Path-Integral Formulation

The path-integral formulation is a formulation of quantum mechanics in which transition amplitudes are calculated by integrating over possible histories of a system. Each history contributes a complex phase determined by its action, rather than an ordinary probability. For a particle, these histories are paths through space over time; in quantum field theory, they are configurations of fields throughout spacetime. When properly defined for the same quantum system, the formulation reproduces the predictions of operator-based quantum mechanics while providing a particularly direct connection with classical dynamics. (journals.aps.org)

Historical development

The approach grew from the connection between the classical Lagrangian and quantum evolution. Paul Dirac’s work on this connection inspired Richard Feynman to investigate short-time transition amplitudes proportional to an exponential of the action. In his 1965 Nobel lecture, Feynman described how this construction led him to the Schrödinger equation and, through repeated composition of short-time amplitudes, to a sum over histories. (nobelprize.org)

Feynman’s paper Space-Time Approach to Non-Relativistic Quantum Mechanics, published on April 1, 1948, presented the formulation systematically, established its equivalence to familiar quantum mechanics, and discussed applications to quantum electrodynamics. (journals.aps.org)

Transition amplitudes and the action

For a particle moving in one dimension, the propagator $K(x_f,t_f;x_i,t_i)$ is the amplitude for evolution between specified initial and final positions. For a time-independent Hamiltonian operator $\hat H$ ,

K(x_f,t_f;x_i,t_i) = \langle x_f|e^{-i\hat H(t_f-t_i)/\hbar}|x_i\rangle .

It evolves a wave function according to

\psi(x_f,t_f) = \int dx_i\, K(x_f,t_f;x_i,t_i)\psi(x_i,t_i).

Here $\hbar=h/(2\pi)$ , where $h$ is the Planck constant. (ocw.mit.edu)

For the Hamiltonian $\hat H=\hat p^2/(2m)+V(\hat x)$ , the corresponding path integral is conventionally written

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},

with

S[x] = \int_{t_i}^{t_f}dt\, \left[\frac{m}{2}\dot x^{\,2}-V(x)\right].

The action is a functional: it assigns a number to an entire path. The integration includes histories that do not satisfy the classical equations of motion. The notation $\mathcal D x$ denotes functional integration, whose normalization and limiting prescription must be specified. (mitocw.ups.edu.ec)

The contributions are amplitudes, so alternatives combine through interference. Schematically, for alternatives with amplitudes $A_a$ ,

P=\left|\sum_a A_a\right|^2,

not $\sum_a|A_a|^2$ . The path integral therefore does not describe an ordinary random selection among classical trajectories. Its sum-over-histories language specifies a calculation of quantum amplitudes, not a requirement that an unmeasured particle possess a definite classical path. (journals.aps.org)

Time slicing and equivalence to operator mechanics

A standard construction divides the time interval into $N$ steps of length $\epsilon=(t_f-t_i)/N$ . Complete sets of position states are inserted between successive evolution operators. Splitting the kinetic and potential operators at short times and evaluating the momentum integrals gives

\begin{aligned} K={}&\lim_{N\to\infty} \left(\frac{m}{2\pi i\hbar\epsilon}\right)^{N/2} \int\prod_{j=1}^{N-1}dx_j\\ &\quad\times \exp\left[ \frac{i}{\hbar} \sum_{j=0}^{N-1} \left( \frac{m(x_{j+1}-x_j)^2}{2\epsilon} -\epsilon V(x_j) \right) \right], \end{aligned}

where $x_0=x_i$ and $x_N=x_f$ . This expression applies to the stated kinetic-plus-potential Hamiltonian, with appropriate assumptions and a convergence prescription. The normalization is essential; the continuum notation is not merely an unweighted sum of paths. (mitocw.ups.edu.ec)

Because the construction begins with the evolution operator, its propagator obeys the Schrödinger equation and the composition law

K(x_f,t_f;x_i,t_i) = \int dx\, K(x_f,t_f;x,t)K(x,t;x_i,t_i).

For a free particle and $\Delta t=t_f-t_i>0$ , the exact result is

K_0 = \sqrt{\frac{m}{2\pi i\hbar\Delta t}}\, \exp\left[ \frac{im(x_f-x_i)^2}{2\hbar\Delta t} \right].

The exponent contains the action of the straight classical path, while the prefactor incorporates the integration over fluctuations. (ocw.mit.edu)

Classical limit and semiclassical approximation

The connection with classical mechanics follows from the stationary-phase approximation. When the action varies on scales large compared with $\hbar$ , rapidly varying phases tend to cancel. Neighborhoods of stationary histories can make the leading contributions:

\delta S[x_{\mathrm{cl}}]=0.

For a differentiable Lagrangian, this condition gives the Euler–Lagrange equations. “Least action” is a conventional name: the relevant action need only be stationary, not necessarily minimal. (ocw.mit.edu)

A semiclassical calculation expands $x=x_{\mathrm{cl}}+\eta$ and retains quadratic fluctuations in $\eta$ . Multiple classical histories may contribute and interfere. Thus the classical limit is not obtained simply by discarding every nonclassical path; even the leading approximation generally includes fluctuation factors and may involve several stationary histories. (ocw.mit.edu)

Imaginary time and statistical mechanics

Under a suitable analytic continuation $t=-i\tau$ , often called a Wick rotation, the oscillatory weight becomes

e^{iS/\hbar}\longrightarrow e^{-S_E/\hbar},

where, for the particle above,

S_E[x] = \int d\tau\, \left[ \frac{m}{2}\left(\frac{dx}{d\tau}\right)^2+V(x) \right].

This Euclidean path integral connects quantum evolution with statistical mechanics. At temperature $T$ , setting $\beta=1/(k_BT)$ , the partition function is

Z=\operatorname{Tr}e^{-\beta\hat H} = \int_{x(0)=x(\beta\hbar)} \mathcal D x\,e^{-S_E[x]/\hbar}.

Taking the trace closes the particle’s imaginary-time path, giving an interval of length $\beta\hbar$ . (web.mit.edu)

For suitable Hamiltonians, Euclidean evolution has a rigorous probabilistic representation through the Feynman–Kac formula, using measures associated with Brownian motion. Typical paths in this representation are not differentiable; the displayed kinetic-action expression must therefore be understood through its limiting or measure-based construction, rather than ordinary differentiation along each path. (arxiv.org)

Quantum fields and different integration variables

For a scalar field $\phi(x)$ , the histories are whole field configurations. Introducing an external source $J(x)$ gives a generating functional of the form

Z[J] = \mathcal N\int\mathcal D\phi\, \exp\left[ \frac{i}{\hbar} \left( S[\phi]+\int d^d x\,J(x)\phi(x) \right) \right],

with vacuum boundary conditions and a suitable convergence prescription understood. Functional derivatives with respect to $J$ generate time-ordered correlation functions. Expanding the interaction part of the action yields perturbation theory, whose terms can be organized as Feynman diagrams. (cambridge.org)

Not every path integral integrates ordinary position coordinates. Phase-space representations integrate positions and momenta and must account for operator ordering. Path integrals for fermions conventionally use Grassmann variables, whose anticommutation encodes fermionic algebra. Their integration is algebraic, not integration against an ordinary probability distribution. (arxiv.org)

Mathematical and computational limitations

The continuum notation conceals substantial mathematical issues. Real-time weights are oscillatory rather than positive, and a formal $\mathcal D x$ does not automatically define a countably additive measure. Time slicing, operator ordering, boundary conditions, and convergence prescriptions are part of the definition. A classical action alone is not always enough to specify a unique quantum theory, particularly for more general Hamiltonians. (arxiv.org)

Imaginary-time representations can support Monte Carlo calculations when the resulting weights permit probabilistic sampling. Negative or complex weights create the sign problem: large cancellations can make statistical estimates exponentially costly. Troyer and Wiese established NP-hardness for a general class of fermionic sign problems. This result concerns a generic solution; it does not rule out efficient methods for particular models or special parameter regimes. (arxiv.org)

References

Space-Time Approach to Non-Relativistic Quantum Mechanicsjournals.aps.org
Richard P. Feynman – Nobel Lecturenobelprize.org
Quantum Theory I, Lecture 10 Notesocw.mit.edu
Quantum Theory I, Lecture 11 Notesmitocw.ups.edu.ec
Quantum Theory I, Lecture 12 Notesocw.mit.edu
MKM_NNPSS_2022_MIT_Lecture1web.mit.edu
Wiener Integration for Quantum Systems: A Unified Approach to the Feynman-Kac formulaarxiv.org
Some practice with the path integral in field theorycambridge.org
PHYS4181 Particle Physics - Phenomenologyippp.dur.ac.uk