[[Mod. note -- This article arrived at the moderation system with
very long lines. I have manually rewrapped these... alas resulting
in some oddities of spacing and indentation. -- jt]]
On Thursday, October 5, 2006 at 4:48:39 PM UTC-5, Jay R. Yablon wrote:
> The metric tensor g_uv is typically understood to be a function of
> spacetime coordinates, that is, g_uv(x^u).
>
> Has anyone ever considered a metric tensor Fourier transformed into
> momentum space,
>
> g_uv(p^u) = int {d^4x exp (-2 pi i p x) g_uv(x^u)}?
>
> And, is there any reason why it might not make sense to consider this?
> Above, int is an integral sign and exp is exponentiation.
>
> Note that the inverse transformation, of course, is:
>
> g_uv(x^u) = int {d^4p exp (+2 pi i p x) g_uv(p^u)}
>
> which includes the factor int d^4p which after reduction becomes int
> (dp^2/p^2) = ln p^2 which, evaluated between a mass m and infinity oo,
> becomes the logarithmically-divergent term ln (oo/m^2) to which a cutoff
> oo-->M^2 is assigned, such that ln (oo/m^2) --> ln (M^2/m^2), which is
> then "swept under the rug" into the definition of the running charge
> during renormalization.
>
> Thanks,
>
> Jay.
> _____________________________
> Jay R. Yablon
> Email: jyablon@nycap.rr.com
Back in grad school, played with the idea.
Consider the metric dS^2=dx^2+dy^2+dz^2-dt^2 (c=1)
dS^2= n_ij dx^i dx^j (n_ij=1 Minkowski metric), (i,j=1,2,3,4)
PDE for this metric is the steady state of
the 2nd order diffusion PDE d_i[d_i f(x_i,T)=d_T f(x_i,T) ('4D+1T')
d_i partial derivative w.r.t. the variables
here T some convenient parameter (along mass, thermodynamic
temperature etc.)
the steady state is the wave equation
d_i[d_i f(x_i,T)=0 ('4D+0')
(here cords are orthogonal Cartesian + time).
The diffusion PDE can be derived from information theory or
maximum entropy methods as D[]+D[b<(x_i)^2>]==0 with
the Shannon measure or the entropy measure (negentropy)
=Integral{ P(x_i,T)lnP(x_i,T) dV} , the maximization of entropy
obtains the Gaussian (for flat spacetime metrics) f= P(x_i,T)=
e^(-x_i ^2)/2DT /Sqrt[4PiDT]^4
and wave forms (sine cosine etc.) for the steady state.
Fourier transform of the Gaussian obtains momenta (energy-momenta) as
P(k_i,T) => Integral{ e^I(x_i.k_i) e^-(x_i ^2)/2DT V_dx_i }
complete the square,
=> Integral{ e^-(x_i - I.n.k_i/2)^2 e^-(k_i ^2 n./4) dVol}
P(k_i,T)=e^-(bb (k_i ^2)) /Z(bb(T))
which solves the diffusion PDE
d_(k_i)[d_(k_i) P(k_i,bb)]= d_bb P(k_i,bb)
which in steady state is d_(k_i)[d_(k_i) P(k_i,bb)]=0
Now from tensor analysis the diffusion tensor (here flat,
Minkowski like) is the inverse of the fundamental metric tensor
(D is now diffusion tensor not the variational symbol used
above) (D_ij)^-1=g_ij ( of the k_i.., where above it would
have been of the x_i)
dK^2= g_ij dk^i dk^j
= dk_x ^2 + dk_y ^2 +..-dk_t ^2 .
and we note that the 'dispersion relation' of the momentum-diffusion
PDE obtained by Fourier transforms Integral{ e^-I(k_i.(x_i-x_i')
e^I.(W_T.bb) ( d_(k_i)[d_(k_i) P(k_i,bb)]= d_bb P(k_i,bb)) dVol_k_i
}
=> let x_i-x_i'=Dx_i
W_T= s.Dx_i^2
while the dispersion relation of the coordinate based diffusion
PDE by the inverse Fourier transforms was
E_T= s' k_i ^2 (thermodynamic energy = kinetic momentum energy(3D) +
temporal energy (1t)) .
constants hbar c D n etc. aside (set to 1 say,) 2-point PDEs
(forward and backwards Fokker-Planck of which the diffusion eqns
are a special case of) solved by
P(x_i,T|x_i',T'), for which diffusion tensors are by tensor analysis
(try change of variables on the backwards F-P for non flat diffusion
tensors for general relativistic case and derived metrics) are
inverse metric tensors (see Risken, haken, Gardiner books or
literature), make a straightforward case for propagation (statistical
mechanics and thermodynamics), and geometry/topology.
And as at least for flat metrics, where propagation is (flat)
constant diffusion tensor valued, x->k Fourier transforms or at
the max entropy state function level (equiv. negentropy=information
theoretic) direct derivation of least biased PDFs that are the
solutions of said Fokker-Planck PDEs, allow Gaussian x to Gaussian
k PDE(x) to PDE(k) transformations, and then also diffusion
tensors for k that obtain k-metric tensors. Thus far shown
Minkowski...for Reimannian tensors of general relativity the
diffusion tensors are nonlinear, and easiest way to rederive and
generalize the above is to work with the backwards F-P PDEs,
change-of-variables (diagonal tensor to unit diagonal tensor)
from x->X (as in P(x)dx=P(X)dX) or from nonlinear diffusion
coefficients to unit or constant diffusion coefficients, and the
perform the tensor analysis that obtains (D_ij)^-1 =g_ij...this
is at least one way to then obtain 'curved momentum' metric
tensors and invariant 'lengths->momenta' metrics as
dK^2= a_1(k_i,T,..)(dk_1)^2 +.....-a_4(k_i,T)(dk_4)^2
corresponding to dS^2=A_1(x_i,T,..)(dx_1)^2+...-a4(x_i,T,..)(dt^2)
these metrics' lengths of general relativity.
However the same issues exist for momentum-energy-topology...the
Einstein equation can be obtained for a g_ij(k_i..) such that
R_ij-(1/2)Rg_ij=E_ij (E_ij=T_ij the stress-energy tensor, but
to not overuse T)...but for momenta the stress energy tensor now
has shear and hydrodynamic stress rescaled such that it is
momentum like and not space-like. And as the point is to describe
curvature and deformation of spacetime, it remains to be seen
if any new simplification and new insights can be obtained from
such descriptions. However (again,) the challenges are a) time
on the same footing as space, so decoupling time from space or
reducing its order to 1st order as opposed to 2nd order b)
recasting curvature metric coefficients or inversely diffusion
coefficients to constants (i.e. curved->to->'flat' solvable,
easily quantizable) seem to be what is needed for 'quantum
gravity' (at least canonical quantum gravity). While tackling
the spacetime x_i version of a-b , it is not clear that k_i
metrics and k_i PDEs will make for an easier a-b.
Mentioned 'quantization', by that is meant the Matsubara-Schroedinger
diffusion-to-quantum-propagators method...i.e. a diffusion
Fokker-Planck PDE can be analytically continued to the complex
plane and retarded/advanced Green's functions quantum thermal
propagators derived:
[d_(x_i)[d_(x_i)-d_T]P(x_i,T|x_i',T')]=delta{x_i-x_i'}delta{T-T'}
Fourier transform P->G:
G(k_i,I.W_T)=1/[I.W_T-(s.k_i)^2]
analytically continue, or 'wick rotate W_T'
G^(r/a) (k_i,I.W_T)=-I f(w)/[w_T-(s.k_i)^2 +/- I.q]
[continued in next message]
--- SoupGate-Win32 v1.05
* Origin: you cannot sedate... all the things you hate (1:229/2)
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