Random Thoughts
Where is Mahndisa? I hope she is industriously writing papers, or at least enjoying a winter holiday. A while back, Mahndisa mentioned the helpful Riemann Hypothesis page. It is full of interesting information on various attempts to prove either the original hypothesis or modern variants.
For example, it would suffice to prove some form of the GUE hypothesis. Recall that GUE stands for Gaussian Unitary Ensemble, just as in Random Matrix theory. Yes folks, that's the same random matrix theory of the matrix models that kneemo was discussing with us a while back. The idea is that the zeroes of the zeta function (rescaled) are distributed like the eigenvalues of large ($N \rightarrow \infty$) random matrices in the ensemble.
In fact, many attempts to prove the Riemann Hypothesis involve matching the zeroes to a physical eigenvalue problem. Now what physical system could possibly have something to say about such a number theoretic problem? One can't help but wonder.
For example, it would suffice to prove some form of the GUE hypothesis. Recall that GUE stands for Gaussian Unitary Ensemble, just as in Random Matrix theory. Yes folks, that's the same random matrix theory of the matrix models that kneemo was discussing with us a while back. The idea is that the zeroes of the zeta function (rescaled) are distributed like the eigenvalues of large ($N \rightarrow \infty$) random matrices in the ensemble.
In fact, many attempts to prove the Riemann Hypothesis involve matching the zeroes to a physical eigenvalue problem. Now what physical system could possibly have something to say about such a number theoretic problem? One can't help but wonder.
posted by Kea | 12:21 PM
9 Comments:
Hi Kea
Interestingly enough, the Riemann zeta function pops up in the calculation of the critical dimension of bosonic string theory (D=26). In attempting to define (and order) the Virasoro operator L_0 (the light-cone Hamiltonian) there arises a sum of positive integers that appears to diverge. This leads one to define L_0 without this sum, hoping later this sum will converge using some cool math trick, to later add (a multiple of) it as an "ordering constant" c to L_0.
It turns out this sum is equivalent to zeta(-1), where zeta(s) is Reimann's zeta function, analytically continued. Explicitly the sum can be written as zeta(-1)=-1/12=1+2+3+4+..., giving our ordering constant the form: c=1/2(D-2)zeta(-1)=-1/24(D-2) where D is the dimension of spacetime.
Later, when requiring a vanishing commutator for the Lorentz charge, one requires that 1-1/24(D-2)=0 and 1/24(D-2)+a=0. These restrictions yield D=26 and c=-1. The c=-1 happens to give the precise shift in the mass squared operator (M^2=-p^2) for the open string spectrum to include massless photon states.
Who'd have guessed analytic number theory would have anything to do with spacetime dimension and light? Cool stuff!
It's neat that Riemann and Lorentz can be applied to reality (unlike concorde cosmologies). These posts show great promise for maths.
Hi kneemo, hi Louise! Yes, I remember the bosonic string dimension business from some lectures of David Gross, that I attended in 1994. Very pretty.
01 16 07
Hey Kea:
Thanks for the kind thoughts. I have been thinking of all of you, but I am swamped in my personal affairs. I have been hammering out ideas for a coupla papers, in addition to trying to get my business of the ground. I began three weeks ago with zero clients. Now I have sixteen!!! Money can enslave a person, but ultimately physics always wins. Take Care:)
Well, good luck with the business, Mahndisa! What is it?
Hi Kea
I have not read your reference F William Lawvere and Robert Rosebrugh 'Sets for Mathematics'.
I have been reading a classic from the Society of Industrial and Applied Mathematics [SIAM]: Tamer Basar and Geert Jan Olsder. 'Dynamic Noncooperative Game Theory', revised 1999 from 1982. The authors refer to this as a type of representation theory.
Since this is mathematics, the language is similar, but not identical to representation theory used in physics.
Some differences include using C* for cost-to-come and G* for cost-to-go,
Similarities include index sets, infinite topological structured sets, mappings and functionals in discrete time.
There is substitution for some of these items in continuous time such as time intervals, Borel sets, trajectory, action and informational topological spaces.
Tme appears to be treated as a duality.
There may or not be stochastic influences.
The Isaacs condition for the Hamiltonian is used.
Types of such games include:
for discrete time -
OL - open loop
CLPS - closed loop perfect state information
CLIS - CL imperfect state
FB - feedback perfect
FIS - feedback imperfect
1DCLPS - one-step delayed CLPS
1DOS - one-step delayed obsevation sharing
for continuous time -
OL
CLPS
eta-DCLPS - eta-delayed DCLPS
MPS - memoryless perfect state
FB
If players are allowed to be entities capable of exchanging enegy quanta or longevity then this might considered enegy economics?
The stochastic game may be consitent with the probablistic nature of QM.
Is phyisics failing to use a valuable tool of representation theory from applied mathematics?
Hi Doug
Good to see you here. I'm not a Game Theory person myself, but it's probably fair to say that Physics is ignoring a lot of good ideas from that quarter - but then physics is ignoring a lot of good ideas from a lot of quarters - this situation, however, will change in the near future due to certain awkward (for some) undeniable observational facts, such as the large number of black holes which now appear to exist.
01 18 07
Hey Kea:
Sorry, I am a private tutor for math and science. I tutor from five years all the way up to whatever age! Quite rewarding and in concert with my other interests. I thought of you so hard tonight when a kid had the problem of solving for x:
Cos^2(x)=cos(x)
hehehehe I said: "Mydear, this is an idempotent relation because it is of the form something squared equals something." I showed her how to solve the problem using that reasoning and she said: "What about the other root?" And I said: "Isn't zero squared the same as zero?" A"h yes" she replied. Then she said: "OH it is 90 degrees AND zero degrees for our answer."
Kea, you haven't even met this girl yet your thoughts have served to teach her very important concepts in mathematical reasoning.
Thank you.
Cool, Mahndisa. She sounds like a bright kid.
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