Mathematical Physics Vol 1

4. Field theory

Mathematical field theory 1 does not study the physical meaning of a variable defined in a given field. Only the general properties of fields are studied, which are then applied in physics and other areas to specific physical fields. Specific fields are studied in different parts of physics, and in this book some examples will be given for illustration purposes only, in order to help in understanding the theory presented. 4.1 Scalar field Consider a set D of points in a n –dimensional Euclidean space E n . If a (real or complex) number y is assigned to each point M ( x 1 , x 2 ,..., x n ) ∈ D by some law, then we say that a scalar (real or complex) function y of n independent variables is defined and we denote it by: The coordinates ( x 1 , x 2 ,..., x n ) of point M are called independent variables , and the set D domain of definition (or simply domain) of the function f of this point. However, in physics and some natural sciences, as well as in engineering, the term "field" is used to denote a part of space (area) in which a physical phenomenon is observed ("felt"). The term" scalar field ", is used here in the mathematical sense, to denote the domain of definition of a scalar function. Thus, hereinafter, the term "field" will be used instead of "domain", and the previous statement can be rephrased as follows: if a function f assigns a scalar (real or complex number) to each point in D then a scalar field is defined in D . Note that the value of the function depends only on the points in space and not on the selected specific coordinate system. Therefore, it should be kept in mind that the value of the function f at any point M ∈ D does not depend on a selected specific coordinate system. However, its functional form depends on the coordinate system. To emphasize this fact, it is also common to 1 In mathematics, the term "field" is also used for an algebraic structure consisting of a set and two operations with certain properties, such as, for example, a set of real numbers with operations of addition and multiplication. y = f ( M ) or y = f ( x 1 , x 2 ,..., x n ) . (4.1)