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Action principles

Action principles, also known as the principle of stationary action or , are a class of variational principles in physics that determine the evolution of physical systems by requiring the —a scalar functional defined as the time integral of the S = \int_{t_1}^{t_2} L(q, \dot{q}, t) \, dt—to be stationary (typically extremized) with respect to infinitesimal variations in the system's trajectory. The Lagrangian L is generally the difference between the kinetic energy T and potential energy V of the system, L = T - V, though it can take more general forms in advanced applications. This stationarity condition yields the equations of motion through the calculus of variations, providing an alternative and often more elegant formulation to Newton's second law for conservative systems. The historical origins of action principles trace back to the , when Pierre-Louis Maupertuis proposed the principle of least in 1744 as a teleological unification of in and mechanical motion, asserting that nature acts to minimize a analogous to "." This idea was mathematically refined by Leonhard Euler and in the mid-18th century, who developed the to derive from extremizing the integral. William Rowan formalized the modern version in 1834, introducing the principle of stationary in terms of the full and extending its scope to and dynamics, which laid the groundwork for and the Hamilton-Jacobi equation. In , action principles provide a powerful framework for systems with multiple , leading to the Euler-Lagrange equations \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = 0, which are equivalent to Newton's laws for conservative forces but generalize readily to constrained systems and non-Cartesian coordinates. Symmetries in the , such as time translation invariance, imply conservation laws through , linking action principles to fundamental conserved quantities like energy and momentum. Beyond mechanics, these principles extend to field theories, where the action is integrated over S = \int \mathcal{L} \, d^4x, deriving equations such as the Maxwell equations for or the in via Hilbert's 1915 variational approach. In , the action underpins the , where the classical path corresponds to the dominant contribution to the quantum amplitude e^{iS/\hbar}.

Core Concepts

Definition of Action

In physics, the S is a scalar functional that quantifies the of a along a in configuration space, defined as the time of the L: S[q(t)] = \int_{t_1}^{t_2} L(q, \dot{q}, t) \, dt, where q(t) represents the describing the system's configuration, \dot{q} = dq/dt are their time derivatives (generalized velocities), and the is taken over the time from t_1 to t_2. This formulation treats the as a path-dependent quantity, varying with different possible trajectories between fixed endpoints, rather than depending on instantaneous forces. The L itself is constructed for conservative mechanical systems as the difference between the T (which depends on velocities) and the V (which depends on positions): L = T - V. For example, in a simple particle system, T = \frac{1}{2} m \dot{q}^2 and V = V(q), making the action a measure of the net energetic evolution over time. This difference form ensures the action captures the balance between inertial and conservative forces in the system's motion. Historically, the mechanical drew motivation from in , which states that light rays follow paths of stationary (least) time between points; Pierre-Louis de Maupertuis extended this analogy in the 1740s to propose a principle of least for particles, unifying optical and mechanical laws under a variational framework. The has dimensions of energy multiplied by time, equivalent to , with the SI unit of (J s). In quantum contexts, this dimension aligns with the reduced Planck's constant \hbar, representing a fundamental quantum of . The stationary posits that physical paths make this functional stationary, yielding the .

Stationary Action Principle

The stationary action principle asserts that among all possible paths a physical system can take between two points in configuration space, the actual path followed is the one for which the action functional S = \int_{t_1}^{t_2} L(q, \dot{q}, t) \, dt, with L denoting the Lagrangian, is stationary. This stationarity condition requires that the first variation of the action vanishes, \delta S = 0, for variations \delta q that fix the endpoints q(t_1) and q(t_2). To derive the from this principle, consider a small variation q(t) \to q(t) + \delta q(t), leading to \delta S = \int_{t_1}^{t_2} \left( \frac{\partial L}{\partial q} \delta q + \frac{\partial L}{\partial \dot{q}} \delta \dot{q} \right) dt = 0. Integrating the second term by parts yields \int_{t_1}^{t_2} \left[ \frac{\partial L}{\partial q} - \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \right) \right] \delta q \, dt + \left[ \frac{\partial L}{\partial \dot{q}} \delta q \right]_{t_1}^{t_2} = 0. Since the boundary term vanishes due to fixed endpoints, the integrand must be zero for arbitrary \delta q, resulting in the Euler-Lagrange equation: \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \right) - \frac{\partial L}{\partial q} = 0. This equation governs the dynamics of the system and was first systematically applied in . The condition \delta S = 0 implies that the action is at an extremum, which may be a minimum, maximum, or , though physical paths often correspond to minima in simple cases. For instance, in the brachistochrone problem, the curve minimizing travel time under —a functional analogous to the action—yields a path, demonstrating a true minimum that accelerates the particle faster initially despite a longer . This example highlights how stationarity captures efficient dynamical paths without assuming a global minimum. By framing diverse physical laws as consequences of varying a single action functional, the stationary action principle unifies the derivation of across , , and beyond, providing a variational foundation that reveals underlying symmetries and conservations.

Lagrangian Formulation

The Lagrangian formulation provides a foundational for describing the of mechanical systems by defining the function L, which is the difference between the T and the V of the system, L = T - V. This form holds in Cartesian coordinates for a single particle of m, where the is \mathbf{r} = (x, y, z) and velocity \dot{\mathbf{r}}, yielding T = \frac{1}{2} m |\dot{\mathbf{r}}|^2 = \frac{1}{2} m (\dot{x}^2 + \dot{y}^2 + \dot{z}^2) and V = V(\mathbf{r}, t), so L = \frac{1}{2} m (\dot{x}^2 + \dot{y}^2 + \dot{z}^2) - V(x, y, z, t). For more complex systems, such as those involving multiple particles or fields, the kinetic energy generalizes to sums over all components, while the potential accounts for interactions. In non-Cartesian coordinate systems, such as polar coordinates (r, \theta) in two dimensions, the Lagrangian adapts to the geometry of the coordinates; for a particle, T = \frac{1}{2} m (\dot{r}^2 + r^2 \dot{\theta}^2) and V = V(r, \theta, t), giving L = \frac{1}{2} m (\dot{r}^2 + r^2 \dot{\theta}^2) - V(r, \theta, t). More generally, in arbitrary generalized coordinates q_i (where i = 1, \dots, n for n degrees of freedom), the Lagrangian takes the form L(q_1, \dots, q_n, \dot{q}_1, \dots, \dot{q}_n, t) = T(q, \dot{q}, t) - V(q, t), with the kinetic energy expressed via the metric tensor of the coordinate system, such as T = \frac{1}{2} \sum_{i,j} g_{ij}(q) \dot{q}_i \dot{q}_j. This coordinate-independent structure facilitates the analysis of systems with rotational or other symmetries. Specific examples illustrate the construction of L. For a free particle unconstrained in Cartesian coordinates with no potential (V = 0), the Lagrangian simplifies to L = \frac{1}{2} m v^2 = \frac{1}{2} m (\dot{x}^2 + \dot{y}^2 + \dot{z}^2), capturing purely inertial motion. In contrast, for a one-dimensional harmonic oscillator subject to a restoring force from a spring with constant k, the Lagrangian is L = \frac{1}{2} m \dot{x}^2 - \frac{1}{2} k x^2, where the quadratic potential term encodes the oscillatory behavior. Constraints in mechanical systems, such as particles confined to surfaces or linkages, are handled by either selecting that inherently satisfy the constraints or incorporating them explicitly via Lagrange multipliers. In the approach, (expressible as f(q, t) = 0) reduce the number of independent q_i from $3N (for N particles in Cartesian space) to the effective , allowing direct substitution into L without additional terms; for instance, a particle on a uses spherical coordinates (\theta, \phi) to enforce the radial constraint. For , Lagrange multipliers \lambda_k are introduced by augmenting the to \tilde{L} = L - \sum_k \lambda_k f_k(q, t), where the f_k = 0 are the constraint equations; the then include terms \frac{\partial f_k}{\partial q_i} that account for constraint forces, solved alongside the multipliers. For non-holonomic constraints, typically linear in velocities as \sum_i a_{k i} \dot{q}_i = 0 and non-integrable, a modified approach using the Lagrange-d'Alembert principle incorporates multipliers such that the equations become \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = \sum_k \lambda_k a_{k i}, with virtual displacements satisfying the constraints. The formulation connects to through a Legendre transform, which shifts the description from velocities \dot{q}_i to momenta p_i = \frac{\partial L}{\partial \dot{q}_i}; the H(q, p, t) is then defined as H = \sum_i p_i \dot{q}_i - L(q, \dot{q}, t), expressed in terms of q_i and p_i after solving for \dot{q}_i(p). This transform preserves the dynamics but facilitates phase-space analysis and quantization. With the constructed, the stationary action principle—requiring the action \int L \, dt to be stationary—yields the via the Euler-Lagrange equations.

Classical Principles

Hamilton's Principle

Hamilton's principle, formulated by William Rowan Hamilton in 1834, posits that the path taken by a physical system between two fixed points in time evolves such that the action integral is stationary. Mathematically, this is expressed as \delta \int_{t_1}^{t_2} L \, dt = 0, where L is the Lagrangian of the system, typically L = T - V with T as kinetic energy and V as potential energy, and the variation \delta is taken over paths with fixed endpoints q(t_1) and q(t_2) at specified times t_1 and t_2. This principle reformulates the stationary action concept by enforcing fixed temporal boundaries and endpoint configurations, ensuring the actual trajectory extremizes the action without altering the initial and final states. Unlike broader stationary action formulations that may allow variations in endpoints or time, Hamilton's version directly yields the Euler-Lagrange equations for systems with prescribed boundary conditions. From this variational statement, the canonical momentum conjugate to a generalized coordinate q is defined as p = \frac{\partial L}{\partial \dot{q}}, obtained by considering the first-order variation of the action and integrating by parts, where terms vanishing at fixed endpoints lead to the momentum expression. For time-independent Lagrangians (i.e., L not explicitly dependent on t), the energy function h = \sum_i \dot{q}_i \frac{\partial L}{\partial \dot{q}_i} - L is conserved along the trajectory, as its total time derivative vanishes: \frac{dh}{dt} = 0. This conserved quantity often corresponds to the total energy E = T + V for standard mechanical systems. A illustrative example is a single particle of mass m in a potential V(\mathbf{r}), with Lagrangian L = \frac{1}{2} m \dot{\mathbf{r}}^2 - V(\mathbf{r}). Applying yields the Euler-Lagrange equations \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{r}_j} \right) = \frac{\partial L}{\partial r_j}, \quad j = x,y,z, which simplify to m \ddot{\mathbf{r}} = -\nabla V(\mathbf{r}), recovering Newton's second law of motion. Here, the momentum is \mathbf{p} = m \dot{\mathbf{r}}, and for time-independent V, the total energy E = \frac{p^2}{2m} + V(\mathbf{r}) is conserved.

Maupertuis' Principle

Pierre-Louis Maupertuis formulated the principle of least action in 1744 as a teleological argument unifying the laws of ray optics and mechanics, positing that nature minimizes a "quantity of action" defined as the product of mass, velocity, and path length. This early version, presented to the Académie des Sciences, emphasized that physical systems follow paths of least action to embody nature's economy and simplicity. Maupertuis' principle states that for a with fixed total energy E, the true minimizes the abbreviated , given by the variational condition \delta \int p \, dq = 0, where the integral is over paths connecting fixed endpoints in configuration space q, and p = \frac{\partial L}{\partial \dot{q}} is the derived from the L = T - V for conservative potentials V. This formulation fixes the energy E = T + V along varied paths, focusing on spatial trajectories rather than time parametrization. Geometrically, Maupertuis' principle interprets classical trajectories as geodesics in a on the configuration space, equipped with the Jacobi-Maupertuis metric ds^2 = 2(E - V) dq^i dq^j g_{ij}, where g_{ij} is the metric; the shortest path under this metric corresponds to the least action trajectory at fixed . This connection, refined by Jacobi, transforms the dynamical problem into finding geodesics, highlighting the 's role in viewing as . The principle applies exclusively to conservative systems where the Lagrangian and potential are time-independent, ensuring energy conservation without explicit temporal variations; it determines the spatial path but not the timing of motion along it. Unlike Hamilton's principle, which varies full time-dependent trajectories between fixed times, Maupertuis' approach varies energy-constrained paths in space, making it a precursor to more general variational methods.

Conservation Laws via Noether's Theorem

Noether's theorem provides a systematic link between continuous symmetries in the formulation of and corresponding conservation laws. Formulated by in her 1918 paper, the theorem states that if the L(\mathbf{q}, \dot{\mathbf{q}}, t) of a system is invariant under a continuous group of transformations, up to a total time derivative, then there exists a associated with that . This invariance applies to the action S = \int_{t_1}^{t_2} L \, dt, where variations of the path \mathbf{q}(t) under the symmetry leave S unchanged except for boundary terms. To derive the conserved quantity, consider an infinitesimal symmetry transformation \delta q^i = \epsilon K^i(\mathbf{q}, \dot{\mathbf{q}}, t), where \epsilon is an infinitesimal parameter and K^i generates the . For the action to be invariant, the variation of the must satisfy \delta L = \frac{dF}{dt} for some function F(\mathbf{q}, \dot{\mathbf{q}}, t). Substituting into the Euler-Lagrange equations \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}^i} \right) - \frac{\partial L}{\partial q^i} = 0 and integrating by parts yields the \frac{d}{dt} \left( \sum_i \frac{\partial L}{\partial \dot{q}^i} K^i - F \right) = 0. In simpler cases, such as transformations independent of velocities (\delta q = \epsilon f(q)) where \delta L = 0, this reduces to \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} f \right) = 0, directly implying the conservation of the generalized conjugate to the direction. Prominent examples illustrate the theorem's power in classical mechanics. Time-translation invariance, where L has no explicit time dependence (\delta L = 0 under t \to t + \epsilon), leads to conservation of the Hamiltonian H = \sum_i \dot{q}^i \frac{\partial L}{\partial \dot{q}^i} - L, representing total energy. Spatial translation symmetry, \delta q^i = \epsilon (constant shift), yields conservation of linear momentum p_i = \frac{\partial L}{\partial \dot{q}^i}. Rotational invariance under \delta q^i = \epsilon \epsilon_{ijk} q^j (infinitesimal rotation) conserves angular momentum l_i = \sum_j q_j p_k - q_k p_j (with appropriate indices). These conserved quantities arise directly from the underlying symmetries of the physical laws encoded in the action. The theorem's reliance on the stationary action principle ensures that symmetries of the correspond to invariances of the physical paths, as variations around the true preserve the action's stationarity. This unifies diverse laws, revealing their origin in the geometric structure of the action rather than ad hoc assumptions.

Field and Relativistic Principles

Action in

In , the action principle extends the variational framework from discrete particles to continuous fields, where the system's configuration is described by field variables \phi^a(x) defined over . Analogous to the particle case, S is defined as the of a \mathcal{L}(\phi^a, \partial_\mu \phi^a, x) over a volume: S = \int d^4 x \, \mathcal{L}(\phi^a, \partial_\mu \phi^a, x), with the integral typically taken over Minkowski space in relativistic theories. The principle of stationary action requires \delta S = 0 for variations \delta \phi^a that satisfy appropriate boundary conditions, yielding the equations of motion for the fields. Varying the action leads to the Euler-Lagrange equations for fields, which generalize the particle equations by accounting for the infinite in the . For a \phi(x), the variation \delta S includes terms from \delta \mathcal{L} and total , resulting in: \partial_\mu \left( \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \right) - \frac{\partial \mathcal{L}}{\partial \phi} = 0. This equation must hold at every point, enforcing local dynamics. The derivation assumes the density depends on the and its first , ensuring a well-posed variational problem. A key example is the Klein-Gordon action for a free \phi, describing massive spin-0 particles in classical theory: S = \int d^4 x \left[ \frac{1}{2} \partial_\mu \phi \partial^\mu \phi - \frac{1}{2} m^2 \phi^2 \right], where the is \eta^{\mu\nu} = \mathrm{diag}(1, -1, -1, -1). Applying the Euler-Lagrange yields the Klein-Gordon : (\partial_\mu \partial^\mu + m^2) \phi = 0. This illustrates how the action encodes relativistic invariance and through the potential term. For the electromagnetic field, the action is constructed from the field strength tensor F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu, where A_\mu is the : S = -\frac{1}{4} \int d^4 x \, F_{\mu\nu} F^{\mu\nu}. The Euler-Lagrange equations then produce the source-free equations: \partial_\mu F^{\mu\nu} = 0, with the Bianchi identity \partial_\lambda F_{\mu\nu} + \partial_\mu F_{\nu\lambda} + \partial_\nu F_{\lambda\mu} = 0 following from the antisymmetry of F_{\mu\nu}. Including sources via minimal coupling modifies the action to yield the full inhomogeneous equations. Boundary conditions are essential for the to yield unique solutions, as the variation \delta S generates surface terms of the form \int_{\partial V} d^3 \Sigma_\mu \, \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \delta \phi, where \partial V is the . These terms vanish if the field variations satisfy \delta \phi|_{\partial V} = 0, such as fixed values on initial and final hypersurfaces or decay at , ensuring the action is without additional constraints.

Relativistic Actions for Particles

In , the action principle for a free is formulated to ensure , extending the classical stationary action principle to motions approaching the . The proper time action is given by S = -mc \int ds, where m is the rest mass of the particle, c is the , and ds = c \, d\tau = \sqrt{dt^2 - (dx^2 + dy^2 + dz^2)/c^2} is the infinitesimal along the worldline in Minkowski with (+, -, -, -). This form, first proposed by , represents the invariant interval and ensures that the action is a , as the \tau is invariant under Lorentz transformations. An equivalent parameterization in terms of t yields S = -mc^2 \int \sqrt{1 - v^2/c^2} \, dt, where \mathbf{v} = d\mathbf{x}/dt is the three-velocity. The corresponding to this action is L = -mc^2 \sqrt{1 - v^2/c^2}. Applying the Euler-Lagrange equations from the \delta S = 0 produces the relativistic . Specifically, the canonical is \mathbf{p} = \partial L / \partial \mathbf{v} = \gamma m \mathbf{v}, where \gamma = 1 / \sqrt{1 - v^2/c^2} is the , recovering the relativistic for a . This formulation bridges to while preserving the action's reparameterization invariance and Lorentz symmetry. The Lorentz invariance of guarantees that the resulting are consistent across inertial frames, as the depends only on the invariant . For instance, varying leads to the p^\mu = m u^\mu, where u^\mu = dx^\mu / d\tau is the satisfying u^\mu u_\mu = c^2. The time component of the conserved yields the total E = \gamma m c^2. To derive this explicitly, perform the Legendre transform to obtain the : with \mathbf{p} = \gamma m \mathbf{v}, solve for v = |\mathbf{p}| c^2 / \sqrt{(E_0)^2 + (|\mathbf{p}| c)^2} (where E_0 = m c^2), leading to H = \mathbf{p} \cdot \mathbf{v} - L = \sqrt{(m c^2)^2 + (p c)^2} = \gamma m c^2, confirming the relativistic expression.

Einstein-Hilbert Action in General Relativity

The provides the variational foundation for , encoding the dynamics of through the curvature of four-dimensional . Formulated by in November 1915 as part of a unified theory of gravitation and , it posits that the equations arise from extremizing an integral over . This approach complemented Albert Einstein's contemporaneous derivation of the field equations via direct geometric arguments. The pure gravitational part of the action is expressed as S_{\rm EH} = \frac{c^4}{16\pi G} \int R \, \sqrt{-g} \, d^4x, where R denotes the Ricci scalar (twice-contracted ), g = \det(g_{\mu\nu}) is the determinant of the Lorentzian g_{\mu\nu}, c is the , and G is the Newtonian ; the prefactor ensures consistency with the Newtonian limit. The complete action incorporates fields via an additional term S_{\rm matter}, yielding the total action S = S_{\rm EH} + S_{\rm matter}. The T_{\mu\nu}, which sources gravity, emerges from the of the matter action with respect to the inverse metric: T_{\mu\nu} = -\frac{2}{\sqrt{-g}} \frac{\delta S_{\rm matter}}{\delta g^{\mu\nu}}. Varying the total action with respect to the and imposing the stationarity condition produces the : R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}, where R_{\mu\nu} is the Ricci tensor; these equations relate to the of and . Einstein first stated this form on November 25, 1915, resolving prior inconsistencies in his theory and enabling predictions such as the perihelion of Mercury. The Palatini formulation offers an equivalent yet distinct variational route, treating the metric g_{\mu\nu} and the \Gamma^\lambda_{\mu\nu} as independent variables in the . Variation with respect to the connection enforces its compatibility with the , yielding the torsion-free \Gamma^\lambda_{\mu\nu} = \frac{1}{2} g^{\lambda\sigma} (\partial_\mu g_{\nu\sigma} + \partial_\nu g_{\mu\sigma} - \partial_\sigma g_{\mu\nu}); subsequent variation then recovers the standard . Although an early precursor appeared in Attilio Palatini's 1919 work on principles for gravitational equations, the independent variation method—now termed "Palatini's method"—was systematically introduced by Einstein in 1925 during his pursuit of unified field theories. This approach highlights the geometric structure of and facilitates extensions to theories with non-metric-compatible connections.

Quantum Principles

Path Integral Formulation

The provides a quantum mechanical generalization of the classical action principle, where the transition amplitude between states is obtained by summing contributions from all possible paths, each weighted by a involving the classical action. In 1933, proposed this approach in his seminal paper, suggesting that the quantum amplitude for a particle to propagate from an initial q_i at time t_i to a final q_f at time t_f could be represented as an over paths q(t) connecting these points. Specifically, the kernel or is given formally by K(q_f, t_f; q_i, t_i) = \int \mathcal{D}q(t) \, \exp\left( \frac{i}{\hbar} \int_{t_i}^{t_f} L(q, \dot{q}, t) \, dt \right), where L is the Lagrangian, the integral is a functional integral over paths, and the classical action S = \int L \, dt appears in the phase. This formulation bridges classical and quantum mechanics by elevating the single extremal path of Hamilton's principle to a superposition over all paths. In the classical limit as \hbar \to 0, the reduces to the , where the rapidly oscillating phases cause destructive interference among non-classical paths, leaving the dominant contribution from the path that extremizes the action—precisely the classical trajectory satisfying . This semiclassical recovery demonstrates how the quantum sum over paths naturally incorporates the classical principle as the leading-order term in the \hbar expansion. The functional measure \mathcal{D}q(t) in the path integral requires careful definition, typically via a time-slicing regularization where continuous paths are approximated by discrete chains of positions, leading to a \prod_k dq_k in the limit of time steps, normalized such that the satisfies the composition rule for and unitarity. This normalization ensures the total probability is preserved, often fixed by requiring the free-particle to match the known Gaussian form, though the measure's precise form can include factors depending on the coordinates. A key challenge in evaluating path integrals arises from the oscillatory nature of the integrand \exp(i S / \hbar), which lacks and complicates rigorous mathematical definition in . To overcome this, practitioners often employ a to time (t \to -i \tau), transforming the integral to \int \mathcal{D}q \, \exp(-S_E / \hbar) with a positive-definite S_E, which converges better and facilitates computations, though the rotation's validity relies on properties of the amplitude.

Feynman's Sum-Over-Paths

Richard Feynman's sum-over-paths formulation provides an alternative to the standard approach in , interpreting the evolution of a quantum system as a superposition of amplitudes from every conceivable between initial and final configurations. This method emphasizes the role of the classical action in determining phase factors for each , leading to effects that govern quantum probabilities. The idea emerged from Feynman's efforts to reformulate using variational principles akin to those in . It was initially explored in his 1942 PhD thesis, "The Principle of Least Action in ," completed under the supervision of at , where he proposed summing over paths weighted by the action to derive transition amplitudes. The formulation was rigorously developed and published in his 1948 paper, "Space-Time Approach to Non-Relativistic ," in Reviews of Modern Physics, marking a foundational contribution to the technique. At the heart of Feynman's approach is the , or , which gives the for a particle to propagate from position x_1 at time t_1 to x_2 at t_2: \langle x_2, t_2 | x_1, t_1 \rangle = \int_{x(t_1)=x_1}^{x(t_2)=x_2} \exp\left( \frac{i S[x(t)]}{\hbar} \right) \mathcal{D}[x(t)], where S[x(t)] is the classical functional along the x(t), \hbar is the reduced Planck's constant, and \mathcal{D}[x(t)] represents the measure over all paths connecting the endpoints. This expression sums complex amplitudes over an infinite number of paths, with each contribution's magnitude typically normalized to and its dictated by the divided by \hbar. The measure \mathcal{D}[x(t)], as defined in the broader framework, ensures the integral is well-posed by integrating over infinitesimal path variations. The full at a later time is then obtained by convolving the initial with this , reproducing the time-dependent . A concrete illustration of this formalism appears in the case of a non-relativistic , where the is L = \frac{1}{2} m \dot{x}^2 and the is S = \int_{t_1}^{t_2} \frac{1}{2} m \dot{x}^2 \, dt. To derive the , Feynman discretized the time interval into N slices of duration \epsilon = (t_2 - t_1)/N, representing paths as a sequence of positions x_k at times t_k = t_1 + k \epsilon. The then becomes a over these positions, with the action approximated as a of kinetic terms: \exp(i S / \hbar) \approx \prod_{k=1}^N \exp\left( i \frac{m (x_{k} - x_{k-1})^2}{2 \hbar \epsilon} \right). Each factor is a , and performing the integrations sequentially from x_1 to x_2 yields, in the limit N \to \infty, the exact : \langle x_2, t_2 | x_1, t_1 \rangle = \sqrt{ \frac{m}{2 \pi i \hbar (t_2 - t_1)} } \exp\left( \frac{i m (x_2 - x_1)^2}{2 \hbar (t_2 - t_1)} \right). This result precisely matches the free-particle solution to the time-dependent Schrödinger equation, confirming the equivalence of the sum-over-paths method to canonical quantization. The power of Feynman's sum-over-paths lies in its natural incorporation of quantum interference, where the total amplitude arises from the coherent superposition of path contributions. Each path's phase \exp(i S / \hbar) causes paths with similar actions to add constructively, while those with rapidly varying phases cancel out. Near the classical path—where the action is stationary, satisfying \delta S = 0 as in Hamilton's principle—neighboring paths have nearly identical phases, leading to enhanced constructive interference and the dominant contribution to the integral. This stationary-phase mechanism explains the emergence of classical trajectories in the limit \hbar \to 0, as quantum fluctuations become negligible and the sum is effectively restricted to paths close to the least-action path. Such interference effects underscore why quantum particles do not follow definite paths but exhibit wave-like behavior, with probabilities given by the squared modulus of the total amplitude.

Schwinger's Variational Principle

Schwinger's quantum action principle, developed by in 1951, extends the classical to and by treating quantum dynamics through variations of an functional. This approach incorporates external sources J coupled to the dynamical fields, enabling the systematic generation of quantum correlation functions without relying on perturbative expansions from the outset. Unlike classical principles that seek stationary points of the action, Schwinger's formulation uses variations to derive quantum that incorporate commutation relations and field operators inherently. The core of the principle lies in the quantum action S, modified by source terms such as \int J \phi \, d^4x for a \phi. The generating functional Z[J] is defined as the over field configurations weighted by e^{i(S + \int J \phi)}, normalized appropriately, and it encapsulates all vacuum expectation values. Schwinger established that Z[J] = \exp(i W[J]), where W[J] serves as the that generates connected correlation functions. This relation allows W[J] to be obtained via the Legendre transform from Z[J], providing a generating potential for irreducible diagrams in . Correlation functions are extracted through successive functional derivatives of W[J] with respect to the sources J, evaluated at J = 0. For instance, the one-point function is \langle \phi(x) \rangle = \frac{\delta W[J]}{\delta J(x)} \big|_{J=0}, while the connected two-point function follows as \langle \phi(x) \phi(y) \rangle_c = \frac{\delta^2 W[J]}{\delta J(x) \delta J(y)} \big|_{J=0}. Higher-order n-point functions similarly arise from n-th derivatives, offering a complete hierarchy for computing quantum amplitudes. This derivative structure mirrors the role of sources in probing field expectations, with the variations preserving the quantum commutation algebra. By varying the quantum action under the influence of sources, Schwinger's principle yields the Dyson-Schwinger equations, a set of nonlinear functional differential equations that express the quantum equations of motion for the generating functional. These equations take the form \left( \frac{\delta S}{\delta \phi(x)} + J(x) \right) Z[J] = 0 in integrated form, leading to relations like \langle \frac{\delta S}{\delta \phi(x)} \phi(y_1) \cdots \phi(y_n) \rangle = -i \delta(x - y_1) \langle \phi(y_2) \cdots \phi(y_n) \rangle + \text{permutations} for functions. The Dyson-Schwinger equations provide a non-perturbative framework equivalent to the , but derived variationally, allowing solutions for Green's functions through self-consistent closures rather than explicit path summations. This equivalence highlights the principle's power in unifying with classical variational methods.

Analogies and Applications

Optico-Mechanical Analogy

The optico-mechanical analogy traces its origins to , formulated in 1662, which posits that rays propagate between two points along paths that render the stationary. The is defined as the \int_A^B n(\mathbf{r}(s)) \, ds, where n(\mathbf{r}) is the at position \mathbf{r} along the path parameterized by s, and the stationarity condition \delta \int n \, ds = 0 yields the governing equations for ray trajectories. In the , Pierre-Louis Maupertuis extended this variational approach to , drawing a direct analogy between optical and mechanical systems at fixed . Maupertuis' principle states that particle trajectories minimize the abbreviated action \int \mathbf{p} \cdot d\mathbf{q}, where \mathbf{p} is the ; here, the magnitude of p = mv (with mass m and speed v) serves as the mechanical counterpart to the n in , leading to analogous refraction laws such as for conservation at interfaces: mv_1 \sin \theta_1 = mv_2 \sin \theta_2. William Rowan Hamilton formalized this optico-mechanical analogy in the 1830s by establishing a mathematical equivalence between in and Maupertuis' principle in through a shared variational framework. introduced the , a potential solving a , which unifies ray paths in (orthogonal to wavefronts) with particle trajectories in (orthogonal to surfaces of constant action), interpreting the integrand as inverse in or in . In modern contexts, this analogy underpins the ray optics approximation, where the eikonal equation |\nabla S|^2 = n^2(\mathbf{r}) emerges directly from the stationary action principle for the optical length, with S as the eikonal function representing the phase. This high-frequency limit of wave optics parallels the classical limit of quantum mechanics, facilitating geometric descriptions of light propagation in inhomogeneous media.

Applications in Chemistry and Optics

In , action principles underpin the description of molecular reaction paths by identifying trajectories that extremize the action functional, providing a variational framework for locating minimum energy paths between reactants and products. This approach aligns with the intrinsic reaction coordinate, which traces the path of steepest descent in the , enabling the prediction of reaction mechanisms without exhaustive sampling of configuration space. methods extend this to quantum tunneling in , where the Euclidean action is minimized along bounce trajectories in , yielding rate constants for low-temperature reactions such as proton transfer in enzymes or hydrogen abstraction in hydrocarbons. These semiclassical approximations, derived from formulations, capture quantum corrections to classical barrier crossing, with applications demonstrating enhanced accuracy over traditional Eyring theory for tunneling reactions. In , employs functionals to model dynamics under external perturbations, such as fields, by stationary points of a time-dependent that incorporates the Kohn-Sham orbitals and interaction kernels. This formalism resolves causality issues in earlier derivations, allowing simulations of ultrafast processes like charge migration in biomolecules or in solar cells, where the ensures the density evolves self-consistently from initial conditions. Feynman path integrals, rooted in the action principle, facilitate the computation of vibrational spectra in by summing over quantum paths to evaluate correlation functions, particularly for anharmonic effects in liquids like or . For instance, ring-polymer variants of path integrals reproduce and Raman spectra with quantum delocalization, as seen in simulations of in where approximations yield frequencies within 10 cm⁻¹ of experiment. In , serves as a analogous to the action, stating that light rays follow paths of stationary optical length (integral of along the trajectory), from which emerges as the condition for at interfaces. This least-time criterion governs ray tracing in heterogeneous media, such as atmospheric mirages or fiber , where deviations from stationarity lead to non-physical paths. Variational methods extend this to lens design, optimizing surface profiles to minimize aberrations by extremizing functionals that combine wavefront error and manufacturing constraints, as in progressive addition lenses where error functionals balance near- and far-vision zones. Such approaches, solving fourth-order elliptic equations, have enabled aberration-free designs with reduced by up to 50% compared to empirical methods.

Modern Applications in Condensed Matter and Beyond

In , action principles underpin effective theories for describing transitions and phenomena. The Ginzburg-Landau functional serves as a for modeling near the critical temperature, where the order parameter ψ represents the superconducting wavefunction, and the functional F[ψ] = ∫ [α|ψ|^2 + (β/2)|ψ|^4 + (1/(2m*))|(-iℏ∇ - 2eA)ψ|^2 + (B^2)/(8π)] dV minimizes to yield the for the superconducting state. This phenomenological approach, extended to dynamical actions in time-dependent Ginzburg-Landau , captures vortex and nonequilibrium behaviors in type-II superconductors. Similarly, Chern-Simons actions describe the topological response in insulators, where the S_CS = (k/4π) ∫ a ∧ da + (2/3) a ∧ a ∧ a, with k the Chern number, encodes the quantized Hall and protects edge states in topological insulators. In time-reversal invariant topological insulators, a higher-dimensional Chern-Simons term arises from dimensional reduction, classifying Z_2 invariants and axion-like magnetoelectric effects. In , the provides a manifestly diffeomorphism-invariant formulation for the propagation of bosonic strings in a target . The action is given by S = -\frac{1}{4\pi \alpha'} \int d^2\sigma \, \sqrt{h} \, h^{ab} \partial_a X^\mu \partial_b X^\nu \, G_{\mu\nu}(X), where X^μ(σ^a) are embedding coordinates, h^{ab} is the metric, and G_{\mu\nu} is the target ; varying this action yields the while allowing to the conformal gauge for quantization. This formulation reveals the of 26 for bosonic strings and facilitates the computation of string scattering amplitudes via path integrals over configurations. Recent advances leverage action principles for simulations on quantum hardware, particularly through formulations. In , Monte Carlo methods adapted to variational quantum circuits enable efficient simulation of quantum many-body systems, such as non-Markovian dynamics in open , by discretizing into Trotterized evolutions that mitigate sign problems. For instance, in 2025, -based algorithms adapted to variational quantum circuits were used for preparation in , demonstrating superior performance over the for small molecules on systems up to 6 qubits. The AdS/CFT correspondence further employs action principles to equate the on-shell gravitational action in , S_grav = (1/(16πG)) ∫ √-g (R + 2Λ) d^{d+1}x + boundary terms, with the generating functional of a on the boundary, enabling holographic computations of entanglement and transport coefficients. Beyond physics, action principles appear in minimal models of biological systems, particularly pathways. The principle of stationary action is invoked to analyze folding , where the folding "action" along conformational paths is minimized subject to energetic constraints, akin to least-action trajectories in , ensuring rapid convergence to native states without kinetic traps. In Go-like models, this manifests as funnel-shaped landscapes where the effective action penalizes deviations from minimal , guiding simulations of folding in proteins like chymotrypsin inhibitor. These approaches highlight universal variational principles in self-organizing biological processes.

Historical Development

Early Precursors (17th-18th Century)

The origins of action principles in physics trace back to the 17th century, beginning with advancements in optics that emphasized extremal paths. In 1662, Pierre de Fermat formulated the principle of least time, asserting that light travels between two points along the path that minimizes the time taken, thereby deriving the law of refraction independently of Descartes' earlier work. This optical insight laid foundational groundwork for variational ideas in mechanics, suggesting that natural processes follow economical routes akin to an optico-mechanical analogy where light rays parallel particle trajectories. Building on such notions, introduced the concept of a "quantity of action" in 1686, defined as the product of , , and distance (m v s), which he proposed as a superior to Descartes' conservation of motion (m v). In his "Briefve demonstration de vne nouveau principe de physique" and subsequent 1687 publications, Leibniz critiqued Cartesian for failing to account for empirical collision outcomes, arguing that his quantity of action better reflected the efficiency and conservation inherent in nature's operations. This measure, later termed "action" in some contexts, represented an early quantitative precursor to variational principles by prioritizing a holistic product over simple . The principle gained prominence in the mid-18th century through , who in 1744 explicitly proposed the principle of least as a teleological law governing all natural phenomena, stating that "Nature, in her operation, always takes the path that is the simplest and most economical." Presented in his memoir "Accord de différentes loix de la nature qui avoient jusqu'ici paru incompatibles," applied this to and , positing that the path of a system minimizes the of along the , thereby unifying diverse laws under an economy of effort. His formulation, inspired by Fermat and Leibniz, framed as \int m v \, ds, emphasizing nature's frugality and sparking debates on versus final causes. Concurrently, Leonhard Euler advanced these ideas mathematically in his 1744 treatise Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, which systematized the to solve problems of extremal paths. Euler derived general equations for functionals like \int L(x, y, y') \, dx, enabling the optimization of curves such as brachistochrones and geodesics, and extended Maupertuis' to broader mechanical systems without relying on . His geometric and intuitive approach, avoiding advanced infinitesimals, established variational methods as a rigorous tool for identifying nature's preferred paths, influencing subsequent formalizations in physics.

19th-Century Formalization

In the late 18th century, Joseph-Louis Lagrange formalized the variational approach to mechanics in his seminal work Mécanique Analytique, published in 1788. This treatise unified the principles of mechanics through analytical methods, eschewing geometric figures and direct references to forces in favor of generalized coordinates and the calculus of variations. By building on the principle of virtual work—stating that for systems with frictionless constraints, the total virtual work done by constraint forces is zero for any compatible virtual displacement—Lagrange combined it with d’Alembert's principle to derive equations of motion without explicitly invoking forces, thereby establishing a rigorous framework for dynamics. Building on this foundation, William Rowan Hamilton advanced the variational method in his papers "On a General Method in Dynamics" (1834) and its sequel (1835), where he introduced the principle of varying action. Hamilton defined a characteristic function V = \int 2T \, dt, with T as kinetic energy, to link initial and final states of motion through partial differential equations, enabling a unified treatment of dynamics that emphasized optical analogies and point-mass models. In the 1835 paper, he further developed canonical transformations using generalized coordinates and momenta p_i = \partial T / \partial \dot{q}_i, along with the principal function S = \int (T + U) \, dt for conservative systems, laying the groundwork for what became known as Hamilton's principle. These innovations provided a powerful tool for perturbation methods in celestial mechanics and transformed the variational formulation into a more symmetric and general form. Carl Gustav Jacob Jacobi extended these ideas in his Vorlesungen über Dynamik, delivered as lectures at the in the winter semester of 1842–1843 and published posthumously. Jacobi reformulated the principle of least action in a form, demonstrating that varying only one endpoint (initial or final) of the path suffices to derive the , thereby simplifying Hamilton's original approach and addressing concerns about stationary points beyond minima. This contribution enhanced the applicability of the to multi-degree-of-freedom systems and integrated it more seamlessly with . The 19th-century developments in action principles also began influencing electromagnetism, as seen in Hermann von Helmholtz's early attempts in his 1847 memoir "Über die Erhaltung der Kraft" (On the Conservation of Force). Helmholtz analyzed velocity-dependent forces, such as those in magnetic interactions, showing that they could be consistent with energy conservation under specific conditions, paving the way for variational formulations in electrodynamics by linking mechanical principles to electromagnetic phenomena without violating conservation laws.

20th-Century Quantum and Relativistic Advances

In the early 20th century, David Hilbert's pursuit of axiomatizing physical theories, as outlined in his sixth problem from , profoundly influenced the development of action principles in . Motivated by this goal of rigorous mathematical foundations for physics, Hilbert derived an action principle for gravitation in his 1915 paper "Die Grundlagen der Physik," where he proposed a variational approach combining gravitational and electromagnetic fields, leading to field equations equivalent to those of . This work marked a pivotal milestone, with the Einstein-Hilbert action—formulated as the integral of the Ricci scalar density—serving as the foundational for . The integration of action principles into quantum mechanics began with Paul Dirac's 1933 insight, where he suggested interpreting the quantum propagator as a sum over paths weighted by the exponential of the classical action, extending the Lagrangian formalism to quantum transitions. This idea laid the groundwork for path integrals, though Dirac did not fully formalize the summation. Richard Feynman elaborated on this concept during his 1942 PhD thesis at Princeton, developing the as a sum-over-histories approach to non-relativistic , where amplitudes are obtained by integrating over all possible paths with phase factors given by the action. By 1948, Feynman refined this into a space-time approach applicable to , enabling diagrammatic and resolving infinities through . Julian Schwinger advanced quantum action principles in 1951 with his variational formulation for , treating the action as a functional generating transformation functions between quantum states at different times. In his seminal paper "The Theory of Quantized Fields," Schwinger derived from the stationarity of the quantum principle, providing a unified framework for handling divergences and source variations in interacting fields. This approach complemented Feynman's path integrals and became central to the renormalization program in . Post-1970 developments saw the rise of Wilsonian effective actions in , pioneered by Kenneth Wilson's framework, which integrates out high-momentum modes to yield scale-dependent effective Lagrangians capturing low-energy physics. Wilson's 1971 work on introduced the concept of effective actions as flowing functionals under transformations, enabling treatments of quantum field theories and simulations. This perspective revolutionized understanding of completions and behaviors in theories like . In the 2020s, action principles in quantum gravity have incorporated entanglement structures, with proposals for entanglement actions deriving spacetime geometry from quantum information measures. For instance, recent formulations in loop quantum gravity express the Euclidean path integral via an entanglement action involving geometric operators and volume constraints, linking quantum entanglement entropy to gravitational dynamics. These advances build on holographic principles, suggesting entanglement as a fundamental ingredient in emergent gravity actions.