Usually, requiring that the state vector in a quantum theory be invariant under a unitary transformation require other fields (what we call gauge fields) be added that end up being representing physical interactions.
For example, consider the state vector |y>. The norm <y|y> is invariant under local U(1) transformations
(local, as in a function of space-time, is key). Write the state as |y>=v|0> where v is some operator, then saying <y|y> is invariant is like saying v* v is invariant when v -> v e^(-I* T(x,t)).
In physics, a term ~v* v would represent the potential energy of a spring. To represent the kinetic energy of a spring, we need a term like (d^{\mu}v* )d_{\mu}v where the d_{\mu} represent derivatives w.r.t. space and time (\mu is an index which runs from 0 to 3, 1,2,3 are x,y,z, 0 is time). Here, the U(1) transformation does not leave this term valid, because the derivative acts on the e^(-I T(x,t)) too...
However, if one were to make the replacement
d^{\mu} -> d^{\mu} - I e A^{\mu}(x,t)
where when
v -> v e^(-I* T(x,t))
we have that
A^{\mu} -> A^{\mu} - d^{\mu} T(x,t)
one can show (this doesn't have to be obvious!) that then the Lagrangian
~(D^{\mu}v)* D_{\mu}v - v* v
(where D is the new "covariant derivative" we described above) is now invariant under local U(1) transformations. It turns out that this Lagrangian, completely written out, looks a lot like classical electromagnetism, and it is, and A^{\mu} is the "4-vector potential". In fact, A^0 represents the good old electric potential V that I'm sure you nerds are quite familiar with. I'm not sure how enlightening that was, but at least you've seen the start of QED :)
Now, for this "monster group" ... from here [0] it seems that apparently, gravity in 2+1 dimensions seems dual to this particular group, that is, take the quantum state with that operator |y>=v|0> (v is what we call a creation operator in QFT, it "creates" particles mathematically; in string theory, it creates "vibrations on the string") that under monster group transforms, apparently whatever Lagrangian Witten made out of the v's in conformal field theory (something I should note is not my field, so I may be off here) the gauge group that might be needed would represent gravity.
Thanks for the response. I think this is a good answer to the question of "why would a physicist be working on the Monster group" that addresses the core idea of required invariants, and one along the lines that I have heard before (I suppose at this point I should confess to be an ex-physicist, although I never studied QED at anything but the most elementary level).
But, having heard it, I feel wanting. As you have deftly illustrated, starting with some basic requirements about invariants (from special relativity) you can end up at the Lagrangian associated with QED. And you can then (after the fact!) point out that the field in question is the electromagnetic one. Having studying electromagnetism enough to know what you are talking about, I can hear this kind of argument and it seems clever to me. There is some difference in semantics of the states, of course (classical solution of of the confined electromagnetic field coming in quanta, but perhaps not understood as "quantum observables" proper). But if I had not studied these things I would probably just be confused.
And I think this is how I feel when I hear about this advanced quantum gravity stuff (once again, I don't mean to complain about your excellent response, I think it just addresses a different issue then the one I am talking about).
If you are willing to entertain me, I'll try to ask a clarifying question in terms of analogy. If someone asked me "why does the Lagrangian for QED looks like it does", I might say to them that "the speed of light (light being made of electromagnetic fields) is constant in all inertial reference frames (lets hope they understand Galilean relativity) and that this fact forces QED to look that way". This explanation makes reference to the "stuff" that the field is made out of and the observations about invariants (stated in ordinary language) that it must satisfy.
So, now if I ask "why does the Lagrangian for quantum gravity looks like it does", you would say...
For example, consider the state vector |y>. The norm <y|y> is invariant under local U(1) transformations
|y> -> e^(-I* T(x,t))|y>
For
<y|y> -> <y|e^(I* T(x,t))e^(-I* T(x,t))|y> = <y|1|y> = <y|y>.
(local, as in a function of space-time, is key). Write the state as |y>=v|0> where v is some operator, then saying <y|y> is invariant is like saying v* v is invariant when v -> v e^(-I* T(x,t)).
In physics, a term ~v* v would represent the potential energy of a spring. To represent the kinetic energy of a spring, we need a term like (d^{\mu}v* )d_{\mu}v where the d_{\mu} represent derivatives w.r.t. space and time (\mu is an index which runs from 0 to 3, 1,2,3 are x,y,z, 0 is time). Here, the U(1) transformation does not leave this term valid, because the derivative acts on the e^(-I T(x,t)) too...
However, if one were to make the replacement
d^{\mu} -> d^{\mu} - I e A^{\mu}(x,t)
where when
v -> v e^(-I* T(x,t))
we have that
A^{\mu} -> A^{\mu} - d^{\mu} T(x,t)
one can show (this doesn't have to be obvious!) that then the Lagrangian
~(D^{\mu}v)* D_{\mu}v - v* v
(where D is the new "covariant derivative" we described above) is now invariant under local U(1) transformations. It turns out that this Lagrangian, completely written out, looks a lot like classical electromagnetism, and it is, and A^{\mu} is the "4-vector potential". In fact, A^0 represents the good old electric potential V that I'm sure you nerds are quite familiar with. I'm not sure how enlightening that was, but at least you've seen the start of QED :)
Now, for this "monster group" ... from here [0] it seems that apparently, gravity in 2+1 dimensions seems dual to this particular group, that is, take the quantum state with that operator |y>=v|0> (v is what we call a creation operator in QFT, it "creates" particles mathematically; in string theory, it creates "vibrations on the string") that under monster group transforms, apparently whatever Lagrangian Witten made out of the v's in conformal field theory (something I should note is not my field, so I may be off here) the gauge group that might be needed would represent gravity.
[0] http://en.wikipedia.org/wiki/Monstrous_moonshine#Conjectured...