Unifying the Universe

Thinking up a single theory that explains everything within the physical world is what seems to drive several theoretical physicists. From the dynamics of the largest cosmological structures down to the twists and turns of quantum systems at the smallest of scales.

The leading position in the search for a unifying mathematical scheme is arguably occupied by string theory — an overarching theoretical framework that conjectures one-dimensional vibrating strings as the most fundamental components of the Universe.

This article throws some light on another, more controversial contender, namely Garrett Lisi’s Exceptionally Simple Theory of Everything.

Unifying What?

The toughest hurdle along the way towards unification is supposedly epitomized by the quest of consistently marrying quantum mechanics and Albert Einstein’s theory of general relativity across the entire energy spectrum.

The natural laws that predominately describe the behaviour of systems at short distance scales (atomic and subatomic levels) are called quantum mechanics, and they underpin the workings of the Standard Model of Particle Physics, which brings together all currently known particles and their interactions — except for gravity — under one theoretical roof.

In fact, quantum mechanics is only part of the full conceptual framework that underlies the Standard Model, i.e., quantum field theory, which envisages particles as energy fields stretched across space rather than point-like objects. According to this theory, a particle is manifested as a local excitation of its accompanying quantum field. The various fields that are incorporated within the Standard Model are fermions (the matter particles), gauge bosons (the carriers of the forces), and the Higgs field.

Fermions can be subdivided into leptons (the electron, the muon, the tau, and the corresponding neutrinos) and quarks (there are six: up, down, charm, strange, top, and bottom). Moreover, the leptons and quarks are set up in three generations (see Fig. 1) — the masses generally increase as one moves up a generation in ascending order. Given the high stability of first-generation fermions, ordinary matter is made exclusively of electrons, up quarks, and down quarks.

Gauge bosons organize the interactions between fermions — and sometimes among themselves — and are referred to as force carriers since they mediate the forces (the interactions) between fermions. In that respect, the Standard Model accounts for three of the four fundamental forces of nature: electromagnetism (mediated by photons), the weak force (mediated by W and Z bosons), and the strong force (mediated by gluons) — gravity being the fourth one and is not part of the Standard Model.

All the particles in the Standard Model furthermore possess an anti-particle — they have the same quantum properties as the original particle but with opposite electric charge; some particles are their own anti-particle — and most of them come in left- and right-handed forms — the exact form is determined by the way in which an intrinsic property of the particle (the spin) is aligned with its direction of motion.

Finally, the Higgs particle is responsible for endowing all the fermions — excluding the neutrinos — and gauge bosons with mass (and in some cases, a zero mass).

At larger distance scales, the dynamics of physical systems are primarily explained by the concept of gravity, and the natural laws that matter the most towards this end of the spectrum are those described by the theory of general relativity.

In a nutshell, general relativity reveals that the quantity of mass present in a certain volume of space (the energy density) as well as the pressure that particles exert, can bend our four-dimensional spacetime (that is, three spatial dimensions and one time dimension). Conversely, a curved spacetime informs matter which trajectory to follow. The larger the mass or pressure, the stronger the curvature of spacetime.

The effort of merging quantum field theory and general relativity constitutes an as yet insurmountable hurdle when we move to arbitrarily high energies, i.e., at and beyond the Plank scale (the shortest of distances at which our current physical laws are no longer valid), because there is for the moment no agreement on the theory of quantum gravity,

Such a theory would show how gravity behaves at short distances — remember that ever shorter distance scales (the realm of quantum mechanics) are equivalent to ever higher energies and that, as per general relativity, gravity is non-trivial in situations where extremely high energy densities are involved. Two cases where this becomes particularly relevant include the innermost part of a black hole or the moment at which our Universe is created.

Therefore, a theory of everything would be able to coherently fuse quantum field theory with general relativity across the full energy spectrum. In other words, such a unified theory ought to make clear not only how to deal with gravity at quantum scales but also how quantum gravity is joined with the three fundamental forces of the Standard Model.

Let us turn now to Garrett Lisi’s theory of everything.

Back to Our Roots

Garrett Lisi’s Exceptionally Simple Theory of Everything (a.k.a. E₈ Theory) combines all the known fields and their interactions into one geometrical structure, called an E₈ principal bundle connection. This mathematical object together with its curvature clarify how the E₈ smooth manifold geometry — more concretely, the associated E₈ root system — can reproduce all fields and interactions in our familiar four-dimensional spacetime.

Down the Rabbit Hole

In mathematics, a smooth manifold is considered a space of any number of dimensions which can be locally approximated by Euclidean geometry — for instance, the Earth’s surface is a manifold since, locally, we experience it as a flat, two-dimensional plane, even though it really is the surface of a three-dimensional sphere, which falls out of the scope of Euclidean geometry. To be more precise, the underlying geometry of E₈ Theory is a Lie group, which is a smooth manifold that is equipped with group operations, such as multiplication and taking the inverse.

Mathematics furthermore tells us that a root system is a set of vectors (such a set is also called a vector space) that span a finite-dimensional Euclidean space and come with certain geometrical properties, including a particular symmetry which leaves the root system unaltered under reflections (that is, mirror images) — a vector is a physical quantity that exhibits both a direction and magnitude (velocity is an example, unlike temperature, which only has a magnitude and therefore known as a scalar) and in a root system vectors are referred to as roots.

Specifically, the E₈ root system consists of 240 roots of equal length, which span an eight-dimensional Euclidean space. Stated in a different manner, E₈ can be viewed as a geometrical object of 248 dimensions in mathematical space or, alternatively, as an eight-dimensional object containing 248 symmetries. In addition, this root system is deemed irreducible (fundamental, if you will); it cannot be assembled from root systems with a lower dimension.

What is more, being also a simple Lie algebra, which is special kind of vector space embedded within its Lie group, the E₈ root system is classified as one of the five exceptional categories of simple Lie algebras. Hence the tongue-in-cheek name the Exceptionally Simple Theory of Everything.

According to general relativity, a connection is a mathematical object that transports vectors between nearby points of the manifold — more accurately, between the tangent spaces of these points. By relying on these connections, the curvature of the manifold then spontaneously manifests itself by means of the Riemann tensor — a tensor is the generalization of a vector.

If we place the concept of a connection in the context of Lie groups and Lie algebras, then we obtain something named the principal bundle connection, which is the starting point of the E₈ Theory. One way in which this theory reflects mathematical unification is the fact that the geometry of general relativity is absorbed within principal bundle geometry. This means that the notion of curvature within general relativity can be recovered from the equations of E₈ Theory.

If we put the above mathematical machinery to work, this unifying 248-dimensional geometrical framework spits out all the particles and the interactions of the Standard Model as well as the gravitational field, as schematically depicted by Fig. 3.

Lisi’s model also purportedly unveils a symmetric link between the three generations of fermions (see Fig. 1) called triality. This symmetry involves rotations by 120° that leave the E₈ root system invariant, and fermions of different generations are related by what is known as triality partners. Lisi himself admits, however, that the nature of the relationship between physical particles and triality partners is not yet entirely understood.

Another aspect of the E₈ Theory is that it predicts the existence of 20 new particle fields. Two of them (designated as Pati-Salam particles; the two white-coloured circles in Fig. 3) are acting on right-handed fermions whereas the remaining 18 particles (they are scalar fields; the red, blue, and green squares in Fig. 3) come in three generations and interact with both quarks and leptons — these 18 particles might elucidate the differences in mass among the three generations of fermions.

Moreover, it turns out that some of these new particles may qualify as axions, which are a candidate for dark matter — this unknown type of matter would explain, for instance, why rotating galaxies have not yet flown apart. It must be pointed out though that so far none of these new fields have been detected.

Fig. 4 — The coupling constants for the four interactions (relative values indicated between parentheses). The Planck scale sits at the energy levels where all the four forces would combine into a theory of everything (TOE). [EM = electromagnetism; GUT = Grand Unification Theory]. (Source: Adapted from jobilize).

Apart from unifying all fields and interactions, an additional perk that emerges from this all-encompassing geometric description of nature includes the unification of the coupling constants at high energies — these constants represent the relative strength of the four fundamental forces (interactions). Put differently, the forces converge to the same strength and become unified into one single force.

For the electroweak interactions (electromagnetism and the weak force combined) and the strong interactions, this convergence would occur at the grand unification (GUT) scale, whilst we would need to push the energies all the way up to the Planck scale in order to factor in gravity as well (see Fig. 2 and Fig. 4).

What is more, E₈ Theory suggests that the Higgs field gives rise to both a positive cosmological constant and the masses of the particles. The cosmological constant discloses the amount of the energy density of empty space (vacuum), and observations indeed hint at a (small) positive value. Not only that, this constant has been associated with dark energy, which accounts for the accelerated expansion of our Universe.

Here Come the Critics

Notwithstanding this unifying effort, not everyone is enticed by the theory’s explanatory power. The most elaborated criticism is formulated by Jacques Distler and Skip Garibaldi who present a mathematical proof showing that the attempt of embedding gravity and the Standard Model within the E₈ framework fails to deliver a sufficient number of roots (also called weight vectors) to fully reconstruct the three generations of fermions of the Standard Model.

Phrased in another way, Distler and Garibaldi claim that such embedding results in the production of non-chiral matter (also known as mirror fermions), which is not what we observe in our physical reality — chirality means that left and right representations of fermions couple differently to the forces (or simply put: a chiral fermion is not identical to its mirror image).

Fig. 5 — The mirror image of a chiral fermion is not identical to the original fermion, as its spin is reversed. (Source: Adapted from Matthew Francis).

According to Sabine Hossenfelder, before E₈ Theory can be used to make predictions, at least two issues must be smoothened out: a cosmological constant with a too large a value and the absence of a dynamical symmetry breaking mechanism — roughly speaking, symmetry breaking refers to the situation whereby a high-energy symmetric system evolves into more definite, low-energy states, e.g., the Higgs field is responsible for breaking the symmetry of the electroweak interactions and subsequently giving mass to the W and Z gauge bosons (which can be regarded as the definite states of the symmetry breaking).

Hossenfelder also draws attention to the fact that E₈ Theory says nothing about how the concept of gravity would be translated into the language of quantum mechanics. In fact, although the E₈ framework is compatible with the methods of quantum field theory, Lisi acknowledges that establishing a quantum description of his model currently remains an open question.

Some more pushback is expressed — with some voices more pronounced than others — by John Baez, who conveys some doubts about the choice of the mathematical methods put in place to combine fermions and bosons; by Graham Collins, who goes one step further by reporting that the joining of fermions and bosons cannot possibly be done in a Lie group as it lacks the appropriate, detailed mathematical foundation; by Peter Woit, who states that the underlying problems are not being addressed because invoking large Lie algebras just replaces the issue of unification with the question of what breaks the large symmetry; and by Lubos Motl, who idiosyncratically disagrees with the whole approach.

There is even a bachelor student, Kevin Dijkstra, who uses his bachelor thesis (under the supervision of P.M. Visser) to demonstrate the failure of identifying an adequate mathematical map between the Standard Model and the E₈ root system such that the four charges of the three fundamental forces of the Standard Model correspond to a sufficient number of vectors in E₈ — these four charges are the weak isospin charge (related to the weak force), the weak hypercharge (related to the electromagnetic force), and the two colour charges g3 and g8 (related to the strong force).

A Theoretical Renaissance

Nevertheless, there are also researchers who are polishing up the E₈ geometrical model by tweaking and recalibrating different parts of the structure. For instance, Lee Smolin, Simone Speziale, and (obviously) Garrett Lisi confront the concern with dynamical symmetry breaking — one of the main obstacles brought up by Sabine Hossenfelder — making equally sure that gravity can be coherently coupled to the other forces.

To answer to the objection of aggregating fermions and bosons into one mathematical structure, Urs Schreiber and other mathematicians examine the feasibility of using this liaising structure and do in fact not rule out its application. From a different perspective, Lee Smolin explores an alternative approach for the incorporation of fermions within E₈ Theory, based on a specific notion of locality (referred to as disordered locality) that is borrowed from one of the principal candidates for the theory of quantum gravity, i.e., loop quantum gravity.

Roberto Percacci furthermore shores up the consistency of Lisi’s theoretical framework by pinpointing a loophole in an important mathematical theorem (the Coleman-Mandula theorem). That is, since the vacuum state of E₈ Theory — a vacuum state is a state of minimal energy, a.k.a. the ground state — is of a different kind (i.e., de Sitter space) than the one postulated in the assumptions of the theorem (i.e., flat space), the author maintains that at the moment of unification (before symmetry breaking) E₈ Theory falls beyond the scope of the theorem, thus clearing the way for gravity to be unified with the other forces.

In a more general context, Roberto Percacci and Fabrizio Nesti propose that fermions can retain their essential characteristic of chirality even when moving towards higher levels of unification. As this goes straight to the heart of the criticism articulated by Distler and Garibaldi, these findings might help to solidify E₈ Theory.

As a matter of fact, building on the insights of Percacci and Nesti, Garrett Lisi goes on to show in another researcher paper that the act of inserting gravity and the Standard Model into E₈ holds water mathematically. Even though the model still generates mirror fermions (non-chiral matter), Lisi surmises that they could nonetheless exist in nature and possess high masses (which explains why they have not been detected yet) and potentially play a role in the explanation of the existence of the three generations of fermions interconnected via triality symmetry (which is considered rather speculative).

Unifying Reverberations

All these discussions around and reactions to Lisi’s E₈ Theory have also inspired several researchers to develop or expand other unifying theoretical frameworks to describe our physical world. Matti Pitkänen, for one, investigates how the element of unification offered by E₈ Theory could interrelate with the theory of topological geometrodynamics — this is the study that explicates spacetime solely in terms of geometry (this explicit model deviates from general relativity) and wishes to merge the four forces.

While Pitkänen argues that Lisi’s model cannot be saved, it would however bring insights into a particular symmetry between eight-dimensional geometries and eight-dimensional numbers (called octonions).

Another bolt of inspiration, this time striking Shuichiro Teramachi, leads to the design of a unified field theory — published in a paper titled ‘An Exceptionally Beautiful Theory of Everything’, as a pun on the title of Lisi’s original paper — that amalgamates conceptual ideas presented by Garrett Lisi and Cohl Furey.

By leaning on Furey’s octonionic ladder system — which is based on mathematical operators (ladders) used in quantum field theory — and Lisi’s E₈ Theory, Teramachi aims for a unification of the Standard Model and quantum gravity (which would answer one of the criticisms on Lisi’s work enunciated by Sabine Hossenfelder). In his model, the link between fermions and bosons is accomplished by supersymmetry rather than triality (as per Lisi’s theory).

A third unifying idea, elaborated by Raymond Aschheim, posits that fundamental elements of pure information (called trivalent graphs) embedded in a four-dimensional diamond-like geometry give rise to the 240 E₈ roots uncovered in Lisi’s framework, thereby reproducing all particles and interactions. In addition, general relativity can be recovered by weaving loop quantum gravity into this model.

In Aschheim’s information-based lattice structure, the information is carried by bits — which hold a value of either 0 or 1 — that sit at the nodes or connection points within the trivalent graphs. As a result, a space equipped with curvature can be constructed, emulating a gravitational field.

Raymond Aschheim also works with Klee Irwin and Marcelo Amaral to tie aspects of the E₈ unification program with the geometric structure of loop quantum gravity — this structure is referred to as spin foam — with the objective of devising a unified framework for quantum gravity.

Exceptionally Infinite

Perhaps all the efforts hitherto invested to obtain a theory of everything are in vain, if we heed the thoughts of Frank Close, who likens the levels of our physical reality with the layers of an onion and claims that their number might well turn out to be infinite.

Be that as it may, given our exceptionally developed predilection towards finding connections and patterns in the world around us, the human mind’s inquisitiveness will most likely continue to fuel our fascination with a theory of everything.

Let’s see what the next wave of inspiration brings for Garrett Lisi.

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