The Quantum Lagrangian:

The Proca Form

In 1936 the Romanian physicist Alexandru Proca (1897 Oct
16 - 1955 Dec 13) devised an equation that describes a real vector field **A**
whose vibration manifests as a particle of mass m_{0} carrying one Dirac
unit of spin. Following Proca, we want to devise a Lagrangian density that
describes that vector field. In this case the mathematical description of the
vector field serves us as the state function in our quantum calculations. To
make our Lagrangian density conform to the requirements of relativity, we
represent the state function as a four-vector in full tensor form (that is, in
both contravariant and covariant forms),

(Eq’ns 1)

As is conventional, the contravariant form presents the magnetic vector
potential **A** and the electrostatic potential ö as observers would measure
them.

Lagrange’s function L comes from the derivation of the principle of least action through a Lorentz invariant that consists of the inner product of the four-momentum of a particle and the minuscule four-distance that the particle moves in going between two events; that is,

(Eq’n 2)

(In this equation E represents total energy, not to be confused with the electric field intensity, which comes into play in the following analysis.) In the quantum theory we look at the densities of properties, with differential operators usually representing the properties and the square of the state function providing the density. For our present purpose we put Equation 2 into a form that displays the momentum density and the energy density explicitly, so we have

(Eq’n 3)

If we integrate that expression over all space and then over an indeterminate path between two fixed loci, we will obtain a description of the possible actions that a particle enacts in going from one locus to the other. If we then find the minimum value of the action with respect to a variation in the state function, we get an equation that we can solve for a description of the state function. All we need is the Lagrangian density function.

We have assumed an electromagnetic field, so we have an
electric field intensity **E** and a magnetic induction field **B**. We
know that the energy density in those fields conforms to

(Eq’n 4)

If we multiply that expression by dVdt and subtract the result from zero, we have a start on cobbling up a Lagrangian density.

In addition to an energy density, the field also carries a
linear momentum density as a potential. Imagine a spherical shell bearing a
uniform electric charge with a surface density of the charge given as
σ=ε_{0}E,
in which E represents the electric field intensity at the outside surface of the
sphere due to that charge. On a minuscule patch of area da the force acting to
push the patch away from the center of the sphere equals dF=σEda=ε_{0}E^{2}da.
If that force pushes the patch across an increment of spatial distance dr in the
time interval dt, it gives the patch an increment of linear momentum dp=dFdt=ε_{0}E^{2}dadt.
Multiplying that expression by dr gives us the momentum-density term in Equation
3;

(Eq’n 5)

in which dadr=dV.

Adding that term to the energy density term gives us our initial Lagrangian density,

(Eq’n 6)

in which F_{μν}=∂_{μ}A_{ν}-∂_{ν}A_{μ}
represents the electromagnetic field tensor. The extra factor of ½ reflects the
fact that the field tensor is a symmetric 4x4 matrix, so in the square the
momentum and energy densities get counted twice. Thus we have the pure-field
Lagrangian density, but there’s more to the Lagrangian density than that.

If the quantized vibrations of the forcefield appear as
particles with a rest mass of m_{0}, then we must have an additional
energy density term involving those particles. Although the vibrations of the
electromagnetic field, photons, have zero rest mass, other forcefields produce
mass-bearing particles when they jitter and we want a Lagrangian density that
can take them into account. If we apply the quantum energy operation to the
field four-vector, we get

(Eq’n 7)

But the time derivative applied to the four-potential produces the electric
field in four-vector form, ∂A^{ν}/∂t=-E^{ν},
so we have an energy density,

(Eq’n 8)

Subtracting that from our Lagrangian density gives us

(Eq’n 9)

We have one other way in which we can put energy into a forcefield. We place
within the field something that bears a charge that interacts with the field to
produce a force. With a charge density of
ρ
we have a current-density four-vector j^{ν}=(ρ**v**,
ρc).
Because the state function represents a four-potential, the inner product
between that function and the four-current density yields an energy density,
which we subtract from our Lagrangian density to get

(Eq’n 10)

Thus we can cobble up a Lagrangian density for any forcefield. We need only put together a field term, a mass term, and an interaction term. If the field is associated with more than one kind of particle (as in the weak force), there will be multiple mass terms and multiple interaction terms. If we put that Lagrangian density into Equation 3, integrate the result, and vary that second result with respect to variations in the covariant four-potential to minimize the action, then we get Proca’s equation,

(Eq’n 11)

In the case in which m_{0}=0, that equation becomes equivalent to
Maxwell’s first equation and his fourth equation (minus the time derivative
term), the source equations of the electromagnetic field. That fact indicates to
us that the four-current not only responds to the forcefield, but also serves as
a source of at least part of it.

Next we want to create a general Lagrangian density, one that encodes both particles and fields. That will be the subject of the next essay.

eabf