Path integral formulation

The path integral formulation is a description in quantum mechanics that generalizes the stationary action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude.

This formulation has proven crucial to the subsequent development of theoretical physics, because manifest Lorentz covariance (time and space components of quantities enter equations in the same way) is easier to achieve than in the operator formalism of canonical quantization. Unlike previous methods, the path integral allows one to easily change coordinates between very different canonical descriptions of the same quantum system. Another advantage is that it is in practice easier to guess the correct form of the Lagrangian 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 Hamiltonian. Possible downsides of the approach include that unitarity (this is related to conservation of probability; the probabilities of all physically possible outcomes must add up to one) of the S-matrix 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.^[1]

The path integral also relates quantum and stochastic processes, and this provided the basis for the grand synthesis of the 1970s, which unified quantum field theory with the statistical field theory of a fluctuating field near a second-order phase transition. The Schrödinger equation is a diffusion equation with an imaginary diffusion constant, and the path integral is an analytic continuation of a method for summing up all possible random walks.^[2]

The basic idea of the path integral formulation can be traced back to Norbert Wiener, who introduced the Wiener integral for solving problems in diffusion and Brownian motion.^[3] This idea was extended to the use of the Lagrangian in quantum mechanics by Paul Dirac, who gave the ideas that would lead to path integral formulation in his 1933 article.^[4]^[5]^[6] The complete method was developed in 1948 by Richard Feynman.^[7] Some preliminaries were worked out earlier in his doctoral work under the supervision of John Archibald Wheeler. The original motivation stemmed from the desire to obtain a quantum-mechanical formulation for the Wheeler–Feynman absorber theory using a Lagrangian (rather than a Hamiltonian) as a starting point.

Quantum action principle

In quantum mechanics, as in classical mechanics, the Hamiltonian is the generator of time translations. This means that the state at a slightly later time differs from the state at the current time by the result of acting with the Hamiltonian operator (multiplied by the negative imaginary unit, $- i$ ). For states with a definite energy, this is a statement of the de Broglie relation between frequency and energy, and the general relation is consistent with that plus the superposition principle.

The Hamiltonian in classical mechanics is derived from a Lagrangian, which is a more fundamental quantity relative to special relativity. The Hamiltonian indicates how to march forward in time, but the time is different in different reference frames. The Lagrangian is a Lorentz scalar, while the Hamiltonian is the time component of a four-vector. So the Hamiltonian is different in different frames, and this type of symmetry is not apparent in the original formulation of quantum mechanics.

The Hamiltonian is a function of the position and momentum at one time, and it determines the position and momentum a little later. The Lagrangian is a function of the position now and the position a little later (or, equivalently for infinitesimal time separations, it is a function of the position and velocity). The relation between the two is by a Legendre transformation, and the condition that determines the classical equations of motion (the Euler–Lagrange equations) is that the action has an extremum.

In quantum mechanics, the Legendre transform is hard to interpret, because the motion is not over a definite trajectory. In classical mechanics, with discretization in time, the Legendre transform becomes

\varepsilon H=p(t){\big (}q(t+\varepsilon )-q(t){\big )}-\varepsilon L

and

p={\frac {\partial L}{\partial {\dot {q}}}},

where the partial derivative with respect to ${\dot {q}}$ holds $q (t + ε)$ fixed. The inverse Legendre transform is

\varepsilon L=\varepsilon p{\dot {q}}-\varepsilon H,

where

{\dot {q}}={\frac {\partial H}{\partial p}},

and the partial derivative now is with respect to $p$ at fixed $q$ .

In quantum mechanics, the state is a superposition of different states with different values of $q$ , or different values of $p$ , and the quantities $p$ and $q$ can be interpreted as noncommuting operators. The operator $p$ is only definite on states that are indefinite with respect to $q$ . So consider two states separated in time and act with the operator corresponding to the Lagrangian:

e^{i{\big }}.

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