Physics explains how our material Universe works. However, its theories do not currently form a satisfying whole. For example, the theory that describes the fundamental forces and building blocks of our physical world excludes gravity, and we do not comprehend how the Universe was born. As we naturally strive for things to be complete and make sense, we ask that our theoretical conjectures add up too, regardless of the scale we are dealing with.
For this reason, string theory has set the objective of integrating all the current explanations of how our Universe functions, from the very small to the very large, into one coherent Theory of Everything.
Yuval Noah Harari points out in his book ‘Sapiens: A Brief History of Mankind’ that unification is baked into human nature. Historically, he argues, it is the triad of capital, empires and religion that was the unifying driving force behind the global society that we see today. Is it then not only logical that we also pursue the quest of formulating a theory that puts all the physical mechanisms of our Universe under one roof?
But is this even a feasible and realistic goal?
What Is the Problem?
The fate of a Theory of Everything (TOE) lies in the hands of our ability to theoretically pull together all known fundamental forces in our Universe and match this effort with valid experimental evidence.
There are four fundamental forces in nature explaining all interactions between particles — electromagnetism, the weak nuclear force, the strong nuclear force and gravity. The electromagnetic force acts on any particles that carry an electric charge; the weak force comes into play when particles radioactively decay; the strong force holds the protons and neutrons in the atomic nucleus together and equally ensures that the quarks within the protons and neutrons create a tight structure; and gravity is the result of the bending of spacetime due to the presence of a massive object.
Particle physics covers the first three of them, succinctly meshed together in its Standard Model of Elementary Particles. And it is quantum field theory, which builds on quantum mechanics, that describes the behaviour of these subatomic particles.
Fig. 1. The four fundamental interactions in nature: gravity, the strong force, the weak force and electromagnetism. (Source: nobelprize.org).
The most obstinate hitch in designing such overarching theory boils down to the fact that the smallest, discrete scale of reality, in which the probabilistic rules of quantum mechanics hold sway, cannot be coherently coalesced with its largest, continuous scale, in which gravity has the upper hand and most accurately described by Albert Einstein’s classical theory of general relativity.
The academic field of quantum gravity takes great pains to find a mathematically meaningful connection between these two extremes. In other words, this discipline wants to know how gravity behaves within the realm of the very small. Although quantum gravity is gaining traction, it is all but obvious how to go about merging — and whether they should be merged — these two well-established and allegedly juxtaposed physical theories.
Some physicists regard quantum mechanics as more fundamental than general relativity. In his book “Quantum Mechanics: The Theoretical Minimum”, Leonard Susskind states it strongly: “Quantum mechanics is the real description of nature. Classical mechanics, while beautiful and elegant, is nevertheless an approximation.” Others, such as Andrzei Dragan and Artur Ekert, call such statements into question.
Once a quantum theory of gravity is put in place, the final task towards establishing a TOE is to unite it with the three other fundamental forces. String theory addresses these two steps simultaneously.
Before we delve into quantum gravity and string theory, let us first of all understand to a greater depth why theoretical physics up until this day is immersed in her quest for a TOE.
Historical Unifying Trend
The idea that a universal theory or concept underlies our entire physical world dates back arguably to the fifth century B.C., when the pre-Socratic philosophers Democritus and Leucippus introduced a first atomic model. Fast forward to the eleventh century A.D., during which astronomer and mathematician Abu Ali al-Hasan ibn al-Haytham maintained that sunlight exists of small particles moving at a constant velocity — a notion that we still use today.
A next temporal leap lands us into the seventeenth century, a time in which Pierre Gassendi, Robert Boyle, and even René Descartes, propagated mechanical philosophy. According to this line of thought, any natural occurrence could be reduced to the motion of physically interacting particles — mediated by so-called contact forces.
Later in that same century, Isaac Newton synthesized Galileo Galilei and Christiaan Huygens’ conception of uniform gravity, Johannes Kepler’s laws on the orbital motion of planets in our solar system as well as the theory of tides into his law of universal gravitation.
In the first half of the nineteenth century, mathematician Pierre Simon Laplace alluded already with a pinch of foresight — or, if you will, a tad of elusive hope — to an intellectual tour de force that “would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom”. In the second half of that century, James Clerk Maxwell brought together several laws of electricity, magnetism and optics into his theory of electromagnetism.
In the following century, mainly during the interwar period, much scholarly fervour was devoted to working out a classical unified field theory — an endeavour that has never fully been accomplished. Such theoretical puzzle would have arisen from fitting together the pieces of electromagnetism and Einstein’s theory of general relativity. The scientists who spent considerable amounts of time on this dauntless exploration include Hermann Weyl, Theodor Kaluza, Arthur Eddington, Gunnar Nordström, Rudolf Förster alias Bach and Albert Einstein.
Oskar Klein, however, further developed Kaluza’s findings, what culminated in the Kaluza-Klein theory, the harbinger of string theory. In its original form, Kaluza’s theory expanded general relativity with an additional dimension of space, thereby naturally obtaining the equations of electromagnetism (there were unresolved issues though). In order to bring quantum mechanics to the party, Klein conjectured in a next step that the cylinder-shaped fifth dimension was extremely small and wrapped up on itself. So far, no experiment has been able to observe such extra dimensions.
With the discovery of the strong and weak nuclear force, theoretical physics shifted its time and energy away from classical unification — consolidating electromagnetism and gravity — towards the union of these two new forces and electromagnetism. As a TOE requires the incorporation of quantum mechanics from the outset, pursuing the merger of these three forces — the strong and weak nuclear force and electromagnetism — appeared more promising given that their common language is quantum mechanics.
And correcting its course indeed paid off. In 1968, Sheldon Glashow, Abdus Salam and Steven Weinberg combined the weak nuclear and electromagnetic interactions into one force, i.e. the electroweak force, which was experimentally confirmed in 1983 with the discovery of the weak force carrying particles W and Z bosons. All of this entails not only that, at high enough energy, these two exchange particles are ultimately identical, but also that the electromagnetic and weak interactions emanate from a single force — the electroweak force.
If we are to have a complete TOE, what remains open until today is two challenges: fusing the electroweak with the strong nuclear force into the electronuclear force and uniting gravity with the three other fundamental interactions.
Many Grand Unified Theory (GUT) models hypothesize the first challenge, i.e. tying the electroweak to the strong nuclear force. Especially when accompanied by the detection of supersymmetry, GUTs would increase the chances for a TOE to see the daylight. Intriguingly, the lightest supersymmetric particles that are predicted make at the same time very appealing candidates for dark matter.
A prerequisite for the second challenge — joining gravity with the three other forces — is to have before all else a quantum theory of gravity in place, since quantum mechanics constitutes, as mentioned earlier, the common groundwork for the electroweak and strong forces. Mind you, the study of quantum gravity is first and foremost concerned with investigating the quantum behaviour of the gravitational field, not necessarily with bundling together gravity and the remaining forces.
In the next section, we turn our attention to the essential precondition for the second challenge — articulating a theory of quantum gravity.
In view of all of the above, it becomes clear that theoretical physics perseveres with the search for a TOE, precisely because its history reads as if unveiling a converging trend towards an all-encompassing theory. It is simply too compelling to not pursue the quest for its holy grail.
Fig. 2. The four forces separated out moments after the Big Bang. Note that the Large Electron-Positron Collider (LEP) at the European Organization for Nuclear Research (CERN) was dismantled in 2001 and subsequently replaced by the Large Hadron Collider (LHC). (Source: Arip Nurahman).
Quantum Gravity
A quantum theory of gravity aims to consistently describe gravity through the lens of quantum mechanics. Such theory might help clarifying outstanding central questions, including wrapping our heads around the structure of spacetime at the smallest possible scale (the Planck scale), resolving spacetime singularities, i.e. points in space with infinite density (black holes), and coming to grips with the origins of our Universe.
A conundrum immediately arises in the attempt of devising such broad theoretical framework. In general relativity, gravity is caused by a curved and dynamic spacetime (the curvature is due to the varying distribution of energy and momentum in the universe). In contrast, quantum mechanics — and quantum field theory — postulates a flat and fixed spacetime. Coupling the two appears to be a mathematical enigma.
Notwithstanding, there seems to exist some leeway, and it starts with quantum field theory.
Quantum Field Theory and the Graviton
Quantum field theory is an amalgamation of classical field theory, Einstein’s special theory of relativity and quantum mechanics. It does not contemplate Einstein’s general theory of relativity.
According to this theory, virtual force carriers, a sort of exchange particles called bosons, are responsible for moving around energy throughout the various fields. That energy is quantized, meaning that a particle can be defined as a local quantum excitation of the respective field.
Quantum field theory meticulously shows that quantizing fields works fine for three out of the four forces. The force carrier of the electromagnetic field is the photon; for the weak interactions, these field excitations are the W and Z bosons; and the gluons are the exchange particles for the field of strong interactions.
A common entry point for the construction of a theory of quantum gravity consists of following the tenets of quantum field theory: it wishes to quantize the gravitational field.
As a matter of fact, a hypothesized force carrier for the gravitational field is the massless graviton, which we have not yet experimentally discovered contrary to the other exchange particles. In other words, even though we know what gravity does, we are still in the dark when it comes to what it is. Proving the graviton’s existence would indicate that gravity’s fundamental nature is quantum mechanical.
Fig. 3. Overview of theories contributing to a theory of quantum gravity based on the relevant universal constants c (the speed of light), h-bar (the Planck constant h divided by 2π) and G (the gravitation constant). (Source: Technische Universität Wien).
But there is a glitch in this quantum field approach to gravity. It turns out that we cannot find a way around infinities. In technical terms, the theories are not renormalizable. And this is a problem, for we need theories capable of predicting physical properties with concrete values, not infinite values.
To disentangle this geometric brainteaser, some theories adopt a whole different perspective on the problematic mathematical marriage of quantum mechanics and general relativity.
Generally, there are two main strands of thought in the running: loop quantum gravity and string theory. Unlike the former, the latter also takes on the TOE’s second challenge of unifying the four fundamental forces (see section “Historical Unifying Trend”), which is why we will focus on string theory next.
Untying String Theory
To be fair, there is a cornucopia of alternative theories of quantum gravity: causal fermion systems, the noncommutative standard model, causal set theory, emergent/entropic gravity, causal dynamical triangulation, asymptotically safe gravity, twistor theory and the topos theory approach, to name just a few. Still some other theoretical contenders think about chopping space in indivisible quantum units or viewing the Universe as an enormous single system. The former is what loop quantum gravity essentially does.
But back to string theory now.
String theory refines quantum field theory so that gravity is allowed a place in it. The refinement comes down to a geometrical makeover whereby the point particles are replaced by one-dimensional strings. The different modes of vibration of these strings then give rise to the different particles, both the ordinary matter particles (fermions) and the previously discussed force carriers (bosons), including the graviton.
That is to say, string theory, which is a quantum theory, has gravity intrinsically embedded within its theoretical framework.
Fig. 4. A sketch of how fundamental strings relate to ordinary matter. (Source: dreamstime.com).
An older version of string theory has two major issues: it does not comprise all particles (it only admits bosons) and it brings forth a particle with imaginary mass (tachyon). Fixing these incongruities results in a more mathematically stable and coherent solution that naturally yields a novel feature in string theory: supersymmetry. This is why the theory is often referred to as superstring theory. Basically, this characteristic denotes a spacetime symmetric relationship between bosons and fermions at higher energies, whereby a completely new class of particles is forecasted.
Because of the inherent properties of supersymmetry, string theory dictates the existence of six extra spatial dimensions. In this way, the strings can vibrate in the necessary number of dimensions so that the theory can properly spew out all the required particles. At this point, string theory explains our reality with the use of ten dimensions: one temporal dimension, the three familiar, macroscopic spatial dimensions and six hypothesized spatial dimensions.
But as we only experience three dimensions in space, string theory needs to identify a geometric model that accounts for the extra six dimensions. Enters the Calabi-Yau manifold. Leaning on the Kaluza-Klein theory, these dimensions are microscopically compactified on a topological space called the Calabi-Yau manifold and therefore imperceptible at human scale. Put in a different way, at every point in our visible spacetime there exists, invisible to the naked eye, a six-dimensional space in which the strings move around.
The specific choice of topology of these Calabi-Yau manifolds affects the vibrations of the strings and thus the kind of particle that is being manifested. Subsequently, as the additional dimensions can be curled up on themselves in a vast amount of ways, one consequence is that we get an immense string theory landscape of roughly 10⁵⁰⁰ possible candidates for a four-dimensional macroscopic universe, of which ours is just one of them.
Fig. 5. A cross-section of a 6D Calabi-Yau manifold. (Source: plus.maths).
String theory is actually a set of theories, which can be brought under five main categories — Type I, Type IIA, Type IIB, SO(32) heterotic and E₈xE₈ heterotic superstring theory. The differentiating criteria between all of them centralize around the presence of certain symmetries, whether they involve open or closed strings and whether the strings possess orientation.
Beside strings, another indispensable concept in string theory are the physical, extended objects called branes, which are analogous to point particles, yet only extrapolated to higher spatial dimensions. We can say that a point particle is a zero-dimensional brane, and a string a one-dimensional brane. In the case of open strings, string theory tells us that their ends must attach to a special sort of branes, i.e. D-branes, which can be thought of as submanifolds of the Calabi-Yau manifold.
D-branes play a key role in helping us understand how the five superstring theories relate to one another as well as how the fundamental forces in the Standard Model come about.
Fig. 6. A representation of 0-branes, 1-branes and D2-branes. (Source: profmattstrassler.com).
The unification of theory is tilted to the next level with the discovery of two kinds of dualities among those five types of string theory — a duality signifies the event whereby two different theories describe the same physical situation. S-duality points to correspondences between weak- and strong-coupled theories (these couplings reflect the strength of the fundamental force present in particle interactions), whereas T-duality lays bare geometrical equivalences between various theories. Fig. 7 depicts the two dualities relative to the different superstring theories.
These inter- and intra-theoretical correspondences pave the way for a single, overarching fundamental theory. In concrete terms, Type IIA and E₈xE₈ heterotic superstring theory are S-dually connected through an eleven-dimensional theoretical framework at higher energies, i.e. M-theory. At a lower energy scale, M-theory reduces to the theory of supergravity in eleven dimensions. Therefore, undergirded by a series of dualities, M-theory underlies both supergravity and the five superstring theories.
Fig. 7. Overview of the five 10-dimensional string theories as well as the 11-dimensional supergravity theory and the 11-dimensional M-theory. (Source: adapted from alexdoesphysics).
On top of that, there is another geometric duality from which string theory can benefit, technically denoted as the anti-de Sitter/conformal field theory (AdS/CFT) correspondence. This duality is just one particular illustration of a broader class of gauge-gravity correspondences, which suggest a relation between a theory of gravity and quantum field theory. The prevalent example of the AdS/CFT correspondence is the equivalence between Type IIB superstring theory with five-dimensional gravity and the N=4 supersymmetric Yang-Mills theory, which is an approximate quantum field theory that comes in handy to examine the Standard Model of particle physics.
One of the appealing features of the Ads/CFT correspondence is that it enables a possible way out for a concern raised earlier: How to match the mathematics of a curved and dynamic spacetime (as encountered in general relativity to account for gravity) with that of a flat and fixed spacetime (as assumed in quantum field theories)?
String theory/M-theory is arguably the most favourable candidate for a TOE at present, precisely because of all these dualities and correspondences, and because the theory inherently accommodates gravity.
Let us now dissect some of string theory/M-theory’s particular perks as well as its main shortcomings.
Strengths of String Theory
Decaying Infinities
By postulating one-dimensional strings instead of zero-dimensional point particles, the calculations of string theory have overcome several singularities. For one, a stringy Universe successfully deals with the static spacetime infinities emanating from within the topological spaces orbifolds and conifolds, which contribute to constructing a consistent string theory.
What is more, the theory does away with non-renormalizability. To refresh our memories, this mathematical condition refers in this context to the fact that describing gravitation in the language of quantum field theory generates obstinate infinities. It turns out that string theory allows for renormalizability, even with gravity in the picture.
Hence, string theory/M-theory is able to fundamentally explain gravity through gravitons, which mediate the gravitational interaction.
A Quantum Theory of Gravity
This means that string theory contains a theory of quantum gravity. Indeed, the mathematics making up string theory/M-theory has evolved over the years to form an overall consistent theory of quantum gravity, albeit not entirely complete (see below in section ‘Drawbacks of String Theory’) and only in ten or eleven dimensions.
Its consistency trait is to a great extent indebted to much work that has been done on symmetries and dualities, mostly around the early 1980s and the late 1990s, dubbed as the first and second superstring revolution, respectively. The first one marks the discovery of the five superstring theories and their ability to encompass all elementary particles. Building on these findings, it became clear during the second revolution that all of them are in some way interlinked, leading up to the more fundamental M-theory.
Unifying Forces
String theory/M-theory is thus a nominee for a TOE since it unifies quantum gravity with the three other fundamental forces. This convergence is reflected in string theory coughing up not only the matter particles and forces of the Standard Model but also the graviton. The way this proverbial regurgitation works is that the different quantum mechanical vibrating modes of strings propagate the manifestation of different particles (see section ‘Untying String Theory’).
When looking specifically at force carrying particles, a spin-2 massless closed string at a low-energy scale gives rise to a graviton, whereas spin-1 open strings harvest the exchange particles in the Standard Model. As a side note, spin is an essential, quantum mechanical characteristic of a particle, representing the state of an intrinsic magnet. It does not imply that the particle is actually spinning around some axis (which would be the case for orbital angular momentum).
Against the backdrop of this perk, it is useful to mention that string phenomenology is the field of study that builds models of particle physics basing off of the tenets of string theory.
Gravity on Equal Footing
Another advantage of string theory lies in its potential to possibly provide an explanation for one of particle physics’ long-standing puzzles: the hierarchy problem. In a nutshell, that enigma poses the question: Why is the gravitational interaction so much weaker in comparison to the three other fundamental forces?
The last column of Fig. 8 displays their relative strengths. For instance, it shows that the weak force is 10³⁴ times stronger — that is a whopping thirty-four zeros — than gravity and the strong force even 10⁴⁰ times.
One suggested workaround is supersymmetry — to brush up on our knowledge, this designates a symmetry between fermions (ordinary matter) and bosons (force carriers). Under such supersymmetric model, certain particle properties, such as opposite spins, ensure that quantum corrections cancel out, thereby effectively erasing the large discrepancy between gravity and the other forces (for more technical details, read about this here or here).
As we have seen that supersymmetry is innately ingrained within string theory, it then follows that string theory helps resolving the hierarchy problem.
Fig. 8. Characteristics of the four fundamental forces, including their relative strength (last column). (Source: adapted from paper S. Sahoo).
Microscopic View on Black Holes
In the words of Joseph Conlon, black holes embody “trapped regions of space, characterised by electric charge, mass and rate of rotation, from which nothing can escape”, not even light, due to their enormous gravitational pull. This is the fate that awaits matter as soon as it has crossed the black hole’s event horizon, i.e. its surface.
Einstein anticipated the existence and dynamics of black holes on the basis of his theory of general relativity. But what is lacking is their microscopic description. Enters entropy. In statistical mechanics, entropy stands for the number of microscopic configurations — so-called microstates — that a particular macroscopic thermodynamic system can embrace. It is popularly defined as the degree of disorder or randomness of a system.
Jacob Bekenstein and Stephen Hawking managed to link up thermodynamics with black hole mechanics, culminating in the Bekenstein-Hawking entropy. In other words, their specific definition of entropy determines the finite number of black hole microstates. But just what are these microstates in the case of black holes, and how are they counted?
String theory/M-theory demonstrates its strength by shedding light on these questions. For one type of theoretical black holes, the theory is namely able to produce an explicit number of string states (counted with the assistance of D-branes) that corresponds exactly to the number calculated by the Bekenstein-Hawking entropy.
Not All Is Lost in Black Holes
Another related issue confronted by string theory is the black hole information paradox.
Black holes keep all the information about the physical material that they slurped up during their lifetime solely for themselves. All infalling information — matter that has moved across the black hole’s horizon into its singularity — seems lost forever. And even though black holes apparently evaporate, as Stephen Hawking proclaimed, they only emit information in the form of thermal radiation (although not yet experimentally verified), which tells us nothing about the nature of infalling matter.
The paradox lies in the knowledge that the above scenario flies in the face of one of the most established theories in physics: Quantum mechanics postulates that information is never lost. So, it appears that a black hole is a place where gravity clashes with quantum physics.
As a theory of quantum gravity, string theory offers some relief. Calling upon the AdS/CFT equivalence, one solution in string theory tackles this theoretical riddle by insisting that the information about matter pulled into the black hole is not lost — it sits on the black hole’s boundary, i.e. the black hole’s event horizon where quantum mechanics reigns, akin to the holographic principle.
Fig. 9. Schematic view of the AdS/CFT correspondence and the holographic principle. (Source: J.D. Bekenstein).
Nevertheless, this account does not iron out one troubling aspect of the information paradox. Once the black hole has entirely vapoured away, the infalling matter has also simply vanished with it. What happened to the information that was residing on the black hole’s horizon?
Another solution, i.e. fuzzball theory, claims to have come up with an answer. By replacing the black hole by a giant fuzzball packed with strings (see Fig. 10), this fuzzy framework eliminates both the singularity and the event horizon. Consequently, the information cannot disappear in a singularity. Rather, while the fuzzball evaporates over time, the information gets nestled within the Hawking radiation and released back into the universe.
Fig. 10. A black hole with an event horizon and a singularity versus a fuzzball. (Source: Jennifer Ouellette (Quantamagazine)).
Beyond the realm of strings, loop quantum gravity equally supplies a remedy, completely doing away with the singularity at the black hole’s centre. Instead, the black hole originally shrinks to form a so-called Planck star, and then bounces back and emerges as a white hole, containing all the information that initially fell into the black hole.
Dark Matter Included
To stay in astrophysical spheres, string theory/M-theory can furthermore potentially enhance our comprehension of dark matter.
To properly grasp the reasons behind the presence and behaviour of galaxies, ordinary matter alone cannot do the job. Experiments reveal that, in order to hold the galaxies together, we need more mass than what we visibly observe. Thus, the missing piece of the puzzle seems to be gravitational effects prompted by an invisible kind of substance: dark matter. But the million-dollar question is: What exactly is dark matter?
There are plenty of candidates to take up the role of the newly discovered matter. As previously brought to the fore (see section “Historical Unifying Trend”), the lightest of the conjectured supersymmetric particles — classified as a weakly interacting massive particle (WIMP) — qualifies as the leading dark matter nominee.
As a result, since supersymmetry naturally flows out of string theory, it will come as no surprise that string theory/M-theory is a suitable theory to fundamentally explain dark matter. Moreover, it may be so that this new particle — manifested through heavy string modes — occupies higher spatial dimensions, as one explanation goes, with gravity providing the metaphorical bridge to our three familiar dimensions in space.
Notwithstanding these promising predictions, we have not been able to directly detect a dark matter particle so far.
Drawbacks of String Theory
Lack of Experimental Evidence
String theory/M-theory has hitherto not been confirmed by experiments, which are at the moment hunting for either cosmic strings, gravitons or supersymmetric particles. Experimental corroboration would be instructive in efficiently guiding theoretical physics in its pursuit for the theory of quantum gravity.
The trouble is that quantum gravitational effects occur at the Planck scale, a trillionth of a trillionth of the diameter of a hydrogen atom. Our earthly observational instrumentation is currently not equipped to inquire at such small scale, because the equivalent required energies far exceed the maximum limits of the present-day generation of particle detectors.
This is why studying the centre of black holes — and, more recently, it is also suggested its near horizon — or the early Universe is so interesting at present: gravity exhibits a strong effect on very small scales. Examining these phenomena might lay bare answers on how to coalesce the concept of gravity with quantum mechanics.
Many researchers have looked in sundry ways for indirect evidence of quantum gravitational effects. In one such approach, identifying the signature of primordial gravitational waves within the Cosmic Background Radiation, Lawrence Krauss and Frank Wilczek assert, would signpost that gravity is quantized. Another idea by John Estes et al. probes radio waves of neutron stars orbiting black holes to accumulate support for quantum gravity. Still another method pursued by Jahed Abedi et al. and Vitor Cardoso et al. consists of scrutinizing gravitational waves engendered by merging black holes.
A totally different technique is adopted by Chiara Marletto and Vlatko Vedral, Tanjung Krisnanda et al. and Sougato Bose et al., who use quantum entanglement correlations of gravitationally interacting objects to infer conclusions on quantum gravity.
Not So Universal?
As for the time being we cannot perform experiments at the energy levels crucial to validate string theory, there is a symmetric property quintessential in physics of which we are not sure that it holds at the scale of quantum gravity: Lorentz invariance. This type of symmetry refers to the notion that the laws of physics remain unchanged irrespective of when, where or by whom they are measured.
Nonetheless, it looks like that there is an alternative way to test this key characteristic by means of photons emitted by rotating neutron stars (pulsars). As a matter of fact, the preliminary results hint to the absence of symmetry breaking, indicating that Lorentz invariance holds for string theory.
Clueless about Fundamental Constants
Another downside of string theory/M-theory is that it fails to exclusively predict the Standard Model. Instead, it churns out a whole plethora of possible universes — earlier specified as the string theory landscape (see section “Untying String Theory”). In technical jargon, every such stable outcome is called a vacuum state.
String theory/M-theory is unable to calculate how to get our particular Universe out of this landscape. And considering that a unique set of fundamental constants comes with every vacuum state, it follows that string theory leaves us practically clueless on how to calculate the value of these constants.
For our Universe, the fundamental constants include the speed of light c, the gravitational constant G, Planck’s constant h, the masses of elementary particles, the cosmological constant Λ, the coupling constants for the forces and the mixing angles of quarks and neutrinos.
Some avow that the vast landscape opens the door to the anthropic principle — the constants have the value that they do because, in this way, they permit intelligent life to exist. Except that, as long as we are unable to explain their value by fundamental laws in physics, others will persist in perceiving these anthropic arguments as a failure of science.
Expanding Universe Unexplained
Let us single out one constant: the cosmological constant in Einstein’s theory of general relativity. This constant is a mathematical representation of dark energy, which is a gravitationally repulsive form of energy and the reason why the expansion of the Universe is speeding up.
The value of dark energy can be pinpointed by calculating the vacuum energy, i.e. the energy density of empty space, were it not for the staggering disagreement between observations and projections from quantum field theory: the value is off by 121 orders of magnitude, i.e. 10¹²¹.
String theory/M-theory does however deliver a cosmological constant, albeit with the wrong sign. To match our accelerating Universe we need a positive cosmological constant, whereas string theory gives us a negative one (on the grounds that string theory relies on an anti-de Sitter space — see section “Untying String Theory” — which comes with a negative cosmological constant). One proposed solution would be incorporating worm holes into string theory.
Against all odds, recent research carried out by Cumrun Vafa et al. casts doubt on whether we actually live in a universe with a positive cosmological constant. In more straightforward terms, the expansion of our Universe, Vafa et al. argue, might gradually slow down, a scenario that corresponds with a negative cosmological constant. If this hypothesis stands the test of time, it would eventually give a huge boost to string theory. Experiments — think of the Simons Observatory — will be conducted in the next few years to gather more evidence.
Fig. 11. Different models of the expanding universe: a decelerating universe, a coasting universe and an accelerating universe. (Source: extremetech.com).
Lingering Singularities
Also cosmological or spacetime curvature singularities, such as the initial singularity at the time of the Big Bang or the singularity at the centre of a black hole (let us disregard fuzzballs for a moment), continue to pose a problem for string theory/M-theory. Makoto Natsuume similarly reverberates: “[E]ven though various mechanisms are found to resolve singularities, it is not clear how to resolve the most important singularities — the Schwarzschild and the big bang singularities.” Note that Karl Schwarzschild was the first to determine the radius of a (non-rotating) black hole.
Concerning black holes, the AdS/CFT correspondence may prove useful to find a way out of this singularity conundrum (see the black hole information paradox in section “Strengths of String Theory”). As to the beginning of our Universe, one version of string cosmology suggests replacing the initial singularity — to reiterate, this is a spacetime event characterized by infinite curvature of the gravitational field — with a phase of maximal, finite curvature.
Incomplete Gravity Model
Despite the fact that string theory/M-theory encapsulates gravity, the conjectured theory of gravity does not come out as Einstein’s four-dimensional theory of general relativity. Instead, string theory produces the Brans-Dicke theory of gravity in ten dimensions.
Only after manipulating its coupling constant (letting it run to infinity) and compactifying the extra six dimensions (usually with the help of the Calabi-Yau manifolds) do we get general relativity. But so far, string theory has not satisfactorily achieved these goals.
Yet, there are accounts claiming that directly deriving general relativity from string theory is doable — see, for example, Huggett (2015) or Huang (2019).
Background Dependence
One important point of criticism has to do with the difference between a dynamic and a fixed spacetime in which theories operate.
Background independence means that a physical theory is independent of the spacetime geometry. That is, the theory does not rely upon the a priori existence of space and time. Rather, its equations yield several possible spacetime geometries. Such theoretical quality is deemed favourable, inasmuch as it would make the theory more fundamental.
General relativity, for instance, is background independent, since the theory describes what space and time are. In contrast, string theory is questionably background dependent, as it assumes the existence of space and time so that it can consider what strings are and how they behave.
Numerous attempts have been undertaken to couple string theory to a background independent design — typically in the context of string field theory — notably by Lee Smolin (2005), Washington Taylor (2006), Itzhak Bars and Dmitry Rychkov (2014) and Olaf Hohm (2018). Even so, Steven Carlip et al. point out that, actually, “it is disputable whether [string theory] can be genuinely formulated in a background-independent […] fashion.”
Unlike string theory/M-theory, loop quantum gravity develops a background independent theory by hypothesizing small quanta of spacetime, which could provide for the workings of gravity within the quantum domain. Other quantum gravity theories, e.g. causal sets and dynamical triangulations, are equally modelled against an independent background.
No End in Sight
In the search for the holy grail, one option is that we will never attain a TOE. In the world of mathematics, Kurt Gödel’s theorems of incompleteness tell us, in broad strokes, that, even if a theory is underpinned by true and consistent axioms, there will always be a statement that falls beyond the explanatory reach of the theory. Put differently, there cannot be a finite number of axioms explaining everything.
Applying Gödel’s thoughts to physics, his theorems basically reassure that the academic discipline of theoretical physics will exist indefinitely, albeit never reaching the proverbial shore. That is to say, in the event that we do obtain a comprehensive theoretical framework, it will always remain incomplete, a point also made by Stephen Hawking.
Connecting the Dots
If we accept Yuval Noah Harari’s premise that the dynamics of unification are historically deep-seated into our human nature, then it is not unlikely that we would continue to ignore Kurt Gödel’s theorems of incompleteness, embrace the unifying trend within the history of theoretical physics and strive for a Theory of Everything.
Perhaps we are missing the point altogether: What if both theories — quantum mechanics and the theory of general relativity — are not so different after all? Theoretical physicist Leonard Susskind, for one, affirms such musings in an interview with his colleague Sean Carroll:
“It was that, that made people think that there was this tension between quantum mechanics and gravity, and that there was a conflict between them. I think it’s turning out the other way. I think the point is that they are so closely connected, so almost the same thing that the idea of quantizing one of them just separates them too much.”
Be it as it may, possibly the most ironic feature of string theory/M-theory — the pinnacle of unification — is that it is touted as the leading contender for a Theory of Everything, no less, when all the while it spawns a vast array of universes with different laws of physics, rendering its original proposition of being an all-encompassing theory for the physical world we live in rather implausible, unless, of course, we expand our definition of ‘the physical world’ and subscribe to the notion that, beyond our own Universe, there are other universes to marvel at, too.
But we will keep that discussion for another time.
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A Brief History of Time Book By Stephen Hawking You can explain the complexities of cosmological physics with an attractive combination of clarity and ingenuity. . . . Theirs is a brain of extraordinary power
Olivier Loose
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