In memory of Peter Higgs

April 15, 2024

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In 2013 the Nobel Prize for Physics was awarded to the Irish theoretical physicist Peter Higgs (recently deceased), who in 1964 conjectured a “mechanism” based on a process known as “spontaneous symmetry breaking”¹ able to explain the origin of the mass of the gauge bosons of the electroweak interaction, and more generally, of all elementary particles.

Historical introduction – The quantum revolution

Quantum mechanics is that part of physics that studies the behavior of matter at subatomic scales. It was formulated in the first half of the twentieth century and triggered an authentic conceptual revolution with broad technological implications. Just think that personal computers and smartphones were only possible thanks to the development of “new physics”.

The block diagram in Figure 1 summarizes the main stages that led from the conceptual building of classical mechanics to that of quantum mechanics (QM). It is good to observe that the second does not contradict the first, but simply expands it towards a new class of phenomena that are inexplicable in the first. In this evolutionary process, we did not consider the Planck Hypothesis (black body) which represented the start-up of the quantum revolution.

Figure 1: The main stages that led to the formulation of quantum mechanics. — Figure 1: The main stages that led to the formulation of quantum mechanics

In the block of Figure 1 called “Classical Mechanics”, we read three distinct but equivalent formulations. In the following considerations, the Lagrangian formulation is important. Precisely, any classical particle can be described through a Lagrangian function (or Lagrangian) which in the simplest cases takes the form L = T − V where T and V are the kinetic energy and the potential energy respectively. On this function, it is possible to set a differential equation whose integration leads to the equation of motion of the particle. The advantage of this formulation lies in the fact that it is also valid in the relativistic context and extends to scalar/vector/tensor fields understood as mechanical systems with an infinite number of uncountable degrees of freedom.

In the next block, we find the OQT (Old Quantum Theory), thanks to the famous De Broglie Hypothesis (1923), according to which the motion of a particle of mass m is equivalent to the propagation of a wave called a matter wave. In this way, it was possible to explain the energy levels of the hydrogen atom. They corresponded to stationary states or rather stationary “electronic waves”, therefore not propagating. This correspondence is guaranteed by the relationship E = hν which links the energy to the frequency of the waves through the Planck constant h.

In 1925 the physicists Goudsmit and Uhlenbeck introduced the intrinsic angular momentum of the electron (spin). Despite its name, it is a quantity without a classical analogue, and its exact location would be in the block called Quantum Mechanics (QM). However, Goudsmit and Uhlenbeck’s arguments were phenomenological within OQT.

Returning to 1923, the question burning lips at the time was: what is the wave equation whose solutions are the matter waves conjectured by De Broglie? In 1925, Erwin Scrho¨dinger found the answer, setting up the equation of the same name whose solutions ψ (x, t) described the propagation of matter waves. However, an interpretative problem arose because in general, the aforementioned solutions take on complex values and do not belong to the set of real numbers, as instead happens for the solutions of the D’Alembert equation which governs the propagation of electromagnetic waves.

Schrodinger circumvented the problem by assigning a physical meaning to the real quantity |ψ|² which would thus represent the density of electric charge transported by the particle-wave. But this interpretation was destroyed in 1932 by the discovery of the neutron, for which it was impossible to associate an electric charge density since this particle is electrically neutral. The puzzle was boldly solved by Max Born who reinterpreted the quantity |ψ|² in statistical terms: |ψ (x, t)|² is the probability density of finding the particle at time t in an elementary volume centered at x. In our schematization (Figure 1) we have thus added the last block in which the Born statistical interpretation represents the building block of the Copenhagen Interpretation or the orthodox interpretation of QM.

Classification of particles based on spin

In the QM framework, there is no place for classical quantities such as energy, momentum, angular momentum, etc. or rather, these quantities become observable, as they are intrinsically linked to the measurement process. Mathematically, observables are represented by special objects called Hermi- tian operators, which verify the important property of returning real values in a given measurement process. These operators act on vectors as elements of abstract spaces that mathematicians call Hilbert spaces and which include all the possible wave functions ψ solutions of the Schrodinger equation relating to the quantum system under study. Technically, the elements ψ are “vectors” of the Hilbert spaces. Since the real quantity |ψ (t)|² is linked to a probability, we must necessarily require its conservation by temporal evolution, i.e., independence from the variable t.

Mathematically this is achieved through a unitary transformation performed in the Hilbert space to which ψ (t). belongs. Unitary transformations are equivalent to rotations performed in a Euclidean space and are represented by particular matrices of order n called unitary matrices, whose setU (n) takes on the algebraic structure of a non-abelian group for the usual row-by-column product operation (matrix multiplication). In particular, we are referring to unimodular matrices, i.e., unitary matrices of determinant +1.

Regarding the spin angular momentum, particles are classified into two large families: fermions (half-integer spin) and bosons (integer or zero spin). For example, the electron is a fermion since it has spin 1/2, and for spin 1/2 systems the corresponding symmetry is SU (2) where 2 denotes the dimension of the Hilbert space (actually it is a unit space since Hilbert spaces are infinite-dimensional) associated with the system. The group SU (2) is a subgroup of U (2) since its elements are the zero-trace unimodular matrices. Here S means “special”. The generators of SU (2) are the well-known Pauli matrices².

The second quantization formalism

Historically, the step represented by the last block of Figure 1 is known as the first quantization and allows the description of the non-relativistic subatomic world. We are therefore talking about particles (electrons, protons, etc.) which are nevertheless in a bound state. Think of atomic and molecular spectroscopy, or solid-state physics with its extraordinary technological implications (semiconductors, superconductors, etc.). However, this approach is not able to explain the behavior of relativistic systems, such as those involving scattering processes.

At the time there was therefore a need for a new formalism as an evolution of the first. The first step towards the second quantization which would have incorporated Special Relativity was the famous Dirac equation which led to the discovery of the positron (the antiparticle of the electron). Electrons and positrons annihilate if they interact, as for any massive pair (particle, antiparticle). In the relativistic domain, the reverse process is also defined: if there is sufficient energy it is possible – under appropriate conditions – to create particle-antiparticle pairs, thanks to the famous relation E = mc². It follows that in such a theory we are forced to define quantum fields as primary entities, while particles are the “excited states” of such fields. Phenomenologically, what we call a particle is an epiphenomenon of the corresponding underlying quantum field.

This consists of the second quantization formalism, the novelty of which is that the wave function ψ (x, t) is no longer a simple scalar field or an element of the corresponding Hilbert space, where the Hermitian operators representing the observables of the quantum system under study, but it becomes an operator which we denote by Ψ (x) and which depends on the generic point-event x = (x⁰, x¹, x², x³) of the Minkowski spacetime. The creation/annihilation of particles as excited (or de-energized) states of the Ψ field is achieved through particular non-Hermitian operators, called creation operators and destruction operators.

A notable example is given by quantum electrodynamics (QED) which describes the interaction of fermions without an internal structure (e.g. the electron) and the electromagnetic field in turn represented by a 4-potential A^µ (x) with µ = 0, 1, 2, 3. A typical interaction process is given by electron-antielectron annihilation which gives rise to two photons.

Gauge theories and symmetries

The interaction between a fermionic field Ψ and the electromagnetic field A^µ is usually described in the Lagrangian formalism. In particular, a “lagrangian density” is constructed L, i.e., a lagrangian per unit of “volume” of spacetime. This formalism is typical of both classical and quantum fields since a field is a mechanical system whose number of degrees of freedom is infinite and uncountable. Hence the need to define a lagrangian density.

Definition 1 – A transformation of the type:

Is a gauge transformation if it leaves the lagrangian density unchanged.

Note that the most general gauge transformations are non-local, in the sense that a transformation law of the type (1) depends on the spacetime event point.

Definition 2 – A lagrangian density equipped with gauge transformations defines a Gauge Theory.

In any gauge theory, there are also bosonic fields Φ (x) which in turn can admit gauge transformations. More precisely, the bosonic interaction fields are called gauge fields and the corresponding quanta are called gauge bosons. Being interaction fields, these bosons transport (or mediate, as they say in jargon) the interaction considered. For example, photons are the gauge bosons of the electromagnetic field. Incidentally, photons have zero mass and spin 1. Gauge theories satisfy particular symmetries which are represented by unitary transformations that we talked about in section (2).

The Standard Model of elementary particles

According to the Standard Model, all matter is made up of quarks and leptons (fermions without an internal structure) that interact via four forces:

Electromagnetic interaction.
Gravitational interaction.
Weak interaction.
Strong interaction.

A typical example of force 1. is given by atoms whose electrons are electrostatically bound to the nucleus. Similarly, in a molecule, the component atoms are held together by the Coulomb field. The weak interaction concerns some radioactive decay processes and involves leptons. The strong interaction binds quarks giving rise to protons and neutrons; thus also in the atomic nucleus neutrons and protons are linked by the strong interaction.

The great challenge of fundamental theoretical physics consists in unifying the four fundamental interactions into a single superforce. In this framework, the fundamental interactions are nothing more than different aspects of a single dominant interaction in the primordial universe, whose expansion triggered a succession of phase transitions that determined the distinction between the individual interactions.

The range of an interaction X is determined by the mass m(X) of the gauge bosons. If m(X) = 0 the interaction is long-range. This is the range of the electromagnetic interaction since it is mediated by the photon, a boson of zero mass and spin s = 1 (vector boson). The gravitational field is long-range, so a boson of zero mass and spin 2 is mediated by the graviton (if it exists) because the corresponding classical field is a tensor field of rank 2 (spacetime metric of General Relativity).

The strength of an interaction X is measured by the coupling constant g_X For QED:

That is, the fine structure constant.

Hadrons are strongly interacting particles (protons, neutrons, pions,…). The strong interaction is mediated by eight massive vector bosons, called gluons. The strong interaction is described by a gauge theory called quantum chromodynamics (QCD).

Notation 3: In the description of a force/interaction we note a notable difference between the classical mechanics approach and that of QFT. While in the first the force is introduced in a Newtonian F = ma or equivalently Lagrangian/Hamiltonian way, in QFT a force between particles is nothingother than an exchange of gauge bosons, therefore essentially of particles. For example, according to classical electrodynamics, two electrons repel each other as they are subjected to a repulsive force which is expressed according to Coulomb’s law, but which can however be put in the form F = ma, to be able to determine the equation of motion of a single electron. In QFT/QED however, the two electrons exchange photons, and as a result their trajectories diverge.

Spontaneous symmetry breaking – The Higgs mechanism

The weak interaction is mediated by the vector bosons W ⁽⁺⁾, W ⁽⁻⁾, and Z⁽⁰⁾ which quantize the corresponding fields W ^(±) (x), Z⁽⁰⁾ (x), as demonstrated by Glashow, Salam and Weinberg in the period 1966-1972. It is a short-range interaction because the aforementioned bosons have non-zero mass.

For an energy E > 10² GeV, the electromagnetic interaction and the weak interaction are unified in the electroweak interaction, whose lagrangian density has the symmetry of the group SU (2)×U (1) where × expresses the direct product of individual groups. The electroweak interaction is mediated by four massless vector bosons, called intermediate vector bosons: W₁, W₂, W₃, and B₀. Precisely, Wein-berg and Salam introduced four vector gauge fields W₁ (x), W₂ (x), W₃ (x), and B₀(x) quantized by the bosons which, being massless, determined a long-range interaction. Incidentally, if E > 10² GeV the leptons are also massless.

For energies E < 10² GeV the so-called spontaneous symmetry breaking occurs for which the corresponding lagrangian density is no longer invariant for SU (2)×U (1). Following this, the vector fields W ^(±) (x) , Z⁽⁰⁾ (x) are reconstructed through a linear combination of the fields W₁ (x) , W₂ (x) , W₃ (x) , B₀ (x); the bosons W ⁽⁺⁾, W ⁽⁻⁾, and Z⁽⁰⁾ acquire mass, while photon γ remains “discovered” (i.e., massless) as the gauge boson of the electromagnetic interaction. In such a scenario, leptons gain mass.

Spontaneous symmetry breaking is a physical process conjectured by Goldstone (1961), developed by Higgs (1964)¹ (Figure 2), Kibble (1967), and later by Weinberg and Salam. In this process the resulting ground state of the “new” system is infinitely degenerate, i.e. there are infinitely many physically distinct states with the same energy. However, in the final configuration, there will be only one system state.

Figure 2: In this suggestive image the lagrangian density of the Higgs field appears. — Figure 2: In this suggestive image the lagrangian density of the Higgs field appears

Two elementary examples can make the concept clearer: ferromagnetic substances and the crystalline state of solids. Ferromagnets are characterized by a critical temperature T_c (Curie temperature) such that for T > T_c, the magnetic dipoles are randomly oriented following high thermal agitation. It is a disordered but highly symmetric state since there is no privileged direction along which the individual dipoles align. In other words, the distribution of dipoles is isotropic and the resulting symmetry is rotational or what is the same, spherical. By cooling the ferromagnet to T < T_c, dipoles orient themselves along a given direction. In this new configuration, there is therefore a privileged direction and the resulting symmetry is manifestly cylindrical, which is a weaker symmetry than the spherical one. We conclude that the cooling of the system has resulted in a spontaneous symmetry breaking, and it is important to highlight that this occurs in correspondence with a phase transition (from a disordered but symmetric phase to an ordered phase but with a lower degree of symmetry). The analogy with the electroweak theory is evident: the infinite degenerate states resulting from the symmetry breaking are the analogue of the infinite possible directions of orientation of magnetic dipoles for T < T_c and in the final configuration one and one single direction.

Likewise for the crystalline state of solids. Here the critical temperature is the melting temperature T_f of the solid: for T > T_f the distribution of the ions is random, therefore isotropic, and a rotational/spherical symmetry is therefore reconstructed. For T < T_f the system recomposes the crystalline state with consequent (spontaneous) symmetry breaking. Among the infinite possible crystalline states, only one is realized.

Even in the Higgs mechanism, it is the temperature that determines the symmetry breaking. Precisely, in the primordial universe when the temperature dropped (following expansion) to T = 10² GeV (temperature in energy units) a phase transition occurred with spontaneous symmetry breaking which separated the weak interaction from the electromagnetic one³. From the above, the Higgs mechanism returns the mass to the vector bosons of the weak interaction. This occurs through an unprecedented scalar field called the Higgs field, mediated by the boson of the same name which is manifestly spin 0.

Technically, this is not a fifth interaction (which, by the way, would violate Occam’s Razor Principle). The Higgs field is a kind of quintessence that fills space, and it is a constant field. The particles interact with this field through a coupling term that depends on the particle. The energy E involved in the particle-Higgs field interaction process translates into the mass of the particle itself m = E/c². Using a suggestive metaphor from M.J.C. Veltmann, we can say that particles acquire mass through the Higgs field in the same way that pieces of blotting paper absorb ink. Pieces of paper of varying sizes absorb equally variable quantities of ink. In this allegory, the pieces of paper represent particles, while the ink is the Higgs field.

The Higgs field is necessarily scalar. If it were vector (spin 1 of the Higgs boson) the mass of the particles would be somehow associated with the rotations in physical space. We propose the following suggestive example: if the Higgs boson had spin s = 1, a ballerina rotating on itself would become a system with variable mass depending on the rotation angle. That is, it could gain weight by rotating in a given direction, and lose weight by reversing the direction of rotation.

Experimental verification

For several decades the Higgs boson remained confined to theoretical physics. The experimental verification of the boson took place in April 2012 following the ATLAS and CMS experiments carried out at CERN⁴. The following year Peter Higgs and Franc¸ois Englert were awarded the Nobel Prize for Physics.