This is to the Orleanders and Spudd clones who scratch their heads in dismay when confronted with magnets, electric motors and such (any contributions from more science-wise types is "naturally" welcomed, I don't presume to be the world's greatest explainer):

Seeing a phrase like "broken symmetry" in some article about fundamental particle physics or quantum theories is a bit puzzling. How do symmetries get "broken" and what does it have to do with forces?

A simple example of symmetry-breaking and a force being "transmitted" is available through the (somewhat prosaic but still cool) science of pendulum dynamics - and what happens when you couple pendulums together.

Two pendulums with a roughly equal length, suspended "near" each other (say a distance less than half their combined length), and coupled, with a weak spring (say, a length of elastic string or a thin rubber band, or a wound length of steel wire - anything with a spring constant), connected from one to the other pendulum, will demonstrate the principles of broken symmetry and the transmission of a force.
Rigid pendulums would probably work better in this example - so actually building a coupled pendulum means thinking about what to use, and how much room, which ceiling to bang nails into, and so on. There are other ways to achieve a coupling without directly tying a bit of elastic string or attaching a spring to both lengths, but a direct connection is more obvious.
Two pendulum clocks next to each other on a wall are coupled, for instance - through the wall itself, which has a certain "springiness" or elasticity, but anyways.

A coupled pendulum such as this has two stable modes, or ways it can move, they both either swing in phase - left and right simultaneously, or 180 degrees out of phase - left and right opposing swings.

If you stop one pendulum from swinging and leave the other, then the swinging "half" of the coupled system will slowly transfer momentum through the weak coupling to the other half. The first pendulum will eventually stop swinging, and the initially stationary one will be swinging with the displacement the first had. Then the process reverses, as the second "oscillator" transfers momentum back to the first.

The two stable or symmetrical modes of oscillation, or dynamics, are broken when one of the oscillators (pendulums), is stopped from swinging (the symmetry-breaking bit).

A broken symmetry leaves a "force", which pushes on the other motionless pendulum until it has "absorbed" all the momentum from the oscillator, leaving it motionless (with zero momentum).

(this still needs something, but I can't think what it is just now...)

A broken symmetry leaves a "force", which pushes on the other motionless pendulum until it has "absorbed" all the momentum from the oscillator, leaving it motionless (with zero momentum).

This isn't right...electromagnetism is unbroken, yet it still produces a force.

What do you mean electromagnetism is "unbroken" exactly? Electromagnetism, to a physicist is either a single "field" with two apparently separate parts, that are actually two "ways the field can move" or oscillate.
Or to a layman it's magnets picking up bits of steel, or electric charges that zap you when you close the car door on a warm day, apart from all the other things we do with electricity and its partner magnetism?

My understanding of "force" is something that appears or has to be added when a symmetry is broken. But enlighten me here, where does the argument fall over with EM?

How about several balls suspended like a pendulum, you've seen them, as they swing and collide, their momentum is stopped and energy and momentum is transmitted through 2 or 3 balls to the last on hit. Something like that?

Just trying to illustrate that there are symmetries, or "balanced forces" and you can make them asymmetrical, is all. An easy way to see what a symmetry is, and how to bust it, is with coupled pendulums, but the principle extends to any kind of coupled oscillatory, or harmonic motion.

So the EM field is "balanced" by having two opposing charges, two kinds of "motion" electric and magnetic, and so electric force and magnetic force, that appear - but only if there's a bit of "unbalancing".

One way to "unbalance" the EM field, or introduce asymmetry, is with electronics - coils (inductors), capacitors (charge separators), and resistors, are all "passive" components that rotate the electric and magnetic parts of the EM field, so producing, controlling and amplifying (scaling) oscillations - is really all electronic circuits do. A generator or DC motor is a circuit too.

P.S. The example of a Newton's cradle, where the two outer balls alternate between swinging, and transferring all their momentum through the central row of balls, so they stop swinging, is also a coupled oscillator.
The coupling is the central row of balls (which could be any kind of solid object - it just has to couple the momentum of the swinging pendulum or oscillator to the stationary one on the other end. Because the coupling is rigid rather than elastic, the transfer is immediate rather than gradual). There aren't any symmetrical or stable states available, like with two freely swinging weights, because neither oscillator can complete an oscillation - so there's only the asymmetrical mode allowed for, or available to a Newton's cradle.
So what would happen to one of these things if, instead of a row of metal objects suspended in the middle as a coupling, there was a chunk of rubber, or a spring?

Another term for "stable oscillatory mode" is stationary state. Atoms have stationary states.

P.S. I'm pretty sure Yang and Lee got a Nobel for "discovering" the asymmetry in left-handed right-handed particles, but that's a lot more complicated than an asymmetrical coupled oscillator - although the principle is much the same.
However the broken symmetry in left and right-handed particles means the universe is full of matter, instead of antimatter.
Mass exists because of a broken symmetry. Yang and Lee verified that asymmetries are what make things like "force" and "mass" observable, which is a pretty deep insight.

Well, I think I'll go have a beer.

That beer must have erased something, I can't seem to see what the answer is to a question about dynamics of rigid bodies.

For the curious: if you hold a bike-wheel by its axle (mounting it in a shortish length of galvanised pipe helps), and extend your arm, nothing much happens - your arm gets tired because of downward acceleration of it + the wheel towards the centre of the planet.

If you spin the wheel (or get someone to spin it), a force appears, as the spinning wheel precesses in a gravitational field. You feel a force (torque) trying to rotate your arm around.

OK, one for the boffins:

How do you explain gyroscopic precession as the appearance of a force due to an asymmetry? Or how does (giving a wheel) angular momentum break a symmetry, and what is the symmetry?

I had a wine, and i am still fathoming what you are at ;)

What do you mean electromagnetism is "unbroken" exactly? Electromagnetism, to a physicist is either a single "field" with two apparently separate parts, that are actually two "ways the field can move" or oscillate. But enlighten me here, where does the argument fall over with EM?Physicists use 'broken' to mean a different thing to what you describe.

For instance the electromagnetic gauge is U(1). You can change the phase of a field which has electromagnetic charge and the system doesn't change overall. It's got a phase symmetry, \psi and e^{i\theta}\psi obey all the same equations. It's 'unbroken' but it's still a force.

Then you have things like the electroweak force. In an unbroken electroweak system, you find that you have 4 fields which can be swapped amongst themselves in a particular way. Lots of nice symmetry. Imagine a cube, you can rotate it around a number of ways and it still looks like a cube, you cannot tell the difference wether you do nothing, rotate it 90 degrees about one axis of symmetry or 180 degrees about another axis. It's got loads of symmetry. Now suppose I cut off one of the corners of the cube. Then it's obvious when I rotate the cube, there's an asymmetry in the system which tells me that the new, rotated, states aren't the same as the original one. The symmetry is at best reduced, at worst destroyed.

That is broken symmetry. Pre breaking, electroweak has 4 fields which transform under SU(2). After symmetry breaking these juggle around and form 4 specific linear combinations. Three of them pick up mass and one doesn't. Three vector bosons and a photon. Symmetry is broken but there's a force.

The SU(3) symmetry of gluons isn't broken and it definitely works.
Yang and Lee verified that asymmetries are what make things like "force" and "mass" observable, which is a pretty deep insight.Can you be a bit more specific please? CP violation in weak decays can account for baryogenesis and/or leptogenesis but masses and forces preexisted the differential decay due to CP violation. After all, matter and antimatter still attract and repel one another or themselves, irrespective of what some of them might decay into.

CP violation in weak decays can account for baryogenesis and/or leptogenesis but masses and forces preexisted the differential decay due to CP violation. After all, matter and antimatter still attract and repel one another or themselves, irrespective of what some of them might decay into."Broken symmetry" and "force", are meant to be more or less interchangeable then? I thought equal amounts of left and right handed particles were created in the first condensation - which should have promptly annihilated each other, so why does the universe contain matter (- rhetorical)?

Or to put it another way - physicists, expecially the high-energy ones, look for symmetries and asymmetries, and try to explain them. Or they build a theory and see what it needs to make it asymmetric, or what breaks the symmetry?

That's what I thought "symmetry-breaking" was about: an explanation for an apparent force, like electrostatic force, or magnetic force (why does a bit of iron inside a wire coil experience a torque when the coil is "energized" with a current?). You appear to be relating the concept to only QFT, and the gauge symmetries...?

Maxwell's theories show the symmetrical relation between the electric and magnetic forces, which wasn't spotted and explained "correctly" until Einstein came along and understood it's a case of a Lorentz symmetry group. Then the implications of that led to special and general relativity.

I can be more specific about my Yang & Lee statement, if you can be more specific about how the phase symmetry in U(1) is "still a force" - phase isn't a force...? I thought a force was because of an asymmetry (in a field), like I think I say a few times in the above?

A coil with a current through it is a symmetry-breaking rotation, because a magnetic field around a coil has a broken symmetry, like a permanent magnet - so a force appears. There's another way to describe what happens, thanks to Ampere, Laplace, Maxwell et al, but it's an example of how a local symmetry can be made asymmetrical, and a force is the result - like the coupled pendulum example, which illustrates inertial symmetry.

Any ideas about how a precessing gyroscope experiences a force, but a gyroscope with no angular momentum doesn't? (I know about rotating inertial frames and all, but how is it explained as an asymmetry? Because a non-rotating wheel is more symmetrical in a rotating inertial field?)

P.S. Another version of a local symmetry made asymmetrical, is "gauge choice". Hang on - that should be a gauge choice is a choice of symmetry as a local one. Then when the symmetry is locally "broken", there's a leftover bit in the equation - a force.
"The phase of an EM field" might be a candidate for a gauge, as a local or global symmetry, say in some experiment or other.

Right, here's a fairly readable reference (no equations, anyway); this one's about symmetry-breaking, or "spontaneous" symmetry-breaking, so let's go to:
Spontaneous symmetry breaking (SSB) occurs in a situation where, given a symmetry of the equations of motion, solutions exist which are not invariant under the action of this symmetry without any explicit asymmetric input (whence the attribute “spontaneous”). A situation of this type can be first illustrated by means of simple cases taken from classical physics.

Consider for example the case of a linear vertical stick with a compression force applied on the top and directed along its axis. The physical description is obviously invariant for all rotations around this axis. As long as the applied force is mild enough, the stick does not bend and the equilibrium configuration (the lowest energy configuration) is invariant under this symmetry.

When the force reaches a critical value, the symmetric equilibrium configuration becomes unstable and an infinite number of equivalent lowest energy stable states appear, which are no longer rotationally symmetric but are related to each other by a rotation.
The actual breaking of the symmetry may then easily occur by effect of a (however small) external asymmetric cause, and the stick bends until it reaches one of the infinite possible stable asymmetric equilibrium configurations.

In substance, what happens in the above kind of situation is the following: when some parameter reaches a critical value, the lowest energy solution respecting the symmetry of the theory ceases to be stable under small perturbations and new asymmetric (but stable) lowest energy solutions appear. The new lowest energy solutions are asymmetric but are all related through the action of the symmetry transformations.
In other words, there is a degeneracy (infinite or finite depending on whether the symmetry is continuous or discrete) of distinct asymmetric solutions of identical (lowest) energy, the whole set of which maintains the symmetry of the theory.

In quantum physics SSB actually does not occur in the case of finite systems: tunnelling takes place between the various degenerate states, and the true lowest energy state or “ground state” turns out to be a unique linear superposition of the degenerate states.
In fact, SSB is applicable only to infinite systems — many-body systems (such as ferromagnets, superfluids and superconductors) and fields — the alternative degenerate ground states being all orthogonal to each other in the infinite volume limit and therefore separated by a “superselection rule” (see for example Weinberg, 1996, pp. 164-165).

Historically, the concept of SSB first emerged in condensed matter physics.

The prototype case is the 1928 Heisenberg theory of the ferromagnet as an infinite array of spin 1/2 magnetic dipoles, with spin-spin interactions between nearest neighbours such that neighbouring dipoles tend to align.
Although the theory is rotationally invariant, below the critical Curie temperature Tc the actual ground state of the ferromagnet has the spin all aligned in some particular direction (i.e. a magnetization pointing in that direction), thus not respecting the rotational symmetry. What happens is that below Tc there exists an infinitely degenerate set of ground states, in each of which the spins are all aligned in a given direction.

A complete set of quantum states can be built upon each ground state. We thus have many different “possible worlds” (sets of solutions to the same equations), each one built on one of the possible orthogonal (in the infinite volume limit) ground states.

To use a famous image by S. Coleman, a little man living inside one of these possible asymmetric worlds would have a hard time detecting the rotational symmetry of the laws of nature (all his experiments being under the effect of the background magnetic field). The symmetry is still there — the Hamiltonian being rotationally invariant — but “hidden” to the little man.
Besides, there would be no way for the little man to detect directly that the ground state of his world is part of an infinitely degenerate multiplet. To go from one ground state of the infinite ferromagnet to another would require changing the directions of an infinite number of dipoles, an impossible task for the finite little man (Coleman, 1975, pp.141-142). ...[I]n the infinite volume limit all ground states are separated by a superselection rule.

The same picture can be generalized to quantum field theory (QFT), the ground state becoming the vacuum state, and the role of the little man being played by ourselves.

This means that there may exist symmetries of the laws of nature which are not manifest to us because the physical world in which we live is built on a vacuum state which is not invariant under them.
In other words, the physical world of our experience can appear to us very asymmetric, but this does not necessarily mean that this asymmetry belongs to the fundamental laws of nature. SSB offers a key for understanding (and utilizing) this physical possiblity.
--plato.stanford.edu

"Broken symmetry" and "force", are meant to be more or less interchangeable then? No. Infact, I specifically said otherwise. What about the sentence "It's 'unbroken' but it's still a force." did you not get?
which should have promptly annihilated each other, so why does the universe contain matterThe weak force doesn't respect chiral symmetry. It'd still be a force if it did.
Or to put it another way - physicists, expecially the high-energy ones, look for symmetries and asymmetries, and try to explain them. Or they build a theory and see what it needs to make it asymmetric, or what breaks the symmetry?They examine a system and look for symmetries. The symmetries constrain what things are allowed in the symmetry.

For instance, in quantum field theory Lorentz symmetry is required. That means your description must not change when you do a Lorentz transformation, so your equations must be 'Lorentz scalars'. This constrains the terms allowed in your Lagrangian.
That's what I thought "symmetry-breaking" was about: an explanation for an apparent force, like electrostatic force, or magnetic force (why does a bit of iron inside a wire coil experience a torque when the coil is "energized" with a current?). You appear to be relating the concept to only QFT, and the gauge symmetries...?The magnetic force is a component of the electromagnetic force, it's not an 'apparent force'.
Maxwell's theories show the symmetrical relation between the electric and magnetic forces, which wasn't spotted and explained "correctly" until Einstein came along and understood it's a case of a Lorentz symmetry group. Then the implications of that led to special and general relativity.The Lorentz transformations properties of EM was known before Einstein. Hence why it's called a Lorentz transformation and not an Einstein transformation.

Besides, the electromagnetic force is a bit strange. In 4 space-time dimensions there's 3 E components and B components. In other numbers of space-time dimensions, they are different. For instance in 5 space-time dimensions there's 4 E's and 6 B's.
I can be more specific about my Yang & Lee statement, if you can be more specific about how the phase symmetry in U(1) is "still a force" - phase isn't a force...? I thought a force was because of an asymmetry (in a field), like I think I say a few times in the above?The phase isn't a force, the exchange of field quanta is the force. The phase choice gives things like electromagnetic charge conservation.
Any ideas about how a precessing gyroscope experiences a force, but a gyroscope with no angular momentum doesn't? (I know about rotating inertial frames and all, but how is it explained as an asymmetry? Because a non-rotating wheel is more symmetrical in a rotating inertial field?)You seem to be thinking about all of this completely incorrectly.

Breaking symmetries doesn't imply a force. Symmetries imply conserved quantities. Translational symmetry -> momentum conservation. Rotational symmetry -> angular momentum conservation. U(1) symmetry in EM -> charge conservation. Breaking a symmetry removes the conserved quantity. CP is violated, no conservation of baryon number, you get an excess of one kind of matter.

A gyroscope which is spinning has non-zero angular momentum. To conserve this, it must precess.

What about the sentence "It's 'unbroken' but it's still a force." did you not get?
The bit about what exactly is "unbroken", and what's "still a force". The EM field isn't a force, and nor is phase symmetry a force.
Force is something you observe because of an asymmetry in a field - an inertial field, an electric potential field - a temperature gradient is an asymmetrical field.
The magnetic force is a component of the electromagnetic force, it's not an 'apparent force'.I would say that, more exactly, the magnetic field is a component of the EM field. The magnetic force is definitely something "apparent" about this field.
It's apparent when you produce a rotation (or transform) in the EM field, that is a local asymmetry (say, by "stopping" one of a coupled pair of "swinging pendulums", or by moving a charge through a coiled conductor, or across a capacitor) - the local asymmetry "reveals" a force, or a rotationally symmetric global relation changes to an asymmetric local relation.
Breaking symmetries doesn't imply a force. Symmetries imply conserved quantities.
That's just another way of saying global and local symmetry choices reveal a force at work. To have something that changes, you also need something invariant.

Breaking the symmetry makes the force apparent - like the conservation of angular momentum in a rotating inertial frame.
A force is observed pulling on your arm (rotating it), because the spinning object has angular momentum and is inertially coupled to a rotating body with angular momentum (but how does it exchange momentum by rotating your arm?).
It's actually a bigger coupled oscillating system (like the two pendulums).
How does your explanation reveal an asymmetry or explain a force with: "it must precess". Why "must" it?
Where are we in this text book? Obviously you're somewhere near the last few chapters, and I've barely read the first. But are we on the same page with individual concepts of gauge symmetry, and symmetry-breaking?

[W]e have the resulting close connection between the notion of symmetry, equivalence and group: a symmetry group induces a partition into equivalence classes. The elements that are exchanged with one another by the symmetry transformations of the figure (or whatever the “whole” considered is) are connected by an equivalence relation, thus forming an equivalence class.

The group-theoretic notion of symmetry is the one that has proven so successful in modern science. ... The way in which the regularity of the whole emerges is dictated by the nature of the specified transformation group.

Summing up, a unity of different and equal elements is always associated with symmetry, in its ancient or modern sense; the way in which this unity is realized, on the one hand, and how the equal and different elements are chosen, on the other hand, determines the resulting symmetry and in what exactly it consists.

The definition of symmetry as “invariance under a specified group of transformations” allowed the concept to be applied much more widely, not only to spatial figures but also to abstract objects such as mathematical expressions — in particular, expressions of physical relevance such as dynamical equations.

Moreover, the technical apparatus of group theory could then be transferred and used to great advantage within physical theories.--plato.stanford.edu

The Lorentz transformations properties of EM was known before Einstein. Einstein was slow, apparently to pick up tensor calculus, but derived Lorentz theory as part of his relativity ideas.
He saw that Maxwell's EM model didn't need an ether to move through, it just needed to be re-formalised as a Lorentz group, or as his version of the Lorentz form.

As I said rather vaguely earlier with: "a case of a Lorentz group".

And someone needs to explain what was meant with:
In an unbroken electroweak system, you find that you have 4 fields which can be swapped amongst themselves in a particular way.
The Yang-Mills fields that describe vector bosons...?
Pre breaking, electroweak has 4 fields which transform under SU(2). After symmetry breaking these juggle around and form 4 specific linear combinations. Three of them pick up mass and one doesn't. Three vector bosons and a photon. Symmetry is broken but there's a force.
What is the symmetry-breaking, or how is it broken, you're implying the Higgs coupling? And what about this force, what is it, the "weak" force?
The SU(3) symmetry of gluons isn't broken and it definitely works.
You're going to have to expand what you mean with: "isn't broken"...?

Well, just to sort of come out and say it: the idea of a connection still has the same kind of meaning to a physicist as anyone else. The idea of a coupling, as a kind of connection is a bit more exact (there's a lot of mathematical equations, or a formalism, that have to go along with it).

Sometimes a possible connection (a correlation) is observed, but it depends what's being looked for, and how. Sometimes the idea of a possible connection leads to models (abstractions) that end up fitting the real thing pretty closely. Another idea to connect to quantum gauge theories is "approximation", as the phrase "up to some gauge", that someone posted elsewhere implies.