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Keep $\lambda$ for the cubic coupling and $j$ for the external source. Expanding the interaction part of the scalar field path integral yieldsA functional derivative of the source exponential inserts $i\varphi (y)$. In particular, ${\delta }_j(y{)}^3{e}^{i\int j\varphi }=-i\varphi (y{)}^3{e}^{i\int j\varphi }$, so $-i\lambda \varphi (y{)}^3$ can be replaced by $\lambda {\delta }_j(y{)}^3$. The functional derivatives commute and can be taken outside the free path integral. Resumming the exponential series givesThis is a formal perturbation series; a real cubic scalar potential is unbounded and the expression does not assert a stable nonperturbative vacuum state.

To evaluate the free Gaussian path integral, use integration by parts to writeThe inverse kernel isCompleting the square with $\varphi =\chi -K^{-1}j$ changes the exponent to $i\int \chi K\chi /2-i\int j{K}^{-1}j/2$. The translation-invariant functional measure makes the remaining Gaussian path integral precisely $Z[0;0]$, henceFor $ϵ>0$, the exponent includes $-ϵ\int {\varphi }^2/2$, which damps the regulated finite-dimensional integrals. The Feynman i-epsilon prescription displaces the positive-energy pole below the real $p^0$ axis and the negative-energy pole above it, selecting the time-ordered vacuum expectation value. Indeed the Feynman propagator is ${\Delta }_F=iD$. Thus $iϵ$ specifies both convergence and the vacuum state boundary conditions, with $ϵ\to 0^{+}$ taken after integration.

Split the time-evolution operator into $N$ factors with $ϵ=T/N$, and insert $N-1$ position completeness relations between them. Each factor is evaluated by inserting momentum states normalized by $⟨x|p⟩=(2\pi \hslash {)}^{-1/2}{e}^{ipx/\hslash }$:Multiplying the $N$ kernels and integrating the intermediate positions produces the stated time-sliced path integral, including its $N/2$ normalization power. For a free particle this Gaussian short-time kernel is exact, so the convolution identity already holds at finite $N$. Oscillatory integrals are defined with a vanishing convergence regulator, consistently fixing the square-root phase.

Measure intermediate time $u$ from $t^{\prime }$. The classical path with the fixed endpoints isIt satisfies $m{\ddot{x}}_c=-\alpha$. Direct integration of $m{\dot{x}}_c^2/2-\alpha x_c$ givesFor an arbitrary path write $x=x_c+\eta$ with $\eta (0)=\eta (T)=0$. Integrating the cross term by parts gives $-\int (m{\ddot{x}}_c+\alpha )\eta du=0$. Thus $S[x]=S_c+\int m{\dot{\eta }}^{2}du/2$ exactly. The fluctuation path integral is the free endpoint-fixed one, so its determinant supplies $\sqrt{m/(2\pi i\hslash T)}$. Multiplication by $\theta (T)e^{i{S}_c/\hslash }$ reproduces the previous kernel. This exact classical-path factorization of a quadratic path integral is possible because a linear potential introduces no additional quadratic fluctuation term.

Use $\hslash =1$. Taking the trace of the imaginary-time time-evolution operator identifies the initial and final position, giving the path integralThe measure includes integration over the common endpoint. The paths are real periodic bosonic paths on the imaginary-time circle, not paths with fixed zero endpoints and not only classical trajectories. The kinetic measure is defined by time slicing; typical contributing paths need not be differentiable. For the free expansion assume $m,\kappa >0$. The interacting partition function requires a potential bounded below; otherwise the following formulas have the meaning of a formal perturbation series about the free harmonic oscillator.

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.

Authored UC San Diego lecture notes. Section2.3, printedpp.29–38, discusses superfluid phase dynamics;§§3.1–3.2, printedpp.51–61, construct bosonic and fermionic coherent-state path integrals. These are conceptual sources, not claimed worked answers to this exam.

Riccardo Rattazzi developed a path-integral account of quantum mechanics that connects interference, operator evolution and statistical methods. His notes derive the path integral from quantum time evolution, work out the free-particle propagator and treat Gaussian fluctuations.^[ref]

The weight $e^{iS/\hslash }$, with $S$ the action functional, is an amplitude with a phase, not a classical probability assigned to each path. Combining amplitudes before taking probabilities produces interference. Analytic continuation to Euclidean time then connects suitable quantum calculations with statistical-mechanical formulations, under the required convergence and boundary assumptions.

Richard Feynman formulated quantum mechanics as a sum over possible particle histories. In the path integral approach, each history contributes a probability amplitude with wave phase determined by its classical action, schematically $e^{iS/\hslash }$. Adding these probability amplitudes before taking their squared magnitude retains quantum interference. His 1948 paper establishes agreement with the Schrödinger equation, giving another formulation of the same theory and a powerful way to reorganize calculations.^[ref]

The thermal rotor path integral traces paths closed modulo $2\pi$. Lift them to the line and sum endpoints differing by $2\pi m$. In inverse-energy time $0<\tau <\beta$, the kinetic exponent is $I\int {\dot{\varphi }}^{2}d\tau /(2{\hslash }^2)$. Integer $m$ counts trajectory windings, not angular-momentum eigenstates.

For a Witten index,the path integral of supersymmetric quantum mechanics has periodic fermionic boundary conditions. They differ from the antiperiodic conditions for an ordinary thermal partition function. Under suitable trace and regulator assumptions, positive-energy bosonic and fermionic partners cancel, permitting a short-time evaluation from the surviving zero modes. A symmetry insertion instead imposes the twisted boundary conditions of an equivariant supertrace.

For the supersymmetric oscillator with bosonic energies $E_n^B=n$ and fermionic energies $E_n^F=n+1$ in units with unit oscillator gap,Pairing before taking $\beta \to 0$ preserves the single unpaired ground state. Taking separate short-time limits of the two traces gives divergent quantities and obscures that cancellation. A continuum spectrum or ill-defined trace requires additional analysis; supersymmetry alone does not justify exchanging all limits and integrals.

Reviews of Modern Physics 20, 367–387. The original path integral formulation weights histories by their classical action phase. doi.org/10.1103/RevModPhys.20.367

The vacuum expectation value of a time ordering of quantum field insertions, for example $⟨0|T\{\varphi (x)\varphi (y)\}|0⟩$. The vacuum path integral with the Feynman i-epsilon prescription computes these functions. For fermion fields the ordering includes a minus sign for each odd permutation. Differentiating a time-ordered correlation function can differentiate its ordering step functions and produce Ward identity contact terms.

Reflecting an interval repeatedly across its endpoints maps a reflected path to a free path in an image interval. A return trajectory from $q\in (0,L)$ ends at $q+2rL$ after an even number of wall reflections, or $-q+2rL$ after an odd number. An infinite square well has reflection amplitude $-1$, so the corresponding Dirichlet heat kernel on an interval is the difference of direct and mirrored free kernels. Individual straight paths illustrate the geometry; the path integral sums all time-sliced paths with the same endpoints.

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.

Localizing an infinitesimal symmetry inside a normalized path integral turns current conservation into a distributional identity. The variation of each inserted field contributes a delta function at that insertion. The displayed sign uses $\delta S=-\int ({\partial }_{\mu }{ϵ}^a)j_a^{\mu }$; the opposite current convention reverses the left side. Measure anomalies or noninvariant boundary states can add terms.

A scalar field with no spacetime coordinates reduces its path integral to an ordinary integral. It preserves Wick contractions and Feynman-diagram symmetry factors, making it a useful model of perturbative quantum field theory.