Lectures on String Theory

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If one tries to construct the quantum mechanics of relativistic strings one finds
that mathematically consistent theory exists if and only if the dimension of spacetime
where string propagates is 26. The number 26 was named the “critical dimension”.
On the one hand, it was pretty remarkable and unexpected to find an example
of physical theory which puts restrictions on space-time where it is defined. On the
other hand, it was certainly not clear why a theory that shared at least some qualitative
features with hadronic physics should exist in 26 dimensions only. A subsequent
discovery of QCD (Quantum Chromo Dynamics) as the most appropriate candidate
to describe the theory of strong interactions led to a considerable loss of interest to
string theory.
In 1974, Scherk and Schwarz came up with a proposal to completely alter the
view on string theory. They suggested to consider the massless spin two particle
absent in the hadronic world as the graviton – the quantum of the gravitational
interaction. Indeed, this particle neatly fits the properties of the graviton – string
theory predicts that this particle interacts according to the standard laws of General
Relativity. Gravitational interactions have a natural scale, called the Planck mass,
which is around 10 19 GeV. This is a huge number in comparison with characteristic
energies of hadronic physics, 100 − 200 MeV. Thus, according to their view, string
theory could provide the unifying description of all the particles and matter forces,
including gravity and it operates on a new fundamental scale.
Even if one accepts that quantum mechanics of relativistic strings can be defined
in the unusual number 26 of the space-time dimensions, another problem arises. Such
string does not contain fermionic degrees of freedom and it predicts the existence of
a particle with the negative mass squared: m 2 < 0. Such a particle, tachyon, is a
source of instability and its existence indicates that either the theory is ill-defined
or it is formulated around a “wrong” ground state, or as physicists say, around a
“wrong” vacuum. Critical dimension, tachyon and absence of fermions were the
puzzling features the string theory had to face.
The status of string theory changed again with the discovery of supersymmetry.
All universe is made of two fundamental types of particles: bosons and fermions.
Fermions constitute all the matter and bosons mediate interactions of the matter
particles. Supersymmetry is a new type of symmetry between bosons and fermions
(Wess and Zumino 1974). Many physicists hope today that supersymmetry could
provide an underlying principle for unification of all interactions.
The first success in incorporating supersymmetry in string theory was achieved
in 1971 by Ramond, who constructed a string analogue of the Dirac equation (the
spinning string). Shortly afterwards, Neveu and Schwarz constructed a new bosonic
string theory. They realized that the two constructions were different facets of a
single theory - an interacting superstring theory containing Neveu and Schwarz’s