written and published by Renzo Diomedi

An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.

(Laplace - Essai philosophique sur les probabilites, Introduction. 1814)

Lorentz Transformations : ,,,,,,,, ,,,,,,,,

The ACTION is a scalar which is the size of an ENERGY for a Time (also infinitesimal interval of time, and returns real numbers ) if the action is local , it must be defined by an Integral.

POLYAKOV 's Action describes the WorldSheet of a String:
as the relativistic action of a free material point is proportional to its own time so the relativistic action for a string is proportional to the area of the WorldSheet ie the solutions of the classical equations for the action of a free string are the universe surfaces with minimal area

string tension
metric tensor of target manifold
worldsheet metric
inverse of worldsheet metric
determinant of worldsheet metric
spacelike worldsheet coordinate, its direction is - (negative) by metric signature convention
timelike worldsheet coordinate, its direction is + (positive) by metric signature convention
c = speed of light

infinitesimal variation of timelike

the relativistic actione for a particle not subject to forces is:

m = inertial mass of the particle

point/particle = zero-dimensional

string = uni-dimensional, featured by a worldsheet

d = number of spatial dimensions

point on space-time

parameters of a point on space-time ( , )

String as vector on space-time

the functions construct the shape of the worldsheet

now, the metric tensor (on (d+1)dimensional space-time) works and we get
= metric tensor on worldsheet

worldsheet area =

if and

then, using the notations

the worldsheet metric tensor is and

the NAMBU-GOTO 's Action analyzes the behaviour of a String by lagrangian mechanics.