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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?
Quantum Gravity: Why so Difficult?
• 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
Quantum Gravity: Why so Difficult?
• The Problem
• Infinities when Calculating Probabilities
• Basically, it all Translates Into a Normalization
Problem, i.e.

2
    N   x  dx  1

Where Does the Problem Come From?
• Let’s Start Slowly From the Beginning
Quantum Gravity: Why so Difficult?
• Normalization
• The Statistical Interpretation of Quantum
Mechanics Says that,
  x
2
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
2

    N   x  dx  1

Finding the Coefficient N so that this is True is
Called Normalizing the Wavefunction
Quantum Gravity: Why so Difficult?
• Time Independent Perturbation Theory
• You Take a Hamiltonian Corresponding to an Exact
Solution to the Schrodinger Equation and Perturb
It.
Hˆ  Hˆ (0)   Hˆ '
Then you Expand in Terms of Power Series and
Cutoff at some order.
 
 0
 
1
 
2
 2
• But You Originally Normalized for the Solved
Wavefunction. You Have to Pick a New N – You
have to Renormalize

2
    N   x  dx  1

Quantum Gravity: Why so Difficult?
• Time Independent Perturbation Theory II
• What if One of the Terms is Infinite?
 
 0
 
1
 2
  ,  
2
2
 2

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
Quantum Gravity: Why so Difficult?
• What is Quantum Field Theory?
• Recall that Maxwell’s Equations in Free Space
Imply the Existence of Electromagnetic Waves
1 2 E
 E 2 2
 t
2
• Imagine a Rectangular Resonant Cavity with a
Standing Wave in the z direction. The E and B
Fields Could be
E  z   E0 cos  kz  xˆ ,
B  z   B0 sin  kz  yˆ
• Make a Hamiltonian from the Energy Density (in
1-D)


1
H     0 E 2  B 2 dz
z
0 

Quantum Gravity: Why so Difficult?
• 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 Gravity: Why so Difficult?
• 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 Gravity: Why so Difficult?
• 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.
Quantum Gravity: Why so Difficult?
• Some More Quantum Mechanics
• Recall the Schrodinger Equation
2

 2
i

V 
2
t
2m x
• It Was Solved by Separation of Variables and We
Found that the Time Dependence Was
  t   eiHt
• This Unitary Operator Governs the Amplitude to go
from One Place to Another
• Now a Feynman Story
Quantum Gravity: Why so Difficult?
• 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.
Quantum Gravity: Why so Difficult?
• The Path Integral
• Dirac Derived the Path Integral Representation of
the Feynman Sum over Paths Statement.
qF e
 iHt
qI   Dq  t e
i
T
0
1

dt  mq 2 
2

• It Turns Out That if You Write the Hamiltonian for
the Harmonic Oscillator as H, you get
qF e
 iHt
qI   Dq  t e
i
T
0
1

dt  mq 2 V  q  
2

You Can See That the Lagrangian of the Harmonic
Oscillator Pops Out
Quantum Gravity: Why so Difficult?
• The Path Integral (II)
• Sometimes this is written in terms of the action S.
Z   Dq  t eiS
• It’s Not Too Much of a Stretch Now to Add an
Anharmonicity Term
Z   D e
i
T
0 d
4

2
1

x      m 2 2    4  J  
 4!
2 

• The d x Indicates We are Integrating over
Spacetime, and the  Indicates We are Talking
about Fields. The  4!  is the Perturbation Causing
Anharmonicity and J  Represents a Force (think of
pushing on the mattress).
4
4
Quantum Gravity: Why so Difficult?
• Summary
• Quantum Field Theory is No Big Deal.
• All You Have to do is to Evaluate the Path Integral.
Quantum Gravity: Why so Difficult?
• Summary
• Quantum Field Theory is No Big Deal.
• All You Have to do is to Evaluate the Path Integral.
• Oh, Except for One Thing …
Quantum Gravity: Why so Difficult?
• Summary
• Quantum Field Theory is No Big Deal.
• All You Have to do is to Evaluate the Path Integral.
• Oh, Except for One Thing …
• The Integral is Impossible to do
Quantum Gravity: Why so Difficult?
• 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
Quantum Gravity: Why so Difficult?
• Storm Clouds
• It Turns That if you Follow the Rules That Describe
a Meson-Meson Scattering Event,
the 2 Correction

to the Amplitude Diverges. It has the Form
4
 1   1  4
  2   k 4  d k
• 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.
Quantum Gravity: Why so Difficult?
• All Feynman Diagrams with Loops Generate
Terms Which Diverge!
• A Way Out Was Found
• The Process Involves Two Phases
• Regularization
• Renormalization
Quantum Gravity: Why so Difficult?
• Regularization
• We Have Found Integrals of the Form


0
d 4 kF  k 
Which Diverge at Large k
• Why Not Integrate Up to a Large, but Not Infinite
Momentum Parameterized by 

I  I    d 4 kF  k 
0
This is Called Momentum Cutoff
• Justification: We Decide That There is No Reason
Our Theory Must be Valid to Arbitrarily Large
Energies
Quantum Gravity: Why so Difficult?
• Regularization (II)
• The General Solution to the Integral is
1
I  A     B  C  

• 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
Quantum Gravity: Why so Difficult?
• 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  p and  is Actually a Function of 
• Let’s Look at a Real Example
Quantum Gravity: Why so Difficult?
• Renormalization (II)
• Warning, Lots of Equations Coming …
• A Theoretical Meson-Meson Scattering Amplitude
Correction of Order  2 is, After Integrating to a
Cutoff
  2 
 2 
 2 
3
M  i  iC  log 

log

log

O

 





s
t
u





 
2
• To Make Our Lives Easier, We Define
 2 
 2 
 2 
L  log 
  log 
  log 

s
t
u






Where s, t and u are Momenta From the Feynman
Diagram
Quantum Gravity: Why so Difficult?
• 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
2
2
2










2
3
i p  i  iC  log 
  log 
  log 
   O  
 t0 
 u0  
  s0 
• Again, To Make Our Lives Easier, Define
 2 
 2 
 2 
L0  log 
  log 
  log 

s
t
u
 0 
 0 
 0 
Quantum Gravity: Why so Difficult?
• Renormalization (IV)
• If the Measurement is Done, We Predict
iP  i  iC 2 L  O  3 
• Now Solve for i
i  i p  iC 2 L  O  3 
• We also Expect
iC 2 L  iCp2 L
• So, When We Say i We Mean
i  i p  iC p2 L  O  3 
Quantum Gravity: Why so Difficult?
• Renormalization (V)
• We Originally Predicted
M  i  iC 2 L  O  3 
• But We Really Meant That
i  i p  iC p2 L  O  3 
• So, Combining the Two, What We Really Should
Have Written is
M  i p  iC  p2 L  iC  p2 L0  O  3 
• But Notice that
iCP2 L  iCp2 L0  iCp2  L0  L 
Quantum Gravity: Why so Difficult?
• Renormalization (VI)
• What Was  L0  L 
 2 
 2 
 2 
L0  log 
  log 
  log 

s
t
u
 0 
 0 
 0 
 2 
 2 
 2 
L  log 
  log 
  log 

s
t
u






• So
s
t 
u
L0  L  log    log    log  
 s0 
 t0 
 u0 
• Did You See It? My Hands Never Left My Arms!
• The Cutoff is Gone into a Puff of Logic
Quantum Gravity: Why so Difficult?
• 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.
Quantum Gravity: Why so Difficult?
• 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
Quantum Gravity: Why so Difficult?
• 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
 dx   
dx dx  
L
, x     g  x 

d

d

d


 

For a Spacetime With Metric g
Quantum Gravity: Why so Difficult?
• 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
Quantum Gravity: Why so Difficult?
• Don’t Buy the Tickets Quite Yet
• Since the Action is in an Exponential, It Must be
Dimensionless. Recall
1
 4
2
2 2
L      m     J 
 4!
2
• Notice That if Each Piece of the Lagrangian has
Dimension 4
1
 4
2
2
2
L  [1][1]  [1] [1]  [1]  J [1]
 4!
2
• Then  Has Dimension [1]  Has Dimension [1]
and  Has Dimension [0]
• The Coupling Constant is Dimensionless
Quantum Gravity: Why so Difficult?
• Don’t Buy the Tickets Quite Yet (II)
• Remember We Did the Trick Where We Equated 
and  p to Renormalize
• That Was Equating  to  EM (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
Quantum Gravity: Why so Difficult?
• 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
Quantum Gravity: Why so Difficult?
• 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