Path integral formulation

Functional integrals are commonly referred to in theoretical physics as a path integral. This special designation goes back to the Feynman formulation of quantum mechanics, in the case of a movement of a particle from point A to point B are taken into account all possible paths from A to B and not, as in classical mechanics, only the path with the smallest effect.

Path integrals are a fundamental tool in quantum field theory, which can be defined with the help of Grassmann variables path integrals for fermionic fields for a long time. Perturbation theory, renormalization group, etc., are usually formulated in terms of path integrals.

Additionally, you experience path integrals in classical statistical mechanics in the calculation of partition functions as well as in the critical static and dynamic. The formal similarity between quantum field theory and classical statistical mechanics also includes perturbation theory, renormalization group, instantons and other techniques.

Definition

A path integral ( functional integral ) extends over the space of a complex-valued function Φ or real (x ), and not as an ordinary integral over a finite-dimensional space. The coordinate x acts in the path integral only as a continuous index. A precise definition involves the approximation of the function Φ (x ) by the function values ​​Φ (xn ) on a lattice with lattice constant a and the limit a → 0,

The integrand of the integral path is an exponential function, the exponent contains quantum mechanical event that effect integral S, a functional of the function Φ (x). In the case of statistical mechanics, path integrals writes typically in the form

Being referred to as the Hamiltonian. Quantum field theories and field theories of critical dynamics or statics often require a finite lattice constant ( regularization cutoff ). The limit a → 0, N → is executable in this case only after calculation of physical quantities.

Historical, applications, variants

Functional integrals in quantum mechanics by Paul Dirac used as early as 1934 ( in German Physical journal of the Soviet Union Vol.3, 1933, p 64). Feynman developed from the eponymous path integral formulation of quantum mechanics in the 1940s. In the case of point-like particles is incorporated here all possible paths Q (t ) of a particle between two points. In the generalization in quantum field theory is instead on the field configurations Φ (x, t) integrated.

The transition amplitude between two configurations is given by the path integral of exp ( IS / h) with the appropriate boundary conditions. This simple statement can be declared the basic principle of quantum mechanics, the Schrödinger equation is a consequence of it.

In quantum mechanics and quantum field theory, the exponent in the integrand of the path integral is imaginary. In contrast, the exponent of the path integral of classical physics are real. In mathematics, are path integrals and functional integrals of the functional analysis. The convergence behavior and the well- definedness of the path integral is mathematically not fully explored; but it can safely be assumed that the imaginärzeitige formulation can be justified exactly with the Wiener measure in many cases, and that the so-called Wick rotation, an exact correlation between real-valued and imaginary formulation is ( " Statistical Physics and Quantum Field Theory ").

Previously, the term path integral for path integrals was used, but there are risks of confusion here.

Quantum mechanics of point particles

The quantum mechanics of the particle is described by the Schrodinger equation

Where H ( p, q, t) is the Hamiltonian, q is a position in the room and the impulse operator. The Feynman path integral

Extends across the paths Q (t) of the particle and provides for solving ψ ( Q, T ) of the Schrödinger equation, the time t is the solution at the time t '. The constant scaling factor is I.A. uninteresting, is belonging to the Hamiltonian Lagrangian.

In some more compact notation indicates the integral path, the probability that the particle at the time t ' at the point B to see if it was located at the time t A is proportional to | Z (B, A ) | 2

The integral contains only the paths (A, T ) to (B, T '), and it is

Derivation

The transition from the Schrödinger equation for the Feynman path integral does not require quantum mechanics. Rather, other differential equations of similar structure (eg Fokker - Planck equations ) is equivalent to a path integral. The derivation of the path integral for the Schrödinger equation requires only four lines, is instructive, and therefore can be outlined here. For the sake of uniqueness is established that are in all terms of H, the nabla operators left. Provides an integration of the Schrödinger equation for a space dimension over a time interval ε

The other sign of the nabla operator in the second row explained by the fact that the derivatives are available in all terms of the Hamiltonian function right here and act on the δ - function. A partial integration leads back to the first line. Inserting the Fourier integral

Results

This equation gives ψ (Q ', t ε ) as a functional of ψ (q, t). N = ( t' -t) / ε -time iteration provides ψ (Q ', T') in the form of a path integral of q and p,

This " Hamiltonian " form of the path integral in the case of the Schrödinger equation is usually simplified by performing the p- integrals. This is in closed form possible since p occurs only square in the exponent ( due to possible complications in special cases, see ref ). The result is the above Feynman path integral.

In classical physics, you can see the motion of particles ( and, for example, light rays) between two points A, B (Hamilton 's principle ) calculate in space and time with the principle of least action in the context of variational calculus. The effect is the time integral of the difference between kinetic and potential energy ( Lagrangian ) of start time point at which the particle is in A, until the end time point at which the particle is in B. According to Hamilton's principle, the effect of the selected path is an extremum, its variation vanishes. For a free particle potential without a movement results in a straight line from point A to point B. An example in which the path is not a straight line more of a light beam, the media of different optical density has happened (which, with the help can describe a potential in the Lagrangian ), here is the cheapest way ( optical path ) is not a straight more: it comes to the refraction of the light beam.

In quantum mechanics, integrated with a path integral over all possible paths along which the particle from A to B could reach, and weights the paths here with a " phase factor " proportional to the exponential of the imaginary -made and divided by the reduced Planck's constant action functional. One also called the sum of all paths because this is integrated over all paths, albeit with different weights. amplitude is on each path the same, but the phase is determined by the particular effect is different. the classical path is characterized that, for him, the variation of the action according to Hamilton's principle disappears. paths around so wear at approximately the same phase, which results in constructive interference. at more distant paths of the integrand oscillates at effects large compared Planck's quantum of action are ( classical limit ), on the other hand so fast that the contributions of these paths cancel each other out. If, however, the effects as in typical quantum- mechanical systems of the order of Planck's constant, also wear paths in addition to the classical path in the path integral.

In this respect, the Hamilton 's principle for particle trajectories turns out only as a special case of the more general Hamiltonian principle for fields. Formally, the integration over all possible ( generalized ) places by an integration over all possible field configuration is doing in the Feynman formulation substituted, so that the actual role of the path integral for solving wave and field equations becomes clearer, as in the last section was indicated for the Schrödinger equation. This fact can be understood in analogy to the transition from the aforementioned ray optics to wave optics. On the other hand, motivates the modified Hamilton's principle with the replacement of the phase space coordinates by fields the canonical quantization of the Euler -Lagrange field equations, providing a completely operator- treatment of quantum mechanics is possible and so an alternative approach to quantum field theory is created, which was not discussed here.

• Hagen Kleinert Path Integrals in Quantum Mechanics, Statistics and Polymer Physics ", Oxford University Press, 1993 (out of print, online readable here ) Latest English edition. Path integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 4th edition, World Scientific (Singapore, 2006) ( also available online)
• Gert Roepstorff path integrals in quantum physics, Vieweg 1991, 1997 (English translation: Path integral approach to quantum physics - an introduction, Springer 1996)
• Richard P. Feynman, Albert R. Hibbs: Quantum Mechanics and Path Integral, emended edition 2005, Dover Publications, 2010 (Editor Daniel F. Styer, the numerous errors in the output of 1965 corrected )
• Jean Zinn -Justin Path integrals in Quantum Mechanics, Oxford University Press 2005