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
   

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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.
   

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3. What the hell is quantum eld
   theory?
   

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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 → ∞)
   

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5. Philosophiæ Naturalis Principia
   Mathematica
   

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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
   

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

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8. Technology
   Ordinary differential equations (2nd order)
   

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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)
   

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10. Physical systems are lazy
    

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

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12. Technology
    Variational calculus
    Ordinary differential equations
    

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

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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
    

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15. Einstein's nemesis
    

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

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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
    −
    −
    √
    

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18. Wavefunctions
    |particle position⟩ = |position 1⟩ + |position 4⟩ − |position
    1
    3
    −
    −
    √
    1
    3
    −
    −
    √
    1
    3
    −
    −
    √
    ⟨x|ψ⟩ ∼ sin(2π )
    x
    L
    

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

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

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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
    

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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
    

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23. Technology
    Linear algebra
    Partial differential equations ( , )
    ∂
    ∂t
    ∂
    2
    ∂x2
    

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

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

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26. Get your locality on!
    

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27. Information can never travel faster
    than c
    

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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?
    

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

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

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31. Technology
    Partial differential equations
    

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

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33. Waves are particles too!
    

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34. What is a particle?
    A lump of wave
    

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

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36. Uncertainty and the ephemeral
    nature of the interacting vacuum
    

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

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38. Correlation functions
    Stick 'em probes in!
    

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

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40. The Standard model
    QFT applied to particle physics
    

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

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42. To recap
    

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

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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.
    

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

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