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# 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

## 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 ✔

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