Path integral

Path integrals are to Gregor Wentzel , Paul Dirac and especially Richard Feynman declining formulation of quantum mechanics , at a motion of a particle from point when to point all possible paths of to be taken into account and not, as the smallest in classical mechanics, only the path with Effect . ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle B}$

Generalized path integrals integrate via functions as variables and are therefore also referred to as functional integrals . As such, they have long been a fundamental tool in quantum field theory . Fault calculation, renormalization group etc. are i. d. Usually formulated with the help of path integrals.

In addition, path integrals also appear in classical statistical mechanics when calculating sums of state as well as in critical statics and dynamics . The formal commonality between quantum field theory and classical statistical mechanics also includes perturbation calculus, renormalization groups , instantons and other techniques.

History, applications, variants

Path integrals were used in quantum mechanics as early as 1924 by Gregor Wentzel , but the work was largely forgotten after that and remained isolated. More influential were the work of Paul Dirac from 1933 and Dirac's account in his The Principles of Quantum Mechanics . From this, Feynman developed the path integral formulation of quantum mechanics named after him in the 1940s. In the case of point particles, integration takes place over all possible paths of a particle between two points. The generalization in quantum field theory is instead integrated via the field configurations . In its most general version, the path integral can be understood as a computational expression for the transition amplitude in Dirac's abstract Hilbert space formulation of quantum field theory. According to Julian Schwinger's quantum effect principle, this corresponds to the requirement for a stationary, operator-valued quantum effect. ${\displaystyle q(t)}$ ${\displaystyle \textstyle \Phi (x,t)}$

The transition amplitude between two configurations is given by the path integral over with corresponding boundary conditions. This simple statement can be explained as the fundamental principle of quantum mechanics, the Schrödinger equation is a consequence of it. ${\displaystyle \textstyle \exp(\mathrm {i} S/\hbar )}$

In quantum mechanics and quantum field theory, the exponent in the integrand of the path integral is imaginary. In contrast to this, the exponents of the path integrals in classical physics are real. In mathematics, path integrals or functional integrals are part of functional analysis . The convergence behavior and the well-defined nature of the path integral have not been fully researched mathematically; the imaginary-time formulation with the Wiener measure can in many cases be justified exactly and with the so-called Wick rotation there is an exact connection between real-valued and imaginary formulation ("statistical physics or quantum field theory").

Quantum mechanics of point particles

The quantum mechanics of a particle is described by the Schrödinger equation

${\displaystyle {\frac {\partial }{\partial t}}\psi \left(q,t\right)={\frac {-\mathrm {i} }{\hbar }}H\left({\hat {p}},q,t\right)\psi \left(q,t\right),}$

where is the Hamilton function , a position in space and the momentum operator . Feynman's path integral ${\displaystyle H(p,q,t)}$ ${\displaystyle q}$ ${\displaystyle {\hat {p}}=-\mathrm {i} \hbar \nabla _{q}}$

${\displaystyle \psi \left(q^{\prime },t^{\prime }\right)={\mathfrak {\mathcal {N}}}\int {\mathcal {D}}q\exp \left\{{\frac {\mathrm {i} }{\hbar }}\int _{t}^{t^{\prime }}\mathrm {d} t^{\prime \prime }L\left(q,{\dot {q}},t^{\prime \prime }\right)\right\}\psi \left(q,t\right)}$

extends over the paths of the particle and provides the solution at the time for solving the Schrödinger equation at the time . The constant normalization factor is i. A. uninteresting, is the Lagrangian function belonging to the Hamilton function . ${\displaystyle q(t)}$ ${\displaystyle \psi (q,t)}$ ${\displaystyle t}$ ${\displaystyle t^{\prime }}$ ${\displaystyle {\mathfrak {\mathcal {N}}}}$ ${\displaystyle \textstyle L\left(q,{\dot {q}},t\right)=p{\dot {q}}-H\left(p,q,t\right)}$

In a somewhat more compact notation, the path integral says that the probability of finding the particle at the point in time if it was at the point in time is proportional to with ${\displaystyle t^{\prime }}$ ${\displaystyle B}$ ${\displaystyle t}$ ${\displaystyle A}$ ${\displaystyle \textstyle \left|Z(B,A)\right|^{2}}$

${\displaystyle Z\left(B,A\right)={\mathfrak {\mathcal {N}}}\int {\mathcal {D}}q\exp \left({\frac {\mathrm {i} }{\hbar }}S\right).}$

The integral here only contains the paths from to , and it holds ${\displaystyle (A,t)}$ ${\displaystyle (B,t^{\prime })}$

${\displaystyle Z\left(B,A\right)={\mathfrak {\mathcal {N'}}}\int \mathrm {d} q_{c}Z\left(B,C\right)Z\left(C,A\right).}$

Derivation

The transition from the Schrödinger equation to the Wentzel-Feynman path integral does not require quantum mechanics. Rather, other differential equations with a similar structure (e.g. Fokker-Planck equations) are also equivalent to a path integral. The derivation of the path integral to the Schrödinger equation requires only four lines, is instructive, and can therefore be sketched here. For the sake of clarity, it is determined that the Nabla operators are on the left in all terms of . An integration of the Schrödinger equation for a spatial dimension over a time interval provides ${\displaystyle H}$ ${\displaystyle \epsilon }$

${\displaystyle {\begin{aligned}\psi \left(q',t+\epsilon \right)&=\left(1-{\frac {\mathrm {i} \epsilon }{\hbar }}H\left(-\mathrm {i} \hbar \nabla _{q}',q',t\right)\right)\psi \left(q',t\right)+{\mathcal {O}}\left(\epsilon ^{2}\right)\\&=\int _{-\infty }^{\infty }\mathrm {d} q\left(\left(1-{\frac {i\epsilon }{\hbar }}H\left(\mathrm {i} \hbar \nabla _{q},q,t\right)\right)\delta \left(q^{\prime }-q\right)\right)\psi \left(q,t\right)+{\mathcal {O}}\left(\epsilon ^{2}\right).\end{aligned}}}$

The other sign of the Nabla operator in the second line is explained by the fact that the derivatives in all terms of the Hamilton function are on the right here and act on the function. Partial integration leads back to the first line. Insertion of the Fourier integral ${\displaystyle \delta }$

${\displaystyle \delta \left(q'-q\right)=\int _{-\infty }^{\infty }{\frac {\mathrm {d} p}{2\pi \hbar }}e^{{\frac {\mathrm {i} }{\hbar }}p\left(q'-q\right)}}$

results

${\displaystyle {\begin{aligned}\psi \left(q',t+\epsilon \right)&=\int _{-\infty }^{\infty }\mathrm {d} q\int _{-\infty }^{\infty }{\frac {\mathrm {d} p}{2\pi \hbar }}\left(\left(1-{\frac {\mathrm {i} \epsilon }{\hbar }}H\left(p,q,t\right)\right)e^{{\frac {\mathrm {i} }{\hbar }}p\left(q'-q\right)}\right)\psi \left(q,t\right)+{\mathcal {O}}\left(\epsilon ^{2}\right)\\&=\int _{-\infty }^{\infty }\mathrm {d} q\int _{-\infty }^{\infty }{\frac {\mathrm {d} p}{2\pi \hbar }}\exp \left\{{\frac {\mathrm {i} \epsilon }{\hbar }}\left[p{\frac {q'-q}{\epsilon }}-H\left(p,q,t\right)\right]\right\}\psi \left(q,t\right)+{\mathcal {O}}\left(\epsilon ^{2}\right).\end{aligned}}}$

This equation yields as a functional of . One iteration delivers in the form of a path integral over and , ${\displaystyle \textstyle \psi (q^{\prime },t+\epsilon )}$ ${\displaystyle \psi (q,t)}$ ${\displaystyle N=(t^{\prime }-t)/\epsilon }$ ${\displaystyle \textstyle \psi (q^{\prime },t^{\prime })}$ ${\displaystyle q}$ ${\displaystyle p}$

${\displaystyle \psi \left(q^{\prime },t^{\prime }\right)=\lim _{N\rightarrow \infty }\left(\prod _{n=1}^{N}\int {\frac {\mathrm {d} q_{n}\mathrm {d} p_{n}}{2\pi \hbar }}\right)\exp \left\{{\frac {\mathrm {i} }{\hbar }}\sum _{n=1}^{N}\epsilon \left[p_{n}{\dot {q}}_{n}-H\left(p_{n},q_{n},t_{n}\right)\right]\right\}\psi \left(q_{1},t_{1}\right).}$

This “Hamiltonian” form of the path integral is usually simplified in the case of the Schrödinger equation by executing the -integrals. This is possible in a closed form, since the exponent only occurs as a square (because of possible complications in special cases, see Ref.). The result is the Feynman path integral listed above. ${\displaystyle p}$ ${\displaystyle p}$

Quantum field theory

The path integral (functional integral) extends here over the space of a real or complex-valued function , and not like an ordinary integral over a finite-dimensional space. The coordinate only functions as a continuous index in the path integral. A precise definition includes the approximation of the function by the function values on a space lattice with lattice constant as well as the Limes , ${\displaystyle \Phi (x)}$ ${\displaystyle x}$ ${\displaystyle \Phi (x)}$ ${\displaystyle \Phi (x_{n})}$ ${\displaystyle a}$ ${\displaystyle a\to 0}$

${\displaystyle Z=\int _{-\infty }^{\infty }{\mathcal {D}}\phi \exp \left({\frac {\mathrm {i} }{\hbar }}S\left(\phi \right)\right)=\lim _{N\rightarrow \infty }\prod _{n=1}^{N}\int _{-\infty }^{\infty }\mathrm {d} \phi \left(x_{n}\right)\exp \left({\frac {\mathrm {i} }{\hbar }}S\left(\phi \right)\right).}$

The integrand of a path integral is an exponential function; in the case of quantum mechanics, the exponent contains the action integral , a functional of the function . In the case of statistical mechanics, path integrals are usually written in the form ${\displaystyle S}$ ${\displaystyle \Phi (x)}$

${\displaystyle Z=\int _{-\infty }^{\infty }{\mathcal {D}}\phi \exp \left(-{\mathcal {H}}\left(\phi \right)\right),}$

where is referred to as a Hamiltonian. Quantum field theories and field theories of critical dynamics or statics often require a finite lattice constant (regularization, cutoff). In this case, the limit can only be implemented after the physical quantities have been calculated. ${\displaystyle {\mathcal {H}}}$ ${\displaystyle a\to 0,N\to \infty }$

For fermions , Grassmann variables (anti-commuting variables) are used to form path integrals.

Résumé

In classical physics, the movement of particles (and, for example, rays of light) between two points in space and time can be calculated using the principle of the smallest effect (Hamilton's principle) within the scope of the calculus of variations . The effect is the time integral of the difference between kinetic and potential energy ( Lagrangian function ) from the starting time at which the particle is in to the end time at which the particle is in. According to Hamilton's principle, the effect is an extremum for the chosen path, its variation disappears. For a free particle without potential there is a movement on a straight line from a point to a point . An example in which the path is no longer a straight line is that of a ray of light passing media of different optical density (which can be described with the help of a potential in the Lagrangian function), here the cheapest path ( optical path ) is no longer a straight line: the light beam is refracted. ${\displaystyle A,B}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle B}$

In quantum mechanics, one integrated with a path integral over all possible paths on which the particles according could reach, and weighted paths thereby with a "phase factor" proportional to the exponential function of the imaginary made and by the reduced Planck's constant divided effect functional . This is also called the sum of all paths , because it integrates over all paths, albeit with different weights. The amplitude is the same for each path, but the phase , which is determined by the respective effect, is different. The classical path is characterized by the fact that the variation of the effect disappears according to the Hamilton principle . Paths in the vicinity thus contribute in roughly the same phase, which leads to constructive interference. In the case of paths that are further away, the integrand oscillates so quickly for effects that are large compared to Planck's quantum of action (classic limit case) that the contributions of these paths cancel each other out. If, on the other hand, the effects are of the order of magnitude of Planck's quantum of action, as in typical quantum mechanical systems, paths in addition to the classical path also contribute to the path integral. ${\displaystyle A}$ ${\displaystyle B}$

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

Books

• Hagen Kleinert path integrals in quantum mechanics, statistics and polymer physics , Spektrum Akademischer Verlag 1993 (out of print, readable online 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 Integrals, Emended Edition 2005 , Dover Publications, 2010 (Editor Daniel F. Styer, who corrected numerous errors in the 1965 edition) New edition website with additions
• Jean Zinn-Justin Path Integrals in Quantum Mechanics , Oxford University Press 2005
• Harald JW Müller-Kirsten Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral , 2nd ed., World Scientific (Singapore, 2012)

Web links

• Jean Zinn-Justin Path Integral , Scholarpedia

References and footnotes

1. ^ ^{A b} Zinn-Justin, Jean: Quantum field theory and critical phenomena . Clarendon Press, Oxford 1996, ISBN 0-19-851882-X .
2. ↑ Wentzel, Zur Quantenoptik, Z. Physik, Volume 22, 1924, pp. 193-199.
3. ↑ Wentzel, On the quantum theory of the X-ray brake spectrum, Z. f. Physik, Volume 27, 1924, pp. 257-284.
4. ↑ Salvatore Antoci, Dierck-E. Liebscher, Wentzel's path integrals, Int. J. Theor. Phys. Vol. 37, pp. 531-535 (1998) (pdf). Then, in the mid-1960s , Thomas S. Kuhn came across Wentzel's contribution about a letter from Dirac from 1925, who at the time was one of the few (besides Max von Laue ) who noticed Wentzel's work.
5. ^ S. Antoci, DE Liebscher, The third way to quantum mechanics is the forgotten first , Ann. Fondation Louis de Broglie, Vol. 21, pp. 349-367 (1996)
6. ^ Dirac, The Lagrangian in Quantum Mechanics, Physikalische Zeitschrift der Sovietunion Babd 3, 1933, p. 64
7. ^ Feynman, Space-time approach to non-relativistic quantum mechanics, Rev. Mod. Phys., Volume 20, 1948, pp. 367-387.
8. ^ ^{A b c} K. Huang: Quarks Leptons & Gauge Fields . World Scientific, 1982.
9. ↑ The derivation of a path integral to a Fokker-Planck equation can be done according to the same scheme.
10. ↑ An example gives the product of two factors, of which the first is a constant that may diverge towards infinity, while the second factor represents a function that can be differentiated. Then the logarithm of the product is definitely differentiable, whereby the infinite constant is omitted. The addition of a third factor results in the addition of an additional summand when taking the logarithm, etc. ${\displaystyle x}$ ${\displaystyle x}$