= Solution
For the <real scalar field>, the mostly-plus <Minkowski metric> gives a positive kinetic energy. Its <canonical momentum> and <Hamiltonian operator> are
$$
\pi=\frac{\partial\mathcal L}{\partial\dot\phi}=\dot\phi,
\qquad
H=\frac12\int d^3\boldsymbol x\,
\left[\pi^2+(\boldsymbol\nabla\phi)^2+m^2\phi^2\right].
$$
The <Euler-Lagrange field equation> is $(\Box-m^2)\phi=0$, or $(\partial_t^2-\nabla^2+m^2)\phi=0$. <Canonical quantization of a real scalar field> promotes the fields to Hermitian operators and replaces the equal-time <Poisson brackets> by <canonical commutation relations>:
$$
[\phi(t,\boldsymbol x),\pi(t,\boldsymbol y)]=i\delta^{(3)}(\boldsymbol x-\boldsymbol y),
\qquad [\phi(t,\boldsymbol x),\phi(t,\boldsymbol y)]=[\pi(t,\boldsymbol x),\pi(t,\boldsymbol y)]=0.
$$
The <Heisenberg equation of motion> then gives $\dot\phi=\pi$ and $\dot\pi=\nabla^2\phi-m^2\phi$, so the classical field equation remains an operator identity. Products at coincident points need a regulator; equivalently, the field is an <operator-valued distribution>.
Resolve the free field into positive- and negative-frequency <plane waves>. With $E_{\boldsymbol k}=\sqrt{|\boldsymbol k|^2+m^2}$ and $k\cdot x=-E_{\boldsymbol k}t+\boldsymbol k\cdot\boldsymbol x$,
$$
\phi(x)=\int\frac{d^3\boldsymbol k}{(2\pi)^3\sqrt{2E_{\boldsymbol k}}}
\left[a(\boldsymbol k)e^{ik\cdot x}+a^\dagger(\boldsymbol k)e^{-ik\cdot x}\right].
$$
Reality pairs the two terms by Hermitian conjugation. The equal-time <canonical commutation relations> are equivalent to
$$
[a(\boldsymbol k),a^\dagger(\boldsymbol k')]=(2\pi)^3\delta^{(3)}(\boldsymbol k-\boldsymbol k'),
\qquad [a,a]=[a^\dagger,a^\dagger]=0.
$$
For example, the two mixed terms in $[\phi,\pi]$ each supply half the <Dirac delta function>. Thus the normalization $1/\sqrt{2E_{\boldsymbol k}}$ is fixed, not optional. Each independent mode is a <quantum harmonic oscillator>, with $a$ and $a^\dagger$ its <annihilation operator> and <creation operator>.
Choose the <Fock vacuum> by $a(\boldsymbol k)|0\rangle=0$. Repeated <creation operators> construct the <bosonic Fock space>. Substitution into the <Hamiltonian operator> yields
$$
H=\int\frac{d^3\boldsymbol k}{(2\pi)^3}E_{\boldsymbol k}a^\dagger(\boldsymbol k)a(\boldsymbol k)+E_0,
\qquad E_0=\frac V2\int\frac{d^3\boldsymbol k}{(2\pi)^3}E_{\boldsymbol k},
$$
where a finite volume $V$ can be used as a regulator. <Normal ordering> sets the flat-space vacuum reference to zero. The spatial <momentum operator> is
$$
\boldsymbol P=\int\frac{d^3\boldsymbol k}{(2\pi)^3}\boldsymbol k\,a^\dagger(\boldsymbol k)a(\boldsymbol k).
$$
In particular, $[H,a^\dagger(\boldsymbol k)]=E_{\boldsymbol k}a^\dagger(\boldsymbol k)$ and $[\boldsymbol P,a^\dagger(\boldsymbol k)]=\boldsymbol k\,a^\dagger(\boldsymbol k)$. A one-particle state therefore has positive <energy>, <momentum> $\boldsymbol k$, and invariant mass $m$. A scalar field transforms in the trivial <spin> representation, so its particles have spin zero. Because the field is real there is only one set of oscillators: the particle is its own <antiparticle>, with no distinct conserved particle-minus-antiparticle charge. The free theory does conserve its occupation-number sum, though generic scalar interactions need not.
A momentum eigenstate extends throughout space. A localized particle is described by a superposition such as
$$
|f\rangle=\int\frac{d^3\boldsymbol k}{(2\pi)^3}\,f(\boldsymbol k)a^\dagger(\boldsymbol k)|0\rangle,
\qquad\int\frac{d^3\boldsymbol k}{(2\pi)^3}|f(\boldsymbol k)|^2=1.
$$
Its <wave packet> evolves through the phase $e^{-iE_{\boldsymbol k}t}$; a narrow packet has <group velocity> $\boldsymbol v=\boldsymbol k/E_{\boldsymbol k}$, whose magnitude is at most one. Localization and dispersion concern these superpositions, rather than classical trajectories attached to individual field modes.
Relativistic spacetime behavior is encoded by <microcausality>. Directly from the oscillator <commutators>,
$$
[\phi(x),\phi(y)]=\int\frac{d^3\boldsymbol k}{(2\pi)^3\,2E_{\boldsymbol k}}
\left[e^{ik\cdot(x-y)}-e^{-ik\cdot(x-y)}\right].
$$
This is a Lorentz-invariant distribution and vanishes at equal time. Any spacelike separation can be transformed to an equal-time separation, so $[\phi(x),\phi(y)]=0$ for spacelike $x-y$. Thus local operations at spacelike-separated points commute. The vacuum <Wightman function> need not vanish outside the light cone, but that correlation does not transmit a controllable signal. The <Feynman propagator> is
$$
\langle0|T\phi(x)\phi(y)|0\rangle
=\int\frac{d^4p}{(2\pi)^4}\,\frac{-i\,e^{ip\cdot(x-y)}}{p^2+m^2-i0},
$$
which propagates positive <energy> forward and negative <energy> backward in the <time-ordered product>. For causal response one uses the <retarded Green function>, whose support lies in or on the future light cone.
Finally, commuting <creation operators> imply
$$
a^\dagger(\boldsymbol k_1)a^\dagger(\boldsymbol k_2)|0\rangle
=a^\dagger(\boldsymbol k_2)a^\dagger(\boldsymbol k_1)|0\rangle.
$$
The multiparticle state is symmetric under the <particle exchange operator>, which is <Bose statistics>. For a normalized single mode, the occupation states are $(a^\dagger)^n|0\rangle/\sqrt{n!}$ with $n=0,1,2,\ldots$: any number of identical <bosons> can occupy it. There is no <Pauli exclusion principle> for these particles. This construction agrees with the <Spin-statistics theorem> relating integer <spin> to bosonic exchange symmetry under relativistic locality and positive-energy assumptions. It demonstrates the required scalar case without assuming that theorem as the quantization prescription.
\b[The quantized <real scalar field> describes positive-<energy>, mass-$m$, <spin>-zero <bosons> that are their own <antiparticles>; commuting <creation operators> give <Bose statistics>, and local field <commutators> enforce <microcausality>.]
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