In *The Road to Reality* (2004), Roger Penrose writes about a great analogy for **symmetry breaking**. Apparently, this analogy is rather common in the literature. (No, it’s not the thing about the pencil — this one involves an iron ball.) Once again, I find myself agreeing with Penrose about something; it *is* a great analogy.

Symmetry breaking (which can be explicit or spontaneous) is critical in many areas of physics. For instance, it’s instrumental in the Higgs mechanism that’s responsible for the mass of some particles.

The short post is for those interested in physics who (like I) have struggled to understand exactly what symmetry breaking is and why it matters.

**Symmetry** is a property something can have that makes it look indistinguishable when that something (or your point of view of it) is *transformed*. That last word covers a lot of territory: it can mean a movement in some direction, a rotation around an axis, a reflection (as in a mirror), or something abstract or mathematical.

One simple example: We say someone’s face is symmetrical if the left and right sides are mirror images of each other. Specifically, they have a bilateral symmetry along the vertical axis through the center of the face. So, a symmetrical face’s reflection in a mirror is indistinguishable from the face viewed directly.

More precisely, if we *transform* a symmetrical face via a reflection, the result of the transformation is indistinguishable from what we started with. This simple symmetry has only that one transformation. But we typically consider *doing nothing* a valid transformation because the result is indistinguishable from the start. So, there are in fact two transformations in this *symmetry group*.

*Pop Quiz: Why does a mirror reverse left and right but not top and bottom?*

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A square looks the same if rotated 90°, 180°, or 270° (or 0° — doing nothing again). So, the symmetry group for a square has four states, but only two transformations: doing nothing and a 90° rotation. A 180° rotation is twice 90°, and a 270° rotation is three times. (If we’re allowed to flip the square over, we add that transformation and gain four more states.)

In contrast, a sphere or ball can be rotated *any* amount on *any* axis and still look the same. It has an *infinite* number of states, a *continuum* of states. But there are only four transformations: doing nothing (always an option) or a rotation of ** x°** on one of the three orthogonal axes.

In general physics, symmetries in the laws lead directly to important conservation laws. That physics works the same here and over there — a transformation of position does not affect the physics — leads to the conservation of momentum. That physics works the same yesterday, today, and tomorrow, leads to the conservation of energy.

In quantum physics, particles are described by symmetry groups where transformations give different particles with related properties. One example is the Eightfold way, an organizational scheme for hadrons.

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**Symmetry breaking** occurs when something happens that, exactly as the phrase says, breaks the symmetry. For example, a scar on an otherwise symmetrical face breaks the symmetry. The mirror reflection is now distinguishable from the untransformed face.

If we cut one corner of the square, or put a dent in the sphere or ball, then their rotational symmetries are broken. Their rotations are now distinguishable.

These examples, a scar, a clipped corner, a dent, are examples of *explicit* symmetry breaking. An irregularity has been imposed on the system. Explicit symmetry breaking in physics and math usually involves adding terms that perturb the system’s original symmetry.

*Spontaneous* symmetry breaking is more related to the system properties or to very low-level random inputs from the environment. A canonical example is a supercooled fluid where a disturbance in one tiny area acts as a seed to a chain reaction that freezes the fluid in seconds. (A fairly easy experiment that’s fun to watch.)

Another common analogy for spontaneous symmetry breaking is a pencil carefully balanced on its tip. It can, in theory, remain balanced indefinitely, but the reality is that any vibration or movement of air will upset the balance. Because of positive feedback, the more off balance it gets, the greater the off-balance force, so any disturbance causes the pencil to fall in an apparently random direction.

While the pencil was balanced, the system had rotational symmetry around the pencil. We could rotate it, and it would look the same. After the pencil falls, the symmetry is broken, and rotations are distinguishable.

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That was the thing about the pencil. The example Penrose referred to goes something like this:

Imagine a sphere of solid iron. Each iron atom acts like a tiny magnet, but usually each “atomic magnet” is oriented in a random direction, so overall the iron ball is not magnetic. If the iron is heated above 770° Celsius (1418° Fahrenheit), the thermal energy agitates the atoms, randomizing their directions. When the iron cools normally, the random orientations of the atoms become fixed. But if the iron cools *slowly and evenly enough* (an idealized condition hard to achieve), a single “seed” of aligned atoms spreads throughout the sphere. The result is a magnetized iron ball with nearly all the atoms aligned in the same *completely random* direction.

In reality, what usually happens is the cooling is too fast and irregular, so multiple seeds start and expand until they encounter other expanding regions. Boundaries form between regions, so the ball ends up with lots of random regions and no overall magnetism. It’s only when there’s still enough heat energy that the larger region can overwhelm the smaller one that the ball freezes out in a unified state. That makes it a great analogy, but a very difficult experiment to actually pull off.

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What’s cool about this analogy, thought, is that, while a pencil illustrates the basic idea, it doesn’t provide a sense of how all of spacetime could be affected by spontaneous symmetry breaking. With the iron ball, even though it retains its shape and appearance, a fundamental aspect of its nature (the magnetism) arises in consequence of the symmetry breaking.

So, whoever thought of it: good thinking!

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Stay symmetrical, my friends! Go forth and spread beauty and light.

∇

January 24th, 2023 at 10:02 am

It took a long time to get through

The Road to Reality(and I’m going to have to re-read some chapters). Penrose is not an easy read. He’s quite willing to subject his readers to all the math (well, a lot of the math anyway).I’m glad I read it. I was reading his more recent

Fashion, Faith, and Fantasy, but it referred back toTRtRso often that I thought I’d better read it first.Penrose explicitly keeps higher math out of

FF&F, butTRtRmore than makes up for it. I’m seeing now why he makes so many references back to these more detailed accounts. Somewhat different audiences, seeing as how each chapter inTRtRends with exercises for the reader!January 24th, 2023 at 10:10 am

Stuck on the mirror Pop Quiz question? I wrote about this a few years ago.

January 24th, 2023 at 10:56 am

An example of symmetry breaking in quantum mechanics:

Under electroweak unification (which joins the EM and weak forces) there are four massless bosons, but due to symmetry breaking in the very early universe (as the Big Bang fireball cooled enough to “freeze out” certain interactions — in close analogy to the iron ball), those four identical massless bosons become the still massless photon and three massive weak force bosons (the W+, W-, and Z⁰).

Symmetry breaking is required to explain those bosons we actually observe.

January 24th, 2023 at 1:07 pm

“The short post is for those interested in physics ” Ahh, not for me. Just one of those differences, or patterns of discord. I do like symmetry, though. 🙂

January 24th, 2023 at 1:30 pm

Well,… I wouldn’t call it

discordunless it causes conflict. Just normal different interests. Unfortunately, the next two posts will be more of the same. 😜