[FOM] Could spacetime be discrete?

Alasdair Urquhart urquhart at cs.toronto.edu
Mon Jan 16 09:55:13 EST 2006

Several people have brought up the matter of the
"Planck length" in discussion of this question.
My understanding of the idea is roughly as follows.
Quantum field theory works by manipulating divergent
infinite series that in general are mathematically
ill-defined, but can be brought under control because
of special features such as the weakness of
certain interactions.

Joseph Polchinski's text on string theory contains
the following remarks:

"From this dimensional analysis one learns immediately
that the quantum gravitational correction is an
*irrelevant* interaction, meaning that it grows
weaker at low energy, and in particular is negligible
at particle physics energies of hundreds of GeV.
By the same token, the coupling grows stronger
at high energy and at E > M_P perturbation theory
breaks down." (Polchinski Volume 1, p. 3)

Here, M_P is the Planck mass, defined as

	M_P = 1.22 x 10^{19} GeV,

and the Planck length is defined as the inverse of this
in units in which h-slash = c = 1, namely

	M_P^{-1} = 1.6 x 10^{-33} cm.

Thus, the Planck length is the length at which perturbative
calculations break down, and the gravitational force has
to be taken into account.  This length is defined
in purely negative terms as the scale at which the normal
methods of calculation don't work.  I don't see here
any support for the idea of a minimal length, which still
seems highly problematic to me.  However, I'd be glad if
somebody with more knowledge of basic physics would
set me straight.

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