quantum gravity why is it so difficult? craig dowell february 24, 2006

Quantum Gravity

Why is it so Difficult?

Craig Dowell

February 24, 2006

Quantum Gravity: Why so Difficult?

• Why Quantum Gravity?• Two Great Theories Revolutionized Physics

• Quantum Mechanics (verified to great precision)

• General Relativity (verified to great precision)

• Three of Four Fundamental Forces are described by QFT

• Gravity is the Odd Theory Out

• Would be Nice to Know What Happens When Quantum and Gravitational Effects are Both Large?


• Two Paths to a Quantum Theory of Gravity• Start with General Relativity and Rewrite Quantum

Field Theory• Loop Quantum Gravity

• Start with Quantum Field Theory and Reformulate General Relativity

• Quantum Gravity

• String Theory

• I will focus on Quantum Gravity Today• Specifically on Why it is Difficult to Make a Working

Theory


• The Problem

• Infinities when Calculating Probabilities

• Basically, it all Translates Into a Normalization Problem, i.e.

Where Does the Problem Come From?

• Let’s Start Slowly From the Beginning

2

1N x dx


• Normalization• The Statistical Interpretation of Quantum

Mechanics Says that,

Represents the Probability of Finding the Particle at point x at time t.

• The Sum of the Probabilities of the Particle being in all Possible Places must be Unity

Finding the Coefficient N so that this is True is Called Normalizing the Wavefunction

2x

2

1N x dx


• Time Independent Perturbation Theory• You Take a Hamiltonian Corresponding to an

Exact Solution to the Schrodinger Equation and Perturb It.

Then you Expand in Terms of Power Series and Cutoff at some order.

• But You Originally Normalized for the Solved Wavefunction. You Have to Pick a New N – You have to Renormalize

2

1N x dx

(0) 'ˆ ˆ ˆH H H

0 1 22


• Time Independent Perturbation Theory II• What if One of the Terms is Infinite?

The Integral Diverges. There is no N That Can Fix the Problem

• The New Wavefunction is Nonrenormalizable.

• Calculations Using the Wavefunction Would Produce Infinite Probabilities, Which is Nonsense.

• We’ll See That a Similar Problem Causes the Infinities in Quantum Gravity

0 1 2222 ,


• What is Quantum Field Theory?• Recall that Maxwell’s Equations in Free Space

Imply the Existence of Electromagnetic Waves

• Imagine a Rectangular Resonant Cavity with a Standing Wave in the z direction. The E and B Fields Could be

• Make a Hamiltonian from the Energy Density (in1-D)

22

2 2

1 EE

t

0

0

ˆcos ,

ˆsin

E z E kz x

B z B kz y

2 20

0

1z

H E B dz


• What is Quantum Field Theory? (II)• The Resulting Hamiltonian looks like the

Hamiltonian for a Harmonic Oscillator.

• A Single Mode of an Electromagnetic Field in a One-Dimensional Cavity is a Harmonic Oscillator

• Recall Harmonic Oscillators From Quantum Mechanics

• Call the raising operator and use it as a photon creation operator

• Call the lowering operator and use it as a photon annihilation operator.

• The Fourier Components of the Electromagnetic Field are Quantized as Harmonic Oscillators

†


• Quantum Theory of Your Mattress Spring• Imagine the Collection of Harmonic Oscillators as

Springs. Vertices (Blue Rings) are Point Masses

• If You Strike the Mattress, Waves Propagate.

• If You Strike in Two Places, Waves Superpose


• Quantum Theory of Your Mattress Spring (II)• We Want to Model Interacting Particles, Not

Particles that Move Through Each Other.

• Harmonic Oscillators Do Not Allow for Particle Interactions

• Anharmonic Oscillators Allow for Particle Interactions.

• Anharmonic Oscillators Cannot Be Solved Exactly

• Must Resort to Approximation. Think Perturbation Theory.

• That’s Why We Looked at Time Independent Perturbation Theory

• The Correction Terms are Where Those Infinities Will Come In.


• Some More Quantum Mechanics• Recall the Schrodinger Equation

• It Was Solved by Separation of Variables and We Found that the Time Dependence Was

• This Unitary Operator Governs the Amplitude to go from One Place to Another

• Now a Feynman Story

2 2

22i V

t m x

iHtt e


• The Smart-Aleck High-School Physics Student• Double Slit Experiment.

• Amplitude is sum of probabilities of both paths

• What Happens if you Drill Two More Holes• Sum of Probabilities Includes New Holes

• What Happens if you Add Another Screen• Sum of Probabilities Includes Paths Through Holes in

New Screen

• What if You Drill an Infinite Number of Holes in The Screens So That They are No Longer There

• You discover the Feynman Sum over Paths Approach to Quantum Field Theory.


• The Path Integral• Dirac Derived the Path Integral Representation of

the Feynman Sum over Paths Statement.

• It Turns Out That if You Write the Hamiltonian for the Harmonic Oscillator as H, you get

You Can See That the Lagrangian of the Harmonic Oscillator Pops Out

2

0

1

2

Ti dt mq

iHtF Iq e q Dq t e

2

0

1

2

Ti dt mq V q

iHtF Iq e q Dq t e


• The Path Integral (II)• Sometimes this is written in terms of the action S.

• It’s Not Too Much of a Stretch Now to Add an Anharmonicity Term

• The Indicates We are Integrating over Spacetime, and the Indicates We are Talking about Fields. The is the Perturbation Causing Anharmonicity and Represents a Force (think of pushing on the mattress).

iSZ Dq t e

24 2 2 4

0

1

2 4! T

i d x m J

Z D e

4d x

4

4!

J


• Summary• Quantum Field Theory is No Big Deal.

• All You Have to do is to Evaluate the Path Integral.




• Oh, Except for One Thing …




• Oh, Except for One Thing …• The Integral is Impossible to do


• Feynman to the Rescue• Came up With a Process to Approximate the

Solution by Series Solution in Called the Coupling Constant.

• To Keep Track of the Terms in the Expansion, We Draw Little Diagrams.

• Each Part of the Diagram Corresponds to a Rule.

• If You Walk Through the Diagram and Apply the Rules, You Can Write Down a Solution to the Path Integral

• Feynman Diagrams are a Convenient Way of Representing a Double Series Expansion in and

J


• Storm Clouds• It Turns That if you Follow the Rules That Describe

a Meson-Meson Scattering Event, the Correction to the Amplitude Diverges. It has the Form

• This Logarithmic Divergence Happens at Large k (momentum) – So it is Called an Ultraviolet Divergence. It Happens Whenever There is a Loop in a Feynman Diagram.

• If the Correction Diverges, the Amplitude Diverges and we Calculate an Infinite Probability – Nonsense. This is How Those Infinities Happen.

2

44

4

1 1

2d k

k


• All Feynman Diagrams with Loops Generate Terms Which Diverge!

• A Way Out Was Found

• The Process Involves Two Phases• Regularization

• Renormalization


• Regularization• We Have Found Integrals of the Form

Which Diverge at Large k

• Why Not Integrate Up to a Large, but Not Infinite Momentum Parameterized by

This is Called Momentum Cutoff

• Justification: We Decide That There is No Reason Our Theory Must be Valid to Arbitrarily Large Energies

4

0d kF k

4

0I I d kF k


• Regularization (II)• The General Solution to the Integral is

• In the Limit , C Vanishes, B is Independent of and A diverges

• We Have Separated the Integral Into a Piece That is Infinite and a Piece That is Finite

• What If We Could Find a Way to Get Rid of the Infinite Part – Sweep it Under the Rug

• There is a Way Called Renormalization

1I A B C


• Renormalization• We Have a Coupling Constant and a Cutoff

• What Do These Greek Letters Mean• The Coupling Constant is a Real Measurable Value

• The Coupling Constant in Electrodynamics is

• That Shouldn’t Change Depending on the Arbitrary Cutoff we Choose – Who Decides What the Cutoff is and Why

• So if we Change Then Must Change to Compensate

• This Means that in the Real World, There is a Physical Coupling Constant and is Actually a Function of

• Let’s Look at a Real Example

p


• Renormalization (II)• Warning, Lots of Equations Coming …

• A Theoretical Meson-Meson Scattering Amplitude Correction of Order is, After Integrating to a Cutoff

• To Make Our Lives Easier, We Define

Where s, t and u are Momenta From the Feynman Diagram

2

2 2 2

2 3log log logi iCs t u

M O

2 2 2

log log logLs t u


• Renormalization (III)• According to the Theory, If Someone Was to Go

and Actually Physically Measure This Event, She Would Measure a Coupling Constant Predicted By

• Again, To Make Our Lives Easier, Define

2 2 2

2 3

0 0 0

log log logpi i iCs t u

O

2 2 2

00 0 0

log log logLs t u


• Renormalization (IV)• If the Measurement is Done, We Predict

• Now Solve for

• We also Expect

• So, When We Say We Mean

2 3Pi i iC L O

2 3pi i iC L O

i

2 2piC L iC L

2 3p pi i iC L O

i


• Renormalization (V)• We Originally Predicted

• But We Really Meant That

• So, Combining the Two, What We Really Should Have Written is

• But Notice that

2 2 30p p pi iC L iC L M O

2 3p pi i iC L O

2 3i iC L M O

2 2 20 0P p piC L iC L iC L L


• Renormalization (VI)• What Was

• So

• Did You See It? My Hands Never Left My Arms!• The Cutoff is Gone into a Puff of Logic

0L L2 2 2

00 0 0

log log logLs t u

2 2 2

log log logLs t u

00 0 0

log log logs t u

L Ls t u


• Renormalization (VII)• Zee Says,

We started out with two unphysical quantities and their unphysicalness sort of cancelled each other out.

• Blechman Says,

Before you label this as ridiculous, realize that QED has used renormalization all the time, and its results have been tested to as many as fourteen decimal places. That is the best known confirmation of any theory of physics.


• Pause

• My Presentation is About Quantum Gravity

• We Needed to Understand Some Quantum Field Theory Before We Could Understand Why Gravity Didn’t Fit

• Now, on to Quantum Gravity


• In General Relativity Particles Move along Paths Which Extremize Proper Time

• Possible Paths are Described by the Action

• The Conditions for an Extremum of the Action are Lagrange’s Equations

• Particles Follow Geodesics Which Satisfy Lagrange’s Equations, So The Lagrangian is

For a Spacetime With Metric

,dx dx dx

L x g xd d d

g


• How Would One do Quantum Gravity?

• Start with Einstein-Hilbert Action

• Extremize the Proper Time to find the Lagrangian

• Put the Lagrangian into the Feynman Path Integral

• Develop Feynman Rules

• Evaluate the Path Integral

• Regularize

• Renormalize

• Go to Disneyland


• Don’t Buy the Tickets Quite Yet• Since the Action is in an Exponential, It Must be

Dimensionless. Recall

• Notice That if Each Piece of the Lagrangian has Dimension 4

• Then Has Dimension [1] Has Dimension [1] and Has Dimension [0]

• The Coupling Constant is Dimensionless

2 2 2 41

2 4!L m J

2 2 2 41[1][1] [1] [1] [1] [1]

2 4!L J


• Don’t Buy the Tickets Quite Yet (II)

• Remember We Did the Trick Where We Equated and to Renormalize

• That Was Equating to (the coupling constant of the electromagnetic field) Which is Dimensionless

• The Gravitational Coupling Constant is Dimensionful

• Then Has Dimension When We Try to Do the Equivalence Trick in the Series Expansion

p

EM


• Don’t Buy the Tickets Quite Yet (III)

• What Does it Mean to Have an Infinite Series with Terms of Increasing Dimension?

• If You “Cutoff” the Series, You Can Apparently Fiddle with the Resulting Equations to Get Something With a Physical Meaning

• But You Cannot Renormalize

• Quantum Gravity is a Nonrenormalizable Theory

• You Cannot Get Rid of the Infinities in Interactions that Have Loops in The Feynman Diagrams


• Don’t Buy the Tickets Quite Yet (IV)

• Think About an Two Particles Interacting Gravitationally

• Quel Horreur! A Loop. Infinities. Nonsense.

Quantum Gravity: Yes It’s Difficult.

• Quantum Gravity is a Nonrenormalizable Theory That Blows up on Loops

• Gravitational Interactions are Full of Loops

• Zee Says,

“Nonrenormalizable theories evoke fear and loathing in

theoretical physicists.”

• I’ll That to Mean Difficult.

• Quantum Gravity is Difficult and Now We Know Why

quantum gravity why is it so difficult? craig dowell february 24, 2006

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