Physics 234B: String Theory II
Quantum Gravity and Unifications
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Lectures: Tue and Thu, 12:40-2pm.
Discussions: will start after Spring break. The primary time for discussions will be Thu, 3:40-4:30pm. We will also have a few discussions in the secondary time slot, Fri 2:10-3pm. If anyone anticipates having time conflicts during these discussion times, please let me know by email.
Place: 402 Le Conte Hall (both lectures and discussions)
Office: 401 Le Conte Hall.
This year, our focus in the 234B course will be on gravity -- both
classical and quantum, how it emerges from string and superstring theory,
whether it can have an independent life outside of strings, how it enters
into various dualities and unifications across different areas of physics
(from supersymmetric unifications with the Standard Model of particle
physics, to unifications with non-gravitational physics via the AdS/CFT
correspondence and other dualities, to nonrelativistic Lifshitz gravity and
its relation to the Causal Dynamical Triangulations approach to quantum
gravity, to applications in condensed matter physics). We will also discuss
some of the many fundamental open puzzles of gravity.
A slightly more detailed plan for this course is as follows:
1. Survey of (semi)classical gravity.
This Chapter will serve a two-fold purpose: First, it is an introduction into the subject, making sure that we all start on the same page. Secondly, it will give a first glimpse of what are some of the fundamental puzzling features of gravity, from the perspective of many-body physics and quantum field theory, especially from the viewpoint of the renormalization group.
2. Superstring theory as quantum theory of gravity.
Strings and M-theory give us a beautiful, self-consistent and very rich arena for understanding how gravity can be combined with quantum mechanics. We will discuss first the question "Why Strings?", and will see several independent reasons why strings are special among objects of all possible dimensions. Then we will focus on the spacetime supergravity description, and discuss various string-string dualities and unifications.
3. AdS/CFT correspondence and holography.
AdS/CFT correspondence is perhaps the most successful byproduct of string theory. It is a duality, relating a seemingly non-gravitational quantum field theory on one hand, to a system with gravity (such as string theory) on the other. It therefore represents another "unification", of gravity with field theories both relativistic (of interest to particle physicists) and nonrelativistic (of interest to condensed matter and statistical physics). It also represents a constructive example of the (still rather mysterious) "holographic behavior" of quantum gravity, predicted by robust arguments based on the thermodynamic properties of black holes.
4. Effective field theory of gravity.
We will examine relativistic gravity using the framework of low-energy effective field theory. In this picture, the cosmological constant problem arises as the most pressing of all Naturalness problems. I will review the concept of technical Naturalness. We will also discuss the Appelquist-Carazzone decoupling theorem (and whether it should apply to gravity), and take a look at various previous attempts how to solve the cosmological constant problem. Also to be included: the Weinberg-Witten theorem; and at least a brief discussion of the recent "amplitudes" approach, including the color-kinematics duality.
5. Keldysh-Schwinger formalism, QFT out of equilibrium.
One of the best guarded secrets in theoretical physics: Relativistic QFT, as it is usually taught in our high-energy theory courses, is only a fragment of the whole story! It is a simplification of the whole thing, resulting from the assumption that the system has a stable, static and eternal vacuum. However, this assumption -- while OK for scattering particles at colliders -- seems rather dubious for gravity, especially when we take into account that we live in an evolving cosmological universe. The whole story of QFT without the simplifying assumptions about the vacuum is known for historical reasons as the "Keldysh-Schwinger formalism" -- I will review its basics, and take a look at how it is slowly making its way into quantum gravity.
6. Nonrelativistic gravity (often now called "Hořava-Lifshitz gravity"), causal dynamical triangulations.
This is a rather novel approach to making sense of gravity in a quantum theory, and it has attracted a lot of attention. The basic idea is to take QFT and the renormalization group seriously, but Lorentz invariance not so seriously. I will review the basics, including some applications to nonrelativistic AdS/CFT correspondence, and the comparison to the Monte Carlo lattice approach to quantum gravity known as Causal Dynamical Triangulations.
7. Effective field theory of inflation.
Methods of effective field theory have now been successfully applied to inflationary cosmology, a setting in which gravity and its quantum features are at the forefront. We will review the basics of this approach, and point out its connections to some of the previous Chapters discussed in this course.
Clearly, this is a rather ambitious and long list, which means that we will only be spending about two weeks on each of the 7 Chapters. At that pace, I expect that we will have covered Chapters 1 through 4 around Spring break, leaving Chapters 5 through 7 for after the break.
Here I will be posting various comments, references and extra reading material options, on a weekly basis as we go through the material.
Week 1. Before we can quantize gravity we need to understand its basics classically. You can review classical General Relativity by leafing through Sean Carroll's book
S.M. Carroll, Spacetime and Geometry. An Introduction to General Relativity, Addison Wesley, 2004.
This book is now a classic, and your first reference if you need to refresh your memory on anything in classical GR.
Week 2. I mentioned in lecture that relativistic gravity with the general R-squared Lagrangian in 3+1 dimensions is formally asymptotically free, but when interpreted in Minkowski signature it suffers (at least on first inspection) from the presence of propagating ghosts and hence apparent violations of unitarity. Per audience request, here is a reference that discusses renormalization of the higher-derivative gravity and contains further useful references:
R. Percacci, A Short Introduction to Asymptotic Safety, arXiv:1110.6389.
Week 3. Here are some useful or interesting references related to the miscellaneous (semi)classical aspects of gravity reviewed in the lectures. Witten's spinor-based proof of the positive energy theorem in asymptotically flat spacetimes is in
E. Witten, A New Proof of the Positive Energy Theorem, Comm. Math. Phys. 80 (1981) 381.
The closest partial analog in de Sitter space holds only perturbatively, and with stringent boundary conditions at the cosmological horizon, as shown by
L.F. Abbott and S. Deser, Stability of Gravity with a Cosmological Constant, Nucl. Phys. B195 (1982) 76.
A very interesting discussion of some issues (related to the absence of a proper positive energy theorem) in quantum gravity in de Sitter space are in Witten's talk from the Strings 2001 in Mumbai,
E. Witten, Quantum Gravity in De Sitter Space, arxiv:hep-th/0106109.
The Chern-Simons approach to quantum gravity in 2+1 dimensions is discussed in some detail in the review paper
S. Carlip, Quantum Gravity in 2+1 Dimensions: The Case of a Closed Universe, Living Rev. Rel. 8:1 (2005),
A fascinating extension and revision of the older results on quantum gravity in 2+1 dimensions, in view of AdS/CFT, can be found in
A. Maloney and E. Witten, Quantum Gravity Partition Functions in Three Dimensions, arxiv:0712.0155,
and are further extended in various follow-up papers by Alex Maloney and his
The generalization of Penrose's notion of conformal infinity of spacetime, required for more complicated asymptotics such as that of nonrelativistic AdS/CFT dualities, was presented in
P. Hořava and C.M. Melby-Thompson, Anisotropic Conformal Inifinity, arXiv:0909.2841.
Week 4. A very careful treatment of the proper (IR and UV) regularization and renormalization of the free scalar field theory in 1+1 dimensions can be found in Chapter 17 of
M. Stone, The Physics of Quantum Fields (Springer, 1999).
The large N limit is discussed in many textbooks on QFT and stat-mech, including Chapter VII.4 of Tony Zee's book Quantum Field Theory in a Nutshell. The 1+1 dimensional scalar field theory with O(N) symmetry (discussed in class) is solved in the large N limit for example on pages 463-5 of Peskin-Schroeder.
A rather prehistoric (written 15 years before AdS/CFT!), yet illuminating review paper of the idea that large N is actually a sort of classical limit:
L.G. Yaffe, Large N Limits as Classical Mechanics, Rev. Mod. Phys. 54 (1982) 407.
A great comprehensive reference on matrix models, their large N expansion and relation to the branch of string theory known as "noncritical string theory" is in Yu Nakayama Master's Thesis,
Y. Nakayama, Liouville Field Theory -- A Decade after the Revolution, arXiv:hep-th/0402009;
(warning: that review contains much much more than just the parts relevant for our lectures, so please read at your own risk!).
Attempts to generalize from matrices to higher-rank tensors have finally started producing somewhat intriguing results in recent years, under the name of "tensor models" (and to some degree, the closely related area of so-called "group field theory" pioneered by V. Rivasseau and his school); see for example the review
R. Gurau, A Review of the Large N Limit of Tensor Models, arXiv:1209.4295.
Week 5. For the basics of critical strings and superstrings, the book by K. Becker, M. Becker and J.H. Schwarz, String Theory and M-Theory. A Modern Introduction (Cambridge, 2007) represents a great reference. Compared to some other textbooks, it contains many of the more modern developments, including string dualities, M-theory, AdS/CFT, and supersymmetric quantum black hole/brane physics.
A nice outline of Friedan's derivation of the beta-functions in the nonlinear sigma model in 1+1 dimensions is presented in Ch. 3.4 of Volume 1 of the timeless classic Superstring Theory by M.B. Green, J.H. Schwarz and E. Witten (Cambridge, 1987).
Week 6. A comprehensive review of the first years of supergravity is
P. van Nieuwenhuizen, Supergravity, Phys. Rept. 68 (1981) 189.
A classic review of the first decade of supegravity in higher dimensions, and its compactifications to N>1 supergravities in 3+1 dimensions, is
M.J. Duff, B.E.W. Nilsson and C.N. Pope, Kaluza-Klein Supergavity, Phys. Rept. 130 (1986) 1.
At that point, supergravity was taken over completely by string theorists, and the more recent literature is then a part of the string theory literature (see, e.g., the book by Becker, Becker, Schwarz cited above).
There has been a lot of excitement and a lot of work recently on the possibility that N=8 supergravity in 3+1 dimensions may be finite, or at least finite up to some really high order in perturbation theory. Some interesting reviews of this subject are
R. Kallosh, An Update on Perturbative N=8 Supergravity, arXiv:1412.7117,
Z. Bern et al, Amplitudes and Ultraviolet Behavior of N=8 Supergravity, arXiv:1103.1848,
L.J. Dixon, Ultraviolet Behavior of N=8 Supergravity, arXiv:1005.2703,
Z. Bern, J.J.M. Carrasco and H. Johansson, Progress on Ultraviolet Finiteness of Supergravity, arXiv:0902.3765.
The subject is still quite controversial, see for example
T. Banks, Arguments Against a Finite N=8 Supergravity, arXiv:1205.5768.
On the gravity side, basics of General Relativity at the level of 231 are
required. On the quantum side, basics of QFT at the level of 232A would be
a good start, and will effectively be required.
On the other hand, basics of Yang-Mills gauge theories (either 233A or 232B)
would also be useful, but will not be strictly required for students who wish
to sign up for this course. Same with the basics of String Theory -- it
would be useful and nice to have seen strings (at the level of 234A), but it
will not be formally required in order to take this course. If you are
interested in this course and have questions about having all the
prerequisites, feel free to talk to me (in person or via email); I will try
to be flexible, so that we can accommodate as many students in this course
as possible, even those who come from various diverse backgrounds such as
particle phenomenology, cosmology, condensed matter, or pure math.
Homeworks and Reading Assignments
There will be no homework sets for this course. Instead, there will be a
list of reading assignments, where each student gets to choose one from the
list of many important gravity-related research papers. Then, in the second
half of the semester, the students will present their summary of the ideas and
results of their chosen paper during our Discussion Sessions. The final grade
will be based on this presentation. There will be no final exam.