Information—Consciousness—Reality pp 93-138 | Cite as

# The Unification Power of Symmetry

## Abstract

The introduction of a new kind of symmetry ushered in a golden era for theoretical physics. The marriage of this novel gauge theory with quantum field theory culminated in the standard model of particle physics. This is the unified description of all three non-gravitational forces in the universe, a momentous milestone in human knowledge generation. Inspired by this success, physicists hoped for a “theory of everything,” uniting the standard model with general relativity, the theory of gravity. These attempts uncovered five ten-dimensional superstring theories, unified within an overarching eleven-dimensional framework called M-Theory. To this day, the theory of everything remains an elusive dream. Albert Einstein, arguably the most insightful physicists, played a rather tragic role in the history of unification and quantum theory.

*Level of mathematical formality*: high (however, the mathematically involved parts are encapsulated and demarcated by the tags Open image in new window , hence easily bypassed).

All the previous accounts of symmetry have one thing in common: they are all instances of global symmetry principles. This means that these symmetries are unchanged for all points in space-time. The introduction of a new kind of twist on the idea of symmetry in 1918 unlocked even greater powers of this abstract formalism in its propensity to probe reality, paving the way for a novel type of field theory to flourish. Today, the tremendous success of the mathematical framework underlying the standard model, providing a unified and overarching theory of all non-gravitational forces, can be understood to rest on the insights gained from what is known as gauge theory. The idea fueling this novel approach is related to a new kind of symmetry, called gauge symmetry. It is a local symmetry, meaning that its properties are now a function of the space-time coordinates \(x^\mu \). This principle was first fully formulated, independently, by Hermann Weyl and Emmy Noether in the same year (Brading 2002). However, the course of the history of gauge theory, and in parallel the road to unification, would take meandering paths.

## 4.1 Back to Geometry: The Principle of Covariance

Einstein’s theory of general relativity, sketched in ( 3.14), is an extremely elegant and aesthetic physical theory. It is based on two very subtle principles, a physical and a mathematical requirement.

Inertia, the measure of a body’s resistance to acceleration, is encoded in Newton’s second law describing the resulting forceAs a child, the Nobel Prize-winning physicist Richard Feynman asked his father why a ball in his toy wagon moved backward whenever he pulled the wagon forward. His father said that the answer lay in the tendency of moving things to keep moving, and of stationary things to stay put. “This tendency is called inertia,” said Feynman senior. Then, with uncommon wisdom, he added: “But nobody knows why it is true.”

*F*due to the acceleration \(a=\ddot{x}\) reads \(F=m_{\text {i}} a\). The mass term \(m_i\) appearing in this equation is called inertial mass. This is to distinguish it from the mass term appearing in Newton’s law of universal gravitation,

^{1}called gravitational mass \(m_{\text {g}}\). A simple experiment, going back to Newton, is the following. A bucket partly filled with water is hung from a long cord and rotated so many times until the cord becomes strongly twisted. By releasing the bucket, after ensuring the water is at rest, it will rotate in the other direction due to the cord untwisting. Slowly the water begins to rotate with the bucket and as it does so the water moves to the sides of the bucket. In effect, the surface of the water becomes concave. This effect is not due to the water spinning relative to the bucket, as, at some point, the bucket and the water are spinning at the same rate while the surface stays concave. Again, the question of inertia emerges. Why should the surface of the water bulge? What is the origin of this effect? One explanation was proposed by the philosopher and physicist Ernst Mach. He attributed the source of inertia to the whole matter content of the universe, an idea today referred to as Mach’s principle (Misner et al. 1973). This principle guided Einstein in his formulation of general relativity (Penrose 2004, p. 753). In his equivalence principle, Einstein asserted that the gravitational mass \(m_{\text {g}}\) is equivalent to the inertial mass \(m_{\text {i}}\). In other words, the acceleration a body experiences due to its mass being exposed to the pull of the gravitational force, is independent of the nature of the body. The insight leading to the postulation of this principle, Einstein would later call “the happiest thought of my life” (Thorne 1995, p. 97). This thought was the following, quoting Einstein in Thorne (1995, p. 96f.):

In effect, the principle of equivalence states that there is no local way of knowing if one is feeling the effect of gravitational pull or the force due to acceleration. So a free falling observer will not detect any traces of gravity in her local reference frame, and only the laws of special relativity apply. Einstein soon derived two testable consequences of the equivalence principle, namely that gravity bends light, and that the frequency of radiation varies with the strength of gravity (Torretti 1999, p. 290). Unfortunately, it was later shown that Mach’s principle is not actually incorporated in general relativity (Penrose 2004, p. 753), and still today the origins of inertia are puzzling (Matthews 1994). Thus, seemingly obvious and uncontroversial aspects of reality can have very deep and mysterious connotations.I was sitting in a chair in the patent office at Bern, when all of a sudden a thought occurred to me: “If a person falls freely, he will not feel his own weight.”

^{2}must stay unchanged: \(|\varvec{a}| = \sqrt{3} =|\varvec{a}^\prime |\). Covariance may not appear like a particularly profound insight into the workings of nature, as one could argue that theses are common sense requirements for a physical theory. However, the ramifications are far-reaching and profound.Formally, general coordinate transformations in four-dimensional space-time are defined as follows

^{3}in general terms for a vector \(\text {d} x^\mu \) as

^{4}these technicalities are irrelevant for this discussion. It suffices to recall that the metric tensor can be utilized to lower or raise indices, e.g., \(A_\mu = g_{\mu \nu } A^\nu \). The metric also transforms as a second-rank tensor:

Guided by the principles of equivalence and covariance, Einstein was able to formulate the famous geometrodynamic field equations, one of the most aesthetic and accurate physical theories. One experiment confirmed the effect of gravity, as predicted by general relativity, on clocks up to an accuracy of \(10^{-16}\) hertz (Chou et al. 2010). Another experiment measured the “twisting” of space-time, called frame-dragging, due to the rotation of Earth to be \(37.2 \pm 7.2\) milliarcseconds. The theoretical value was calculated to be 39.2 milliarcseconds (Everitt et al. 2011). This amazing accuracy between experiment and theory is only rivaled by the relativistic quantum field theory of electrodynamics, know as quantum electrodynamics, winning Tomonaga, Julian Schwinger, and Feynman a Nobel Prize in 1965. In this theory, the magnetic moment of the electron can be computed. The experimental measurement can be performed with an impressive precision of fourteen digits, in exact correspondence with the theoretical value (Hanneke et al. 2008). For more details on the field equations of general relativity, see Sect. 10.1.2.

## 4.2 The History of Gauge Theory

A key feature of general relativity is that it is a local theory. Only local coordinate systems are meaningful. Christoffel symbols describe the effects of transporting geometrical information along curves in a manifold, allowing coordinate systems to be related to each other. In detail, the value of \(\Gamma _{ {\nu }\mu \lambda }^{\nu }\), defined via the metric tensor at each point in space-time, depends on the properties of the gravitational field, allowing the relative “orientation” of local coordinate systems to be compared. Weyl took this idea to the next level (Moriyasu 1983). He wondered if the effects of other forces of nature could be associated with a corresponding mathematical quantity similar to \(\Gamma _{ {\nu }\mu \lambda }^{\nu }\). Weyl was specifically thinking about electromagnetism.

He embarked on a quest that would eventually reveal “one of the most significant and far-reaching developments of physics in this [20th] century” (Moriyasu 1983, p. 1) in 1918, when he was attempting to derive a unified theory of electromagnetism and gravitation (Weyl 1918). The same year Noether published her famous theorems relating symmetry to conserved quantities, Weyl was independently attempting to explain the conservation of the electric charge with a novel local symmetry. He called the invariance related to this new symmetry *Eichinvarianz* . Although the notion was originally related to invariances due to changes in scale, the English translations of Weyl’s work referred to gauge invariance and gauge symmetry. It would, however, require nearly 50 years for gauge invariance to be rediscovered and reformulated as the powerful theory known today. Indeed, the idea of local gauge symmetry was premature in 1918, where the only known elementary particles were electrons and protons.

*e*denotes the elementary charge. See, for instance de Wit and Smith (2014), Peskin and Schroeder (1995). Unfortunately, it was shown by Einstein and others that Weyl’s gauge theory based on changes in scale had failed—it lead to conflicts with known physical facts (Vizgin 1994; Moriyasu 1983 and Penrose 2004, Section 19.4). The mathematical observation that Maxwell’s equations are gauge invariant was simply seen as an accident, as there was no deeper interpretation of the phenomena able to shed some light on the issue. The potential \(A_\mu \) was just a ghost in the theory.

*C*is the amplitude, \(\varvec{k}\) the wave vector, and \(\omega \) represents the wave’s angular frequency. For details, see, for instance Schwabl (2007). A change of the phase of a wave by the amount \(\lambda \) is related to the transformation \(\exp (i \lambda )\). In quantum mechanics, for the wave function of an electron, this is realized by the transformation

*e*is the elementary charge. Weyl’s essential idea was to interpret the phase of the wave function as the new local variable. In other words, the value \(\lambda \) is promoted to \(\lambda (x^\nu )\) in (4.21). Instead of changes in scale, this new local gauge transformations is now interpreted as changes in the phase of \(\psi (t, \varvec{x})\), encoded via \(\lambda \) at various points in space-time (Weyl 1929). From the explicit form of the gauge transformation (4.21), the covariant derivative and the transformation properties of the gauge fields can easily be derived.The transformation of the derivative of the field is given by

To summarize, the electromagnetic interactions of charged particles can be understood as a local gauge theory, embedded in the deeper framework of quantum mechanics. Just as the \(\Gamma _{ {\nu }\mu \lambda }^{\nu }\) describe how coordinate systems are related to each other in general relativity, the connection between phase values of the wave function at different points is given by \(A_\mu \), just as Weyl had originally envisioned. The link to the global symmetry transformations discussed previously is given by the following. Recalling that ( 3.29) describes the transformation properties of a quantum field under a group action, the formula given in (4.21) can be understood as a special case thereof. If the variable \(\lambda \), parameterizing the symmetry transformations, would be a constant, (4.21) reveals the transformation property of the field \(\psi \) under a globalIt is interesting that something like this can be around for thirty years but, because of certain prejudices of what is and is not significant, continues to be ignored.

*U*(1) symmetry.

^{5}The simple mathematical trick of letting the parameter \(\lambda \) become space-time dependent is responsible for the transition between the global and the local symmetry. In other words, and in the general case where the parameters of the symmetry group are not restricted do being scalars as seen in ( 3.27), the notion of “gauging the symmetry” is the straightforward substitution

*G*, with the group generators represented as matrices \(\text {X}^k\) which satisfy commutation relations ( 3.19), and the parameters \(\theta _k\) are now gauged. Note that (4.21) is a special case of (4.26).

^{6}The covariant derivative is constructed from these fields. Similarly to (4.11) and (4.8)

*g*into the theory

^{7}. The local parameter and the gauge fields are rescaled with this value

*g*appear in the Lagrangian and can be interpreted as the physical coupling strength (de Wit and Smith 2014). This is a number that determines the strength of the interaction associated with the gauge fields. As was seen for the case of electromagnetism, \(g=e\). Essentially, the abstract concepts of the formal representation are enriched by encoding additional measurable aspects of the physical reality domain.

*SU*(2) . However, this specific theory for the strong force failed. It was known from experiments, that the nuclear force only acted on short ranges. Yang and Mills’ theory, however, predicted that the carrier of the force, the gauge field, would be, like the photon in electromagnetism, long-range. This is because there is no way to incorporate gauge invariant mass terms for the gauge field into the Lagrangian (Moriyasu 1983). Nevertheless, this specific kind of gauge theory laid the foundation for modern gauge theory, culminating in the standard model of particle physics. Unfortunately, the potential power inherent in the formal machinery of gauge theories was not anticipated at the time. Indeed, Freeman Dyson would, eleven years after the introduction of Yang-Mills theory, gloomily remark (quoted in Moriyasu 1983, p. 73):

Quantum field theory (Sect. 10.1.1) and gauge theory were each plagued, individually, by major problems. While the issue of quantum field theory was related to a mathematical nightmare, the gauge theory problem was related to symmetry. It was found that any gauge invariant Lagrangian cannot contain mass terms, as they necessarily break covariance. So how can a physical system with mass be described by a gauge theory and still have properties which violate gauge invariance? The mathematical problem was, in detail, related to infinities appearing in the framework. Quantum field theory is based on perturbation theory, the idea of taking the solution to an easier problem and then adding corrections to approximate the real problem. Unfortunately, the perturbation series are divergent, assigning infinite values to measurable quantities.It is easy to imagine that in a few years the concepts of field theory will drop totally out of the vocabulary of day-to-day work in high energy physics.

Quote from Ryder (1996, p. 308). It is ironic, that at a time when experimental physics had entered a golden era, theoretical efforts, after so many promising findings, would dwindle and “the practice of quantum field theory entered a kind of ‘Dark Age”’ (Moriyasu 1983, p. 85). However, due to new technological advances—epitomized by the high-energy particle accelerator—more and more particles were discovered. Simply organizing these was a challenge. As an example, Murray Gell-Mann and others introduced new fermions, they called quarks (Gell-Mann and Ne’eman 1964). Now it was possible to categorize many of the observed particles as being composed of quarks. The quarks themselves are representations of the global symmetry groupIt is obvious that, in order for a field theory to be at all sensible or believable, the problems raised by the divergences must be satisfactorily resolved.

*SU*(3) . Gell-Mann called this classification scheme the Eightfold Way. Although alluding to the Noble Eightfold Path of Buddhism, the reference is “clearly intended to be ironic or humorous” (Kaiser 2011, p. 161).

General references are Peskin and Schroeder (1995), Ryder (1996), Cheng and Li (1996).The qualitative behavior of a quantum field theory is determined not by the fundamental Lagrangian, but rather by the nature of the renormalization group flow and its fixed points. These, in turn, depend only on the basic symmetries that are imposed on the family of Lagrangians that flow into one another. This conclusion signals, at the deepest level, the importance of symmetry principles in determining the fundamental laws of physics.

The solution to the problem of incorporating mass terms into a gauge-invariant theory is discussed in the following section. The details require a journey deep into the undergrowth of the abstract world.

### 4.2.1 The Higgs Mechanism

*SU*(2) invariant form, is derived to be

^{8}the minimum of \(\mathcal {V}\) is shifted. Now there is a local maxima at \(\phi = 0\) and an infinite number of minima appear at

^{9}of the scalar field \(\phi \), such that \(\langle \phi _i \rangle = v\). The new Higgs fields, associated with a Higgs boson, is defined as

*v*. In essence, the Higgs field “plays the role of a new type of vacuum in gauge theory” (Moriyasu 1983, p. 120). Formally, replacing \(\phi _i\) in the appropriate places in the Lagrangian with \(h+v\), yields the much awaited mass terms appearing due to the value

*v*entering the mathematical machinery.

*g*and \(g^\prime \) are the coupling constants introduced in (4.34). The terms \(\tau ^i\) and

*Y*are the generators of the symmetry groups

*SU*(2) and

*U*(1), respectively. Finally, \(W^{i}_{\mu }\) and \(B_\mu \) are the gauge fields associated with the corresponding symmetry groups, and \(i=1,2,3\). The gauge-invariant Lagrangian, containing the field-strength tensors, reads

^{10}Note that (4.41a) and (4.14) are identical expressions. In the next step, the physical boson fields are constructed from the quantities \(W_\mu ^1, W_\mu ^2 , W_\mu ^3, B_\mu \). This yields the \(W^{\pm }\) bosons (\(W^{\pm }_\mu )\), the

*Z*boson (\(Z_\mu \)), and the photon field (\(A_\mu \)). These gauge bosons are the carriers of the electroweak force. As anticipated, these quantum fields receive mass terms, if \(\phi \) is substituted with \(h+v\) from (4.38) in the Lagrangian \(\mathcal {L}_{\text {Higgs}}\), described in ( 3.11) and (4.35). This is the process of spontaneous symmetry breaking and results in

Building on the works of Schwinger, Philip Warren Anderson spelled out the first accounts of what would later become known as the Higgs mechanism (Anderson 1963). He also incorporated the insights gained from superconductivity. There, in the theory of Bardeen et al. (1957), it was realized that the mechanism of breaking the symmetry was associated with the appearance of a new boson (Nambu 1960). These ideas could be systematically generalized within the context of quantum field theory (Goldstone et al. 1962). Anderson grappled with the technicalities related to the Goldstone theorem, which was a final hurdle in the mass generating mechanism. The term “spontaneous symmetry breaking” was introduced in Baker and Glashow (1962), to account for the fact that the mechanism does not require any explicit mass terms in the Lagrangian to violate gauge invariance. The full model was developed in the same year by three independent groups:[...] associating a gauge transformation with a local conservation law does not necessarily require the existence of a zero-mass vector boson.

^{11}Englert and Brout (1964), Higgs (1964), Guralnik et al. (1964). However, the names Higgs mechanism and Higgs boson stuck. Indeed, the Nobel Committee, allowed to nominate a maximum of three people, only awarded François Englert and Higgs, with a Nobel Prize in 2013, after the 2012 discovery at CERN’s LHC (CERN 2013):

A general reference is Gunion et al. (2000). Here the parenthesis closes.[...] today, the ATLAS and CMS collaborations at the Large Hadron Collider (LHC) presented preliminary new results that further elucidate the particle discovered last year. Having analyzed two and a half times more data than was available for the discovery announcement in July [2012], they find that the new particle is looking more and more like a Higgs boson, the particle linked to the mechanism that gives mass to elementary particles. It remains an open question, however, whether this is the Higgs boson of the Standard Model of particle physics, or possibly the lightest of several bosons predicted in some theories that go beyond the Standard Model. Finding the answer to this question will take time.

### 4.2.2 Tying Up Some Loose Ends

Incidentally, Yang-Mills theory also uncovered a new type geometry for physics. This understanding only became apparent in the 1970s, and helped in popularizing gauge theories. Interestingly, this new concept in physics of uniting space-time with an “internal” symmetry space had been proposed by mathematicians at nearly the same time. See, for instance Moriyasu (1983, p. 32), Schottenloher (1995, p. 8). In detail, gauge theories have the topology of a fiber bundle. This means, that at every point in space-time a Lie group *G* is attached; there is an internal symmetry space existing at every space-time coordinate. The group *G* associated with a point \(x^\nu \) is called a fiber. As a particle moves through space-time, it also follows a path through the internal spaces at each point. The gauge transformations describe how the internal spaces at different points can be transformed into each other. The tangent bundle *TM* , described in Sect. 3.1.1, is a specific example of a fiber bundle. More details can be found in Drechsler and Mayer (1977), Nash and Sen (1983), Coquereaux and Jadczyk (1988).

Finally, there is one peculiar historical confusion related to Noether and Weyl. It is a good reminder that the devil, as always, is in the details. Many textbooks and review articles on quantum field theory gloss over the fact, that Noether actually published two theorems in 1918. The first one, famously deals with global symmetries and conserved quantities. However, she also proved a second theorem relating to local symmetry, which, prima facie, has nothing to do with conservation laws. Brading (2002) observes that there is either no, or no detailed, discussion of the second theorem in the literature, for instance O’Raifeartaigh (1997), Vizgin (1994), Kastrup (1984), Moriyasu (1982). Notable exception are Utiyama (1959), Byers (1999), Rowe (1999). As mentioned, Weyl, working on his unified field theory of electromagnetism and gravity in 1918, independently was trying to explain the conservation of the electric charge with the notion of a local symmetry. His results, in effect, can be understood as an application of Noether’s second theorem. The confusion arises, because “the standard textbook presentation of the connection between conservation of electric charge and gauge symmetry in relativistic field theory involves Noether’s first theorem” (Brading 2002, p. 9). Although these books discuss both local and global symmetries, they do not mention her second theorem. Despite the fact that both ways of deriving the conservation of electric charge, employing local or global symmetries, are correct, the text book approach via global symmetry is somewhat misleading. There it is implied that the conservation of charge depends on the Euler-Lagrange equations of motion being fulfilled. Noether’s second theorem, and Weyl’s derivation, yields the conservation law based on local symmetry only, without the necessity of the additional constraint due the equations of motion. See Brading (2002).

## 4.3 The Road to Unification

The road to unification has been a rocky one. Unification is the epitome of human understanding of reality. What appear as independent phenomena, described by fragmented theories, suddenly become united in a unified framework. It is the ultimate act of translation seen in Fig. 2.1: superficially separate properties of the natural world are encoded and merged into a single formal description. In essence, from the multifarious complexity of nature the formal essence is distilled, a unified theory of phenomena. Such an over-aching structure of knowledge has the power to unlock new and unexpected understanding of the workings of nature. This is why, in physics, the ultimate unified field theory describing all fundamental forces and elementary particles is, grandiosely, known as “the theory of everything.” It should be noted, however, that here the context of “everything” excludes emergent complexity, discussed in Chap. 6, and the fact that a conscious entity, the physicist, is doing the inquiring, covered in Chaps. 11 and 14. Nevertheless, this version of the theory of everything tries to explain all observable phenomena related to the fundamental workings of reality. In detail, it should explain all four known forces and describe the behavior of all elementary particles and antiparticles. What this all amounts to can be seen in Fig. 4.1.

In the history of physics there were a few instances where different abstract formalisms representing unrelated aspects of the world could be fused into a single conceptual formalism. For instance, Maxwell’s insight that light was an electromagnetic wave, unifying the fields of optics and electromagnetism. Or the fusion of thermodynamics with statistical mechanics (Gibbs 1884, 1902). In a sense, special relativity can be understood as the merger of electromagnetism with the laws of classical mechanics (Einstein 1905b), and general relativity as the synthesis of inertial and gravitational forces (Einstein 1915).

### 4.3.1 Jumping to Higher Dimensions

*X*,

*Y*are two vector fields on the manifold and \(\nabla _X\) , related to (4.8), computes the covariant derivative of a vector field in the direction of

*X*. The Lie brackets, introduced in ( 3.19), are now also functions of vector fields. For a basis \(\varvec{e}_i\) one finds

^{12}(Kibble 1961; Sciama 1962, 1964). Setting the curvature to zero in the Riemann-Cartan space-time, uncovers Weitzenböck space-time with \(T \ne 0, R= 0\), a variant Einstein would later work on, as detailed in Sect. 4.3.3. More details are found in Gronwald and Hehl (1996). Generalizing the idea of geometrization was the main avenue for unification at the time. A wealth of details on the history of unified field theories, including an extensive bibliography, can be found in Goenner (2004). A shorter version is Goenner (2005).

*M*and

*N*run from 1 to 5, and \(A_\mu \) is the vector potential of (4.12), incorporated with a proportionality factor

*c*. The component \(g_{55} = \phi \) is a new scalar gravitational potential. While promising, this extra-dimensional framework was plagued by inconsistencies. Moreover, could there really be any physical reality at the heart of this idea transcending human perception? Although Kaluza published his work in 1921 (Kaluza 1921), Einstein would remain silent on these matters until 1926. In that year, the physicist Oskar Klein

^{13}reawakened the interest in Kaluza’s ideas (Klein 1926). He not only linked quantum mechanics to the machinery of general relativity in five dimensions, crucially, he was able to give a physical interpretation of the extra dimension. This idea is today known as compactification, or dimensional reduction. If the extra dimension is “curled up” tight enough it becomes undetectable from our familiar slice of reality. Only at sufficiently large energies, the three-dimensional world unveils its richer structure due to the additional compactified dimensions. Today, modern versions of Kaluza-Klein theories can go up to 26 dimensions.

^{14}

*r*gives the “radius” of the fifth dimension. Expanding \(\psi \) in a Fourier series yields

*y*-component of a state with given

*n*as being associated with the momentum \(p=|n| /r\). Thus, for a sufficiently small

*r*, only the \(n=0\) state will appear in the low-energy world we live in. As a result, all observed states will be independent of

*y*

*r*is of the order of the Planck length \(l_p \approx 1.6 \times 10^{-35}\) m, the masses associated with the higher modes (\(n\ne 0\)) would be of the order of the Planck mass \(m_p \approx 2.2 \times 10^{-8}\) kg (Collins et al. 1989, p. 295), removing the effects of the higher-dimensional space-time structure from current technological possibilities. What is today known as Kaluza-Klein theory is in fact an amalgamation of different contributions by both scientists. A detailed account of their various contributions can be found in Goenner and Wünsch (2003).A modern version of Kaluza-Klein theory can, for instance, be found in Kaku (1993), Collins et al. (1989). Now

*G*to become a dynamical variable. This constant appears in Newton’s law of universal gravitation

*F*between two masses, \(m_1\) and \(m_2\), separated at a distance

*r*. It is also featured in Einstein’s field equations of general relativity, sketched at in ( 3.14)

*c*denotes the speed of light in a vacuum. Hence, in Brans-Dicke gravity, the following substitution is made

### 4.3.2 The Advent of String Theory

Quantum gravity has always been a theorist’s puzzle

par excellence.

Experiment offers little guidance except for the bare fact that both quantum mechanics and gravity do play a role in natural law.

All three quotes from Green et al. (2012a, p. 14). In this respect, string theory has a lot to offer and, indeed, ties together some of the ideas emerging from the early attempts in constructing a unified field theory (Green et al. 2012a, p. 14):The real hope for testing quantum gravity has always been that in the course of learning how to make a consistent theory of quantum gravity one might learn how gravity must be unified with other forces.

Nonetheless, string theory was in fact discovered by accident. Edward Witten, arguably the most important contributor to the enterprise, once remarked (quoted in Penrose 2004, p. 888):The earliest idea and one of the best ideas ever advanced about unifying general relativity with matter was Kaluza’s suggestion in 1921 that gravity could be unified with electromagnetism by formulating general relativity not in four dimensions but in five dimensions.

The evolution of this theory also had many twists and turns. Originally, string theory models were proposed to describe the strong nuclear force in the late 1960s, known as dual resonance models. These developments started with (Veneziano 1968). In 1970, it was independently realized by Yoichiro Nambu, Leonard Susskind, and Holger Bech Nielsen, that the equations of this theory should, in fact, be understood as describing one-dimensional extended objects, or strings (Schwarz 2000). The first manifestation of these ideas is known as bosonic string theory, living in 26-dimensional space-time. See, for instance Polchinski (2005a). One year later, a string theory model for fermions was proposed (Ramond 1971; Neveu and Schwarz 1971). However, these theories, aimed at describing hadrons, i.e., composite particles comprised of quarks held together by the strong force, were competing with another theory which was rapidly gaining popularity. By 1973, quantum chromodynamics had become an established and successful theory describing hadrons. It was formulated as a Yang-Mills gauge theory with aIt is said that string theory is part of twenty-first-century physics that fell by chance into the twentieth century.

*SU*(3) symmetry group, capturing the interaction between quarks and gluons, the gauge bosons in the theory. A special property, called asymptotic freedom (Politzer 1973; Gross and Wilczek 1973), was instrumental in developing the theory, winning a Nobel prize in 2004. Unsurprisingly, in the wake of quantum chromodynamics, the string model became an oddity within theoretical physics.

*Strings and Gravity*

*predicting gravity*” (Witten 2001, p. 130). Indeed, as up to then the merger of gravity with quantum physics proved to be such an intractable and elusive puzzle, this was big news:

Again, Witten quoted in Penrose (2004, p. 896). Unfortunately, at the time not many physicists took the idea seriously. It would take another ten years before string theory would experience the next advancement in its evolution: an event that would propel it into the limelight of theoretical physics. After 1984, string theory was transformed into one of the most active areas of theoretical physics. See, for instance Bradlyn (2009), for a chart of the number of string theory papers published per year from 1973 onward, as cataloged by the ISI Web of Science. Or Google’s Ngram Viewer,[...] the fact that gravity is a consequence of string theory is one of the greatest theoretical insights ever.

^{15}which “charts the yearly count of selected

*n*-grams (letter combinations) or words and phrases, as found in over 5.2 million books digitized by Google Inc (up to 2008)”.

^{16}It is also interesting to graph comma-separated phrases in comparison: “string theory, loop quantum gravity.” This clearly illustrates the predominance of string theory over other proposed “theories of everything,” like loop quantum gravity. While string theory is a theory of quantum gravity originating in the paradigm of quantum field theory, loop quantum gravity has its foundation in general relativity. See, for instance, Smolin (2001) for a popular account of the various paths to quantum gravity, and (Giulini et al. 2003) for a technical one. For a general discussion of loop quantum gravity, consult Sect. 10.2.3.

*Supersymmetry*

In the early 1980s it was realized, that by introducing a crucial novel element into the string theory formalism, some pressing problems could be solved. Inadvertently, a powerful new level of descriptive power would emerge. This missing element was associated with a novel symmetry property, called supersymmetry. Historically, it was originally developed as a symmetry between hadrons, namely a symmetry relating mesons (a composition of a quark and an anti-quark) to baryons (made up of three quarks, like the neutron and proton) (Miyazawa 1966). “Unfortunately, this important work was largely ignored by the physics community” (Kaku 1993, p. 663). Only in 1971, a refined version of supersymmetry was independently discovered from two distinct approaches. In the early version of fermionic string theory (Ramond 1971; Neveu and Schwarz 1971) a new gauge symmetry was discovered, from which supersymmetry was derived (Gervais and Sakita 1971). The second approach was based on the idea of extending the Poincaré algebra described in ( 3.52), resulting in the super-Poincaré algebra (Gol’fand and Likhtman 1971). Then, in 1974, the first four-dimensional supersymmetric quantum field theory was developed (Wess and Zumino 1974). Even ten years before it would have a fertilizing effect on string theory, supersymmetry was understood as a remarkable symmetry structure in and of itself, fueling advancements in theoretical physics. In essence, it is a symmetry eliminating the distinction between bosons and fermions. Now matter particles—fermions described by spinors with 720\(^{\circ }\) rotational-invariance—and force mediating particles—the gauge bosons emerging from the covariant derivatives in the Lagrangian, with 360\(^{\circ }\) rotational-invariance—lose their independent existence in the light of supersymmetry. It also turns out that supersymmetry is the only known way to unify internal gauge symmetries with external space-time symmetries, a marriage otherwise complicated by the Coleman-Mandula theorem (Coleman and Mandula 1967). There is, however, a heavy phenomenological price to pay for the mathematical elegance of supersymmetry. The number of existing particles has to be doubled, as each matter fermion and gauge boson must have a supersymmetric partner, conjuring up a mirror world of Fig. 4.1.

*Q*, which converts bosonic states into fermionic ones, and vice versa. Symbolically, \(Q|B\rangle = |F\rangle \).Infinitesimally, supersymmetric transformations

*Q*can be expressed in group theoretic terms described in ( 3.30), similarly to the example given for the Lorentz group in ( 3.30)

*Q*and \(\varepsilon \) is the usual parametrization parameter. From the Poincaré algebra the super-Poincaré algebra can be constructed by adding

*Q*to the old (bosonic) commutation relations seen in ( 3.52). The new (fermionic) sector of the algebra is now given by anticommutation relations for the

*Q*, similar to ( 3.16), which are defined as

*Q*transform themselves as a 2-component Weyl spinor under Lorentz transformations. This means that the usual four-dimensional theory is broken down to two dimensions via the Pauli matrices \(\sigma ^i, i=1,2,3\). Mathematically, the Pauli matrices are related to the Dirac matrices \(\gamma ^\mu \), introduced in Sect. 3.2.2.1, (given in the Weyl representation) as follows

*X*by virtue of the Pauli matrices

^{17}It should be noted that \(\mathcal M \in SL(2,\mathbb {C})\), establishing a relationship between the Lorentz group and

*SL*(2,\(\mathbb {C}\)). See, for instance Sternberg (1999). A Weyl spinors transforms under this representation as

*Supergravity*

However, at the same time, the idea of supersymmetry was uncovering important novel insights. “Perhaps one of the most remarkable aspects of supersymmetry is that it yields field theories that are finite to all orders in perturbation theory” (Kaku 1993, p. 664). This makes the heavy machinery of renormalization redundant. And, as many times before in the history of physics, tinkering with the mathematical formalism would uncovered new ideas and powerful tools that had the power to unlock new and unexpected knowledge. In the early years, supersymmetry was understood as a global symmetry. Taking the promising step of gauging supersymmetry, that is, by reconstructing it as a local gauge symmetry, a new type of gauge theory emerged. This new theory, called supergravity, is a supersymmetric theory inevitably accommodating gravity (Freedman et al. 1976). Only two years after string theory was given a new twist as “theory of everything,” another viable candidate for quantum gravity had been discovered, fascinating the community of theoretical physicists. Not long after the discovery of the elven-dimensional limit to supersymmetry (Nahm 1978), it was realized in Cremmer et al. (1978) “that supergravity not only permits up to seven extra dimensions but in fact takes its simplest and most elegant form when written in its full eleven-dimensional glory” (Duff 1999, p. 1). Supergravity would provide the impetus for a revival of Kaluza-Klein theory. This allowed \(D=11, N=1\) supergravity to be compactified to a \(D=4, N=8\) theory (Cremmer and Julia 1979), where \(N>1\) describes the extended supersymmetry algebra andI didn’t work on string theory itself, although I did play a role in the prehistory of string theory. I was a sort of patron of string theory—as a conservationist I set up a nature reserve for endangered superstring theorists at Caltech, and from 1972 to 1984 a lot of the work in string theory was done there. John Schwarz and Pierre Ramond, both of them contributed to the original idea of superstrings, and many other brilliant physicists like Joel Sherk and Michael Green, they all worked with John Schwarz and produced all sorts of very important ideas.

*D*denotes the dimensions of space-time. In an influential paper, Witten proved that the structure of the associated four-dimensional gauge-group is actually determined by the structure of the isometry group—the set of all distance-preserving maps—of the compact seven-dimensional manifold \(\mathcal {K}\) (Witten 1981). He showed, “what to this day seems to be merely a gigantic coincidence, that seven is not only the maximum dimension of \(\mathcal {K}\) permitted by supersymmetry but the minimum needed for the isometry group to coincide with the standard model gauge group \(SU(3)\times SU(2) \times U(1)\)” (Duff 1999, p. 2). The next steps were the development of \(N=8\) supergravity with

*SO*(8) gauge symmetry in \(D=4\) anti-de Sitter space

^{18}or \(AdS_4\) (De Wit and Nicolai 1982), and its extension to eleven dimensions, compactified on a seven dimensional sphere \(S^7\) which admits an

*SO*(8) isometry (Duff and Pope 1983). Indeed, the compactification from eleven-dimensional space-time to \(AdS_4 \times S^7\) could be shown to be the result of spontaneous compactification (Cremmer and Scherk 1977). These were certainly very promising developments. Indeed, so much so, that a then 38-year-old Stephen Hawking was tempted in 1980, in his inaugural lecture as Lucasian professor of mathematics at the University of Cambridge, England (Hawking 1980), to divine that \(N=8\) supergravity was the definite “theory of everything” (Ferguson 2011). Indeed (as quoted in Ferguson 2011, p. 5):

He [Hawking] said he thought there was a good chance the so-called Theory of Everything would be found before the close of the twentieth century, leaving little for theoretical physicists like himself to do.

*The First Superstring Revolution*

Alas, things turned out quite differently and supergravity did not fulfill its promising claims. “We therefore conclude that, despite the initial optimism, \(N=8\) supergravity theory is not theoretically or phenomenologically satisfactory” (Collins et al. 1989). Perhaps the most damning problem was the reappearance of non-renormalizability. The infinities meticulously removed from quantum field theory returned to render the theory of supergravity useless. Indeed, all known quantum theories of spin-2 particles, meaning the elusive gravitons, are now known to be non-renormalizable. Yet again, gravity and quantum physics refuse to cooperate. That is, as quantum theories of point particles. This opened up a loophole for string theory, as its theoretical machinery never touched the notion of particles and rested on extended one-dimensional, vibrating strings. Some general references are Wess and Bagger (1992), Buchbinder and Kuzenko (1998), Duff (1999).

^{19}An earlier modification to the original Ramond and Neveu-Schwarz models was conjectured to harbor supersymmetry (Gliozzi et al. 1977), which was proved by Green and Schwarz (1981). Unfortunately, these superstring theories appeared to be inconsistent (Alvarez-Gaume and Witten 1984), plagued by anomalies. Then, later in 1984, an avalanche was triggered by some notable developments. For one, a method was found to cancel the anomalies by assigning the gauge group of the theory to be

*SO*(32) or \(E_8 \times E_8\) (Green and Schwarz 1984). Moreover, a new superstring theory was introduced, called heterotic string theory (Gross et al. 1985). The “first superstring revolution” was ignited. In the words of Witten (2001, p. 130):

Also Hawking realized the potential, aligning his prophecy (Ferguson 2011, p. 213f.):Since 1984, when generalized methods of “anomaly” cancellation were discovered and the heterotic string was introduced, one has known how to derive from string theory uncannily simple and qualitatively correct models of the strong, weak, electromagnetic, and gravitational interactions.

To summarize, five consistent string theories have been developed, living in ten-dimensional space-time. In the low-energy limit, they reduce to \(N=1,2\), \(D=10\) supergravity of point particles. String theory is a bizarre contraption. It alludes to outlandish realms of reality, like ten-dimensional space-time and a mirror world of supersymmetric particles laying latent in the undiscovered weaves of the fabric of reality. It is built up of an extraordinarily vast and abstract formal machinery, blurring the borders between mathematics and physics, as was discussed in Sect. 2.1.4. Yet, at its heart, it has a surprisingly simple and colorful intuition attached to it (Greene 2013, p. 146):In June 1990, ten years after his inaugural lecture as Lucasian Professor, I asked him [Hawking] how he would change his Lucasian lecture, were he to write it over again.

Isthe end in sight for theoretical physics? Yes, he said. But not by the end of the century. The most promising candidate to unify the forces and particles was no longer the \(N=8\) supergravity he’d spoken of then. It was superstrings, the theory that was explaining the fundamental objects of the universe as tiny, vibrating strings, and proposing that what we had been thinking of as particles are, instead, different ways a fundamental loop of string can vibrate. Give it twenty or twenty-five years, he said.

String theory has the potential to unify all known forces within a single framework. The theory could support a marriage between gravity and quantum mechanics, by offering a theory of quantum gravity which is not plagued by unwanted infinities. Finally, it can accommodate the symmetries of the standard model. The six extra spacial dimensions are compactified on special geometries, called Calabi-Yau manifolds (Candelas et al. 1985). They are shapes, wrapping the additional dimensions into tiny packages located at each point in four-dimensional space-time, which reside at length scales not accessible to current experimental probes. Basically, Calabi-Yau manifolds are similar to fiber bundles. Moreover, as the strings still vibrate in all ten dimension after compactification, “the precise size and shape of the extra dimensions has a profound impact on string vibrational patterns and hence on particle properties” (Greene 2004, p. 372). As an example, if the Calabi-Yau spaces have a topology with three holes, as a result, there will be three families of elementary particles (fermions) , as seen in Fig. 4.1. Technically, the number of particle generations is one half of the Euler characteristic of the chosen Calabi-Yau manifold (Candelas et al. 1985).What appear to be different elementary particles are actually different “notes” on a fundamental string. The universe—being composed of an enormous number of these vibrating strings—is akin to a cosmic symphony.

General references for string theory are, for instance (Hatfield 1992), (Polchinski 2005a, b), (Green et al. 2012a, b), (Rickles 2014). Examples of non-technical references are Greene (2004, 2013), Randall (2006), Susskind (2006).

Quote from Duff (1999, p. 326). Technical references on M-theory theory are, for instance Duff (1999), Kaku (2000), Rickles (2014). More details on the issues plaguing string theory—and quantum gravity in general—can be found in Sect. 10.2.2. For the notion of AdS/CFT duality, see Sect. 13.4.1.2.Witten put forward a convincing case that this distinction is just an artifact of perturbation theory and that non-perturbatively these five theories are, in fact, just corners of a deeper theory. [...] Moreover, this deeper theory, subsequently dubbed

M-theory, has \(D=11\) supergravity as its low energy limit! Thus the five string theories and \(D=11\) supergravity represent six different special points in the moduli space ofM-theory.

### 4.3.3 Einstein’s Unified Field Theory

In the following years, many physicists and mathematicians started to study the implications and finer details of Kaluza-Klein theory. See Goenner (2004, Section 7.2.4). Albeit promising, the theory ultimately failed. In 1929, the physicists Vladimir A. Fock summarized the situation as follows (Goenner 2004, p. 105):It appears that the union of gravitation and Maxwell’s theory is achieved in a completely satisfactory way by the five-dimensional theory.

Also the reality status and the meaning of the extra dimensions was seen as problematic. Indeed, a little more than a year after his initial publication on the matter, Klein conceded (Goenner 2004, p. 112):Up to now, quantum mechanics has not found its place in this geometric picture [of general relativity] ; attempts in this direction (Klein, [...]) were unsuccessful.

In 1928, Einstein himself took a leading role in the conceptual development of a unified field theory. A new wave of research ensued. Einstein was tinkering with the equations of general relativity and set out to extend the formalism. On the 10th of June of that year, he introduced the idea of teleparallelism, originally calledParticularly, I no longer think it to be possible to do justice to the deviations from the classical description of space and time necessitated by quantum theory through the introduction of a fifth dimension.

*Fernparallelismus*, which allowed the comparison of the direction of a tangent vector at various points in space-time (Einstein 1928a). Technically, the underlying space-time is a Weitzenböck space-time

^{20}(Gronwald and Hehl 1996). Four days later, Einstein published his first attempt of constructing a unified theory of gravitation and electromagnetism (Einstein 1928b).

Quote from Einstein (1928b, p. 224, translation mine). With this, Einstein would embark on a more than two-decade long scavenger hunt, chasing this elusive goal. At the time, he was probably quite upbeat about the project’s future. His intuition as a physicists had been validated by two very unexpected and profound theories of relativity. An additional motivational factor was perhaps also given by the fact that he had disproved Weyl’s attempts at a unified field theory. On the 10th of January 1929, Einstein published an update (Einstein 1929a, p. 1, translation mine):[Since the publication of the 10th of June] I discovered that this theory—at least to a first approximation—yields the field laws for gravitation and electromagnetism easily and naturally. It is thus conceivable, that this theory will supersede the original version of general relativity.

Unfortunately, there would be more clouds on the horizon. Others doubted the validity of the field equations Einstein presented in Einstein (1929a). To such criticism, on the 21st of March 1929, he responded as follows (Einstein 1929b, p. 156, translation mine):Indeed, it was possible to assign the same coherent interpretation to the gravitational and the electromagnetic field. However, the derivation of the field equation from Hamilton’s principle did not lead to a straightforward and unambiguous path. These difficulties intensified under further reflection. Since then, I was however successful in finding a satisfactory derivation of the field equations, which I will present in the following.

Then, on the 9th of January 1930 (Einstein 1930, p. 18, translation mine):In the meantime, I have discovered a possibility to solve this problem in a satisfactory manner, founded on Hamilton’s principle.

Undeterred, Einstein continued with his quest. He was assisted by Walther Mayer, a mathematician specialized in topology and differential geometry. New technical publications followed (Einstein and Mayer 1930, 1931a, b). Einstein knew that his attempts had opened a Pandora’s box of challenges. Indeed, the very notion of teleparallelism, the original seeding insight, had to be abandoned. On the 21st of March 1932, in a letter to Élie Cartan, he observed (as quoted in Goenner 2004, p. 85):A couple of months ago I published an article [...] summarizing the mathematical foundation of the unified field theory. Here I want to recapitulate the essential ideas and also explain how some remarks appearing in previous works can be improved.

By 1932, Einstein had become increasingly isolated in his research. Most physicists considered his attempts to be ultimately futile. Indeed, from 1928 to 1932 Einstein had been faced with criticism by notable scholars, like Hans Reichenbach, a logical positivist philosopher of science, and Weyl (Goenner 2004). But Wolfgang Pauli was most vocal in his criticism. Already on the 29th of September 1929, in a letter to a fellow physicists, Pauli confessed (Goenner 2004, p. 89):[...] in any case, I have now completely given up the method of distant parallelism. It seems that this structure has nothing to do with the true character of space [...].

Then, on the 19th of December 1929, Pauli wrote a direct and blunt letter to Einstein (quoted in Goenner 2004, p. 87):By the way, I now no longer believe one syllable of teleparallelism; Einstein seems to have been abandoned by the dear Lord.

Einstein answered on 24th of December 1929 as follows (Goenner 2004, p. 88):I thank you so much for letting be sent to me your new paper [...], which gives such a comfortable and beautiful review of the mathematical properties of a continuum with Riemannian metric and distant parallelism [...]. Unlike what I told you in spring, from the point of view of quantum theory, now an argument in favor of distant parallelism can no longer be put forward [...]. It just remains [...] to congratulate you (or should I rather say condole you?) that you have passed over to the mathematicians. Also, I am not so naive as to believe that you would change your opinion because of whatever criticism. But I would bet with you that, at the latest after one year, you will have given up the entire distant parallelism in the same way as you have given up the affine theory earlier. And, I do not wish to provoke you to contradict me by continuing this letter, because I do not want to delay the approach of this natural end of the theory of distant parallelism.

In 1931, a collaborator of Einstein published a review article on teleparallelism (Lanczos 1931). It appeared in a journal whose name can be literally translated as “Results in the Exact Sciences.” Pauli, when reviewing the article, sarcastically remarked (Goenner 2004, p. 89):Your letter is quite amusing, but your statement seems rather superficial to me. Only someone who is certain of seeing through the unity of natural forces in the right way ought to write in this way. Before the mathematical consequences have not been thought through properly, it is not at all justified to make a negative judgment.

A summary of Einstein’s work between 1914 and 1932, appearing in theIt is indeed a courageous deed of the editors to accept an essay on a new field theory of Einstein for the “

Resultsin the Exact Sciences.”

*Preußischen Akademie der Wissenschaften*, can be found in Simon (2006).

### 4.3.4 A Brief History of Quantum Mechanics

After 1932, things became quiet around Einstein’s unified field theory. He spent the remaining years up to his death in 1955 publishing articles on the philosophy of science, the history of physics, and special and general relativity. He was also concerned with quantum mechanics, a subject he continued to be displeased with. Ironically, Einstein himself was instrumental in the creation of the theory.

*Quanta*

*Lichtquanten*. These light quanta, known as photons today, each come with the energy

*h*. Max Planck had proposed this relationship to explain the observed frequency spectrum of black-body radiation (Planck 1901), for which he would receive a Nobel prize in 1918. With the radical and revolutionary assumption that radiation is not emitted continuously but in discreet amounts, Planck was able to solve a puzzle, which had baffled physicists at the time: all previous theoretical calculations of black-body radiation resulted in nonsensical, infinite results. “It is a remarkable fact that so simple a hypothesis [\(E=h\nu \)], even if incomprehensible at first sight, leads to a perfect agreement with everything we can observe and measure” (Omnès 1999, p. 138).

For Planck, postulating quanta was an act of despair: “I was ready to sacrifice any of my previous convictions about physics” (quoted in Longair 2003, p. 339). Indeed, he originally believed that the notion of quanta was “a purely formal assumption and [he] really did not give it much though [...]” (Longair 2003, p. 339). Einstein understood the quantum hypothesis literally to explain the photoelectric effect. This work, and not his groundbreaking publications on special and general relativity, would win him a Nobel prize in 1921. Planck and Einstein’s discoveries led to the quantum revolution.

Einstein’s reservations explicitly dealt with the probabilistic and indeterministic nature of quantum theory. For instance, Born’s interpretation of the wave function as a probability amplitude (Born 1926) or, later, Heisenberg’s uncertainty principle (Heisenberg 1927). Einstein still believed that his unified field theory would shed light on these issues, and “that the quantum mechanical properties of particles would follow as a fringe benefit from [the field theory]” (Goenner 2004, p. 8). In the end, his skepticism stemmed from certain philosophical considerations relating to the nature of reality, of which there is an abundance. For instance, the philosopher Charles S. Peirce proposed the theory ofQuantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not bring us closer to the secret of the “old one.” I, at any rate, am convinced that He is not playing with dice.

*tychism*, where he argued that chance and indeterminism are indeed ruling principles in the universe (Peirce 1892)—a direct antithesis to Einstein’s opinion.

*Entanglement*

*A*would instantaneously change the properties of a particle

*B*, regardless of the distance of separation between the two. Einstein felt victorious, as he did not believe such “spooky actions at a distance” (Kaiser 2011, p. 30) could be possible. But alas, things turned out differently. John Stewart Bell was able to furnish a theorem out of the EPR paradox. He proved that non-locality was indeed endemic to quantum mechanics (Bell 1964). The experimental validation was given years later in Freedman and Clauser (1972) and notably by Aspect et al. (1981, 1982a, b). In trying to expose the outlandish nature of quantum physics, Einstein helped to distill one of reality’s most mind-boggling properties: entanglement, a term introduced by Schrödinger in (1935) to account for the “spooky action at a distance.” Although it seems to imply that in some bizarre way reality is simultaneously interconnected with itself, entanglement does not allow actual information to propagate faster than the speed of light. Hence special relativity is not violated and there are no tenable objections form physics against entanglement. Quite to the contrary, today entanglement plays a central role in the emerging fields of quantum computation, quantum information, and quantum cryptography, the cutting-edge of current technological advancements. Indeed (Nielsen and Chuang 2007, pp. 11f.):

Key to this surge in research was a theorem proved in 1982. It goes by the name of the no-cloning theorem (Wootters and Zurek 1982). In a nutshell (Kaiser 2011, p. xxv):Entanglement is a uniquely quantum mechanical

resourcethat plays a key role in many of the most interesting applications of quantum computation and quantum information; entanglement is iron to the classical world’s bronze age. In recent years there has been a tremendous effort trying to better understand the properties of entanglement considered as a fundamental resource of Nature, of comparable importance to energy, information, entropy, or any other fundamental resource.

This property thwarts any attempts to intercept the communication of information, allowing for a 100% secure transmission channel: quantum encrypted communications cannot, by the laws of nature, be tapped without the signal being affected. This promise of perfect security would be the gold standard in an age of information processing and global computer networks. Experiments have demonstrated the proof-of-concept, for instance Poppe et al. (2004). And real-world applications followed (Hensler et al. 2007):[...] the no-cloning theorem stipulates that it is impossible to produce perfect copies (or “clones”) of an unknown arbitrary quantum state. Efforts to copy the fragile quantum state necessarily alter it.

For more details on entanglement and the no-cloning theorem, see Sect. 10.3.2.1.On Thursday, October 11 [2007], the State of Geneva announced its intention to use quantum cryptography to secure the network linking its ballot data entry center to the government repository where the votes are stored. The main goal of this initiative, a world first, is to guarantee the integrity of the data as they are processed.

In the ebb and flow of history one can sometimes lose track of the peculiarities and coincidences leading to a major advancement in science. The popularization of entanglement and the development of the no-cloning theorem are prominent examples of how a very unlikely group of people can end up being responsible for such revolutionary feats: a loose collaboration of physicists, dabbling in psychedelics, Eastern mysticism, parapsychology, and other esoteric concepts. Indeed, it is often hard to appreciate how drastically geopolitics has influence the development of science and how crucial mindsets and culture can be for setting research agendas. See Kaiser (2011).

Some technical aspects of quantum mechanics can be found in the paragraphs encapsulating the following equations: ( 3.24) on p. 78, ( 3.51) on p. 88, and (4.20) on p. 100.

### 4.3.5 Einstein’s Final Years

Einstein, well aware of the pressing conflicts in his approach, was confident that no one else could claim any certainty on the matter either (Einstein 1956, p. 165f.):For the present edition I have completely revised the “Generalization of Gravitation Theory” [Appendix II] under the title “Relativistic Theory of the Non-Symmetric Field.” For I have succeeded—in part in collaboration with my assistant B. Kaufman—in simplifying the derivations as well as the form of the field equations. The whole theory becomes thereby more transparent, without changing its contents.

For a detailed account of all of Einstein’s works, see Schilpp (1970).Is it conceivable that a field theory permits one to understand the atomistic and quantum structure of reality? Almost everybody will answer this question with “no.” But I believe that at the present time nobody knows anything reliable about it. [...] One can give good reasons why reality cannot at all be represented by a continuous field. From the quantum phenomena it appears to follow with certainty that a finite system of finite energy can be completely described by a finite set of numbers (quantum numbers). This does not seem to be in accordance with a continuum theory, and must lead to an attempt to find a purely algebraic theory for the description of reality. But nobody knows how to obtain the basis of such a theory.

In the end, Einstein’s efforts at a unified field theory are reduced to a footnote in history. What has stayed, is Weyl’s idea of local gauge symmetry, Kaluza’s venture into extra dimensions, and Klein’s compactification scheme. Gauge symmetry would reveal itself as the unifying principle behind the standard model of particle physics, as is discussed in the next section. Additional spatial dimensions, the novel symmetry principle called supersymmetry, and compactification are the fundamental building blocks of string theory to this day.In 1950, after 30 years of intensive study, Einstein expounded a new theory that, if proved, might be the key to the universe.

## 4.4 Unification—The Holy Grail of Physics

This was witnessed in the insights gained from invariance: the emergence of conservation laws, presented in Sect. 3.1, and the fundamental physical classification of matter states and particle fields, discussed in Sect. 3.2. Perhaps the epitome of Weinberg’s dream comes in the guise of unification, the theme with which this chapter began. Indeed, Weinberg was himself instrumental in showing how symmetry principles are instrumental tools for crafting a unified theory of all known forces excluding gravity, leading to the standard model of particle physics. Weinberg in (1992, p. 142):We believe that, if we ask why the world is the way it is and then ask why that answer is the way it is, at the end of this chain of explanations we shall find a few simple principles of compelling beauty.

However, before these groundbreaking insights could be uncovered, some obstacles still needed to be removed for quantum field theory and gauge theory to emerge from the “Dark Age.” The single most damning problem was that the mathematics was still plagued by infinities, the demon of non-renormalizability. Almost simultaneously, Weinberg (1967) and Abdus Salam (Salam 1968) “boldly ignored the problem of the ‘non-renormalizable’ infinities and instead proposed a far more ambitious unified gauge theory of the electromagnetic and weak interactions” (Moriyasu 1983, p. 102). They built on the work by Schwinger (1957) and Sheldon Glashow (Glashow 1961) and developed a spontaneously broken gauge theory by incorporating the Higgs mechanism, discussed in Sect. 4.2.1. Nearly a century after Maxwell’s merger of electricity and magnetism, the next step in unifying the forces of nature was in sight: the electroweak interaction. It is based on the gauge group\(SU(2)\times U(1)\) . Salam, Glashow, and Weinberg were awarded the Nobel Prize in Physics in 1979 for this achievement.Symmetry principles have moved to a new level of importance in this [twentieth] century [...]: there are symmetry principles that dictate the very existence of all the known forces of nature.

The strong nuclear force, responsible for the stability of matter, confining the quarks into hadron, was successfully described as a gauge theory of a new quantum charge, called color (Han and Nambu 1965; Greenberg and Nelson 1977). These are new quantum properties carried by the quarks, just like electric charge is a property of some fermions and bosons, recall Fig. 4.1. Hence the term quantum chromodynamics is used to describe this theory. The gauge potential fields are called gluons and mediate the strong interaction between the color charged quarks. The gauge group isHowever, [renormalizability] could not be proved at the time and the general response of the community of physicists to the Weinberg-Salam theory was best described some years later by Sidney Coleman: “Rarely has so great an accomplishment been so widely ignored.”

*SU*(3) and some mathematical tools can be borrowed from the

*SU*(3) classification in the old quark model provided by Gell-Mann and Ne’eman (1964). However, “it is important to keep in mind that neither the theoretical predictions nor the experimental tests of chromodynamics have yet achieved the level of either quantum electrodynamics or the Weinberg-Salam theory” (Moriyasu 1983, p. 122).

Although, basically, the standard model was created by splicing the electroweak theory and the theory of quantum chromodynamics, it ranks as “one of the great successes of the gauge revolution” (Kaku 1993, p. 363). Technically, the standard model is a spontaneously broken quantum Yang-Mills theory describing all known particles and all three non-gravitational forces.

*SU*(3) generators \(\lambda ^\alpha \), with the corresponding coupling constant \(\hat{g}\) and gluon gauge fields \(G_\mu ^\alpha \), are added to (4.39). The scalar Higgs field is responsible for generating the mass terms without violating covariance, as described in Sect. 4.2.1.

General references are Moriyasu (1982, 1983), Collins et al. (1989), Kaku (1993), Peskin and Schroeder (1995), Cheng and Li (1996), Ryder (1996), O’Raifeartaigh (1997).

## Footnotes

- 1.
Seen in (4.55).

- 2.
This is also called the norm of a vector: \(|\mathbf X | := \sqrt{x_1^2 + x_2^2 + x_3^2}\).

- 3.
Infinitesimal quantities are the cornerstone of the notion of derivatives in calculus. The idea being, that, for instance, a small displacement along the x-axis, \(\varDelta x\), is infinitesimally set to the approach zero, yielding \(\text {d}x\), an abstract non-zero quantity.

- 4.
Called covariant and contravariant behavior.

- 5.
*U*(1) is also called the unitary group. - 6.
Note that for every matrix generator \(\text {X}^k\) there is now an associated vector \(A^k_\mu \).

- 7.
Some authors choose \(-ig\), which changes some technical aspects of the equations, for instance, \(c_2=-1/g\).

- 8.
Recall that this corresponds to superluminal tachyons, as seen in ( 3.55).

- 9.
It should be noted, that although \(\phi \) is a scalar and hence has only one component by definition, gauge invariance requires it to transform as a complex doublet representation of

*SU*(2) , in effect assigning it four components. - 10.
It has the value 1 if (

*i*,*j*,*k*) is an even permutation of (1, 2, 3), or −1 if it is an odd permutation, and 0 if any index is repeated. - 11.
Ordered by publication date.

- 12.
This theory is sometimes called Einstein-Cartan gravity, due to Cartan’s early work on the subject (Cartan 1922).

- 13.
Not to be confused with Felix Klein of the

*Erlanger Program*. - 14.
Disguised as bosonic string theory, see Sect. 4.3.2.

- 15.
Found at https://books.google.com/ngrams.

- 16.
- 17.
For a complex matrix \(A_{ij}\), \(A_{ij}^*:= \bar{A}_{ij}^{t}\), where the operator

*t*transposes the matrix and \(\bar{A}_{ij}\) is the complex conjugate of the complex matrix element \({A}_{ij}\). - 18.
A specific symmetric manifold with a curvature.

- 19.
Nowadays, when people refer to string theory they implicitly mean superstring theory.

- 20.

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