Exercise attempting to evaluate the path integral of the Einstein-Hilbert action over the range from -00 to +00
Jay R. Yablon
exercise where I have endeavored to show, mathematically, how to
evaluate the path integral:
Z=${-oo to +oo}Dg exp[i $d^4x sqrt(-g)((1/2k)R+L_m)] (1)
for the Einstein-Hilbert action. This is at the link below:
In section 1, we briefly review the Einstein-Hilbert action and insert
this into the standard Feynman path integral. That is, in section 1, we
simply obtain (1), while laying out the standard, known background
material which supports this action and relates it to the Einstein
equation. Nothing in section 1 is new.
Then, in section 2, we show how to *reparameterize* the matter
Lagrangian in such a way as to allow the path integral (1) to be
mathematically evaluated over the definite field-density range from
negative to positive infinity, using Fourier analysis.
The end result, deduced in (2.25) and generalized for various choices of
"measure" in (2.33), is the *mathematical* deduction that the evaluation
of (2) above over the definite range -oo to +oo is as follows:
Z=${-oo to +oo} D(sqrt(-g)g_ab) exp [i $d^4x sqrt(-g) ((1/2k)R + L_m)]
= delta^(4xoo)[.25 g_ab ((1/2k)R + L_m) ] (2)
where delta^(4xoo) is a 4 x infinite dimensional Dirac delta (Fourier
impulse) function.
The E-H Lagrangian density:
L = (1/2k)R + L_m (3)
therefore moves through the path integration intact, and ends up inside
an impulse function in the space conjugate to spacetime. In this
formulation, the traceless EM field (L=.25 F^uv F_uv) clearly would
contribute, as would any trace matter kT=R.
I want to be very clear on one point: I am saying nothing here or in
this paper about the *physics* that is associated with (2). I am simply
addressing the *mathematical* problem of evaluating the definite
integral (1), and getting a real answer, which, according to what is
deduced here, the mathematics tells us is (2). The only place where
anything other than pure mathematical calculation comes into play, is in
selecting the "measure" for the integration field. Rather than "choose"
one measure over another, I have used what seem to be the four most
reasonable possibilities for the measure, and, as it turns out, they all
lead to the same mathematical outcome.
I will keep my fingers crossed that after several trials that were
errors, perhaps I finally have figured out the right way to do this.
Thanks.
Jay
____________________________
Jay R. Yablon
Email: jya...@nycap.rr.com
co-moderator: sci.physics.foundations
Weblog: http://jayryablon.wordpress.com/
Web Site: http://home.roadrunner.com/~jry/FermionMass.htm
Jay R. Yablon
the "measure problem" which Dr. Carlip raised in one a recent posts in a
parallel thread at
http://groups.google.com/group/sci.physics.research/browse_frm/thread/d15e35b971697965#.
I have therefore posted an update to this exercise at:
What the above now contains, just added, is a detailed discussion of how
to select the measure so that the overall path integral evaluates to
something that is invariant under general coordinate transformations. I
begin to introduce this discussion at equation (2.13), and the upshot is
that the only measure which meet this criteria of a generally-invariant
evaluation of the path integral is sqrt(-g), alone.
In brief, the "measure problem" is resolved by distinguishing between
local coordinate invariance and global coordinate invariance. The goal
is to end up with a result that is globally-invariant, even if the
variable of integration is itself not locally-invariant. One may think
of this approach as one of "hidden invariance." ***This all works,
because in taking a definite integral, even over a coordinate dependent
measure, the measure drops out from the overall expression that results
from evaluating the definite integral.***
In essence, the way this works is that when we do path integrals, we are
forced to select a particular system of coordinates, perform the
definite path integration in this system of coordinates, and obtain a
result. Then, we transform into a different system of coordinates,
again do the path integration, and obtain a second result. The goal is
to find that the second result is the same as the first result.
Repeating this ad infinitum, we choose system after system of
coordinates until we have exhausted every possible coordinate system,
and each time we do the integration, it is the goal to end up with the
same, invariant result. That is, no matter what system of coordinates
we choose, our path integral should yield the same invariant result.
Again, even though any particular choice of measure is a non-covariant
choice, the path integral should evaluate invariantly no matter what
coordinates we choose to represent the measure. This is then a
"globally" covariant result. This is what we mean by "hidden
covariance." This updated post demonstrates that sqrt(-g) is the only
measure which meets this goal and is the only truly invariant choice.
The final result of this effort, is in equation (3.19):
Z = $(-oo to +oo) Dsqrt(-g) exp[i S_EH] = delta^(4xoo)(L_EH) (1)
where the Einstein Hilbert action:
S_EH = $(-oo to +oo) sqrt(-g) L_EH d^4x (2)
and the Einstein Hilbert Lagrangian density is:
L_EH = (1/2 kappa) R + L_matter (3)
If you look closely at (3.19), and think about the Fourier
transformation:
$(-oo to +oo) dx exp[i x w] = delta (w) (4)
it should be self-evident that (1) (which is (3.19) in the linked file)
evaluates to a Dirac delta. In fact, in retrospect, one can omit just
about everything leading up to (3.19), stare at (3.19) closely for a few
moments, and realize that (3.19) is just a disguised form of (4), with a
coordinate-dependent measure that "washes out" no matter what coordinate
system one uses for the integration.
I just realized I am missing some sqrt(2pi) factors on the deltas. I'll
put this in a next draft, but those are non-essential to the main
result.
Jay.
Ken S. Tucker
> Overnight, it became apparent to me that I should specifically discuss
> the "measure problem" which Dr. Carlip raised in one a recent posts in a
>
> I have therefore posted an update to this exercise at:
>
Hi Jay.
For what is being measured, large but finite limits may enhance
the ability to do a solution. We'll use light rays passing by a mass
(that is mathematically variable), and measure light deflection and
the Saphiro Effect, that provides the light-rays path in space time.
With the emitted and then inbound light consisting of a fixed
frequency
and wavelength, (an invariant, defined by (scalar) number of
wavelengths)
charts a quantized 'path' extended to near infinity, though remains
finite,
so a measurement can be predicted.
HTH's
Regards
Ken