5. THE SCALAR FIELD

It is completely in the spirit of general field theory to assume that a quantity that is defined everywhere and at every instant of time exists. This rather vague statement, which can be written in the usual coordinates as

(5.1)

does not exclude the case = 0 in some regions of space-time or -> 0 asymptotically as the spatial coordinates do go to infinity. As mentioned above, the Newtonian theory of gravitation uses precisely such a quantity: the gravitational potential outside a mass M,

(5.2)

where G is the Newtonian gravitational constant G = 0.57 x 10^-8 cm³ g^-1 s^-2. The derivative of the potential determines the force which acts on a body with mass m:

(5.3)

As is well known, when we generalize the Newtonian theory to distributions of mass it takes the form of the Poisson equation

(5.4)

where is the mass density in g cm^-3. For a star, this equation results in a smooth function (without an infinity in the center of the star) inside the star and solution (5.2) outside the star. We now assume that perhaps other potentials, with somewhat different properties (see immediately below), whose strength depends on a new type of special "scalar" charge density q instead of , exist. We assume the existence of an independent scalar field (or fields) of nongravitational origin.

We now apply special relativity. The Newton and Poisson formulations imply that the field at each point in space depends on the position of the mass (or the mass distribution) at the same instant of time. The left- and right-hand sides of the Poisson equations are assumed to be evaluated at the same instant of time. This flagrantly contradicts the fundamental idea that the speed c (the speed of light, 3 x 10¹⁰ cm s^-1) is the limiting speed for all transfers of energy, momentum, or information. In order to make the Poisson equation comply with relativity, we must replace the ² (Laplacian) with a (d'Alembertian):

(5.5)

In mathematics, it is well known that the solutions of< = f with = 0 at t = - propagate from the source toward the future at the speed of light.

By setting = (x, y, z, t), we tacitly assume that it is a four-scalar: in a given place x, y, z at a given instant of time t, it is the same for every observer; it does not change if the observer is moving, i.e., under a Lorentz transformation. This is not as trivial as it seems at first glance. If there is a distribution of electric charge = ne, ⁽⁵⁾ we must ask if the charge is at rest or, in other words, if there is an electric current present as well. Even if there is no current in one frame of reference, a moving observer will observe a current and the values of the charge density will be transformed: ' = / sqrt[1 - ²]. In fact, is the time component of a four-vector and is not a scalar. In contrast, this is not the case for the function . We explicitly assume that is not transformed, so that is a scalar - a four-scalar in Minkowski space. In general, the derivatives of are not scalars. In particular, ð / ðt is the time component and grad is the space component of a four-vector. None of these derivatives are invariant; none of the second derivatives are either, with one important exception. The d'Alembertian is a scalar! It is a scalar in four-dimensional space in the same way that the Laplacian ² is a scalar in three-dimensional space.

Therefore, the right-hand side of the equation (i.e., f) must also be a scalar. This means that the source of the scalar field, the "scalar field charge", is also a scalar. Furthermore, it cannot be a particle volume density!

The second point, which we already used in writing the generalized equation, is that we could put the scalar itself (multiplied by some constant denoted by µ²) in the right-hand side. Even more generally, one could use V(), an arbitrary scalar function V of the scalar . In this case, the scalar field is called a self-interacting scalar field. This V-term is not used in the simplest case, with µ² , because it leads to a linear equation:

(5.6)

In order to come to an understanding of how scalar field theory works, we shall discuss two simple exact solutions. The first is

(5.7)

It may easily be seen that for

(5.8)

and arbitrary amplitude A, (5.7) is a solution to the free-field (f = 0) equation. This is a completely new property that the Newtonian scalar gravitational field does not possess. The new relativistic equation for allows the field to propagate like a wave; this property did not exist in the Newton-Poisson approximation. By standard well-known methods, one can show that the phase velocity

(5.8)

This does not violate relativity theory: the group velocity v_g, i.e., the velocity at which an impulse is propagated, determines the speed of information propagation. As is well known,

(5.10)

The energy density of the classical scalar field is given by the following expression:

(5.11)

The energy density is always positive and behaves like the T₄₄ component of the energy-momentum tensor; this formula like the other versions of the theory (some special V() instead of 1/2 µ^{2 2}) is basic; its cosmological importance will be discussed below. By quantizing the plane wave solution, one finds that the wave can be treated as a collection of particles ("quanta"), where each particle has the following characteristics:

(5.12)

By virtue of the equivalence formula E = mc², we can say that the particles have a mass

(5.13)

so that the term µ introduced in (5.7) is the quantum mechanical rest mass of the particles corresponding to the solution of (5.7) (up to a trivial factor of /c). The waves are longitudinal; the y and z components of grad are not involved in the propagation along the x-axis - compare this with a similar electromagnetic wave, which would be transverse: E_y = H_z = A cos ( t - kx). The scalar field has no intrinsic angular momentum, and the particles are called scalar particles.

We now turn to the second simple solution. Assume a static time-independent point source at the origin, f = B /r. It is easy to find a static (time-independent), spherically symmetric solution for . Equation (5.6) reduces to

(5.14)

The solution to (5.14) is

(5.15)

So the second effect resulting from the insertion of µ in the right-hand side of the basic equation is that the interaction has a cutoff at a distance r₀ = µ^-1; by comparison with the Newton and Coulomb laws,

(5.16)

But once again we must emphasize that the most important qualitative difference (which is due to the fact that the vector electromagnetic potential has a time component) between the electrostatic field and a true scalar field is present even in the case µ = 0. The difference is that equal charges repel one another in the electrostatic case, while equal charges attract one another in the scalar case. A related effect is that a point electric charge has infinite positive energy, while a point scalar charge has infinite negative energy (in the classical theory).

The fact that equal charges attract one another in scalar field theory makes it similar to Newtonian gravitation, but, as mentioned above, detailed study shows that real gravitation (as in celestial mechanics) is not a scalar field.

We shall return to the mechanical picture of the scalar field below (Section 7), after a short historical note (Section 6).

⁵ n is the volume density of electrons; e is their charge. In the general case of different particles, = n_ie_i. Back.