Path integral: mathematical aspects

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Author: Prof. Sergio Albeverio, Bonn University
Author: Dr. Sonia Mazzucchi, Hausdorff Centre for Mathematics, Bonn, Germany

Prof. Sergio Albeverio accepted the invitation on 27 February 2009 (self-imposed deadline: 27 August 2009).

Figure 1: Your introductory color figure is here to "advertise" the article or summarize the main point.

Figure 1: Your introductory color figure is here to "advertise" the article or summarize the main point.

Integrals over spaces of paths or more generally of fields has been introduced as heuristic tools in several areas of physics and mathematics. Mathematically they should be intended as extensions of finite dimensional integrals suitable to cover the applications the heuristic path integrals were originally thought for.

Eponymous naming conventions are: functional integrals, infinite dimensional integrals, field integrals. Feynman path integrals (or functionals), and Wiener path integrals (or integrals with respect to Wiener measures) are special cases. In probability th concept of flat integral also occours. A particular realization of Gaussian path integrals is given by "white noise functionals".

In the article the general terminology "path integrals" will be used.

Probabilistic path integrals

Origins

Wiener, Brownian motion

Wiener measure, Wiener integrals, Brownian motion

Applications

Other Gaussian path integrals

powers of the Laplacian. fields

Path integrals associated with diffusion processes

Path integrals associated with infinitely divisible stochastic processes or infinitely divisible measures

Path integrals associated with stochastic (P)DE's

Stochastic quantization

Asymptotics of path integrals

laplace method

Extensions

discrete spaces, fractals

Applications

Physics Hydrodynamics, Biology, Economics

Feynman path integrals

Origins

In 1948, following a suggestion by Dirac, R. P. Feynman proposed a new suggestive description of the time evolution of the state of a non relativistic quantum particle moving in the $d$ dimensional space under the action of a force field with potential $V$ According to Feynman, the wave function $\psi$ evaluated at the time $t$ in the point $x\in\R^d$ , i.e. the solution of the Schroedinger equation,

(1)

$\left\{ \begin{array}{l} i\hbar\frac{\partial}{\partial t}\psi(t,x)=-\frac{\hbar^2}{2m}\Delta \psi(t,x) +V(x)\psi(t,x)\\ \psi (0,x)=\psi _0(x)\\ \end{array}\right.$

should be given by a "sum over all possible hystories of the system", that is by an heuristic integral over the space of paths $\gamma:[0,t]\to\R^d$ such that $\gamma(0)=x$ :

(2)

$\psi(t,x)=\int e^{\frac{i}{\hbar}S_t(\gamma)}\psi(0, \gamma(0))D\gamma,.$

In the formula above $D\gamma$ denotes a Lebesgue-type measure on the space of paths, $\hbar$ is the reduced Planck constant and $S_t(\gamma)$ is the classical action functional of the system evaluated along the path $\gamma$

$S_t(\gamma)=\int_0^t\frac{m}{2}\dot\gamma(s)^2ds-\int_0^tV(\gamma(s))ds$

Feynman's approach is particularly suggestive as it creates a bridge between the classical Lagrangian description of the physical world and the quantum one, reintroducing in quantum mechanics the classical concept of trajectory, which had been banned by the traditional formulation of the theory. It allows, at least heuristically, to associate a quantum evolution to each classical Lagrangian. Moreover it makes very intuitive the study of the "semiclassical limit" of quantum mechanics, i.e. the study of the behavior of the wave function when the Planck constant $\hbar$ is regarded as a small parameter which is allowed to converge to 0. In fact, when $\hbar$ becomes small, the integrand $e^{\frac{i}{\hbar}S_t(\gamma)}$ behaves as a strongly oscillatory function and, according to an heuristic application of the stationary phase method, the main contribution to the integral should come from those paths which make stationary the phase functional $S _t$ . These, by Hamilton's least action principle, are exactly the classical orbits of the system. Feynman extended this heuristic formulation to the description of the dynamics of more general quantum systems, including the relativistic quantum fields, and introducing an heuristic calculus that works even when rigorous arguments fail.

Mathematical problems

Despite the successfully predicting power of Feynman path integral, it lacks of mathematical rigour. First of all the Lebesgue-type flat measure $D\gamma$ on a space of path is not well defined from a mathematical point of view and cannot be used as ``reference measure, i.e. the measure with respect to which "Feynman measure" has density $e^{i\frac{S_t}{\hbar}(\gamma)}$ .

In 1960 Cameron proved that it is not even possible to construct Feynman measure as a Wiener measure with complex variance, i.e. as limit of finite dimensional approximations of the expression

$\frac{e^{\frac{i}{\hbar}\int_0^t \frac{m}{2}\dot\gamma (s)^2ds}D\gamma}{\int e^{\frac{i}{\hbar}\int_0^t \frac{m}{2}\dot\gamma (s)^2ds}D\gamma}$

as the resulting measure would have infinite total variation. Furthermore, it is possible to see that the resulting complex measure would have infinite total variation even on bounded sets (Even the Lebesgue measure on $\R^n$ has infinite total variation, but its total variation on bounded pluri-intervals is finite).

Since it is not possible to give meaning to the Feynman integral of a function $f$ on the space of path $\gamma$

$I(f)\equiv\int f(\gamma)\frac{e^{\frac{i}{\hbar}\int_0^t \frac{m}{2}\dot\gamma (s)^2ds}D\gamma}{\int e^{\frac{i}{\hbar}\int_0^t \frac{m}{2}\dot\gamma (s)^2ds}D\gamma}$

in terms an integral with respect to a $\sigma$ -additive measure, one can try to define $I(f)$ as a linear continuous functional on a suitable Banach algebra of functions $f$ .

Various approaches

Sequential approach

The present approach is the closer to Feynman's original derivation of its formula and it is largely implemented in the physical literature, also as a practical tool for exactly solvable models (see Grosche Steiner).

The starting point is the Lie-Kato-Trotter product formula, which allows one to write the solution of the SChroedinger equation (1) ion terms of a strong limit:

(3)

$\psi(t) = \lim_{n\to\infty} \left( e^{-\frac{i}{\hbar} \frac{t}{n} V} e^{-\frac{i}{\hbar} \frac{t}{n} H_0} \right)^n\psi(0)$

where $H_0=-\frac{\hbar^2}{2m}\Delta$ is the free quantum Hamiltonian operator, and $V$ is the multiplication operator associated to the potential $V$ . By taking an initial datum $\psi(0)$ belonging to the Schwartz space $S(\R^d)$ and inserting in Equation (3) the formula for the Green function of the quantum free evolution operator $e^{-\frac{i}{\hbar} \frac{t}{n} H_0}$ one gets:

(4)

$\psi(t,x) = \lim_{n\to\infty} \left( 2\pi i\frac{\hbar}{m} \frac{t}{n} \right)^{ -\frac{d n}{2}} \int\limits_{\R^{nd}} e^{-\frac{i}{\hbar} \sum_{j=1}^{n} \left[ \frac{m}{2} \frac{\left( x_j-x_{j-1}\right)^2 }{\left( \frac{t}{n} \right)^2} - V\left( x_j \right)\right] \frac{t}{n}} \phi(x_0) dx_0 \ldots dx_{n-1}$

The latter expression can be interpreted as the finite dimensional approximation of a path integral. Indeed, if $\gamma$ is a continuous trajectory from $[0,t]$ to $\R^d$ , with $\gamma(t)=x$ , let us set $x_j:=\gamma(jt/n)$ , fo r $j=0,\dots n$ . The exponent in the integrand can be interpreted in terms of the Riemann sum of the classical action functional evaluated along the path $\gamma$ :

$S_t(\gamma)= \int_0^t\Big(\frac{m}{2}\dot\gamma^2(s)-V(\gamma(s))\Big)ds = \lim_{n\to\infty}\sum_{j=1}^{n} \left[ \frac{m}{2} \frac{\left( x_j-x_{j-1}\right)^2 }{\left( \frac{t}{n} \right)^2} - V\left( x_j \right) \right] \frac{t}{n}.$

These relations can provide a rigorous definition of Feynman integration in two different ways.

One possibility is the study of a generalization of Trotter product formula 3, which is a special case of the semigroup product formula

(5)

$s - \lim_{n\to\infty} (F(t/n))^n=\exp(tF'(0)),$

where $t\mapsto F(t)$ is a strongly continuous function from $\R$ (or $\R^+$ ) into the space of bounded linear operators on an Hilbert space $\mathcal H$ , while $F'(0)$ has to be interpreted as some operator extension of the strong limit $s - \lim_{t\to 0}t^{-1}(F(t)-I)$ . In particular if $A,B$ are self-adjoint operators in $\mathcal H$ and $F(t)=e^{itA}e^{itB}$ , one gets formally the Trotter product formula:

(6)

$s - \lim_{n\to\infty} (e^{itA/n}e^{itB/n})^n=e^{it(A+B)}$

(where the sum $A+B$ has to be suitably interpreted) Nelson in 1964 applied equation 6 to the rigorous mathematical definition of Feynman path integrals, under the assumption that the potential $V$ belongs to the class considered by Kato (CITATION OR BRIEF DESCRIPTION). Some time later Friedman studied formula 5 in connection with the description of continuous quantum observation.

Another version of the sequential approach is also known as time slicing approximation and consists in the definition of the Feynman integral as the limit of finite dimensional approximations of the form (4), by approximating the paths $\gamma$ with piecewise linear paths or piecewise classical paths. The time slicing approximation, in particular with piecewise polygonal paths, is extensively used in the physical literature not only as a tool for the definition of the Feynman integral, but also as a practical method of computation for particular solvable models

Analytic continuation

One of the first attempts to the rigorous mathematical realization of Feynman path integrals involves analytic continuation of Gaussian Wiener integrals. In fact, by considering a Wiener measure $W_\lambda$ with covariance $\lambda\in\R^+$ , and a suitable functional $f$ on the space $C_t$ of continuous path on the interval $[0,t]$ , the following formula holds:

(7)

$\int_{C_t}f(\omega)dW_\lambda(\omega)=\int_{C_t}f(\sqrt \lambda\omega)dW(\omega)$

If $\lambda$ is complex, the left hand side of () is not well defined, but the right hand side can still be meaningful, provided that the functional $f$ has suitable analyticity and measurability properties. In particular, for $\lambda=i$ , it is the natural candidate for the analytic-continued Wiener integral.

Concerning the application of this functional to the Feynman path integral representation of the solution of the Schroedinger equation, one considers the heat equation with potential

(8)

$- \frac{\partial}{\partial t}u(t,x)=-\frac{1}{2} \Delta_x u(t,x) +V(x)u(t,x), \qquad x\in\R^d$

and the representation of its solutions in term of a Wiener integral, i.e. the Feynman-Kac formula:

(9)

$u(t,x)=\int_{C_t}e^{-\int _0^tV(\omega(s)+x))ds}u(0,\omega(t)+x)dW(\omega).$

By introducing in equations (8) and (9) a real positive parameter $\lambda$ , related to the physical time, or to the mass, or to the Planck constant, and by allowing it to assume complex values, one gets, at least heuristically, for $\lambda=i$ the Schroedinger equation and the functional integral representation of its solution. This procedure can be rigorously implemented under analyticity and slow-growing conditions on the potential and on the initial datum. In partrcular it is possible to consider potentials hich are the sum of a quadratic part plus a bounded perturbation, potential with singularities (Nelason and Doss), potentials with particular polynomial growth (Doss) and potentials with exponential growth that are Laplace transform of measures (Albeverio, Brzezniak and Haba)

Daubechies and Klauder approach

White noise

Parseval

Infinite dimensional oscillatory integrals

Non standard analysis

Poisson processes

p-adici

Applications

Quantum mechanics

Stochastic Schroedinger, measurement theory

QFT

Topological field theory

Staistical mechanics (classical-quantum). Feynman-vernon.

Quantum computing

Chern-Simons (Freed,...). Topological invariants (Kaufmann)

Quantum fluids (?)

Dissipative systems

Asymptotics

Sequential: Kumano-Go and Fukushima. Doss Ben arous Stationary phase

Mitoma su Chern Simon

Trace formula (Blanchard, Brzniak)

Invited by:

Dr. Riccardo Guida, Institut de Physique Théorique; CEA, IPhT; CNRS; Gif-sur-Yvette, France

Retrieved from "http://www.scholarpedia.org/article/Path_integral:_mathematical_aspects"

Categories: Physics | Quantum Mechanics | Mathematical Physics

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