Quantum Field Theory and the Limits of Knowledge

Sean Carroll, Caltech
Quantum Field Theory
and the
Limits of Knowledge

Two claims:
1. The laws of physics underlying everyday life
are completely known.
2. The structure of quantum field theory provides
a warrant for claim 1.

“Laws of physics underlying everyday life”
= The Core Theory
• Quantum field theory in a
4-dimensional spacetime.
• Matter (fermions): quarks,leptons.
• Strong, weak, electromagnetic forces.
• Gravitation = general relativity.
• Higgs field.

Long history of embarrassingly premature triumphalism.
“[We are] probably nearing the limit of all we can know
about astronomy.” – Simon Newcomb, 1888
“The more important fundamental laws and facts of
physical science have all been discovered.”
– Albert Michelson, 1894
“Physics as we know it will be over in six months.”
– Max Born, 1928
There is a 50% chance that “we would find a complete
unified theory of everything by the end of the century.”
– Stephen Hawking, 1980

Perfectly obvious but necessary caveats
We’re nowhere close to understanding the fundamental
theory of everything.
We don’t understand the non-everyday: dark matter,
quantum gravity, the Big Bang…
We don’t fully understand macroscopic aggregations:
condensed matter, chemistry, biology, economics…
Quantum mechanics or quantum field theory could
always be wrong.

Known particles/forces,
general relativity
(Core theory)
Dark matter/energy,
new particles/forces,
hidden sectors
Underlying reality
(theory of everything)
Higher-level
macro-phenomena
of everyday life
Astrophysics,
cosmology

The Core Theory in more detail:
Quantum Mechanics
Think of “configurations,”
e.g. the location x of a particle.
Assign a complex number to
every possible configuration.
That describes a quantum state: a “wave function” Ψ(x)
that lives in a very-high-dimensional Hilbert space.
Schrödinger evolution equation:
x
x
Ψ(x)

Measurements in Quantum Mechanics
But we don’t “see” the wave function.
Measurements return some specific value of the
configuration (or other observable).
Probability of measurement outcome = |wave function|2
.
After measurement, wave function “collapses” (becomes
suddenly concentrated on observed outcome).
Seems absurd. But – good enough to successfully
predict the outcome of every experiment ever done.

(Some) Observables are Quantized
Standard example: Simple Harmonic Oscillator.
Particle moving in a potential ,
where x is the position and ω is the frequency.
Energy is quantized
into discrete levels:

Quantum Field Theory
QFT is not a successor/alternative to QM; it’s just
a particular QM model, with a particular Hamiltonian.
Namely: “configurations” are “values of (relativistic)
fields throughout space.” E.g. φ(x).
The quantum state (wave function) is a complex
amplitude for each possible field configuration, Ψ[φ(x)].
Examples: electromagnetic field, electron field,
top quark field, gravitational field (metric), etc.

Particles from fields
Each mode acts like a simple harmonic oscillator!
Energy levels = number of particles.
Wavelength = 1/momentum.
Indeed, relativity+QM+particles QFT.
Decompose oscillating field into a sum of “modes”
of different wavelengths (Fourier transform):
= +
+ …+

Interactions
Particle interactions are encoded in Feynman diagrams.
= +
+ + …

Adding up virtual particles
Every particle has a
momentum, and total
is conserved at
each vertex.
When there are loops,
momentum “flowing
through the loop” (q)
is arbitrary, and gets
summed over.
Result is often infinite.

don’t need to worry
about what happens here
Ken Wilson: organize QFT by energy/length scale
Remember: energy & momentum ~ 1/(wavelength).
IR
UV
Λ
(“cutoff”
energy
scale)
long
wavelengths/
low energies
short
wavelengths/
high energies

Think of your theory as only describing energies below
the ultraviolet cutoff scale Λ.
I.e., only include wavelengths longer than 1/Λ.
Result is an effective field theory below Λ.
Effective Field Theory

All diagrams with N legs contribute to an interaction
term (in Lagrangian) between N particles.
There are an infinite number of terms in
EFT equations of motion…
φ4
φ8
φ6

Both the field φ and the cutoff Λ have units of energy,
and the Lagrangian governing interactions is (energy)4
.
So schematically we have:
Higher-order terms are negligible at low energy (<< Λ).
Only a finite number of relevant/marginal interactions.
… but only a finite number of terms matter
“relevant” “marginal” “irrelevant”

At energies below Λ, an EFT can be a complete theory.
Above Λ, new phenomena can kick in.
E.g. Fermi theory of weak interactions Standard Model.
Effective field theories tell us their regime of applicability:
below the ultraviolet cutoff Λ.
Fermi coupling

“We haven’t quantized gravity,” but I’m treating
gravity like a perfectly ordinary effective field theory.
Because it is – as long as gravity is weak (far from
black holes, Big Bang, etc.).
In terms of curvature parameter R, interactions look like
Here on Earth, 1st
term is 1050
times bigger than 2nd
.
Quantum Gravity?

A given effective field theory with cutoff Λ could have
many “ultraviolet completions” at higher energies.
That’s why it’s hard to do experiments relevant to
quantum gravity: we expect Λ ~ Eplanck ~ 1015
ELHC.
Multiple realizability
loop quantum gravity string theory dynamical triangulations

general relativity
(Core theory)
Dark matter/energy,
hidden sectors
Underlying reality
Higher-level
emergent phenomena
of everyday life
Astrophysics,
cosmology
Underlying physics only influences us via Core Theory.

What about new particles/forces?
strongly
interacting
light/
long range/
low energy
heavy/
short range/
high energy
weakly
interacting
accessible
inaccessible
known
knowns
known
unknowns
Unknown unknowns = violations of QFT itself.

QFT puts very tight
constraints on new phenomena.
time
new particle
new
interaction
If a new particle can
interact with ordinary
particles:
Then that particle
can be created in
high-energy collisions.
“Crossing symmetry.”

Constraints on new particles
As-yet-undiscovered
particles must be either:
1. very weakly interacting,
2. too heavy to create, or
3. too short-lived to detect.
In any of those cases, the new particle would
be irrelevant to our everyday lives.

To be relevant to everyday physics, any new forces
must interact with protons, neutrons, electrons,
and/or photons.
Experiments are ongoing (torsion balances) to
search for new, weak, long-range forces.
Two ways to hide:
1. weak interactions, or
2. very short ranges.
Constraints on new forces

Strength(relativetogravity)
Range [Long et al. 2003; Antoniadis 2003]
Experimental limits on new forces
Ruled Out
Allowed
new
gravitational-
strength
force
(10-36
E&M)

general relativity
(Core theory)
Dark matter/energy,
hidden sectors
Underlying reality
Higher-level
emergent phenomena
of everyday life
Astrophysics,
cosmology
New particles/forces are too heavy/weak to influence us.

gravity
other forces matter Higgs
quantum mechanics spacetime
Punchline:
the laws of physics underlying everyday experience.
Other phenomena are too massive or weakly-coupled to
have any impact on the particles of which we are made.

• Astrology is not correct.
Implications of the Core Theory
• You can’t bend spoons with your mind.
• The soul does not survive the body.

3. Accessible deviations from textbook QM.
(Hidden variables, spontaneous/induced collapse.)
Loopholes?
2. Breakdown of QFT itself. E.g. non-local constraints/
interactions from quantum gravity (holography).
1. New forces with environment-dependent couplings.
4. Divine intervention.

general relativity
(Core theory)
Dark matter/energy,
hidden sectors
Underlying reality
Higher-level
emergent phenomena
of everyday life
Astrophysics,
cosmology

Quantum Field Theory and the Limits of Knowledge

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