The Feynman Path Integral

Derivation

The Feynman Path Integral

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Statement

For a non-relativistic particle with Hamiltonian \( \hat{H} = \frac{\hat{p}^2}{2m} + V(\hat{x}) \), the position-space propagator \( K(x_f,t_f;x_i,t_i) = \langle x_f | e^{-i\hat{H}(t_f-t_i)/\hbar} | x_i \rangle \) equals a sum over all paths \( x(t) \) joining \( x_i \) at \( t_i \) to \( x_f \) at \( t_f \), each weighted by \( e^{iS[x]/\hbar} \) where \( S[x]=\int_{t_i}^{t_f} L\,dt \) is the classical action; in the limit \( \hbar \to 0 \) the sum is dominated by the stationary-action path, recovering classical mechanics.

Why it matters

The path integral gives a formulation of quantum mechanics built entirely from the classical Lagrangian and action, without ever writing an operator equation. It makes the classical limit transparent: the phase \( S/\hbar \) becomes enormous and rapidly oscillating as \( \hbar \to 0 \), so only paths near \( \delta S = 0 \) survive, and Hamilton's principle emerges as the ridge of constructive interference.

Beyond pedagogy, it is the working language of quantum field theory, statistical mechanics (via the Euclidean rotation \( t \to -i\tau \)), instanton and tunnelling calculations, and gauge theory. Symmetries of the action map directly onto Ward identities, and the measure encodes anomalies.

Assumptions

Time-independent Hamiltonian of the form \( \hat{H}=\hat{T}(\hat{p})+\hat{V}(\hat{x}) \).If \( \hat{H} \) depends on time, the single exponential \( e^{-i\hat{H}T/\hbar} \) must be replaced by a time-ordered product; if kinetic and potential parts do not separate additively the Gaussian momentum integral below no longer closes in elementary form.

The Trotter product formula converges: \( e^{-i(\hat{T}+\hat{V})\varepsilon/\hbar} = e^{-i\hat{V}\varepsilon/\hbar}e^{-i\hat{T}\varepsilon/\hbar} + O(\varepsilon^2) \).If \( \hat{T} \) and \( \hat{V} \) cannot be split with controlled error (e.g. unbounded-below or badly singular potentials), the slice-by-slice factorisation is not justified and the limit \( N\to\infty \) need not reproduce \( e^{-i\hat{H}T/\hbar} \). Rigour: for self-adjoint semibounded \( \hat{H} \) the strong-limit form (Trotter–Kato) holds; the naive product can fail for potentials without a common core.

The oscillatory Gaussian \( \int dp\, e^{-i a p^2} \) is defined by analytic continuation \( a \to a - i0^+ \).If the \( i\epsilon \) prescription (equivalently, a small negative imaginary part of the time, or Euclidean rotation) is dropped, the momentum integral is only conditionally convergent and the propagator's phase and prefactor become ambiguous.

Cartesian coordinates and a flat measure \( \prod_j dx_j \).In curvilinear coordinates or on a curved manifold the naive product measure acquires ordering/Jacobian corrections (a \( \frac{\hbar^2}{8m}R \)-type term), so the phase is no longer simply \( S[x]/\hbar \).

Derivation

\[ K = \langle x_f | e^{-i\hat{H}T/\hbar} | x_i \rangle, \qquad T \equiv t_f - t_i \]

Definition of the propagator as the position matrix element of the evolution operator (prior result: time-evolution operator). A

\[ e^{-i\hat{H}T/\hbar} = \left( e^{-i\hat{H}\varepsilon/\hbar} \right)^{N}, \qquad \varepsilon = \frac{T}{N} \]

Split evolution into \( N \) equal slices; legal because \( \hat{H} \) is time-independent so the exponents commute and add. A

\[ K = \int \left( \prod_{j=1}^{N-1} dx_j \right) \prod_{j=0}^{N-1} \langle x_{j+1} | e^{-i\hat{H}\varepsilon/\hbar} | x_j \rangle \]

Insert a resolution of identity \( \int dx_j\,|x_j\rangle\langle x_j| = \hat{1} \) between each pair of factors, with \( x_0=x_i,\ x_N=x_f \). A

\[ \langle x_{j+1} | e^{-i\hat{H}\varepsilon/\hbar} | x_j \rangle = \langle x_{j+1} | e^{-i\hat{V}(\hat{x})\varepsilon/\hbar} e^{-i\hat{p}^2\varepsilon/2m\hbar} | x_j \rangle + O(\varepsilon^2) \]

Trotter split of the short-time exponential; the error is \( O(\varepsilon^2) \) per slice because \( [\hat{V},\hat{T}]\varepsilon^2 \) is the leading correction. B

\[ e^{-i\hat{V}(\hat{x})\varepsilon/\hbar}|x_{j+1}\rangle\ \text{acts left as}\ e^{-iV(x_{j+1})\varepsilon/\hbar}\langle x_{j+1}| \]

\( \hat{x} \) is diagonal in the position basis, so \( V(\hat{x}) \) acts by its eigenvalue \( V(x_{j+1}) \); a c-number can be pulled out of the matrix element. B

\[ \langle x_{j+1} | e^{-i\hat{p}^2\varepsilon/2m\hbar} | x_j \rangle = \int \frac{dp}{2\pi\hbar}\, e^{ip(x_{j+1}-x_j)/\hbar}\, e^{-ip^2\varepsilon/2m\hbar} \]

Insert momentum completeness \( \int \frac{dp}{2\pi\hbar}|p\rangle\langle p| = \hat{1} \) with \( \langle x|p\rangle = e^{ipx/\hbar} \); \( \hat{p}^2 \) is now diagonal. B

\[ \int \frac{dp}{2\pi\hbar}\, e^{-i\varepsilon p^2/2m\hbar}\, e^{ip\,\Delta x_j/\hbar} = \left( \frac{m}{2\pi i \hbar \varepsilon} \right)^{1/2} \exp\!\left[ \frac{i m (\Delta x_j)^2}{2\hbar\varepsilon} \right] \]

Complete the square and evaluate the Fresnel (oscillatory Gaussian) integral with \( \varepsilon \to \varepsilon - i0^+ \); here \( \Delta x_j \equiv x_{j+1}-x_j \). C

\[ \langle x_{j+1} | e^{-i\hat{H}\varepsilon/\hbar} | x_j \rangle = \left( \frac{m}{2\pi i \hbar \varepsilon} \right)^{1/2} \exp\!\left\{ \frac{i\varepsilon}{\hbar}\left[ \frac{m}{2}\left(\frac{\Delta x_j}{\varepsilon}\right)^2 - V(x_{j+1}) \right] \right\} \]

Combine the kinetic Gaussian (Step 7) with the potential phase (Step 5); the bracket is the discrete Lagrangian evaluated on slice \( j \). B

\[ K = \lim_{N\to\infty} \left( \frac{m}{2\pi i \hbar \varepsilon} \right)^{N/2} \int \left(\prod_{j=1}^{N-1} dx_j\right) \exp\!\left\{ \frac{i}{\hbar} \sum_{j=0}^{N-1} \varepsilon\left[ \frac{m}{2}\left(\frac{\Delta x_j}{\varepsilon}\right)^2 - V(x_{j+1}) \right] \right\} \]

Multiply the \( N \) short-time kernels and collect the prefactors and the sum in a single exponent. B

\[ \sum_{j=0}^{N-1} \varepsilon\left[ \frac{m}{2}\left(\frac{\Delta x_j}{\varepsilon}\right)^2 - V(x_{j+1}) \right] \xrightarrow[N\to\infty]{} \int_{t_i}^{t_f} \left[ \frac{m}{2}\dot{x}^2 - V(x) \right] dt = S[x] \]

As \( \varepsilon\to 0 \), \( \Delta x_j/\varepsilon \to \dot{x} \) and the Riemann sum tends to the time integral of the Lagrangian, i.e. the classical action functional. C

\[ K(x_f,t_f;x_i,t_i) = \int_{x(t_i)=x_i}^{x(t_f)=x_f} \mathcal{D}x\; e^{\,iS[x]/\hbar}, \qquad \mathcal{D}x \equiv \lim_{N\to\infty}\left(\frac{m}{2\pi i\hbar\varepsilon}\right)^{N/2}\prod_{j=1}^{N-1}dx_j \]

Absorb the measure and limit into the formal symbol \( \mathcal{D}x \); this is the Feynman sum over paths. B

\[ K \approx \sum_{\text{cl}} A_{\text{cl}}\, e^{\,iS_{\text{cl}}/\hbar}, \qquad \left.\frac{\delta S}{\delta x(t)}\right|_{x_{\text{cl}}} = 0 \ \Longleftrightarrow\ \frac{d}{dt}\frac{\partial L}{\partial \dot{x}} - \frac{\partial L}{\partial x}=0 \]

Stationary-phase evaluation as \( \hbar\to 0 \): off-stationary paths interfere destructively, leaving the stationary-action (classical) paths (prior result: Hamilton's principle). C

Result

\[ K(x_f,t_f;x_i,t_i) = \int_{x_i}^{x_f} \mathcal{D}x\; \exp\!\left( \frac{i}{\hbar}\int_{t_i}^{t_f} L(x,\dot{x})\,dt \right) \;\xrightarrow[\hbar\to 0]{}\; A\,e^{\,iS_{\text{cl}}/\hbar} \]

Reading. The amplitude to go from \( (x_i,t_i) \) to \( (x_f,t_f) \) is obtained by giving every conceivable trajectory the same modulus but a phase equal to its action in units of \( \hbar \), then adding. Neighbouring paths near an extremum of \( S \) share almost the same phase and add constructively; far from an extremum they cancel. As \( \hbar\to 0 \) only the classical path survives, and Hamilton's principle \( \delta S = 0 \) is the statement that this path is where the phases line up.

Units check. The exponent must be dimensionless: \( [S]=[\,\text{energy}\times\text{time}\,]=\mathrm{J\,s} \) and \( [\hbar]=\mathrm{J\,s} \), so \( S/\hbar \) is a pure number. In one dimension \( K \) must carry the dimension of \( \delta(x) \), i.e. \( \mathrm{length}^{-1} \), since \( K\to\delta(x_f-x_i) \) as \( T\to 0 \). The prefactor \( \left(\frac{m}{2\pi\hbar\varepsilon}\right)^{1/2} \) has dimension \( \left[\frac{\mathrm{kg}}{\mathrm{J\,s}\cdot\mathrm{s}}\right]^{1/2}=\left[\frac{\mathrm{kg}}{\mathrm{kg\,m^2}}\right]^{1/2}=\mathrm{m}^{-1} \). Consistent.

Limiting cases

Free particle \( (V=0) \): the path integral is a product of Gaussians and gives \( K=\left(\frac{m}{2\pi i\hbar T}\right)^{1/2}\exp\!\left[\frac{im(x_f-x_i)^2}{2\hbar T}\right] \), with \( S_{\text{cl}}=\frac{m(x_f-x_i)^2}{2T} \).
Classical limit \( (\hbar\to 0) \): stationary phase collapses the sum onto \( x_{\text{cl}} \); Newton's second law and Hamilton's principle re-emerge.
Quadratic actions (free particle, harmonic oscillator, constant force): the fluctuation integral is exactly Gaussian, so \( K=A(t)\,e^{iS_{\text{cl}}/\hbar} \) is exact, not merely semiclassical (Van Vleck formula is exact).
Short time \( (T\to 0) \): the phase in the kinetic term dominates and \( K\to\delta(x_f-x_i) \), recovering the initial condition of the propagator.
Euclidean rotation \( (t\to -i\tau) \): \( iS/\hbar \to -S_E/\hbar \) gives a real, convergent weight \( e^{-S_E/\hbar} \) — the bridge to statistical mechanics and the partition function \( Z=\oint \mathcal{D}x\,e^{-S_E/\hbar} \).

Breaks when

Non-separable or velocity-dependent Hamiltonians (magnetic fields, \( \hat{p}\hat{x} \) cross terms, relativistic \( \sqrt{p^2c^2+m^2c^4} \)): the Trotter split of Step 4 and the elementary momentum Gaussian of Step 7 fail, and operator-ordering ambiguities enter the discrete action. A midpoint prescription is then mandatory.
Curved configuration space or curvilinear coordinates: the flat product measure \( \prod dx_j \) is wrong; a metric-dependent measure plus a \( \frac{\hbar^2}{8m}R \) quantum potential appears, so the weight is no longer just \( e^{iS/\hbar} \).
Singular or unbounded-below potentials (attractive \( 1/r^2 \), \( -x^4 \)): the Trotter limit may not converge to \( e^{-i\hat{H}T/\hbar} \), and the naive continuum action must be regularised.
Spin and Grassmann (fermionic) degrees of freedom: there is no position-basis path integral of this bosonic form; one needs coherent-state or Grassmann-valued paths instead.

Failure modes

Dropping the measure prefactor. Students quote \( K=\int\mathcal{D}x\,e^{iS/\hbar} \) but forget the \( (m/2\pi i\hbar\varepsilon)^{N/2} \) normalisation, then get a dimensionally wrong or divergent free-particle propagator.
Treating the paths as differentiable. The dominant contributing paths are nowhere-differentiable (like Brownian trajectories); assuming \( \dot{x} \) exists path-by-path leads to wrong scaling — one must keep \( \Delta x_j \sim \sqrt{\varepsilon} \), not \( \Delta x_j \sim \varepsilon \).
Confusing "least action" with "stationary action". The classical path extremises, not minimises, \( S \); past a conjugate (focal) point it is a saddle, and each such point contributes a Maslov phase \( e^{-i\pi/2} \) that students omit.
Ignoring the \( i\epsilon \) / \( i0^+ \) prescription. Evaluating \( \int dp\,e^{-iap^2} \) as if convergent gives a wrong sign or missing factor of \( i \) in the prefactor.
Ordering \( V(x_j) \) vs \( V(x_{j+1}) \) carelessly. For separable \( V \) the endpoint choice is \( O(\varepsilon) \) and washes out, but for velocity-dependent couplings the midpoint value \( V\!\left(\tfrac{x_j+x_{j+1}}{2}\right) \) is required for a Hermitian \( \hat{H} \).

Discussion

The path integral reorganises quantum mechanics around interference of histories rather than evolution of a state vector. Every equation of the operator formalism can be recovered from it: differentiating \( K \) with respect to \( t_f \) reproduces the Schrödinger equation, insertion of \( x(t) \) into the measure generates correlation functions, and the composition law \( K(3,1)=\int dx_2\, K(3,2)K(2,1) \) is nothing but completeness of position states re-expressed as gluing of paths. It is the same physics viewed from the Lagrangian rather than the Hamiltonian side.

The classical limit is its most illuminating feature. Because the weight is a pure phase \( e^{iS/\hbar} \) of unit modulus, no single path is "more probable"; classicality is an emergent interference effect. Where \( S \) changes by many multiples of \( \hbar \) between neighbouring paths, contributions swirl through all phases and cancel; only in the immediate neighbourhood of \( \delta S=0 \) do phases add. Thus Hamilton's principle is not an independent postulate but the condition for constructive interference, and the width of the surviving tube of paths scales as \( \sqrt{\hbar} \), quantifying quantum fluctuations about the classical trajectory.

The connection to the "waves" and "chance" threads is direct. The single-slit and double-slit amplitudes are literally \( \sum_{\text{paths}} e^{iS/\hbar} \) over the geometric paths through each aperture, and Born's rule \( P=|K|^2 \) turns the interfering amplitudes into probabilities. The "energy" thread enters through the action \( S=\int(T-V)dt \) and through the Euclidean rotation, which converts the oscillatory quantum weight into the Boltzmann weight \( e^{-\beta H} \) of statistical mechanics upon identifying imaginary-time period with inverse temperature.

At the level of field theory the same construction, promoted to functional integration over field configurations \( \phi(x) \), gives the generating functional \( Z[J]=\int\mathcal{D}\phi\,e^{i(S[\phi]+\int J\phi)/\hbar} \), from which Feynman diagrams arise as the perturbative expansion of \( \log Z \). The measure \( \mathcal{D}\phi \) is where subtle quantum effects live: its non-invariance under a classical symmetry is precisely an anomaly (Fujikawa's derivation of the chiral anomaly), and gauge redundancy must be removed by Faddeev–Popov determinants. None of this is visible in the naive continuum symbol \( \int\mathcal{D}x\,e^{iS/\hbar} \); it is encoded in the careful \( N\to\infty \) limit derived above.

Common misconceptions. The path integral does not say the particle "takes all paths simultaneously" as a physical fact — it is a computational rule for an amplitude; only \( |K|^2 \) is measurable. Also, the continuum expression \( \int\mathcal{D}x\,e^{iS/\hbar} \) is shorthand: the object with mathematical meaning is the time-sliced limit, and manipulations that ignore the measure or the \( i\epsilon \) prescription can give nonsense.

Worked examples

Free-particle propagator amplitude for an electron

Compute \( K \) and the classical action for an electron travelling \( \Delta x = x_f-x_i \) in time \( T \), with \( V=0 \). B

\[ S_{\text{cl}} = \frac{m\,(\Delta x)^2}{2T}, \qquad K = \left(\frac{m}{2\pi i\hbar T}\right)^{1/2} e^{\,iS_{\text{cl}}/\hbar} \]

Symbolic result: quadratic action, so the Gaussian path integral is exact. B

\[ m = 9.11\times10^{-31}\,\mathrm{kg},\quad \Delta x = 1.0\times10^{-9}\,\mathrm{m},\quad T = 1.0\times10^{-15}\,\mathrm{s} \]

Insert numbers (nanometre in a femtosecond, atomic-scale motion). A

\[ S_{\text{cl}} = \frac{(9.11\times10^{-31})(1.0\times10^{-9})^2}{2(1.0\times10^{-15})} = 4.56\times10^{-40}\,\mathrm{J\,s} \]

Numerator \( 9.11\times10^{-49} \), divide by \( 2\times10^{-15} \). A

\[ \frac{S_{\text{cl}}}{\hbar} = \frac{4.56\times10^{-40}}{1.055\times10^{-34}} = 4.3\times10^{-6}\ \text{rad} \]

Phase is tiny: this electron is deep in the quantum regime, far from many-\( \hbar \) classical phases. B

\[ K = \left(\frac{9.11\times10^{-31}}{2\pi i(1.055\times10^{-34})(10^{-15})}\right)^{1/2} e^{\,i(4.3\times10^{-6})} \approx 3.7\times10^{9}\,e^{-i\pi/4}\,\mathrm{m}^{-1/2}\cdot\mathrm{m}^{-1/2} \]

Reading. The prefactor magnitude is \( |m/2\pi\hbar T|^{1/2}\approx 3.7\times10^{9}\,\mathrm{m}^{-1} \); the \( i^{-1/2}=e^{-i\pi/4} \) supplies the universal free-particle phase, and the classical action contributes a negligible extra phase. Because \( S_{\text{cl}}/\hbar \ll 1 \), the trajectory is not classically sharp — quantum spreading dominates on these scales.

Units check. \( S_{\text{cl}} \) in \( \mathrm{J\,s} \), \( S_{\text{cl}}/\hbar \) dimensionless, \( K \) in \( \mathrm{m}^{-1} \) (1D). Consistent.

Classical action of a macroscopic ball, showing \( \hbar\to 0 \) dominance

Compute \( S_{\text{cl}}/\hbar \) for a \( 0.10\,\mathrm{kg} \) ball moving \( 1.0\,\mathrm{m} \) in \( 0.50\,\mathrm{s} \) under gravity, to show why only the classical path counts. B

\[ S_{\text{cl}} = \int_0^T \left(\tfrac{1}{2}m\dot{x}^2 - mgx\right)dt \sim \tfrac{1}{2}m\bar{v}^2\,T, \qquad \bar{v} = \frac{\Delta x}{T} \]

Order-of-magnitude: the kinetic term sets the scale of the action. B

\[ m = 0.10\,\mathrm{kg},\quad \Delta x = 1.0\,\mathrm{m},\quad T = 0.50\,\mathrm{s}\ \Rightarrow\ \bar{v}=2.0\,\mathrm{m\,s^{-1}} \]

Insert numbers. A

\[ S_{\text{cl}} \sim \tfrac{1}{2}(0.10)(2.0)^2(0.50) = 0.10\,\mathrm{J\,s} \]

Evaluate the kinetic scale of the action. A

\[ \frac{S_{\text{cl}}}{\hbar} = \frac{0.10}{1.055\times10^{-34}} \approx 9.5\times10^{32}\ \text{rad} \]

A path deviation changing \( S \) by even \( 10^{-33}\,\mathrm{J\,s} \) shifts the phase by \( \sim\!10\,\mathrm{rad} \), so neighbouring non-classical paths cancel. C

\[ \frac{S_{\text{cl}}}{\hbar} \approx 9.5\times10^{32} \gg 1 \]

Reading. The action is \( 32 \) orders of magnitude larger than \( \hbar \). Any path departing from \( x_{\text{cl}} \) by a macroscopically tiny amount already changes the phase by many full turns, so all such paths interfere destructively. Stationary phase is overwhelmingly sharp: the ball follows its Newtonian trajectory, and the quantum tube of width \( \sim\!\sqrt{\hbar} \) is utterly unmeasurable.

Units check. \( S_{\text{cl}} \) in \( \mathrm{J\,s} \), ratio to \( \hbar \) dimensionless. Consistent with the \( \hbar\to 0 \) classical-limit statement.

Problems

Starting from the sliced form in Step 9 with \( V=0 \), show by induction on \( N \) that the free-particle propagator is \( K=\left(\frac{m}{2\pi i\hbar T}\right)^{1/2}\exp\!\left[\frac{im(\Delta x)^2}{2\hbar T}\right] \).
Solution
Combine two adjacent free slices: \( \int dx_1 \left(\frac{m}{2\pi i\hbar\varepsilon}\right)\exp\!\left[\frac{im}{2\hbar\varepsilon}\big((x_2-x_1)^2+(x_1-x_0)^2\big)\right] \). The exponent is quadratic in \( x_1 \); completing the square gives a Gaussian whose integral is \( \left(\frac{m}{2\pi i\hbar\varepsilon}\right)^{1/2}\left(\frac{2\pi i\hbar\varepsilon}{m}\cdot\frac{1}{2}\right)^{1/2}\times(\text{residual}) \). Carrying it out yields \( \left(\frac{m}{2\pi i\hbar\,2\varepsilon}\right)^{1/2}\exp\!\left[\frac{im(x_2-x_0)^2}{2\hbar(2\varepsilon)}\right] \): one slice of double duration. By induction \( n \) slices give total duration \( n\varepsilon \), so after \( N \) slices \( K=\left(\frac{m}{2\pi i\hbar N\varepsilon}\right)^{1/2}\exp\!\left[\frac{im(\Delta x)^2}{2\hbar N\varepsilon}\right] \). With \( N\varepsilon=T \) this is the stated result — and it is \( N \)-independent, so the limit is trivial.
For the free particle, verify explicitly that \( K\to\delta(x_f-x_i) \) as \( T\to 0^+ \).
Solution
Write \( K=\left(\frac{m}{2\pi i\hbar T}\right)^{1/2}e^{im(\Delta x)^2/2\hbar T} \). This is a nascent-delta Gaussian of the form \( \frac{1}{\sqrt{2\pi\sigma^2}}e^{-(\Delta x)^2/2\sigma^2} \) with imaginary variance \( \sigma^2 = i\hbar T/m \). As \( T\to 0 \), \( |\sigma|\to 0 \), the phase oscillates infinitely fast for any \( \Delta x\neq 0 \) (contributions cancel), while the normalisation \( \int K\,d(\Delta x)=1 \) is preserved (Fresnel integral). Hence \( K\to\delta(\Delta x) \), which is exactly the initial condition \( \langle x_f|x_i\rangle=\delta(x_f-x_i) \) that the evolution operator must satisfy at \( T=0 \).
A particle of mass \( m=1.0\times10^{-26}\,\mathrm{kg} \) travels \( \Delta x = 5.0\times10^{-8}\,\mathrm{m} \) in \( T=2.0\times10^{-12}\,\mathrm{s} \) with \( V=0 \). Compute \( S_{\text{cl}} \) and \( S_{\text{cl}}/\hbar \), and state whether the motion is quantum or classical.
Solution
\( S_{\text{cl}}=\frac{m(\Delta x)^2}{2T}=\frac{(1.0\times10^{-26})(5.0\times10^{-8})^2}{2(2.0\times10^{-12})} \). Numerator: \( (1.0\times10^{-26})(2.5\times10^{-15})=2.5\times10^{-41} \). Divide by \( 4.0\times10^{-12} \): \( S_{\text{cl}}=6.25\times10^{-30}\,\mathrm{J\,s} \). Then \( S_{\text{cl}}/\hbar = 6.25\times10^{-30}/1.055\times10^{-34}\approx 5.9\times10^{4} \). Since \( S_{\text{cl}}/\hbar \gg 1 \), the phase spans tens of thousands of radians and the motion is effectively classical, though far less extreme than a macroscopic body.
Show that the Euclidean rotation \( t\to -i\tau \) turns the weight \( e^{iS/\hbar} \) into \( e^{-S_E/\hbar} \) with \( S_E=\int d\tau\left[\frac{m}{2}\left(\frac{dx}{d\tau}\right)^2+V(x)\right] \), and hence that the harmonic-oscillator partition function is \( Z=\oint\mathcal{D}x\,e^{-S_E/\hbar} \) over \( \tau \)-periodic paths.
Solution
Under \( t=-i\tau \), \( dt=-i\,d\tau \) and \( \dot{x}=dx/dt=i\,dx/d\tau \). The action \( iS/\hbar=\frac{i}{\hbar}\int dt\left(\frac{m}{2}\dot{x}^2-V\right) \) becomes \( \frac{i}{\hbar}\int(-i\,d\tau)\left(\frac{m}{2}(i x')^2 - V\right)=\frac{1}{\hbar}\int d\tau\left(-\frac{m}{2}x'^2 - V\right)=-\frac{1}{\hbar}S_E \), where \( x'=dx/d\tau \) and \( S_E=\int d\tau\left(\frac{m}{2}x'^2+V\right) \) is positive (a genuine energy). The weight is now the real, damped \( e^{-S_E/\hbar} \). Identifying imaginary time with inverse temperature via period \( \beta\hbar \) (so \( x(\tau)=x(\tau+\beta\hbar) \)) gives the quantum-statistical trace \( Z=\mathrm{Tr}\,e^{-\beta\hat{H}}=\oint\mathcal{D}x\,e^{-S_E/\hbar} \) over periodic paths — the Feynman–Kac bridge to statistical mechanics.
For a general Lagrangian expand a path as \( x(t)=x_{\text{cl}}(t)+\eta(t) \) with \( \eta(t_i)=\eta(t_f)=0 \). Show that to second order the propagator factorises as \( K=e^{iS_{\text{cl}}/\hbar}\,F(t_f,t_i) \), and identify what \( F \) depends on.
Solution
Expand \( S[x_{\text{cl}}+\eta]=S_{\text{cl}}+\left.\frac{\delta S}{\delta x}\right|_{\text{cl}}\!\cdot\eta+\frac{1}{2}\left.\frac{\delta^2 S}{\delta x^2}\right|_{\text{cl}}\!\cdot\eta^2+\dots \) The linear term vanishes because \( x_{\text{cl}} \) satisfies \( \delta S=0 \) (Euler–Lagrange). Substituting into \( K=\int\mathcal{D}x\,e^{iS/\hbar} \) and changing integration variable from \( x \) to \( \eta \) (a shift, Jacobian 1, same zero boundary conditions), \( K=e^{iS_{\text{cl}}/\hbar}\int\mathcal{D}\eta\,\exp\!\left[\frac{i}{2\hbar}\int\!\!\int \eta\,\frac{\delta^2 S}{\delta x^2}\Big|_{\text{cl}}\eta\right] \). The remaining Gaussian fluctuation integral \( F \) depends only on the second variation of the action evaluated on the classical path — i.e. on \( m \), on \( V''(x_{\text{cl}}(t)) \), and on the endpoints \( t_i,t_f \) — but not on \( x_i,x_f \) except through \( x_{\text{cl}} \). For quadratic \( V \) the expansion truncates exactly, giving the exact Van Vleck result \( K=F(t_f-t_i)\,e^{iS_{\text{cl}}/\hbar} \); for general \( V \) it is the leading semiclassical (WKB) approximation, with \( F \) related to \( \left(\frac{\partial^2 S_{\text{cl}}}{\partial x_f\partial x_i}\right)^{1/2} \).