Since the putative discovery of the Higgs boson this past summer, I have heard and read multiple attempts at explaining what exactly this discovery means. They usually go along the lines of “The Higgs mechanism gives mass to particles by acting like molasses in which particles move around …” More sophisticated accounts will then attempt to explain that the Higgs boson is an excitation in the Higgs field. However, most of the explanations I have encountered assume that most people already know what mass actually is and why particles need to be endowed with it. Given that my seventh grade science teacher didn’t really understand what mass was, I have a feeling that most nonphysicists don’t really have a full appreciation of mass.
To start out, there are actually two kinds of mass. There is inertial mass, which is the resistance to acceleration and is mass that goes into Newton’s second law of and then there is gravitational mass which is like the “charge” of gravity. The more gravitational mass you have the stronger the gravitational force. Although they didn’t need to be, these two masses happen to be the same. The equivalence of inertial and gravitational mass is one of the deepest facts of the universe and is the reason that all objects fall at the same rate. Galileo’s apocryphal Leaning Tower of Pisa experiment was a proof that the two masses are the same. You can see this by noting that the gravitational force is given by
where is the universal gravitational constant, is the mass of object 1 (e.g. earth), is the mass of the second object, and is the distance between the objects. Thus if we try to compute the acceleration of object 2 in the presence of object 1 we get the equation
and the ‘s cancel out. Thus the gravitational acceleration on earth is always the same and given that movements on the surface of the earth is small compared to , which is measured to the center of the earth, then we get the usual gravitational acceleration m/s/s. My seventh grade teacher mistakenly believed that two objects fall at the same rate because the earth’s mass was so much bigger than the object’s mass.
The fact that gravitational mass is equal to inertial mass was a great mystery until Einstein showed why in the theory of general relativity. What Einstein observed was that if you were in an elevator, you wouldn’t know if you were in outer space being accelerated by a rocket, or sitting on the surface of the earth or conversely you wouldn’t know the difference between falling in a gravitational field or sitting in outer space. Hence, the two must be equivalent. However, given that you are not a point object, in actuality you could actually know the difference because on the earth you would notice that the downward acceleration is not exactly vertical but points towards the center of the earth unlike on a rocket. This is technically called tidal forces and working out these tidal forces under various conditions forms the basis of Einstein’s general relativity equations. I once remember reading a letter to the editor in the magazine Popular Science claiming that Einstein was wrong because we could detect tidal forces in the elevator experiment. This was a perfect example of how having a little knowledge is a dangerous thing.
Now back to the Higgs boson and what it means for endowing particles with (intertial) mass. I will save the details of the Higgs mechanism for a future post and simply sketch out what it means for a quantum particle to have mass. Everyone knows Einstein’s famous equation from special relativity: . What this means is that the rest mass of a particle has (is) energy. The more complete form of the equation is where is the momentum of the particle. A massless particle, like a photon, has no rest mass and its energy is given by its momentum. In quantum mechanics, particles are actually waves, and all waves are described by a wave equation. The Schrodinger equation is the wave equation for a nonrelativistic particle. For a wave, the energy is given by the frequency of the wave and the momentum is given by the wave number (i.e. inverse wave length). We can thus write the relativistic energy equation as
This is called the dispersion relation. We can always reconstruct a wave equation from a dispersion relation by inverse Fourier transforming and assigning and (for one spatial dimension) for a wave function . This then gives us the wave equation
This is called the Klein-Gordon equation and doesn’t actually pertain to the particles you know and love. Schrodinger actually derived this equation first but abandoned it because probability was not conserved. It makes a come back in quantum field theory for what are called scalar particles. So we see that if , the wave would travel with speed . However, for nonzero mass, the group velocity of the wave would depend on wavelength as well as always being less than . A wave packet would spread out and disperse. This is what the physicists mean when they say the Higgs mechanism is like molasses and slows particles down. It adds a term proportional to to the wave equation and adds dispersion. Now, this is still a gross simplification as the standard model of particle physics is a theory for quantum fields and not particles like I have described here. Instead of a wave function, the pertinent mathematical object is the Lagrangian of the quantum field. The wave equation is the equation of motion (Euler-Lagrange equation) for this Lagrangian obtained by extremizing the action. The mass comes in as the addition of a term to the Lagrangian that is proportional to the field squared, which yields a term proportional to the field in the equation of motion. In the standard model, the intermediate vector bosons of the weak force start out with zero mass like the photon. They acquire mass by coupling to a Higgs field, which adds the quadratic term to the Lagrangian. The mechanism is actually the same as what happens in the Meissner effect of superconductivity where the electromagnetic field “acquires a mass”. I will try to make all of this clear in a future post.