It is often the case in scientific developments that new insights are preceded by periods of confusion. The discovery of the theoretical mechanism of spontaneous local symmetry breaking is no exception to that. There were actually (at least) two sources of misinformation.

First, there appeared to be almost consensus about the limited usefulness of quantised relativistic field theories; it was agreed that they would form a dangerous path to follow, full of internal contradictions and traps. After all, in such theories, particles were represented as strictly point-like, whereas everyone knew that particles must have more complicated internal structures than that. Point-like particles would generate divergent integrals for the forces between them, and only in very exceptional cases, such as quantum electrodynamics, could the divergences be cancelled out by what was called ‘renormalisation procedures’, but these were generally thought to be suspect: these are theories that carry diseases, which can be ‘swept under the rug’, but then they leave behind structures with internal inconsistencies. Thus, all approaches based on field theories were suspect. Papers in which authors made use of arguments that vaguely looked like quantised field theories often began with extensive apologies to the readers, begging them not to take such resemblances too seriously.

This was one of the reasons why Yang–Mills theoriesReference Yang and Mills ^{1 ,} Reference Shaw ² were left mostly untouched in a corner. Yang and Mills had developed a grandiose idea, a generalisation of Maxwell’s theory of electromagnetism, which allowed light quanta themselves (photons) to carry charges. These theories were based on non-Abelian gauge groups. The gauge group that could be associated with ordinary electromagnetism was an Abelian group.

But there was a second confusion, concerning the mechanism of spontaneous symmetry breaking. Whenever a system exhibits some global symmetry, such as baryon number conservation, isospin, or chiral isospin, one may consider two types of symmetry breaking. A symmetry is broken explicitly when local interactions occur that do not respect the symmetry. Rotation symmetry can be broken by an external magnetic field, baryon symmetry can be broken when baryons are allowed to decay or be created, and so on. The energy or mass spectrum, which would be degenerate because of the symmetry, then splits apart, yielding particles with different mass values.

A symmetry can also be broken spontaneously. This happens when not the interactions, but the solutions to the equations break the symmetry. A famous example is chiral symmetry, the symmetry that allows left-rotating and right-rotating massless particles to be separately conserved. The symmetry is believed to be valid, only slightly explicitly broken – but the chiral particles have masses, and so their chiral charges are not conserved at all. It was understood that this symmetry breaking happened because of the presence of very light – nearly massless – particles of negative parity, the pions. This led to the Nambu–Goldstone theorem.Reference Nambu ^{3 ,} Reference Goldstone ⁴ It can be proved quite generally that: if a symmetry is spontaneously broken, there will be massless particles.

For chiral isospin symmetry, this theorem worked perfectly; it is spontaneously broken, and the fact that pions are not entirely massless can be understood by noting that there are weak forces that break chiral symmetry explicitly. What made the situation more complicated was that the theorem was easy to understand in the language of quantised field theories, but it had to be understood to be independent of quantum fields, as the latter were considered to be suspect.

The real confusion at this point was that the theorem was believed to also hold for cases where it hadn’t been proven: spontaneously broken local symmetries. The idea of spontaneously breaking the local symmetries of electromagnetism and the Yang–Mills theory had been proposed several times, by Higgs,Reference Higgs ^{5 ,} Reference Higgs ⁶ Brout and EnglertReference Englert and Brout ⁷ and Guralnik, Hagen and Kibble.Reference Guralnik, Hagen and Kibble ⁸ The latter had looked carefully at the fate of various Abelian and non-Abelian local groups and subgroups. Peter Higgs had observed that the associated scalar field, the Higgs field, would only partly be represented by particles; the rest of the field components would be taken up by the photons or gauge bosons, which would be turned into massive particles. The surviving scalar field particles would also all be massive.

Now this violated the Nambu–Goldstone theorem. ‘Where are the massless Goldstone particles?’ critics would ask. Since Higgs, and the other authors mentioned, could not answer that question, the idea of ‘spontaneously breaking local symmetries’ was not taken up. It should have been noticed that the same mechanism would explain superconductivity,Reference Anderson ⁹ where indeed there are no massless particles.

The real reason why the Nambu–Goldstone theorem does not apply to local gauge theories is that the name was chosen incorrectly: local gauge symmetries can neither explicitly nor spontaneously be broken! A better name for what happens is the ‘Brout–Englert–Higgs–Guralnik–Hagen–Kibble (BEH) mechanism’. Here, the vacuum state is entirely symmetric under the local gauge symmetry, but in order to describe the quantum theory one must perform a gauge-fixing procedure, such that only a small subset of the gauge orbits is included. This gauge fixing procedure applies to the vacuum as well as to states containing particles, such that no massless particles can emerge.

This situation is quite in contrast with what happens with global symmetries. However, if we take the limit where the gauge coupling parameter g tends to zero, we recover the global gauge symmetry. The particles that had previously been absorbed by the gauge bosons to make them massive are now liberated again and emerge as massless particles – the Nambu–Goldstone particles. These aspects of the particle system were not quite understood by some of the researchers, which caused others to turn their backs on such theories, suspecting them to be inconsistent.

For local gauge theories, there are three different types of physical realisations, none of which have to do with symmetry breaking:

1. (1) The massless phase: this is the phase realised in the familiar Maxwell system, and for the non-Abelian case it is the phase described in Chen-Ning Yang and Robert Mills’ original paper.Reference Yang and Mills ¹ The symmetry is quite explicit in the particle spectrum.

2. (2) The BEH phase: this is the phase that in the limit g→0 displays spontaneous symmetry breaking, which is usually meant to be the fact that the lowest-energy state, the vacuum, transforms into a different vacuum state under the symmetry transformations in question. When g ≠ 0, the vacuum is unique again. This has everything to do with the boundary conditions of the systems: global symmetries affect the boundary, while local symmetries do not have to refer to the boundaries. The particle mass spectrum is split; the local gauge symmetry is not manifest.

3. (3) The confinement phase: this phase may be regarded as the electromagnetic dual of the BEH phase; it becomes a global gauge theory not when the electric coupling constant g _e tends to zero but its magnetic dual, the magnetic coupling constant g _m =4π/g _e . In the usual description of this phase, there is a mass gap, just as in the ordinary BEH phase. When precision is asked for, it appears to be extremely difficult to prove that there is a mass gap, but physical arguments for this are very strong.Reference Faddeev ¹⁰

In short, as soon as a symmetry may be regarded as a local symmetry, the action of an element of this gauge group, which may have a very tiny support, can never be strong enough to cause the vacuum state not to be invariant. Furthermore, explicit breaking of such a symmetry would render it totally useless as a mode of describing particle states. It is the local invariance that turns a subset of the physical field degrees of freedom into ghost fields; they are to be removed from the physically observable part of the particle spectrum, and hence do not take part in the dynamical rules. This would become invalid in the presence of the tiniest conceivable explicit breaking of the symmetry, which is why one never attempts to break local symmetries explicitly.

The history of the various events and discoveries that led from the earliest insights into these matters to the full development of the Standard Model has been recounted many times and we need not repeat it here. According to my personal recollection, the earliest descriptions of these developments mentioned Higgs as the father of the Higgs particle. Afterward, issues were raised making historians realise that he should share the honour with Englert and Brout. The papers by Guralnik, Hagen and Kibble came very shortly after these first publications, but I remember Thomas Kibble being too much of an gentleman to engage himself with any such sort of battle. It is not quite clear whether his name would have been attached more firmly to the BEH mechanism had he been a person with a more outspoken character. In any case, he and his group were the first to investigate more details about the group theoretical aspects of this mechanism in the non-Abelian case.

Instead, I now give an imaginary history. If the discoveries had followed a different line, the discovery of the renormalisable algebraic structure of the Standard Model could have proceeded as follows. According to this sequence of arguments, we would have arrived at a lesson that may well be applicable to gravitation, and investigators could have concluded that today’s theory of gravitation is much closer to being renormalisable than usually asserted.

Let us consider the established knowledge of the electric and weak forces somewhere in the 1960s.Reference ‘t Hooft ^{11 ,} Reference ‘t Hooft ¹² We start with the non-renormalisable 4-fermion interaction term that had been identified experimentally:

1. (1) Suppose we rewrote the ‘experimentally observed’ 4-fermion term, for simplicity written as ½ J _μ J _μ , using an auxiliary field A _μ , such that one has −½ A _μ ² + J _μ A _μ in the Lagrangian. We use auxiliary ‘schizon’ fields φ to describe all terms that break the internal SU(2) ⊗ U(1) symmetry.

2. (2) We notice that this modifies the appearance of the Lagrangian, but it does not modify the physics, so the theory is still not renormalisable.

3. (3) Now, however, we see that this is because the kinetic terms for the fields A _μ and φ are missing. We notice that the internal SU(2) ⊗ U(1) symmetry can be regarded as a local one.

4. (4) Suppose we collected all divergent Feynman diagrams with external A _μ and φ lines. Their divergences have the form of the missing kinetic terms; clearly they need counter terms in order to renormalise the theory.

5. (5) If the procedure was carried out sufficiently carefully, we would have found that the local SU(2) ⊗ U(1) symmetry is respected, and therefore the required kinetic terms must take the form of the Yang–Mills action.

6. (6) We add these Yang–Mills terms, as well as the covariant kinetic term for the φ field, and notice that nothing more is needed for the renormalisability of the theory.

Indeed, the φ fields that had to be added are the Higgs field(s). We have the BEH mechanism at work here; it came quite naturally.

The amazing thing is that today’s formulation of the Einstein–Hilbert (EH) action is actually very similar to a BEH theory, and the only reason for gravity not to be renormalisable, as it stands today, is that the theory isn’t finished. We should simply have further pursued the path outlined above. What emerges then is a view of the gravitational force that is not well-known but, in this author’s opinion, it deserves more attention than it gets (see, however, Refs Reference Mannheim13 and Reference Mannheim14).

The BEH mechanism comes into the picture when we split gravity’s fundamental field, the metric tensor g _μν (x), which has ten independent components, into a volume part, basically represented by the determinant of the metric tensor, and a remainder g _μν ¹ with only nine independent parts. The field Ω(x) representing the determinant can now be arranged to be the only field that has a dimension of mass, whereas everything else, both in the Einstein–Hilbert action for gravity and the Standard Model, can be arranged to be dimensionless. In such a setting, Ω(x) plays the role of the Higgs field. By performing local conformal transformations, Ω can be tuned to stay equal to 1, its ‘vacuum expectation value’, while the other nine components can be reduced to just five independent ones by a rearrangement of coordinates (‘gauge fixing’).

One notices that, so far, the theory consisting of the combination ‘Standard Model’ and ‘EH action’ was merely reparametrised, and, being non-renormalisable from the start, continues to be non-renormalisable. Now, however, we can follow the above procedure in a stepwise fashion that had been so successful in lifting the Standard Model itself towards a renormalisable system. Steps 1 and 2 have already been carried out by our introduction of the variable Ω.

Arriving at Step 3, we see that all diagrams with external lines only consisting of g _μν ¹ fields produce quartic divergences, for which the theory in its present stage does not offer any counter terms. Step 4 says that we should collect these terms and carefully examine their symmetry structure while respecting all symmetries of the original theory. The most noticeable symmetry is local conformal symmetry. Since this symmetry is respected everywhere in the theory, the divergences must form local representations of the same symmetry. According to Step 5, this means that there exists exactly one term one can add to the original action as a counter term: the square of the Weyl tensor.

According to Step 6, it is imperative to add this term to the original action, after which we can use it to renormalise away all infinities. It is not just a question of taste to add such a term; the term grows all by itself as a consequence of higher order (one-loop) quantum corrections. So it must be there. What kind of theory do we now get? The resulting theory is indeed renormalisable. The propagator of the g _μν ¹ field obtains a 1/k ⁴ contribution, which makes the action power-counting-renormalisable. However, the theory still is very dubious.

Let us see what kind of physics would result from this theory. By concentrating on plane waves in the 3-direction, one can disentangle the propagators and see where the poles are. What is the nature of the particles it represents? First, one finds all the particles of the Standard Model, unchanged. Also, the propagator for the graviton stays as it is, a particle with mass 0 and spin 2. But we find a new particle, an object with spin 2 and a mass M that is determined by the new Weyl-squared action. It has the 5-fold degeneracy expected for a spin-2 object. However, it is a particle with negative metric. It has been called gravitello in Refs Reference ‘t Hooft11 and Reference ‘t Hooft12.

It is easy to see why the negative metric is inevitable. The 1/k ⁴ behaviour of the g ¹ propagator, necessary for renormalisability, results from the difference between the massless and the massive spin-2 propagators. One of these two (the massive one) must come with the wrong sign. This should be sufficient to dismiss the resulting theory. With negative metric, the probability to produce a gravitello should be negative. Alternatively, one could argue that the energy of the gravitello, both for the mass term and for the kinetic term, should be negative. This would make the entire universe unstable against formation of unlimited amounts of gravitello particles together with divergent amounts of energy in the form of ordinary matter and gravitons (the massless graviton carries positive energy, in spite of the attraction caused by gravitational fields). It has been argued that the negative metric problem can be cured,Reference Mannheim ^{13 ,} Reference Mannheim ¹⁴ but similar arguments during the days preceding the Standard Model did not hold up.Reference Lee and Wick ¹⁵

In this author’s view, the theory can only be cured by attaching fundamental new physics to the gravitello field. This has not yet been successful. One could imagine that, after unravelling the gravitello field from the gravitational field, one could find Einstein’s equation, which in its simplest form (with c=1) reads

$$G_{{\mu \nu }} \, {\equals}\, 8\pi \, GT_{{\mu \nu }} $$

where now we obtain

$$T_{{\mu v}}^{{{\rm matter}}} {\plus}T_{{\mu v}}^{{{\rm gravitello}}} \, {\equals}\, 0$$

with $T_{{\mu v}}^{{{\rm gravitello}}} $ being negative. This could be a way to ensure that the gravitello merely serves as an anvil to represent the new gravity equations. As stated, however, we have not yet been successful in obtaining a completely self-consistent theory along these lines. One difficulty, which has always been there, is that our view of the gravitational interactions cannot be complete without a clear image of what happens under gravitational collapse. Black holes must be accommodated for in a fully integrated theory.Reference ‘t Hooft ¹⁶