marcus
Jul15-04, 09:30 PM
quantum gravity research seems to be undergoing some changes
so just because this function comes up today doesnt mean it
will be around for eternity, but that said, look at this function
L(x) = \sum_{p = semi-integer > 0}\frac{1}{x^\sqrt{p(p+1)}}
It comes up in connection with Black Hole entropy, and LQG.
The Immirzi parameter (a useful number in LQG) can be found
by solving the equation
L(x) = 1/2
what does this function look like? Would it be convenient for anyone at PF to plot it? has anyone seen this function before. I have not.
It is vaguely reminding me of the Riemann zeta, but it really is not anything like the Riemann zeta
you sum over all half-integer spins-----1/2, 1, 3/2, 2, 5/2,.....
and you take the geometric mean of successive pairs of spins
and use that as an exponent for x.
--------------------------
suppose you solved numerically and discovered the X such that
L(X) = 1/2
(do we know that a solution exists? is it unique?)
then if you took the natural logarithm of X and divided by 2 pi
that would be what Christoph Meissner at Warsaw (and Jerzy L, and friends) are saying is the right Immirzi number.
maybe even without the quantum gravity connection this function
L(x) is a nice function. or who knows maybe it is just some throwaway
function with a boring shape.
like 1/x3/2
Oh, if you could find an analytic expression for this function, or rather its inverse, then you would have a research paper
following on the coattails of Meissner gr-qc/0407052
but no one will find such an analytic expression because Poles are known to be very good at that sort of thing and Meissner would have already made great efforts to do it and couldnt. so it is a kind of novelty, a nice-looking
easy to define function which Poles cannot solve.
so just because this function comes up today doesnt mean it
will be around for eternity, but that said, look at this function
L(x) = \sum_{p = semi-integer > 0}\frac{1}{x^\sqrt{p(p+1)}}
It comes up in connection with Black Hole entropy, and LQG.
The Immirzi parameter (a useful number in LQG) can be found
by solving the equation
L(x) = 1/2
what does this function look like? Would it be convenient for anyone at PF to plot it? has anyone seen this function before. I have not.
It is vaguely reminding me of the Riemann zeta, but it really is not anything like the Riemann zeta
you sum over all half-integer spins-----1/2, 1, 3/2, 2, 5/2,.....
and you take the geometric mean of successive pairs of spins
and use that as an exponent for x.
--------------------------
suppose you solved numerically and discovered the X such that
L(X) = 1/2
(do we know that a solution exists? is it unique?)
then if you took the natural logarithm of X and divided by 2 pi
that would be what Christoph Meissner at Warsaw (and Jerzy L, and friends) are saying is the right Immirzi number.
maybe even without the quantum gravity connection this function
L(x) is a nice function. or who knows maybe it is just some throwaway
function with a boring shape.
like 1/x3/2
Oh, if you could find an analytic expression for this function, or rather its inverse, then you would have a research paper
following on the coattails of Meissner gr-qc/0407052
but no one will find such an analytic expression because Poles are known to be very good at that sort of thing and Meissner would have already made great efforts to do it and couldnt. so it is a kind of novelty, a nice-looking
easy to define function which Poles cannot solve.