Spontaneous symmetry breaking (SSB) is an important phenomenon in quantum physics, appearing in many areas.

There are two ways in which a symmetry which is present in a model on the level of the Lagrangean will fail to manifest itself in the quantum states. One is quantum anomaly, and the second is SSB. However, whereas quantum anomaly destroys also the conservation laws associated with the symmetry via Noether's theorem (at least on the local level), spontaneously broken symmetries still correspond to a conserved local current.

SSB can apply both to global and local (gauge) symmetry, and both to continuous and discrete symmetry.

In case of a global symmetry group G, SSB occurs when the groundstate of model |0> fails to be G-invariant. In general, it would be invariant under a certain subgroup H of G. As G is a symmetry group, for any element g of G, g|0> is also a groundstate of the model. Therefore, we necessarily obtain multiple groundstates. Due to the cluster decomposition principle, those different groundstate will necessarily lie in different superselection sectors. Therefore, any element g of G that doesn't also belong to H transforms the states of one superselection sector into the states of another. As a result, when we limit our view to a single superselction sector (what in reality we have to do), the symmetry transformation vanishes.

One might ask in what kind circumstances would |0> fail to be G-invariant. In fact, it is easy to produce examples of this kind. The simplest thing is a scalar field phin with a potential V(phin) s.t. V is G invariant however its minima aren't. The classical example of this sort is the so-called Mexican hat potential

V(phin) = (SUMnphin2 - a2)2

where a is some real constant. If phin is an N-dimensional vector (i.e. the index n takes the values 1..N), V is invariant under the full N-dimensional rotation group SO(N) (and even O(N), but we may ignore this for our purpose), however, its minima (the points satisfying

SUMnphin2 = a2

are only symmetric under an SO(N-1) subgroup.

Of course, in reality what matters is the quantum potential, rather than potential appearing in the Lagrangean. Therefore, quantizations may results in destroying or creating SSB. The later case is sometimes called "radiative symmetry breaking". Note that quantization may lead to SSB via other mechanisms also, such as the appearance of an unsymmetric quadratic fermion vacuum condensate (in case there are fermions in the model). This is usually called "dynamical symmetry breaking". At any rate, it is comparatively easy to prove that in our example one would get SSB for a sufficiently high value of a.

An important phenomenon associated with the SSB of global continuous symmetry is the appearance of so-called Goldstone bosons. Those are the massless particles corresponding to excitations of the field along the directions tangent to the vacuum manifold. The later is the set of groundstates resulting from SSB, having the geometry of the quotient space G/H. For the Mexican hat example, G/H is the (N-1)-sphere.

SSB of local (gauge) symmetry doesn't lead to multiple vacua as gauge symmetry doesn't act on the physical states anyway. However, it may still happen unsymmetric vacuum expectation values appear in gauge theory (to produce an example, just couple a "Mexican hat" type field to a gauge field). In such cases we can fix the gauge symmetry by demanding the field that develops the expectation values to choose everywhere the same point on the "vacuum manifold" G/H. In such a model, no Goldstone bosons appear (as we just "gauged away" the excitations which produced them before), however, the gauge boson (which otherwise has to be massless) gains mass! This may be demonstrated by substituting the vacuum expectation values into the Lagrangean: the result is a mass-term for the gauge field.

It's important to emphasize that although I used the language of quantum field theory most way through this write-up, the concepts described apply in exactly the same way to models of the kind found in solid state physics. A few real-life examples of SSB would be:

- The chiral symmetry of the quark field in quantum chromodynamics. It is the reason chiral symmetry is not manifest in the particle multiplets, although the chiral currents are still conserved.

- The rotational symmetry of space in the Heisenberg (XXX) model of ferromagnets. It is the reason spontaneous magentization appears within each domain, and therefore also the reason for the hysteresis loop.

- The electromagnetic U(1) gauge symmetry in superconductors. There the Cooper-pairs result in the appearance of the vacuum expectation values and all the special properties of superconductors (such as zero resistance and magnetic flux quantization) can be traced back to the SSB.

- The breaking of the electroweak SU(2) X U(1) group down to the U(1) group of electromagnetism (in our notations, G = SU(2) X U(1), H = U(1)) in the Glashow-Weinberg-Salam model.

When the universe was born, was there you and I,
Evaporated in the cosmic soup with all the story yet to be?
Or was it when the big heat died and the symmetry broke,
And we were but eyes in the bright solid light,
Knowing our reach someday
Was gravitation, attracting everything,
When I all needed was you.
My first equation ever, to touch,
To take you in arms imagined as electric forces,
Fields united in grand theories science has yet to write,
Made of words never yet spoken.
God made us love,
Animate will to annihilate,
Matter I with you anti.
So we struggle with the calculus
Knowing that as comes the next universe
and the next
So will you and I,
In the soul illuminating flash of E from M
Emerge as the brilliant symmetry
Detonates to glistening light shards,
Leaving us writing in words never yet spoken.

What they say is that the universe began as a singular explosion - something not of time and space, but a point outside of this, in n-dimensional heaven somewhere - the whole universe compact as a pea. Everything that is and was and will be, crushed and compacted, weighing nothing because there was no gravity to measure, the same time hot as Uriel's heart and cold as the dark side of the moon.

Our universe was small and hot and expanded - not an expansion as we know. Not an inflating party balloon filling with helium and taking up space we could have used for something else - but expanding as space itself is created, from Shiva's infinite nothing.

They say that back when the universe was very young - which itself is hard to say because time had yet materialized - when it was young it was hot and the same in all the directions. Everywhere was the same as everywhere and symmetry ruled.

Our universe cooled and expanded as it aged and at some point, it cooled enough for particles to form. Out of God's own hot soupy birth fluid lumps formed. These became the subatomic particles - the fermions and the bosons that applied the forces between them.

When it cooled further the particles joined into atoms, and later the atoms to molecules which became the stuff of the stars and planets.

As the beautiful boring symmetry of our birth chilled and grew it fell apart we emerged along with all the light and asteroids and rockfish.

And things were now different in all directions.

So they say.

Those they, who say such things.

There are equations for all of this. Dear one I have studied them during my life here with you. They are beautiful. They are confusing. They are terrifying.

Einstein, who refined the gravity handed to him by Newton - added a lambda to his tensor calculus that seemed perfect enough without it. Later he called it his biggest mistake. After he died, nobody thought it was a mistake, this lambda, this stuff that reverses gravity and pushes apart everything and has since the beginning. This lambda that suggests everything we are, everything we can see, is only thirty percent of everything that is.

That leaves seventy percent out of our reach. That leaves us alone in a mysterious universe where most of the stuff God created was given to someone else and hidden from us, yet tantilizingly detectable by our telescopes and colliders and brains, great joker God. Knew it would drive us nuts.

Dear one, we know it's there but we can't see it. Can't touch it.
Dark matter, dark energy.

Such boring and mysterious words for something that is most of everything.

Were you to fall into a black hole, and was I to watch, I would see you falling forever, until time itself ended.

You would simply never get there during the lifetime of this universe.

On the other hand, you'd just fall in as if you jumped off a kitchen stool. You'd get there in no time, never realizing that the whole of everything was over. This is relativity, general as it can be. And we can talk about the terrible gravitation that pulled you into constituent atoms as you crossed the event horizon to the spot where time and space merge and nothing we know operates.

What this means, is that everything that has fallen into black holes everywhere is still falling in - and always will, as far as we're concerned.

Everything that has gone in is there, falling in never reaching - is the same as something ever speeding up toward the speed of light. If we look hard enough into our telescopes we'll see that.

Now, the universe is expanding, and the way the math works, the universe is expanding faster, the farther away from us you are. In fact, no matter where you are everybody sees the universe expanding away from himself faster the farther away something is. Our math and our telescopes and our Nobel Prizes tell us this.

So then when you look into space, the farther things are, the faster they're receding into universal nothingness. And at some particular distance, things are moving away from us at almost the speed of light - which means the sight of them is always lost to us. And we can imagine things that might be beyond that speed-of-light distance, and those things we can never know.

It's as if we're inside a black hole, watching ourselves fall in,

Time warping as we fall.

To us, every day goes by. We wake in the morning. Go to work. Peel gum from our shoes. Order our beers and leave tips for the wait staff. All the while things move farther and farther, moving slower and slower, never getting to the there they're heading toward. Slower and slower. Until the universe ends.

But fear not, dear heart - we are not in a black hole. We know this because we have detected the remnants of God's symmetric breath from which we emerged. This is the three-degree Kelvin background radiation that surrounds us like a mottled blanket. It is not uniform in temperature or polarization. It is asymmetric, as are we.

If things had remained symmetric, nothing at all would have happened, and we would not be.

So this cosmic radiation reminds us we were created. It reminds us that we coalesced from something absolutely brilliant and perfect.

Looking outward in all directions, and seeing light itself that cannot escape reminds us we are in this big here that was assembled.

And you can be pure like a scientist and suggest no reason for any of it, because creation theories cloud your vision,

You can be a mystic and see the hand of a creator in all of it because missing God clouds your vision,

Or you can deny any of it, is.

This is the territory in which we find ourselves awake, we who emerged when the symmetry broke,

And you burst forth alight to love me, and I you.

Now go to sleep, my love. There's nothing to fear. It's late and your mother and I are going to bed ourselves.

Hold close your questions, and remember your dreams.

It will all be here tomorrow.

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