In undergraduate electrodynamics courses, it is generally just a given. The polarisation vector is perpendicular to the k-vector, meaning the electric (or magnetic) field vector oscillates in a plane perpendicular to the direction of propagation of the wave. A consequence of this is that electromagnetic polarisation can be described with a 2 element vector (at least in the fully polarised case). We expect a vector in 3D space to have 3 components, but confining it to a plane reduces the dimensionality.

One approach to answering this question is to appeal to Maxwell's equations. One can derive the electromagnetic wave equation, write down the general form of the solution and show that it can only be a solution if the polarisation vector is perpendicular to the k-vector.^[1] This is of course a reasonable answer, but I don't find it satisfying because it doesn't give an intuitive feel for why this has to be the case.

What about the quantum mechanical view? The classical polarisation states map onto the projections of the spin vector, labelled m_s, at least in terms of the degrees of freedom. Photons are spin-1 bosons, so we expect to find 3 values of m_s, (+1,0,-1); we even call spin-1 states ‘triplet’ states based on this assumption. But it turns out that for photons this is not the case. Photons have no m_s = 0 spin state, just m_s = +/-1, labelling right and left circular polarisations. The two quantum polarisation states of a photon may be familiar from Alice and Bob's communications using quantum cryptography.

The conclusion seems to be that whether viewed from the classical or quantum perspective the details of electromagnetic wave polarisation are consistent with a transverse wave. This is perhaps not surprising, but it still begs the question, why? A clue can be found when we look outside electromagnetism at the broader electroweak theory. In electroweak theory, there are 4 gauge bosons, all spin-1, of which the photon is one. The other 3 gauge bosons are the uncharged Z⁰ and the charged W⁺ and W^-, together responsible for mediating the weak nuclear force. These latter 3 gauge bosons have polarisation vectors that are not confined to the plane perpendicular to the k-vector, which means that the waves describing them are not, in general, transverse waves.^[2]

Why would polarisation be different in this way for the photon and its weak force mediating cousins? The most obvious difference is the rest mass, which is zero for the photon and non-zero for the others. A consequence of the photon having zero rest mass is that it travels at the speed of light as observed from any inertial frame. Let's do a thought experiment: consider watching a photon fly by at the speed of light. Now imagine that the photon is holding a metre ruler oriented along its direction of travel. What would the length of the metre ruler be as measured by you? We know that lengths along the direction of travel relative to an observer are contracted according to l = l₀/γ.

$$l = l_0\sqrt{1-v^2/c^2}$$

For v = c, the expression in the square root will be equal to zero and hence the observed length of the metre ruler will be contracted to zero, as would any length measured along the direction of travel. The components of vectors transform like the corresponding coordinates in a given space, so in fact the component of any vector along the direction of travel will similarly be contracted to zero. Another way to think about this is that the plane perpendicular to the k-vector, into which the polarisation vector is constrained, is the whole of 3-dimensional space, length-contracted to zero size along the direction of travel!

Like any really satisfying explanation, this is almost obvious once you see it, but in discussing this with other physicists, it appears to me that this is not a widely appreciated explanation. There may or may not be practical benefits to this insight, but it does give an intuitive understanding of the phenomenon.

1. Classical Electrodynamics, 3^rd Ed. (1999), John David Jackson, 047130932
   See the discussion of plane waves in a non-conducting medium, pages 295-297.
2. The Ideas of Particle Physics, 4^th Ed. (2020), James Dodd and Ben Gripaios, 978-1108727402
   See the discussion of the consequences of vector (i.e. spin-1) bosons with non-zero rest mass on page 99.

How would you respond to the assertion that energy is the time component of momentum?^[1][2] It might not at first glance seem reasonable; they are dimensionally different, after all. Also, we spent years learning about them and nobody ever mentioned that they could be the same thing. So what would it take to convince you of this assertion? The ideal approach is to start with a simple statement that everyone can agree on and then derive the conclusion through a set of logical steps. So let's give it a try.

According to special relativity, we live in a 4 dimensional universe with one time dimension and three spatial dimensions. To identify a location within that 4-dimensional space, a 4 component position vector is needed, the ‘4-position’. Such a location is referred to as an event, because it identifies both a time and a place. The spatial components used will depend on the chosen coordinate system; for the present discussion, we will use cartesian coordinates, resulting in a 4-position (t, x, y, z). This approach uses the Minkowski 4-vector formalism. These calculations are usually done using units in which c = 1, making time and space dimensionally equivalent, but if you would prefer to use SI units then the 4-position is (ct, x, y, z).

Now let's take the 4-position vector and apply a Fourier transform to it. The Fourier transform is a linear operator, so it applies component-wise, with no mixing between components. So the Fourier transform of the 4-position is a 4-vector whose components are simply the Fourier transform pairs of the coordinates, (ω, k_x, k_y, k_z). This is called the 4-wavevector and it is the frequency space counterpart of the 4-position: a vector frequency, for which the time component is temporal frequency and the spatial components are spatial frequency.

A frequency space equivalent of the Lorentz transformations could be defined: in the same sense that different observers in different inertial frames will disagree on what is time and what is space, they will also disagree on what is ω and what is k. Now all we have to do is take our 4-wavevector and multiply it by ħ to get another 4-vector, for which the time component is ħω and the spatial components are ħk, explicitly: (E, p_x, p_y, p_z), the 4-momentum, with energy as the time component.

Of course, it is in a sense an arbitrary decision to call it momentum; we could just as well say that momentum provides the spatial components of the 4-energy, or invent some new name altogether (conventionally and unimaginatively it is referred to as “energy-momentum”). And of course, different observers in different inertial frames will disagree on what is energy and what is momentum. We spent years learning about energy and momentum and yet no one ever thought to tell us that they were ultimately the same thing.

Some may complain about the statement that energy and the other components of momentum are “the same thing” and with some justification. There are geometric differences described by Minkowski geometry; in particular, timelike vectors have real magnitudes and spacelike vectors have imaginary magnitudes (the square of the line element, ds², is negative for spacelike vectors). This does feel like a valid challenge, but remember that the Lorentz transformations tell us that observers in different inertial frames will disagree on what is time and what is space. So if you accept the principle of relativity (which we do), that there is no priviliged rest frame so all inertial frames are equivalent, then it is difficult to argue that time and space are not the same thing. They are different as observed from a given inertial frame, but somehow globally have to be the same thing. Any suggestions on untangling this semantic knot would be very welcome.

None of this should prevent us from agreeing that energy is the time component of momentum. And now that we are (hopefully) aligned on that, we can start to ask other interesting questions. Time is the time component of position and energy is the time component of momentum, but what about all the other vectors; don't they have time components? And the answer is yes, they do! There is a set of underlying connections between physical quantities that we may have hitherto considered entirely separate.

Where to begin? The next vector most people ask about is the force, which is natural given its importance for dynamical calculations. This is an easy one, due to its known relationship with momentum. According to Newton's 2^nd law, for a force, F, and momentum, p, we have, $$\mathbf{F} = \frac{d\mathbf{p}}{dt}$$ So the spatial components of the 4-force are just the time-derivatives of the spatial components of the 4-momentum. The time component of the 4-force is then the time derivative of the energy, otherwise known as power. The power dissipated in the application of a force is the time component of the force and again we could define relationships like the Lorentz transformations relating these components seen from the perspectives of observers in different inertial frames. There is a subtlety here, because we have to ask, whose version of time are we differentiating with respect to? The correct answer is that we differentiate with respect to proper time, which means time as measured on a clock following the path through spacetime of the point at which the force is applied. Proper time is represented by τ, where τ = tγ. So the correct 4-force is γ(v)(P, F_x, F_y, F_z), where P is the power transferred in the application of the force.

More interesting connections come from electromagnetism, which has from the beginning been our best window into special relativity, due to the fundamentally relativistic nature of light. Consider the source of the electromagnetic field in Maxwell's equations. The source terms include either ρ or J, the charge density and current density respectively. The components of the current density, J, can be thought of as flux of electric charge along each of the 3 spatial dimensions. Similarly, the charge density, ρ can be thought of as the flux of electric charge along the time dimension. This turns out to be a more general result, important in the definition of the stress-energy tensor of general relativity: flux in time is equivalent to density in space. The units can be seen to match with the expected factor of c, $$\textnormal{Cm}^{-3}\times \textnormal{ms}^{-1} = \textnormal{Cm}^{-2}\textnormal{s}^{-1}.$$

Another interesting example from the world of electromagnetism is the connection between the scalar and vector potentials. The scalar potential, φ is the familiar potential from electrostatics, used to calculate the electric field, E, $$\mathbf{E} = -\mathbf{\nabla}\phi$$ The vector potential, A, is used to calculate the magnetic field, $$\mathbf{B} = \mathbf{\nabla}\times \mathbf{A}$$ By now, it probably comes as no surprise to hear that the scalar and vector potentials provide the time and space components respectively of a 4-potential. As expected the units are different by a factor of velocity.

It is interesting to ponder the different levels of mental gymnastics needed to integrate these relationships into our world view. The idea that time and space are fundamentally the same thing is just baffling, entirely contradictory to our experience. Accepting energy as a component of momentum feels like less of a jump, given that the definition of energy has always been something of an enigma, although it remains perplexing that we were never taught this during years of studying physics. And then there is the simple fact that charge and current are the same thing, viewed from different inertial frames, which is just obvious.

1. Six not so easy pieces, 4^th Ed. (2011), Richard P. Feynman, 978-0465025268
   See the discussion of the energy-momentum 4-vector on page 102.
2. The Road to Reality: A Complete Guide to the Laws of the Universe, 4^th Ed. (2007), Roger Penrose, 0224044478
   See the discussion of 4-vectors on page 433 and the following pages.

If you haven't read the article, “Is energy a component of momentum?”, then I highly recommend doing so before continuing.

In 1686, Isaac Newton published his universal law of gravitation, in which mass is the source of gravity. When Einstein published his work on special relativity in 1905, the world discovered that mass and energy are actually the same thing. In 1907, Minkowski published his work on special relativity, including the startling insight that energy is the time component of momentum. So where does that leave us? Is momentum the source of gravity?

Fortunately Einstein continued thinking about the problem and in 1916 he astonished the world with his general theory of relativity. The source of the gravitational field in general relativity is indeed the energy-momentum 4-vector, but there is more to it than that. The simplest way to understand this is through comparison with the source of the electromagnetic field in Maxwell's equations. Here we will take advantage of the illuminating, but often under-appreciated parallels between gravity and electromagnetism.

The source of the electromagnetic field is of course electric charge, q; however, it is not q but ρ and J that appear as the sources in Maxwell's equations, the charge density and current density respectively. Having hopefully read the article, “Is energy a component of momentum?”, you will already know that ρ and J provide the time and spatial components respectively of the current density 4-vector. Current density is simply flux of electric charge (charge per unit time per unit area) and flux in time corresponds with density in space.^[1]

How does it look if we apply the same principle to gravity, treating energy-momentum as the gravitational “charge”? An obvious difference is the vector nature of gravitational charge, so we expect a tensor source term representing the flux of the 4 components of energy-momentum along each of the 4 dimensions of space-time. $$ \require{colortbl}\require{color} \left(\begin{array}{cccc} \rowcolor{yellowgreen} \cellcolor{salmon} T^{00} & T^{01} & T^{02} & T^{03} \\ \rowcolor{yellow} \cellcolor{yellowgreen} T^{10} & \cellcolor{skyblue} T^{11} & T^{12} & T^{13} \\ \rowcolor{yellow} \cellcolor{yellowgreen} T^{20} & T^{21} & \cellcolor{skyblue} T^{22} & T^{23} \\ \rowcolor{yellow} \cellcolor{yellowgreen} T^{30} & T^{31} & T^{32} & \cellcolor{skyblue} T^{33} \\ \end{array}\right) $$ This is what is referred to as the stress-energy tensor, for reasons that will become clear. The elements, T^μν, represent the fluxes of the four components of energy-momentum along each of the four dimensions of space-time. Here the index μ labels the flux directions and ν labels the components of energy-momentum, with the values 0-3 identifying time and the 3 spatial axes, which we will call x, y and z, respectively.

Starting with the top row, then, the first element is the flux of energy along the time axis, also known as energy density (red). This component on its own corresponds with Newtonian gravity, with mass (energy) density as the source. The remaining three elements of the first row are the densities of the 3 spatial components of linear momentum (green). The concept of density of momentum is not immediately intuitive, but it turns out that density of the k^th component of linear momentum is identically equal to the flux of energy along the k-axis (green): T^0k = T^k0 for k = 1-3.

There remains to describe the 3 × 3 matrix representing the flux of the 3 spatial components of momentum along each of the 3 spatial axes. But what is flux of momentum? Flux is flow of something per unit time, per unit area. We know that momentum per unit time is force and force per unit area is pressure, or more generally stress (pressure being the isotropic special case of stress). So the the remaining 3 × 3 elements of the stress-energy tensor are in fact the stress tensor, familiar from the analysis of deformation in materials. Flux of the i^th-component of momentum along the i-axis corresponds with the i^th-component of force due to pressure (blue). Flux of the i^th-component of momentum along the j-axis, with i ≠ j, corresponds with a component of shear stress (yellow).

It might come as a surprise to learn that the stress tensor, extended into 4 dimensions, is the source of gravity; it certainly surprised me. Nonetheless, the connection is entirely real. In conditions we normally encounter, the contribution of pressure/stress to gravitational attraction is negligible; however, in the interior of a large star undergoing gravitational collapse, the extreme pressure, which might be expected to support the star, adds a significant additional gravitational pull, leading to runaway gravitational collapse.^[2] It all feels a far cry from Newton's not entirely universal law of gravitation.

1. The Road to Reality: A Complete Guide to the Laws of the Universe, 4^th Ed. (2007), Roger Penrose, 0224044478
   See the discussion of the interpretation of the energy-momentum tensor and the analogy with the sources in Maxwell's equations on page 456.
2. The Road to Reality: A Complete Guide to the Laws of the Universe, 4^th Ed. (2007), Roger Penrose, 0224044478
   See the discussion of the implications of pressure as a source of the gravitational field for a collapsing star on page 463.

The choice of potatoes is of course arbitrary. When I first starting discussing this with people, I used the mass of the Earth as an example, but I decided to use something more immediately familiar, hence potatoes. In particular, I wanted to make clear that the mass I am talking about is truely the mass we measure (indirectly via weight) when we put the potatoes on a balance. In fact, any material composed of atoms and molecules would do.

Ultimately this is a question about the masses of protons and neutrons, which make up the overwhelming majority of the masses of atoms and molecules. So we can boil down the title question to the perhaps more tractable question: what fraction of the masses of protons and neutrons are due to the rest masses of their constituent quarks?

The clue is in the fact that protons and neutrons have almost identical masses, even though their quark composition is different.^[1] The neutron mass, m_n, is greater than the proton mass, m_p, by approximately 1 part in 1000. But the neutron is composed of two down-quarks and an up-quark, while the proton is composed of two up-quarks and a down-quark, and the down-quark mass, m_d, is approximately double the up-quark mass, m_u.

• \(m_\textnormal{n} = 939.56542052(54)\textnormal{ MeV}\)
• \(m_\textnormal{p} = 938.27208816(29)\textnormal{ MeV}\)
• \(m_\textnormal{d} = 4.7^{+0.5}_{-0.3}\textnormal{ MeV}\)
• \(m_\textnormal{u} = 2.2^{+0.5}_{-0.4}\textnormal{ MeV}\)

The neutron and proton masses are very precisely measured quantities, but the quark masses are not directly observable and have to be inferred through processes that result in a large fractional uncertainty. Nonetheless, the ratio m_u/m_d is thought to be in the region of 0.5, with reasonable confidence. Added note: a lattice QCD approach demonstrated in 2010 appears to have dramatically reduced the uncertainty on the light quark masses, finding a ratio of 0.420.^[2]

For the purpose of illustration, let's try the calculation using a ratio of precisely 0.5 and further assuming that the contribution on top of the elementary particle rest masses, Δ, is equal for protons and neutrons. The rationale for this latter assumption is that the strong force is symmetric under exchange of protons and neutrons, only electromagnetism breaks the symmetry. I do realise that this does not fully motivate the assumption, because it assumes that the strong force is the dominant effect, which has not been shown. It does turn out to be the case and it simplifies the calculation, so I hope you can forgive me. $$m_\textnormal{d} = 2m_\textnormal{u}$$ $$ m_\textnormal{n} = m_\textnormal{u} + 2m_\textnormal{d} + \Delta = 5m_ \textnormal{u} + \Delta $$ $$ m_\textnormal{p} = 2m_\textnormal{u} + m_ \textnormal{d} + \Delta = 4m_\textnormal{u} + \Delta $$ We can now use the known m_n/m_p ratio to establish the relationship between Δ and m_u. $$m_\textnormal{n} = cm_\textnormal{p}$$ $$5m_\textnormal{u} + \Delta = c(4m_\textnormal{u} + \Delta)$$ $$ \Delta = \frac{5 - 4c}{c-1} m_\textnormal{u} \equiv k m_\textnormal{u} $$ We are now ready to address the title question, calculating the fraction \(\mathcal{F}\) of a kg of potatoes composed of particle rest mass. We will use the further assumption that potatoes contain an equal number of protons and neutrons (a good approximation for the light elements found in organic matter). $$ \mathcal{F} = \frac{9m_\textnormal{u}}{9m_\textnormal{u} + 2\Delta} = \frac{9}{9+2k} $$ For a neutron mass greater than the proton mass by 1 part in 1000, as observed, we have c = 1.001, corresponding to k = 996 and hence \(\mathcal{F} = 9/2001\), or approximately 0.5%. Explicitly stated, if we add up the rest masses of all the elementary particles in a kg of potatoes, they only contribute on the order of 1% of the actual mass!

So where does the rest of the mass come from? We know that all mass is energy, so we're just looking for sources of energy. We want to add up all the kinetic and potential energy in the potatoes (yes, they are all part of the mass). We generally share an inertial frame with potatoes while weighing them, but there is still kinetic energy in the thermal vibrations within the potatoes; however, this will not be a significant contribution at any remotely plausible temperature. Potential energy is associated with interactions between particles and fields. It turns out that the overwhelming majority of the mass of a kg of potatoes is due to the potential energy of the colour charged quarks and gluons interacting with the strong field within the protons and neutrons.^[3] It is worth remembering this when you hear that interaction with the Higgs field is the “origin of mass”.

1. Where does mass come from?
   Symmetry Magazine, 12-May-2016
   Accessed 30-Mar-2022
2. Precise Charm to Strange Mass Ratio and Light Quark Masses from Full Lattice QCD
   C. T. H. Davies, C. McNeile, K. Y. Wong, E. Follana, R. Horgan, K. Hornbostel, G. P. Lepage, J. Shigemitsu, and H. Trottier (HPQCD Collaboration)
   Phys. Rev. Lett. 104, 132003 – Published 31 March 2010
3. The Road to Reality: A Complete Guide to the Laws of the Universe, 4^th Ed. (2007), Roger Penrose, 0224044478
   See the comment on the strong interaction contribution to hadron masses on page 645.

All mass is energy and all energy is mass, as Einstein famously showed, with his revelation that E = mc². In spite of its status as the most recognisable equation of all time, it is easy to overlook the implications of this relationship. If mass is energy, then the sources of mass are simply the sources of energy, including all kinetic and potential energy.

Consider a ball flying through the air. Its mass as observed from your inertial frame will be larger than that of the same ball at rest, due to the kinetic energy, K, of its motion by K/c². The ball at rest will have a larger mass than the same ball at absolute zero of temperature by the amount of the thermal kinetic energy stored in the vibrations of the molecules making up the ball. Einstein’s 1905 article, “On the electrodynamics of moving bodies”^[1], that introduced special relativity to the world, used this as an example, showing that the mass of a quantity of gas increases with temperature by the amount of its thermal kinetic energy.

Potential energy is also a source of mass. A compressed (or stretched) spring stores elastic potential energy, U, and the mass of the spring is greater than the mass of the uncompressed spring by U/c². The same effect can be seen in the microscopic view in atomic binding energy. A hydrogen atom has a lower mass than the combined mass of its constituent proton and electron. Imagine the bond between proton and electron as a spring in which potential energy is stored as they are pulled apart; it is not a harmonic potential, but there is a restoring force. The mass of the hydrogen atom increases as additional potential energy is stored. The greatest mass achievable occurs when the proton and electron are separated to infinity.

The binding energy of a system of particles is equivalent to the work done on the system to assemble the particles from infinity at a constant speed (any acceleration would require additional work to overcome the inertia). Consider a pair of electrons with infinite separation: work must be done on the system to bring them together, overcoming their mutual repulsion, adding energy and increasing the mass of the system. Now consider an electron and a proton with infinite separation: they have a mutual attraction and hence they do the work to bring themselves together, losing energy and hence decreasing the mass of the system. A more rigorous, although ultimately equivalent way to see this is to consider the work, W, needed to maintain the constant velocity against the mutual attraction that would tend to make them accelerate. In that case the applied force, F, would point outwards, in the opposite direction to the displacement line element, ds, along the path, \(\mathcal{P}\), making the work negative: $$ W = \int_\mathcal{P} \mathbf{F}.d\mathbf{s} $$ This explains why the maximally stretched case for the hydrogen atom, which is the maximum mass case, is equal in mass to the rest masses of the isolated proton and electron. For a bound system, the potential energy must be lower than that of the isolated constituents, otherwise it would not be bound.

If all mass is energy then you might expect that massless particles had zero energy, but in fact this is just confusing terminology. Photons are conventionally referred to as massless particles, even though they have inertia and exert a gravitational attraction, the two defining properties of mass. In fact, they have zero rest mass, which tells us that in the rest frame of a photon it would have zero mass (and hence also zero energy), which means that the energy of a photon is entirely kinetic energy. If you were to accelerate into a photon’s rest frame then it would cease to exist! It is no coincidence then that nature conspires to make it impossible to share an inertial frame with a photon, because they travel at the speed of light with respect to any inertial frame.

In an undergraduate course on special relativity, students typically don’t spend much time thinking about the implications of E = mc². The equation is just an identity between two quantities and is not a practical way to perform calculations. The energy-momentum relationship is much more practical tool: $$ E = \sqrt{p^2c^2+m_0^2c^4} $$ Nonetheless, they are both expressions for total energy and hence they are equivalent. The only assumptions needed to derive one from the other are the definition of momentum, p = mv = γm₀v, and the relativistic mass formula, m = m₀γ. $$ E^2 = m^2v^2c^2 + m^2\left(1 - \frac{v^2}{c^2}\right)c^4 $$ $$ E^2 = m^2c^4 $$ $$ E = mc^2 $$

The take-home message is simple. All energy of any form has inertia and exerts a gravitational attraction. All energy is mass and all mass is energy.

1. On the electrodynamics of moving bodies [English translation]
   A. Einstein
   Annalen Phys. 17 891-921 (1905)
   

If you haven’t already read “What is mass?” and “What fraction of a kg of potatoes is particle rest mass?”, then I suggest doing so before continuing. Readers who have read the aforementioned pieces may have noticed an implicit contradiction. For the sake of clarity, I will summarise the contradictory conclusions here before attempting to explain and resolve the contradiction.

In “What is mass?” we considered the case of binding energy in a hydrogen atom. Defining binding energy as the work done to assemble the constituent particles from infinity at constant velocity (any acceleration would require additional work to overcome inertia) gives a negative binding energy for the attractive 1/r potential in a hydrogen atom. As a result, the mass of the atom is lower than the isolated proton and electron. In “What fraction of a kg of potatoes is particle rest mass?” we showed that the masses of the constituent (valence) quarks in protons and neutrons, and hence in all matter composed of atoms and molecules, is only on the order of 1% of the total mass.

By the definition of binding energy given above, a bound system must have lower mass than its constituent particles at infinite separation. And yet a proton is a stable bound system of particles with mass far higher than the rest masses of its constituent particles. In pondering this conundrum, I always felt that the resolution was likely to come from improving my understanding of the strong interaction, given that electromagnetic interactions in a hydrogen atom are much more familiar. Nonetheless, the root of the conundrum seemed to go beyond any specific force and relied simply on the fact that force points down the gradient of potential energy: if energy (and hence mass) is lower at infinite separation, then surely the interaction must be repulsive?

An important distinction between protons (or other hadrons) and systems bound by the electromagnetic force is that the constituent quarks have never been observed in isolation. This fact has led to a hypothesis that quarks may be permanently confined within hadrons.

There is convincing experimental evidence for a linear rise in potential energy, rising above the rest energy of the isolated constituents, as the quarks are separated. This evidence comes from the so-called charmonium spectrum.^[1][2] Charmonium is a heavy meson composed of a charm quark and charm antiquark. The ground state of charmonium is the famous J/ψ meson first observed in 1974, with a mass of 3.0969 GeV/c². As particle accelerators achieved ever higher energies, additional new mesons were observed that were ultimately understood to be excited states of charmonium. Representing these states graphically in terms of energy and angular momentum produces something that looks reminiscent of the familiar atomic hydrogen energy-level diagram, but with clear distortions. The distribution of the energy levels is consistent with a function for potential energy, U that combines an attractive 1/r potential at short range with a linear rise in r at larger separations, the so-called Cornell potential: $$ U = -\frac{4}{3}\frac{\alpha_s}{r} + \sigma r $$ Where α_s is the quantum chromodynamics (QCD) running coupling (the 1/r term here is explicitly part of the strong interaction and is not related to the attractive electromagnetic potential that exists independently of this) and σ is referred to as the QCD string tension. When it was developed in the 1970s, the Cornell potential was entirely emipical, based on a fit to the observed charmonium energy levels; however, the form has since been computed from first principles using lattice QCD. Note that the radial separation in the ground state is already sufficient to be dominated by the linear term, giving a positive contribution to total energy from the strong force potential energy.^[3]

For an intuitive description of what this means for the behaviour of the strong field, theorists developed the concept of the ‘flux tube’, flux here referring to classical field lines. The electric field lines emerging from a proton point radially outwards. The electric field around a proton points outwards along a field line, with a magnitude proportional to the local density of field lines. This is consistent with the 1/r² dependence of electric field strength because the density of field lines is normalised over area on a sphere of radius r, which is proportional to r². The corresponding strong force field lines, connecting colour charged particles, are stretched into a bundle of parallel lines as the charges are separated, forming a tube with a constant density of field lines. This behaviour results from the non-Abelian nature of quantum chromodynamics, leading to self-interacting gauge bosons; the colour-charged gluons all attract each other, leading to the compact flux tube geometry. The constant field implied by the flux tube description results in a constant force, independent of separation. A constant force corresponds with potential energy varying linearly with separation, consistent with the charmonium spectrum, and equivalently a constant rate of work done (at constant velocity) per unit distance.

It is expected that once the total energy rises to twice the J/ψ meson energy, an additional charm quark-antiquark pair will be produced, resulting in a pair of ground state charmonium mesons. Charm quarks have a large rest mass (1.275 GeV/c²), greater than the mass of the proton and more than 500 times that of the up quark, so current particle accelerators have not achieved the energies required for this. Beyond the limit of the observed charmonium spectrum, there is currently no observational evidence for the behaviour of the potential energy curve. Above the pair production threshold it may not be observable even in principle. Note that high mass of the charm quark makes charmonium amenable to calculations based on well-understood, non-relativitic quantum mechanics, a fact that been of great value to theorists trying to derive understanding of the basic interactions of quarks.

So where does all this leave us with the title question? If the potential energy curve can be taken to drop to zero (equal to the constituent rest masses) at infinity, then the contradiction can be resolved with a simple redefinition of binding energy: the work done to assemble the particles from the potential energy maximum at constant velocity; for the hydrogenic case this redefinition makes no difference. For the hypothesised linear rise to a large positive energy before dropping to zero at infinity, the bound state is necessarily a metastable state, subject to tunnelling through the potential barrier to leave free quarks. Indeed, any hadron that did make its way past the potential barrier, either through tunnelling or acquisition of sufficient kinetic energy to overcome the barrier, would experience a repulsive interaction. The fact that no free quarks have ever been observed may simply reflect a vanishingly low tunnelling rate due to a high potential barrier.

An alternative interpretation is that the potential energy curve just keeps on rising, meaning that there really is no way to escape the potential well because the potential energy rises to infinity at infinite separation. This interpretation would be consistent with the permanently confined quark hypothesis. There is no violation of conservation of energy here; the potentially unbounded rise in energy is only accessible to the extent of the work that can be done on the system in pushing it apart. The case of continuously rising potential energy does not require any redefinition of binding energy, but it loses the satisfying identification of infinite separation with free elementary particles.

Between these two limiting cases, there are intermediate cases, for which the system asymptotes to some finite energy at large separation. In particular, this includes the monotonic case, with no repulsive interaction, just an attractive force that asymptotes to zero, which would constitute a stable bound state. All these cases are equally consistent with experimental observations to date. I find the case of dropping to zero at infinite separation the most intuitively plausible, but I don’t claim that as any kind of evidence.

Has the contradiction been resolved? There are clearly still gaps in our knowledge and we are faced with the unpalatable prospect of truly unknowable aspects of the physics. Nonetheless, I feel that at its core the contradiction has been resolved: a system will remain effectively bound if the potential energy barrier at larger separations is sufficiently high, whatever the form of the potential as the separation tends to infinity. The conclusions drawn for the hydrogenic case in “What is mass?” were based on the assumption that the whole of the potential energy curve for a bound system must be negative and we have good evidence that that is not the case for strong force interactions.

1. The Ideas of Particle Physics, 4^th Ed. (2020), James Dodd and Ben Gripaios, 978-1108727402
   See chapter 35 Quarks and Charm on page 171 and the following pages.
2. Nonstandard heavy mesons and baryons: Experimental evidence
   S. L. Olsen, T. Skwarnicki and D. Zieminska
   Rev. Mod. Phys. 90 015003 (2018)
3. QCD forces and heavy quark bound states
   G. S. Bali
   Phys. Rep. 343 [1-2] 1-136 (2001) [arXiv]

I have heard and read various descriptions of how the Higgs field generates particle rest mass.^[1][2][3] It is a complex topic, so it is perhaps unsurprising that none of them struck me as particularly intuitive, but I do feel that there is a simple logical argument that can motivate the principle from some basic, well-founded assumptions. Here I have used the qualifier particle rest mass to indicate that we are talking about an entity with no observed internal degrees of freedom. Composite bodies, such as molecules, mangos and mountains can still have kinetic energy in the rest frame due to internal vibrations. Subsequent use of the term rest mass in this piece should be interpreted as particle rest mass.

The argument I shall present here is based on three founding assumptions that I hope and expect will be uncontroversial. The assumptions are as follows:

1. E = mc²
   
   All energy is mass and all mass is energy. If you haven’t thought much about the implications of this, then I strongly recommend reading the piece, “What is mass?” before continuing. In searching for the source of particle rest mass, we are searching for a source of energy consistent with the properties of rest mass.


2. All energy is either kinetic or potential energy
   
   Bulk motion, heat and (less obviously) light are all examples of kinetic energy. Chemical, elastic, gravitational and nuclear are examples of potential energy. Some are combinations of the two, such as sound and other compressive waves, where energy is alternately stored as potential energy by compression and released as kinetic energy by rarefaction. Given that rest mass by definition exists in the rest frame of a particle, rest mass cannot be kinetic energy and therefore must be potential energy.


3. Potential energy exists due to the interactions of particles with fields
   
   If an electron (or other charged particle) is placed in an electric field, it has an associated potential energy that will be reflected by an increase in particle mass. Because rest mass exists even in the absence of all previously known fields, we can infer that there must be another, previously unknown field that is the source of rest mass.

What would the properties of this field be?

• It must permeate all space

  Rest mass exists everywhere: an isolated electron, far from any other particles that could be the source of a field, would still have its rest mass.

• It must take the same value everywhere

  Particle rest mass is an invariant quantity, seen to take the same value for a given particle everywhere in space, irrespective of environment or inertial frame.

• It must have zero spin

  Non-zero spin would give the field a polarisation that would locally break the observed isotropy of empty space, just as an electric field breaks the isotropy of space near an electric charge.

• It must have a coupling proportional to particle rest mass

  Particle rest mass is the particle coupling to the field (the “charge”) multiplied by the constant field value, so the coupling effectively is the rest mass.

It turns out that such a field does exist and it is what we call the Higgs field. While most fields have a minimum of energy (the ground state) at zero field, the Higgs field has the famous Mexican hat potential, with a large energy at zero field. The minimum energy ground state (actually a locus of degenerate ground states) is associated with a non-zero field value. In the first moments after the big bang the Higgs potential had a ground state at zero field and all particles had zero rest mass. As the universe expanded, the form of the Higgs potential changed to that which exists today, causing the field to decay to a non-zero field ground state, breaking the symmetry that existed in the early universe and giving particles the rest masses we now observe. As expected, a field that permeates all of space and takes the same value everywhere. Note that the potential energy associated with the Higgs field never applies a force: force points down the gradient of potential energy and there are no gradients in the Higgs field because it takes the same value everywhere.

The famous Higgs boson is a quantum excitation of the Higgs field and observations of particle collisions producing Higgs bosons at the CERN Large Hadron Collider confirm that it is indeed a spin zero, scalar particle. The Higgs boson is highly unstable, with a lifetime on the order of 10^-22 s, but its presence can be inferred from observations of the decay products. If particle coupling to the Higgs field essentially is rest mass, then it might be expected that the dominant decay mode would be to the particle with the largest rest mass, i.e. the top quark. In fact, this decay mode is energetically forbidden because the top quark has a larger rest mass than the Higgs. It further transpires that production of quarks in Higgs decays must create quark-antiquark pairs, so the top-antitop pair is well out of reach. The next largest rest mass is the bottom quark and in fact the dominant decay mode of the Higgs is to a bottom-antibottom pair.^[4]

So is generation of rest mass via interactions with the Higgs field complicated? This explanation glosses over a lot of details, but I have found it a useful way to gain some intuitive understanding of the basic principle. An important take-home message is that rest mass is just another form of potential energy that happens to be generated through interactions with a field that takes the same value everywhere. In that sense, it can be seen as a unifying and simplifying discovery.

1. The Ideas of Particle Physics, 4^th Ed. (2020), James Dodd and Ben Gripaios, 978-1108727402
   See section 21.3 on page 106, Spontaneous Breaking of Local Symmetry - the Higgs Mechanism
2. Where does mass come from?
   D. Kwon
   Symmetry Magazine, 12-May-2016 (Accessed 10-Oct-2022)
3. The Higgs mechanism for undergraduate students
   G. Organtini
   Nuclear and Particle Physics Proceedings 273–275 2572–2574 (2016)
4. The Ideas of Particle Physics, 4^th Ed. (2020), James Dodd and Ben Gripaios, 978-1108727402
   See section 41.2 on page 204, Decays of the Higgs Boson.

An important parameter studied by cosmologists is the ratio of radiation to matter in the energy content of the universe. Matter is traditionally described as a substance that has mass, which can be confusing because, by mass-energy equivalence, we know that all energy is mass and all mass is energy (for further discussion of this, see the piece “What is mass?”). A better definition casts radiation as composed of relativistic particles (v ∼ c), while matter is composed of non-relativistic particles (v ≪ c). Radiation is then taken to include photons and neutrinos, while matter includes atoms and molecules, gas and dust.

Why is this an interesting distinction to make from the point of view of cosmology? Cosmology is concerned with gravity and a given quantity of energy will impart the same gravitational attraction whether it is matter or radiation. The difference is important when the region of space within which the energy is contained expands, and we know that all space is expanding. For a region of space containing matter and characterised by a length parameter, a, the energy (mass) density will drop off like 1/a³ as the region expands. This is unsurprising and is essentially just the definition of density as a quantity normalised by volume. For the same region of space containing the same total energy composed of radiation, the energy density will drop off like 1/a⁴ as the region expands.^[1]

Why the difference? The additional factor of 1/a is due to the redshift as the wavelength of the radiation increases. Consider a distribution of photons in the space; the density of photons drops off like 1/a³, while the energy, E, of each photon is inversely proportional to the wavelength, λ: $$ E = \frac{h}{\lambda} $$ Where h is Planck's constant and λ ∝ a. So the evolution of the expanding universe under the influence of the mutual gravitational attraction of its contents will depend on the ratio of radiation to matter, which will itself evolve, decreasing as the universe expands because the energy density of radiation drops off faster than the energy density of matter.

The formal calculation of the dependence of energy density on a comes from the Friedmann–Lemaître–Robertson–Walker equations, based on an exact solution of the Einstein field equations of general relativity for a homogenous and isotropic universe. For a single component universe (single equation of state) the dependence of energy density on scale factor a goes like $$ \varepsilon = \varepsilon_0 a^{-3(1+w)}, $$ where the dimensionless parameter w captures the material properties of the contents of the universe, as determined by the equation of state, P = εw, where P is pressure. The equation of state is a concept from thermodynamics and for most physicists the familiar example will be the ideal gas equation, $$ PV = NkT, $$ relating the pressure, P, volume, V, and temperature, T, for a number of particles, N, and where k is Boltzmann's constant. Using the substitutions N = M/m (number of particles is the ratio of total mass to mass per particle) and recognising that E = Mc² and ε = E/V, we can derive for material obeying the ideal gas equation, $$ w = \frac{kT}{mc^2} $$ Reassuringly, for baryonic matter (think atoms and molecules) this gives the expected w = 0 result, equivalent to ε ∝ a^-3, because thermal kinetic energy at any remotely plausible temperature is negligible compared with the rest energy.

All this makes sense and provides a satisfying answer to the title question. Nonetheless, it still leaves open the question of precisely what defines whether a particular source of energy will contribute to matter or radiation. What characteristic or characteristics determine whether it contributes to the matter or radiation content of the universe? The standard description of matter as a substance with mass suggests that the crucial underlying difference between the particles comprising radiation and those comprising matter is rest mass. But it can’t be solely particle rest mass (Higgs mass) that contributes to matter because that is only a very small fraction even of baryonic matter (see the piece “What fraction of a kg of potatoes is particle rest mass?” for more details).

I don't yet have an answer to the above question but I feel there are two natural hypotheses:

1. Radiation is any energy, kinetic or potential, moving at relativistic speed
2. Radiation is all kinetic energy, whether relativistic or not

These two hypotheses are probably observationally identical because the relativistic contents of the universe are dominated by kinetic energy and anything with a significant fraction of potential energy will not move at relativistic speeds. For the case of photons the total potential energy is identically zero, so the energy of a photon is purely kinetic energy. There are various ways to see why this must be the case, but probably the simplest is just to recognise that photons carry no charge of any kind and so do not couple to any fields. Another argument for why photon energy must be kinetic energy recognises that nonzero potential energy would give photons the potential to increase their speed, which is not possible for a particle travelling at the speed of light. Neutrino energy is taken to be part of the radiation content of the universe, even though they do couple to the Higgs field and so have a (very small) Higgs mass, as demonstrated by the observation of neutrino flavour oscillations - recognised by the award of the 2015 Nobel Prize in Physics. So this leaves open the question of whether neutrino potential energy contributes (a very small amount) to the matter or radiation content of the Universe.

There is perhaps a clue from this calculation for a relativistic electron gas, which suggests that the equation of state includes a quadratic dependence on velocity such that only electrons travelling very close to the speed of light would contribute to radiation. This result seems to support hypothesis 1 but I feel that there is more to learn on this point.

1. Lecture notes 19: Nucleosynthesis Measuring Cosmological Parameters
   V. H. Hansteen
   University of Oslo AST1100 Lecture Notes, 31-Oct-2006 (Accessed 14-Aug-2023)
2. Lecture notes 18: Cosmology
   V. H. Hansteen
   University of Oslo AST1100 Lecture Notes, 31-Oct-2006 (Accessed 14-Aug-2023)