Grand Unified Theory

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Grand_Unified_Theory

 

Beyond the Standard Model
Simulated Large Hadron Collider CMS particle detector data depicting a Higgs boson produced by colliding protons decaying into hadron jets and electrons

A Grand Unified Theory (GUT) is a model in particle physics in which, at high energies, the three gauge interactions of the Standard Model comprising the electromagnetic, weak, and strong forces are merged into a single force. Although this unified force has not been directly observed, the many GUT models theorize its existence. If unification of these three interactions is possible, it raises the possibility that there was a grand unification epoch in the very early universe in which these three fundamental interactions were not yet distinct.

Experiments have confirmed that at high energy the electromagnetic interaction and weak interaction unify into a single electroweak interaction. GUT models predict that at even higher energy, the strong interaction and the electroweak interaction will unify into a single electronuclear interaction. This interaction is characterized by one larger gauge symmetry and thus several force carriers, but one unified coupling constant. Unifying gravity with the electronuclear interaction would provide a more comprehensive theory of everything (TOE) rather than a Grand Unified Theory. Thus, GUTs are often seen as an intermediate step towards a TOE.

The novel particles predicted by GUT models are expected to have extremely high masses—around the GUT scale of ${\displaystyle 10^{16}}$ GeV (just a few orders of magnitude below the Planck scale of ${\displaystyle 10^{19}}$ GeV)—and so are well beyond the reach of any foreseen particle collider experiments. Therefore, the particles predicted by GUT models will be unable to be observed directly, and instead the effects of grand unification might be detected through indirect observations such as proton decay, electric dipole moments of elementary particles, or the properties of neutrinos. Some GUTs, such as the Pati–Salam model, predict the existence of magnetic monopoles.

While GUTs might be expected to offer simplicity over the complications present in the Standard Model, realistic models remain complicated because they need to introduce additional fields and interactions, or even additional dimensions of space, in order to reproduce observed fermion masses and mixing angles. This difficulty, in turn, may be related to an existence of family symmetries beyond the conventional GUT models. Due to this, and the lack of any observed effect of grand unification so far, there is no generally accepted GUT model.

Models that do not unify the three interactions using one simple group as the gauge symmetry, but do so using semisimple groups, can exhibit similar properties and are sometimes referred to as Grand Unified Theories as well.

Unsolved problem in physics:

Are the three forces of the Standard Model unified at high energies? By which symmetry is this unification governed? Can the Grand Unification Theory explain the number of fermion generations and their masses?

History

Historically, the first true GUT which was based on the simple Lie group SU(5), was proposed by Howard Georgi and Sheldon Glashow in 1974. The Georgi–Glashow model was preceded by the semisimple Lie algebra Pati–Salam model by Abdus Salam and Jogesh Pati, who pioneered the idea to unify gauge interactions.

The acronym GUT was first coined in 1978 by CERN researchers John Ellis, Andrzej Buras, Mary K. Gaillard, and Dimitri Nanopoulos, however in the final version of their paper they opted for the less anatomical GUM (Grand Unification Mass). Nanopoulos later that year was the first to use the acronym in a paper.

Motivation

The supposition that the electric charges of electrons and protons seem to cancel each other exactly to extreme precision is essential for the existence of the macroscopic world as we know it, but this important property of elementary particles is not explained in the Standard Model of particle physics. While the description of strong and weak interactions within the Standard Model is based on gauge symmetries governed by the simple symmetry groups SU(3) and SU(2) which allow only discrete charges, the remaining component, the weak hypercharge interaction is described by an abelian symmetry U(1) which in principle allows for arbitrary charge assignments. The observed charge quantization, namely the postulation that all known elementary particles carry electric charges which are exact multiples of one-third of the "elementary" charge, has led to the idea that hypercharge interactions and possibly the strong and weak interactions might be embedded in one Grand Unified interaction described by a single, larger simple symmetry group containing the Standard Model. This would automatically predict the quantized nature and values of all elementary particle charges. Since this also results in a prediction for the relative strengths of the fundamental interactions which we observe, in particular the weak mixing angle, grand unification ideally reduces the number of independent input parameters, but is also constrained by observations.

Grand unification is reminiscent of the unification of electric and magnetic forces by Maxwell's theory of electromagnetism in the 19th century, but its physical implications and mathematical structure are qualitatively different.

Unification of matter particles

For an elementary introduction to how Lie algebras are related to particle physics, see Particle physics and representation theory.

 

Schematic representation of fermions and bosons in SU(5) GUT showing 5 + 10 split in the multiplets. Neutral bosons (photon, Z-boson, and neutral gluons) are not shown but occupy the diagonal entries of the matrix in complex superpositions

SU(5)

Main article: Georgi–Glashow model

The pattern of weak isospins, weak hypercharges, and strong charges for particles in the SU(5) model, rotated by the predicted weak mixing angle, showing electric charge roughly along the vertical. In addition to Standard Model particles, the theory includes twelve colored X bosons, responsible for proton decay.

SU(5) is the simplest GUT. The smallest simple Lie group which contains the standard model, and upon which the first Grand Unified Theory was based, is

${\displaystyle {\rm {SU(5)\supset SU(3)\times SU(2)\times U(1)}}}$.

Such group symmetries allow the reinterpretation of several known particles, including the photon, W and Z bosons, and gluon, as different states of a single particle field. However, it is not obvious that the simplest possible choices for the extended "Grand Unified" symmetry should yield the correct inventory of elementary particles. The fact that all currently known matter particles fit perfectly into three copies of the smallest group representations of SU(5) and immediately carry the correct observed charges, is one of the first and most important reasons why people believe that a Grand Unified Theory might actually be realized in nature.

The two smallest irreducible representations of SU(5) are 5 (the defining representation) and 10. In the standard assignment, the 5 contains the charge conjugates of the right-handed down-type quark color triplet and a left-handed lepton isospin doublet, while the 10 contains the six up-type quark components, the left-handed down-type quark color triplet, and the right-handed electron. This scheme has to be replicated for each of the three known generations of matter. It is notable that the theory is anomaly free with this matter content.

The hypothetical right-handed neutrinos are a singlet of SU(5), which means its mass is not forbidden by any symmetry; it doesn't need a spontaneous symmetry breaking which explains why its mass would be heavy.

SO(10)

Main article: SO(10) (physics)

The pattern of weak isospin, W, weaker isospin, W', strong g3 and g8, and baryon minus lepton, B, charges for particles in the SO(10) Grand Unified Theory, rotated to show the embedding in E₆.

The next simple Lie group which contains the standard model is

${\displaystyle {\rm {SO(10)\supset SU(5)\supset SU(3)\times SU(2)\times U(1)}}}$.

Here, the unification of matter is even more complete, since the irreducible spinor representation 16 contains both the 5 and 10 of SU(5) and a right-handed neutrino, and thus the complete particle content of one generation of the extended standard model with neutrino masses. This is already the largest simple group which achieves the unification of matter in a scheme involving only the already known matter particles (apart from the Higgs sector).

Since different standard model fermions are grouped together in larger representations, GUTs specifically predict relations among the fermion masses, such as between the electron and the down quark, the muon and the strange quark, and the tau lepton and the bottom quark for SU(5) and SO(10). Some of these mass relations hold approximately, but most don't (see Georgi-Jarlskog mass relation).

The boson matrix for SO(10) is found by taking the 15 × 15 matrix from the 10 + 5 representation of SU(5) and adding an extra row and column for the right-handed neutrino. The bosons are found by adding a partner to each of the 20 charged bosons (2 right-handed W bosons, 6 massive charged gluons and 12 X/Y type bosons) and adding an extra heavy neutral Z-boson to make 5 neutral bosons in total. The boson matrix will have a boson or its new partner in each row and column. These pairs combine to create the familiar 16D Dirac spinor matrices of SO(10).

E₆

Main article: E6 (mathematics)

In some forms of string theory, including E₈ × E₈ heterotic string theory, the resultant four-dimensional theory after spontaneous compactification on a six-dimensional Calabi–Yau manifold resembles a GUT based on the group E₆. Notably E₆ is the only exceptional simple Lie group to have any complex representations, a requirement for a theory to contain chiral fermions (namely all weakly-interacting fermions). Hence the other four (G₂, F₄, E₇, and E₈) can't be the gauge group of a GUT.

Extended Grand Unified Theories

Non-chiral extensions of the Standard Model with vectorlike split-multiplet particle spectra which naturally appear in the higher SU(N) GUTs considerably modify the desert physics and lead to the realistic (string-scale) grand unification for conventional three quark-lepton families even without using supersymmetry (see below). On the other hand, due to a new missing VEV mechanism emerging in the supersymmetric SU(8) GUT the simultaneous solution to the gauge hierarchy (doublet-triplet splitting) problem and problem of unification of flavor can be argued.

GUTs with four families / generations, SU(8): Assuming 4 generations of fermions instead of 3 makes a total of 64 types of particles. These can be put into 64 = 8 + 56 representations of SU(8). This can be divided into SU(5) × SU(3)_F × U(1) which is the SU(5) theory together with some heavy bosons which act on the generation number.

GUTs with four families / generations, O(16): Again assuming 4 generations of fermions, the 128 particles and anti-particles can be put into a single spinor representation of O(16).

Symplectic groups and quaternion representations

Symplectic gauge groups could also be considered. For example, Sp(8) (which is called Sp(4) in the article symplectic group) has a representation in terms of 4 × 4 quaternion unitary matrices which has a 16 dimensional real representation and so might be considered as a candidate for a gauge group. Sp(8) has 32 charged bosons and 4 neutral bosons. Its subgroups include SU(4) so can at least contain the gluons and photon of SU(3) × U(1). Although it's probably not possible to have weak bosons acting on chiral fermions in this representation. A quaternion representation of the fermions might be:

${\displaystyle {\begin{bmatrix}e+i{\overline {e}}+jv+k{\overline {v}}\\u_{r}+i{\overline {u_{r}}}+jd_{r}+k{\overline {d_{r}}}\\u_{g}+i{\overline {u_{g}}}+jd_{g}+k{\overline {d_{g}}}\\u_{b}+i{\overline {u_{b}}}+jd_{b}+k{\overline {d_{b}}}\\\end{bmatrix}}_{\rm {L}}}$

A further complication with quaternion representations of fermions is that there are two types of multiplication: left multiplication and right multiplication which must be taken into account. It turns out that including left and right-handed 4 × 4 quaternion matrices is equivalent to including a single right-multiplication by a unit quaternion which adds an extra SU(2) and so has an extra neutral boson and two more charged bosons. Thus the group of left- and right-handed 4 × 4 quaternion matrices is Sp(8) × SU(2) which does include the standard model bosons:

${\displaystyle {\rm {SU(4,H)_{L}\times H_{R}=Sp(8)\times SU(2)\supset SU(4)\times SU(2)\supset SU(3)\times SU(2)\times U(1)}}}$

If ${\displaystyle \psi }$ is a quaternion valued spinor, ${\displaystyle A_{\mu }^{ab}}$ is quaternion hermitian 4 × 4 matrix coming from Sp(8) and ${\displaystyle B_{\mu }}$ is a pure imaginary quaternion (both of which are 4-vector bosons) then the interaction term is:

${\displaystyle {\overline {\psi ^{a}}}\gamma _{\mu }\left(A_{\mu }^{ab}\psi ^{b}+\psi ^{a}B_{\mu }\right)}$

Octonion representations

It can be noted that a generation of 16 fermions can be put into the form of an octonion with each element of the octonion being an 8-vector. If the 3 generations are then put in a 3x3 hermitian matrix with certain additions for the diagonal elements then these matrices form an exceptional (Grassmann-) Jordan algebra, which has the symmetry group of one of the exceptional Lie groups (F₄, E₆, E₇ or E₈) depending on the details.

${\displaystyle \psi ={\begin{bmatrix}a&e&\mu \\{\overline {e}}&b&\tau \\{\overline {\mu }}&{\overline {\tau }}&c\end{bmatrix}}}$

${\displaystyle [\psi _{A},\psi _{B}]\subset J_{3}(O)}$

Because they are fermions the anti-commutators of the Jordan algebra become commutators. It is known that E₆ has subgroup O(10) and so is big enough to include the Standard Model. An E₈ gauge group, for example, would have 8 neutral bosons, 120 charged bosons and 120 charged anti-bosons. To account for the 248 fermions in the lowest multiplet of E₈, these would either have to include anti-particles (and so have baryogenesis), have new undiscovered particles, or have gravity-like (spin connection) bosons affecting elements of the particles spin direction. Each of these possess theoretical problems.

Beyond Lie groups

Other structures have been suggested including Lie 3-algebras and Lie superalgebras. Neither of these fit with Yang–Mills theory. In particular Lie superalgebras would introduce bosons with the wrong statistics. Supersymmetry however does fit with Yang–Mills.

Unification of forces and the role of supersymmetry

The unification of forces is possible due to the energy scale dependence of force coupling parameters in quantum field theory called renormalization group "running", which allows parameters with vastly different values at usual energies to converge to a single value at a much higher energy scale.

The renormalization group running of the three gauge couplings in the Standard Model has been found to nearly, but not quite, meet at the same point if the hypercharge is normalized so that it is consistent with SU(5) or SO(10) GUTs, which are precisely the GUT groups which lead to a simple fermion unification. This is a significant result, as other Lie groups lead to different normalizations. However, if the supersymmetric extension MSSM is used instead of the Standard Model, the match becomes much more accurate. In this case, the coupling constants of the strong and electroweak interactions meet at the grand unification energy, also known as the GUT scale:

${\displaystyle \Lambda _{\text{GUT}}\approx 10^{16}\,{\text{GeV}}}$.

It is commonly believed that this matching is unlikely to be a coincidence, and is often quoted as one of the main motivations to further investigate supersymmetric theories despite the fact that no supersymmetric partner particles have been experimentally observed. Also, most model builders simply assume supersymmetry because it solves the hierarchy problem—i.e., it stabilizes the electroweak Higgs mass against radiative corrections.

Neutrino masses

Since Majorana masses of the right-handed neutrino are forbidden by SO(10) symmetry, SO(10) GUTs predict the Majorana masses of right-handed neutrinos to be close to the GUT scale where the symmetry is spontaneously broken in those models. In supersymmetric GUTs, this scale tends to be larger than would be desirable to obtain realistic masses of the light, mostly left-handed neutrinos (see neutrino oscillation) via the seesaw mechanism. These predictions are independent of the Georgi–Jarlskog mass relations, wherein some GUTs predict other fermion mass ratios.

Proposed theories

Several theories have been proposed, but none is currently universally accepted. An even more ambitious theory that includes all fundamental forces, including gravitation, is termed a theory of everything. Some common mainstream GUT models are:

minimal left-right model — SU(3)C × SU(2)L × SU(2)R × U(1)B-L Georgi–Glashow model — SU(5) SO(10) Flipped SU(5) — SU(5) × U(1) Pati–Salam model — SU(4) × SU(2) × SU(2) Flipped SO(10) — SO(10) × U(1)

Trinification — SU(3) × SU(3) × SU(3) E6 331 model chiral color

Not quite GUTs:

Technicolor models Little Higgs String theory Causal fermion systems

M-theory Preons Loop quantum gravity Causal dynamical triangulation theory

Note: These models refer to Lie algebras not to Lie groups. The Lie group could be [SU(4) × SU(2) × SU(2)]/Z₂, just to take a random example.

The most promising candidate is SO(10). (Minimal) SO(10) does not contain any exotic fermions (i.e. additional fermions besides the Standard Model fermions and the right-handed neutrino), and it unifies each generation into a single irreducible representation. A number of other GUT models are based upon subgroups of SO(10). They are the minimal left-right model, SU(5), flipped SU(5) and the Pati–Salam model. The GUT group E₆ contains SO(10), but models based upon it are significantly more complicated. The primary reason for studying E₆ models comes from E₈ × E₈ heterotic string theory.

GUT models generically predict the existence of topological defects such as monopoles, cosmic strings, domain walls, and others. But none have been observed. Their absence is known as the monopole problem in cosmology. Many GUT models also predict proton decay, although not the Pati–Salam model; proton decay has never been observed by experiments. The minimal experimental limit on the proton's lifetime pretty much rules out minimal SU(5) and heavily constrains the other models. The lack of detected supersymmetry to date also constrains many models.

• Proton Decay. These graphics refer to the X bosons and Higgs bosons.
• 

  Dimension 6 proton decay mediated by the X boson ${\displaystyle (3,2)_{-{\frac {5}{6}}}}$ in SU(5) GUT

• 

  Dimension 6 proton decay mediated by the X boson ${\displaystyle (3,2)_{\frac {1}{6}}}$ in flipped SU(5) GUT

• 

  Dimension 6 proton decay mediated by the triplet Higgs ${\displaystyle T(3,1)_{-{\frac {1}{3}}}}$ and the anti-triplet Higgs ${\displaystyle {\bar {T}}({\bar {3}},1)_{\frac {1}{3}}}$ in SU(5) GUT

Some GUT theories like SU(5) and SO(10) suffer from what is called the doublet-triplet problem. These theories predict that for each electroweak Higgs doublet, there is a corresponding colored Higgs triplet field with a very small mass (many orders of magnitude smaller than the GUT scale here). In theory, unifying quarks with leptons, the Higgs doublet would also be unified with a Higgs triplet. Such triplets have not been observed. They would also cause extremely rapid proton decay (far below current experimental limits) and prevent the gauge coupling strengths from running together in the renormalization group.

Most GUT models require a threefold replication of the matter fields. As such, they do not explain why there are three generations of fermions. Most GUT models also fail to explain the little hierarchy between the fermion masses for different generations.

Ingredients

A GUT model consists of a gauge group which is a compact Lie group, a connection form for that Lie group, a Yang–Mills action for that connection given by an invariant symmetric bilinear form over its Lie algebra (which is specified by a coupling constant for each factor), a Higgs sector consisting of a number of scalar fields taking on values within real/complex representations of the Lie group and chiral Weyl fermions taking on values within a complex rep of the Lie group. The Lie group contains the Standard Model group and the Higgs fields acquire VEVs leading to a spontaneous symmetry breaking to the Standard Model. The Weyl fermions represent matter.

Current status

There is currently no hard evidence that nature is described by a Grand Unified Theory. The discovery of neutrino oscillations indicates that the Standard Model is incomplete and has led to renewed interest toward certain GUT such as SO(10). One of the few possible experimental tests of certain GUT is proton decay and also fermion masses. There are a few more special tests for supersymmetric GUT. However, minimum proton lifetimes from research (at or exceeding the 10³⁴-10³⁵ year range) have ruled out simpler GUTs and most non-SUSY models. The maximum upper limit on proton lifetime (if unstable), is calculated at 6 x 10³⁹ years for SUSY models and 1.4 x 10³⁶ years for minimal non-SUSY GUTs.

The gauge coupling strengths of QCD, the weak interaction and hypercharge seem to meet at a common length scale called the GUT scale and equal approximately to 10¹⁶ GeV (slightly less than the Planck energy of 10¹⁹ GeV), which is somewhat suggestive. This interesting numerical observation is called the gauge coupling unification, and it works particularly well if one assumes the existence of superpartners of the Standard Model particles. Still it is possible to achieve the same by postulating, for instance, that ordinary (non supersymmetric) SO(10) models break with an intermediate gauge scale, such as the one of Pati–Salam group.

Ultra unification

In 2020, a proposed theory called ultra unification tries to bring together the Standard Model and grand unification, particularly for the models with 15 Weyl fermions per generation, without the necessity of right-handed sterile neutrinos by adding new gapped topological phase sectors consistent with the nonperturbative global anomaly cancellation and cobordism constraints (especially from the baryon minus lepton number B−L, the electroweak hypercharge Y, and the mixed gauge-gravitational anomaly such as a Z/16Z class anomaly). Gapped topological phase sectors are constructed via symmetry extension, whose low energy contains unitary Lorentz invariant topological quantum field theories (TQFTs), such as four dimensional noninvertible, five dimensional noninvertible, or five dimensional invertible entangled gapped phase TQFTs. Alternatively, there could also be right-handed sterile neutrinos, gapless unparticle physics, or some combination of more general interacting conformal field theories, to cancel the mixed gauge-gravitational anomaly. In either case, this implies a new high-energy physics frontier beyond the conventional zero dimensional particle physics that relies on new types of topological forces and matter, including gapped extended objects such as line and surface operators or conformal defects, whose open ends carry deconfined fractionalized particle or anyonic string excitations. A physical characterization of these gapped extended objects require extensions of mathematical concepts such as cohomology, cobordism, or categories into particle physics.

Theory of everything

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Theory_of_everything 

Beyond the Standard Model
Simulated Large Hadron Collider CMS particle detector data depicting a Higgs boson produced by colliding protons decaying into hadron jets and electrons

A theory of everything (TOE or ToE), final theory, ultimate theory, theory of the world or master theory is a hypothetical, singular, all-encompassing, coherent theoretical framework of physics that fully explains and links together all physical aspects of the universe. Finding a theory of everything is one of the major unsolved problems in physics. String theory and M-theory have been proposed as theories of everything. Over the past few centuries, two theoretical frameworks have been developed that, together, most closely resemble a theory of everything. These two theories upon which all modern physics rests are general relativity and quantum mechanics. General relativity is a theoretical framework that only focuses on gravity for understanding the universe in regions of both large scale and high mass: stars, galaxies, clusters of galaxies etc. On the other hand, quantum mechanics is a theoretical framework that only focuses on three non-gravitational forces for understanding the universe in regions of both small scale and low mass: sub-atomic particles, atoms, molecules, etc. Quantum mechanics successfully implemented the Standard Model that describes the three non-gravitational forces – strong nuclear, weak nuclear, and electromagnetic force – as well as all observed elementary particles.

General relativity and quantum mechanics have been thoroughly proven in their separate fields of relevance. Since the usual domains of applicability of general relativity and quantum mechanics are so different, most situations require that only one of the two theories be used. The two theories are considered incompatible in regions of extremely small scale – the Planck scale – such as those that exist within a black hole or during the beginning stages of the universe (i.e., the moment immediately following the Big Bang). To resolve the incompatibility, a theoretical framework revealing a deeper underlying reality, unifying gravity with the other three interactions, must be discovered to harmoniously integrate the realms of general relativity and quantum mechanics into a seamless whole: the theory of everything is a single theory that, in principle, is capable of describing all phenomena in the universe.

In pursuit of this goal, quantum gravity has become one area of active research. One example is string theory, which evolved into a candidate for the theory of everything, but not without drawbacks (most notably, its lack of currently testable predictions) and controversy. String theory posits that at the beginning of the universe (up to 10⁻⁴³ seconds after the Big Bang), the four fundamental forces were once a single fundamental force. According to string theory, every particle in the universe, at its most microscopic level (Planck length), consists of varying combinations of vibrating strings (or strands) with preferred patterns of vibration. String theory further claims that it is through these specific oscillatory patterns of strings that a particle of unique mass and force charge is created (that is to say, the electron is a type of string that vibrates one way, while the up quark is a type of string vibrating another way, and so forth).

Name

Initially, the term theory of everything was used with an ironic reference to various overgeneralized theories. For example, a grandfather of Ijon Tichy – a character from a cycle of Stanisław Lem's science fiction stories of the 1960s – was known to work on the "General Theory of Everything". Physicist Harald Fritzsch used the term in his 1977 lectures in Varenna. Physicist John Ellis claims to have introduced the acronym "TOE" into the technical literature in an article in Nature in 1986. Over time, the term stuck in popularizations of theoretical physics research.

Historical antecedents

Antiquity to 19th century

Many ancient cultures such as Babylonian astronomers, Indian astronomy studied the pattern of the Seven Classical Planets against the background of stars, with their interest being to relate celestial movement to human events (astrology), and the goal being to predict events by recording events against a time measure and then look for recurrent patterns. The debate between the universe having either a beginning or eternal cycles can be traced back to ancient Babylonia. Hindu cosmology posits that time is infinite with a cyclic universe, where the current universe was preceded and will be followed by an infinite number of universes.  Time scales mentioned in Hindu cosmology correspond to those of modern scientific cosmology. Its cycles run from our ordinary day and night to a day and night of Brahma, 8.64 billion years long. 

The natural philosophy of atomism appeared in several ancient traditions. In ancient Greek philosophy, the pre-Socratic philosophers speculated that the apparent diversity of observed phenomena was due to a single type of interaction, namely the motions and collisions of atoms. The concept of 'atom' proposed by Democritus was an early philosophical attempt to unify phenomena observed in nature. The concept of 'atom' also appeared in the Nyaya-Vaisheshika school of ancient Indian philosophy.

Archimedes was possibly the first philosopher to have described nature with axioms (or principles) and then deduce new results from them. Any "theory of everything" is similarly expected to be based on axioms and to deduce all observable phenomena from them.

Following earlier atomistic thought, the mechanical philosophy of the 17th century posited that all forces could be ultimately reduced to contact forces between the atoms, then imagined as tiny solid particles.

In the late 17th century, Isaac Newton's description of the long-distance force of gravity implied that not all forces in nature result from things coming into contact. Newton's work in his Mathematical Principles of Natural Philosophy dealt with this in a further example of unification, in this case unifying Galileo's work on terrestrial gravity, Kepler's laws of planetary motion and the phenomenon of tides by explaining these apparent actions at a distance under one single law: the law of universal gravitation.

In 1814, building on these results, Laplace famously suggested that a sufficiently powerful intellect could, if it knew the position and velocity of every particle at a given time, along with the laws of nature, calculate the position of any particle at any other time:

An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.

— Essai philosophique sur les probabilités, Introduction. 1814

Laplace thus envisaged a combination of gravitation and mechanics as a theory of everything. Modern quantum mechanics implies that uncertainty is inescapable, and thus that Laplace's vision has to be amended: a theory of everything must include gravitation and quantum mechanics. Even ignoring quantum mechanics, chaos theory is sufficient to guarantee that the future of any sufficiently complex mechanical or astronomical system is unpredictable.

In 1820, Hans Christian Ørsted discovered a connection between electricity and magnetism, triggering decades of work that culminated in 1865, in James Clerk Maxwell's theory of electromagnetism. During the 19th and early 20th centuries, it gradually became apparent that many common examples of forces – contact forces, elasticity, viscosity, friction, and pressure – result from electrical interactions between the smallest particles of matter.

In his experiments of 1849–50, Michael Faraday was the first to search for a unification of gravity with electricity and magnetism. However, he found no connection.

In 1900, David Hilbert published a famous list of mathematical problems. In Hilbert's sixth problem, he challenged researchers to find an axiomatic basis to all of physics. In this problem he thus asked for what today would be called a theory of everything.

Early 20th century

In the late 1920s, the new quantum mechanics showed that the chemical bonds between atoms were examples of (quantum) electrical forces, justifying Dirac's boast that "the underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known".

After 1915, when Albert Einstein published the theory of gravity (general relativity), the search for a unified field theory combining gravity with electromagnetism began with a renewed interest. In Einstein's day, the strong and the weak forces had not yet been discovered, yet he found the potential existence of two other distinct forces, gravity and electromagnetism, far more alluring. This launched his thirty-year voyage in search of the so-called "unified field theory" that he hoped would show that these two forces are really manifestations of one grand, underlying principle. During the last few decades of his life, this ambition alienated Einstein from the rest of mainstream of physics, as the mainstream was instead far more excited about the emerging framework of quantum mechanics. Einstein wrote to a friend in the early 1940s, "I have become a lonely old chap who is mainly known because he doesn't wear socks and who is exhibited as a curiosity on special occasions." Prominent contributors were Gunnar Nordström, Hermann Weyl, Arthur Eddington, David Hilbert, Theodor Kaluza, Oskar Klein (see Kaluza–Klein theory), and most notably, Albert Einstein and his collaborators. Einstein searched in earnest for, but ultimately failed to find, a unifying theory (see Einstein–Maxwell–Dirac equations).

Late 20th century and the nuclear interactions

In the twentieth century, the search for a unifying theory was interrupted by the discovery of the strong and weak nuclear forces, which differ both from gravity and from electromagnetism. A further hurdle was the acceptance that in a theory of everything, quantum mechanics had to be incorporated from the outset, rather than emerging as a consequence of a deterministic unified theory, as Einstein had hoped.

Gravity and electromagnetism are able to coexist as entries in a list of classical forces, but for many years it seemed that gravity could not be incorporated into the quantum framework, let alone unified with the other fundamental forces. For this reason, work on unification, for much of the twentieth century, focused on understanding the three forces described by quantum mechanics: electromagnetism and the weak and strong forces. The first two were combined in 1967–68 by Sheldon Glashow, Steven Weinberg, and Abdus Salam into the electroweak force. Electroweak unification is a broken symmetry: the electromagnetic and weak forces appear distinct at low energies because the particles carrying the weak force, the W and Z bosons, have non-zero masses (80.4 GeV/c² and 91.2 GeV/c², respectively), whereas the photon, which carries the electromagnetic force, is massless. At higher energies W bosons and Z bosons can be created easily and the unified nature of the force becomes apparent.

While the strong and electroweak forces coexist under the Standard Model of particle physics, they remain distinct. Thus, the pursuit of a theory of everything remains unsuccessful: neither a unification of the strong and electroweak forces – which Laplace would have called 'contact forces' – nor a unification of these forces with gravitation has been achieved.

Modern physics

Conventional sequence of theories

A theory of everything would unify all the fundamental interactions of nature: gravitation, the strong interaction, the weak interaction, and electromagnetism. Because the weak interaction can transform elementary particles from one kind into another, the theory of everything should also predict all the various different kinds of particles possible. The usual assumed path of theories is given in the following graph, where each unification step leads one level up on the graph.

Theory of everything

Quantum gravity

Space Curvature

Electronuclear force (Grand Unified Theory)

Standard model of cosmology

Standard model of particle physics

Strong interaction SU(3)

Electroweak interaction SU(2) x U(1)Y

Weak interaction SU(2)

Electromagnetism U(1)EM

Electricity

Magnetism

In this graph, electroweak unification occurs at around 100 GeV, grand unification is predicted to occur at 10¹⁶ GeV, and unification of the GUT force with gravity is expected at the Planck energy, roughly 10¹⁹ GeV.

Several Grand Unified Theories (GUTs) have been proposed to unify electromagnetism and the weak and strong forces. Grand unification would imply the existence of an electronuclear force; it is expected to set in at energies of the order of 10¹⁶ GeV, far greater than could be reached by any currently feasible particle accelerator. Although the simplest grand unified theories have been experimentally ruled out, the idea of a grand unified theory, especially when linked with supersymmetry, remains a favorite candidate in the theoretical physics community. Supersymmetric grand unified theories seem plausible not only for their theoretical "beauty", but because they naturally produce large quantities of dark matter, and because the inflationary force may be related to grand unified theory physics (although it does not seem to form an inevitable part of the theory). Yet grand unified theories are clearly not the final answer; both the current standard model and all proposed GUTs are quantum field theories which require the problematic technique of renormalization to yield sensible answers. This is usually regarded as a sign that these are only effective field theories, omitting crucial phenomena relevant only at very high energies.

The final step in the graph requires resolving the separation between quantum mechanics and gravitation, often equated with general relativity. Numerous researchers concentrate their efforts on this specific step; nevertheless, no accepted theory of quantum gravity, and thus no accepted theory of everything, has emerged. It is usually assumed that the theory of everything will also solve the remaining problems of grand unified theories.

In addition to explaining the forces listed in the graph, a theory of everything may also explain the status of at least two candidate forces suggested by modern cosmology: an inflationary force and dark energy. Furthermore, cosmological experiments also suggest the existence of dark matter, supposedly composed of fundamental particles outside the scheme of the standard model. However, the existence of these forces and particles has not been proven.

String theory and M-theory

Unsolved problem in physics:

Is string theory, superstring theory, or M-theory, or some other variant on this theme, a step on the road to a "theory of everything", or just a blind alley?

Since the 1990s, some physicists such as Edward Witten believe that 11-dimensional M-theory, which is described in some limits by one of the five perturbative superstring theories, and in another by the maximally-supersymmetric 11-dimensional supergravity, is the theory of everything. There is no widespread consensus on this issue.

One remarkable property of string/M-theory is that extra dimensions are required for the theory's consistency. In this regard, string theory can be seen as building on the insights of the Kaluza–Klein theory, in which it was realized that applying general relativity to a five-dimensional universe (with one of them small and curled up) looks from the four-dimensional perspective like the usual general relativity together with Maxwell's electrodynamics. This lent credence to the idea of unifying gauge and gravity interactions, and to extra dimensions, but did not address the detailed experimental requirements. Another important property of string theory is its supersymmetry, which together with extra dimensions are the two main proposals for resolving the hierarchy problem of the standard model, which is (roughly) the question of why gravity is so much weaker than any other force. The extra-dimensional solution involves allowing gravity to propagate into the other dimensions while keeping other forces confined to a four-dimensional spacetime, an idea that has been realized with explicit stringy mechanisms.

Research into string theory has been encouraged by a variety of theoretical and experimental factors. On the experimental side, the particle content of the standard model supplemented with neutrino masses fits into a spinor representation of SO(10), a subgroup of E8 that routinely emerges in string theory, such as in heterotic string theory or (sometimes equivalently) in F-theory. String theory has mechanisms that may explain why fermions come in three hierarchical generations, and explain the mixing rates between quark generations. On the theoretical side, it has begun to address some of the key questions in quantum gravity, such as resolving the black hole information paradox, counting the correct entropy of black holes and allowing for topology-changing processes. It has also led to many insights in pure mathematics and in ordinary, strongly-coupled gauge theory due to the Gauge/String duality.

In the late 1990s, it was noted that one major hurdle in this endeavor is that the number of possible four-dimensional universes is incredibly large. The small, "curled up" extra dimensions can be compactified in an enormous number of different ways (one estimate is 10⁵⁰⁰ ) each of which leads to different properties for the low-energy particles and forces. This array of models is known as the string theory landscape.

One proposed solution is that many or all of these possibilities are realised in one or another of a huge number of universes, but that only a small number of them are habitable. Hence what we normally conceive as the fundamental constants of the universe are ultimately the result of the anthropic principle rather than dictated by theory. This has led to criticism of string theory, arguing that it cannot make useful (i.e., original, falsifiable, and verifiable) predictions and regarding it as a pseudoscience. Others disagree, and string theory remains an active topic of investigation in theoretical physics.

Loop quantum gravity

Current research on loop quantum gravity may eventually play a fundamental role in a theory of everything, but that is not its primary aim. Loop quantum gravity also introduces a lower bound on the possible length scales.

There have been recent claims that loop quantum gravity may be able to reproduce features resembling the Standard Model. So far only the first generation of fermions (leptons and quarks) with correct parity properties have been modelled by Sundance Bilson-Thompson using preons constituted of braids of spacetime as the building blocks. However, there is no derivation of the Lagrangian that would describe the interactions of such particles, nor is it possible to show that such particles are fermions, nor that the gauge groups or interactions of the Standard Model are realised. Utilization of quantum computing concepts made it possible to demonstrate that the particles are able to survive quantum fluctuations.

This model leads to an interpretation of electric and colour charge as topological quantities (electric as number and chirality of twists carried on the individual ribbons and colour as variants of such twisting for fixed electric charge).

Bilson-Thompson's original paper suggested that the higher-generation fermions could be represented by more complicated braidings, although explicit constructions of these structures were not given. The electric charge, colour, and parity properties of such fermions would arise in the same way as for the first generation. The model was expressly generalized for an infinite number of generations and for the weak force bosons (but not for photons or gluons) in a 2008 paper by Bilson-Thompson, Hackett, Kauffman and Smolin.

Other attempts

Among other attempts to develop a theory of everything is the theory of causal fermion systems, giving the two current physical theories (general relativity and quantum field theory) as limiting cases.

Another theory is called Causal Sets. As some of the approaches mentioned above, its direct goal isn't necessarily to achieve a theory of everything but primarily a working theory of quantum gravity, which might eventually include the standard model and become a candidate for a theory of everything. Its founding principle is that spacetime is fundamentally discrete and that the spacetime events are related by a partial order. This partial order has the physical meaning of the causality relations between relative past and future distinguishing spacetime events.

Causal dynamical triangulation does not assume any pre-existing arena (dimensional space), but rather attempts to show how the spacetime fabric itself evolves.

Another attempt may be related to ER=EPR, a conjecture in physics stating that entangled particles are connected by a wormhole (or Einstein–Rosen bridge).

Present status

At present, there is no candidate theory of everything that includes the standard model of particle physics and general relativity and that, at the same time, is able to calculate the fine-structure constant or the mass of the electron. Most particle physicists expect that the outcome of ongoing experiments – the search for new particles at the large particle accelerators and for dark matter – are needed in order to provide further input for a theory of everything.

Arguments against

In parallel to the intense search for a theory of everything, various scholars have seriously debated the possibility of its discovery.

Gödel's incompleteness theorem

A number of scholars claim that Gödel's incompleteness theorem suggests that any attempt to construct a theory of everything is bound to fail. Gödel's theorem, informally stated, asserts that any formal theory sufficient to express elementary arithmetical facts and strong enough for them to be proved is either inconsistent (both a statement and its denial can be derived from its axioms) or incomplete, in the sense that there is a true statement that can't be derived in the formal theory.

Stanley Jaki, in his 1966 book The Relevance of Physics, pointed out that, because any "theory of everything" will certainly be a consistent non-trivial mathematical theory, it must be incomplete. He claims that this dooms searches for a deterministic theory of everything.

Freeman Dyson has stated that "Gödel's theorem implies that pure mathematics is inexhaustible. No matter how many problems we solve, there will always be other problems that cannot be solved within the existing rules. […] Because of Gödel's theorem, physics is inexhaustible too. The laws of physics are a finite set of rules, and include the rules for doing mathematics, so that Gödel's theorem applies to them."

Stephen Hawking was originally a believer in the Theory of Everything, but after considering Gödel's Theorem, he concluded that one was not obtainable. "Some people will be very disappointed if there is not an ultimate theory that can be formulated as a finite number of principles. I used to belong to that camp, but I have changed my mind."

Jürgen Schmidhuber (1997) has argued against this view; he asserts that Gödel's theorems are irrelevant for computable physics. In 2000, Schmidhuber explicitly constructed limit-computable, deterministic universes whose pseudo-randomness based on undecidable, Gödel-like halting problems is extremely hard to detect but does not at all prevent formal theorys of everything describable by very few bits of information.

Related critique was offered by Solomon Feferman and others. Douglas S. Robertson offers Conway's game of life as an example: The underlying rules are simple and complete, but there are formally undecidable questions about the game's behaviors. Analogously, it may (or may not) be possible to completely state the underlying rules of physics with a finite number of well-defined laws, but there is little doubt that there are questions about the behavior of physical systems which are formally undecidable on the basis of those underlying laws.

Since most physicists would consider the statement of the underlying rules to suffice as the definition of a "theory of everything", most physicists argue that Gödel's Theorem does not mean that a theory of everything cannot exist. On the other hand, the scholars invoking Gödel's Theorem appear, at least in some cases, to be referring not to the underlying rules, but to the understandability of the behavior of all physical systems, as when Hawking mentions arranging blocks into rectangles, turning the computation of prime numbers into a physical question. This definitional discrepancy may explain some of the disagreement among researchers.

Fundamental limits in accuracy

No physical theory to date is believed to be precisely accurate. Instead, physics has proceeded by a series of "successive approximations" allowing more and more accurate predictions over a wider and wider range of phenomena. Some physicists believe that it is therefore a mistake to confuse theoretical models with the true nature of reality, and hold that the series of approximations will never terminate in the "truth". Einstein himself expressed this view on occasions. Following this view, we may reasonably hope for a theory of everything which self-consistently incorporates all currently known forces, but we should not expect it to be the final answer.

On the other hand, it is often claimed that, despite the apparently ever-increasing complexity of the mathematics of each new theory, in a deep sense associated with their underlying gauge symmetry and the number of dimensionless physical constants, the theories are becoming simpler. If this is the case, the process of simplification cannot continue indefinitely.

Lack of fundamental laws

There is a philosophical debate within the physics community as to whether a theory of everything deserves to be called the fundamental law of the universe. One view is the hard reductionist position that the theory of everything is the fundamental law and that all other theories that apply within the universe are a consequence of the theory of everything. Another view is that emergent laws, which govern the behavior of complex systems, should be seen as equally fundamental. Examples of emergent laws are the second law of thermodynamics and the theory of natural selection. The advocates of emergence argue that emergent laws, especially those describing complex or living systems are independent of the low-level, microscopic laws. In this view, emergent laws are as fundamental as a theory of everything.

The debates do not make the point at issue clear. Possibly the only issue at stake is the right to apply the high-status term "fundamental" to the respective subjects of research. A well-known debate over this took place between Steven Weinberg and Philip Anderson.

Impossibility of being "of everything"

Although the name "theory of everything" suggests the determinism of Laplace's quotation, this gives a very misleading impression. Determinism is frustrated by the probabilistic nature of quantum mechanical predictions, by the extreme sensitivity to initial conditions that leads to mathematical chaos, by the limitations due to event horizons, and by the extreme mathematical difficulty of applying the theory. Thus, although the current standard model of particle physics "in principle" predicts almost all known non-gravitational phenomena, in practice only a few quantitative results have been derived from the full theory (e.g., the masses of some of the simplest hadrons), and these results (especially the particle masses which are most relevant for low-energy physics) are less accurate than existing experimental measurements. Even in classical mechanics there are still unsolved problems, such as turbulence, although the equations have been known for centuries. The theory of everything would almost certainly be even harder to apply for the prediction of experimental results, and thus might be of limited use.

A motive for seeking a theory of everything, apart from the pure intellectual satisfaction of completing a centuries-long quest, is that prior examples of unification have predicted new phenomena, some of which (e.g., electrical generators) have proved of great practical importance. And like in these prior examples of unification, the theory of everything would probably allow us to confidently define the domain of validity and residual error of low-energy approximations to the full theory.

The theories generally do not account for the apparent phenomena of consciousness or free will, which are instead often the subject of philosophy and religion.

Infinite number of onion layers

Frank Close regularly argues that the layers of nature may be like the layers of an onion, and that the number of layers might be infinite. This would imply an infinite sequence of physical theories.

Impossibility of calculation

Weinberg points out that calculating the precise motion of an actual projectile in the Earth's atmosphere is impossible. So how can we know we have an adequate theory for describing the motion of projectiles? Weinberg suggests that we know principles (Newton's laws of motion and gravitation) that work "well enough" for simple examples, like the motion of planets in empty space. These principles have worked so well on simple examples that we can be reasonably confident they will work for more complex examples. For example, although general relativity includes equations that do not have exact solutions, it is widely accepted as a valid theory because all of its equations with exact solutions have been experimentally verified. Likewise, a theory of everything must work for a wide range of simple examples in such a way that we can be reasonably confident it will work for every situation in physics.