Path integral: mathematical aspects

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Sergio Albeverio and Sonia Mazzucchi (2011), Scholarpedia, 6(1):8832. doi:10.4249/scholarpedia.8832 revision #137047 [link to/cite this article]
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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 the concept of flat integral also occurs. A particular realization of Gaussian path integrals is given by "white noise functionals".

In the present article the mathematical theory of Feynman path integrals will be presented, while the theory and the applications of path integrals of probabilistic type, as Wiener integrals, will be presented in Probabilistic integrals: mathematical aspects.

Feynman path integrals should not be confused with the "Feynman integrals" occurring in the study of terms arising in perturbation expansions in quantum field theory, which are associated with Feynman diagrams. The latter are finite dimensional complex integrals, and are discussed in particular in relation to renormalization.


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 Schrödinger equation, \[ \left\{ \begin{array}{l} i\hbar\frac{\partial}{\partial t}\psi(t,x)=-\frac{\hbar^2}{2m}\Delta \psi(t,x) +V(x)\psi(t,x)\\ \psi (0,x)=\psi _0(x)\\ \end{array}\right. \] should be given by a "sum over all possible histories of the system", that is by an heuristic integral over the space of paths \(\gamma:[0,t]\to\R^d\) such that \(\gamma(0)=x\ :\) \[ \psi(t,x)=\int_\Gamma e^{\frac{i}{\hbar}S_t(\gamma)}\psi(0, \gamma(0))D\gamma. \] In the formula above \(D\gamma\) denotes a Lebesgue-type measure on the space \(\Gamma\) of paths, \(\hbar\) is the reduced Planck constant, \(m\) is the mass of the particle 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 real-valued 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 extrapolation of the stationary phase method to the path integral case, 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 used it to deduce procedures (Feynman rules, Feynman diagrams) that permit to perform computations which yield numbers even when rigorous arguments fail. Since then heuristic Feynman path integrals have become the basis of much of contemporary physics (including quantum fields, in particular gauge fields), and has stimulated the development of many areas of mathematics.

Mathematical problems

Despite the successfully predicting power of the concept of Feynman path integral, it lacks of mathematical rigour. First of all the Lebesgue-type flat measure \(D\gamma\) on a space of paths is not defined from a mathematical point of view and cannot be used as a reference measure, i.e. a 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's measure" as a Wiener measure with a 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, even on bounded sets in path space (as opposite to the case of Lebesgue measure on \({\mathbb R}^n\ ,\) which has infinite total variation, but its total variation on bounded measurable subsets of \({\mathbb R}^n\) is finite).

Since it is not possible to give meaning to the Feynman integral \(I(f)\) of a function \(f\) on the space \(\Gamma\) of paths \(\gamma\) \[ I(f)\equiv\int f(\gamma)\frac{e^{\frac{i}{\hbar}\int_0^t \frac{m}{2}\dot\gamma (s)^2ds}D\gamma}{\int e^{\frac{i}{\hbar}\int_0^t \frac{m}{2}\dot\gamma (s)^2ds}D\gamma} \] in terms of an integral with respect to a \(\sigma\)-additive (complex-valued) measure, one can try to define \(I(f)\) as a linear continuous functional on a suitable linear space of functions \(f\ .\)

Various approaches

Sequential approach

This approach is the closest to Feynman's original derivation of its formula and it is largely implemented in the physical literature, also as a practical tool for performing computations in exactly solvable models, see, e.g., Grosche and Steiner (1998).

The starting point is the Lie-Kato-Trotter product formula, which allows one to write the solution \(\psi\)of #schroedinger in terms of a strong limit in the \(L^2({\mathbb R}^d)\) space: \[\tag{1} \psi(t) = \lim_{n\to\infty} \left( e^{-\frac{i}{\hbar} \frac{t}{n} V} e^{-\frac{i}{\hbar} \frac{t}{n} H_0} \right)^n\psi(0), \]

where \( H_0=-\frac{\hbar^2}{2m}\Delta \) is the free quantum Hamiltonian operator, and \(V \) is the multiplication operator in \(L^2({\mathbb R}^d)\) associated to the potential \(V \ .\) By taking an initial datum \( \psi(0) \) belonging, e.g., to the Schwartz space \( S(\R^d) \) and inserting in Equation (1) the formula for the Green function of the quantum free evolution operator \( e^{-\frac{i}{\hbar} \frac{t}{n} H_0} \ ,\) \[ e^{-\frac{i}{\hbar} \frac{t}{n} H_0} (x,y)=\left(\frac{2\pi i\hbar t}{mn}\right)^{-d/2}e^{-\frac{i}{\hbar} \frac{m}{2t/n} |x-y|^2}, \] one gets: \[\tag{2} \psi(t,x) = \lim_{n\to\infty} \left( 2\pi i\frac{\hbar}{m} \frac{t}{n} \right)^{ -\frac{d n}{2}} \int\limits_{\R^{nd}} e^{-\frac{i}{\hbar} \sum_{j=1}^{n} \left[ \frac{m}{2} \frac{\left( x_j-x_{j-1}\right)^2 }{\left( \frac{t}{n} \right)^2} - V\left( x_j \right)\right] \frac{t}{n}} \phi(x_0) dx_0 \ldots dx_{n-1}. \]

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 \(x_j:=\gamma(jt/n)\ ,\) for \(j=0,\dots n\ .\) The exponent in the integrand can be interpreted in terms of the Riemann sum of the classical action functional evaluated along the path \(\gamma\ :\) \[ S_t(\gamma)= \int_0^t\Big(\frac{m}{2}\dot\gamma(s)^2-V(\gamma(s))\Big)ds = \lim_{n\to\infty}\sum_{j=1}^{n} \left[ \frac{m}{2} \frac{\left( x_j-x_{j-1}\right)^2 }{\left( \frac{t}{n} \right)^2} - V\left( x_j \right) \right] \frac{t}{n}. \] These relations can provide a rigorous definition of Feynman integration in two different ways.

One possibility is the study of a generalization of Trotter's product formula (1), which is a special case of the semigroup product formula \[\tag{3} s - \lim_{n\to\infty} (F(t/n))^n=\exp(tF'(0)), \]

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: \[ s - \lim_{n\to\infty} (e^{itA/n}e^{itB/n})^n=e^{it(A+B)} \] (where the sum \(A+B\) has to be suitably interpreted). Nelson (1964) applied the latter equation to the rigorous mathematical definition of Feynman path integrals, under the assumption that the potential \(V\) belongs to the class considered by Kato (see the book of Johnson and Lapidus for a detailed discussion). Some time later Friedman (1971/72) studied (3) in connection with the description of continuous quantum observations (see also A.D. Sloan (1981) and the references in the book by S. Albeverio, R. Høegh-Krohn and S. Mazzucchi (2008).

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 given in (2), by approximating the paths \(\gamma\) with piecewise linear paths or piecewise classical paths (i.e. paths which piecewise solve the classical Newton equation of motion). 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 path integral, but also as a practical method of computation for particular solvable models, see, e.g. the book by Grosche and Steiner and the book by Kleinert. Rigorous versions of the time slicing approximation with piecewise classical paths have been given by D. Fujiwara and coworkers.

Analytic continuation

One of the first attempts to the rigorous mathematical realization of Feynman path integrals involved 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 paths on the interval\([0,t]\ ,\) the following formula holds: \[ \int_{C_t}f(\omega)dW_\lambda(\omega)=\int_{C_t}f(\sqrt \lambda\omega)dW(\omega). \] If \(\lambda\) is complex, the left hand side 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 analytically-continued Wiener integral.

Concerning the application of this functional to the Feynman path integral representation of the solution of the Schrödinger equation, one considers the heat equation with potential \[ - \frac{\partial}{\partial t}u(t,x)=-\frac{1}{2} \Delta_x u(t,x) +V(x)u(t,x), \qquad x\in\R^d \] and the representation of its solutions in term of a Wiener integral. This is given by the Feynman-Kac formula: \[ u(t,x)=\int_{C_t}e^{-\int _0^tV(\omega(s)+x))ds}u(0,\omega(t)+x)dW(\omega). \] This formula is valid. e.g., if \(V\) is bounded and continuous, but also when \(V\) is fairly arbitrary and lower bounded, see, e.g., Johnson and Lapidus.

By introducing in the heat equation and in the corresponding Feynman-Kac formula 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 Schrödinger 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 particular it is possible to consider potentials which are sums of a quadratic part plus a bounded potential with singularities (Nelson (1964), Doss (1980)), potentials with particular polynomial growth (Doss (1980), Albeverio and Mazzucchi (2009), Albeverio, Khrennikov and Smolyanov (1999), Grothaus, Streit and Vogel (2009)) and potentials with exponential growth that are Laplace transforms of measures (Albeverio, Brzeźniak and Haba (1998), Kuna, Streit and Westerkamp (1998)).

Daubechies and Klauder's approach

An alternative way to define Feynman path integrals by means of convergent Wiener integrals has been proposed by I. Daubechies and J. Klauder between 1982 and 1985. Let us consider coherent states \(\phi_{q,p}\ ,\) defined by \[ \phi_{q,p}=e^{i(pQ-qP)}\phi_0, \] where \(Q,P\) are respectively the quantum position and momentum operators and \(\phi_0\) is the ground state of an harmonic oscillator in the \(d\)-dimensional Euclidean space. The matrix elements of the unitary evolution operator \(U(t)=e^{-\frac{i}{\hbar}Ht}\) between two coherent states (written for simplicity in the case where \(\hbar=1\) and \(d=1\)) should be formally be given by the phase space Feynman path integral: \[\tag{4} \langle \phi_{q'',p''},U(t)\phi_{q',p'}\rangle={\mathcal N} \int e^{\frac{i}{2}\int_0^t(p(s)\dot q(s)-q(s)\dot p(s))ds-i\int_0^tH(p(s),q(s))ds}dpdq, \]

where \((q(s),p(s))_{s\in[0,t]}\) represents a generic path in the phase space, while \(H:{\mathbb R}^2\to{\mathbb R}\) is defined as the matrix element of the quantum Hamiltonian operator with respect to \(\phi_{q,p}\ ,\) i.e. \[ H(q,p)= \langle \phi_{q,p},H \phi_{q,p}\rangle, \qquad (q,p)\in{\mathbb R}^2, \] and \({\mathcal N}\) represents a normalization constant. The heuristic expression (4) is defined in terms of the following Wiener integral \[\tag{5} \int e^{\frac{i}{2}\int_0^t(p(s)d q(s)-q(s)d p(s))-i\int _0^t h(p(s),q(s))ds}dW_\nu(p,q) , \]

obtained by inserting into (4) an extra factor \[ e^{-\frac{1}{2\nu}\int_0^t(\dot p(s)^2+\dot q(s)^2)ds }, \] representing formally the density of a Wiener measure (on the space of paths \((q(s),p(s))_{s\in[0,t]}\) in the phase space) with diffusion constant \(\nu>0\ .\) The exponent \(\int_0^t(p(s)\dot q(s)-q(s)\dot p(s))ds\) is replaced by \(\int_0^t(p(s)d q(s)-q(s)d p(s))\) and interpreted as a (Ito or Stratonovich) stochastic integral. The function \(H:{\mathbb R}^2\to {\mathbb R} \) is replaced with \(h:{\mathbb R}^2\to {\mathbb R} \ ,\) given by \[\tag{6} h(p,q)=\exp\big(-\frac{1}{2}(\partial _p^2+\partial_q^2)\big)H(p,q),\qquad (p,q)\in{\mathbb R}^2. \]

The matrix elements \(\langle \phi_{q'',p''},U(t)\phi_{q',p'}\rangle \) are given by the following limit \[ \langle \phi_{q'',p''},U(t)\phi_{q',p'}\rangle =\lim_{\nu\to\infty}2\pi e^{\nu t/2}\int e^{\frac{i}{2}\int_0^t(p(s)d q(s)-q(s)d p(s))-i\int _0^th(p(s),q(s))ds}dW_\nu(p,q), \] where \(W_\nu\) is the product of two independent Wiener measures (one in \(p\) and one in \(q\)) with diffusion constant \(\nu\) pinned at \(p',q' \) for \(s=0\) and at \(p'',q'' \) for \(s=t\ .\) This formula is valid for all self-adjoint Hamiltonian operators \(H\) on \(L^2({\mathbb R})\) for which the linear span of the harmonic oscillator eigenstates is a core and such that \[ H=\int h(p,q)P_{p,q}\frac{dpdq}{2\pi}, \] where \(P_{p,q}:L^2({\mathbb R})\to L^2({\mathbb R})\) for \((p,q)\in{\mathbb R}^2\) is the projection operator \[ P_{p,q}(\psi)=\phi_{q',p'}\langle \phi_{q',p'},\psi\rangle, \qquad\psi\in L^2({\mathbb R}). \] One has also to impose that for all \(\alpha>0\) the bound \[ \int_{{\mathbb R}^2} | h(p,q)|^2e^{-\alpha(p^2+q^2)}dpdq <\infty \] is satisfied. The class of operators satisfying these condition includes all Hamiltonians that are polynomial in \( P\) and \( Q\ .\) The same technique has also been applied to systems with spin.

White noise

The white noise approach realizes the heuristic Feynman integrand , namely \(e^{\frac{i}{2}\int_0^t\dot\gamma(\tau)^2d\tau}\ ,\) as an infinite dimensional distribution and the Feynman path integral \(\int e^{\frac{i}{2}\int_0^t\dot\gamma(\tau)^2d\tau}f(\gamma)d\gamma\) as a distributional pairing with a suitable test function \(f\) on path space. As the heuristic Lebesgue measure \(d\gamma\) is not well defined on an infinite dimensional space of paths, it has to be replaced by a Gaussian measure. More precisely the underlying measure space is the d-dimensional white noise space \((S'_d,{\mathcal B}, \mu)\ ,\) where \( S'_d\) is the space of vector-valued Schwartz distributions \(S'_d:=S({\mathbb R})\otimes {\mathbb R}^d\ ,\) \(\mu \) is the Gaussian measure on the \(\sigma\)-algebra \({\mathcal B}\) generated by the cylindrical sets in path space, defined via Minlos' theorem by its characteristic functional \[ \xi\in S_d\mapsto \int_{S'_d}e^{i\langle \xi,\omega\rangle }d\mu(\omega)=e^{-\frac{|\xi|^2}{2}}, \qquad |\xi|^2=\int_{\mathbb R} |\xi (\tau)|^2d\tau, \] where \(\langle \xi,\omega\rangle\) is the distributional pairing between an element \(\xi\) belonging to the nuclear space \(S_d:=S({\mathbb R})\otimes {\mathbb R}^d\ ,\) \(S({\mathbb R})\) being the space of Schwartz test functions, and \(\omega \in S'_d\ .\) Formally, the elements \(\omega \in S'_d\) represents the velocities of the Brownian paths, as a version of the \(d-\)dimensional Brownian motion is given by \[ B_\tau(\omega):=(\langle 1_{[0,\tau]}\otimes e_1,\omega\rangle,\dots,1_{[0,\tau]}\otimes e_d,\omega\rangle), \] where \(\{e_1,...,e_d\} \) is a basis of \({\mathbb R}^d \) and \( 1_{[0,\tau]}\) is the characteristic function of the interval \( [0,\tau]\ ,\) \( \tau>0\ .\)

By considering the space \(L^2(\mu):=L^2(S'_d,{\mathcal B},d\mu) \ ,\) it is possible to construct in a completely analogous way as for the finite dimensional case the Gelfand triple \[ (S_d)\subset L^2(\mu)\subset (S_d)'. \] The elements of \((S_d)'\) are called white noise (or Hida) distributions, while the elements in \((S_d)\) are the corresponding test functions (both relative to infinite dimensional real spaces. For a detailed treatment of this topic see, e.g., the books by T. Hida, H.H. Kuo, J. Potthoff, L. Streit (1993), N. Obata (1994) and H.H. Kuo (1996).

An Hida distribution \(\Phi\in (S_d)'\) can be uniquely characterized by its T-transform, an infinite dimensional analogue of the Fourier transform, that is the functional \(T\Phi:S_d\to{\mathbb C}\) defined by \[ \xi\in S_d\mapsto T\Phi(\xi):=\langle\langle e^{i\langle \xi,\,\cdot\,\rangle},\Phi\rangle\rangle, \] where \(\langle\langle e^{i\langle \xi,\,\cdot\,\rangle},\Phi\rangle\rangle\) denotes the distributional pairing between \(e^{i\langle \xi,\,\cdot\,\rangle}\in(S_d)\) and \(\Phi\in(S'_d)\ .\)

A basic characterization theorem allows one to identify the functionals that are T-transforms of Hida distributions. Indeed it has been proved by J. Potthoff and L. Streit in 1991 that a functional \(F:S_d\to{\mathbb C}\) is the T-transform of a unique Hida distribution iff it has the following properties:

  • For all \(\xi,\eta\in S_d\) the mapping

\[ z\in {\mathbb R}\mapsto F(\xi+z\eta)\in{\mathbb C} \] has an analytic continuation to \({\mathbb C}\) as an entire function.

  • There exist some positive constants \(a,b\) and a continuous norm \(\|\;\|\) on \(S_d\) such that for all \(z\in{\mathbb C}\ ,\) \(\xi\in S_d\ ,\)

\[ |F(z\xi)|\leq ae^{b|z|^2\|\xi\|^2}. \]

Extensions of the characterization theorem have been given by Kondratiev (1992), followed by various authors, see, e.g., Kondratiev, Leukert, Potthoff, Streit, and Westerkamp (1996).

The fundamental ideas for the definition of Feynman path integrals in terms of white noise calculus were presented by Hida and Streit (1983). In particular the Feynman integrand for the free particle \(e^{\frac{i}{2}\int_0^t\dot\gamma(\tau)d\tau}\) can be realized as the Hida distribution, which can be written as \(I_0(x,t;x_0,t_0)(\omega)=Ne^{\frac{i+1}{2}\int_0^t \omega(\tau)^2ds}\delta(\gamma(0)-y) \ ,\) where the paths \(\gamma\) are modeled by \(\gamma(\tau)=x-\int_\tau^t\omega(\sigma)d\sigma\ ,\) \(N \) stands for normalization and \(\delta(\gamma(0)-y) \) fixes the initial point of the path.

These construction techniques allow one to handle more general potentials, such as the (time dependent) harmonic oscillator, the Fourier and Laplace transforms of bounded measures and some potentials with polynomial growth, see, e.g., the work by M. de Faria, J. Potthoff, L. Streit (1991), D.C. Khandekar and L. Streit (1992), A. Lascheck, P. Leukert, L. Streit, W. Westerkamp (1993), M. Grothaus, D.C. Khandekar, J.L. da Silva, L. Streit (1997), T. Kuna, L. Streit, W. Westerkamp (1998), M. Grothaus, L. Streit, A. Vogel (2009).

Important applications of the white noise approach include the mathematical Chern-Simons models of topological quantum fields, following Atiyah-Witten basic ideas.

Parseval duality

This approach was introduced by K. Itô in 1961 and further systematically and extensively developed by Albeverio and Høegh-Krohn in the 70s. The paths \(\gamma\) over which the Feynman integration is performed are represented by the elements of a real separable Hilbert space \( \mathcal H\ ,\) with norm \(|\gamma|^2=\int_0^t\dot\gamma(\tau)^2 d\tau\ ,\) while the heuristic Feynman path integral \(\int e^{\frac{i}{2}\int_0^t\dot\gamma(\tau)^2 d\tau}f(\gamma)d\gamma\) is realized as an infinite dimensional Fresnel integral on the Hilbert space \( \mathcal H\ ,\) written as \(\tilde \int_{\mathcal H} e^{\frac{i}{2}|\gamma|^2}f(\gamma)d\gamma \ .\) The latter is defined for the functions \(f:{\mathcal H}\to{\mathbb C}\) that are Fourier transforms of complex bounded variation measures \(\mu_f\) on \(\mathcal H\ ,\) that is functions of the form: \[\tag{7} f(\gamma)=\int _{\mathcal H}e^{i\langle \gamma,\eta\rangle }d\mu_f(\eta), \]

and it is defined as the right hand side of the following Parseval equality: \[\tag{8} \tilde \int_{\mathcal H} e^{\frac{i}{2}|\gamma|^2}f(\gamma)d\gamma= \int_{\mathcal H} e^{-\frac{i}{2}|\gamma|^2}d\mu_f(\gamma). \]

It is possible to prove that the functions \(f\) as in (7) form a Banach algebra \({\mathcal F}({\mathcal H}) \ ,\) where the norm of a function \(f\) is the total variation of the corresponding measure \(\mu_f\ ,\) and the Fresnel integral is a linear continuous functional on \({\mathcal F}({\mathcal H}) \ .\)

Main applications of this approach include the development of detailed method of stationary phase in infinite dimensions, with applications to the study of the relations between quantum and classical mechanics on \(\R^d\) (with potentials are Fourier transforms of bounded complex measures on \(\R^d\ ,\) scattering theory and the construction of Abelian Chern-Simons model, see the book by Albeverio, Høegh-Krohn and Mazzucchi (2008).

Infinite dimensional oscillatory integrals

This approach is an extension and generalization of the Parseval duality approach. It was introduced by Elworthy and Truman in 1984 and further developed by Albeverio and Brzezniak in the 90s. The main idea is the generalization to an infinite dimensional setting of the definition and the properties of the classical oscillatory integrals \[ \int_{{\mathbb R}^n}e^{\frac{i}{\hbar}\Phi(x)}f(x)dx, \] where \(\Phi:{\mathbb R}^n\to {\mathbb R} \) is the phase function, \( f:{\mathbb R}^n\to {\mathbb C} \) is a measurable function ("amplitude") and \(\hbar \) a real positive parameter. Well known examples of such integrals are the Fresnel integrals \[ \int_{{\mathbb R} }e^{\frac{i}{\hbar}|x|^2}f(x)dx, \] and the Airy integrals \[ \int_{{\mathbb R} }e^{\frac{i}{\hbar}x^3}f(x)dx, \] both appearing. e.g., in the theory of wave diffraction. Following Hörmander, oscillatory integrals can be defined, even in the case where the function \(f\) is not summable, as limits of a sequence of regularized integrals. More precisely, by introducing a regularizing function \( \phi\in S({\mathbb R}^n) \) such that \( \phi(0)=1 \ ,\) the integral is defined as the limit (when this exists and it does not depend on \( \phi\)): \[ \int_{{\mathbb R}^n}e^{\frac{i}{\hbar}\Phi(x)}f(x)dx:= \lim_{\epsilon \to 0}\int_{{\mathbb R}^n}e^{\frac{i}{\hbar}\Phi(x)}f(x)\phi (\epsilon x)dx. \] General classes of phase and amplitude functions for which the oscillatory integral is well defined are presented in Hörmander and Albeverio and Mazzucchi (2005).

This definition can be generalized to the case where the integration is performed on an infinite dimensional real separable Hilbert space \( \mathcal H\ .\) By considering a sequence \( \{ P_n \}_{n\in{\mathbb N}}\) of projection operators, such that \( dim( P_n {\mathcal H})=n\ ,\) \( P_n{\mathcal H}\subset P_{n+1}({\mathcal H})\) and \( \lim_{n\to\infty }P_n x\to x\) for all \( x\in{\mathcal H} \ ,\) an infinite dimensional oscillatory integral is defined as the limit of the finite dimensional approximations (when the limit exist and does not depend on the sequence \( \{ P_n \}_{n\in{\mathbb N}}\)): \[ \int_{{\mathcal H} }e^{\frac{i}{\hbar}\Phi(x)}f(x)dx:=\lim_{n\to\infty}\frac{\int_{P_n{\mathcal H} }e^{\frac{i}{\hbar}\Phi(P_nx)}f(P_nx)dP_nx}{\int_{P_n{\mathcal H} }e^{\frac{i}{\hbar}\Phi(P_nx)} dP_nx }, \] where the integrals on the right hand side are meant as finite dimensional oscillatory integrals. In the case where the phase function \( \Phi\) is a quadratic form, the integral is also called infinite dimensional Fresnel integral. The complete characterization of the largest class of Fresnel integrable functions is still an open problem, even in finite dimensions, but it is possible to find interesting subsets of it, as the Fresnel algebra. Indeed, given any function \( f:{\mathcal H}\to {\mathbb C} \) verifying (7) for some \( \mu_f \ ,\) it is possible to prove that it is Fresnel integrable and its infinite dimensional Fresnel integral is given by the Parseval equality, i.e. (8) (that in this setting is a theorem and not a definition as it was in the approach described in the previous section).

The proof of equalities of Parseval's type for infinite dimensional oscillatory integrals has been extended by Albeverio and Mazzucchi in 2005 to cover the case where the phase function is a polynomial whose leading term has degree 4 (this case is of particular relevance in physics, a polynomial interaction of 4th order being typical for Lagrangian quantum field theory).

The Feynman path integral representation for the solution of the Schrödinger equation in the case where the potential and the initial datum are Fourier transforms of a complex bounded variation measure on \({\mathbb R}^d \) has been realized as an infinite dimensional oscillatory integral on the Hilbert space \({\mathcal H}_t \) of absolutely continuous paths \( \gamma:[0,t]\to{\mathbb R}^d\) with weak derivative \( \dot\gamma\in L^2({\mathbb R}^d)\ ,\) and, e.g., fixed final point \(\gamma(t)=0 \ ,\) endowed with the inner product \[ \langle\gamma_1,\gamma_2\rangle=\int_0^t\dot \gamma_1(\tau)\dot\gamma_2(\tau)d\tau \ .\] (The case of a fixed initial point \(\gamma(0)=0 \) can be treated analogously). By assuming that \(V,\psi_0\in{\mathcal F}({\mathbb R}^d)\ ,\) it is possible to prove that the following infinite dimensional oscillatory integral on \( {\mathcal H}_t\) \[ \int_{{\mathcal H}_t}e^{\frac{i}{2\hbar}|\gamma|^2}e^{-\frac{i}{\hbar}\int_0^tV(\gamma(\tau)+x)d\tau}\psi_0(\gamma(0)+x)d\gamma \] is well defined and is a representation for the solution \(\psi (t,x)\) of the Schrödinger equation. Analogous results have been obtained by Albeverio and Mazzucchi in 2005 for potentials of polynomial type with quartic growth, including also the case where the potential can depend explicitly also on time.

Non standard analysis

An alternative approach to the rigorous mathematical definition of Feynman path integrals makes use of nonstandard analysis and is described in the book by Albeverio et al. (1986). As in the sequential approach, the starting point is the finite dimensional approximation of the Feynman path integral by means of piecewise linear paths, by means of (2): \[ \phi(t,x) = \lim_{n\to\infty} \left( 2\pi i \frac{\hbar}{m} \frac{t}{n} \right)^{ -\frac{d n}{2}} \int_{{\mathbb R}^nd} e^{\frac{i}{\hbar}\sum_{j=1}^{n} \left[ \frac{m}{2} \frac{\left( x_j-x_{j-1}\right)^2 }{\left( \frac{t}{n} \right)^2} - V\left( x_j \right)\right] \frac{t}{n}} \phi(x_0) dx_0 \ldots dx_{n-1}. \] The leading idea is the extension of the expression \[ \left( 2\pi i \frac{\hbar}{m} \frac{t}{n} \right)^{ -\frac{d n}{2}} \int_{{\mathbb R}^nd} e^{\frac{i}{\hbar}\sum_{j=1}^{n} \left[ \frac{m}{2} \frac{\left( x_j-x_{j-1}\right)^2 }{\left( \frac{t}{n} \right)^2} - V\left( x_j \right)\right] \frac{t}{n}} \phi(x_0) dx_0 \ldots dx_{n-1} \] from \(n\in {\mathbb N}\) to a nonstandard hyperfinite infinite \(n\in \,^*{\mathbb N}\ .\) The result is just an internal quantity. For a suitable class of potentials its standard part can be shown to exist and to solve the Schrödinger equation. Even if this approach provides a very suggestive realization of the Feynman path integrals, it has not been systematically developed yet, some contributions have been provided by T. Nakamura (1991) and K. Loo (2000). Recently F.S. Herzberg (2010) has initiated an interesting approach to the Feynman path integral following Nelson's radically elementary mathematics.

Poisson processes

This approach has been originally proposed by V. P. Maslov and A. M. Chebotarev in 1979 and further developed by S. Albeverio, Ph. Blanchard, Ph. Combe, R. Høegh-Krohn, R. Rodriguez, M. Sirugue and M. Sirugue-Collin in the 80s. Some recent results and new applications have been given by V.N. Kolokoltsov.

The main point is the definition of the Feynman path integral construction of the solution to the Schrödinger equation in momentum representation: \[ \left\{ \begin{array}{l} \frac{\partial}{\partial t}\tilde\psi(p)=-\frac{i}{2}p^2\tilde \psi(p)-i V(-i\nabla_p)\tilde\psi(p)\\ \tilde\psi (0,p)=\tilde\phi(p),\\ \end{array}\right. \] \(t\geq 0, p\in\R^d\ ,\) in terms of the expectation with respect to the probability measure underlying a Poisson process on \(\R^d\ .\)

The potentials \(V\) which can be handled by this method are those which are Fourier transforms of complex bounded measures on \({\mathbb R}^d\) (which we already mentioned under Parseval approach): \[ V(x)=\int_{{\mathbb R}^d}e^{i kx}d\mu _v(k),\qquad x\in{\mathbb R}^d. \] In fact for any complex bounded variation measure \(\mu \) on \({\mathbb R}^d\ ,\) there exist a positive finite measure \(\nu\) and a complex-valued measurable function \(f\) such that \(\mu (d k)=f(k)\nu(d k)\ .\) Without loss of generality one can assume that \(\nu(\{ 0\})=0\ ,\) as otherwise the condition can be fulfilled by a translation of the potential. The measure \(\nu\) is then a finite Lévy measure and one can consider the Poisson process having Lévy measure \(\nu\) (see e.g. the book by P. Protter (1990) for these concepts). This process has almost surely piecewise constant paths. A typical path \(P\) on the time interval \([0,t]\) is defined by a finite number of independent random jumps \(\delta_1,...,\delta_n\ ,\) distributed according to the probability measure \(\nu/\lambda_\nu\ ,\) with \(\lambda_\nu=\nu({\mathbb R}^d)>0\ ,\) occurring at random times \(\tau_1,...\tau_n\ ,\) distributed according to a Poisson measure with intensity \(\lambda_\nu\ .\)

Under the assumption that \(\tilde\phi(p)\) is a bounded continuous function, it is possible to prove that the solution \(\tilde\psi\) of the Schrödinger equation can be represented by the following path integral: \[ \tilde\psi(t,p)=e^{t\lambda_\nu}{\mathbb E}_p^{[0,t]}[e^{-\frac{i}{2}\sum_{j=0}^n(P_j,P_j)(\tau_{j+1}-\tau_{j})\prod_{j=1}^n(-if(\delta_j))}\tilde\phi (P(t))], \] where the expectation is taken with respect to the measure associated to the Poisson process and the sample path \(P(\cdot)\) is given by \[ P(\tau)=\left\{ \begin{array}{l} P_0=p,\quad 0\leq\tau<\tau_1\\ P_1=p+\delta_1,\quad \tau_1\leq\tau<\tau_2\\ ...\\ P_n=p+\delta_1+\delta_2+...+\delta_n,\quad \tau_n\leq\tau\leq t.\\ \end{array}\right. \]

This approach has also been successfully applied to other quantum systems, such as Fermi systems, relativistic quantum systems described by the Klein Gordon equation and Dirac equation, and certain quantum field theoretical models, see, e.g. the papers by Ph. Combe, R. Høegh-Krohn, R. Rodriguez, M. Sirugue and M. Sirugue-Collin (1980, 1981, 1982).


P-adics quantum theory is an approach to quantum theory where the underlying field of real numbers (resp. complex numbers) get replaced by the non-Archimedean field \({\mathbb Q}_p\) of p-adic numbers (relative to a prime number p) or, respectively, of some complex extensions of it. The non-Archimedean character of \({\mathbb Q}_p\ ,\) which, by the way, is just as \(\mathbb R\) is the closure of the rational \(\mathbb Q\) with respect to a norm (namely the p-adic norm instead of the usual Euclidean norm), makes that the analysis based on it, namely "p-adic analysis" presents several new features, which makes simpler, e.g., the discussion of convergence of series or, more generally, presents some advantageous features of discreteness which are not present in the reals. From this point of view an extension of quantum mechanics (and, more generally, physics) from the usual setting to a p-adic one has some good motivation. this was made clear originally by work of Volovich and Vladimirov in connection with the possible microscopic structure of space-time, and then applied to many areas of physics first, see e.g. the works by Volovich and Vladimirov (1989-1994), and then to other sciences, see, e.g., the works by A.Yu. Khrennikov, e.g. a book, published in 2004, on information dynamics in cognitive, psychological, social and anomalous phenomena.

Feynman path integrals (and related probabilistic integrals) have been discussed both in relation to quantum mechanics over a p-adic space resp. p-adic space-time, with \(\mathbb C\) valued wave function, as well as with \(\mathbb Q\)-valued wave functions. For the former let us mention that an analogue of Wiener measure on certain Banach spaces oven non-Archimedean local fields has been constructed by Satoh (1994). Markov processes over non-Archimedean fields have been studied by S.N. Evans (1989), S. Albeverio and W. Karwowski (1990), H. Kaneko (2004), A.N. Kochubei (2001), T. Yasuda (2000), V.S. Vladimirov, I.V. Volovich, E.I Zelenov (1994), Khrennikov (2004).

A.N. Kochubei and M.R. Sait-Ametov (2004) have extended methods of Euclidean quantum field theory to the p-adic case and constructed non Gaussian measures corresponding to suitable polynomial interactions. An analogue of Feynman-Kac formula has been discussed, e.g., in work by T. Digernes, V.S. Varadarajan and D. E. Weishart (2008). Extensions to adelic spaces (in a way combining the cases with underlying field \(\mathbb R\) and the ones with underlying space \({\mathbb Q}_p\ ,\) for any p) have been discussed by A. Blair (1994) and B. Dragovich and coworkers(2009) (also in connection with problems of heuristic approaches to areas like cosmology, string theory, and quantum gravity). Further extensions concerned analogy of Poisson-Maslov measures and processes, in work by O.G. Smolyanov and N.N. Shanarov (2008). Whereas all these approaches are for probabilistic type integrals, work by A. Khrennikov handles the definition of Feynman path integrals for \(\mathbb C\)-valued resp. \({\mathbb Q}_p\)-valued quantum mechanics over p-adic spaces resp. superspaces, much in the spirit of the duality approach based on the Parseval formula (generalized functions over a commutative number field). It remains to be seen how useful the elaboration of these approaches might be in connection with applications to physics.


Quantum mechanics

The primary goal of any approach to the mathematical definition of Feynman path integrals is the realization of the representation () of the solution of the Schrödinger equation () in term of a well defined functional integral. Analogously, there exists also a Feynman path integral representation for the fundamental solution of the Schrödinger equation, obtained formally by replacing in the Cauchy problem () the initial datum \(\psi_0\) with the \(\delta \) distribution.

Interesting results have been obtained by using the methods we mentioned. The first problem which has been solved by means of different methods is the description of the harmonic oscillator, with possibly time dependent frequency. The inclusion of a linear term dependent on time is natural in the framework of white noise analysis, but has also been done by means of the Parseval duality approach and the approach by infinite dimensional oscillatory integrals. The case of an anharmonic oscillator potential, with an anharmonic perturbation \(V(x)\ ,\) \(x\in{\mathbb R}^d\ ,\) belonging to the class the Fourier transforms of complex measures on \({\mathbb R}^d\) can be rather easily handled by means of all approaches. Unfortunately this class of potentials excludes some potentials of physical interest. The case of unbounded perturbations that are Laplace transforms of measures, including the Morse potential, has been studied by means of white noise analysis and of analytically continued Wiener integrals. Remarkably the treatment of these potentials is perturbative in a rigorous sense, as the corresponding Dyson series is convergent. The rigorous treatment of "non-perturbative potentials" presents additional difficulties and only some particular cases have been studied. The existing results include potentials with singularities, as the Coulomb potential (Nelson (1964), Doss (1980)) and polynomial potentials with particular degree (Albeverio and Mazzucchi (2005), Doss(2010), Grothaus, Streit and Vogel (2009)). Remarkably, as pointed out for the first time by Nelson, the Feynman path integral formulation can provide a non ambiguous construction of the quantum dynamics even in cases where it is not uniquely determined by the traditional methods as the quantum Hamiltonian operator is not essentially self-adjoint. This has been exemplified in details for the case of a quartic oscillator with the "wrong" sign by Mazzucchi (2008).

Stochastic Schrödinger equation, measurement theory

Feynman path integrals are a flexible tool and can provide a functional integral description for the time evolution of a large class of quantum system. Interesting examples can be found in the theory of continuous quantum measurement.

Indeed some heuristic Feynman path integral formulae for the description of the dynamic of a quantum particle submitted to the continuous measurement of its position have been proposed. An interesting example has been provided by Mensky's formula, where the wave function of the particle, if the measured trajectory is the path \(\omega(s)_{s\in [0,t]}\ ,\) is given by the restricted path integral: \[ \psi (t,x,\omega)= \int _{ \{\gamma(t)=x \} }e^{\frac{i}{\hbar}S_t(\gamma)}e^{-\lambda\int_0^t(\gamma (s)-\omega(s))^2ds}\phi(\gamma (0))D\gamma, \] where \(\lambda\in{\mathbb R}^+\) is a real positive parameter which is proportional to the accuracy of the measurement. Heuristically Mensky's formula suggests that, as an effect of the correction term \(e^{-\lambda\int_0^t(\gamma (s)-\omega(s))^2ds}\) due to the measurement, the paths \(\gamma \) giving the main contribution to the integral () are those closer to the observed trajectory \(\omega\ .\)

An alternative description of the same physical phenomenon is given in terms of a class of stochastic Schrödinger equations, as the Belavkin equation: \[ \left\{ \begin{array}{l} d\psi(t,x)=-\frac{i}{\hbar}H\psi(t,x)dt- \frac{\lambda}{2}x^2\psi(t,x) dt+ \sqrt\lambda x \psi(t,x) d B(t)\\ \psi(0,x)=\psi_0(x) \quad\quad\quad (t,x)\in [0,T] \times {\mathbb R}^d, \end{array} \right. \] where \(H\) is the quantum mechanical Hamiltonian operator, \(B\) is a \(d-\)dimensional Brownian motion, \(dB(t)\) is the Ito differential and \(\lambda >0\) is a coupling constant, which is proportional to the accuracy of the measurement.

Rigorous Feynman path integral representations for the solution of Belavkin equation and consequently rigorous realizations of Mensky's formula, in the case where the potential appearing in the Hamiltonian \(H\) is the Fourier transform of a complex measure on \({\mathbb R}^d\ ,\) have been obtained in terms of infinite dimensional oscillatory integrals by Albeverio, Kolokolt'sov and Smolianov (1996/97) and by Albeverio, Guatteri and Mazzucchi (2003). Related results have been described in Kolokolt'sov's book and in Exner's book.

Quantum field theory

Heuristic Feynman path integrals are commonly used by physicists as a tool for formulating contemporary theories of quantum fields, gauge fields, quantum gravity, and various approaches to quantum gravity (loop quantum gravity, string theory). The gap which exists between heuristics and rigor concerning Euclidean path integrals in relation to these areas is not unexpectedly perhaps, present also between heuristics and rigor concerning Feynman path integrals in these areas. There are some areas, like quantum fields on curved manifolds, or quantum gravity, where a direct relativistic Feynman path integral approach might be in principle closer to a not existing "real theory" than to a Euclidean approach, since, except for special space-times, there is no natural way to perform analytic continuation from a Euclidean to a relativistic approach. Up to now rigorous approaches to Feynman path integrals for relativistic quantum fields are limited to models with space and ultraviolet cut-offs (i.e. with interaction limited to a bounded region of space and with a regularization to avoid divergences due to the singular nature of the fields, as already expected from the free-field case). See the book by Albeverio. Høegh-Krohn and Mazzucchi (2008), where the case of bounded continuous regularized interactions with space-cut-off is treated. the removal of the cut-offs has been achieved for such models in the Euclidean framework, but only in space-time dimension 2, see the work by Albeverio and Høegh-Krohn (1973) and of Fröhlich and Seiler (1976).

Topological quantum field theory

A particularly interesting application of Feynman's integration can be found in topological quantum field theory, precisely in Chern-Simons theory. The basic functional integral in this case is still of the following form \[ I^\Phi(f)\equiv\int_\Gamma e^{i\Phi(\gamma)}f(\gamma)d\gamma \] but in this case the integration is performed on a space \(\Gamma\) of geometric objects, i.e. on the space of connection 1-forms on the principal fiber bundle over a 3-dimensional manifold \(M\ ,\) with compact structure Lie group \(G\) (the "gauge group"). The phase function \(\Phi\) is the Chern-Simons action functional: \[ \Phi(\gamma)\equiv\frac{k}{4\pi} \int_M \Big(\langle \gamma\wedge d\gamma\rangle -\frac{2}{3}\langle \gamma\wedge [\gamma\wedge \gamma]\rangle \Big), \] where \(\gamma\) denotes a \(g-\)valued connection 1-form, \(g\) being the Lie algebra of compact Lie group \(G\ .\) \(\Phi\) is metric independent. The function \(f\) to be integrated is given by \[\tag{9} f(\gamma):=\prod_{i=1}^n Tr(Hol(\gamma,l_i))\in{\mathbb C}, \]

where \((l_1,...,l_n)\ ,\) \(n\in{\mathbb N}\ ,\) are loops in \(M\) whose arc are pairwise disjoint and \(Hol(\gamma,l)\) denotes the holonomy of \(\gamma\) around \(l\ .\) According to a conjecture by Witten and Schwartz the integral \(I^\Phi(f)\) should represent a topological invariant. In particular, if \(G=SU(2)\) and \(M=S^3\ ,\) \(I^\Phi(f)\) should give the Jones polynomial, if \(G=SU(n)\) the Homfly polynomials, while if \(G=SO(n)\) \(I^\Phi(f)\) should give the Kauffman polynomials.

The Chern-Simon functional integral has been rigorously mathematically realized as Fresnel integral resp. white noise integral (in the Abelian case) by Albeverio and Schäfer in (1995) resp. by Leukert and Schäfer in (1996), or as a white noise distribution by Albeverio and Sengupta in (1997). The construction of the functions \(f\) of the form given in (9) and the corresponding derivations of the topological invariants (at least for \(M\) of the form \(S^1\times \Sigma\ ,\) \(\Sigma\) being a 2-manifold) are particularly technical and we refer to the original work by A. Hahn (2004,2008). First steps in a large \(k\) (i.e. "semiclassical") expansion, of great relevance in topology due to the heuristic relation with Vassiliev's invariants of knots, have been initiated by Albeverio and Mitoma (2009).

Statistical mechanics (classical-quantum). Feynman-Vernon functional.

Feynman path integrals for expectation with respect to temperature states of the harmonic oscillator have been first discussed in the book Albeverio, Høegh-Krohn and Mazzucchi (2008). The discussion has been pursued in the Euclidean framework for quantum and classical statistical mechanics with interaction in the recent book by Albeverio, Kondratiev, Kozitsky and Röckner (2009). A discussion of the semiclassical limit has been initiated by Albeverio and Høegh-Krohn and continued in Albeverio, Kondratiev, Kozitsky and Röckner(2009).

A Feynman path integral description of quantum open systems, that is of systems interacting with an external environment, has been introduced for the first time by Feynman and Vernon. By modeling the environment in terms of a many body quantum system and by tracing out its degrees of freedom, a heuristic path integral formula for the reduced density matrix of the open system was proposed. In the case of a quantum mechanical particle moving in \(\R^d\) and interacting with an environment described by the variable \(R\in\R^N\) the formula looks as follows: \[ \rho^r(t)(x,y)=\int\rho_t(x,y,R,R)dR = \int\int\rho(\gamma(0),\gamma'(0)) e^{\frac{i}{\hbar}\big(S_t(\gamma)-S_t(\gamma')\big)}F(\gamma,\gamma')d\gamma d\gamma', \] where the integral is taken over the path \(\gamma,\gamma':[0,t]\to{\mathbb R}^d\) such that \(\gamma(t) =x\ ,\) \(\gamma'(t) =y\ ,\) \(x,y\) being coordinates of the particle, \(S_t\) is the classical action functional, while \(F\) is called influence functional and describes the influence of the external environment on the open system. It contains basically the information on the interaction of the open system with the environment as well as the state of the latter. In the case where the environment is an N-dimensional system of harmonic oscillators with frequency \(\Omega>0\) at thermal equilibrium at temperature \(T>0\ ,\) and the interaction potential with the \(d\)-dimensional quantum system is of the form \(V(x,R)=xCR\ ,\) \(x\in{\mathbb R}^d\ ,\) \({\mathbb R}\in{\mathbb R}^N\) (\(C\) being a linear application from \({\mathbb R}^N\to{\mathbb R}^d\ ,\) i.e. an \(N\times d\) matrix, the influence functional \(F\) can be explicitly computed and assumes the following form: \[ F(\gamma,\gamma',x,y)= e^{\frac{i}{2\hbar}\int_0^tC^T(\gamma(s)+x-\gamma'(s)-y)\Omega_B^{-1}\int_0^s\sin(\Omega(s-r))C^T(\gamma(r)+x+\gamma'(r)+y)d rd s} \] \[ e^{-\frac{1}{2\hbar}\int_0^tC^T(\gamma(s)+x-\gamma'(s)-y)\Omega^{-1}\coth\!\left(\frac{\hbar \Omega}{2kT}\right)\int_0^s\cos(\Omega(s-r))C^T(\gamma(r)+x-\gamma'(r)-y)d rd s}. \] This heuristic formalism has been widely applied to the description of several physical systems, as in the Caldeira-Leggett model of the quantum Brownian motion.

Formulae () and () have been rigorously mathematically realized in terms of infinite dimensional oscillatory integrals, see Albeverio, Cattaneo, Di Persio and Mazzucchi (2007).

Quantum computing

The rigorous study of problems of quantum computing is very challenging. Most of the methods developed so far concern the case of systems with finite dimensional state space, see, e.g., Fei, albeverio, cabello, Jing, Goswami (2010). Feynman path integrals for such systems have been discussed (without applications to quantum computing) in work by E.C. Thomas (2000) One of the problems of the construction of quantum computers concerns a better understanding of the phenomena of decoherence, due to interaction of the interesting system with a surrounding noisy medium. The work by Albeverio, Cattaneo, Di Persio and Mazzucchi (2007) discusses this problem by Feynman path integrals . Quantum computation, with its problem of constructing suitable entangled states, has also been brought in contact, somewhat naturally, with topological problems of entanglement, and Chern-Simons model of topological quantum fields has been brought to discuss such problems, in work by L.H. Kauffman and J.J. Lomonaco (2010), E. Dennis, A. Kitaev and J. Preskill (2002), and by C. Nayak, S.H. Simon, A. Stern, M. Freedman, S. Das Sarma (2008).

Since Chern-Simons model has been constructed rigorously by Feynman path integrals, it is natural to try to bring this construction in connection with the mentioned approach to entanglement via topological quantum field theory models.

Dissipative systems

Quantum mechanics with a complex valued potential has been discussed as a model for simple dissipative systems, and a systematic discussion in terms of rigorous Feynman path integrals (and related techniques) has been presented in P. Exner's book. Further developments in this direction have been discussed in work by S. Tcheremchantsev (1983), A. de Bivar-Weinholtz and M.L. Lapidus (1990), S. Albeverio and Z. Brzeźniak (1995), G.W Johnson and M.L. Lapidus (2000).


In the analytic study of finite dimensional oscillatory integrals , i.e. \[ I(\epsilon):=\int_{{\mathbb R}^n}e^{\frac{i}{\epsilon}\Phi(x)}f(x)dx, \] with \(\Phi:{\R^n}\to{\mathbb R}\) and \(f:{\R^n}\to {\mathbb C}\ ,\) a particular interest is given to the study of their asymptotic behavior in the limit where \(\epsilon\to 0\ .\) This interest is due to the fact that such asymptotics gives, on one hand, the possibility to compute approximately the integral, on the other hand it relates the integral with mathematical and physical problems of interest in themselves. The fundamental tool for the study of the asymptotics of the integral () is the stationary phase method, originally introduced by Stokes and Kelvin in the description of wave phenomena. More recent investigations can be found in the work of V. P. Maslov and coworkers, in connection with the study of the semiclassical limit of quantum mechanics, and in the work by L. Hörmander, in connection with the theory of Fourier integral operators and the study of partial differential equations. According to the stationary phase method, when \(\epsilon\) can be considered very small, the integrand \(e^{\frac{i}{\epsilon}\Phi(x)}\) oscillates very fast, in such a way that the contributions to the integral coming from the positive and the negative parts of the oscillations annul each other, and the only regions of \({\mathbb R}^n\) giving a non vanishing contribution to the value of the integral are the neighbourhoods of the stationary points of the phase function \(\Phi\ ,\) i.e. the points \(x_c\) satisfying the equation \[ \Phi'(x)=0, \] where \(\Phi'\) stands for the gradient of \(\Phi\ .\) In the case where the Hessian \(\Phi''(x_c)\) has a non trivial kernel (i.e. \(\det\Phi''(x_c)\neq 0\) ) and the critical points are isolated, one easily gets an expansion of \(I(\epsilon)\) in powers of \(\epsilon\) as a sum of contributions coming from the single critical points, the leading contribution being given by complex Gaussian integrals (with phase function \(\frac{1}{2}(x,\Phi''(x_c)x)\)). In the case where the phase \(\Phi\) has some degenerate critical points \(x_c\) (i.e. \(\det\Phi''(x_c)= 0\) ) the study of the asymptotics of \(I(\epsilon)\) becomes more technical and the theory of unfoldings of singularities plays a crucial role (which systematically characterizes normal forms of phase functions around critical points).

Interesting considerations follow from an heuristic application of the same method to the study of the asymptotics of the Feynman path integral \[ I(\hbar):=\int e^{\frac{i}{\hbar}S_t(\gamma)}\psi_0(\gamma)d\gamma, \] in the semiclassical limit, i.e. when the Planck constant \(\hbar\) can be considered negligible and plays the role of the small parameter \(\epsilon\ .\) The role of the phase function \(\Phi\) is played by the action functional \(S_t\ ,\) and, according to Hamilton's least action principle, its stationary points are exactly the classical orbits of the system.

A rigorous mathematical formulation of these ideas, that is the implementation of an infinite dimensional version of the stationary phase method for the study of the asymptotics of functional integral of Feynman's type is a rather difficult task. Once the Feynman integral () is mathematically realized in terms of a well defined functional, it is even hard to verify whether it admits an asymptotic expansion at all. Some rather technical results have been obtained basically in the case of non degenerate phase functions in the framework of Fresnel or infinite dimensional oscillatory integrals by Albeverio and Høegh-Krohn, by Albeverio, Boutet de Monvel-Berthier and Brzeźniak (1995) and by Rezende (1985). Analogous, but less detailed, results have been obtained in the study of analytically continued Wiener integrals by Azencott and Doss (1985) and by Ben Arous and Castell (1996) and in the sequential approach by Kumano-Go and Fujiwara.

If the phase \(S_t\) presents some degenerate critical paths, the problem is more difficult. Only some special cases of this situation have been studied, see e.g. Albeverio and Brzeźniak (1993), by reducing the study of the degeneracy on a finite dimensional subspace of the space of paths.

In the study of the solution of the Schrödinger equation it is physically meaningful to consider an initial datum \(\psi_0\) of the following form \[ \psi_0(x)=e^{\frac{i}{\hbar}S_0(x) }\phi_0(x),\quad x\in{\mathbb R}^d \] where \(S_0\) is real and \(S_0,\phi_0\) are independent of \(\hbar\ .\) This initial data corresponds to an initial particle distribution \(\rho_0(x)=|\phi_0|^2(x)\) and to a limiting value of the probability current \(J_{\hbar=0}=S_0'(x)\rho_0(x)/m\ ,\) giving an initial particle flux associated to the velocity field \(S_0'(x)/m\ .\) By imposing some technical regularity conditions on the potential \(V\) as well as on the functions \(S_0,\phi_0\) in addition to the natural assumption of smoothness, it is possible to prove that the solution of the Schrödinger equation has an asymptotic expansion in powers of \(\hbar \ ,\) whose leading term is the sum of the values of the functions \[ \Big| \det\Big(\Big( \frac{\partial \bar\gamma_k^{(j)}}{\partial y_l^{(j)}}(y^{(j)},t)\Big)\Big) \Big|^{-1/2}\Big( e^{-\frac{i}{2}\pi m^{(j)}}e^{-\frac{i}{\hbar}S}e^{-\frac{i}{\hbar}S_0} \phi_0\Big)(\bar\gamma ^{(j)}) \] taken at the points \(y^{(j)}\) such that a classical particle starting at \(y^{(j)}\) at time zero with momentum \(\nabla S_0(y^{(j)})\) is in \(x\) at time \(t\ .\) \(S(\bar\gamma ^{(j)})\) is the classical action along this classical path \(\bar\gamma^{(j)}\) and \(m^{(j)}(\bar\gamma ^{(j)})\) is the Maslov index of the path \(\bar\gamma^{(j)}\ ,\) i.e. \(m^{(j)}\) is the number of zeros of \(\det\Big(\Big( \frac{\partial \bar\gamma_k^{(j)}}{\partial y_l^{(j)}}(y^{(j)},\tau)\Big)\Big)\) as \(\tau\) varies on the interval \((0,t)\ .\)

Another suggestive application of the stationary phase method to the study of the functional integral representation of quantum mechanical quantities is the investigation of the semiclassical behavior as \(\hbar\downarrow 0\) of the trace of the Schrödinger group \(Tr[e^{-\frac{i}{\hbar}Ht}] \) studied by Albeverio, Blanchard and Høegh-Krohn (1980) and by Albeverio, Boutet de Monvel-Berthier and Brzeźniak (1996). This provides a rigorous proof of the (fixed-time) Gutzwiller's trace formula, connecting the semiclassical asymptotics for \(\hbar \downarrow 0\) of the trace of the Schrödinger group \(Tr(e^{-\frac{i}{\hbar}Ht})\) to the classical periodic orbits of the system. One can look at this as a quantum analogue of Selberg's trace formula, relating the trace of the heat kernel on manifolds of constant negative curvature with a sum over contributions associated with the periodic geodesics. The interest in this kind of relations has been renewed in recent years since, according to the theory of quantum chaos, the type of distribution of the energy eigenvalues of a given quantum mechanical system should reflect the one of the underlying classical system, namely whether it is integrable, resp. chaotic. The trace formula has also interesting relations with certain problems of number theory (in fact, e.g., for the Laplacian on the torus it is expressed by theta functions, which, via a Mellin transform, are related to the zeta function, see, e.g., Albeverio, Blanchard, and Høegh-Krohn (1982)).

Recently, in a paper by Albeverio and Mitoma (2009), asymptotic methods have been applied to a regularized version of the [[#Topological quantum field theory|Chern-Simon topological field theory}} and have given interesting results on Vassiliev invariants, in the direction of rigorously establishing heuristic results obtained from a "perturbative version" of Chern-Simon models.

Fermionic, non commutative, supersymmetric path integrals

In connection with quantum physics involving Fermi particles a formalism of "non commutative" Feynman path integrals has been developed. It is more algebraic in character than the one of "commutative" Feynman path integrals we have been discussed, partly however it uses the path integrals we have been discussing extending them into a "noncommutative world". This is also the case of path integrals discussed in connection with supersymmetric theories (putting boson and Fermi particles on "equal footing"). We do not enter in detail about these extensions, which would deserve a separate treatment, and limit ourselves to give a few references, e.g. the work by A. Rogers (1987), O.G. Smolyanov and E.T. Shavgulidze (1989), A. Inomata and G. Junker (1994), R. Léandre and A. Rogers (2006).


  • Albeverio, S; Blanchard, Ph and Høegh-Krohn, R (1982). Feynman path integrals and the trace formula for the Schrödinger operators. Comm. Math. Phys. 83(1): 49-76. doi:10.1007/bf01947071.
  • Albeverio, S and Brzeźniak, Z (1993). Finite-dimensional approximation approach to oscillatory integrals and stationary phase in infinite dimensions. J. Funct. Anal. 113(1): 177-244. doi:10.1006/jfan.1993.1051.
  • Albeverio, S; Hahn, A and Sengupta, A N (2004). Rigorous Feynman path integrals, with applications to quantum theory, gauge fields, and topological invariants. Stochastic analysis and mathematical physics (SAMP/ANESTOC 2002). World Sci. Publishing River Edge, NJ: 1-60.
  • Albeverio, S and Høegh-Krohn, R (1967). Oscillatory integrals and the method of stationary phase in infinitely many dimensions, with applications to the classical limit of quantum mechanics. Invent. Math. 40(1): 59-106. doi:10.1007/bf01389861.
  • Albeverio, S and Mazzucchi, S (2005). Generalized Fresnel integrals. Bull. Sci. Math. 129(1): 1--23. doi:10.1016/j.bulsci.2004.05.005.
  • Albeverio, S and Mazzucchi, S (2005). Feynman path integrals for polynomially growing potentials. J. Funct. Anal. 221(1): 83--121. doi:10.1016/j.jfa.2004.07.014.
  • Albeverio, S; Høegh-Krohn, R and Mazzucchi, S (2008). Mathematical theory of Feynman path integrals. An Introduction. 2nd and enlarged edition. Lecture Notes in Mathematics 523. Springer-Verlag, Berlin.
  • Cameron, R H (1960). A family of integrals serving to connect the Wiener and Feynman integrals. J. Math. and Phys. 39: 126--140.
  • Dirac, P A M (1933). The Lagrangian in quantum mechanics. Physik. Z. Sowjetunion 3: 64. Reprinted in Selected Papers on Quantum Electrodynamics, Schwinger J ed., Dover (New-York 1958)
  • Doss, H (1980). Sur une résolution stochastique de l'équation de Schrödinger à coefficients analytiques. Comm. Math. Phys. 73 (3): 247--264. doi:10.1007/bf01197701.
  • Elworthy, D and Truman, A (1984). Feynman maps, Cameron-Martin formulae and anharmonic oscillators. Ann. Inst. H. Poincaré Phys. Théor. 41(2): 115--142.
  • Kumano-go, N and Fujiwara, D (2008). Feynman path integrals and semiclassical approximation. RIMS Kokyuroku Bessatsu B5: 241--263.
  • Feynman, R P (1948). Space-Time Approach to Non-Relativistic Quantum Mechanics. Review of Modern Physics 20(2): 367-387. doi:10.1103/revmodphys.20.367.
  • Friedman, C N (1971/72). Semigroup product formulas, compressions, and continual observations in quantum mechanics. Indiana Univ. Math. J. 21: 1001--1011.
  • Hida, T; Kuo, H H; Potthoff, J and Streit, L (1993). White noise. An infinite-dimensional calculus. Mathematics and its Applications, 253. Kluwer Academic Publishers Group, Dordrecht.
  • Itô, K (1961). Wiener integral and Feynman integral. Proc. Fourth Berkeley Symposium on Mathematical Statistics and Probability. California Univ. Press, Berkeley 2: 227-238.
  • Itô, K (1967). Generalized uniform complex measures in the Hilbertian metric space with their applications to the Feynman path integral. Proc. Fifth Berkeley Symposium on Mathematical Statistics and Probability. California Univ. Press, Berkeley 2 (1): 145-161.
  • Johnson, G W and Lapidus, M L (2000). The Feynman integral and Feynman's operational calculus. Oxford University Press, New York.
  • Mazzucchi, S (2009). Mathematical Feynman Path Integrals and Applications. World Scientific Publishing, Singapore.
  • Nelson, E (1964). Feynman integrals and the Schrödinger equation. J. Math. and Phys. 5: 332-343. doi:10.1063/1.1704124.

Further reading

  • DeWitt-Morette C (ed), et al (1995). Special Issue on Functional Integration. J. Math. and Phys. 36 (5): 2135-2564.
  • Albeverio, S; Høegh-Krohn, R; Fenstad, J E and Lindstrøm, T (1986). Nonstandard methods in stochastic analysis and mathematical physics. Academic Press, Inc., Orlando, FL.
  • Albeverio, S; Kondratiev, Y; Kozitsky, Y and Roeckner, M (2009). The statistical mechanics of quantum lattice systems. A path integral approach. European Mathematical Society (EMS), Zürich.
  • Atiyah, M F (1979). Geometry of Yang-Mills fields. Lezioni Fermiane. Scuola normale superiore, Pisa.
  • Berezin, F A and Shubin, M A (1991). The Schrödinger equation. Kluwer Academic Publishers Group, Dordrecht.
  • Bytsenko, A A; Cognola, G; Elizalde, E; Moretti, V and Zerbini, S (2003). Analytic aspects of quantum fields. World Scientific Publishing Co., River Edge, NJ.
  • Cartier, P and DeWitt-Morette , C (2006). Functional integration: action and symmetries. Cambridge University Press, Cambridge.
  • Exner, P (2000). Open quantum systems and Feynman integrals. D. Reidel Publishing Co., Dordrecht.
  • Faddeev, L D and Slavnov, A A (1991). Gauge Fields. Introduction to quantum theory. (2nd edition). Addison-Wesley Publishing Company, T.
  • Feynman, R P and Hibbs, A R (1965). Quantum Mechanics and Path Integrals. McGraw-Hill Companies, New York.
  • Grosche, C and Steiner, F (1998). Handbook of Feynman path integrals. Springer, Berlin, Heidelberg.
  • Hörmander, L (2003). The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis. Springer-Verlag, Berlin.
  • Huang, K (1998). Quantum field theory. From operators to path integrals. ohn Wiley & Sons, Inc., New York .
  • Klauder, J R (2000). Beyond conventional quantization. Cambridge University Press, Cambridge .
  • Kleinert, H (1995). Path Integrals in Quantum Mechanics, Statistics and Polymer Physics. World Scientific, Singapore.
  • Kolokoltsov, V N (2000). Semiclassical analysis for diffusions and stochastic processes. Springer-Verlag, Berlin.
  • Lapidus, M L (2008). In search of the Riemann zeros. Strings, fractal membranes and noncommutative spacetimes. American Mathematical Society, Providence, RI.
  • Leandre, R (2006). Path Integrals in Noncommutative Geometry. Encyclopedia of Mathematical Physics. Elsevier: 8-12.
  • Maslov, V P and Fedoryuk, M V (1981). Semiclassical approximation in quantum mechanics. D. Reidel Publishing Co., Dordrecht-Boston, Mass..
  • Mensky, M V (2000). Quantum measurements and decoherence. Models and phenomenology. Kluwer Academic Publishers, Dordrecht.
  • Parisi, G (1988). Statistical field theory. Benjamin/Cummings Publishing Co., Reading, MA.
  • Popov, V N (1983). Functional integrals in quantum field theory and statistical physics. D. Reidel Publishing Co., Dordrecht-Boston, Mass..
  • Rivers, R J (1990). Path integral methods in quantum field theory. Cambridge University Press, Cambridge .
  • Roepstorff, G (1994). Path integral approach to quantum physics. An introduction. Springer-Verlag, Berlin.
  • Rovelli, C (2004). Quantum gravity. Cambridge University Press, Cambridge.
  • Schulman, L S (2005). Techniques and applications of path integration. Dover Publications, US.
  • Various, authors (). Path integrals from meV to MeV: Proceedings of International Conferences. World Scientific Publishing Co., River Edge, NJ.

See also

Path integral, Principle of least action

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