| action (physics) | |
|---|---|
| Name | Action |
| Dimension | ML^2T^-1 |
| Siunit | J⋅s |
| Baseunits | kg⋅m^2⋅s^-1 |
| Introduced | Pierre Louis Maupertuis; formalized by William Rowan Hamilton |
action (physics)
Action in physics is a scalar quantity defined as the time integral of a Lagrangian along a trajectory; it plays a central role in both classical mechanics and quantum mechanics as the generator of dynamics and quantum phase. In the context of Quantum Physics the action determines interference via the phase factor exp(iS/ħ) and underlies formulations such as the path integral formulation and canonical quantization.
In mechanics the classical action S[γ] for a path γ(t) between fixed endpoints is defined by the integral S[γ] = ∫_{t1}^{t2} L(q,ẋ,t) dt, where L is the Lagrangian of the system. The first mention of action in modern form is associated with Pierre Louis Maupertuis and later formalized by Joseph-Louis Lagrange and William Rowan Hamilton. Action encodes dynamics compactly and serves as a bridge between variational principles, symmetry, and conservation laws. In quantum contexts S appears in the phase factor exp(iS/ħ) used by Richard Feynman in the path integral and relates to the Hamilton–Jacobi equation in semiclassical approximations.
The principle of least action (more precisely stationary action) states that the physical trajectory makes the action stationary under small variations of the path with fixed endpoints. Hamilton's principle produces the Euler–Lagrange equation and yields equivalent results to Newtonian mechanics and Hamiltonian mechanics. The action framework generalizes readily to systems with constraints via Lagrange multipliers and to continuum systems leading to action functionals for classical field theory such as the Maxwell equations derived from the Maxwell Lagrangian.
Feynman's path integral formulation expresses the quantum propagator as a sum over histories, each weighted by the complex phase exp(iS/ħ). This connects quantum amplitudes to classical action: paths near stationary action contribute coherently, reproducing classical dynamics in the ħ → 0 limit via the method of stationary phase. The path integral framework is central to modern treatments by Richard Feynman, and it underpins perturbative techniques used in quantum electrodynamics and quantum chromodynamics as implemented at facilities such as CERN and Fermilab.
The action enters the quantum phase of the propagator K(x_b,t_b;x_a,t_a) = ∫ D[x(t)] exp(iS[x]/ħ). Interference between contributions is governed by phase differences Δφ = ΔS/ħ, so classical action differences on the order of Planck's constant ħ lead to observable quantum interference in systems such as double-slit experiment, Josephson junctions, and Bose–Einstein condensate interferometry. Semiclassical tools like the WKB approximation, Gutzwiller trace formula, and Van Vleck determinant rely on the action to connect spectra of quantum systems to underlying classical orbits studied in classical chaos.
In quantum field theory (QFT) fields are governed by action functionals S[φ] whose stationary conditions yield field equations. The path integral over field configurations leads to generating functionals used to compute correlation functions and S-matrix elements in theories such as Quantum electrodynamics and Yang–Mills theory. Regularization and renormalization procedures (e.g., dimensional regularization, renormalization groups from Kenneth Wilson) act on the action or effective action Γ[φ]. Classical actions like the Einstein–Hilbert action in general relativity couple to quantum approaches in semiclassical gravity and approaches such as effective field theory.
Noether's theorem links continuous symmetries of the action to conserved currents: invariance of S under time translations implies conservation of energy, spatial translations give conservation of momentum, and rotational symmetry leads to conservation of angular momentum. Gauge symmetries of the action underlie conserved charges in electromagnetism and non-Abelian gauge theories; these principles guide model building at institutions like Institute for Advanced Study and in collaborations such as the LHC experiments. In quantum settings symmetry constraints on the action determine selection rules, anomalies, and the structure of renormalized theories studied by Gerard 't Hooft and others.
Action has SI units of Joule second (J⋅s), the same dimension as Planck constant ħ and h. The scale set by ħ ≈ 1.054×10^−34 J⋅s defines the crossover between quantum and classical behavior: when typical action S ≫ ħ classical approximations hold; when S ~ ħ quantum interference is significant. Typical microscopic actions include atomic transitions characterized by energies from NIST data and timescales from spectroscopy; macroscopic quantum phenomena such as superconductivity and superfluidity manifest collective actions that can still be compared to ħ via coherence volumes and effective action estimates. Dimensionless actions S/ħ guide semiclassical expansions and numerical methods such as lattice lattice gauge theory computations performed on supercomputers by collaborations at Brookhaven National Laboratory and CERN.
Category:Classical mechanics Category:Quantum mechanics Category:Physical quantities