= Solution
Use <natural units> and the <Minkowski metric> $\eta=\operatorname{diag}(1,-1,-1,-1)$. A <real scalar field> assigns a real variable $\phi(t,\mathbf x)$ to each spatial point. Its <Lagrangian density> can be taken to be
$$
\mathcal L=\frac12\partial_\mu\phi\,\partial^\mu\phi-V(\phi)=\frac12\dot\phi^2-\frac12|\nabla\phi|^2-V(\phi),\qquad S[\phi]=\int d^4x\,\mathcal L.
$$
The <principle of stationary action> gives the <Euler-Lagrange equation> $\Box\phi+V'(\phi)=0$. With $V(\phi)=m^2\phi^2/2$, this is the <Klein-Gordon equation>. An additional nonlinear part of $V$ describes interactions.
The <canonical momentum> is $\pi=\partial\mathcal L/\partial\dot\phi=\dot\phi$. The <Legendre transform in mechanics> gives the <canonical Hamiltonian density of a real scalar field>
$$
\mathcal H=\pi\dot\phi-\mathcal L=\frac12\pi^2+\frac12|\nabla\phi|^2+V(\phi),\qquad H=\int d^3x\,\mathcal H.
$$
The <Hamiltonian> equations $\dot\phi=\pi$ and $\dot\pi=\nabla^2\phi-V'(\phi)$ recover the same field equation. In <canonical quantization>, the fields become operators satisfying the equal-time <canonical commutation relations>
$$
[\widehat\phi(t,\mathbf x),\widehat\pi(t,\mathbf y)]=i\delta^3(\mathbf x-\mathbf y),\qquad[\widehat\phi,\widehat\phi]=[\widehat\pi,\widehat\pi]=0.
$$
A spatial lattice makes the analogy with many coupled quantum-mechanical coordinates precise. Each lattice field value is a coordinate, with its own <conjugate momentum>. The <path integral> is another representation of the same quantum evolution.
To see its origin, first consider one coordinate with $H=p^2/(2M)+V(q)$. Split a time interval into $N$ steps of length $\varepsilon$ and insert position and momentum resolutions of the identity. The short-time kernel is
$$
\langle q_{j+1}|e^{-i\varepsilon\widehat H}|q_j\rangle=\int\frac{dp_j}{2\pi}\exp\left\{ip_j(q_{j+1}-q_j)-i\varepsilon\left[\frac{p_j^2}{2M}+V(q_j)\right]\right\}+O(\varepsilon^2).
$$
Multiplying the kernels and integrating over intermediate positions gives the <phase-space path integral>
$$
K(q_f,t_f;q_i,t_i)=\int\mathcal Dq\,\mathcal Dp\,\exp\left[i\int_{t_i}^{t_f}(p\dot q-H(p,q))\,dt\right].
$$
The endpoints of $q$ are fixed. The momentum integrals are <Gaussian integrals>; completing the square produces the <configuration-space path integral>
$$
\boxed{K(q_f,t_f;q_i,t_i)=\int_{q_i}^{q_f}\mathcal Dq\,e^{iS[q]},\qquad S[q]=\int_{t_i}^{t_f}\left(\frac M2\dot q^2-V(q)\right)dt.}
$$
At finite slicing its normalization contains $(M/(2\pi i\varepsilon))^{N/2}\prod_{j=1}^{N-1}dq_j$. This fixes the composition law and the initial delta-function kernel. One sums over all paths, not merely solutions of the classical equation. Restoring $\hbar$ replaces the weight by $e^{iS/\hbar}$; stationary phase explains the emergence of classical trajectories.
For the field, use <scalar field configuration eigenstates> $|\varphi\rangle$, satisfying $\widehat\phi(\mathbf x)|\varphi\rangle=\varphi(\mathbf x)|\varphi\rangle$. Insert their completeness relations on every time slice. This gives
$$
\langle\varphi_f|e^{-i\widehat H(t_f-t_i)}|\varphi_i\rangle=\int\mathcal D\phi\,\mathcal D\pi\,\exp\left[i\int d^4x\,(\pi\dot\phi-\mathcal H)\right].
$$
The endpoint field configurations are fixed. Integrating the Gaussian momentum variables leaves the <scalar field path integral> $\mathcal N\int\mathcal D\phi\,e^{iS[\phi]}$. The <functional measure> means a regulated product over the field variables. A spacetime lattice or another <ultraviolet cutoff> makes this product finite before the continuum limit; interacting continuum calculations may require <renormalization>. The oscillatory Minkowski weight is an amplitude, not a positive <probability density>.
For <vacuum expectation values>, the boundaries must select the vacuum rather than arbitrary field configurations. Long imaginary-time evolution suppresses excited states: $e^{-TH}|n\rangle=e^{-TE_n}|n\rangle$, so after normalization only the lowest-energy component remains as $T\to\infty$. This is <vacuum projection by imaginary time>. The corresponding <Feynman i-epsilon prescription> in the real-time integral specifies the vacuum boundary conditions and the poles of the propagator. With $t=-i\tau$, the <Euclidean path integral> has the weight $e^{-S_E}$, where
$$
S_E=\int d\tau\,d^3x\left[\frac12(\partial_\tau\phi)^2+\frac12|\nabla\phi|^2+V(\phi)\right].
$$
It is often a useful regulated starting point; analytic continuation returns the vacuum time-ordered quantities.
Introduce a classical source $J(x)$ and define the <normalized vacuum generating functional>
$$
Z[J]=\frac{\int\mathcal D\phi\,\exp i\left(S[\phi]+\int d^4x\,J(x)\phi(x)\right)}{\int\mathcal D\phi\,e^{iS[\phi]}},\qquad Z[0]=1,
$$
with the same vacuum prescription in numerator and denominator. A <functional derivative> brings down $i\phi(x)$. The order of the time slices makes the operator insertion time-ordered. Thus <source differentiation inserts time-ordered field operators>:
$$
\boxed{\langle\Omega|T\{\widehat\phi(x_1)\cdots\widehat\phi(x_r)\}|\Omega\rangle=\left.\frac1{i^r}\frac{\delta^rZ[J]}{\delta J(x_1)\cdots\delta J(x_r)}\right|_{J=0}.}
$$
The denominator removes vacuum diagrams and gives normalized expectation values. It is essential that these are <time-ordered products>; differentiating this vacuum functional does not directly give every possible operator ordering.
The free theory illustrates the method. Its quadratic kernel is $K=-\Box-m^2$ with the vacuum pole prescription, and completing the square gives the <Gaussian evaluation of a free scalar generating functional>
$$
Z_0[J]=\exp\left[-\frac12\int d^4x\,d^4y\,J(x)\Delta_F(x-y)J(y)\right],\qquad \Delta_F(x-y)=\int\frac{d^4p}{(2\pi)^4}\frac{i\,e^{-ip\cdot(x-y)}}{p^2-m^2+i0}.
$$
Two source derivatives give $\Delta_F=\langle\Omega|T\{\widehat\phi(x)\widehat\phi(y)\}|\Omega\rangle$. Higher derivatives give all pairings, the content of <Wick theorem>. For an interaction $S_{\rm int}$, one may use <path-integral perturbation by source derivatives>:
$$
Z[J]=\frac{\exp\left(iS_{\rm int}\left[\frac1i\frac\delta{\delta J}\right]\right)Z_0[J]}{\left.\exp\left(iS_{\rm int}\left[\frac1i\frac\delta{\delta J}\right]\right)Z_0[J]\right|_{J=0}}.
$$
Expanding this expression generates <Feynman diagrams> and their <Wick contractions>. The <connected generating functional> $W[J]=-i\log Z[J]$ retains connected contributions; in particular $\delta W/\delta J=\langle\phi\rangle_J$. These functionals turn the computation of field-operator expectations into source differentiation of an ordinary regulated integral.
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