# Unification of the forces?

1. Feb 15, 2012

### tinto99

Ok so I'm a bit confused. We have a confirmed electroweak theory - electromagnetism and the weak nuclear force were unified through the gauge principle and the fact that they merge into one force at high enough energies.

Merging the strong nuclear force with electroweak would give us a Grand Unified Theory, of the electronuclear force. What I don't understand though is this - the strong nuclear force is also described by gauge theory. We have also confirmed that at even higher energies (even earlier moments of our universe), we have an electronuclear force.

So why is the strong nuclear force not regarded as being unified with the electroweak force? Is it that we know that they are unified at high energies, it's just we can't explain why (e.g. the fact that baryons have to be able to convert into leptons, something we can't account for at the moment?)

I just don't understand what it is that means we don't have a GUT. It seems to me that that we've already described how the three forces are unified by the gauge principle, and know that they're different because of symmetry breaking.

If someone could enlighten me that'd be great, thanks :)

2. Feb 15, 2012

### jambaugh

Before we even unify weak and em theory we still have a gauge theory with distinct gauge groups for each force:
$$G_{w+e+c} = SU(2)_{weak} \times U(1)_{charge} \times SU(3)_{color}$$

In addition the weak gauge symmetry is broken so that only a residual $U(1)_{z-isospin}$ component remains at normal temperatures.

The electro-weak unification involves a combined
$$G_{ew+c} = U(2)_{electro-weak}\times SU(3)_{color}$$
gauge group. Now the type of group didn't really change: $U(2)=SU(2)\times U(1)$ but the meaning of the center did: $U(2)_{ew}=SU(2)_{isospin}\times U(1)_{hyper-charge}$
then the e-m group is a correlated combination of the hyper-charge and residual isospin groups after symmetry breaking. The mechanism predicts to some extent how weak charge correlates to electrical charge.

One may imagine however that the hyper-charge appendage is itself a residual of some other gauge group say the part of an electro-color unification, (U(1)xSU(3)=U(3)).

A fully unified theory would combine both electro-weak and color groups in a larger (edit: simple) group e.g. SU(5) which has SU(2)xU(1)xSU(3) subgroup.

When we use semi-simple groups the coupling constants for each simple factor is determined independently. If we can combine them into a single unification group, it will have a single coupling constant. The presumption is that the unification group, has itself undergone some symmetry breaking mechanism leaving the semi-simple product group of symmetries.

 SU(5) unification (Georgi Glashow model) makes some almost right predictions however it also predicts proton decay (a mechanism would exist in this larger symmetry to convert quarks to leptons). Many experiments have been done searching for protonic decay (watching huge numbers of protons {in water} over a long period of time. So far no significant evidence of protonic decay has been observed to my knowledge. The very existence of conventional matter given the age of the universe implies protons must have very long half-lives. The lower bound quoted in wikipedia is > 10^33 years...[edit2 based of experiments] an unfathomably long time.

Last edited: Feb 15, 2012
3. Feb 16, 2012

### tinto99

Thank you very much for your reply, although I'm only 15 and so have little knowledge of the notation you're using (I'm also a bit confused by the isospin and hypercharge parts). Is there any way you could simplify it a bit?

And as for the proton decay - I understand they haven't seen one in 25 years of searching? Lee Smolin states in his book 'The Trouble with Physics' that we should give up the SU(5) unification? My knowledge of the gauge principle and symmetry breaking doesn't span far beyond what he says in this book, sorry. Is supersymmetry an alternative to grand unification?

4. Feb 16, 2012

### jambaugh

SU stands for "special unitary group" and "U" for unitary group. These can be represented mathematically by matrices. The unitary group U(n) is the group of n by n unitary matrices, complex matrices whose conjugate transpose yields their inverse. Every unitary group has a central U(1) subgroup consisting of the unit complex numbers times the identity matrix. It is central in that it doesn't interact with the other group elements, they commute. Thus we can factor a unitary group into two parts: U(n) = SU(n)xU(1).

The "special" in SU means we have removed this central U(1) component.

Semi-simple groups are products of simple groups: G = S1 x S2 x ....

Eli Cartan classified the simple Lie groups (continuous groups) into 4 main infinite classes along with a finite number of exceptional cases.
There are the special orthogonal groups SO(n) of rotations in n-dimensions, where n is even and n is odd. These define two classes. They can be represented by real orthogonal matrices (transpose = inverse).

There are the special unitary groups SU(n)
There are the symplectic groups Sp(2n) which are a bit involved.

Each of these classic group types can be though of as preserving some form on the vector space upon which they act. Orthogonal groups preserve a symmetric form ( (u|v) = (v|u) like a dot product). Unitary groups preserve a Hermitian form ( <u|v> = complex conjugate of <v|u> ). Symplectic groups preserve a symplectic form ([u|v] = - [v|u]).

"Isospin" is the name for the gauge degree of freedom associated with the weak force. Hyper-charge is like electromagnetic charge (and indeed contributes to EM charge).

I should also mention that the classification of simple groups gets more complicated when you allow for indefinite forms. For example there are pseudo-orthogonal simple groups SO(p,n) p for "positive dimension" "n for negative dimension". In particular the group of Lorentz transformations in Relativity is the group SO(3,1) (3 space + 1 time).

How the gauge group relate to particle spectra is when we look at the specific representations. The same Lie group may have different matrix representation acting on spaces of different dimensions. For e.g. the spatial rotation group SO(3) and more generally the Lorentz group SO(3,1) there are spinor, vector, and higher rank tensor representations. The spinor representation relates to the spin of fermions. Bosons typically have a vector representation (indicated by their polarization vector).

With the unitary gauge groups we look at the fundamental representation and the adjoint representation (matrices acting on vectors and matrices acting on each other) which gives us the fermionic and bosonic particle spectrum. Hence the color group SU(3) transforms 3 dimensional color (3 quark colors) and 8 adjoint elements (the 8 gluons) in distinct representations.

To learn more study up on Linear Algebra and Abstract Algebra.

Here's a [STRIKE]good[/STRIKE] great book on the subject, easy to read and not requiring too much technical background: In Search for the Ultimate Building Blocks by Gerard t'Hooft.

5. Feb 16, 2012

### tinto99

I haven't learnt about matrices or groups yet but will certainly look into them, thanks again

6. Feb 16, 2012

### The_Duck

Tinto, the short answer to your question is we have no experimental evidence for any of the many proposed theories that would actually unify the electroweak and the strong interactions into one "grand unified theory." We have *not* confirmed that any such unification held in the first moments of the universe--rather the idea of a GUT is compelling enough that many *believe* that some sort of unification held then.

Grand unified theories take the two distinct gauge theories that describe the electroweak and the strong interactions and place them within a larger gauge theory. If this is actually the case it means that at high energies, for instance, gluons and photons and W and Z bosons are all basically the same particle. Obviously this is not the case at low energies. GUTs also invariably predict new particles and processes, but the new particles are so massive and the new processes are so rare that we have not been able to observe them, if they exist. (We have been able to rule out some proposed GUTs by refuting the predictions they make along these lines).

7. Feb 16, 2012

### tinto99

Does the fact that we can make experimental observations that confirm the electroweak theory mean that the symmetry between electromagnetism and the weak nuclear force isn't completely broken even at our low temperatures? And even if we haven't confirmed it, is it more or less an accepted fact that at extremely high temperatures and energies, the electroweak and strong forces do merge into one? Is it just that we're having trouble describing how this would work? Does the lack of evidence about proton decay throw SU(5) grand unification out the window?

And on a sort of separate note, and I have no idea about this as I saw it on a website and wasn't sure - do we require supersymmetry for the forces to merge into one at high enough temperatures? I think I heard that with our standard model, the ranges and strengths of the forces would narrowly miss out merging into one another even at extremely high temperatures.

8. Feb 16, 2012

### The_Duck

Electroweak symmetry is broken, but the fact that it holds at high temperatures has consequences at low temperatures, where it is broken. For example it requires certain mathematical relationships among the rates of various weak and electromagnetic processes that we can measure and confirm.

"Accepted fact" is I think too strong a word; there is no experimental evidence, just a number of good theoretical motivations. There's no reason the universe has to implement this idea. The situation is, it's possible to construct many different kinds of GUTs that are consistent with current observations--we know how to write down a variety of plausible GUTs--but there are no positive experimental results to distinguish which of these is right, or even to indicate that grand unification occurs; all we have are negative results ruling some theories out. So there is no Grand Unified Theory in the sense that there is no *one* grand unified theory.

If you haven't seen it already, the Wikipedia page address some of your questions, include the possible role of supersymmetry.

9. Feb 18, 2012

### tinto99

ok thanks i'll check out the wikipedia page

10. Feb 18, 2012

### juanrga

No really. There is not real unification into a single interaction because as one of the interactions continue being mediated by its own boson.