What is quantum field theory?

Invited talk at the gathering of the Society of Physics Students (at UT Austin). Intended as an introduction, for physics undergraduate students with varying levels of familiarity with classical and quantum mechanics, but no prior knowledge of field theory.

Transcript

1. What is quantum eld theory? From wooden blocks to the building blocks of nature @ Siva Swaminathan SPS
2. Quantum eld theory is a framework to describe systems which evolve quantum mechanically (unitarity) have many degrees of freedom (excitations) which respect locality (causality) That's basically it.
3. What the hell is quantum eld theory?
4. Let's back up for a bit... Classical Mechanics Quantum Mechanics Classical Field theory Quantum Field theory h → 0 reducing DOFs (c → ∞) h → 0 reducing DOFs (c → ∞)
5. Philosophiæ Naturalis Principia Mathematica
6. Newton's laws What is a force? That thing which causes acceleration. What is mass? That thing which tells you how hard it is to produce acceleration. = m (actually ) F ⃗ a⃗ dp ⃗ dt
7. "Degrees of freedom" In how many different ways can it move?
8. Technology Ordinary differential equations (2nd order)
9. How to do physics? Guess the differential equation describing a system Specify the state (initial conditions) Predict dynamics (solving the differential equation) { , } x i x ˙i x(t)
10. Physical systems are lazy
11. Principle of "least" stationary action S = ∫ dt L( , ) x i x ˙i Action is the price paid by a physical system
12. Technology Variational calculus Ordinary differential equations
13. Symmetries = Conservation laws (Noether) Conserved quantities help specify/constrain states Think of symmetries as "frame invariance"
14. How to do physics? Guess a Lagrangian Fix and Solve for Derive the equations of motion and conserved quantities L(x, ) x ˙ x i x f x ˙i (Euler-Lagrange) ( ) − = 0 d dt ∂L ∂x ˙ ∂L ∂x eg: for mechanical systems L = KE − PE
15. Einstein's nemesis
16. Heisenberg's uncertainty principle Cannot specify the position/momentum exactly! DAFUQ!
17. Superpositions Enlarge your notion of a system's state |state of cat⟩ ∼ |dead cat⟩ + |alive cat⟩ |pointing direction⟩ = |X⟩ + |Y⟩ − |Z⟩ 1 3 − − √ 1 3 − − √ 1 3 − − √
18. Wavefunctions |particle position⟩ = |position 1⟩ + |position 4⟩ − |position 1 3 − − √ 1 3 − − √ 1 3 − − √ ⟨x|ψ⟩ ∼ sin(2π ) x L
19. Measurements "Collapse of the wavefunction" Physicists don't discuss this in polite company.
20. In a random world, what are the observables? How do you characterize random variables?
21. Democrazy Everything that can happen should be considered. Z ∼ ∫ [Dx(t)]e iS/ℏ Every possibility gets to vote (with strength ) Contributions will interfere (reinforce/cancel) e iS/ℏ
22. How to do physics Specify an initial quantum state Find Find the probability amplitude to evolve to putative Unitarity : |initial state⟩ ⟨observable⟩ = ⟨state|observable|state⟩ |final state⟩ ∑ = 1 p f
23. Technology Linear algebra Partial differential equations ( , ) ∂ ∂t ∂ 2 ∂x2
24. All the magic of quantum mechanics is arguably a consequence of linearity/unitarity
25. Oh, by the way... Spookiness ✗ Nonlocality ✗ Democracy ✔ Determinism ✗ Predictability ✔
26. Get your locality on!
27. Information can never travel faster than c
28. Local degrees of freedom A eld is a set of values: a quantity at each point of space (and time) How do those degrees of freedom in uence each other?
29. Examples Diffusion, Network of springs (sound) Electromagnetism, General relativity
30. Essentially, a theory of waves Free -vs- Interacting Linear -vs- Nonlinear
31. Technology Partial differential equations
32. How to do physics? Guess the correct PDE Specify appropriate initial/boundary conditions Solve the PDE!
33. Waves are particles too!
34. What is a particle? A lump of wave
35. What are the analogies between particle and wave behaviour?
36. Uncertainty and the ephemeral nature of the interacting vacuum
37. What are the observables? Think of modeling the weather
38. Correlation functions Stick 'em probes in!
39. Scattering amplitudes It's like trying to collide together two mechanical watches, to understand their structure, by observing the gears that y out.
40. The Standard model QFT applied to particle physics
41. What are the fundamental degrees of freedom? Particles are just different manifestations of energy
42. To recap
43. Classical Mechanics Quantum Mechanics Classical Field theory Quantum Field theory h → 0 reducing DOFs (c → ∞) h → 0 reducing DOFs (c → ∞)
44. Quantum eld theory is a framework to describe systems which evolve quantum mechanically (unitarity) have many degrees of freedom (excitations) which respect locality (causality) That's basically it.
45. Thank you "The rst principle is that you must not fool yourself — and you are the easiest person to fool." -- Richard Feynman