The Klein–Gordon equation failed as the equation of one particle and succeeds as the equation of a field. Treat $\hat{\phi}(\mathbf{r},t)$ as a field in a box of volume $V$. Its plane waves $e^{i\mathbf{k}\cdot\mathbf{r}}$ are its modes, and the Klein–Gordon equation gives each the frequency $\omega_{k}$ with $\hbar\omega_{k}=\sqrt{\hbar^{2}k^{2}c^{2}+m^{2}c^{4}}$. Quantise each mode as an oscillator: the field becomes an operator, a sum over modes of $\hat{a}_{\mathbf{k}}$ times the positive-frequency wave and $\hat{a}^{\dagger}_{\mathbf{k}}$ times the negative-frequency one. The energy becomes $\sum_{\mathbf{k}}\hbar\omega_{k}(\hat{a}^{\dagger}_{\mathbf{k}}\hat{a}_{\mathbf{k}}+\tfrac12)$ and the momentum $\sum_{\mathbf{k}}\hbar\mathbf{k}\,\hat{a}^{\dagger}_{\mathbf{k}}\hat{a}_{\mathbf{k}}$, so the one-quantum state $\hat{a}^{\dagger}_{\mathbf{k}}\ket{0}$ has energy $\sqrt{\hbar^{2}k^{2}c^{2}+m^{2}c^{4}}$ and momentum $\hbar\mathbf{k}$ — exactly a relativistic particle of mass $m$. The negative-frequency solutions that troubled the one-particle reading now multiply $\hat{a}^{\dagger}$: they make a quantum of positive energy rather than describe a state of negative energy. A complex field has two kinds of quanta with opposite charges, particles and antiparticles, and the Klein–Gordon density becomes their charge, negative when antiparticles outnumber particles, as a charge may be. And the fields at two points separated by a spacelike interval commute: nothing done at one can affect a measurement at the other.

A sentence that is nonsense as a statement about one person can be sense as a statement about a crowd. The Klein–Gordon equation was nonsense about one particle and is sense about a field and its quanta.

Spacetime around a point $x$ at the centre, time up and one direction of space across: the light cone, dashed, divides the points $y$ that a signal can join to $x$, above and below it, from those it cannot, to the sides. At every $y$ to the sides, the field commutes with the field at $x$.

A real field obeying the Klein–Gordon equation in a box of volume $V$ is quantised mode by mode, with constants $N_{k}$ and h.c. the Hermitian conjugate of the term before it:

• The constants $N_{k}$ are fixed so that $\hat{H}=\sum_{\mathbf{k}}\hbar\omega_{k}(\hat{a}^{\dagger}_{\mathbf{k}}\hat{a}_{\mathbf{k}}+\frac12)$; the momentum is (stated).

Both operators are sums of number operators, taken normal-ordered so that the vacuum has zero energy and momentum. The number operator of a mode, commuted with the creation operator of the same mode, gives that creation operator back, and with any other mode it commutes. So the quantum carries $\hbar\omega_{k}$ and $\hbar\mathbf{k}$, which obey the energy–momentum relation of a particle of mass $m$.

From $[\hat{a}_{\mathbf{q}},\hat{a}^{\dagger}_{\mathbf{k}}]=\delta_{\mathbf{q}\mathbf{k}}$.

The normal-ordered vacuum has $\hat{H}\ket{0}=0$.

The same steps, with $\hbar\mathbf{k}$ in place of $\hbar\omega_{k}$.

The frequency of the Klein–Gordon modes: the energy–momentum relation of a particle of mass $m$.

A one-quantum state of the scalar field has energy $\hbar\omega_{k}=\sqrt{\hbar^{2}k^{2}c^{2}+m^{2}c^{4}}$ and momentum : a particle of mass .

The negative-frequency part of the quantised field multiplies a creation operator: it makes a quantum of positive energy.

For a complex field the Klein–Gordon density becomes the charge operator: particles count positive and antiparticles negative.

The momentum of the field is $\sum_{\mathbf{k}}\hbar\mathbf{k}\,\hat{a}^{\dagger}_{\mathbf{k}}\hat{a}_{\mathbf{k}}$: each quantum in mode carries .

Fields at two points separated by a spacelike interval commute, so a measurement at one cannot affect a measurement at the other.

Great work! You now know how to:

• quantise the Klein–Gordon field mode by mode
• show that its quanta are relativistic particles of mass $m$
• say what becomes of the negative frequencies and the density