Path integral formulation

The path integral formulation is a description in <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a> that generalizes the <a href="/facts/Stationary_action_principle/QaBTpoKz">stationary action principle</a> of <a href="/facts/Classical_mechanics/ykG8upsN">classical mechanics</a>. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or <a href="/facts/Functional_integral/VBGTnDzq">functional integral</a>, over an infinity of quantum-mechanically possible trajectories to compute a <a href="/facts/Probability_amplitude/ryPgJvs8">quantum amplitude</a>.
This formulation has proven crucial to the subsequent development of <a href="/facts/Theoretical_physics/Cs2Y12rQ">theoretical physics</a>, because manifest <a href="/facts/Lorentz_covariance/K5AEuGKg">Lorentz covariance</a> (time and space components of quantities enter equations in the same way) is easier to achieve than in the operator formalism of <a href="/facts/Canonical_quantization/AKHX2JQn">canonical quantization</a>. Unlike previous methods, the path integral allows one to easily change <a href="/facts/Coordinates/Mtd0q4SP">coordinates</a> between very different <a href="/facts/Canonical_coordinates/9g2SFFRz">canonical</a> descriptions of the same quantum system. Another advantage is that it is in practice easier to guess the correct form of the <a href="/facts/Lagrangian_(field_theory)/p9BCUz57">Lagrangian</a> of a theory, which naturally enters the path integrals (for interactions of a certain type, these are coordinate space or Feynman path integrals), than the <a href="/facts/Hamiltonian_(quantum_mechanics)/5XwK2HmD">Hamiltonian</a>. Possible downsides of the approach include that <a href="/facts/Unitarity/x2gXDAsS">unitarity</a> (this is related to conservation of probability; the probabilities of all physically possible outcomes must add up to one) of the <a href="/facts/S-matrix/NffJvlpp">S-matrix</a> is obscure in the formulation. The path-integral approach has proven to be equivalent to the other formalisms of quantum mechanics and quantum field theory. Thus, by deriving either approach from the other, problems associated with one or the other approach (as exemplified by Lorentz covariance or unitarity) go away.
The path integral also relates quantum and <a href="/facts/Stochastic/uMbgL94r">stochastic</a> processes, and this provided the basis for the grand synthesis of the 1970s, which unified <a href="/facts/Quantum_field_theory/VoewsNJV">quantum field theory</a> with the <a href="/facts/Statistical_field_theory/EeYlVGoB">statistical field theory</a> of a fluctuating field near a <a href="/facts/Second-order_phase_transition/eLP2BY8R">second-order phase transition</a>. The <a href="/facts/Schr%25C3%25B6dinger_equation/3csgSIWF">Schrödinger equation</a> is a <a href="/facts/Diffusion_equation/l3CUuLqF">diffusion equation</a> with an imaginary diffusion constant, and the path integral is an <a href="/facts/Analytic_continuation/aBCuEgvt">analytic continuation</a> of a method for summing up all possible <a href="/facts/Random_walk/08v0jmfv">random walks</a>.
The path integral has impacted a wide array of sciences, including <a href="/facts/Polymer_physics/01sWSKtL">polymer physics</a>, quantum field theory, <a href="/facts/String_theory/HUJFIPYH">string theory</a> and <a href="/facts/Cosmology/yM19zM7Y">cosmology</a>. In physics, it is a foundation for <a href="/facts/Lattice_gauge_theory/90xxuLpx">lattice gauge theory</a> and <a href="/facts/Quantum_chromodynamics/3cA2ctyb">quantum chromodynamics</a>. It has been called the "most powerful formula in physics", with <a href="/facts/Stephen_Wolfram/pJTChkGB">Stephen Wolfram</a> also declaring it to be the "fundamental mathematical construct of modern quantum mechanics and quantum field theory".
The basic idea of the path integral formulation can be traced back to <a href="/facts/Norbert_Wiener/cVMRAHCU">Norbert Wiener</a>, who introduced the <a href="/facts/Wiener_integral/KPh7vYd7">Wiener integral</a> for solving problems in diffusion and <a href="/facts/Brownian_motion/Th0g0wkr">Brownian motion</a>. This idea was extended to the use of the <a href="/facts/Lagrangian_(field_theory)/p9BCUz57">Lagrangian</a> in quantum mechanics by <a href="/facts/Paul_Dirac/6bt5yRdk">Paul Dirac</a>, whose 1933 paper gave birth to path integral formulation. The complete method was developed in 1948 by <a href="/facts/Richard_Feynman/32vGFrtB">Richard Feynman</a>. Some preliminaries were worked out earlier in his doctoral work under the supervision of <a href="/facts/John_Archibald_Wheeler/I1A5t0Fd">John Archibald Wheeler</a>. The original motivation stemmed from the desire to obtain a quantum-mechanical formulation for the <a href="/facts/Wheeler%25E2%2580%2593Feynman_absorber_theory/uq5gthkA">Wheeler–Feynman absorber theory</a> using a <a href="/facts/Lagrangian_(field_theory)/p9BCUz57">Lagrangian</a> (rather than a <a href="/facts/Hamiltonian_(quantum_mechanics)/5XwK2HmD">Hamiltonian</a>) as a starting point.

Path integral formulation open-in-new

Path integral formulation