PROBLEM -

A Scalar Field: In this problem you are to work out some of the properties of a Lorentz scalar field φ(x) (as opposed to the Lorentz 4-vector field Aµ(x) that we have been studying). Assume that this field obeys the superposition principle, so that the action is quadratic in the field, just as for the case with electromagnetism.

Lorentz invariance and superposition then uniquely determine the action for the scalar field and its interaction with a particle of mass m to be

S = S_f + S_m + S_mf = ∫d⁴x (½∂_µφ∂^µ? - ½M²φ²) - mc ∫ds - λ ∫ds φ(x)

where λ is the analogue of the charge of the particle (and is called the "Yukawa coupling"), M is a constant with units of inverse length, and the ∫ds integrals are over the world line of the particle, as usual. The first term in S_f is quadratic in the fields and contains two derivatives; it is the scalar analogue of the F^µνF_µν term in electromagnetism. However, gauge invariance forbids a term M²A^µA_µ in electromagnetism; since the theory of a scalar field has no gauge invariance, we are free to add a term M²φ² to the Lagrange density.

(a) Show that the equation of motion for the particle in an external field φ(x) is

(mc + λφ) du^µ/ds = λ (∂^µφ - u^µuν∂_νφ).

Compare this with the Lorentz force law and comment on the similarities and differences.

Note that if φ(x) = v is constant (over all of space time), the effect of the field is equivalent to increasing the mass of the particle by λv/c. This is a manifestation of the famous "Higgs mechanism" in particle physics, whereby a scalar field which is nonzero in the vacuum gives mass to elementary particles.

(b) Now consider the equations of motion for the scalar field φ(x). Imposing the usual conditions that φ(x) → 0 at spatial infinity, use the variational principle to show that the equation of motion for the scalar field is

(∂_µ∂^µ + M²)φ(x) = -λ_ρ(x)

where ρ(x) = ∫ds δ⁽⁴⁾(x - x(τ )). This equation is known as the Klein-Gordon equation (with a source ρ(x)).

(c) Consider a particle at rest at the origin, ρ(x) = δ⁽³⁾(x^→). Show that the solution to the scalar field equation of motion is

φ(r) = -(λ/4π)e^-Mr/r.

This is known as a "Yukawa potential." Note that the M = 0 limit of this expression yields the familiar Coulomb potential; for nonzero M, the force is effectively zero for distances much larger than 1/M. This is why we don't see any long-range forces due to scalar fields in nature: for the Higgs field, 1/M is approximately 10^-18 m, or about 10^-3 of the radius of the proton. Another scalar field in nature is responsible for the strong nuclear force between protons and neutrons. Since this force has a range of approximately the size of the proton, Yukawa concluded in the 1930's that the force was mediated by a scalar field with 1/M ∼ 10^-15 m. In the quantum theory, this allowed him to predict the existence of a massive scalar particle with mass hM/c ∼ 10^-28 kg, which turned out to be a subatomic scalar particle called the pion.

(d) Note that we could have chosen positive or negative signs for both the "kinetic term" ∂^µφ∂_µφ and the "mass term" -M²φ² in the action for φ(x).

However, we will see shortly that the energy density carried by a field is given by T⁰⁰, the 00 component of the energy-momentum tensor, defined by

T^µν = (∂L/∂(∂_νφ))∂^µφ - g^µνL

where L is the Lagrange density of the field. Find the energy density of a free scalar field φ in terms of φ(x) and its derivatives. Argue from T⁰⁰ that if we had chosen different signs for the kinetic and mass terms in the action the theory would not make physical sense.