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CHAPTER 1
Types of Fields

Our task in this book is to discuss the mathematical techniques which are useful in the calculation and analysis of the various types of fields occurring in modern physical theory. Emphasis will be given primarily to the exposition of the interrelation between the equations and the physical properties of the fields, and at times details of mathematical rigor will be sacrificed when it might interfere with the clarification of the physical background. Mathematical rigor is important and cannot be neglected, but the theoretical physicist must first gain a thorough understanding of the physical implications of the symbolic tools he is using before formal rigor can be of help. Other volumes are available which provide the rigor; this book will have fulfilled its purpose if it provides physical insight into the manifold field equations which occur in modern theory, together with a comprehension of the physical meaning behind the various mathematical techniques employed for their solution.

This first chapter will discuss the general properties of various fields and how these fields can be expressed in terms of various coordinate systems. The second chapter discusses the various types of partial differential and integral equations which govern fields, and the third chapter treats of the relation between these equations and the fundamental variational principles developed by Hamilton and others for classic dynamics. The following few chapters will discuss the general mathematical tools which are needed to solve these equations, and the remainder of the work will be concerned with the detailed solution of individual equations.

Practically all of modern physics deals with fields: potential fields, probability fields, electromagnetic fields, tensor fields, and spinor fields.

Mathematically speaking, a field is a set of functions of the coordinates of a point in space. From the point of view of this book a field is some convenient mathematical idealization of a physical situation in which extension is an essential element,  i.e., which cannot be analyzed in terms of the positions of a finite number of particles. The transverse displacement from equilibrium of a string under static forces is a very simple example of a one-dimensional field; the displacement y is different for different parts of the string, so that y can be considered as a function of the distance x along the string. The density, temperature, and pressure of a fluid transmitting sound waves can be considered as functions of the three coordinates and of time.  Fields of this sort are obviously only approximate idealizations of the physical situation, for they take no account of the atomic properties of matter. We might call them  material fields.

Other fields are constructs to enable us to analyze the problem of  action at a distance, in which the relative motion and position of one body affects that of another. Potential and force fields, electromagnetic and gravitational fields are examples of this type. They are considered as being  ``caused'' by some piece of matter on some test body at the point in question. In some cases the theory can be modified so as to take the quantum rules into account in a more or less satisfactory way.

Finally fields can be constructed to ``explain'' the quantum rules. Examples of these are the Schroedinger wave function and the spinor fields associated with the Dirac electron. In many cases the value of such a field at a point is closely related to probability.   For instance the square of  the Schroedinger wave function is a measure of the probability that an elementary particle is present.  Present quantum field theories suffer from many fundamental difficulties and so constitute one of the frontiers of theoretical physics.

In most cases considered in this book fields are solutions of partial differential equations, usually second-order, linear equations, either homogeneous or inhomogeneous.  The actual physical situation has often to be simplified for this to be so, and the simplification can be justified on various pragmatic grounds. For instance, only the ``smoothed-out'' density of a gas is a solution of the wave equation, but this is usually sufficient for a study of sound waves, and the much more tedious calculations of the actual motions of the gas molecules would not add much to our knowledge of sound.

   The Procrustean tendency to force the physical situation to fit the requirements of a partial differential equation results in a field which is both more regular and irregular than the ``actual'' conditions. A solution of a differential equation is more smoothly continuous over most of space and time than is the corresponding physical situation, but it usually is also provided with a finite number of mathematical discontinuities which are considerably more ``sharp'' than the ``actual'' conditions exhibit. If the simplification has not been too drastic, most of the questions which can be computed from the field will correspond fairly closely to the measured values. In each case, however, certain discrepancies between calculated and measured values will turn up, due either to the ``oversmooth'' behavior of the field over most of its extent or the presence of mathematical discontinuities and infinities in the computed field, which are not present in ``real'' life. Sometimes these discrepancies are trivial, in that the inclusion of additional complexities in the computation of the field to obtain a better correlation with experiment will involve no conceptual modification of the theory; sometimes the discrepancies are far from trivial, and a modification of the theory to improve the correlation involves fundamental changes in concept and definitions. An important task of the theoretical physicist lies in distinguishing between trivial and nontrivial discrepancies between theory and experiment.

One indication that  fields are often simplifications of physical reality is that fields often can be defined in terms of a limiting ratio of some sort. The density field of a fluid  which is transmitting a sound wave is defined in terms of the ``density at a given point'', which is really the limiting ratio between the mass of the fluid in a volume surrounding the given point and the magnitude of the volume, as this volume goes to  ``zero''.  The electric intensity ``at a point'' is the limiting ratio between the force on a test charge at the point and the magnitude of the test charge as this magnitude goes to ``zero''.  The value of the square of the Schroedinger wave function is the limiting ratio between the probability that the particle is in a given volume surrounding a point and the magnitude of the volume as the volume is shrunk to ``zero'', and so on.  A careful definition of the displacement at a ``point'' of a vibrating string would also utilize the notion of a limiting ratio.

These mathematical platitudes are stressed here because the technique of the limiting ratio must be used with caution when defining and calculating fields.  In other words the terms ``zero'' in the previous paragraph must be carefully defined in order to achieve results which correspond to ``reality''.  For instance the volume which serves to define the density field for a fluid must be reduced several orders of magnitude smaller than the cube of the shortest wavelength  of transmitted sound in order to arrive at a ratio that is a reasonably accurate solution to the wave equation.  On the other hand, this volume must not be reduced to a size commensurate with atomic dimensions, or the resulting ratio will lose its desirable properties of  smooth continuity and would not be a useful construct. As soon as this limitation is realized, it is not difficult to understand that the description of a sound wave in terms of a field which is a solution of the wave equation is likely to become inadequate if the ``wavelength'' becomes shorter than interatomic distances.

In a similar manner we define the electric field in terms of  a test charge which is small enough so that it will not affect the distribution of the charges ``causing''  the field. But if the magnitude of the test charge is reduced until it is the same order of magnitude as the electric charge, we might expect the essential atomicity of charge to involve us in difficulties (although this is not necessarily so).

In some cases the limiting ratio can be carried to magnitudes as small as we wish. The probability fields of wave mechanics are as ``fine-grained'' as we can imagine at present.

 

1.1. Scalar Fields


When the field under consideration turns out to be a simple number, a single function of space and time, we call it a  scalar field.  The displacement  of a string or a membrane from equilibrium is a scalar field. . . .