Deriving the Klein-Gordon Equations: and, Why They Don't Matter.

Tight-binding Hamiltonian heuristic diagram.

The Klein-Gordon equations are equations of motion that can be derived from the first quantization of the energy-mass relation. Below, this derivation is made explicit, and it's sparse applicability is discussed.

Klein-Gordon usually isn't even mentioned in the canonical physicists' education until one discusses the derivation of the Dirac equation, and then it's usually passed over and forgotten about. Later, one might encounter the Klein-Gordon field in a (typical) introductory quantum field theory class as a first example. But, why is a seemingly good equation so often passed over and never treated in detail? And where does it even come from?

The Klein-Gordon equations can be easily be derived from an attempt at first quantization of Einstein's famous mass energy relation:

\[ E^2 = p^2 + m^2 \]
but they can also be derived from a free (non-interacting) scalar field Lagrangian density as the respective Euler-Lagrange equation. Below, both of these approaches are discussed. After, we examine it's limited scope of applicability; as well as present a short discussion of it's historical significance.

Deriving Klein-Gordon

Quantizing Einstein's Energy-Momentum Relation

Say you're an early 20th century physicist and you have two things at the forefront of your mind (and you're not Dirac): Einstein says everything satisfies the E-M relation;

\begin{align} E^2 & = p^2 + m^2\\ \end{align}
and the hot new thing is to (first) quantize systems, by assuming the following equalities.
\begin{align} p\rightarrow -i\vec{\nabla} &\quad\quad\quad E\rightarrow i\frac{\partial }{\partial t}\\ \end{align}
Now, you're seeking a quantized, relativistic equation, so you naturally subsitute to arrive at the Klein-Gordon Equations:
\begin{align} -\frac{\partial^2 }{\partial t^2} &= -\nabla^2 +m^2 \end{align}
which we may multiply by a 'prospective' wavefunction \(\phi\) (we will soon, after the next section, explain why this cannot be a wavefunction but instead must represent something else):
\begin{align} -\frac{\partial^2 \phi}{\partial t^2} &= -\nabla^2\phi +m^2\phi \end{align}

A Scalar Free Field's Action

Now let's take an entirely different approach. Imagine we'd like to model the dynamics of a massive, scalar free field variable. Since it's a free field, we can take the potential energy to be zero and hence we should only have kinetic terms.

Recall again from the E-M relation that the kinetic energy is equal to the difference of the total energy and the rest energy (\(E_k = E^2-m^2\)). Hence, for a field of mass density \(m\), the rest energy density squared is \(m^2\phi^2/2\) (assuming a real field) and the total energy density is (in terms of a four-vector derivative with implicit Minkowski metric) \(\partial^{\mu}\phi\partial _{\mu}\phi/2\). This leaves us with a Lagrangian density that is essentially just this kinetic energy density of the field, as below.

\[ {\large \mathscr{L}_{\scriptsize KG}}=\frac{1}{2}\partial ^{\mu}\phi^*\partial_{\mu}\phi - \frac{1}{2}m^2 \phi^*\phi \]
Of course, this Lagrangian density should be integrated over all points of the field (that is, all of time and space), to give us the corresponding action.
\[ \large \mathcal{S}_{\scriptsize KG}={\scriptsize \frac{1}{2}}\int\Big[\partial ^{\mu}\phi^*\partial_{\mu}\phi - m^2 \phi^*\phi\Big]\ dx_{\mu} \]
Of course, to get a handle on the dynamics of the system, we then solve for the Euler-Lagrange (field) equations according to:
\[ \partial_{\mu}\Bigg(\frac{\partial \mathscr{L}}{\partial\big(\partial_{\mu}\phi\big)}\Bigg)=\frac{\partial\mathscr{L}}{\partial \phi} \]
Which again yield the now familiar Klein-Gordon equations:
\[ \partial_{\mu}\phi\partial^{\mu}\phi -m^2\phi=0 \]


While the Klein-Gordon equations are easy to derive and seem to promise covariance, their applicability is empirically tested and proven false in most scenarios. It simply doesn't work for most matter we interact with (like electrons, photons, protons, etc.).

However, it can be applicable to spin-less quantum systems. This is especially relevant now with the supposed existence of the Higgs Boson, which correpsonds to a scalar field and thus would be/is an elementary particle governed by the Klein-Gordon equations.

The problem really is that the conserved quantity admitted by the continuous U(1) symmetry doesn't allow for a Noether current that can be interpreted as a probability density since it's non-positive definite. It simply can't account for the dynamics of Fermionic systems, which require to be modelled as spinors.

Historical Significance

The Klein-Gordon approach was essentially the first attempt at a relativistic wave equation for quantum wavefunctions. Apparently, Schrodinger even derived it before the Schrodinger equation, but discarded it due to empirical failures (didn't work applied to hydrogen).

Further Resources

For more detail pertaining to these equation's history, see this article: Equation with the many fathers. The Klein–Gordon equation in 1926.