Symmetries, forces and time constriants

[Boof-head]

Isospin is conserved, it has an exact symmetry; NN bound states is "N states bound to N states", it's a general statement about total spin, decomposed into three terms.

 

Isospin is exact in SU(5), the simplest rep is all in right-handed isospin particles; these imply the existence of left-handed particles, and the subgroups that break SU(5) have them. Again, mass is explained by the Georgi-Glashow model as a function of the distance scales, if the mechanism the Higgs "supplies" to it had a place it would explain the mass heirarchy.

[Erasmus00]

Isospin is conserved, it has an exact symmetry;

 

No, it isn't. It is conserved under strong interactions, but not weak interactions. Its a symmetry of strong interactions, thats it.

 

NN bound states is "N states bound to N states", it's a general statement about total spin, decomposed into three terms.

 

Particles bind together, not states? If N objects are bound to N objects, doesn't that mean that 2N objects are all bound together? Also, isospin does not add or subtract to spin. They are unrelated, completely different things. Isospin has nothing to do with angular momentum.

 

Again, mass is explained by the Georgi-Glashow model as a function of the distance scales, if the mechanism the Higgs "supplies" to it had a place it would explain the mass heirarchy.

 

This sentence doesn't make much sense. The higgs mechanism is used in SU(5) same as in every GUT. The Higgs fields necessary to break SU(5) are written on page 2 of their original 1974 paper, as I said.

 

When you say "function of the distance scales" it makes it sound like you are talking about renormalization effects, which do run the couplings in these theories, but don't introduce mass.

[Boof-head]

No, it isn't. It is conserved under strong interactions, but not weak interactions. Its a symmetry of strong interactions, thats it.

Okay, so isospin is only related to SU(3), then? This is what you appear to be saying. If this is what you are saying, can you explain how weak and electromagnetic forces have distinct spin which is not related to isospin?

 

You have said this, you see, which has prompted the question

isospin does not add or subtract to spin. They are unrelated, completely different things. Isospin has nothing to do with angular momentum.

 

Does this mean isospin is always conserved, as an inexact symmetry rather than, as I say, an exact one? Since you work with particle physics this should be a no-brainer.

 

The Higgs mechanism for SU(5) can wait, unless you also think you can explain what it is in SU(5) and how it explains the masses of these particles. Do you know WHAT the mechanism actually is and do you know how it explains the masses

 

Note: I have carefully underlined some words, to help with the problem. We are talking here about a theory (the Higgs field) that explains mass, so how does it, in say a sentence or two?

 

And here is the full "quote" about the NN states of, I'm sure, bound "particles", in some "spin-space" or other. I can't quite recall if it's from some lecture notes I copied or just from a thread somewhere. It isn't mine, and the original or at least my copy of it looks like this:

NN bound states

[math] \psi_{total} = \psi_{space} [/math] x [math] \psi_{spin-momentum} [/math] x [math] \psi_{isospin} [/math]

 

[math] \psi_{isospin} [/math] is antisymmetric under exchange of degrees of freedom.

 

P.E.P. exchange implies [math] \psi_{total} [/math] is antisymmetric for all exchange interactions.

 

[math] \psi_{space} [/math] can be shown to be symmetric [math] ( l = 0 ) [/math], and spin is also symmetric (spin-1 or spin-0).

[Erasmus00]

Okay, so isospin is only related to SU(3), then? This is what you appear to be saying. If this is what you are saying, can you explain how weak and electromagnetic forces have distinct spin which is not related to isospin?

 

Spin is defined by how particles transform under Lorentz (SO(3,1) symmetry). Spin 1/2 particles transform in the spinor representation of SO(3,1) which is (more or less) SU(2)xSU(2). This is unrelated to isospin.

 

Isospin is an approximate symmetry observed in nature. It was originally an SU(2) symmetry relating protons (+1/2 isospin) and neutrons (-1/2 isospin). We now assign isospin to up quarks and down quarks. Now, a weak interaction can turn a proton into a neutron, and so isospin is not conserved. BUT isospin is conserved under strong interactions.

 

The Higgs mechanism for SU(5) can wait, unless you also think you can explain what it is in SU(5) and how it explains the masses of these particles. Do you know WHAT the mechanism actually is and do you know how it explains the masses

 

It works the same way in SU(5) it does in the standard model. A set of scalar fields (that live in the adjoint rep of SU(5)) take a diagonal expectation value which then breaks SU(5) into SU(3)xSU(2)xU(1). From here, things follow the standard model, with SU(2)xU(1) breaking down to U(1), and all particles acquiring masses through coupling to the higgs field.

 

And here is the full "quote" about the NN states of, I'm sure, bound "particles", in some "spin-space" or other. I can't quite recall if it's from some lecture notes I copied or just from a thread somewhere. It isn't mine, and the original or at least my copy of it looks like this:

 

I can't make sense of it. If it means neutron-neutron bound state, isospin cannot be antisymmetric, because neutrons have the same isospin.

[Boof-head]

Well, I have a vague recollection of the source and I'm on his trail; if he's commited a "spin-conservation" offense, he will be dealt with (by the cosmic equation)

 

However, he is doing a PhD in string theory, so I'm a little anxious about being shot in the foot.

 

P.S. You have outlined how the Higgs "gives mass" to particles in the SU(5) formalism; can you outline how it determines the heirarchy and predicts the masses of quarks and leptons, or the observation (experimental) that there are three families in the heirarchy?

 

(I already know it doesn't do this, so how is it part of the SU(5) GUT, in that case? Is it a disjoint union, in fact, of two distinct fields? So that the mechanism is still not incorporated 'fully' - I vaguely recall reading something like this)

[Boof-head]

The Higgs field must be scalar or it would break CPT; you would weigh more in one direction than another.

Analogously the electron has broken isospin - the electron doesn't care which direction it travels. Isospin is conserved because of antisymmetric exchanges, in the P.E.P region of SU(2). Pauli algebra is a subgroup which ignores the electron's left-handedness as a partner of the positron's right-handed isobaric spin-momentum. The outline above is for total spin in NN particles, a general view of the breakdown.

[Erasmus00]

The Higgs field must be scalar or it would break CPT;

 

Its would break all of Lorentz symmetry if not a scalar.

 

Analogously the electron has broken isospin - the electron doesn't care which direction it travels.

 

Electrons don't have any isospin, and isospin has nothing to do with direction of travel. The electron does have charge under the weak force, which is unbroken.

 

The outline above is for total spin in NN particles, a general view of the breakdown.

 

What is an NN particle? Two neutrons?

[Boof-head]

Electrons don't have any isospin, and isospin has nothing to do with direction of travel

Ok, can you explain positrons, why are these the antiparticle of the electron?

 

Here's a few googlers for "NN bound states"; when you've looked, can you explain why there are more than 270k of them? Well, that ain't that big and if you add "QFT" it's less than 50k

NN bound states - Google Search

[Erasmus00]

Ok, can you explain positrons, why are these the antiparticle of the electron?

 

What does this have to do with isospin? Also, is this question "why do electrons have anti-particles?" The answer is that to allow for creation and annihilation we need particles and anti-particles.

 

As to N-N bound states, I was confused by nuclear physics nomenclature. Its a nucleon-nucleon bound state, not a neutron-neutron bound state.

[Boof-head]

My grasp of electron-positron symmetry is that it's the same as quark-antiquark symmetry; because of isospin. [except for the two directions of angular momentum]

 

Leptons have handedness, so do quarks. How are electrons transformed to positrons? In SU(5) a left-handed positron is equivalent to a right-handed one "plus" a right-handed antineutrino; there are no left-handed antineutrinos in SU(5). Isospin for SU(5) is represented by the right-left antisymmetry of lepton chirality. In SU(2) exchanges the isopin is conserved but doesn't take part in Pauli exchanges (you can't rotate isospin).

 

SU(3) and SU(2) have antisymmetric isospin, they are disjointly connected in SU(5); where they connect there is a 2x3 and a 3x2 intersection of 12 intermediary exchanges: the X particles.

 

As to N-N bound states, I was confused by nuclear physics nomenclature. Its a nucleon-nucleon bound state, not a neutron-neutron bound state.

Right, it's "N states bound to N states" an inner product, except one of them can be "N anti-states"; it's from HEP and QFT theories (as you may have discerned).

[Erasmus00]

My grasp of electron-positron symmetry is that it's the same as quark-antiquark symmetry; because of isospin. [except for the two directions of angular momentum]

 

What symmetry is this? I can't figure out what you are trying to say.

 

Leptons have handedness, so do quarks. How are electrons transformed to positrons?

 

With a charge conjugation operator?

 

Lets get this completely straight-isospin represents up quarks to down quarks. That is it. The full right handed/left handed dirac quarks.

 

Weak isospin relates left handed doublets to other left handed doublets (up left and down left, etc).

 

SU(3) and SU(2) have antisymmetric isospin

 

This doesn't make any sense.

 

Right, it's "N states bound to N states"

 

Its specifically one nucleon bound to one nucleon (a proton-neutron bound state in nature).

[Boof-head]

Lets get this completely straight-isospin represents up quarks to down quarks. That is it. The full right handed/left handed dirac quarks.

Conclusion: "isospin is restricted to only SU(3) exchanges between quarks." Correct?

Weak isospin relates left handed doublets to other left handed doublets (up left and down left, etc).

Weak isospin is distinct, you don't see left or right-handed particles in SU(2), the weak force?

 

Since a non-scalar Higgs would break Lorentz symmetry, how, or is this related to CPT symmetry? Surely Lorentz symmetry requires T at least?

[Boof-head]

What positrons and electrons "have to do with isospin" is that they are mirrored versions of each other.

They have the same mass - symmetry #1.

They have the same charge, but opposite in sign - antisymmetry #1

They have the same spin, but opposite in sign (direction) - antisymmetry #2

 

Leptons form the axes in SU(5); An axis relates a charged lepton to a neutral partner which is the corresponding neutrino/antineutrino. Since these are only isospin-symmetric as left and right handed single particles rather than pairs, the neutrinos represent SU(5) isospin when they "pair" with a charged lepton.

The electron/positron symmetry occurs in nature with the same representation: 2 leptons, a left-handed electron + right-handed positron (antielectron).

 

The absence of left-handed anti-neutrinos and right-handed neutrinos is the isospin symmetry. This is because there are only two ways to construct a lepton axis; if the other 2 neutrinos existed there would be four kinds of axis, and at least twice as many particle interactions.

 

So isospin is 'broken' in SU(5) as a global symmetry for the actual particles and exchanges (32 -2), a total of 30 to account for. As I've noted, SU(3) is isospin invariant, quarks can change spin by changing color. SU(2) is not, weak isospin is because of this broken lepton symmetry.

 

If the other (missing) neutrinos were around, space would look different; total spin is conserved because spacetime is spin-symmetric and SU(5) is not.

Spacetime is not affected by lepton isospin, this does not imply that charged leptons don't have any; the U(1) group and Pauli's su(2) don't account for it, because spacetime does.

 

Hence, an interaction must account for time, and is why Feynman diagrams show electrons moving forward, and positrons backward along the "spacetime" direction, or arrow of time. Feynman's diagrams and their antisymmetric "time exhange" for particle/antiparticle implies that time exists because SU(5) is broken (at greater than about the 10^-29 cm scale)

[Boof-head]

Spacetime parameters

 

Einstein's spacetime uses formal models (Gaussian quadratics, Riemann geometry, Lorentz, Maxwell) to parameterize space and time. That is, it re-codes space,time -> spacetime. In which mass is a boundary for energy, and energy is a boundary for mass.

The only way to parameterize them is to parameterize space and time; these are assumed to be continuous, so spacetime is a product (not a sum). Discrete space and time both 'vanish' except, as with any Boole product, one of them is recoverable. We copy one or the other in order to 'retain' something.

 

Implication is usually framed in propositional yes/no arguments of the form IF ... THEN ...; these imply summation and inversion, as IF a or b THEN a,b.

 

Inversion is IF a THEN not b, IF a THEN not a is clearly "invalid" but represents an inversion of a, or it says "a can be inverted". This implies that "I am" means "I am not".

Boolean logic lets you invert a, by using reciprocation and unitary equivalence, or a/a =1. so that (not a)/a = -1 and we're consistent. With just positive 1, and 0 this doesn't work out. The existence of -1 is implied by Boolean (universal) logic, and so the sq. root of -1 must also exist.

 

A direct product is represented propositionally as IF a and b THEN ab; which can be restated with IF not a or not b THEN not ab, or, IF not (not a and not b) THEN ab.

 

Therefore we imagine that "the number 1 exists" so we can invert it and imagine that universal (time) logic does too. We need a way to generate a sequence, given predicates (general parametrization); we need sequentiality. "Flow" requires that a potential vanishes from disjoint union products, such as of 'space' and 'time'.

 

Flow is because one or the other 'disappears' when spacetime appears in our epistemologic frame.