Sean Carroll, Caltech

Quantum Field Theoryand the

Limits of Knowledge

Two claims:

1. The laws of physics underlying everyday lifeare completely known.

2. The structure of quantum field theory providesa 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 knowabout 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 fundamentaltheory 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-levelmacro-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 theconfiguration (or other observable).

Probability of measurement outcome = |wave function|2.

After measurement, wave function “collapses” (becomessuddenly 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 quantizedinto discrete levels:

Quantum Field Theory

QFT is not a successor/alternative to QM; it’s justa 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 amomentum, and totalis conserved ateach vertex.

When there are loops,momentum “flowing through the loop” (q)is arbitrary, and getssummed over.

Result is often infinite.

don’t need to worryabout what happens here

Ken Wilson: organize QFT by energy/length scale

Remember: energy & momentum ~ 1/(wavelength).

IR

UV

Λ(“cutoff”energyscale)

longwavelengths/low energies

shortwavelengths/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 interactionterm (in Lagrangian) between N particles.

There are an infinite number of terms inEFT 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 treatinggravity 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 havemany “ultraviolet completions” at higher energies.

That’s why it’s hard to do experiments relevant toquantum gravity: we expect Λ ~ Eplanck ~ 1015 ELHC.

Multiple realizability

loop quantum gravity string theory dynamical triangulations

Known particles/forces,general relativity

(Core theory)

Dark matter/energy,new particles/forces,

hidden sectors

Underlying reality(theory of everything)

Higher-levelemergent phenomena

of everyday life

Astrophysics,cosmology

Underlying physics only influences us via Core Theory.

What about new particles/forces?

stronglyinteracting

light/long range/low energy

heavy/short range/high energy

weaklyinteracting

accessible

inaccessible

knownknowns

knownunknowns

Unknown unknowns = violations of QFT itself.

QFT puts very tightconstraints on new phenomena.

time

new particle

newinteraction

If a new particle caninteract with ordinaryparticles:

Then that particlecan be created inhigh-energy collisions.

“Crossing symmetry.”

Constraints on new particles

As-yet-undiscovered particles must be either:

1. very weakly interacting,2. too heavy to create, or3. too short-lived to detect.

In any of those cases, the new particle wouldbe irrelevant to our everyday lives.

To be relevant to everyday physics, any new forcesmust 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

Stre

ngth

(rel

ativ

e to

gra

vity

)

Range [Long et al. 2003; Antoniadis 2003]

Experimental limits on new forces

Ruled Out

Allowed

newgravitational-

strengthforce

(10-36 E&M)

Known particles/forces,general relativity

(Core theory)

Dark matter/energy,new particles/forces,

hidden sectors

Underlying reality(theory of everything)

Higher-levelemergent 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.

Known particles/forces,general relativity

(Core theory)

Dark matter/energy,new particles/forces,

hidden sectors

Underlying reality(theory of everything)

Higher-levelemergent phenomena

of everyday life

Astrophysics,cosmology