Symmetries, forces and time constriants

[Boof-head]

This is one for the boffins of quantum cosmology:

 

Why is a symmetry-breaking important in gauge theories, and why are 'broken' symmetries important? Are asymmetries in general, important to notions of physical motion - force interactions - and what do constraints,or 'time constraints' have to do with it?

 

We have explanations for 3 known forces, one of which is long-range; the second known long-range force, gravity requires a symmetry-breaking interaction; what is gravitational symmetry and what are we looking for in that respect?

 

Starter for 10?

[Boof-head]

One of the things you learn about in QM is particle spin; then chirality comes in, which is quite a bit trickier to understand.

Isospin symmetry (not spin) is 'broken' for the weak and strong force interactions; chirality has two flavours. Isospin at the strong and weak level is fundamentally different, and is why we 'get' photons which are not spin-oriented - this orientation is restricted to fermions with mass. Mass is what breaks the isospin symmetry, when quarks and leptons 'acquire mass', or a vacuum expectation value.

 

This is all explained by higher symmetry groups with subgroups: SU(5), SU(6) GUTs have subgroups SU(3), SU(2), and U(1) embedded in their algebraic structure. The subgroups break the higher symmetries when mass is acquired by particles (vertices/edges) so the rotations are constrained. This constraint also 'generates' time, so to speak, or "time appears because of the symmetry-breaking"...

[Boof-head]

Here's a symmetry I find is very much a theme in theoretical physics and in applied physics. Every working theory has to be built around 3 distinct things, 1) a structure, a complex or lattice of some kind, possibly an undirected graph, but with an ordering, either long or short range (or both) because of its geometry; 2) a space of vectors in the neighborhood, that can interact with the structure, or 'energize' parts of it; 3) an algebra of states, that depend on the interactions between different parts or neighborhoods.

 

This is, I think, more or less what's known as a spectral triple is. Spectral theory is, as the name implies, a theory that describes an algebraic spectrum, like the EM one, whose structure is 'matter that radiates' and whose vectors are massless (but not 'matterless'). The spectral triple here is G, h and c. When you 'spin' this with time, you get G(t) or massive bodies, h(t) or quantum time-dependence (usually a Schrodinger Hamiltonian, the perturbation operator) and c(t); the last is also an algebra of states, which are all distances in a 4-dimensional structure.

 

G(t) deforms this other space-like structure, presumably somewhere sufficiently (in large enough time t, G will become a large structure - the algebra of G is to add matter together and form a product) that G will 'reduce' the other two members in the three, or compress the range they can act over; the algebra of h is to quantize the smallest spaces or structures that can form, or add together. These algebras are disjoint, but all three in the triple (G,h,c) meet at a vertex.

 

If you collapse this 3-degree vertex by reducing G,h to G(h), then you have a duple instead, or (G(h),c); we have a general solution because of blackbody thermal spectra and GR; the invariant velocity, c, constrains the spectrum of its algebra to geodesics over a time-surface, so we can't be certain what happens with 'just h' when we build a (wave-)mechanical 'time-function', because G just isn't in the frame and we have to use charge and spin in fermionic circuits, after we remove them from the thermal spectrum that G(h) is. We have to build a 'disjoint space' which is much colder than most of the remaining vacuum, and has much stronger or more tightly curled fields around it.

 

So G,h and c are fundamentally distinct (obviously), and we can do 'simple' or general kinds of embedding of one in the other, since each is a kind of function; G generates Avogadrian or molar bodies of mass (matter), h generates quantum states, by evolving them in a disjoint matter field (Schrodinger-Heisenberg quantized), and c generates distances in Einstein-spacetime.

[Boof-head]

We have more than a handful of theories that describe unified forces; they all have 'problems' of varying degrees of importance so none of them can be said to be complete - there may actually be a fundamental reason that we will never have a complete theory but will only be able to compose partial theories (explanations) for the existence of matter, the force interactions, expansion, inflation and the unexplained dark matter/energy that 'must exist' if our observations are on the mark.

 

Increasingly the largest views are being associated with the smallest; cosmology and particle physics have merged at some level, yet other researchers are saying high-energy large-scale accelerator science is dead, the LHC might be the last one we build because we wlll develop smaller, more efficient technology. Wake-field accelerators and associated tech, are believed by some to be the new direction. So if we can hang together as, you know, a species, and not screw things up so we can stay where we are a bit longer and not have to migrate to Mars (one or two of us), we will almost certainly develop new ideas and new theories that get us closer to the underlying structure (related to the large-scale) of space and time.

 

On that note, I composed this ditty:

 

"On the first of the cosmos, a D-brane gave to me

A Higgs field with symmetries, three

 

On the 2nd of the cosmos, my Higgs field gave to me

Two leptons, in a quark tree

 

On the 3rd of the cosmos, the Higgs field symmetry

Broke into SU(2)xU(1), and SU(3)..."

 

-- The New Plasmas

[Erasmus00]

Maybe it would be good to start by being concrete about definitions. Much of the above appears to be somewhat nonsensical.

 

What we mean by symmetry breaking is that the physical system in a theory does NOT have a symmetry that the laws of physics have. Consider- the electromagnetic interactions have a translation symmetry. If I move the whole system, the electromagnetic force does not change. Liquids also have this translational symmetry, we can move liquids around and not tell we've moved them.

 

Now, solids do NOT have this symmetry. If we have a crystal lattice, only discrete translations are allowed (one crystal unit, or at least an integer number of crystal units). When a system goes from solid to liquid, we say a symmetry has been broken.

 

This is important, because the fact that the laws are symmetric but the system isn't has lots of important consequences. Solids support shear vibrations but liquids don't- this is because of the broken symmetry.

 

Now, a gauge symmetry is a special type of symmetry described by parameters that can have a different value at each point in space. In quantum mechanics, vibrations in these symmetry parameters lead to massless particles. IF, however, the symmetry is broken in the system then the quantum mechancial particles can be massive. This is necessary to describe W and Z bosons.

 

BUT, we do not need any symmetry breaking for gravity- the graviton is expected to be massless, and other than relativity's symmetry (Lorentz symmetry) is has no special symmetry.

[Boof-head]

When a system goes from solid to liquid, we say a symmetry has been broken.

Ok, I can agree with that. Can you explain why you say this then:

What we mean by symmetry breaking is that the physical system in a theory does NOT have a symmetry that the laws of physics have.

You mean by this that a phase-change from solid to liquid, doesn't have a symmetry that "laws of physics have" ??

 

Symmetry-breaking is a mechanism; the mechanism in SU(2) is weak vector bosons which appear because of isospin symmetry. Isospin symmetry is higher than the SU(2) or SU(3) symmetries.

The concept of spontaneous symmetry breaking resolves the question of weak-charge conservation. At an energy much greater than 100 GeV where the SU(2)xU(1) symmetry is observed directly, the mass of a quark or a lepton is negligible; the handedness ... is therefore essentially fixed, and so the weak charge is effectively conserved. At low energy, ... the weak charge is not conserved but can disappear into the vacuum when a massive particle changes handedness [(direction)].

[Boof-head]

And now, this:

 

I've been wanting to explore the notions of 4-dimensional spaces (manifolds), and how to get to the Riemann hypersphere; there is another hyperdimensional sphere that's used in QM, the Bloch sphere, however this is in a fundamentally different Hilbert space. Reconciling the two models is the subject of a lot of mathematical research. The new theories are mostly experiment-free and the 'results' are consistent mathematical models.

 

Start with the 3-sphere.

 

[math] S^3 = \{(x_0,x_1,x_2,x_3) \in \mathbb R^4: {x_0}^2 + {x_1}^2 + {x_2}^2 + {x_3}^2 = 1\} [/math]

 

If we pair the [math] x_i [/math], as [math] (x_0,ix_1) \rightarrow (z) [/math], which takes [math] \mathbb R^2 \rightarrow \mathbb C^2 [/math], then

 

[math] S^3 = \{ (z_1,z_2) \in \mathbb C^2: |z_1|^2 + |z_2|^2 = 1 \} [/math]

 

or [math] S^3 = \{ q \in \mathbb H: |q| = 1 \} [/math], where [math] \mathbb H [/math] is Hamilton's quaternions, then S is a smooth manifold, a closed embedded submanifold of [math] \mathbb R^4 [/math], the Lie group is Sp(1) or U(1,H).

 

A 4-dimensional manifold with an algebraic structure, which has a Euclidean rep (a real version), called spacetime.

[Erasmus00]

Ok, I can agree with that. Can you explain why you say this then:You mean by this that a phase-change from solid to liquid, doesn't have a symmetry that "laws of physics have" ??

 

Not the phase change, the solid. As I state in my previous post- electromagnetic interactions have continuous translation symmetries. Solid crystals have only discrete symmetry. The laws (electic attraction) have symmetry the ground state (the solid) do not. The symmetry is broken.

 

Symmetry-breaking is a mechanism; the mechanism in SU(2) is weak vector bosons which appear because of isospin symmetry. Isospin symmetry is higher than the SU(2) or SU(3) symmetries.

 

What do you mean by higher? And in the electro-weak theory the SU(2) bosons don't break the theory, the scalar field does. This is important- if a vector or fermion field broke the SU(2) symmetry, it would also break Lorentz symmetry.

 

Also, in the post just before this one, you spend time describing a 3-sphere, which doesn't appear relevant to symmetry breaking. Also, a 3-sphere is not equivalent to spacetime.

[Boof-head]

We seem to be talking past each other. The SU(2) symmetry-breaking to SU(2)xU(1) is because of the weak-charge.

The vector bosons are field 'interactions'; as you were a little vague about the phase change from solid to liquid, I was vague about "by the weak vector bosons"; the fact they exist is a broken symmetry - they wouldn't be around if it wasn't. The theory is incomplete because it doesn't explain how particles 'gain' mass; it assumes mass is tied to distance scales.

 

This is the domain of recent quantum-gravity type string theories and their derivatives; according to the 80s view: "The structure of the vacuum spontaneously breaks the [weak] symmetry, giving mass to the three carriers of the weak force [the Z, W+, and W-] but not the photon." The photon is like the trivial case in a larger 'algorithm' with a recurrence relation. Except it isn't a timed algorithm, it's a "massed" one - steps are distance scales; we assign isotopic spin to the highest, unbroken symmetry group and embed SU(2), SU(3), and U(1) in it.

 

The hypersphere and hyperspaces are so a representation of spacetime, this doesn't mean "are equivalent to spacetime"; spacetime is a 4-dimensional manifold. Each dimension is equivalent in Minkowski-spacetime, but this is broken in 3+1 dimensions for a local observer. What's your version of a 4-dimensional spacetime model? Why would you not use a hypersphere, specifically in this version?

 

If a manifold is Riemannian or even pseudo-Reimannian, does it represent Riemann space which is what Einstein thought when he concocted SR/GR?

 

P.S. Since we're into nonsensical, I still think your initial statement is nonsensical; A theory which explains physical interactions, phase changes say, does have the symmetry that real physical solids and liquids have. But you've said they don't, viz: "the physical system in a theory does NOT have a symmetry that the laws of physics have".

 

A theory describes a physical system; these systems obey (symmetrical) physical laws, in real time and space; would you care to reconsider your nonsensical statement?

[Erasmus00]

We seem to be talking past each other. The SU(2) symmetry-breaking to SU(2)xU(1) is because of the weak-charge.

 

In the weak interaction, SU(2)xU(1) breaks to just U(1). In (as yet unproven) guts, either SU(5) or SO(10) break down to SU(2)xU(1).

 

I was vague about "by the weak vector bosons"; the fact they exist is a broken symmetry - they wouldn't be around if it wasn't.

 

They would, they would just be massless.

 

The theory is incomplete because it doesn't explain how particles 'gain' mass; it assumes mass is tied to distance scales.

 

It does explain how particles gain mass- through coupling to the Higgs field which developed a constant value throughout space.

 

Except it isn't a timed algorithm, it's a "massed" one - steps are distance scales; we assign isotopic spin to the highest, unbroken symmetry group and embed SU(2), SU(3), and U(1) in it.

 

Isotopic spin is the approximate symmetry of swapping u and down quarks. Its a global (ungauged symmetry) that doesn't really participate in symmetry breaking (being ungauged). It has nothing to do with broken/unbroken symmetries.

 

 

Also, the only unbroken symmetries at our energy scale are SU(3) color (QCD) and U(1) (QED).

 

The hypersphere and hyperspaces are so a representation of spacetime; spacetime is a 4-dimensional manifold.

 

But a 3-sphere is a 3 dimensional manifold, we only need 3 coordinates to identify a point on its surface.

 

Each dimension is equivalent in Minkowski-spacetime, but this is broken in 3+1 dimensions for a local observer.

 

Minkowski space explicitly treats time differently. The metric diagonal is (-1,1,1,1). Time is clearly singled out.

 

Further, there is no reason to assume spherical geometry. Minkowski space is pseudo-Euclidean. GR does admit 4-sphere type solutions for the universe as a whole, I guess.

 

edit:

P.S. Since we're into nonsensical, I still think your initial statement is nonsensical; A theory which explains physical interactions does have the symmetry that real physical solids and liquids have. But you've said they don't, viz: "the physical system in a theory does NOT have a symmetry that the laws of physics have".

 

A theory describes a physical system; these systems obey (symmetrical) physical laws, in real time and space; would you care to reconsider your nonsensical statement?

 

The statement is meaningful, and I clearly explain what I mean. I even gave an example. The laws of electromagnetism have a continuous translational symmetry. Liquids have a continuous translational symmetry. Solids DO NOT have a continuous translational symmetry.

 

When the system has LESS symmetry (in our case, a solid has only discrete symmetries) then the laws describing the system(em) then a symmetry has broken. So, when a system undergoes a phase transition from liquid to solid, a symmetry breaks. When it goes from solid to liquid, the symmetry is restored.

[Boof-head]

Isotopic spin is the approximate symmetry of swapping u and down quarks. Its a global (ungauged symmetry) that doesn't really participate in symmetry breaking (being ungauged). It has nothing to do with broken/unbroken symmetries.

No, it has an exact symmetry.

Quarks and vector bosons have isospin too, so isospin does have "something to do with" broken symmetries. Massless vector bosons would not participate in any exchanges, so "wouldn't be around", except "in principle", whatever that could possibly mean.

 

Time has an opposite sign, in M-space, so that WE can single it out. Spacetime is not concerned, it sees a distance. The sign is required, because the universe has observers in it, who require it.

 

I think I might give up at this point. You say the 3-sphere is 3-dimensional; is a 2-sphere 2 dimensional and a 1-sphere 1-dimensional? And the meaning of your statement seems to me to be: "a description of physical symmetries does not have the symmetries of physical symmetries." (??)

[Erasmus00]

No, it has an exact symmetry.

Quarks and vector bosons have isospin too, so isospin does have "something to do with" broken symmetries.

 

See the wikipedia page on isospin, or Georgi's group theory book, or Halzen and Martin's Subatomic book. Isospin is a pretty good symmetry, but only approximate. If up and down quarks had the same mass, it would be a good symmetry. The SU(3) flavor symmetry would be good if ups,downs and stranges had the same mass.

 

Massless vector bosons would not participate in any exchanges, so "wouldn't be around", except "in principle", whatever that could possibly mean.

 

Thats not true- photons are massless and interact with charged matter. Gluons are massless and they hold our nucleons together.

 

Time has an opposite sign...so that WE can single it out. Spacetime is not concerned, it sees a distance.

 

In the manifold, the time direction is distinguished. The metric clearly indicates its different. Spacetime "sees" a distance but it computes distance along the time direction as negative and along any of the 3 spatial dimensions as positive.

 

You say the 3-sphere is 3-dimensional; is a 2-sphere 2 dimensional and a 1-sphere 1-dimensional?

 

Yes, thats why they were given these names! An n-sphere is n dimensional!

 

And the meaning of your statement seems to me to be: "a description of physical symmetries does not have the symmetries of physical symmetries." (??)

 

No, a physical system (a solid) that doesn't have the full set of symmetries of the underlying theory (electricity and magnetism). I don't know how to be any clearer?

[Boof-head]

Here is how I understand (but perhaps I'm like, totally lost here) the symmetries in the SU(5) gut.

U(1) and SU(2) are exact symmetries, both describe interactions between massive particles.

U(1) 'ignores' isospin. SU(2) is gauged by weak-charge and isospin.

 

You can compose an su(2) algebra, or SU(2) group with left-handed particles (leptons); there are no right-handed neutrinos or left-handed antineutrinos in SU(5). The isospin points "up", so is an exact symmetrical associative 'algebra of states' over SU(2)/su(2). When the cross-product is taken, or SU(2)xU(1), this is broken - isospin is conserved, and always points "up", since the leptons are left-handed; The asymmetry is seen between massive electrons and "massless" neutrinos - the nodes used in the topographic rep. Electrons exchange a [math] W_0 + V_0 [/math], as [math] \gamma + Z_0 [/math], and the "breaking" is that the neutrinos exchange them as [math] Z_0 [/math]. In that sense, the photon is the representative of the electroweak breaking - the remainder after all the other exact symmetries are accounted for, by the theory and by the actual exchanges.

 

Although it's more than 30 years old, it is a highly successful explanation; but lacks the important Higgs mechanism. It simply assumes the particles gain a VEV, but ties the effective masses to distance scales.

[Boof-head]

Spacetime "sees" a distance but it computes distance along the time direction as negative and along any of the 3 spatial dimensions as positive.

I would say "we compute a distance", spacetime computes that distances in space are the same as distances in time. This is why time has an opposite sign, so we can recover it after a transformation into a distance in the Lorentz/Minkowski basis.

 

P.S.

In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space. Just as an ordinary sphere (or 2-sphere) is a two dimensional surface that forms the boundary of a ball in three dimensions, a 3-sphere is an object with three dimensions that forms the boundary of a ball in four dimensions.

3-sphere - Wikipedia, the free encyclopedia

 

Correction: the SU(2) symmetry is inexact (except for the direction, or handedness which is exact) because weak charge is not conserved. Mr Georgi did say in that quote that it 'vanishes' when an electron changes direction, because electrons have handedness or chirality, apart from their spin angular momentum.

 

He also published this:

The answer [for the massive vector bosons, and massless photons] that is now favored is that the underlying force is indeed symmetrical, and in some hypothetical initial state, all the carriers of the weak force would be massless. What is not symmetrical is the quantum-mechanical vacuum...

Which suggests we need a theory of the quantum vacuum. We need to quantize spacetime.

[Erasmus00]

Here is how I understand (but perhaps I'm like, totally lost here) the symmetries in the SU(5) gut.

U(1) and SU(2) are exact symmetries, both describe interactions between massive particles.

U(1) 'ignores' isospin. SU(2) is gauged by weak-charge and isospin.

 

I can't understand what you are saying. First, we need to realize that in an SU(5) (or any GUT) there are two scales, the GUT breaking scale and weak breaking scale.

 

In SU(5), SU(5) is a gauged symmetry with a U(1) subgroup, an SU(3) subgroup, and an SU(2) subgroup. Something analogous to the Higgs field breaks SU(5) down to SU(3)xSU(2)xU(1). These are our gauge symmetries. Not to be confused with other global symmetries that are accidental (in the sense they aren't built in explicitly). These accidental symmetries include isospin and Gell-man's flavor symmetry.

 

At a lower scale, another higgs field acquires a vev, which breaks SU(2)xU(1) down to just U(1).

 

You can compose an su(2) algebra, or SU(2) group with left-handed particles (leptons)...

 

Isospin is NOT the same thing as spin! Leptons don't carry isospin, which is a meson/baryon quantum number. Spin is related to how things transform under Lorentz symmetry. Since Lorentz symmetry is exact, total angular momentum is conserved, and if your fermions are massless, spin is conserved.

 

Although it's more than 30 years old, it is a highly successful explanation; but lacks the important Higgs mechanism. It simply assumes the particles gain a VEV, but ties the effective masses to distance scales.

 

It has a higgs; only the higgs can take a vev without breaking Lorentz symmetry.

 

Now, from your own wikipedia quote on the 3 sphere

 

In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space. Just as an ordinary sphere (or 2-sphere) is a two dimensional surface that forms the boundary of a ball in three dimensions, a 3-sphere is an object with three dimensions that forms the boundary of a ball in four dimensions

 

3-spheres are 3 dimensional manifolds. 2-spheres, two dimensional, etc.

 

The easiest way to see this is that the 1-sphere is a circle, which requires one coordinate (the angle phi). The 2-sphere requires two angles, usually the spherical polar angles (theta,phi). By extension, the 3 sphere requires 3 angles, so on.

[Boof-head]

Yes, you've just explained that the surface of a sphere is 2-dimensional; The surface of a 3-sphere is 3-dimensional, which tells us the 3-sphere must be 4-dimensional in volume.

 

For some reason, this reminds me of something else with 4-dimensional "volume".

Isospin is NOT the same thing as spin! Leptons don't carry isospin, which is a meson/baryon quantum number. Spin is related to how things transform under Lorentz symmetry. Since Lorentz symmetry is exact, total angular momentum is conserved, and if your fermions are massless, spin is conserved.

Where have I said isospin and fermion spin is the same thing, though? Answer: I haven't said this.

 

Do you know when the Higgs mechanism for SU(5) was added to the original? It's just that he doesn't mention any such thing in the 1981 article, some years after the theory was first published. He talks about the mass heirarchy problem and scaling.

 

How's this?

For NN bound states, isospin is conserved under all exchanges of degrees of freedom. Lepton-baryon exchanges imply that total spin (isospin + fermion spin-momentum + space) is antisymmetric for all exchanges. Space can be shown to have symmetric exchanges (l = 0), so that either 1) fermion spin-1/2 cancels (has positive and negative spin), or 2) particles are spin-1.

[Erasmus00]

Yes, you've just explained that the surface of a sphere is 2-dimensional; The surface of a 3-sphere is 3-dimensional, which tells us the 3-sphere must be 4-dimensional in volume.

 

Right, but when a mathematician or physicists says n-sphere, they mean the surface. The manifold is the surface.

 

Where have I said isospin and fermion spin is the same thing, though? Answer: I haven't said this.

 

You had talked of handedness and leptons, which made it sound very much like you were talking about spin, and not isospin.

 

Do you know when the Higgs mechanism for SU(5) was added to the original? It's just that he doesn't mention any such thing in the 1981 article, some years after the theory was first published. He talks about the mass heirarchy problem and scaling.

 

Its in the original 1974 paper, on page 3.

 

For NN bound states, isospin is conserved under all exchanges of degrees of freedom.

 

What do you mean by NN bound stay? Neutron-neutron? If thats true, where is the lepton for lepton-baryon exchange?