Feynman Path Integral Formulation

1.10 String Theory 41
of the string, provided of course one understands how to formulate the theory in a
consistent and calculable way.
The concept of a relativistic string (see Fig. 1.3) originated in the late sixties in
the context of hadron physics, where it provided a very useful phenomenological
description of certain peculiarities of strong interaction amplitudes. Later on three
main motivations for studying string theories emerged: the search for a description
of quark confinement in terms of gluon strings, a model of grand unification based
on superstrings (Schwarz, 1982), and a description of the three-dimensional Ising
model in terms of some sort of fermionic string (Polyakov, 1979).
One of the first concrete field-theoretic models of a string was given in (Nambu,
1969; Goto, 1971). The usual description introduces world-sheet coordinates σ and
τ, defined on the two-dimensional surface swept out by the time evolution of the
string. When the string is embedded in a d-dimensional space, points on the string
world sheet are assigned coordinates X μ (σ,τ) with μ = 1...d. In analogy with the
action for a relativistic point particle, which is proportional to the proper time, and
therefore to the length of the world line, I = ∫ τ f
τ i
dτ, the simplest action for such a
string is the total area of the world sheet, I = ∫ dA. One can re-write this quantity
by introducing an induced metric g ab on the worldsheet,
∂ X μ ∂ X ν
g ab (σ,τ) =η μν
∂σ a ∂σ b , (1.192)
with σ 1 ≡ τ and σ 2 ≡ σ, and η μν the flat metric in d dimensions. Then the twodimensional
volume element is dA = √ gd 2 σ with g = −det(g ab ), and one has
I =
∫
S
d 2 σ √ g . (1.193)
In a gravity language what one has so far is essentially a cosmological constant term.
In terms of the variables Ẋ = ∂X/∂τ and X ′ = ∂X/∂σ one has
I[X] = 1 √
2πα
∫S
′ d 2 σ (Ẋ · X ′ ) 2 − (Ẋ) 2 (X ′ ) 2 , (1.194)
where a coupling constant α ′ has been introduced, having dimensions of an area,
or of an inverse mass squared. The quantity T 0 = 1/2πα ′ is often referred to for
obvious reasons as the string tension: a string of spatial size R and time extent T will
have an energy per unit length of value T 0 . In the following it will be convenient to
re-absorb such a coupling into a re-definition of the X variables.
One clear distinction that appears early on in this picture is between open (describing
geometrically, in their time evolution, a sheet) and closed strings (described
by a tube). Another important property of the string action is its invariance under
reparametrizations of the world sheet coordinates, X μ (σ,τ) → X μ [ f (σ,τ)]. These
can be considered as two dimensional diffeomorphism acting within the surface;
they express an invariance of the area action under σ a coordinate redefinitions.