(ebook - physics) Quantum Field Theory - An Introduction to String Theory.pdf

Lecture 1
Quantum Field Theories: An
introduction
The string theory is a special case of a quantum field theory (QFT). Any QFT deals
§ £
with smooth maps ¢¡¤£¦¥ of Riemannian manifolds, the dimension of is
the dimension of the theory. We also have an action function ¨ defined on the set
£
§
Map © of smooth maps. A QFT studies integrals

$%

)(+*
&-,
©'& ()
! #"
Map
(+*
.
Here &-, stands for some measure on the space of paths, is a parameter (usually
%
¡ £
/§
0¥21
very small, Planck constant) and Map © is an insertion function. The
number 65798/: should be interpreted as the probability amplitude of the contribution
;¡<£=¥¦§

of the map43 to the integral. The integral

>0?A@
$ED
& ()
4
BC
Map
is called the partition function of the theory. In a relativistic QFT, the space £ has a

#HI
KJKJJ4
/H¤
Lorentzian metric of signature ©GF . The first coordinate is reserved for time,
the rest are for space. In this case, the integral () is replaced with

> @
%

G(;* J
65798/:
&N,
©M& ()
4  3
Map 7L
£ ¡£P¥¦§
&
Let us start with a O -dimensional theory. In this case is a point, so is
§ ¡<§T¥U1
¨
a point QSR and is a scalar function. The Minkowski partition function
of the theory is an integral

>V@
D
J
WB98/:
Q ()
3
7L
Following the Harvard lectures of C. Vafa in 1999, let us consider the following
example:
1
2 LECTURE 1. QUANTUM FIELD THEORIES: AN INTRODUCTION
Example . Recall the integral expression for the X -function:
=[ e[
@ ***@dc
D D

J
XY©'Z ()
\^])_ ] \f]Gg/_ ]
4`Kba 4`Kbaih

kj
O
This integral is convergent for Re ©MZ but can be meromorphically extended to the
\
whole plane with poles at ZlRSm¤n . We have
o
@ @ ***@ts u
HdoB

p
oB
o
q

J
c
XY©'Z ZBXY©MZ XY© Xr©
@
v
By substituting s in (), we obtain the Gauss integral:
] ]