What is quantum eld theory?
From wooden blocks to
the building blocks of nature
@
Siva Swaminathan SPS
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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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What the hell is quantum eld
theory?
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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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Philosophiæ Naturalis Principia
Mathematica
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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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"Degrees of freedom"
In how many different ways can it move?
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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Physical systems are lazy
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Principle of "least" stationary action
S = ∫ dt L( , )
x
i
x
˙i
Action is the price paid by a physical system
Symmetries = Conservation laws
(Noether)
Conserved quantities help specify/constrain states
Think of symmetries as "frame invariance"
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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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Einstein's nemesis
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Heisenberg's uncertainty principle
Cannot specify the position/momentum exactly!
DAFUQ!
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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
−
−
√
Measurements
"Collapse of the wavefunction"
Physicists don't discuss this in polite company.
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In a random world, what are the
observables?
How do you characterize random variables?
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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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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
All the magic of quantum mechanics is arguably a
consequence of linearity/unitarity
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Oh, by the way...
Spookiness ✗
Nonlocality ✗
Democracy ✔
Determinism ✗
Predictability ✔
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Get your locality on!
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Information can never travel faster
than c
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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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Examples
Diffusion, Network of springs (sound)
Electromagnetism, General relativity
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Essentially, a theory of waves
Free -vs- Interacting
Linear -vs- Nonlinear
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Technology
Partial differential equations
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How to do physics?
Guess the correct PDE
Specify appropriate initial/boundary conditions
Solve the PDE!
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Waves are particles too!
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What is a particle?
A lump of wave
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What are the analogies between
particle and wave behaviour?
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Uncertainty and the ephemeral
nature of the interacting vacuum
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What are the observables?
Think of modeling the weather
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Correlation functions
Stick 'em probes in!
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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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The Standard model
QFT applied to particle physics
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What are the fundamental degrees of
freedom?
Particles are just different manifestations of energy
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To recap
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Classical Mechanics Quantum Mechanics
Classical Field theory Quantum Field theory
h → 0
reducing DOFs
(c → ∞)
h → 0
reducing DOFs
(c → ∞)
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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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Thank you
"The rst principle is that you must not fool yourself — and
you are the easiest person to fool." -- Richard Feynman