6.4 Type I string theory

\setlength{\epigraphwidth}{.92\textwidth} \epigraph{Nevertheless, due to the inneratomic movement of electrons, atoms would have to radiate not only electromagnetic but also gravitational energy, if only in tiny amounts. As this is hardly true in nature, it appears that quantum theory would have to modify not only MAXWELLIAN electrodynamics, but also the new theory of gravitation.\footnotemark{}}{Albert Einstein (1916)~^[1]}

\footnotetext{It is the 1st mention of Quantum Gravity. But it can be resolved with just ``QM in classical curved spacetime’’ without even needing EFT of Gravity (low energy quantum gravity). We can show that under a Schwarzschild potential sourced by the Nucleus, just like in ordinary QM coupled to the classical electromagnetic field, an electron has stationary bound states (ensuring the stability of atoms) ^[2].}

Gravity couples to the stress-energy tensor. So, gravity couples to all things physical. Except for gravity, we know that everything else in the physical world is fundamentally quantum in nature. It is probably not possible or at least inelegant¹ to consistently couple classical gravity to quantum fields. So, we are compelled to quantize gravity.

Since it couples to everything, in principle, every calculation that we make to predict something about reality must include corrections coming from quantum gravity. For example, if you have a scalar field theory, we know how to solve the QFT if we neglect the scalar field interaction with gravity, but there will always be corrections due to the interaction between the scalar field and gravity. You can do semiclassical physics (QFT in curved spacetime), but there will still be quantum gravity corrections. These quantum gravity effects become large when the particles’ scattering energies become comparable to the Planck scale, since gravity couples to energy. But, in practice, quantum gravity effects are notoriously hard to detect in experiments. The Planck length is $\sim 10^{16}$ times smaller than the smallest scale that our species probed. If we consider only gravity experiments, then the smallest scale probed as of now is ^[3] \& they measured $30\times10^{-18}N$ force, which in Planck units is $\sim 10^{-61}$. In a loose sense, any approach to quantum gravity is a theory of everything since quantum gravity is necessary to understand anything exactly. But usually, the term theory of everything is reserved for theories like string theory where the theory is hoped to be sufficient to explain all physical phenomena.

We have figured out how to quantize fields with spin $0,\frac{1}{2},1$. But quantizing the spin $2$ gravitational quantum field turned out to be hard or impossible due to the perturbative nonrenormalizability of gravity. It is unlikely that there is a nontrivial UV fixed point for gravity; see section asymptotic. So, it seems highly likely that the framework of QFT is insufficient for quantum gravity, and we have\href{https://www.youtube.com/watch?v=8TGalu36BHA}{… to go… even further beyond!}

\setlength{\epigraphwidth}{0.77\textwidth} \epigraph{I learned very early the difference between knowing the name of something and knowing something.}{Richard Feynman}

When I started these notes, I knew the names of most topics related to quantum gravity. My goal now is to properly understand them instead of just knowing the names.

I do not want to discuss history. But check ^[4] for history. There are many interesting historical facts like how string theory was originally developed to explain the strong force instead of quantum gravity. \subsection{Problems} ^[5]. For problems specific to string theory, check open.
Some of the many problems that are encountered in quantum gravity are:

Perturbative nonrenormalizability:~^[6] The mass dimension of Newton’s constant $G_N$ is $2-D$. For $D>2$, it is negative, making the theory nonrenormalizable, see GRasPerturbativeQFT.
Unitarity of black hole evaporation: The unitarity of quantum physics is in tension with semiclassical gravity, see TheInformationParadox.
Cosmological constant problem:~^[7] The observed value is 120 orders of magnitude smaller than the naive expectations from QFT. Is there a theoretical explanation for the extremely small value? Or do we have to lower our expectations and be satisfied with anthropic arguments within a landscape/multiverse?
Causality and chronology protection conjecture:~^[8] At the level of GR as an EFT, we can already see that the causal structure of gravity has uncertainties/fluctuations uncertainty. Even more complicatedly, many approaches are pointing towards spacetime emerging from something more fundamental; how does causality appear in such a case? Is Hawking’s \href{https://en.wikipedia.org/wiki/Chronology_protection_conjecture}{chronology protection conjecture}, which says that time travel is impossible macroscopically and causality remains sacred, valid in the correct quantum gravity theory?
Singularities and cosmic censorship:~^[9] Can singularities exist? If they exist, is Penrose’s \href{https://en.wikipedia.org/wiki/Cosmic_censorship_hypothesis}{cosmic censorship conjecture} true? String theory is well-defined at singularities, for e.g., when we study orbifolds and orientifolds. And it’s not clear if some version of cosmic censorship can hold true in string theory.
Breakdown of EFT:~^[10] The famous adage that ``General Relativity is incompatible with quantum theories” is false, and the quantum version of General Relativity makes sense as an EFT, see GREFT. But this EFT description breaks down at the so-called `species scale’. This is also related to the UV/IR mixing in the next section.
Failure of classical spacetime: ^[11] If we try to probe a distance smaller than the Planck scale, the momentum and energy will be so large that this region will collapse into a black hole due to the Heisenberg uncertainty principle. So, the classical spacetime that we understand well breaks down at the Planck scale. What is the nature of quantum spacetime from which the classical spacetime emerges? Is it noncommutative geometry NG or something else otherNG?
Problem of time:~^[12] There are various problems of time. Time in ordinary (spacetime) QFT is a label. In the worldline formalism Worldline, it is a dynamic scalar field. What will be the nature of time in quantum gravity after we include things like 1) fluctuations in light cones 2) emergence of spacetime 3) background independent formulations into our consideration?
Holographic description: Does the fact that black hole entropy scales like area really imply that quantum gravity must have a lower dimensional field theory description? Is holography limited to asymptotically AdS spacetimes (AdS/CFT)? Or is it a more generic property?
Framework: All approaches to quantum gravity indirectly use the framework of QFT or GR. For example, worldsheet string theory is $2D$ CFT coupled to 2D gravity, and holography is $D\geq 2$ CFT. Is there a more natural mathematical framework for quantum gravity instead of QFT?

\setlength{\epigraphwidth}{.85\textwidth} \epigraph{Quantum gravity is notoriously a subject where problems vastly outnumber results.}{Sidney Coleman (1989)}

\etocsettocdepth.toc{subsection}

\subsection{Expected properties} These are some expected properties of quantum gravity.

Unitarity: Unitarity is generally needed to make sense of time evolution in quantum theories. It was recently speculated that unitarity should be replaced with the weaker isometry~^[13].
No local observables: See section NoLocalObservables.
UV complete:~^[14] Unlike (most) QFTs, quantum gravity is not expected to be an effective theory. There won’t be a UV cutoff beyond which the theory will be invalid, and new degrees of freedom will appear. There won’t be new physics beyond the Plank scale. So, it should not have any UV divergences. String theory remains the only realistic theory that has high evidence that it is UV finite, see UVfiniteness. Some toy models of quantum gravity, like the dual of the SYK model SYK, are UV finite, but they are not realistic. Most (realistic) alternatives to string theory are also hoped to be UV finite, but they have not provided enough evidence for UV finiteness, see section 4 in ^[15]. Asymptotic gravity asymptotic is slightly different; if that theory exists, then it is UV finite, but the theory probably doesn’t exist. If we assume that topologically massive gravity is UV finite, then Hořava–Lifshitz gravity Horava is UV finite ^[16].
IR complete: ^[17] Although IR divergences are normally considered to be less serious than ultraviolet ones, they certainly cannot be ignored.
UV/IR Mixing: ^[18] This is intimately related to the `Breakdown of EFT’ in the previous section. Unlike in traditional EFTs, quantum gravity theories have black holes, and if you go deeper into the UV by giving more energy to a small region, then you create a black hole by gravitational collapse, whose size grows deeper into the IR as you give more energy. This suggests that in quantum gravity, low-energy phenomenology might be impacted by the UV physics.
Background independence:~^[19]. Some approaches, like LQG LQG, CST CST, and CDT CDT, are manifestly background-independent. Some versions of string theory, like the worldsheet perturbation theory and the BFSS/IKKT Matrix theory, are not background-independent. This problem of string theory is somewhat solved by the AdS/CFT correspondence, see section 6.2 in ^[20] and also by String Field Theory, see the ``Background independence’’ chapter in ^[21].
Holography: ^[22]. Historically, holography was first argued~^[23] based on black hole thermodynamics without invoking string theory.
Breakdown of classical spacetime: Explained in previous section.
Causality: Chronology protection conjecture is expected to be true in quantum gravity.
Black hole entropy: Quantum Gravity should give a microscopic explanation for $S_{BH}$. String theory explains the thermodynamic Bekenstein-Hawking entropy of extremal supersymmetric black holes by giving a microscopic statistical interpretation, see Statistical. This string theoretic derivation matches the Bekenstein-Hawking entropy at leading order and also provides higher-order quantum corrections. Although Loop Quantum Gravity can explain black hole entropy even without needing assumptions like extremal or supersymmetric, their derivation needs fixing the Immirzi parameter so that the proportionality constant matches with the $\frac{1}{4}$ in the Bekenstein-Hawking formula, making the LQG derivation (see LQGBlackHole) much less impressive than the string theoretic derivation.
Unique quantization: It’s not like the universe started as a classical theory, and then someone quantized it. But in physics, we always start with classical theories and then quantize them. Unlike generic classical theories, we expect the final theory to have a unique quantization (it can be expressed in various physically inequivalent formalisms). For example, we expect there to be a unique quantum theory for 11D SUGRA, which is called M-theory.
No global symmetries: See swampland.
Completeness of spectrum: See swampland.
Weak gravity conjecture: See swampland.
Moduli space is non-compact and simply connected: See swampland.
Infinite tower of states (Distance conjecture): See swampland.
Finiteness and string lamppost: See swampland.
Uncertainty in the causal structure and light cones: See uncertainty.
No quantum metric: ^[24]
No test mass limit: ^[25]
**$\Lambda$ and $G$ are not running parameters **: ^[26]
Decoupling: ^[27]
Uniqueness \& No Free Parameters: We don’t expect the final theory to be deformable into many other theories that are also consistent. General relativity can be deformed into any theory that is a subset of \href{https://en.wikipedia.org/wiki/Horndeski
Incompleteness ~^[28]: Physical theories are axiomatic theories. For example, the Ehlers-Pirani-Schild formalism is an axiomatic form of General Relativity. Because of Gödel’s 2nd incompleteness theorem, any axiomatic framework has some undecidable problems. See Tachikawa’s paper ^[29] for undecidable problems in QFT where he constructs a Wess-Zumino model which breaks supersymmetry if and only if ZFC is consistent, which can neither be proved nor disproved. But the final theory is expected to give precise solutions to every physically relevant situation. So, we expect that such problems are not physically possible, and the subset of questions that are physically relevant/possible is decidable.

\subsubsection{No local observables} ~^[30]

Every physical theory from Newton has diffeomorphism invariance, as explained in the beginning of the General relativity chapter in my \href{http://ksr.onl/FP}{FP notes}. For example, Newtonian gravity is a gauge-fixed version of the Newton–Cartan theory. Any local quantity like a scalar $R(x)$ is not diffeomorphism invariant under $x\to x’$. This fact is true in any physical theory. So, in principle, we should always work with diffeomorphism invariant observables, and there are no local diffeomorphism invariant observables. In various theories, this problem is resolved in various ways:

Non-dynamical spacetime theories: When the space(time) is non-dynamical and is $3D$ Euclidean or $4D$ Minkowski, we can always globally gauge fix this freedom by choosing global Cartesian coordinates. After spacetime coordinates are gauge-fixed, we can work with local non-diffeomorphism invariant objects. We no longer care about diffeomorphism invariance.

Examples: Every theory except relativistic gravity theories. Once we gauge fix Newton–Cartan theory to obtain Newtonian gravity, we can work with local observables. In QFT, we gauge fix the metric to Minkowski metric, and after the gauge fix, we can work with local observables.
Dynamical spacetime theories: But, when relativistic gravity is present, the background is dynamical and we cannot globally gauge fix to a nice global coordinate system. ``It is important to realize that it is dynamical gravity that is crucial for this conclusion, and not just the reparameterization invariance of the theory” from ^[31]. Local observables in relativistic gravity are forced to exist in the asymptotic boundary of space-time because diffeomorphism gauge transformations die off asymptotically; in the Minkowski case, observables are S-matrices, and in AdS, the observables are boundary observables (which behave like well defined CFT operators, see AdS/CFT Holography). Only in the asymptotically AdS case it is well understood what the observables are. Even though they are not diffeomorphism invariant, local bulk degrees of freedom do exist. Note that much of what is written below is not specific to quantum gravity but is even true for observables in General Relativity.
Local observables: Asymptotic local observables that are very close to the boundary are well-defined because diffeomorphism gauge transformations die off asymptotically. But what about the local degrees of freedom that are not near the asymptotic boundary? Even if they are not near the boundary, we can still define local operators using the help of gravitational dressing~^[32]. Check ^[33] for many references.

Examples: In HKLL reconstruction, we consider local operators that are not near the asymptotic conformal boundary; gravitational dressing is often implicit.
Nonlocal diffeomorphism invariant observables: When relativistic gravity is present, we can also work with observables that are not near the boundary by considering non-local diffeomorphism invariant observables. These are not independent observables and are completely fixed by the local observables at the asymptotic conformal boundary.

Example: The dual of mutual information. It is a well-defined non-local diffeomorphism invariant quantity. Note that the surfaces (\& their coordinates) are not diffeomorphism invariant, but their areas are diffeomorphism invariant. Unlike normal von Neumann entropy, see HEE, the mutual information is IR finite because the divergences in areas cancel out. It is also UV finite (in terms of matter fields, not in terms of gravity) if we consider quantum extremal surfaces QES instead of classical extremal surfaces. So, this observable is not at the asymptotic conformal boundary, but it is still well-defined. But, of course, it is not independent because the entanglement between asymptotic local observables determines it.

\subsection{QFT in curved spacetime} \etocsettocdepth.toc{paragraph}

^[34]
Even though Hawking radiation has not yet been experimentally observed, almost every reasonable physicist believes in its validity. So, this section could be, in principle, moved to my \href{http://ksr.onl/FP}{FP notes}. But it is highly relevant to quantum gravity.

\subsubsection{Unruh radiation in flat space} \subsubsection{Hawking radiation}

Membrane paradigm:

\subsubsection{The information paradox} \paragraph{The small corrections theorem}~
^[35] \paragraph{AMPS firewall}~\

Cosmological information paradox: ^[36]

\subsection{GR as a QFT}

\subsubsection{Perturbative} ^[37] and section 2 of ^[38] and sections 1 to 6 of ^[39]. \subsubsection{Euclidean gravitational path integral} ^[40] \subsubsection{Gravitational instantons} ^[41]
\subsection{GR as an EFT} Sections 7 to 12 of ^[42] and ^[43] and section 3 of ^[44]

\subsubsection{Universal quantum non-relativistic correction} \subsubsection{Uncertainty in the causal structure \& light cones} ^[45]
The precise causal structure of different quantum gravity proposals is different, as shown in fig:fuzzylightcone. Here, we consider the leading order correction from EFT that is independent of the UV completion of the theory ^[46].

Break down of EFT

~^[47]^[48] We will come to this later in the Swampland section.

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See ^[49] for an example. ↩