Quantum Field Theory and Generalized Functions
In the following lectures I wish to review the present state of quantum field theory following the approach usually called “asymptotic quantum field theory”. Since certain mathematical developments, especially from the theory of generalized functions, wil
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F. ROHRLICH Department of Physics, Syracuse University Syracuse, USA
In the following lectures I wish to review the present state of quantum field theory following the approach usually called "asymptot ic quantum field theory". Since certain mathematical developments, especially
from
the theory of generalized functions,will be an important prerequisit, I shall proceed as follows. In order to avoid interrupting the development of the physical theory by mathematical asides, I shall first present several primarily mathematical topics which will then be used in the second part of these lectures for the development of the physical theory. The outline is therefore as follows: A. Mathematical Preliminaries 1. Generalized Functions 2. Analytic Functionals and the Class ~n of Ge-
r
neralized Functions 3. The Free Field Equation KNa
=0
4. Operator Derivatives 5. The Operators P
l 2
and Their Algebra
t Lecture given at the VI. Internationalen Universitäts-
P. Urban (ed.), Special Problems in High Energy Physics wochen f.Kernphysik,Schladming,26 February-11 March 1967. © Springer-Verlag Wien 1967
229
B.
Asymptotic Quantum Field Theory 6. Causality and the GLZ Expansion 7. The Pugh Equation 8. The Current Formalism 9. 10.
Feynman Diagrams and Boundary Conditions Summary
References
A. Mathematical Preliminaries
1. Generalized Functions
In this lecture I want to define a class of analytic functionals which is a generalization of the well-known functions t.(x), t.R(x), t.C(x), etc. of standard quantum field theory. The latter are actually not functions in the usual sense, but generalized functions. They can be given a precise mathematical meaning only concomitant with a space of sufficiently well-behaved functions, a test function space. Generalized Functions are continuous linear functionals T defined on· a test function space
~.
Under suitable conditions these T also define a
space. It is called
~·,
Example 1: Let ~ = i diffe~entiable
the conjugate space to
~.
, the space of all infinitely
functions (in one dimension for the sake
of this example) which fall off faster than any power. The space 1' is the space of all tempered distributions. More precisely,
'
o
( 1. 1)
=X (a), the space of all infinite-
ly differentiable functions which vanish outside
lxl~a.
The conjugate space of all continuous linear functionals on J((a) is denoted by J(R)
•
While this is an advantage in certain respects it is a disadvantage in the study of the interrelations betn
ween the llr for different r
As analytic functionals
the topological nature of the paths relative to the two singularities in p-space specifies the linear combination of one path in terms of others. This feature is lost if one works with tempered distributions in all four variables. Details of the subject matter discussed here can be found in reference 22 where also further references to distribution theory are given. 3. The Free Field Equation Kn+ 1 a(x) = 0
The Cauchy initial value problern of this equation can be obtained in a rather standa
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