum field theory, containing no particles.

-The φ(x)φ(y) are the quantum field operators at spacetime points x and y. They create or annihilate particles at these points.

-The T (time-ordering operator) rearranges the field operators so that the 1 with the later time coordinate, acts 1st.

So the physical meaning describes the probability amplitude for a particle to propagate from spacetime point y to x. So if a particle is created at y, this function tells us the probability amplitude for it to be detected at x.

The 1st inkling that quantum field theory had was ultraviolet divergence problems, which led to the analysis of perturbative calculations of self-energy and vacuum polarization contributions. In the 1st phase, the divergences were explicitly connected to the ultraviolet problems in classical electrodynamics (1920s). Early efforts to extract finite results from ultraviolet divergent terms were widely regarded with suspicion. The development of covariant perturbation theory (early 1940s) for quantum electrodynamics marked a significant systemization of early efforts that led to the 1st physical justification for renormalization.