Upgrade to Pro — share decks privately, control downloads, hide ads and more …

What is quantum field theory?

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.

Siva Swaminathan

November 23, 2015
Tweet

More Decks by Siva Swaminathan

Other Decks in Science

Transcript

  1. What is quantum eld theory?
    From wooden blocks to
    the building blocks of nature
    @
    Siva Swaminathan SPS

    View full-size slide

  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.

    View full-size slide

  3. What the hell is quantum eld
    theory?

    View full-size slide

  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 → ∞)

    View full-size slide

  5. Philosophiæ Naturalis Principia
    Mathematica

    View full-size slide

  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

    View full-size slide

  7. "Degrees of freedom"
    In how many different ways can it move?

    View full-size slide

  8. Technology
    Ordinary differential equations (2nd order)

    View full-size slide

  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)

    View full-size slide

  10. Physical systems are lazy

    View full-size slide

  11. Principle of "least" stationary action
    S = ∫ dt L( , )
    x
    i
    x
    ˙i
    Action is the price paid by a physical system

    View full-size slide

  12. Technology
    Variational calculus
    Ordinary differential equations

    View full-size slide

  13. Symmetries = Conservation laws
    (Noether)
    Conserved quantities help specify/constrain states
    Think of symmetries as "frame invariance"

    View full-size slide

  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

    View full-size slide

  15. Einstein's nemesis

    View full-size slide

  16. Heisenberg's uncertainty principle
    Cannot specify the position/momentum exactly!
    DAFUQ!

    View full-size slide

  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



    View full-size slide

  18. Wavefunctions
    |particle position⟩ = |position 1⟩ + |position 4⟩ − |position
    1
    3



    1
    3



    1
    3



    ⟨x|ψ⟩ ∼ sin(2π )
    x
    L

    View full-size slide

  19. Measurements
    "Collapse of the wavefunction"
    Physicists don't discuss this in polite company.

    View full-size slide

  20. In a random world, what are the
    observables?
    How do you characterize random variables?

    View full-size slide

  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

    View full-size slide

  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

    View full-size slide

  23. Technology
    Linear algebra
    Partial differential equations ( , )

    ∂t

    2
    ∂x2

    View full-size slide

  24. All the magic of quantum mechanics is arguably a
    consequence of linearity/unitarity

    View full-size slide

  25. Oh, by the way...
    Spookiness ✗
    Nonlocality ✗
    Democracy ✔
    Determinism ✗
    Predictability ✔

    View full-size slide

  26. Get your locality on!

    View full-size slide

  27. Information can never travel faster
    than c

    View full-size slide

  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?

    View full-size slide

  29. Examples
    Diffusion, Network of springs (sound)
    Electromagnetism, General relativity

    View full-size slide

  30. Essentially, a theory of waves
    Free -vs- Interacting
    Linear -vs- Nonlinear

    View full-size slide

  31. Technology
    Partial differential equations

    View full-size slide

  32. How to do physics?
    Guess the correct PDE
    Specify appropriate initial/boundary conditions
    Solve the PDE!

    View full-size slide

  33. Waves are particles too!

    View full-size slide

  34. What is a particle?
    A lump of wave

    View full-size slide

  35. What are the analogies between
    particle and wave behaviour?

    View full-size slide

  36. Uncertainty and the ephemeral
    nature of the interacting vacuum

    View full-size slide

  37. What are the observables?
    Think of modeling the weather

    View full-size slide

  38. Correlation functions
    Stick 'em probes in!

    View full-size slide

  39. Scattering amplitudes
    It's like trying to collide together two mechanical watches,
    to understand their structure, by observing the gears that
    y out.

    View full-size slide

  40. The Standard model
    QFT applied to particle physics

    View full-size slide

  41. What are the fundamental degrees of
    freedom?
    Particles are just different manifestations of energy

    View full-size slide

  42. Classical Mechanics Quantum Mechanics
    Classical Field theory Quantum Field theory
    h → 0
    reducing DOFs
    (c → ∞)
    h → 0
    reducing DOFs
    (c → ∞)

    View full-size slide

  43. 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.

    View full-size slide

  44. Thank you
    "The rst principle is that you must not fool yourself — and
    you are the easiest person to fool." -- Richard Feynman

    View full-size slide