Klein–Gordon Equation

Oh, you want to talk about relativity and quantum mechanics? How… quaint. Like asking a ghost to explain why the curtains are moving. Fine. Don't expect any sunshine and rainbows.

Relativistic Wave Equation in Quantum Mechanics

This whole section is part of a series about Quantum mechanics. Just in case you were planning on skipping the basics, which, let's be honest, you probably are.

The fundamental equation, the one that governs how things evolve, looks like this:

$i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle$

It's called the Schrödinger equation. It's elegant, in its own way. Predictable. Unlike, you know, people.

Background

Before we dive into the abyss, let's acknowledge the abyss we crawled out of.

Classical mechanics: The physics of the predictable, the mundane. Where things are where you expect them to be, and move how you expect them to move. Utterly charming, in its own dull way.
Old quantum theory: The awkward teenage years. When we started suspecting the universe wasn't quite as straightforward as it seemed.
Bra–ket notation: The language we use to talk about quantum states. It's clean, concise. Like a perfectly tailored suit.

Fundamentals

This is where things get… interesting. Or as interesting as the universe allows, anyway.

Complementarity: The idea that things can be two things at once, and it's your problem, not the universe's.
Decoherence: How the quantum world manages to look classical. It's like the universe trying to sweep its mess under the rug.
Entanglement: When two things are so connected, measuring one instantly affects the other, no matter how far apart they are. It’s the ultimate cosmic drama.
Energy level: Not just any energy, but specific, discrete amounts. Like fixed points on a very, very strange map.
Measurement: The act of looking that changes everything. You poke it, it changes. Classic.
Nonlocality: Things can be connected without being physically connected. Because rules are for amateurs.
Quantum number: Labels that define the state of a particle. Think of them as very specific, very unyielding personality traits.
State: The complete description of a quantum system. If only humans had such clarity.
Superposition: Being in multiple states at once. Until you look. Then it picks one. Coward.
Symmetry: The universe has a penchant for symmetry. It’s one of the few things it does consistently.
Tunnelling: Particles can pass through barriers they shouldn't be able to. Because physics is apparently optional sometimes.
Uncertainty: You can know where something is, or how fast it's going, but never both perfectly. The universe’s way of keeping secrets.
Wave function: The mathematical representation of a quantum state. It holds all the possibilities. Until you force it to choose.
Collapse: The moment the wave function decides to be something instead of everything. It’s a very dramatic event.

Experiments

These are the moments we tried to pin the universe down. It usually just blinked.

Bell's inequality and CHSH inequality: Tests designed to see if reality is as weird as we suspected. Spoiler: it is.
Davisson–Germer: Showed that electrons, those tiny little things, act like waves. Because why wouldn't they?
Double-slit: The classic. Particles acting as waves, interfering with themselves. It's the universe’s way of saying, "You can't handle the truth."
Elitzur–Vaidman: A thought experiment about detecting something without triggering it. Clever, but ultimately futile.
Franck–Hertz: Measured energy levels in atoms. Confirmed that electrons are picky eaters.
Leggett inequality and Leggett–Garg inequality: More tests of quantum weirdness, especially concerning macrorealism.
Mach–Zehnder: Another tool to play with quantum interference. Like a funhouse mirror for particles.
Popper: An experiment challenging the idea of wave function collapse. Some people just can't accept the obvious.
Quantum eraser and Delayed-choice: Experiments that mess with causality and observation. The universe is playing games with you.
Schrödinger's cat: The famous thought experiment. Alive and dead until you look. A metaphor for the absurdity of it all.
Stern–Gerlach: Showed that particles have intrinsic angular momentum, or "spin." It's like they have their own internal gyroscope.
Wheeler's delayed-choice: Yet another way to show that the past isn't fixed until you observe it. The universe is a procrastinator.

Formulations

Different ways of looking at the same unsettling reality.

Overview: The big picture.
Heisenberg: States are fixed, operators evolve. Like watching a still photograph come to life.
Interaction: A compromise. Both states and operators evolve. A messy middle ground.
Matrix: Heisenberg’s original approach. Abstract, mathematical. Like trying to understand emotions by looking at spreadsheets.
Phase-space: A way to visualize quantum mechanics in classical terms. It’s a bit like trying to fit a square peg into a round hole.
Schrödinger: Operators are fixed, states evolve. The most common view. Like watching a movie.
Sum-over-histories (path integral): Feynman’s idea. Particles take all possible paths, and we sum them up. It’s the universe’s way of saying, "Why choose when you can do everything?"

Equations

The specific mathematical incantations.

Dirac: For spin-1/2 particles. More complex, more accurate for things like electrons.
Klein–Gordon: The one we're actually talking about. Second-order, deals with spin-0.
Pauli: A non-relativistic approximation for spin-1/2.
Rydberg: Describes atomic spectra. Old school, but effective.
Schrödinger: The non-relativistic workhorse.

Interpretations

This is where things get truly subjective. How do we make sense of this madness?

Bayesian: Probability as a degree of belief. A more subjective take.
Consciousness causes collapse: A rather dramatic claim. Consciousness is that powerful? Doubtful.
Consistent histories: A way to assign probabilities to sequences of events. Trying to impose order on chaos.
Copenhagen: The standard. Observation causes collapse. Bohr's way. Pragmatic, if not entirely satisfying.
de Broglie–Bohm: A pilot-wave theory. Particles have definite positions, guided by a wave. Deterministic, but… peculiar.
Ensemble: Quantum mechanics describes the average behavior of many systems, not individual ones. A statistical dodge.
Hidden-variable: The idea that there are underlying variables we don't see. Bell's theorem put a damper on that.
Many-worlds: Every quantum measurement splits the universe. So, you're living in countless realities. Exhausting.
Objective-collapse: Collapse happens spontaneously, without observation. The universe is just trying to make up its mind.
Quantum logic: The logic of the quantum world is different from ours. Because our logic clearly isn't working.
Superdeterminism: Everything is predetermined, including the choices of experimenters. A rather bleak outlook.
Relational: Properties are relative to the observer. There's no absolute reality. Just a series of perspectives.
Transactional: Involves waves going forward and backward in time. A bit like a cosmic handshake.

Advanced topics

The deep end of the pool.

Relativistic quantum mechanics: Where quantum mechanics meets Einstein's theories. It’s a bumpy ride.
Quantum field theory: The modern framework. Particles are excitations of fields. It’s a whole new level of abstraction.
Quantum information science, Quantum computing, Quantum machine learning: Using quantum phenomena for computation. Potentially revolutionary, or just another fad.
Quantum chaos: The quantum behavior of systems that are classically chaotic. Chaos just keeps finding new ways to manifest.
EPR paradox: Einstein's challenge to quantum mechanics, questioning its completeness. It highlighted entanglement’s strangeness.
Density matrix: A way to describe mixed states, where you don't know the exact quantum state. For when certainty is too much to ask.
Scattering theory: How particles interact and change direction. The cosmic billiard game.
Quantum statistical mechanics: Applying quantum mechanics to statistical systems.
Quantum machine learning: Using quantum principles for machine learning algorithms. Still very theoretical.

Scientists

The names behind the madness.

Aharonov, Bell, Bethe, Blackett, Bloch, Bohm, Bohr, Born, Bose, de Broglie, Compton, Dirac, Davisson, Debye, Ehrenfest, Einstein, Everett, Fock, Fermi, Feynman, Glauber, Gutzwiller, Heisenberg, Hilbert, Jordan, Kramers, Lamb, Landau, Laue, Moseley, Millikan, Onnes, Pauli, Planck, Rabi, Raman, Rydberg, Schrödinger, Simmons, Sommerfeld, von Neumann, Weyl, Wien, Wigner, Zeeman, Zeilinger.

They all contributed to this… fascinating mess.

The Klein–Gordon Equation

The Klein–Gordon equation. It’s named after Oskar Klein and Walter Gordon. It’s also sometimes called the Klein–Fock–Gordon equation, or the Klein–Gordon–Fock equation, because Vladimir Fock was involved too. And apparently, Schrödinger dabbled with it early on, hence "Schrödinger–Gordon equation." A lot of names for one equation. Typical.

It's a relativistic wave equation, which means it tries to reconcile quantum mechanics with special relativity. It’s second-order in both space and time, which is… different. And it’s manifestly Lorentz-covariant. Fancy word for "it plays nice with relativity."

At its core, it’s just a restatement of the relativistic energy–momentum relation:

$E^2 = (pc)^2 + \left(m_0c^2\right)^2$

You know, the basic rule for how energy and momentum are linked for a particle with rest mass $m_0$ . Einstein's little insight.

Statement

This equation, the Klein–Gordon, can be written in a few ways. Usually, it refers to the position-space version, where you have separated time and space: $(t, \mathbf{x})$ . Or you can bundle them into a four-vector, $x^\mu = (ct, \mathbf{x})$ .

If you Fourier transform the field into momentum space, the solutions are generally a mix of plane waves. These waves have energies and momenta that follow the dispersion relation from special relativity.

The equation itself depends on the metric signature convention you're using. The common ones are $\eta_{\mu\nu} = \text{diag}(\pm 1, \mp 1, \mp 1, \mp 1)$ . It’s like choosing which side of the bed to get out of. Doesn't fundamentally change the day, but it changes how you write it down.

Here's the Klein–Gordon equation in normal units, with the metric signature $\eta_{\mu \nu} = \text{diag}(\pm 1, \mp 1, \mp 1, \mp 1)$ :

Position space

With $x^\mu = (ct, \mathbf{x})$ , the equation looks like:

$\left(\ {\frac {1}{c^{2}}}{\frac {\ \partial ^{2}}{\ \partial t^{2}\ }}-\nabla ^{2}+{\frac {\ m^{2}c^{2}\ }{\hbar ^{2}}}\ \right)\ \psi (\ t,\mathbf {x} \ )=0$

This $\psi(t, \mathbf{x})$ is your wave function. It's a superposition of these plane waves:

$\psi (\ t,\mathbf {x} \ )=\int \left\{\ \int e^{\mp i\left(\ \omega t-\mathbf {k} \cdot \mathbf {x} \right)}~\psi (\ \omega ,\mathbf {k} \ )\;{\frac {\ \mathrm {d} ^{3}k\ }{~\left(2\pi \hbar \right)^{3}\ }}\ \right\}{\frac {\ \mathrm {d} \omega \ }{\ 2\pi \hbar \ }}\$

The energies and momenta are linked by that relativistic rule:

$E^{2}-\mathbf {p} ^{2}c^{2}=m^{2}c^{4}$

Fourier transform

We often represent the wave function in momentum space, $\psi(\omega, \mathbf{k})$ , where $\omega = E/\hbar$ and $\mathbf{k} = \mathbf{p}/\hbar$ .

Momentum space

With $p^\mu = (E/c, \mathbf{p})$ , it's cleaner:

$p^{\mu}p_{\mu} = \pm m^2 c^2$

Or, in four-vector notation:

$(\Box + \mu^2) \psi = 0$ , where $\mu = \frac{mc}{\hbar}$

Here, $\Box = \pm \eta^{\mu\nu}\partial_{\mu}\partial_{\nu}$ is the wave operator. $\nabla^2$ is the Laplace operator. $c$ is the speed of light, and $\hbar$ is the Planck constant. They often clutter things up, so we frequently switch to natural units where $c = \hbar = 1$ .

Here’s the same equation in natural units, again with the metric signature $\eta_{\mu \nu} = \text{diag}(\pm 1, \mp 1, \mp 1, \mp 1)$ :

Position space

With $x^\mu = (t, \mathbf{x})$ :

$\left(\ \partial _{t}^{2}-\nabla ^{2}+m^{2}\right)\ \psi (\ t,\mathbf {x} \ )=0$

And its Fourier representation:

$\psi (\ t,\mathbf {x} \ )=\int \left\{\ \int e^{\mp i\ \left(\ \omega \ t\ -\ \mathbf {k} \cdot \mathbf {x} \ \right)}\;\psi (\ \omega ,\mathbf {k} \ )\ {\frac {\mathrm {d} ^{3}k}{\ \left(2\pi \right)^{3}}}\ \right\}{\frac {\mathrm {d} \omega }{\ 2\pi \ }}\$

The energy-momentum relation simplifies:

$E^{2}-\mathbf {p} ^{2}=m^{2}$

Momentum space

With $p^\mu = (E, \mathbf{p})$ :

$(\Box + m^2) \psi = 0$

And the Fourier transform:

$\psi (\ x^{\mu }\ )=\int e^{-i\ p_{\mu }x^{\mu }}\ \psi (\ p^{\mu }\ )\;{\frac {\ \mathrm {d} ^{4}p\ }{\;\left(2\pi \right)^{4}\ }}\$

The momentum constraint is now:

$p^{\mu} p_{\mu} = \pm m^2$

Unlike the Schrödinger equation, this one gives you two values for $\omega$ (energy) for each $k$ (momentum). One positive, one negative. You have to carefully separate these positive and negative frequency parts to get something that describes a relativistic wavefunction.

For the time-independent case, it becomes:

$\left[\ \nabla ^{2}-{\frac {\ m^{2}c^{2}}{\ \hbar ^{2}}}\ \right]\ \psi (\ \mathbf {r} \ )=0$

This looks a lot like the homogeneous screened Poisson equation. It's also equivalent to:

$\hat{p}^{\mu}\hat{p}_{\mu}\psi = m^2c^2\psi$

Where the momentum operator is:

$\hat{p}^{\mu}=i\hbar {\frac {\partial }{\ \partial x_{\mu }\ }}=i\hbar \left(\ {\frac {\partial }{\ \partial (ct)\ }},-{\frac {\partial }{\ \partial x\ }},-{\frac {\partial }{\ \partial y\ }},-{\frac {\partial }{\ \partial z\ }}\ \right)=\left(\ {\frac {\hat {E}}{c}},\mathbf {\hat {p}} \ \right)$

Relevance

It's important to understand this equation not just as a wave equation for a classical field, but as something that can be quantized. When you do that, you get a quantum field whose excitations are spinless particles.

Its theoretical weight is comparable to the Dirac equation, though it has its own quirks. The solutions describe fields that are scalar or pseudoscalar. In particle physics, you can incorporate electromagnetic interactions, leading to scalar electrodynamics. It's not the most useful for particles like pions, but it's foundational.

There's a version for a complex scalar field that’s crucial for the Higgs Boson. And in condensed matter physics, it's a handy approximation for spinless quasi-particles.

It can be re-written to look like a Schrödinger equation, but it's two coupled first-order equations in time. The solutions have two components, hinting at a charge degree of freedom in relativity. It conserves a quantity, but it’s not positive definite, so the wave function can't be a simple probability amplitude. Instead, that conserved quantity is interpreted as electric charge, and the norm squared of the wave function becomes a charge density. It describes spinless particles with positive, negative, and zero charge.

Any component of a free Dirac equation solution is also a solution to the free Klein–Gordon equation. The Klein–Gordon equation, while initially conceived as a single-particle equation, can't sustain a consistent relativistic one-particle theory. Relativity implies particle creation and annihilation beyond certain energy thresholds. So, it's more of a building block for quantum field theory.

Solution for Free Particle

Let’s take the Klein–Gordon equation in natural units, with the mostly-plus signature:

$(\Box + m^2) \psi(x) = 0$

where $\eta_{\mu\nu} = \text{diag}(+1, -1, -1, -1)$ . We solve it with a Fourier transform:

$\psi(x) = \int \frac{\mathrm{d}^4p}{(2\pi)^4} e^{-ip\cdot x} \psi(p)$

Plugging this in and using the orthogonality of complex exponentials, we get the dispersion relation:

$p^2 = (p^0)^2 - \mathbf{p}^2 = m^2$

This means the momenta must lie on shell. We get positive and negative energy solutions:

$p^0 = \pm E(\mathbf{p})$ , where $E(\mathbf{p}) = \sqrt{\mathbf{p}^2 + m^2}$ .

With some constants $C(p)$ , the solution becomes:

$\psi(x) = \int \frac{\mathrm{d}^4p}{(2\pi)^4} C(p) \delta((p^0)^2 - E(\mathbf{p})^2) e^{ip\cdot x}$

It’s common to separate out the negative energies and focus on positive $p^0$ :

$\psi(x) = \int \frac{\mathrm{d}^4p}{(2\pi)^4} \frac{\delta((p^0)^2 - E(\mathbf{p})^2)}{2E(\mathbf{p})} (A(p)e^{-ip\cdot x} + B(p)e^{+ip\cdot x})\theta(p^0)$

This simplifies further by performing the $p^0$ -integration, picking only the positive frequency part from the delta function:

$\psi(x) = \int \left.{\frac {\mathrm {d} ^{3}p}{(2\pi )^{3}}}{\frac {1}{2E(\mathbf {p} )}}\left(A(\mathbf {p} )e^{-ip\cdot x}+B(\mathbf {p} )e^{+ip\cdot x}\right)\right|_{p^{0}=+E(\mathbf {p} )}$

This is your general solution. Since the initial Fourier transform used Lorentz invariant quantities like $p\cdot x = p_\mu x^\mu$ , this solution is also Lorentz invariant. If you don't care about Lorentz invariance, you can absorb the $1/2E(\mathbf{p})$ factor into the coefficients $A(p)$ and $B(p)$ .

History

The equation carries the names of Klein and Gordon. They proposed it in 1926 as a description for relativistic electrons. Fock independently arrived at it shortly after Klein. Others like Kudar, de Donder, van den Dungen, and de Broglie were also in the vicinity.

Turns out, modeling the electron's spin required the Dirac equation. But the Klein–Gordon equation is spot on for spin-0 relativistic particles, like the pion. And then, in 2012, CERN announced the discovery of the Higgs boson. It's a spin-zero particle, making it the first observed elementary particle described by this equation. We're still sussing out if it's the Standard Model Higgs or something… stranger.

Erwin Schrödinger himself considered it first, back in 1925, looking for an equation for de Broglie waves. He even wrote a manuscript on it for the hydrogen atom. But it predicted the fine structure incorrectly. He wisely submitted his non-relativistic equation instead.

In 1926, Fock generalized Schrödinger's equation to include magnetic fields and velocity-dependent forces, independently deriving this equation. Klein and Fock used methods from Kaluza and Klein. Fock also figured out the gauge theory for the wave equation.

The free particle solution is simple: plane waves.

Derivation

Let’s start with the non-relativistic energy equation:

$\frac{\mathbf{p}^2}{2m} = E$

Quantizing this gives the non-relativistic Schrödinger equation for a free particle:

$\frac{\mathbf{\hat{p}}^2}{2m}\psi = \hat{E}\psi$

where $\mathbf{\hat{p}} = -i\hbar\mathbf{\nabla}$ is the momentum operator and $\hat{E} = i\hbar\frac{\partial}{\partial t}$ is the energy operator.

The problem is, the Schrödinger equation isn't relativistically invariant. It breaks down at high speeds.

So, we try to use the relativistic energy-momentum relation:

$E = \sqrt{\mathbf{p}^2c^2 + m^2c^4}$

If we just insert the quantum operators:

$\sqrt{(-i\hbar\mathbf{\nabla})^2c^2 + m^2c^4}\,\psi = i\hbar \frac{\partial}{\partial t}\psi$

This square root of an operator is tricky. Dirac found it hard to include electromagnetic fields in a Lorentz-invariant way with this. It's also nonlocal.

Klein and Gordon took a different approach. They squared the energy relation:

$\mathbf{p}^2c^2 + m^2c^4 = E^2$

Quantizing that gives:

$\left((-i\hbar\mathbf{\nabla})^2c^2 + m^2c^4\right)\psi = \left(i\hbar\frac{\partial}{\partial t}\right)^2\psi$

Which simplifies to:

$-\hbar^2c^2\mathbf{\nabla}^2\psi + m^2c^4\psi = -\hbar^2\frac{\partial^2}{\partial t^2}\psi$

Rearranging:

$\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\psi - \mathbf{\nabla}^2\psi + \frac{m^2c^2}{\hbar^2}\psi = 0$

See? No imaginary numbers this time. Works for real or complex fields.

We can write this using the Minkowski metric $\eta^{\mu\nu}$ (with signature diag(−c², 1, 1, 1)):

$-\eta^{\mu\nu}\partial_\mu\,\partial_\nu\psi \equiv \frac{1}{c^2}\frac{\partial^2}{\partial t^2}\psi - \mathbf{\nabla}^2\psi$

So, the Klein–Gordon equation becomes covariant:

$(\Box + \mu^2)\psi = 0$ , where $\mu = \frac{mc}{\hbar}$ and $\Box = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2$ .

This operator $\Box$ is the wave operator. Today, we see this as the relativistic field equation for spin-0 particles. And, as mentioned, any component of a free Dirac equation solution also satisfies the free Klein–Gordon equation. This extends to higher spins via the Bargmann–Wigner equations. In quantum field theory, every component of every quantum field has to obey the free Klein–Gordon equation. It’s a universal starting point.

Klein–Gordon Equation in a Potential

We can generalize the equation to include potentials $V(\psi)$ :

$\Box \psi + \frac{\partial V}{\partial \psi} = 0$

The original Klein–Gordon equation is the case $V(\psi) = M^2 \bar{\psi}\psi$ .

A common potential in interacting theories is the $\phi^4$ potential for a real scalar field $\phi$ :

$V(\phi) = \frac{1}{2} m^2 \phi^2 + \lambda \phi^4$

Higgs Sector

The core of the Higgs boson sector in the Standard Model uses a Klein–Gordon field, denoted $H$ . It's a gauge theory, so while $H$ transforms trivially under the Lorentz group, it’s a $\mathbb{C}^2$ -valued vector under the $\text{SU}(2)$ part of the gauge group. So, it's technically a vector field $H: \mathbb{R}^{1,3} \rightarrow \mathbb{C}^2$ , but we call it scalar because of its Lorentz transformation properties.

The potential is:

$V(H) = -m^2 H^\dagger H + \lambda (H^\dagger H)^2$

This is a generalization of the $\phi^4$ potential, but with a crucial difference: it has a circle of minima. This is key to spontaneous symmetry breaking in the Standard Model.

Conserved U(1) Current

A complex Klein–Gordon field $\psi(x)$ has a $\text{U}(1)$ symmetry. If you transform $\psi(x) \mapsto e^{i\theta}\psi(x)$ and $\bar{\psi}(x) \mapsto e^{-i\theta}\bar{\psi}(x)$ (where $\theta$ is a constant), the equation remains unchanged. Noether's theorem tells us this means there’s a conserved current $J^\mu$ :

$J^{\mu}(x) = \frac{e}{2m}\left(\,{\bar {\psi }}(x)\partial ^{\mu }\psi (x)-\psi (x)\partial ^{\mu }{\bar {\psi }}(x)\,\right)$

This current satisfies $\partial_{\mu}J^{\mu}(x)=0$ . We can verify this using the Klein–Gordon equations for $\psi$ and $\bar{\psi}$ :

$(\Box + m^2)\psi(x) = 0$ $(\Box + m^2)\bar{\psi}(x) = 0$

Multiplying the first by $\bar{\psi}$ and the second by $\psi$ , then subtracting, gives:

$\bar{\psi}\square \psi - \psi \square {\bar {\psi }} = 0$

Or, in index notation:

$\bar{\psi}\partial_{\mu}\partial^{\mu}\psi - \psi \partial_{\mu}\partial^{\mu}{\bar {\psi }} = 0$

This is exactly what you need to show that $\partial_{\mu}J^{\mu}(x)=0$ for the current defined above. This $\text{U}(1)$ symmetry can be gauged, leading to scalar QED.

Lagrangian Formulation

The Klein–Gordon equation can also be derived using variational calculus, as the Euler–Lagrange equation from the action:

$S = \int \left(-\hbar ^{2}\eta ^{\mu \nu }\partial _{\mu }{\bar {\psi }}\,\partial _{\nu }\psi -M^{2}c^{2}{\bar {\psi }}\psi \right)\mathrm {d} ^{4}x$

In natural units, with the mostly-minus signature, the actions are simpler:

For a real scalar field $\phi$ of mass $m$ : $S=\int d^{4}x\left({\frac {1}{2}}\partial ^{\mu }\phi \partial _{\mu }\phi -{\frac {1}{2}}m^{2}\phi ^{2}\right)$

For a complex scalar field $\psi$ of mass $M$ : $S=\int d^{4}x\left(\partial ^{\mu }\psi \partial _{\mu }{\bar {\psi }}-M^{2}\psi {\bar {\psi }}\right)$

From the Lagrangian density, we can derive the stress–energy tensor. For the complex field in natural units:

$T^{\mu \nu }=2\partial ^{\mu }{\bar {\psi }}\partial ^{\nu }\psi -\eta ^{\mu \nu }(\partial ^{\rho }{\bar {\psi }}\partial _{\rho }\psi -M^{2}{\bar {\psi }}\psi )$

Integrating the time-time component $T^{00}$ over all space shows that both positive and negative frequency plane-wave solutions can be associated with particles of positive energy. This isn't true for the Dirac equation.

The stress-energy tensor components are conserved currents associated with space-time translations $x^{\mu }\mapsto x^{\mu }+c^{\mu }$ . Thus, $\partial_{\mu}T^{\mu\nu}=0$ (on-shell). Integrating $T^{0\nu}$ over space gives conserved quantities: total energy for $\nu=0$ and total momentum for $\nu=i \in \{1,2,3\}$ .

Non-relativistic Limit

Classical Field

To get the non-relativistic limit ( $v \ll c$ ) of a classical Klein–Gordon field $\psi(x, t)$ , we factor out the oscillatory rest mass energy term:

$\psi(x,t) = \phi(x,t) e^{- \frac{i}{\hbar}mc^2t}$

where $\phi(x,t) = u_E(x) e^{- \frac{i}{\hbar}E't}$ . The kinetic energy $E' = E - mc^2$ is much smaller than $mc^2$ in the non-relativistic limit ( $v \ll c$ ). This means:

$i\hbar \frac{\partial \phi}{\partial t} = E'\phi \ll mc^2\phi$ $(i\hbar)^2 \frac{\partial^2\phi}{\partial t^2} = (E')^2\phi \ll (mc^2)^2\phi$

Substituting this into the Klein–Gordon equation:

$c^{-2}\partial_t^2\psi = \nabla^2\psi - (\frac{mc}{\hbar})^2\psi$

yields:

$-\frac{1}{c^2}\left(i\frac{2mc^2}{\hbar}\frac{\partial \phi}{\partial t} + \left(\frac{mc^2}{\hbar}\right)^2\phi \right)e^{- \frac{i}{\hbar}mc^2t} \approx \left(\nabla^2 - \left(\frac{mc}{\hbar}\right)^2\right)\phi \,e^{- \frac{i}{\hbar}mc^2t}$

Canceling the exponential and the mass term, we get:

$i\hbar \frac{\partial \phi}{\partial t} = -\frac{\hbar^2}{2m}\nabla^2\phi$

This is a classical Schrödinger field.

Quantum Field

For a quantum Klein–Gordon field, the non-relativistic limit is more complicated due to the non-commutativity of field operators. The creation and annihilation operators effectively decouple and behave like independent quantum Schrödinger fields.

Scalar Electrodynamics

You can make a complex Klein–Gordon field $\psi$ interact with electromagnetism in a gauge-invariant way. You replace the ordinary derivatives with gauge-covariant derivatives:

$D_{\mu}\psi = (\partial_{\mu} - ieA_{\mu})\psi$ $D_{\mu}\bar{\psi} = (\partial_{\mu} + ieA_{\mu})\bar{\psi}$

where $A_\mu$ is the electromagnetic four-potential. Under a local $\text{U}(1)$ gauge transformation $\psi \mapsto e^{i\theta(x)}\psi$ , the derivative transforms as $D_\mu \psi \mapsto e^{i\theta} D_\mu \psi$ . The gauge field transforms as:

$A_{\mu} \mapsto A_{\mu} + \frac{1}{e}\partial_{\mu}\theta$

The covariant derivative transforms correctly:

$D_{\mu}\psi \mapsto e^{i\theta}D_{\mu}\psi$

In natural units, the Klein–Gordon equation becomes:

$D_{\mu}D^{\mu}\psi - M^2\psi = 0$

This coupling only works for complex Klein–Gordon fields, not real ones, because only the complex field has the ungauged $\text{U}(1)$ symmetry.

The action for scalar QED is:

$S=\int d^{4}x\,\left(-{\frac {1}{4}}F^{\mu \nu }F_{\mu \nu }+D^{\mu }\psi D_{\mu }{\bar {\psi }}-M^{2}\psi {\bar {\psi }}\right)$

where $F_{\mu \nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}$ is the electromagnetic field strength tensor.

This is classical scalar quantum electrodynamics, or scalar QED.

Scalar Chromodynamics

You can extend this to a non-abelian gauge theory using a group $G$ , coupling the scalar Klein–Gordon action to a Yang–Mills Lagrangian. The field is technically vector-valued, but we still call it scalar based on its Lorentz transformation properties.

Let's use $G = \text{SU}(N)$ . Under a gauge transformation $U(x) \in \text{SU}(N)$ , the scalar field $\psi$ transforms as:

$\psi(x) \mapsto U(x)\psi(x)$ $\psi^{\dagger}(x) \mapsto \psi^{\dagger}(x)U^{\dagger}(x)$

The covariant derivative becomes:

$D_{\mu}\psi = \partial_{\mu}\psi - igA_{\mu}\psi$ $D_{\mu}\psi^{\dagger} = \partial_{\mu}\psi^{\dagger} + ig\psi^{\dagger}A_{\mu}^{\dagger}$

where the gauge field $A_\mu$ transforms as:

$A_{\mu} \mapsto UA_{\mu}U^{-1} - \frac{i}{g}\partial_{\mu}UU^{-1}$

This $A_\mu$ is a matrix-valued field. The chromomagnetic field strength is:

$F_{\mu \nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu} + g(A_{\mu}A_{\nu} - A_{\nu}A_{\mu})$

The action for scalar QCD is:

$S=\int d^{4}x\,\left(-{\frac {1}{4}}{\text{Tr}}(F^{\mu \nu }F_{\mu \nu })+D^{\mu }\psi ^{\dagger }D_{\mu }\psi -M^{2}\psi ^{\dagger }\psi \right)$

Klein–Gordon on Curved Spacetime

In general relativity, gravity is handled by replacing partial derivatives with covariant derivatives. The Klein–Gordon equation, in the mostly-plus signature, becomes:

$0 = -g^{\mu \nu }\nabla _{\mu }\nabla _{\nu }\psi + \frac{m^2c^2}{\hbar^2}\psi$

This can be expanded using the Christoffel symbols $\Gamma^{\sigma}_{\mu\nu}$ (which represent the gravitational field):

$0 = -g^{\mu \nu }\partial _{\mu }\partial _{\nu }\psi + g^{\mu \nu }\Gamma ^{\sigma }{}_{\mu \nu }\partial _{\sigma }\psi + \frac{m^2c^2}{\hbar^2}\psi$

Or equivalently:

$\frac{-1}{\sqrt{-g}}\partial _{\mu }\left(g^{\mu \nu }\sqrt {-g}\partial _{\nu }\psi \right)+{\frac {m^{2}c^{2}}{\hbar ^{2}}}\psi =0$

where $g^{\alpha\beta}$ is the inverse metric tensor, $g$ is its determinant, $\nabla_\mu$ is the covariant derivative, and $\Gamma^{\sigma}_{\mu\nu}$ is the Christoffel symbol.

In natural units, this is:

$\nabla^a\nabla_a\Phi - m^2\Phi = 0$

This also has an action formulation on a spacetime manifold $M$ :

For a real scalar field $\Phi$ : $S=\int _{M}d^{4}x\,{\sqrt {-g}}\left(-{\frac {1}{2}}g^{ab}\nabla _{a}\Phi \nabla _{b}\Phi -{\frac {1}{2}}m^{2}\Phi ^{2}\right)$

For a complex scalar field $\Psi$ : $S=\int _{M}d^{4}x\,{\sqrt {-g}}\left(-g^{ab}\nabla _{a}\Psi \nabla _{b}{\bar {\Psi }}-M^{2}\Psi {\bar {\Psi }}\right)$

Remarks

Spinless particles like pions are unstable and interact strongly. The interaction term in the Hamiltonian is not simple.
The Sine–Gordon equation is a notable example of an Integrable system.
To merge quantum mechanics and special relativity, you need a multi-particle theory – quantum field theory. The Klein–Gordon equation reappears as the equation for components of all free quantum fields.

Steven Weinberg famously omitted relativistic wave mechanics from his quantum mechanics lectures, calling its usual presentation "profoundly misleading." Others, like Walter Greiner, prefer the long, historical route, finding it pedagogically necessary. The free solutions of these original equations still matter in quantum field theory; they form the basis of the Fock space and allow us to express quantum fields.

There. It’s all there. The facts, the equations, the history. Don't expect me to hold your hand through it. If you can't make sense of this, it's not the equations that are the problem.