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A configuration-space path integral integrates histories of the coordinates with weight $e^{iS}$. Fixed endpoint conditions specify a transition amplitude. For a quadratic kinetic energy it follows by Gaussian momentum integration from the phase-space path integral. Time-slicing fixes the normalization, and a regulator gives meaning to the otherwise formal functional measure.

For the quantum harmonic oscillator $H=\hslash \omega (a^{†}a+1/2)$, its normal symbol is $H_N=\hslash \omega (\bar{\psi }\psi +1/2)$. The Weyl ordering symbol of $a^{†}a$ is $\bar{\psi }\psi -1/2$, so $H_W=\hslash \omega \bar{\psi }\psi =p^2/(2m)+m{\omega }^2{q}^2/2$. Switching from adjacent-label coherent-state time slicing to a midpoint phase-space path integral must retain this ordering correction. Both correctly normalized constructions give the thermal partition function of a quantum harmonic oscillator $Z=[2\sinh (\beta \hslash \omega /2){]}^{-1}$; retaining the normal-symbol constant after switching prescriptions would count the zero-point shift twice.

Use natural units and the Minkowski metric $\eta =\mathrm{diag}(1,-1,-1,-1)$. A real scalar field assigns a real variable $\varphi (t,\mathbf{x})$ to each spatial point. Its Lagrangian density can be taken to beThe principle of stationary action gives the Euler-Lagrange equation $\square \varphi +V^{\prime }(\varphi )=0$. With $V(\varphi )=m^2{\varphi }^2/2$, this is the Klein-Gordon equation. An additional nonlinear part of $V$ describes interactions.

The canonical momentum is $\pi =\partial \mathcal{L}/\partial \dot{\varphi }=\dot{\varphi }$. The Legendre transform in mechanics gives the canonical Hamiltonian density of a real scalar fieldThe Hamiltonian equations $\dot{\varphi }=\pi$ and $\dot{\pi }={\nabla }^2\varphi -V^{\prime }(\varphi )$ recover the same field equation. In canonical quantization, the fields become operators satisfying the equal-time canonical commutation relationsA 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 $\epsilon$ and insert position and momentum resolutions of the identity. The short-time kernel isMultiplying the kernels and integrating over intermediate positions gives the phase-space path integralThe endpoints of $q$ are fixed. The momentum integrals are Gaussian integrals; completing the square produces the configuration-space path integralAt finite slicing its normalization contains $(M/(2\pi i\epsilon ){)}^{N/2}{\prod }_{j=1}^{N-1}d{q}_j$. This fixes the composition law and the initial delta-function kernel. One sums over all paths, not merely solutions of the classical equation. Restoring $\hslash$ replaces the weight by $e^{iS/\hslash }$; stationary phase explains the emergence of classical trajectories.

For the field, use scalar field configuration eigenstates $|\phi ⟩$, satisfying $\hat{\varphi }(\mathbf{x})|\phi ⟩=\phi (\mathbf{x})|\phi ⟩$. Insert their completeness relations on every time slice. This givesThe endpoint field configurations are fixed. Integrating the Gaussian momentum variables leaves the scalar field path integral $\mathcal{N}\int \mathcal{D}\varphi e^{iS[\varphi ]}$. 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: $e^{-TH}|n⟩=e^{-T{E}_n}|n⟩$, 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 $e^{-S_E}$, whereIt 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 functionalwith the same vacuum prescription in numerator and denominator. A functional derivative brings down $i\varphi (x)$. The order of the time slices makes the operator insertion time-ordered. Thus source differentiation inserts time-ordered field operators: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 $K=-\square -m^2$ with the vacuum pole prescription, and completing the square gives the Gaussian evaluation of a free scalar generating functionalTwo source derivatives give ${\Delta }_F=⟨\Omega |T\{\hat{\varphi }(x)\hat{\varphi }(y)\}|\Omega ⟩$. Higher derivatives give all pairings, the content of Wick theorem. For an interaction $S_{\mathrm{int}}$, one may use path-integral perturbation by source derivatives: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 $\delta W/\delta J=⟨\varphi {⟩}_J$. These functionals turn the computation of field-operator expectations into source differentiation of an ordinary regulated integral.

A phase-space path integral is defined as a limit of finite-dimensional integrals, not by assigning a classical derivative to every path. Choose $t_1=0<t_2<\cdots <t_{N+1}=T$, let $\Delta t_i=t_{i+1}-t_i$, and set $q_{N+1}=q_f$. On slice $i$ the precise prescription isThe remaining Hamiltonian term must have a compatible operator-ordering prescription. For example, evaluate $H$ at $(p_i,(q_{i+1}+q_i)/2)$ for midpoint/Weyl ordering. A prepoint prescription $H(p_i,q_i)$ defines a corresponding ordering instead. This choice matters for a general mixed $H(p,q)$; the separable kinetic-plus-potential Hamiltonian in the next part admits the usual Trotter prescription.

This is time slicing of a phase-space path integral. Integrate the intermediate $q_i$ and the slice momenta and only then take ${\max }_i\Delta t_i\to 0$. Typical paths of the Euclidean path integral need not be differentiable; the finite difference is the meaning of the printed $\dot{q}$. For a fixed-endpoint kernel the initial coordinate is fixed as well, whereas propagation of a wavefunction includes an integral over that initial coordinate.

For the quantum harmonic oscillator, set $\mu =0$ and $\Omega =\hslash \omega$. The normal symbol is $H_N(\bar{\psi },\psi )=\Omega (\bar{\psi }\psi +1/2)$, so the thermal coherent-state time slicing givesThe connection with a phase-space path integral uses the canonical real coordinatesTheir derivative term satisfies $\bar{\psi }\dot{\psi }=-ip\dot{q}/\hslash$ up to total derivatives that vanish for periodic paths. The normal and Weyl symbols of a harmonic oscillator must be distinguished: in midpoint phase-space time slicing, the Weyl ordering symbol of $a^{†}a$ is $\bar{\psi }\psi -1/2$. Thus the appropriate midpoint Hamiltonian is $H_W=p^2/(2m)+m{\omega }^2{q}^2/2$, with no additional constant. The change from the adjacent-label normal prescription to the midpoint prescription includes this ordering correction.

In physical imaginary time $u=\hslash \tau$, the resulting phase-space path integral isGaussian momentum integration in a phase-space path integral produces the usual oscillator configuration-space path integral. An exact check follows directly from the coherent kernel $⟨{\psi }^{\prime }|e^{-\beta \Omega a^{†}a}|\psi ⟩=\exp (e^{-\beta \Omega }{\bar{\psi }}^{\prime }\psi )$:This is the thermal partition function of a quantum harmonic oscillator. Keeping the $\Omega /2$ normal-ordering constant a second time after switching to the Weyl symbol would double count the zero-point energy.

A phase-space path integral represents a quantum transition kernel by integrating both coordinates and conjugate momenta with action $\int (p\dot{q}-H)dt$. It follows from inserting position and momentum completeness relations between short-time evolution operators. For a quadratic momentum dependence the momentum variables can be integrated by a Gaussian integral.