The Quantum Structure of Space and Time (293 pages)

Mathematical Structures 157

4.7.2 Michael Atiyah: Beyond string theory?

String theory (and M-theory) is a remarkably sophisticated mathematical structure,
as yet incomplete. While it has led to exciting new results in mathematics it is not
yet clear what shape the final physical theory will take. Perhaps it will only be a
modest extension of what we now have, but perhaps it will have to undergo some
radical reformulation.

As we know the great aim is to combine quantum theory with gravity. String
theory maintains the formal apparatus of quantum theory unchanged but modifies
Einstein’s theory of gravitation. Perhaps both sides need modification?

I confess to being a believer in Occam’s razor, in which the simple solution is

always preferred. Although string theory is an impressive structure it still lacks
the overall simplicity that we should aim at.

We may need radically new ideas and I think it is worth investigating whether

retarded differential equations should be seriously considered at a fundamental

level. This idea has been put forward by Raju [l], although my ideas are somewhat
different.

Consider as a simple example the linear equation for the function x(t).

k(t) + kx(t - r) = 0 r>O (1)
where r, k are fixed constants. Such an equation can be solved for t > 0 with initial

data being a function on the interval [-r, 01. This is very different from the usual

differential equations of mathematical physics (for which r = 0), which have been
our paradigm since Isaac Newton.

While this equation makes sense for x(t) in Minkowski space, with t being proper

time along the trajectory, it is not clear how to extend it to wave-propagation in

a relativistic manner. For this purpose note that t - t - r has infinitesimal

generator -r-, so that translation by -r is just exp -r-. Guided by this

we can consider formally the relativistically invariant retarded Dirac operator (as
suggested to me by G. Moore).

where D is the Dirac operator in Minkowski space. In the non-relativistic limit D

reduces to - YO - a where yo is the 4 x 4 matrix (i -:) , showing that we have a

retarded equation for positive energy and an advanced equation for negative energy.

( 3

d

at

i D - mc + k exp (--TD) (2)

c dt

There are problems in interpreting exp (-rD) which I will pass over, but applied

to the plane wave solution

exp (*) (:)