though i'm unclear about the U1 part. — Olivier5
U(1) is just the simplest possible symmetry group. It is the symmetry of a rotating circle. And nothing is more symmetric than a circular object.
If you have a sphere, it always looks the same no matter how you rotate it. That is why the Greek Atomists imagined atoms as little spheres - the simplest material form.
A triangle (or tetrahedron) would have a more complex symmetry. The smallest turn of a triangle makes a visible difference. You can see right away something has moved. It is only after a 120 degree rotation that the triangle maps back on to itself as if nothing in fact changed.
Compare that to spinning a circular disc - one that has no marks to give the game away. Nothing visible ever changes no matter how furiously it is turned. The disc could be standing still for all you can tell.
Photons - as avatars of electromagneticism - have this simplest rotational symmetry. A sine wave is the trace carved out by letting a disc roll for a length by a mark on the circumference. So a photon - understood as a ray with a frequency - is just the simplest way to break the simplest state of symmetry.
It makes use of the two irreducible freedoms of nature under Noether's Theorem - rotational and translational symmetry. A photon rotates once and rolls one length - as the minimal definition of its existence.
At the Planck scale, such an electromagnetic event - a U(1)-expressing rotation + roll that marks a single wave-like beat of "hot action", something energetic happening - clearly happens in an unusual place.
Being confined to a spin and roll limited to a single Planck distance, it would also be the shortest, hence hottest, frequency event to ever exist. And energy being matter, it would also be the most gravitationally massive possible material event - so would curve the spacetime around it to a black hole extreme.
So it all becomes self defining. To break the simplest symmetry takes the simplest asymmetry - the combination of a spin and a roll that creates the mark, the trace, that is a spacetime-filling and energetic event. A single hot beat. The heat of that event defines the size of that spacetime (due to gravitational closure). And the size of that available spacetime in turn defines the heat that that even must have (due to the severest shortening of its frequency).
Ah, good point: there is a mathematical limit in terms of mass to infinite spliting, a limit that is equal to 0 mass, just as there is a solution in the form if a mathematical limit in Zeno's paradox. — Olivier5
So what I have just described is different in that instead the zero is about the zero sum game by which we can get "something from nothing" due to a symmetry-breaking that is based on a reciprocal balancing act.
A photon expresses the world of the circle. We can't tell if a circle is rotating. So that means that if reality is constrained by a generalised demand for maximum symmetry, then the ultimate best solution to that demand is to arrive at the shape of a circle. It is most stable shape in that it always must look the same.
A circle has translational symmetry as well because - without the help of outside reference marks - we can't tell if it is rolling along. This is the standard relativity. Motion is only detectable if the symmetry of the reference frame is broken in some way.
And as I say, putting a dot on the edge of a circle free to rotate + roll then counts as the most minimal mark, the simplest symmetry-breaking. The result is the "energetic event" of a spacetime frame that now contains a single sine wave.
We thus have a toy world described in reciprocal limits. There is both near perfect symmetry (U(1)) and near perfect symmetry-breaking - the dot on the circumference that reveals the still unconstrained "Noether" freedoms of the ability to spin, the ability to roll. The ability to thus mark an empty space constrained to perfect circularity with a sine wave event that the constraints can't in fact eliminate. And what can't be eliminated, must happen.
Real physics is more complex as the real Big Bang could not access the great simplicity of a U(1) world so directly. It actually had to constrain all the other possible symmetries - the many higher or more complex symmetries of group theory - and remove them from the fray first.
That created the shower of other particles with more complex rotational actions – the particles of SU(3) and SU(2) symmetries, according to the Standard Model. And even to get to U(1) perfection involves the Higgs kluge that cracks SU(2).
But the basic picture of what reality is seeking to achieve is to arrive at its greatest state of simplicity - as defined by the complementary limits of a perfect U(1) symmetry broken by a matchingly-perfect least form of asymmetry. The slightest blemish on the Cosmic cheek.
:grin:
The Heat Death tells us that this perfection is where we will arrive in the future.
Given the discovery of Dark Energy (a new unexplained ingredient in the story), we at least know that the Universe is coasting towards a destiny where spacetime will be best described by a reciprocal U(1) structure of holographic event horizons and their "as cold as possible" black body radiation. That is, a de Sitter cosmology.
Spacetime will be devoid of matter. Blackholes will have gobbled up all remaining gravitating matter and spat it out as electromagnetic radiation. So spacetime will be empty with an average temperature of zero degrees K. But it will also be filled with the even radiance of a cosmic bath of photons produced by the quantum holographic mechanism -
the Unruh effect.
These would be photons that - in effect - span the width of the visible universe in just a single wavelength. Their frequency would be measured in multi-billions of lightyears. A single rotation + roll that spans the gap that the speed of light can transverse.
So spot the connection. The beginning of spacetime - the Big Bang - and the Heat Death are mirror images.
Both are defined by that single U(1) based rotation + roll deal. Except at the Big Bang, the spacetime extent is the smallest possible, making the energy of the frequency as hot as possible. And at the Heat Death, it all has unwound to arrive at the complementary state of the largest possible spacetime extent and thus the coldest possible photon, the lowest possible energy wavelength.
Simplicity is always the goal. But because complexity has to be constrained first - all the other available higher symmetry states have to be got rid of along the way - it is only by the end of time that U(1) perfection (in terms of a simple circle and its irreducible symmetry breakings) is achieve.
That is why it isn't turtles all the way down. Existence is a push towards the limiting extreme that is simplicity. And that push is self-terminating in that the constraints (an insistence on arriving at maximal symmetry) contains within it the terminating thing which are those irreducible symmetry breakings.
Every kind of difference can be eliminated by U(1) circular symmetry, except a rotation and a roll. So already, the necessary blemish is built in to break that symmetry (in the simplest way).
Reality can go no further as there is no further splintering of the system arrived at. The constraining towards a symmetrical nothingness gets hung up on an irreducible grain of being. Things can go that far and no further - leaving reality as the coldest-possible fizzle of holographic event horizon radiation. Photons with the physical wavelength of the visible universe - that sea I speak of.
Charlie Lineweaver at ANU has written a bunch of decent papers about all this.
And as a caveat, Dark Energy remains a fly in the ointment. It is necessary to explain why spacetime expansion will get truncated by the de Sitter event horizon mechanism. But we need some further bit of machinery - another Higgs-like field or irreducible entanglement - to fold that into the final theory of everything.
As someone once said, explanations ought to be as simple as possible. But not too simple.
U(1) is the simplest possible story. But getting there was not a simple process as all other symmetries had the possibility of being the case. And the way they would then interact and entangle with each other becomes part of the story of where things actually wound hung up in practice.
The world of quark/SU(3) symmetry and lepton/SU(2) symmetry, plus the Higgs mechanism, is how we are all still hung up at that more complex level of things at the moment. The Universe is still breaking its way down through all those entanglements along the ultimate path.
The more complex symmetries have more complex spin states - chiral spin. And they thus have their own equivalent irreducible rotational symmetries. Higher level Noether freedoms that can't be eliminated directly.
By rights, in a symmetric world, matter ought to be annihilated by anti-matter leaving only radiation. But these complex spins produce uneven outcomes. So some matter survives. Quarks can then protect themselves by forming triplet structures like protons and neutrons.
And so that complexity could last forever. Proton and neutron crud messing up the empty perfection of a cooling and expanding void. A flood of ghostly neutrinos as well, messing up reality with their pointless SU(2) weak force interactions.
So long as black holes perform as advertised - hoovering up the crud and evaporating it into photons - the universe can get there in the end. SU(3) and SU(2) will be rendered relic memories. Maybe surprisingly, the Cosmos will arrive at the mathematically ultimate state of simplicity in terms of its symmetry - and the symmetry-breaking events, the holographic U(1) photons, needed to reveal that that symmetry in fact "exists".
The edge of the disc has to be marked to reveal the world within which it can rotate + roll. The blemish is needed to complete the deal that conjures "something from nothing".