文档介绍：MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY
1. Generalities on quantum field theory
. Classical classical mechanics, we study the motion of a particle. This motion is
described by a (vector) function of one variable, q = q(t), representing the position of the particle as a
function of time. This function must satisfy the Newton equation of motion,
q¨ = −U (q),
where U the potential energy, and the mass of the particle is 1. Another way to express this law of
motion is to say that q(t) must be a solution of a certain variational problem. Namely, one introduces
the Lagrangian
q˙2
L(q)= − U (q)
2
(the difference of ic and potential energy), and the action functional
b
S(q)= L(q)dt
a
(for some fixed a<b). Then the law of motion can be expressed as the least action principle: q(t)
must be a critical point of S on the space of all functions with given q(a)andq (b). In other words,
the Newton equation is the Euler-Lagrange equation for the solution of the variational problem defined
by S.
Remark name “least action principle” comes from the fact that in some cases (for example
when U ≤ 0) the action is not only extremized but also minimized at the solution q(t). In general,
however, it is not the case, and the trajectory of the particle may not be a minimum, but only a saddle
point of the action. Therefore, the law of motion is better formulated as the “extremal (or stationary)
action principle”; this is the way we will think of it in the future.
Remark often consider solutions of Newton’s equation on the whole line rather than
on a fixed interval [a, b]. In this case, the naive definition of an extremal does not make sense, since the
S q L q dt
action integral ( )= R ( ) is improper and in general diverges. Instead, one makes the following
“correct” definition: a function q(t)onR is an extremal of S if the expression
d ∂L ∂L
| L q sε dt