We should be able to really make some progress with string theory, one way or another.

I’m burdened by the fact that I can visualize it, but I can’t explain what I visualize because I understand math at an intuitive level – I know the answer, but I can’t explain how I got it, which is actually pretty worthless in upper level math.

I learned a little group theory as a grad student in solid-state physics (20 years ago). For a massive HW assignment I had to take a messy 40 x 40 matrix and tame it with the tools of group theory. The Prof. said, “I’m giving you three weeks to do it because you really need four!” — he wasn’t kidding. So, I’ll pass on the 453,060 x 453,060 matrix!

The significance of group theory is that it is a mathematical framework that allows one to understand the role of symmetry in geometries and systems of equations.[*]

For a physicist, symmetries are particularly useful because of Noether’s theorem, which states that for any symmetry of a physical system there must be some quantity that is conserved.

For example, if the laws of physics are the same at all times (i.e., if running a given experiment identically at two different times must give the same result) it turns out that the conserved quantity is the total energy of the system (aka the First Law of Thermodynamics).

Similarly, if the laws of physics are the same everywhere in the universe you get the conservation of linear momentum (also called just “momentum”). And, if the laws of physics are the same regardless of the orientation of the experiment (i.e., there is no preferred direction in space) then you get the conservation of angular momentum.

The tie to string theory? My understanding is that you decide what quantities have to be conserved if string theory is to work, then determine the symmetries that give rise to conserved quantities. Evidently the group that contains all required symmetries for superstrings is E8. If all of the details of E8 were worked out, then one would have an arsenal of mathematical relationships to employ in trying to understand the ramifications of string theory.

And that is why this is a big deal.

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[*] How do geometries and systems of equations relate, you ask? Equations can be considered as describing a geometrical object. E.g., y = 3x + 4 describes a particular line in a plane. If you want to generically describe all lines you write y = ax + b, where a and b are constants whose values determine which particular line is under consideration.

(If you recall your H.S. algebra and geometry, you might remember that a = the slope of the line and b = the point where the line crosses the y-axis.)

Any statements you can make that are true for all values of a and b are therefore statements that are true about all lines. Conversely, any statement you can make that are true about all lines are statements that are true about all equations of the form y = ax +b.

For example, I might need to solve two simultaneous equations:
[Eq. 1] y = ax + b
[Eq. 2] y = cx + d

For a given set of constants a, b, c, and d, there might be one point {x,y} which satisfies both equations. That particular pair of values {x, y} is the solution to the simultaneous equations.

It is possible that there is no solution for a particular set of constants a, b, c, and d. Is there an easy way to find out if a solution exists?

Remember that these equations describe straight lines. Two lines on a plane have to cross at some point {x, y} — unless they are parallel. (If they cross, the coordinates {x, y} of the crossing point is the solution to the simultaneous equations.)

If two lines are parallel, how do their equations relate? Well, parallel lines must have the same slope, so the value of “a” in Eq. 1 must be the same as the value of “c” in Eq. 2.

So, if I have two equations like Eq. 1 & 2, but a = c, then there is no {x, y} pair that satisfies both equations.

Group theory allows the argument above to be formulated as a set of mathematical relationships, rather than as an exercise in logic.

My understanding from Lisa Randall’s book is that there is something of a truce at the moment regarding string theory. The objection from its critics is not that it cannot be proved, but that it cannot be observed experimentally at any energies that can be… or are ever likely to be… created in any collider or similar apparatus. String theorists construct abstract models, then look for resemblance to well-established aspects of known reality. Sometimes they find them. But they cannot do experiments with the strings themselves. Anything that gives string theorists more powerful analytical tools is very much to their benefit, as they have no experimental tools.

You can’t map lands you’ve never seen.

If the string guys are happy, maybe I’ll be less confused sometime in the future.