With Minneapolis in the news these days (unfortunately), my stream of thought went to one of the (fun) times that I went there, as CEO of *SmartOps*, to meet with customers (like *Medtronic*, who have also been in the news lately).

As I was early for the meeting, I decided to spend some time at an airport lounge (*United*, as I recall) before heading over to *Medtronic*.

Guess who walks in right after me?

Cameron Diaz!

No makeup, and just wearing T-shirt and jeans, flats (not high-heels). Still:

There is Something about Mary!

Since these days I am digging deeper into *Quantum Field Theory* (QFT), I wanted to better understand what the big deal was about the *Higgs Boson* (aka “God Particle”) that attracted a lot of media attention in 2012:

Physicists Find Elusive Particle Seen as Key to Universe

Let me begin by modifying one of the most memorable dialogues from the movie *The Princess Bride*:

Those words you keep using?

Symmetry. Mass.

I don’t think they mean what you think they mean.

If I asked you, a layperson,

which is more symmetric, gas (water vapor, say) or ice cube,

how would you answer?

Because you and I think about *gas as this featureless thingy* that has no structure, let alone possessing symmetry, and ice cube, being *so well-shaped*, we think, if rotated by certain specific amounts, like 90 degrees, will look the same, and so there you have the answer:

Ice Cube.

Wrong!

Why?

What we are thinking of is called *discrete* symmetry.

The gas, however has a *much deeper* symmetry, called

*continuous symmetry*.

What?

Take (a spherical bowl of) gas and rotate it anyway you want and the gas inside will still “look” the same featureless thingy like it did before, indicating that there is no way you can know if you have rotated it, and so is considered to be so much more symmetric than ice cube that only retains its original look — fools you when it is rotated as if it has not been rotated — at only a certain discrete number of carefully chosen rotations.

The mathematics to study continuous symmetry was developed by **Sophus Lie**, and the abstract objects under investigation and analysis are called *Lie Groups*.

A *group* is a collection of (in this case, infinite) entities that satisfy certain criteria of *composition* (could be *multiplication* or *addition* or any other *operation* that we can define between the members).

If you compose two members of a group, the result is a member in the group.

There is a unique member, denoted by IDENTITY, which when composed with any member does not change that other member.

Composition is ASSOCIATIVE.

For every member, there is an INVERSE member, that, when they are composed with each other, results in the IDENTITY member.

Therefore:

The set of integers is **not** a group under *multiplication* as composition while the set of rational numbers is a group (with ONE as the IDENTITY).

The set of integers is a group if the composition is *addition*, with ZERO being the IDENTITY.

If additionally, composition is *commutative*, then the Group is an *Abelian group,* named after **Henrik Abel**.

Let me now connect this abstract mathematical concept of Lie groups to our example of ice cube and water vapor being rotated.

The set of all continuous rotations is a Lie Group.

(As are the set of space, and time, *translations*.)

To mathematically study continuous symmetry, we use Lie Groups.

Lie Groups have *generators* (“infinitesimal transformations arbitrarily close to IDENTITY”) that can be used to create every member of the group.

The essence of a continuous symmetry is encoded in the generator.

The simplest Lie Group has one generator, a complex number, which can also be represented as a vector, representing rotations on a plane, called unitary group U(1).

A more complex Lie Group, called special unitary group SU(3), representable by 3×3 matrices that are unitary and have positive unit determinant, has 3*3-1=8 generators.

Eight! What should this remind you of?

The generators, in general, do *not* commute. (Think about rotations in two different axes: pick up a book and try it!) Many interesting groups are *non-Abelian*.

The algebra created by the commutative relations of the generators of the group is called *Lie Algebra*.

It should come as no great surprise that:

Lie Groups and Lie Algebra are natural candidates to study continuous symmetry in quantum field theory!

Let us have some fun.

How many types of gluons are there?

Eight.

Quantum chromodynamics (QCD), with quarks and gluons, is studied using SU(3), a Lie Group with eight generators (“gluons are the basis states of the Lie algebra”).

Surely You’re Kidding, Professor Tayur!

No.

Do you want to discuss another word that you and I think we certainly know very well?

Mass.

What does symmetry have to do with mass?

Everything.

Yes, like the mass of a billiard ball, in classical physics, that is used to teach us a lot of concepts like conservation of mass and conservation of momentum and conservation of energy and all that.

There is a deep connection between continuous symmetry and conservation laws, thanks to the work of **Emmy Noether**.

Rotational symmetry implies conservation of angular momentum.

Translation (in space) symmetry implies conservation of linear momentum.

Translation (in time) implies conservation of Energy.

Conservation of electric charge? Of Baryon number. Same thing. There is a continuous symmetry either in *spacetime* (like the three examples above) or *intrinsic* to an *abstract field space* that would correspond to a conservation law (like *global shift* of the field, or *phase shift* as examples).

Back to mass. You and I may think that mass is an *intrinsic feature* of the physical object in question.

That is, “the object has a mass” actually makes perfect sense to us.

Even without gravity, which does not exist in the context of QFT anyway, it is the *inertial mass* that we are talking about (“something that resists force”).

If you divide the object in your hand into pieces, and that into pieces, and so on, and on, until you cannot subdivide anymore, you get to elementary particles.

You would think that this elementary particle, say an electron, has mass.

No.

Mass is not an intrinsic property of an elementary particle.

Time to recall a line from the movie, *The Matrix*:

There is no spoon.

All elementary particles in *Quantum Field Theory* (QFT) are massless and travel at the speed of light, like photons.

If you have not yet caught on, let me explicitly say it:

The generator of a specific U(1) group corresponds to a photon.

That is because the elementary particles, which are in QFT just excitations of a quantized field (remember my previous post about *bobbing monkey on a spring mattress*), one for each type of elementary particle, are *plane waves*, that never dissipate, and can travel forever, which are

interpreted as being *massless*.

Then, how come, when we measure them in our real world, some of them, like electrons, appear to have mass?

When they interact with a

special field, through the interaction term in the Lagrangian, the resulting wave equation will have an extra-term (“Yukawa”), which will lead to a different dispersion equation, and this implies that the solution is not a plane wave, and so will dissipate as it travels, and the rate at which it dissipatesis interpreted as mass.

What is the name of this “special” field?

You guessed it:

Higgs.

When the symmetry of an electron field is *spontaneously broken* , due to interaction with Higgs Field, whose excitations are called Higgs Bosons, the electron “gains” mass.

Mass is just deformed energy.😏

This was theorized in 1964. It was experimentally confirmed in 2012.

The 2013 Nobel Prize in Physics was awarded jointly to **François Englert and Peter W. Higgs**

for the theoretical discovery of a mechanism that contributes to our understanding of the origin of mass of subatomic particles, and which recently was confirmed through the discovery of the predicted fundamental particle, by the ATLAS and CMS experiments at CERN’s Large Hadron Collider.

This is just one more in the list of Physics Nobels (1933; 1965; 1969; 1979; 2004; 2008) that have gone to theoretical physicists that conceive of elementary particles as excitations, created and annihilated through interactions among quantum fields.

Putting it my *maximally inverse *framing:

The Nobel Prize is the Fields Medal for Theoretical Physicists!😏

Fascinating. Knowledge. Your discussion is humbling.

Nicely put together. Enjoyed reading it.

Thanks, interesting.