Abstract
Path integrals are a ubiquitous tool in theoretical physics. However, their use is sometimes hindered by the lack of control on various manipulations–such as performing a change of the integration path–one would like to carry out in the light-hearted fashion that physicists enjoy. Similar issues arise in the field of stochastic calculus, which we review to prepare the ground for a proper construction of path integrals. At the level of path integration, and in arbitrary space dimension, we not only report on existing Riemannian geometry-based approaches that render path integrals amenable to the standard rules of calculus, but also bring forth new routes, based on a fully time-discretized approach, that achieve the same goal. We illustrate these various definitions of path integration on simple examples such as the diffusion of a particle on a sphere.
Acknowledgments
We thank C. Aron, D. Barci, R. Chetrite, R. Cont, P.-M. Déjardin, H. W. Diehl, P. Drummond, Z. González-Arenas, H. J. Hilhorst, H. K. Janssen, G. S. Lozano, A. Rançon, S. Renaux-Petel, and F. A. Schaposnik for very helpful discussions.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Notes
1 This is a classical fact of stochastic calculus which, for completeness, is explained in Section 2.1.3. The equivalence in Equation (Equation3(3) (3) ) means that the distribution of the two processes is the same at all times–when starting from the same initial condition.
2 Importantly, when writing the time-discretized version of the integrals [Citation39], we see that the difference between the left- and right-hand-side of Equation (Equation100(100) (100) ) is of order , validating the use of Equation (Equation100(100) (100) ) in the exponential of (Equation99(99) (99) ).
3 Indeed, denoting and the noise amplitude and the volume measure of the process , we have and thus .
4 The prefactor in (Equation115(115) (115) ) allows one to eliminate a constant pre-exponential factor in .
5 In Equation (Equation117(117) (117) ), the convention we use is to have a prefactor that encodes the fact that the propagator of Equation (Equation110(110) (110) ) transforms like a scalar, and that both the measure and the action possess this same property, see Equations (Equation119(119) (119) ) and (Equation120(120) (120) ). Equivalently to (Equation117(117) (117) ), one can also write that the probability that belongs to a domain at time is
6 Note that, as will be discussed at the end of Section 4.2, choosing a different convention for the discretization of the path-integral measure, one can arrive at a different expression of the Lagrangian in which applying the stochastic chain rule in the Itō and in the Hänggi–Klimontovich case leads to a correct computation, as recently shown in Ref. [Citation52].
7 See Sec. 4 of Ref. [Citation32] (in which ), where due to typos, one reads instead of the correct expression .
8 For a function of the variable , we denote by the partial derivative with respect to , while we keep for the derivative with respect to of a function . The notations and will also be used in some cases for readability purposes.
9 Note that in prefactor of the exponential of the propagator, the cubic substitution rule is independent of its prefactor and takes the general form (in the spirit of Refs. [Citation33,Citation85]). The rule (Equation176(176) (176) ) can also be formally inferred by expanding the exponential of its l.h.s., using the above substitution rule (valid in prefactor) and re-exponentiating; but the complete justification of Equation (Equation176(176) (176) ) is the one provided in the present section.
10 In Ref. [Citation39] Equation (Equation74(74) (74) ) has a typo and should read: .
11 In dimension one, this becomes: which corrects Equation (88) of Ref. [Citation39] which had a calculation mistake.
12 We recall that the relation between the scalar invariant propagator and the actual propagator of the process is , see Equation (Equation110(110) (110) ).
13 We recall that is read as , see e.g. Equation (Equation64(64) (64) ) in one dimension.
14 For an operator , the fraction appearing e.g. in (Equation279(279) (279) ) is understood as a series .
15 The quantity of Equation (Equation328(328) (328) ) corresponds, in the arXiv v1 Ref. [Citation117], to the integrand of the last line of Equation (E.22) multiplied by (with in our settings), which gives . This is the same result above but when changing variables from u to x.