How Feynman tried to quantize
gravity and discovered ghosts,
and other stories
Hirosi Ooguri

Aspen Center for Physics Colloquium
23 August 2018
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How Feynman tried
to quantize gravity

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"There's a certain irrationality to
any work in gravitation, so it's
hard to explain why you do any of it."based on a tape‐recording of Feynman's lecture "Quantum Theory of Gravitation"at the Conference on Relativistic Theories of Gravitation organized by Leopold Infeld
in Poland in July 1962.
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Dirac and Feynman
in the Jabłonna Palacenear Warsaw, Poland
in July 1962.

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In the 1962 lecture in Poland, Feynman presented
what is now known as the Feynman Tree Theorem.
In quantum theory, probabilities of all possible
outcomes always sum up to be one.
From this assumption, Feynman derived a remarkable
set of relations between classical amplitudes andquantum amplitudes in non‐gravitational theory.
tree = classical
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Feynman explaining his tree theorem
in Jabłonna, Poland in July 1962.

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Feynman Tree Theorem
Relations between classical and quanum amplitudes"I assumed the tree theorem to be true, and used it in reverse."

He discovers that these relations do not hold in
gravity with virtual gravitons in quantum processes.
"something is fundamentally wrong"
based on a tape‐recording of Feynman's lecture"Quantum Theory of Gravitation" in Poland in July 1962.
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Gell‐Mann suggested him to study the Yang‐Mills
theory, and Feynman found the same problem.
"if you have two examples of the same disease,then there are many things you don't worry about.""The Yang‐Mills case took me about a day."
However, the gravitational case was harder.
"I tried again and again and was never able to do it; and it was finally put
on a computing machine. ... I think it's the first problem in algebra that
I know of that was done on a machine that has not been done by hand."based on a tape‐recording of Feynman's lecture"Quantum Theory of Gravitation" in Poland in July 1962.
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Feynman noticed that his tree theorem also holds
in the gravitational case if an "artificial dopey particle"
is included in virtual quantum processes.
" I only got this completely straightened out a week before I came here.
I do not claim that this method of quantization can be obviously
and evidently carried out on to the next order. "based on a tape‐recording of Feynman's lecture"Quantum Theory of Gravitation" in Poland in July 1962.
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Feynman noticed that his tree theorem also holds
in the gravitational case if an "artificial dopey particle"
is included in virtual quantum processes.
" I only got this completely straightened out a week before I came here.
I do not claim that this method of quantization can be obviously
and evidently carried out on to the next order. "His "dopey particle" proposal has turned out tobe essential in quantizing Yang‐Mills and gravity.
based on a tape‐recording of Feynman's lecture"Quantum Theory of Gravitation" in Poland in July 1962.
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1967

De Witt, "Quantum Theory of Gravity. I, II, III"
Faddeev and Popov, "Feynman Diagrams for the Yang‐Mills Field"

1972't Hooft and Veltman, "Regularization and Renormalization of Gauge Fields"

1973

Gross and Wilczek, "Ultraviolet Behavior of Nonabelian Gauge Theories"
Politzer, "Reliable Perturbative Results for Strong Interactions?"
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Variations on a Theme of Feynman Tree Theorem

12/64Feynman's notes on gravitationin one of the 93 Feynman boxes
at the Caltech Archives.
This folder also contains a manuscript for
his talk at the 1962 conference in Poland.

Feynman Exhibition(May 11 ‐ December 14)
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A young theorist sent his annotated paper to Feynman.
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A young theorist sent his annotated paper to Feynman.
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Begin a digression

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Ludwig Dimitrevich Faddeev:"I saw a huge folder of Feynman's
notes on the Bethe ansatz.
I would like to know why he wasinterested in integrable models.
Could you find the notes?"
Bamberg, Germany (March 2008)

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Ansatz: an attempt at a solution followed by a demonstration that
it is correct, rather than a solution derived using a known approach.
For the Heisenberg model of L spin in a 1D circle, Hans Bethe (1931)
started with the ferromagnetic ground state and considered asubspce with n spins overturned. Going to the momentum basis,

Bethe assumed,
The momenta then must satisfy,

This gives 2^L energy eigenstates.
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Feynman's blackboard
when he passed awayin February, 1988.

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Feynman's blackboard
when he passed awayin February, 1988.

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Feynman:"I got really fascinated by these (1 + 1)‐dimensional models
that are solved by the Bethe ansatz and how mysteriously
they jump at you and work and you don't know why.
I am trying to understand all this better."
Asia‐Pacific Physics News (June/July 1988),
published posthumously.

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10 page memo for his lunch
time talk on the Bethe ansatz
in January, 1987.

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Ludwig Dimitrevich Faddeev:"No, these are not what I saw!
I saw a very thick notebook."

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Faddeev decided to takeit upon himself to findthe missing notebooks.

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Faddeev decided to takeit upon himself to findthe missing notebooks.

But, the notebook he saw
was not at the Archives.

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Where is the notebook
Faddeev saw?

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Elias Kiritsis (Caltech 1988 Ph.D.)
When I arrived at Caltech in the fall of 1984, Feynman was "away" from
Lauritsen. I was told that he was spending his time in the computer science
department, as he was interested in "quantum computation", and his
famous work on the subject dates from that era.
In 1986, he suddently he reappeared in the Physics department with a new"hobby" that he immediately shared with all people. His intuition was thatto solve QCD you must define the right variables like in integrable models.
He started spending time with us, and having often lunch with us as he wastrying to share his excitement and make us interested.

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Alexios Polychronakos (Caltech 1987 Ph.D.)
Last time I saw Feynman was when I visited Caltech Dec 1987 ‐ Jan 1988.
He passed soon after that, and I did not visit Caltech again until sometime
late in 1988, which was when his secretary, the late Helen Tuck, gave mea folder with his notes, thinking that I should have them. The folder is stillin my possession.
Since Faddeev apparently saw the hand‐written notes in question "just after
the death of Prof. Feynman," there is a small chance that he might be referringto the ones I have, although they are on Calogero‐Sutherland models andnot on the Bethe ansatz.

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Sandip Trivedi (Caltech 1990 Ph.D.)
Feynman's idea was to use the progress in integrability in two dimensions
to study fragmentation or hadronisation in QCD.
Consider a high energy scattering process which gives rise to a quark or gluon
moving at high energy that fragments as it moves away from the interaction
region. There is one preferred direction along which the quark or gluon moves
where the momentum is high and the physics is ``hard", and two transverse
directions where it is ``soft".
Feynman's idea was to separate these directions. Since high energy physics is
simple in QCD due to asymptotic freedom, the hard physics would be easy todeal with. This would then leave the two transverse directions where the physics
is soft and non‐trivial, and perhaps here the progress in integrability could be
used to understand the process of fragmentation.
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Sandip Trivedi (Caltech 1990 Ph.D.)
The first and perhaps most important lesson, for a beginning graduate
student like me, was just how thorough and careful Feynman was.
Feynman explained the six‐vertex model to us, and his instructions were
for us to try and solve it without looking at Baxter's solution, already in print.
"We gotta know how to solve every
problem that has been solved."

Occasionally the discussions would extend over dinner ‐ for which Feynman
often generously paid. I remember one great dinner at a Mexican restaurant
on Colorado Boulverd where he explained the spin statistics theorem, using
a belt which he took off from his pant.
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Sandip Trivedi (Caltech 1990 Ph.D.)

On the six‐vertex model:

One day Feynman called us in to his office. His eyes were shinning.
``I have solved it !" he exclaimed.
``Get me some multi‐coloured chalk !" he yelled to his secretary Helen.
What followed was pure Feynman genius, in equal measure physics andshowmanship of the highest order.
The key was to look at the lattice not in the standard fashion, as we had
been doing, but at 45 degrees. Then, due to the particle number conservation,a solution easily presented itself in a few lines of algebra.
31/64Sandip Trivedi (Caltech 1990 Ph.D.)

On the eight‐vertex model:This was more complicated because there was no particle number conservation,and particles could be created and destroyed. Feyman was optimistic as ever.
``I invented all the tricks on how to deal with the creation and destruction ofparticles in QED," he said to us with a wink. ``We will surely get there".
Sadly, Feynman fell quite ill soon thereafter. Our meetings grew infrequent.

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Sandip Trivedi (Caltech 1990 Ph.D.)
I do have some notes of Feynman which are a treasured possession of mine.
During his final illness, Feynman instructed Helen Tuck to share his notes
with the students. We were amazed how meticulous and detailed they are.
I show the notes to my students to inspire them.
When I look back one of the things that strikes me is just how remarkablea place Caltech was, and continues to be.
I could come in as a beginning graduate student and soon thereafter
start working with one of the greatest scientists of the 20th century.
Where else does one get an opportunity like this, so easily!

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Feynman's vision is being partially realized
in supersymmetric gauge theories.

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end of the digression

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Is non‐renormalizability
a problem with gravity?

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It is often said that, since the Einstein theory,is not renormalizable and cannot be used
to compute quantum gravity effects reliably.

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It is often said that, since the Einstein theory,is not renormalizable and cannot be used
to compute quantum gravity effects reliably.
"Whether it is a truly significant
objection to a theory, I don't know."
Feynman Lectures on Gravitation (1962 ‐ 63)
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However, ....

Kenneth G. Wilson (1936 ‐ 2013)
1961 Caltech PhD
1982 Nobel Prize
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For example, the pion theory in nuclear physics,

is also not renormalizable.
Neverthelss, we can used it to calculate quantum
effects reliably as far as energy scale is less than F ,and they have been verified experimentally.
It is an effective theory of the more fundamental QCD.
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John Preskill:
I spoke with Feynman a number of times about renormalization theory during
the mid‐80s. I was surprised how the effective field theory viewpoint did notcome naturally to him.
Mark Wise gave a talk about chiral Lagrangian, and Feynman thought Mark wasfoolishly doing perturbation theory in the pion‐nucleon scattering. I explainedthat the calculation was really an expansion in the pion momentum, andtherefore justified when the pions are sufficiently soft. He understood quickly,
but I was surprised he needed my prompting to catch on.
It was funny in a way. I would tell Feynman things that I knew were common
lore, and he would be receptive, but he seemed unfamiliar with the ideas.

41/64Sidney Coleman:[Feynman] was like the guy that climbs Mt. Blanc barefoot just to show it could
be done. A lot of things he did were to show, you didn't have to do it that way,
you can do it this other way. And the other way, in fact, was not as good as
the first way, but it showed he was different.
I am sure Dick thought of that as a virture, as noble. I don't think it's so.
I think it's kidding yourself. Those other guys are not all a collection of yo‐yos.
Sometimes it would be better to take the recent machineary they have built
and not try to rebuilt it, like reiventing the wheel.
I know people who are in fact very original and not cranky but have not done as
good physics as they could have done because they were more concerned at a
certain juncture with being original than being right. Dick could get away with a
lot because he was so goddam smart. He really could climb Mt. Blanc barefoot.
from "Genus" by James Gleick
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Einstein gravity is also an effective theory.
As in the pion theory, the Einstein gravity can be used to makereliable predictions including quantum effects, provided energy
and momenta are below its cutoff scale (threshold above which
a more fundamental theory, such as string theory, is required).
For example:☆ Hawking radiation from a black hole
☆ Cosmic microwave background fluctuations

caused by quantum effects during the inflation☆ Corrections to the Newton potential:

relativity effectquantum effect
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However, ....

Gravity is Different.

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Gravity is Different:(1) Not all low energy theories of gravity can bederived from consistent quantum theories.
(2) The physical world is hierarchial. The explorationof shorter distances has led us to more and more
fundamental laws of nature. This hierarchy willterminates once we complete quantum gravity.

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Remarkably, there is a theory
which successfully quantize
gravity and which contains
all ingredients for a realistic
model of particle physics.

Superstring Theory
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Michael Green and John Schwarz:
Crucial breakthroughs were made in August 1984 at the Aspen Center
for Physics.
While walking to one of the workshop seminars, JHS remarked to MBG
that there might be a gauge group for which the two contributions cancel.
At the end of the seminar MBG said to JHS "SO(32)," which was the correct result.
ACP Science Histories

The 20th anniversary
Celebration in Aspenon 12 August 2004
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Swampland Question
Given an effective theory of gravity, how can
one judge whether it is realized as a low energy
appropximation to a consistent quantum theory
such as string theory?
Vafa: hep‐th/0509212;
Vafa + Ooguri: hep‐th/0605264

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Constraints on Symmetry
Symmetry has played important roles in physics(1) In identifying and formulating
fundamental laws of nature(2) In using these laws to understandand predict dynamics and phases of matters.

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Symmetry can be deceiving:
Two seemingly different microscopic Lagrangians
with different gauge symmetries and different matter
contents can describe the same quantum system.
"Duality"
Equivalencen can be
between full Hilbert spaces
or about their low energy limits(such as in the Seiberg dualities).
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Symmetry can be deceiving:Global symmetry is well‐defined andis independent of which Lagrangian
description you use.

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Symmetry can be deceiving:Global symmetry is well‐defined andis independent of which Lagrangian
description you use.
However, it has been argued that a
consistent quantum theory of gravity
does not have global symmetry.

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Standard argument against
global symmetry in quantum gravity:
If there is a continuous global symmetry G, we can combinea large number of G‐charge matters to make a black holein an arbitrary large representations of G.
Let it Hawking‐radiate, keeping its mass > the Planck mass.
The Hawking radiation is G‐blind (If G were a gauge symmetry,the radiation would have charge inbalance). The dimensionof the G representation exceeds the number of states allowedby the Bekenstein‐Hawking entropy formula.
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We have refined and proven these conjectures in AdS/CFT.
《 work in progress with Daniel Harlow 》(1) Any global symmetry in AdS is inconsistent with locality of CFT.
(2) A compact (discrete or continuous) symmetry G in CFT
corresponds to a gauge symmetry with the same G in AdS.
(3) In a gravitational theory with gauge group G, there must be
physical states in every finite dimensional irreducible unitary
representation in G.
+ with some additional assumption:(4) Internal global symmetry of CFT is compact.
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Holographic Principle:
Gravitational theory in Anti‐de Sitter
space (AdS) is equivalent to Conformal
Field Theory (CFT) at the boundary.

AdS = CFTThe correspondence enables us toprove theorems about quantum
gravity by translating them into
statements on conformal field theory.

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Recently, it was discovered that local excitationsin the gravitational theory have a special type ofquantum entanglement, similar to the one used
for quantum error correcting codes, with
sub‐systems sharing quantum secret keys.
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Quantum gravity cannot have symmetry
If a gravitational theory in has symmetry,

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Quantum gravity cannot have symmetry
If a gravitational theory in has symmetry,
there must be an operator transforming
a local excitation into another.
However, a symmetry generating
operator cannot change quantum
error correcting properties ofquantum states.

Contradiction
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Weak Gravity Conjecture

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Weak Gravity Conjecture
If a low energy theory contains both gravity andelectromagnetism, the theory must also containa particle with charge Q and mass m, such that:

Arkani‐Hamed, Motl, Nicolis, Vafa: hep‐th/0601001
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unless protectedby supersymmetry

We proposed this as a stronger version of the conjectureand showed that it restricts types and masses of neutrinos
Vafa + Ooguri: 1610.1533

Our idea has been explored further in recent papers, leading toconstraints on particle physics models beyond the Standard Model.
Ibanez, Martin‐Lozano, Valenzuela: 1706.05392,1707.05811
Hamada, Shiu: 1707.06326
Gonzalo, Herraez, Ibanez: 1803.08455

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Are de Sitter vacua in the Swampland?
The Dine‐Seiberg problem (1985) poses a question to any
attempt to construct of meta‐stable vacua with zero or positive
vacuum energy in the weakly coupled regime of string theory.
The problem can be sharpened by applying recent insightsfrom string theory [to appear with E. Palti, G. Shiu, C. Vafa].

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Without gravity, any low energy effective theory
can be embedded in a consistent high energy theory.
With gravity, the situation is different, and there are
non‐trivial constraints on gravitational low energy
effective theories (swampland conditions).
With more work, we may be able to extract
predictions on observable phenomena in low energy
from consistency of quantum gravity in high energy.
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"And so I 'm stuck to have to continue thisinvestigation, and of course you all appreciatethat this is the secret reason for doing any work,no matter how absurd and irrational and academic
it looks; we all realize that no matter how smalla thing is, if it has physical interest and is thought
about carefully enough, you're bound to think ofsomethig that is good for something else."
based on a tape‐recording of Feynman's lecture"Quantum Theory of Gravitation" in Poland in July 1962.
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