I can answer some of that, compact, but there's a few assumptions in there that don't make a lot of sense to me.
I'm rather suspicious of that line of argument since for a long time string theory people were thinking 6 compactified dimensions were necessary -- wasn't till ~1995 that Schwarz (I think it was) convinced everyone 7 were needed.
compact said:
They state that SU(3)xSU(2)xU(1) is contained in the symmetry group, but why are we so interested in this, as opposed to something else in the symmetry group? It's certainly crucial here that the above group is a gauge group, but what is it the gauge group for and what (if any) local variable does the above gauge group correspond to? I'm assuming that knowing this would make it clear why this is a "truly obvious" thing to look at.
It turns out that by requiring the Lagrangian of a quantum field theory be invariant under local gauge transformations (ie, of the field) of some (particular finite-dim representation of a) symmetry/Lie group, the form of the Lagrangian, and hence the quantum field theory, is almost uniquely determined. Up to a slew of constants, of course. (It's also assumed the field theory is renormalizable.)
By choosing SU(3)xSU(2)xU(1) as the fundamental gauge group of nature, and using the measured values of particle masses, coupling constants, mixing matrices, etcm you get the Standard Model of field theory -- the SU(3) local gauge symmetry corresponds to the strong interaction, and the SU(2)xU(1) to the electroweak interaction. I think you might need to add in the Higgs field or just its vacuum expectation value by hand, but I'm not sure.
For some reason though they want the spacetime manifold to obey these symmetries. That would seem like it maybe could somehow be justified were they global symmetries -- if we really lived in R3xS1 space, with position along the S1 dim somehow corresponding to QED phase, we would see a global U(1) symmetry there. But for local gauge symmetries it doesn't seem to make any sense...
Why are we looking for a manifold of minimum dimension? At the end of the day we're just going to pick out the dimensions corresponding to a non-compact symmetry group, so who cares about having extra dimensions so long as the Minkowski symmetry is still present?
Presumably the idea is if we want to to make the local gauge symmetries of our fields into symmetries of extra spatial dimensions, we need at least this many of them. But why we would want/need to do that...?
Any why does it matter if the other dimensions are compact? I'm sure there's many other ways we could decompose this M4xB symmetry group of minimal dimension into a product two groups with distinct topological properties. How do we know that this is the one we're interested in? That is, I'll agree that it appears promising that when we "peel off" the non-Minkowski part of the group, the resulting group is compact, but how do we know that this is really what we want to be looking at and not just some mathematical coincidence?
I'm not sure what you mean here. If we live in a spacetime manifold M4xB, B needs to be very small in order to correspond to reality, where local spacetime looks like M4 to us. The specific requirement of compactness I know comes out of the physics of QFT/string theory somehow.
It's not clear to me why the manifold couldn't be some more complex thing that wasn't decomposable into a direct product of M4 and something else though. Is that what you're saying?
Originally posted by VelocideX
People are right in saying that all spatial dimensions must be mutually perpendicular (to be strictly correct, they are orthogonal), but the problem is that time is not a spatial dimension and therefore cannot be perpendicular in any traditional sense....
Clearly you can consider spacetime as a four dimensional space of events, eg containing points (x,y,z,t). You don't need to have an inner product (and hence a notion of 'perpendicular') on a space to define dimensionality -- all you need is a topology, which clearly exists for spacetime (just R4).
Originally posted by JTNOLA5211
Why arent they observable to us ? Can we see if we looked for them or are they purely mathmatical?
So these dimensions are pruely spatial and are technically observable to us? Care to explain?
Im curious.....
The idea is that they're "rolled up" to an incredibly small size, so we don't notice them.
