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We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances. Therefore, to the same natural effects we must, so far as possible, assign the same causes. Isaac Newton

Behind the concept of mass

Wigner’s approach or Higgs mechanism?

Sunday 24 July 2011 (updated Tuesday 2 August 2011)

The Large Hadron Collider (LHC) at CERN is the most powerful high energy collider in History. It collects a lot of data that physicists are presently analyzing. We expect the LHC to answer some fundamental open questions about the present Standard Model of particles. Two fundamental questions have explicitly motivated the construction of the LHC and some of it detectors: does the Higgs particle exist? Is supersymmetry a symmetry of Nature?

I would like to address in this article the question of the origin of the masses of elementary particles. Mass is one of the greatest problems in today theoretical physics. The well-known puzzle of the missing mass for 90% of our visible universe worries a lot of astrophysicists and high energy physicists.

In his general theory of relativity, Einstein solved one of the first intriguing problems concerning mass. Newton introduced two concepts of mass: the inertial mass which relates acceleration and force, and the gravitational mass which is at the origin of the gravitational force. Einstein takes as a principle that these two masses are in fact the same not only in values (as experimentally observed), but also in essence. This leads naturally to a theory which extends at the same time the theory of gravitation elaborated by Newton and the principle of relativity of movement proposed by Galileo Galilei.

Concerning the origin of the masses of the elementary particles, the situation is quite strange. Two very different approaches are proposed nowadays, and they are far from being equivalent in their substance.

Wigner’s approach and the Poincaré group

One of the most fascinating contribution of mathematical physics to theoretical physics took place in 1939 with the publication by Eugene Wigner of his now famous paper: “On unitary representations of the inhomogeneous Lorentz group”, Annals of Mathematics 40 (1), 149–204 (1939). The “inhomogeneous Lorentz group” is now called the Poincaré group.

In this paper, Wigner classifies irreducible representations of the Poincaré groups and finds the remarkable following result: for massive particle, irreducible representations of the Poincaré group are labeled by two quantities. The first one is the spin of the particle (integer or half-integer) which appears because the rotation group is a subgroup of the Poincaré group. The second quantity is no less than the mass, which appears because translations in space-time are in this group.

To fully understand this achievement, I must elaborate a little about all these technical terms.

After quantum mechanics was formulated in the 1920’s, it was recognized that this theory was deeply connected to, and more importantly structured by, some symmetry groups acting on Hilbert spaces. For instance, the group of ordinary rotations in the $3$-dimensional space plays a very important role in this formalism. Spin, which is an intrinsic property of particles, was understood as label of vectors in irreducible representations of the group $SU(2)$. This group is a two-fold covering of the $3$-dimensional rotation group $SO(3)$.

In the conception proposed by Wigner, the mass becomes an intrinsic property of particles, very like spin and charges. From a pure mathematical point of view, this mass can be any non zero real number. Physically, only positive real numbers are considered.

The case of zero mass is a little bit different because then the spin of the particle is replaced by the so-called helicity, a property specific to massless particles.

This mathematical achievement was considered as a great discovery, because it gave a very natural status to the mass, now associated to an intrinsic property of elementary particles. It so deeply structured minds that the physical terminology of “elementary particle” is nowadays associated to the mathematical concept of “irreducible representation”, those that are considered in the paper by Wigner.

Quantum Field Theory relies heavily of this result to encode and manage properties of elementary particles. In this theory, a “one-particle state” is explicitly defined by a set of numbers, among them the mass, the spin (or the helicity), the charges for the known fundamental interactions, and the parity, a quantity related to the (full) Lorentz group.

The Higgs mechanism

A crucial problem about masses appeared in the 60’s when it became clear that the unification of some fundamental interactions (electromagnetism and weak interactions) would require the use of gauge field theories. Some of the gauge bosons which carry the weak interaction should experimentally have non zero masses, but the mathematical structure of gauge field theories prevents such a non zero mass.

The Higgs mechanism is an ad hoc construction in the Standard Model which permits gauge boson particles to get masses even if the underlying theory is a gauge field theory.

It is worth mentioning that the Higgs mechanism should be properly called the Brout–Englert–Higgs mechanism, or even the Higgs–Brout–Englert–Guralnik–Hagen–Kibble mechanism... In the following, I will continue to call it the Higgs mechanism, keeping in mind the contribution of all the other physicists mentioned here.

In 1933, Enrico Fermi proposed to describe weak interactions (beta decay) using contact interactions between particles. But this theory could not accommodate for various experimental facts.

A new model was then proposed, based on the principle of local gauge symmetries. In this scheme, interactions are mediated by particles called gauge bosons which are modeled by Yang-Mills fields. These gauge bosons are coupled to matter fields via a so-called minimal coupling. This mathematical structure is a generalization of the one used to model electromagnetism whose gauge boson is the massless photon.

Because the weak interaction is a short distance interaction, the gauge bosons of the weak interaction must have non zero masses. This follows from ideas initiated by Yukawa in the 1930’s. But if one strictly follows the principle of local gauge symmetries, this was not possible without adding some other technical complications.

Then enters the ad hoc solution. From a technical point of view, in order to combine gauge principle with massive gauge bosons, physicists where led to introduce auxiliary fields (scalar fields). These fields are used in a tricky procedure to add masses to massless gauge particles. In fact, matter fields are also involved in this mechanism and they also acquire their mass through this mechanism.

The Higgs mechanism is inspired by previous considerations in condensed matter physics. The key point is the spontaneous symmetry breaking procedure which changes and combines some terms in the Lagrangian of the model: some fields are recombined to produce massive particles, like electrons, neutrinos, quarks, $W^\pm$, $Z^0$… The photon itself emerges after this step and it is the only massless particle at the end.

The main consequences of this approach are the following:

  • all particles of the Standard Model are massless particle at the very beginning of the construction, before the Higgs mechanism plays its role;
  • all non zero masses of elementary particles which emerge after the symmetry breaking mechanism (matter particles and bosons) are generated through this mechanism.

In other words, in the Standard Model of particle physics, masses are not attached intrinsically to the fundamental fields of the theory. Mass is an emergent quantity.

Two is one too many

It is very unlikely that two different, and in essence nonequivalent, descriptions of the same physical property can be maintained in the same theory. In fact, it is more than unlikely: it is shocking!

Even if by chance God plays dice, it would not be in his habits to play roulette at the same time…

Classical mechanics offers a lot of different descriptions for the same physical situation. From a mathematical point of view, one can use the Newtonian, the Lagrangian or the Hamiltonian formalism. We know how to relate them because they are equivalent in generic situations.

On the other hand, there is a well-known situation where it was long thought that two nonequivalent descriptions were needed: in Quantum Mechanics the duality between particles and waves has worried a lot of physicists. But the actual mathematical theory of Quantum Mechanics explains that one needs to replace particle and wave by something which is neither one nor the other: a state in a Hilbert space. Depending on the physical situation under study, such a state can reveal itself as a particle or as a wave and this explains the “duality”.

The problem of the definition of the mass is quite embarrassing. Contrary to classical mechanics, we are not dealing with some different mathematical descriptions which could be hopefully reunited in a new mathematical modelling. Contrary to the “old quantum mechanics”, we are not dealing with some wrong physical terminologies which could be hopefully outmatched by a new concept in a new physical theory.

We are dealing with two simple ideas of mass: on one hand mass is an intrinsic property shared by particles, in this sense it is a concept similar to charge and spin; on the other hand it is an emerging quantity which takes one value or another “by accident”.

As said before, elementary particles, in the sense of Quantum Field Theory, get their non zero masses through the Higgs mechanism. But non elementary particles (particles composed of elementary particles, the proton and the neutron for instance) get their mass essentially through the famous relation $E = mc^2$. The energy involved is the binding energy which maintains together the elementary particles inside the composite particle. It is now a fact that the mass of the proton and the neutron is not an intrinsic property but an emerging quantity.

We cannot expect an intrinsic property related in a rigid way to some group theoretical structures to be obtained as an algebraic combination of coupling constants put by hand in the model. Indeed, these coupling constants are not constrained in the Standard Model. We can hope that in the future they will be deduced from a principle. But in the Higgs mechanism approach the mass will always be a recombined emerging quantity and not an input of the theory. If a group is to be used to “classify” masses, then this group will never describe a fundamental symmetry which Nature “uses” at a very basic level. At most it can only describe an emerging symmetry.

What we are really waiting for with the LHC experiments is to answer the following question: is mass an intrinsic property of elementary particle as in the Wigner’s approach, or is it a more phenomenological concept emerging one way or another, for instance in a Higgs-like mechanism?

Well, the LHC could tell us that both propositions are wrong: an entirely new physics should then be required…

Practical consequences

If the LHC answers this question by telling us that mass is an emerging concept, then physicists will have to abandon the Wigner’s approach. In doing so, they will have to discard the Poincaré group as a fundamental group of Nature.

In fact, because in the Higgs approach masses do not play an important role, the Standard Model does not really support the full Poincaré group as a group of symmetry. Only the $SL(2,\mathbb {C})$ group (a two-fold covering of the connected component of the unit of the Lorentz group) is implemented, and the Poincaré group can be used for recombined fields obtained after the symmetry breaking, when left and right Weyl spinor fields are arranged into full Dirac spinor fields with mass.

In General Relativity, the Poincaré group is meaningless in the generic situation where the space-time manifold does not get enough global symmetries: the translations are missing. In favorable situations, translations are encoded with the help of some (global) Killing vector fields.


If one has to renounce to use the Poincaré group, then Axiomatic Quantum Field Theory will be in trouble in its foundations.

This theory was developed in the 1950’s as a tentative to construct a well founded mathematical scheme for Quantum Field Theory. As one of its “axioms”, this theory poses that elementary particles are modeled as vectors in irreducible representations of the Poincaré group, so that the Wigner’s approach is at the heart of this theory.

Unfortunately, this approach to Quantum Field Theory has never succeeded in its ambitions to give us a practical structure to predict or to compute something new. One of its successes concerns the PCT theorem which can be proved rigorously using its axioms. The “P” and “T” symmetries are not captured by the $SL(2,\mathbb {C})$ group used in the Standard Model because they relate the four connected components of the Lorentz subgroup of the Poincaré group.

This axiomatization of Quantum Field Theory has been elaborated before the construction of the Standard Model and before the systematic use of (local) gauge symmetries. It is a pity that gauge bosons have never been implemented in a satisfactory way in this theory.

We can hope that the LHC could tell us why this theory has failed until now and that it could give us new insights on how to cure it…


The second consequence I would like to mention concerns the physics of elementary particles. As mentioned before, the LHC has been build to answer two open questions raised by theoreticians. The first one is about the Higgs particle and more generally the Higgs mechanism. The second one concerns supersymmetry.

For the moment, supersymmetry is a pure speculation which raises a lot of problems and solves only a few. No experimental evidence has been found yet to confirm it. On the contrary, the “space of possibilities” (space of parameters) is shrinking months after months for now 2 or 3 years (thanks to combined results from the LEP, the FermiLab and now the LHC).

The argumentation developed in the following does not relies on these technical considerations. It is not based on any computation and experimental results. It is founded on logic and principles.

Supersymmetry is a “new” symmetry based on an extension of the Poincaré group: it is encoded in some spinor representations of a chosen extension of the Poincaré algebra. So that if the LHC forces us to discard the Poincaré group as a fundamental symmetry of Nature, then supersymmetry would be automatically discarded also. This is a noway escape.

Higgs mechanism and supersymmetry are logically incompatible: they suppose two nonequivalent concepts of mass which any reasonable physicist should find untenable to maintain in the same theory.

Supersymmetry could be conceptually compatible with the Higgs mechanism if it were not based on an extension of the Poincaré group. Is there another formulation of supersymmetry which could bypass the use of the Poincaré group? I am not an expert to answer this question.

If I were a theoretical physicist involved at the same time in Higgs particle phenomenology and in supersymmetry model building, I would really consider to prepare myself for another quest…


Finally, if the mass is not an intrinsic property of particles, like other charges are, then gravity should no longer be considered on the same foot as the electroweak and the strong interactions discovered so far in particle physics.

Indeed, the actual model of these interactions considers as a fundamental ingredient the notion of charge. This charge is the parameter which governs the amplitude of the minimal coupling between gauge bosons and matter fields. For the moment, these charges are taken to be intrinsic properties of particles.

The charge associated to the gravity interaction is believed to be the mass. If mass is no more a fundamental property but an emerging quantity, then gravity cannot be a “fundamental interaction”. It should be considered as an emerging theory. By “emerging theory”, I mean a theory obtained as a limit, an approximation, a particular sector or something similar from a more fundamental theory. For instance, thermodynamics is an emerging theory for which statistical physics is the fundamental theory.

A direct consequence of this fact is that the construction of a quantum version of this theory may involve a very different approach to the one used to quantize gauge field theories so far considered in particle physics. As an illustration, let me recall that the “quantum version” of thermodynamics is obtained from the quantum version of statistical physics, which is not strictly speaking a “canonical quantization” of “classical statistical physics”. It required a lot of subtle steps to be elaborated. The direct quantization of thermodynamics would have been a wrong idea.

Another possibility could be that charges considered in “fundamental interactions” are no more intrinsic properties of particles, and that they should be viewed as emerging coupling constants like the mass. Obviously, this would require a complete change of paradigm in today physics.

Conclusion

In order to answer new questions, science often brings in the game new ways to look at old concepts.

For a long time we have believed that mass was an intrinsic property of the fundamental constituents of matter. The Higgs mechanism forces us to take a very different look at this old conception of mass. Now we are waiting for the verdict of the experiment. Before this arbitration, we can draw some important conclusions about our models.

With the help of experimental facts, logic and principles should be used with perspicacity to discard inconsistency and redundancy in our present theoretical scheme. This would benefit the foundations on which we can elaborate our future theories.