Path integral

A path integral represents a quantum amplitude or correlation function as an integral over field histories weighted by the exponential of the action.

For a Hamiltonian operator with kinetic term ${\hat{p}}^2/2$, insert position-basis resolutions between Trotter product formula factors. The measure is the limit of $(2\pi i\Delta {)}^{-M/2}{\prod }_{j=1}^{M-1}d{q}_j$, and the exponent contains ${\sum }_j[(q_j-q_{j-1}{)}^2/(2\Delta )-\Delta V(q_{j-1})]$. Its continuum notation is $\int \mathcal{D}q{e}^{iS[q]}$, with fixed endpoint values. A damping prescription defines the oscillatory limit.

A quadratic path integral becomes a product of Gaussian integrals after diagonalizing its fluctuation operator. One real coordinate per mode gives an inverse square root of the functional determinant; one complex coordinate, or two real coordinates, gives its inverse. A regulator and normalization are essential: determinant ratios are often meaningful before an absolute determinant is defined.

For a quadratic potential energy, shifting a path to its classical solution leaves an exactly quadratic fluctuation action with homogeneous Dirichlet boundary conditions. The first variation vanishes by the Euler-Lagrange equation. The Gaussian path integral prefactor is an inverse square root of a regulated functional determinant and does not depend on the endpoints. At conjugate times the ordinary formula must instead be read by distributional continuation.

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.

A Gaussian auxiliary-field identity replacing a quadratic expression in a composite field by a linear coupling to a new field. Completing the square in $M^2/(2U)-Mj$ yields an exponential $e^{U{j}^2/2}$ after integrating $M$, with normalized Gaussian prefactors. It converts suitable quartic field interactions into bilinear couplings and permits fermionic fields to be integrated out.

For two Grassmann spin components, $\mathbf{j}=\bar{\psi }\mathbf{\sigma }\psi$ satisfies ${\mathbf{j}}^2=-6{\bar{\psi }}_{\uparrow }{\bar{\psi }}_{\downarrow }{\psi }_{\downarrow }{\psi }_{\uparrow }$. A real vector auxiliary field with cost ${\mathbf{M}}^2/(2U)$ and coupling $-\mathbf{M}\cdot \mathbf{j}$ therefore reproduces the repulsive $3U$ contact action. All three spin components are essential to that normalization.

Insert the coherent-state resolution of identity between short-time evolution operators. For a normally ordered Hamiltonian, the kernel uses the next bra label and current ket label: $⟨{\psi }_{j+1}|e^{-ϵH}|{\psi }_j⟩=\exp ({\bar{\psi }}_{j+1}{\psi }_j-ϵH({\bar{\psi }}_{j+1},{\psi }_j)+O({ϵ}^2))$. Together with the Gaussian measure this yields the first-order derivative term $\bar{\psi }{\partial }_{\tau }\psi$. The adjacent-label prescription retains operator ordering information that a formal continuum expression alone can hide.

The thermal trace closes bosonic coherent-state paths periodically and fermionic coherent-state paths antiperiodically: $\psi (\beta )=\zeta \psi (0)$ and $\bar{\psi }(\beta )=\zeta \bar{\psi }(0)$, with $\zeta =1$ for bosons and $-1$ for fermions. The minus sign follows from the fermionic coherent-state trace, not from choosing a different Hamiltonian. These boundary conditions determine the allowed Matsubara frequencies.

An antiperiodic free fermion field with inverse propagator ${\partial }_{\tau }+\xi$ has a thermally normalized determinant $1+e^{-\beta \xi }$. The factor counts empty and occupied states of one level. Its logarithmic Matsubara sum is meaningful only with a regularization fixing the determinant normalization; an unqualified infinite product is not convergent.

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.

An interaction depending on the field can be represented by substituting $(1/i)\delta /\delta J$ into its action and acting on the free generating functional. Expanding that exponential produces Feynman diagrams, contractions and vertex weights. Normalize by the resulting value at $J=0$ to cancel vacuum diagrams.

Long imaginary-time evolution multiplies an energy eigenstate by $e^{-T{E}_n}$. After normalization it projects onto the lowest energy component with nonzero initial overlap, or onto the lowest-energy subspace if the vacuum is degenerate. This provides vacuum boundary conditions for a path integral and is related to the Feynman i-epsilon prescription.

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.

A classical path joining prescribed endpoints is a stationary point of the action, satisfying the Euler-Lagrange equation and the endpoint boundary conditions. Expanding the path integral about each such path separates its classical phase from its quadratic fluctuation determinant. There may be multiple classical paths or none within a chosen real saddle class; the semiclassical propagator must respect that saddle structure rather than assuming a unique trajectory for every potential.

For $H=p^2/(2m)+V(q)$, time slicing gives a Euclidean path integral over paths with the specified endpoints and action $S_E=\int [m{\dot{q}}^2/2+V(q)]dt$. Each step contributes normalization $(m/(2\pi \hslash \delta ){)}^{1/2}$. This normalization fixes the units and endpoint factors of the kernel, which differ from amplitudes between normalized localized wavepackets.

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.

Integrating each momentum in a quadratic Hamiltonian yields the configuration-space kinetic action and its time-slice normalization. A raw $dp$ measure gives $\sqrt{2\pi m/\Delta t}$; the normalized Fourier measure $dp/(2\pi )$ gives $\sqrt{m/(2\pi \Delta t)}$. Dropping the time dependence changes the quantum-mechanical propagator.

For $H=p^2/(2m)+V(q)$, time slices of length $\delta$ use $N$ momentum measures $d{p}_j/(2\pi \hslash )$ and $N-1$ intermediate coordinate measures. Each momentum Gaussian integral gives $[m/(2\pi i\hslash \delta ){]}^{1/2}\exp [im(q_j-q_{j-1}{)}^2/(2\hslash \delta )]$. The resulting configuration integral has normalization $[m/(2\pi i\hslash \delta ){]}^{N/2}$. A common endpoint convention for $V$ and the Trotter product formula identify both limits with the same operator transition kernel; a damping or imaginary-time prescription defines the oscillatory integrals.

// Target: perturbative-quantum-field-theory.bigb

The first-order term means ${\sum }_i{p}_i(q_{i+1}-q_i)$ in a finite-dimensional integral. Specify the endpoints and an ordering prescription, then take the maximum time step to zero. Typical Euclidean paths are not classically differentiable, so the discrete prescription is essential.

A path integral with weight $e^{-S_E}$ uses the Euclidean action obtained by Wick rotation. Bosonic convergence requires appropriate positivity and boundary conditions.

An instanton is a localized finite-action classical solution in Euclidean spacetime, used in a semiclassical description of quantum tunnelling and other nonperturbative effects. In one-dimensional quantum mechanics it can connect distinct degenerate potential minima over infinite Euclidean time. A quartic double-well instanton is an elementary example; a Yang-Mills instanton is a gauge-field example with different geometric structure.

A Euclidean bounce is a finite-action Euclidean saddle associated with decay of a false vacuum. It returns to the same metastable vacuum at the two ends of Euclidean time, instead of joining two degenerate stable vacua. Its characteristic negative fluctuation direction produces the imaginary part associated with a decay rate; the leading semiclassical suppression is $e^{-S_B/\hslash }$. Translation zero modes are treated as collective coordinates. It should not be confused with the stable double-well tunnelling instanton whose repeated crossings give a real level splitting.

The orientation-reversed Euclidean trajectory of an instanton. In a symmetric double well it goes from the right vacuum to the left vacuum and has the same positive action as the forward crossing. Its topological orientation changes sign, not its Euclidean action.

When individual instantons are much narrower than their typical separation, multi-crossing paths factor at leading semiclassical order. Integrating $n$ ordered centers over time $\tau$ contributes ${\tau }^n/n!$. In a double well, crossings alternate direction and even/odd counts yield hyperbolic cosine/sine amplitudes. This approximation requires large action relative to $\hslash$, matched local well structure and control of corrections from interactions and local anharmonicity.

For a one-dimensional quantum tunnelling instanton, the prefactor combines nonzero Gaussian fluctuation eigenvalues with the translation zero-mode Jacobian. Here $L_I=-{\partial }_t^2+V^{\prime \prime }(q_I)/m$, $L_0=-{\partial }_t^2+{\omega }^2$, and the prime excludes the zero mode. Common endpoint regularization and reference normalization define the determinant ratio. The resulting $K$ has frequency units, while $K{e}^{-S_I/\hslash }$ is the dilute crossing rate. This determinant/collective-coordinate structure is also discussed in Dunne, Sulejmanpasic and Ünsal's treatment; unequal neighbouring well frequencies require care and need not share the symmetric-double-well result.

Differentiating the Euclidean classical equation for an instanton gives a zero eigenfunction proportional to its time derivative. Its norm satisfies $\mu \int {\dot{q}}_I^{2}d\tau =S_I$, so replacing the Gaussian zero-mode integral by the center coordinate gives $\sqrt{S_I/(2\pi \hslash )}d{\tau }_0$. The nonzero modes belong in the primed functional determinant. This removes the false divergence without deleting the center's integration volume.

For $V(q)=m{\omega }^2(q^2-a^2{)}^2/(8a^2)$, the zero Euclidean-energy condition is $m{\dot{q}}^2/2=V(q)$. Integrating gives the displayed instanton from $-a$ to $a$. Its action is ${\int }_{-a}^a\sqrt{2mV(q)}dq=2m\omega a^2/3$. The arbitrary center $t_0$ supplies a translation zero mode, treated as a collective coordinate in the instanton fluctuation prefactor.

The quartic double-well instanton has $L_I=-{\partial }_{\tau }^2+{\omega }^2[1-3{\mathrm{sech}}^2(\omega \tau /2)/2]$. For a positive regulator $\epsilon$ and $r=2\sqrt{1+\epsilon /{\omega }^2}$, the common infinite-line determinant ratio is $(r-1)(r-2)/[(r+1)(r+2)]$. Its zero-mode factor is $\epsilon /(12{\omega }^2)$ to leading order, giving the displayed primed ratio. Therefore the instanton fluctuation prefactor is $K=\omega \sqrt{6S_I/(\pi \hslash )}$ and the double-well tunneling splitting is $2\hslash K{e}^{-S_I/\hslash }$.

With a time-slice regulator, inserting field-coordinate resolutions of identity in evolution kernels produces a bosonic phase-space path integral; integrating the canonical momentum gives the configuration-space action. For fermions, resolutions in fermionic coherent states instead produce a Berezin integral and the first-order kinetic term. Vacuum endpoint projection supplies the Feynman i-epsilon prescription. Both constructions reproduce the same free oscillator correlators and their Wick theorem expansions, and thus agree order by order in a regulated perturbation series.

A thermal path integral represents the canonical partition function by paths periodic in Euclidean time of circumference $\beta$. For $H=p^2/2+V(x)$, its formal action is $S_E[x]={\int }_0^{\beta }({\dot{x}}^2/2+V(x))d\tau$. The periodic boundary condition results from taking the trace of the evolution kernel. The continuum measure requires a regularization and normalization; finite-dimensional measures make both choices explicit.

Use an orthonormal real Fourier series basis on a circle of length $\beta$, with ${\nu }_r=2\pi r/\beta$. Retain the constant coefficient and the sine/cosine pair for each $1\le r\le N$. The oscillator action becomes $\frac{1}{2}{\omega }^2{q}_0^2+\frac{1}{2}{\sum }_r({\omega }^2+{\nu }_r^2)(q_r^2+s_r^2)$. A product of real Gaussian integrals gives the displayed functional determinant, with $A_N$ independent of $\omega$. Ratios for two frequencies converge regardless of $A_N$. The absolute limit requires its normalization: $A_N={\beta }^{-1}{\prod }_r{\nu }_r^2$ gives the thermal partition function of a quantum harmonic oscillator in the limit. Multiplying this choice by an oscillating positive sequence preserves all frequency ratios but destroys the absolute limit.

The functional measure is the formal integration measure over field configurations in a path integral. Its transformation under a symmetry can preserve the corresponding Ward identity or produce a quantum anomaly.