Path integral

A path integral represents quantum evolution or correlation functions as an integral over histories. Dividing time into short steps, inserting resolutions of the identity and multiplying short-time kernels gives the familiar oscillatory weight $e^{iS/\hslash }$, together with a measure and boundary data.

The weight is a complex amplitude, not an ordinary probability assigned independently to each classical trajectory. Histories interfere, and stationary-phase configurations reproduce classical equations when the approximation is controlled. A Euclidean path integral uses imaginary time and a decaying action weight under suitable convergence conditions, helping study tunnelling and thermal states. A formal continuum expression must retain a definition of its discretization, operator ordering and normalization where they affect the result.

An operator equation in a path integral $\mathcal{O}(x)=0$ asserts that insertions of $\mathcal{O}(x)$ vanish in every time-ordered correlation function with the other insertion points separated from $x$. Derivatives of time ordering can produce Ward identity contact terms at coincidence. For example, conservation of a Dirac current gives a vanishing separated-point divergence, while a charged insertion contributes a delta-function contact term fixed by its transformation. A vanishing one-point expectation value alone is insufficient.

In a bosonic coherent-state path integral, coherent-state resolutions between imaginary-time evolution factors turn a normal-ordered grand Hamiltonian $K=H-\mu N$ into its complex symbol. Coherent-state overlaps supply the first-order temporal term. A thermal trace makes the fields periodic in imaginary time. The chemical potential couples to number, and interactions retain their normal-ordering convention.

In a bosonic coherent-state path integral, normal ordering determines which fields occur on the two neighboring time slices. For unnormalized coherent states, the short-step kernel obeyswhere $K$ is the normal-ordered grand Hamiltonian and $ϵ$ has inverse-energy units. Creation operators supply the later conjugate field and annihilation operators the earlier field.

The overlaps yield the temporal derivative term in the continuum action, but replacing the two times by one too early can lose equal-time ordering information. Retaining the discrete prescription fixes contact terms and the meaning of occupation insertions. A continuum expression is therefore used together with its time-slice convention, especially when evaluating products at coincident times.

Bosonic traces impose periodic complex fields and inverse determinants. Fermionic traces impose antiperiodic Grassmann fields and determinants. The distinction encodes quantum statistics, not a freely chosen boundary condition.

Physical imaginary time runs over $[0,\beta \hslash ]$ with an action divided by $\hslash$. Rescaling it by $\hslash$ gives a variable in $[0,\beta ]$ and the operator ${\partial }_{\tau }+\epsilon$. Matsubara eigenvalues in the latter variable have energy units.

With exact one-slice weight $q=e^{-\beta \epsilon /M}$ and periodic shift matrix $P$, $\det (I-qP)=1-q^M$. This finite-dimensional identity fixes the normal-ordered thermal determinant without an ambiguous unregulated frequency product.

An ordinary normal-ordered coherent-state Gaussian for energy $\epsilon a^{†}a$ gives $(1-e^{-\beta \epsilon }{)}^{-1}$. The physical oscillator with zero-point energy $\epsilon /2$ requires the factor $e^{-\beta \epsilon /2}$. A symmetric determinant prescription may encode that factor implicitly; it must not be counted twice.

For diffusion constant $D>0$, Gaussian increments have variance $2D\Delta t$. Their convolution gives a path weight proportional to $\exp [-\sum (\Delta x{)}^2/(4D\Delta t)]$ with measure factor $(4\pi D\Delta t{)}^{-M/2}$ for $M$ slices. The continuum kinetic action is notation for this limiting measure; typical Brownian paths have no ordinary derivative.

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.

The quadratic fluctuation eigenvalues are $(n\pi /\mathcal{T}{)}^2+{\omega }^2$. Dividing by the free-particle determinant gives ${\prod }_{n\ge 1}[1+(\omega \mathcal{T}/n\pi {)}^2{]}^{-1/2}=\sqrt{\omega \mathcal{T}/\sinh (\omega \mathcal{T})}$. The short-time free kernel fixes the absolute normalization.

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.

For the auxiliary-field convention with cost $M^2/(2U)$ and coupling $-M\bar{\psi }{\sigma }_z\psi$, the homogeneous saddle point equation equates M to U times spin polarization per site. M has energy units. It is not itself a magnetic moment, and its coefficient changes if the contact interaction or density-of-states spin convention is changed. The exact three-component decoupling must be distinguished from selecting one constant spin direction as a mean-field approximation.

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.

Insert coherent-state completeness between short factors of $e^{-\beta (H-\mu N)}$. The adjacent bra and ket labels determine the normal-ordered symbol $H_N({\bar{\psi }}_{j+1},{\psi }_j)$, while overlaps generate the derivative term. Bosonic labels are complex and periodic in imaginary time; fermionic labels are independent Grassmann variables and antiperiodic. The continuum shorthand is defined by that temporal ordering and measure, not an arbitrary substitution of commuting fields for operators.^[ref]

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 formal product over fermionic Matsubara frequencies is divergent before its normalization and time-slice prescription are fixed. The coherent-state thermal trace fixes the single-level factor to $1+e^{-\beta \xi }$, counting its empty and occupied states. A symmetrically regularized product can instead expose $2\cosh (\beta \xi /2)$; the additional factor $e^{-\beta \xi /2}$ restores the physical empty-state energy convention. Dropping that factor can incorrectly shift the grand potential and particle occupation.

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.

For positive mass $m$ and angular frequency $\omega$, the Euclidean action $S_E=\frac{m}{2}{\int }_0^{\mathcal{T}}({\dot{x}}^2+{\omega }^2{x}^2)d\tau$ with $x(0)=x(\mathcal{T})=x$ has the unique minimizing path displayed. The Euler-Lagrange equation is ${\ddot{x}}_0={\omega }^2{x}_0$, and the positive second variation proves uniqueness. On this path the integrand is $\frac{m}{2}d(x_0{\dot{x}}_0)/d\tau$, giving $S_E[x_0]=m\omega x^2\tanh (\omega \mathcal{T}/2)$. Increasing the interval of imaginary time lets its interior approach the minimum of the potential energy even while both endpoints remain fixed.

A local $d$-dimensional quantum field at finite temperature can often be represented by an anisotropic classical-looking action in $d+1$ dimensions, with a finite extra imaginary-time extent. The zero-temperature limit makes that extent infinite. A real nonnegative statistical weight is an additional requirement: fermionic signs and imaginary terms can obstruct interpretation as an ordinary classical probability distribution.

Taking the trace of the position-space imaginary-time kernel identifies its endpoints and integrates over the common position. The resulting closed paths have imaginary-time length $\beta \hslash$. Bosonic coherent fields are periodic, while fermionic coherent fields are antiperiodic. This temporal condition follows from the trace, independently of the spatial boundary conditions.

For $H=p^2/(2m)+V(x)$ and a smooth potential on scales larger than the thermal de Broglie wavelength ${\lambda }_T=\sqrt{2\pi \beta {\hslash }^2/m}$, the diagonal thermal kernel approaches $⟨x|e^{-\beta H}|x⟩\simeq {\lambda }_T^{-d}{e}^{-\beta V(x)}$. Integration over its diagonal gives the classical phase space integral with one factor $(2\pi \hslash {)}^{-1}$ per coordinate-momentum pair. Exchange and boundary corrections must be small; a positive imaginary-time weight by itself does not make an arbitrary quantum system classical.

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.

A collective coordinate labels a family of physically distinct saddle configurations, such as the center ${\tau }_0$ of an instanton. Differentiating the saddle with respect to that coordinate gives a zero mode of the quadratic fluctuation operator. An ordinary Gaussian integral over its coefficient would be divergent, because moving along the family costs no quadratic action.

Instead, change variables from the zero-mode coefficient to the collective coordinate and include the associated Jacobian. For a one-dimensional instanton with $m\int {\dot{q}}_I^{2}d\tau =S_I$, the standard normalized measure is $\sqrt{S_I/(2\pi \hslash )}d{\tau }_0$. The remaining fluctuations contribute a primed functional determinant, excluding this zero mode. Integrating over the allowed center positions supplies the physical observation-time factor rather than deleting the mode and losing that factor.

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.

Return amplitudes sum even numbers of well-to-well events, while transfer amplitudes sum odd numbers. A dilute event gas gives cosh and sinh factors and a doublet splitting $2\hslash A{e}^{-S_0/\hslash }$.

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.