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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.

Expanding a configuration-space path integral about a classical trajectory $q_c$ with fixed endpoints gives $S[q_c+f]=S[q_c]+\frac{1}{2}\int f{\mathcal{O}}_{c}fdt+O(f^3)$, where $f$ obeys homogeneous Dirichlet boundary conditions. The Gaussian functional integral contributes $(\det {\mathcal{O}}_c{)}^{-1/2}$ with the phase prescribed by continuation of the quantum-mechanical propagator. Relative to the free operator ${\mathcal{O}}_0=-m{d}^2/d{t}^2$, the prefactor is $D_0(T)[\det {\mathcal{O}}_c/\det {\mathcal{O}}_0{]}^{-1/2}$. In a general potential the operator, and thus the prefactor, depends on the endpoints through $q_c$. For a quadratic potential $V^{\prime \prime }$ is constant and all higher fluctuation terms vanish, so the approximation is exact away from singular focusing times. At a zero mode the simple determinant formula requires a limiting prescription or a different treatment of the zero mode.

On $[0,T]$ with vanishing endpoint fluctuations, the sine-mode eigenvalues are $(\pi n/T{)}^2-{\omega }^2$. Divide by the free eigenvalues and use the sine infinite product. The ratio determines the harmonic oscillator transition kernel once the free configuration-space path integral is normalized. Zeros correspond to caustics where the fluctuation operator has a zero mode.

Use canonical quantization with $[\hat{q},\hat{p}]=i\hslash$, $\hat{p}=-i\hslash {\partial }_q$, and $\hat{H}={\hat{p}}^2/(2m)+V(\hat{q})$. Assume a real potential energy with the regularity and lower-bound conditions needed for a self-adjoint Hamiltonian and its Trotter product formula; singular potentials require their domain and limiting prescription to be specified separately. The transition amplitude is $K(\beta ,\alpha ;T)=⟨\beta |e^{-iT\hat{H}/\hslash }|\alpha ⟩$.

Take $\delta =T/N$, $q_0=\alpha$, $q_N=\beta$, and choose the left-endpoint potential energy convention. The precise time-sliced phase-space path integral isThe endpoint positions are fixed and the momenta are unconstrained. Its continuum notation iswhere the functional measure means the displayed limit, not an unspecified product of unnormalized measures.

At each slice, complete the square in momentum. With the square-root branch selected by continuation from positive Euclidean time,Thus the time-sliced equivalence of phase-space and configuration-space path integrals holds already at finite $N$:This defines the configuration-space path integralThere are $N$ Gaussian prefactors but only $N-1$ coordinate integrations. Omitting that distinction would change the kernel normalization. The paths integrated in the limit need not be differentiable; the kinetic action notation stands for the lattice difference expression.

To identify this with the operator amplitude, normalize $⟨q|p⟩=(2\pi \hslash {)}^{-1/2}{e}^{ipq/\hslash }$. The kernel of one product factor is exactlyInsert $N-1$ position resolutions of the identity between factors. Their product is precisely $K_N$. The Trotter product formula then givesso both functional integrals represent the same canonical kernel. One can define the finite integrals first at $T=-i\tau$, $\tau >0$, where the Gaussian momentum and coordinate kernels converge, and then continue to the real-time boundary value. Alternatively use an equivalent oscillatory damping prescription. Operator convergence fixes the distributional kernel limit; it need not imply pointwise convergence at every endpoint for every admissible potential energy. There is no operator-ordering ambiguity for this separated kinetic-plus-potential Hamiltonian once the common slicing convention is fixed.

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.

For the Hamiltonian $p^2/(2m)+V(q)$, the real-time configuration-space path integral iswith its measure defined by time slicing. Wick rotation gives the Euclidean path integral and the thermal trace.

For the infinite square well on $(0,L)$, the Dirichlet boundary conditions select the normalized energy eigenstatesThere is no $n=0$ state: the corresponding sine is identically zero. Taking the trace in this orthonormal basis gives the canonical partition function

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.

For $H={\hat{p}}^2/(2m)+V(\hat{x})$, divide elapsed time $t$ into $J$ slices $\Delta t=t/J$ and apply the Trotter product formula. Inserting complete position and momentum bases gives the regulated phase-space path integralThe endpoint values are $x_0=x_i$, $x_J=x_f$. Each Gaussian momentum integration in a phase-space path integral can be performed explicitly, leaving the configuration-space path integralIts measure is the time-sliced limit of $(m/(2\pi i\hslash \Delta t){)}^{J/2}{\prod }_{j=1}^{J-1}d{x}_j$, with the usual infinitesimal convergence prescription. The integral sums amplitudes, not positive classical probabilities. Operator ordering remains part of the time-slice definition. Expanding a short-time kernel recovers the Schrödinger equation; composing kernels reproduces the time-evolution operator.

In the classical limit, rapid phase cancellation suppresses paths away from stationary action. Varying $S$ with fixed endpoints gives $m\ddot{x}=-V^{\prime }(x)$. The semiclassical result sums the classical-path phases multiplied by fluctuation functional determinants; for a quadratic Hamiltonian the fluctuation integral is Gaussian and this expansion is exact.

For quantum statistical mechanics, continue $t=-i\tau$ by a Wick rotation. The Euclidean action and thermal trace giveThe trace closes position-space paths. In a many-particle trace one also sums over endpoint permutations with the appropriate exchange signs. In a coherent state field formulation, bosonic fields are periodic and fermionic coherent states instead give antiperiodic Grassmann fields, hence bosonic and fermionic Matsubara frequencies $2\pi n/(\beta \hslash )$ and $(2n+1)\pi /(\beta \hslash )$. ^[ref]

The quantum-to-classical imaginary-time mapping views a local $d$-dimensional quantum problem as a classical-looking field theory in $d+1$ dimensions, with imaginary-time extent $\beta \hslash$. A genuine classical probability interpretation additionally requires a nonnegative weight; fermion signs and Berry terms can obstruct it.

For a smooth potential in the high-temperature or small-$\hslash$ limit, imaginary-time paths fluctuate only over the thermal de Broglie wavelength ${\lambda }_T=\sqrt{2\pi \beta {\hslash }^2/m}$. If this is short compared with the potential's variation scale, the diagonal free kernel givesThis classical phase-space limit of the thermal kernel is the classical phase space partition function for one degree of freedom. A dilute identical-particle gas also needs its $1/N!$ factor; when exchange is appreciable, classical statistics fails. As a check, a single quantum harmonic oscillator gives $Z=[2\sinh (\beta \hslash \omega /2){]}^{-1}\to 1/(\beta \hslash \omega )$, precisely the classical phase-space integral.

The scalar field path integral generalizes the configuration-space path integral by replacing finitely many coordinates with field values. Its regulated measure integrates one field variable at every spacetime lattice point. Fixed boundary fields give transition kernels; vacuum boundary conditions and sources give the normalized vacuum generating functional.