special relativity was on firm observational and theoretical footing,
it was appreciated that the Schrödinger equation of quantum mechanics
was not Lorentz invariant, therefore quantum mechanics as it was so
successfully developed in the 1920s was not a reliable description of
nature when the system contained particles that would move at or near
the speed of light.
problem is that the Schrödinger equation is first order in time
derivatives but second order in spatial derivatives. The Klein-Gordon
equation is second order in both time and space and has solutions representing
particles with spin 0:
Dirac came up with "square root" of Klein-Gordon equation
using matrices called "gamma matrices", and the solutions
turned out to be particles of spin 1/2:
where the matrix hmn
is the metric of flat spacetime. But the problem with relativistic quantum
mechanics is that the solutions of the Dirac and Klein-Gordon equation
have instabilities that turn out to represent the creation
and annihilation of virtual particles from essentially empty
Further understanding led to the development of relativistic
quantum field theory, beginning with quantum
electrodynamics, or QED for short,
pioneered by Feynman, Schwinger and Tomonaga in the 1940s. In quantum
field theory, the behaviors and properties of elementary particles can
calculated using a series of diagrams, called Feynman
diagrams, that properly account for the creation and annihilation
of virtual particles.
The set of the Feynman diagrams for the scattering of two electrons
The straight black lines represent electrons.
The green wavy line represents a photon,
or in classical terms, the electromagnetic field between the two electrons
that makes them repel one another. Each small black loop
represents a photon creating an electron and a positron, which then
annihilate one another and produce a photon, in what is called a virtual
process. The full scattering amplitude is the sum
of all contributions from all possible loops of photons, electrons,
positrons, and other available particles.
The quantum loop calculation comes with a very big problem. In order
to properly account for all virtual processes in the loops, one must
integrate over all possible values of momentum, from zero momentum to
infinite momentum. But these loop integrals for an particle of spin
J in D dimensions take the approximate form
If the quantity 4J + D - 8 is negative, then the integral behaves fine
for infinite momentum (or zero wavelength, by the de Broglie relation.)
If this quantity is zero or positive, then the integral takes an infinite
value, and the whole theory threatens to make no sense because the calculations
just give infinite answers.
The world that we see has D=4, and the photon has spin J=1. So for the
case of electron-electron scattering, these loop integrals can still
take infinite values. But the integrals go to infinity very slowly,
like the logarithm of momentum, and it turns out that in this case,
the theory can be renormalized so that
the infinities can be absorbed into a redefinition
of a small number of parameters in the theory, such as the mass
and charge of the electron.
Quantum electrodynamics was a renormalizable theory, and by the 19402,
this was regarded as a solved relativistic quantum theory. But the other
known particle forces -- the weak nuclear force that makes radioactivity,
the strong nuclear force that hold neurons and protons together, and
the gravitational force that holds us on the earth -- weren't so quickly
conquered by theoretical physics.
In the 1960s, particle physicists reached towards something called a
dual resonance model in an attempt to describe the strong nuclear force.
The dual model was never that successful at describing particles, but
it was understood by 1970 that the dual models were actually quantum
theories of relativistic vibrating strings and displayed very
intriguing mathematical behavior. Dual models came to be called string
theory as a result.
But in 1971, a new type of quantum field theory came on the scene that
explained the weak nuclear force by uniting it with electromagnetism
into electroweak theory, and it was
shown to be renormalizable. Then similar wisdom was applied to the strong
nuclear force to yield quantum chromodynamics,
or QCD, and this theory was also renormalizable.
Which left one force -- gravity -- that couldn't be turned into a renormalizable
field theory no matter how hard anyone tried. One big problem was that
classical gravitational waves carry spin J=2, so one should assume that
a graviton, the quantum particle that carries the gravitational force,
has spin J=2. But for J=2, 4 J - 8 + D = D, and so for D=4, the loop
integral for the gravitational force would become infinite like the
fourth power of momentum, as the momentum in the loop became infinite.
And that was just hard cheese for particle physicists, and for many
years the best people worked on quantum gravity to no avail.
But the string theory that was once proposed for the strong interactions
contained a massless particle with spin J=2.
In 1974 the question finally was asked: could
string theory be a theory of quantum gravity?
The possible advantage of string theory is that the analog of a Feynman
diagram in string theory is a two-dimensional smooth surface, and the
loop integrals over such a smooth surface lack the zero-distance, infinite
momentum problems of the integrals over particle loops.
In string theory infinite momentum does not even mean zero distance,
because for strings, the relationship between distance and momentum
is roughly like
The parameter a'
(pronounced alpha prime) is related to the string
tension, the fundamental parameter of string theory, by the relation
The above relation implies a minimum observable length for a quantum
string theory of
The zero-distance behavior which is so problematic in quantum field
theory becomes irrelevant in string theories, and this makes string
theory very attractive as a theory of quantum gravity.
If string theory is a theory of quantum gravity, then this minimum length
scale should be at least the size of the Planck length, which is the
length scale made by the combination of Newton's constant, the speed
of light and Planck's constant
although as we shall see later, the question of length scales in string
theory is complicated by string duality, which can relate two theories
with seemingly different length scales.