planck's length and planck's time: there is plenty of room at the bottom

feynman is once said to have remarked "there is plenty
of room at the bottom". today while
listening to prof. b's quantum mechanics (QM) lectures, i found new meaning to
what feynman said. in engineering and physics, there is a technique called
dimensional analysis, i.e. when you don't quite understand the physics of the
problem, you identify some characteristic parameters (which are usually a
length, duration of time, or speed of something). bunch them all up together
and do a simple analysis on their dimensions.
this usually reveals something about fundamental behavior of the system
with the exception of some numerical constants.






prof. b's point was that we use mass, length, and time as
the fundamental units because we are classical creatures; but in QM and
gravity, branches of physics that describe the very small and the very large,
it is better to describe the world in terms of fundamental physical constants:
the planck's constant h (which is
related to the smallest unit of energy) with dimensions M.L²T^-1,
the speed of light c with dimensions
L.T^-1, and the gravitational constant G
with dimensions L³M^-1T^-2.




the basic idea in dimensional analysis is to combine h,c, and G with unknown, yet to be determined exponents to yield length and
time. i'll illustrate this with length.




length:


h^αc^βG^γ
= L (1)


inserting the dimensional forms for h,c, and G we get,


(M.L²T^-1) ^α (L.T^-1)
^β(L³M^-1T^-2)
^γ = L (2)


one can
now group powers of M, L, and T together and match them with the corresponding
terms on the right hand side which has length (raised to the power one) and
mass and time (raised to zero). thus
this gives three equations for the unknown exponents α,β, and γ. solving them
one gets α, γ = 1/2 and β = -3/2; i.e. L = (hG)^1/2/c^3/2. this is a fundamental length scale called the
planck's length. when one inserts the
numerical values of h, c, and G (in SI units) into this length scale, one gets
a value of 1.6*10^-35 m. a
similar exercise for time yields a time scale known as the planck's time which
has a value of 5.38*10^-44 s.




the implications of the planck length and time are intriguing and startling. physicists
think that both time and length are ultimately granular! it's not clear even for
example whether length and time scales smaller than these have any physical
meaning! right now we can probe length scales on the order of about 10^-17m.
that's a good 18 orders of magnitude larger than the planck length. similarly
the smallest time interval that has been successfully measured is 1.2*10^-17
s, a duration 37 orders larger than the planck time. feynman probably had this
in mind when he said, "there is plenty of room at the bottom".

formatting got f*ed up. will try to fix it in a subsequent post.