Path integrals and stochastic calculus

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.

PACS:

• 05.40.-a
• 02.50.-r
• 05.10.Gg
• 03.65.Dband

Keywords:

• theoretical physics
• path integrals
• stochastic calculus

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 (3(3) $\frac{\mathrm{d}x}{\mathrm{d}t}\stackrel{\alpha }{=}f\left(x\right)+g\left(x\right)\eta ⟺\frac{\mathrm{d}x}{\mathrm{d}t}\stackrel{{\alpha }^{\prime }}{=}f\left(x\right)+(\alpha -{\alpha }^{\prime })g^{\prime }\left(x\right)g\left(x\right)+g\left(x\right)\eta$(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 [39], we see that the difference between the left- and right-hand-side of Equation (100(100) ${\int }_{\begin{array}{c}{t}_0\\ \alpha =0\end{array}}^{t_{\mathrm{f}}}\mathrm{d}t\frac{\mathrm{d}x}{\mathrm{d}t}f\left(x\right)={\int }_{\begin{array}{c}{t}_0\\ \alpha \end{array}}^{t_{\mathrm{f}}}\mathrm{d}t\frac{\mathrm{d}x}{\mathrm{d}t}f\left(x\right)-\alpha {\int }_{t_0}^{t_{\mathrm{f}}}\mathrm{d}t{f}^{\prime }\left(x\right).$(100) ) is of order $\sqrt{\mathrm{\Delta }t}$, validating the use of Equation (100(100) ${\int }_{\begin{array}{c}{t}_0\\ \alpha =0\end{array}}^{t_{\mathrm{f}}}\mathrm{d}t\frac{\mathrm{d}x}{\mathrm{d}t}f\left(x\right)={\int }_{\begin{array}{c}{t}_0\\ \alpha \end{array}}^{t_{\mathrm{f}}}\mathrm{d}t\frac{\mathrm{d}x}{\mathrm{d}t}f\left(x\right)-\alpha {\int }_{t_0}^{t_{\mathrm{f}}}\mathrm{d}t{f}^{\prime }\left(x\right).$(100) ) in the exponential of (99(99) $\begin{array}{rl}& \mathbb{P}(x_{\mathrm{f}},t_{\mathrm{f}}|x_0,t_0)=〈\exp (-\frac{1}{2}{\int }_{t_0}^{t_{\mathrm{f}}}\mathrm{d}tf(x{)}^2+{\int }_{\begin{array}{c}{t}_0\\ \alpha =0\end{array}}^{t_{\mathrm{f}}}\mathrm{d}t\frac{\mathrm{d}x}{\mathrm{d}t}f\left(x\right))〉,\end{array}$(99) ).

3 Indeed, denoting $G^{\mu i}$ and $\sqrt{\mathrm{\Omega }}$ the noise amplitude and the volume measure of the process $\mathbf{u}\left(t\right)$, we have $G^{\mu i}=\frac{\mathrm{\partial }{U}^{\mu }}{\mathrm{\partial }{x}^{\nu }}{g}^{\nu i}$ and thus $\sqrt{\omega \left({\mathbf{x}}_{\mathrm{f}}\right)}=|det\frac{\mathrm{\partial }{U}^{\mu }}{\mathrm{\partial }{x}^{\nu }}\left({\mathbf{x}}_{\mathrm{f}}\right)|\sqrt{\mathrm{\Omega }\left({\mathbf{u}}_{\mathrm{f}}\right)}$.

4 The prefactor $(2\pi \mathrm{\Delta }t{)}^{\frac{d}{2}}$ in (115(115) ${\mathrm{e}}^{-\mathcal{S}\left[\mathbf{x}\right(t\left)\right]}=\underset{N\to +\mathrm{\infty }}{lim}\prod _{k=0}^{N-1}{\left(2\pi \mathrm{\Delta }t\right)}^{\frac{d}{2}}{\mathbb{K}}_{\mathrm{\Delta }t}({\mathbf{x}}_{k+1},t_{k+1}|{\mathbf{x}}_k,t_k).$(115) ) allows one to eliminate a constant pre-exponential factor in ${\mathbb{K}}_{\mathrm{\Delta }t}$.

5  In Equation (117(117) ${\mathbb{P}}_{\mathbf{x}}({\mathbf{x}}_{\mathrm{f}},t_{\mathrm{f}}|{\mathbf{x}}_0,t_0)\stackrel{\mathfrak{d}}{=}\sqrt{\omega \left({\mathbf{x}}_{\mathrm{f}}\right)}{\int }_{\mathbf{x}\left(t_0\right)={\mathbf{x}}_0}^{\mathbf{x}\left(t_{\mathrm{f}}\right)={\mathbf{x}}_{\mathrm{f}}}\mathcal{D}\mathbf{x}\exp \{-{\int }_{t_0}^{t_{\mathrm{f}}}\mathrm{d}t{\mathcal{L}}_{\mathfrak{d}}^{\mathbf{x}}[\mathbf{x}\left(t\right),\dot{\mathbf{x}}\left(t\right)\left]\right\}$(117) ), the convention we use is to have a prefactor $\sqrt{\omega \left({\mathbf{x}}_{\mathrm{f}}\right)}$ that encodes the fact that the propagator $\mathbb{K}$ of Equation (110(110) $\mathbb{K}({\mathbf{x}}_{\mathrm{f}},t_{\mathrm{f}}|{\mathbf{x}}_0,t_0)=\frac{\mathbb{P}({\mathbf{x}}_{\mathrm{f}},t_{\mathrm{f}}|{\mathbf{x}}_0,t_0)}{\sqrt{\omega \left({\mathbf{x}}_{\mathrm{f}}\right)}},$(110) ) transforms like a scalar, and that both the measure $\mathcal{D}\mathbf{x}$ and the action possess this same property, see Equations (119(119) $\mathcal{D}\mathbf{x}=\mathcal{D}\mathbf{u},$(119) ) and (120(120) ${\mathcal{L}}_{\mathfrak{d}}^{\mathbf{x}}[\mathbf{x},\dot{\mathbf{x}}]={\mathcal{L}}_{\mathfrak{d}}^{\mathbf{u}}[\mathbf{u},\dot{\mathbf{u}}]={\mathcal{L}}_{\mathfrak{d}}^{\mathbf{u}}\left[\mathbf{U}\right(\mathbf{x}),(\mathrm{\partial }\mathbf{U}/\mathrm{\partial }\mathbf{x})\cdot \dot{\mathbf{x}}],$(120) ). Equivalently to (117(117) ${\mathbb{P}}_{\mathbf{x}}({\mathbf{x}}_{\mathrm{f}},t_{\mathrm{f}}|{\mathbf{x}}_0,t_0)\stackrel{\mathfrak{d}}{=}\sqrt{\omega \left({\mathbf{x}}_{\mathrm{f}}\right)}{\int }_{\mathbf{x}\left(t_0\right)={\mathbf{x}}_0}^{\mathbf{x}\left(t_{\mathrm{f}}\right)={\mathbf{x}}_{\mathrm{f}}}\mathcal{D}\mathbf{x}\exp \{-{\int }_{t_0}^{t_{\mathrm{f}}}\mathrm{d}t{\mathcal{L}}_{\mathfrak{d}}^{\mathbf{x}}[\mathbf{x}\left(t\right),\dot{\mathbf{x}}\left(t\right)\left]\right\}$(117) ), one can also write that the probability that ${\mathbf{x}}_{\mathrm{f}}$ belongs to a domain $\mathcal{X}$ at time $t_{\mathrm{f}}$ is $\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{b}({\mathbf{x}}_{\mathrm{f}}\in \mathcal{X},t_{\mathrm{f}}|{\mathbf{x}}_0,t_0)\stackrel{\mathfrak{d}}{=}{\int }_{\mathbf{x}\left(t_0\right)={\mathbf{x}}_0}^{\mathbf{x}\left(t_{\mathrm{f}}\right)\in \mathcal{X}}\mathcal{D}\mathbf{x}\exp \{-{\int }_{t_0}^{t_{\mathrm{f}}}\mathrm{d}t{\mathcal{L}}_{\mathfrak{d}}^{\mathbf{x}}[\mathbf{x}\left(t\right),\dot{\mathbf{x}}\left(t\right)\left]\right\}.$

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. [52].

7 See Sec. 4 of Ref. [32] (in which $D=\frac{1}{4}$), where due to typos, one reads $-\frac{1}{12}{r}^4{\dot{\varphi }}^4\mathrm{d}t$ instead of the correct expression $-\frac{1}{6}{r}^2{\dot{\varphi }}^4\mathrm{d}{t}^2$.

8 For a function $\mathbf{X}\left(\mathbf{u}\right)$ of the variable $\mathbf{u}$, we denote by ${\mathrm{\partial }}_{u^{\mu }}{X}^{\alpha }$ the partial derivative with respect to $u^{\mu }$, while we keep ${\mathrm{\partial }}_{\mu }{U}^{\alpha }$ for the derivative with respect to $x^{\mu }$ of a function $\mathbf{U}\left(\mathbf{x}\right)$. The notations $\frac{\mathrm{\partial }{X}^{\alpha }}{\mathrm{\partial }{u}^{\mu }}$ and $\frac{\mathrm{\partial }{U}^{\alpha }}{\mathrm{\partial }{x}^{\mu }}$ 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 $\frac{\mathrm{\Delta }{x}^{\mu }\mathrm{\Delta }{x}^{\nu }\mathrm{\Delta }{x}^{\rho }}{\mathrm{\Delta }t}\doteq {\omega }^{\mu \nu }\mathrm{\Delta }{x}^{\rho }+{\omega }^{\mu \rho }\mathrm{\Delta }{x}^{\nu }+{\omega }^{\nu \rho }\mathrm{\Delta }{x}^{\mu }$ (in the spirit of Refs. [33,85]). The rule (176(176) $A_{\mu \nu \rho }^{\left(3\right)}\frac{\mathrm{\Delta }{x}^{\mu }\mathrm{\Delta }{x}^{\nu }\mathrm{\Delta }{x}^{\rho }}{\mathrm{\Delta }t}\doteq 3A_{\mu \nu \rho }^{\left(3\right)}{\omega }^{\mu \nu }\mathrm{\Delta }{x}^{\rho }+3A_{\mu \nu \rho }^{\left(3\right)}{A}_{\alpha \text{β}\sigma }^{\left(3\right)}{\omega }^{\mu \alpha }{\omega }^{\nu \text{β}}{\omega }^{\rho \sigma }\mathrm{\Delta }t,$(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 (176(176) $A_{\mu \nu \rho }^{\left(3\right)}\frac{\mathrm{\Delta }{x}^{\mu }\mathrm{\Delta }{x}^{\nu }\mathrm{\Delta }{x}^{\rho }}{\mathrm{\Delta }t}\doteq 3A_{\mu \nu \rho }^{\left(3\right)}{\omega }^{\mu \nu }\mathrm{\Delta }{x}^{\rho }+3A_{\mu \nu \rho }^{\left(3\right)}{A}_{\alpha \text{β}\sigma }^{\left(3\right)}{\omega }^{\mu \alpha }{\omega }^{\nu \text{β}}{\omega }^{\rho \sigma }\mathrm{\Delta }t,$(176) ) is the one provided in the present section.

10 In Ref. [39] Equation (74(74) ${\omega }^{\mu \nu }\left(\mathbf{x}\right)=g^{\mu i}\left(\mathbf{x}\right)g^{\nu j}\left(\mathbf{x}\right){\delta }_{ij}.$(74) ) has a typo and should read: $A^{\left(3\right)}\mathrm{\Delta }{x}^3\mathrm{\Delta }{t}^{-1}\doteq 3A^{\left(3\right)}2Dg(x{)}^2\mathrm{\Delta }x+3[A^{\left[3\right)}2Dg(x{)}^2\mathrm{\Delta }x{]}^2$.

11 In dimension one, this becomes: $\begin{array}{rl}& -\frac{1}{2}g(x+\alpha \mathrm{\Delta }x{)}^2\frac{\mathrm{\Delta }{x}^2}{\mathrm{\Delta }t}\doteq \mathrm{M}\mathrm{C}\mathrm{R}-(1-2\alpha )\frac{X^{″}\left(u\right)}{X^{\prime }\left(u\right)}\mathrm{\Delta }u+\frac{(1-2\alpha {)}^2}{2}\frac{G\left(u{)}^2{X}^{″}\right(u{)}^2}{X^{\prime }(u{)}^2}\mathrm{\Delta }t\\ & -\frac{1-3\alpha (1-\alpha )}{2}\frac{G\left(u{)}^2{X}^{‴}\right(u)}{X^{\prime }\left(u\right)}\mathrm{\Delta }t+\frac{3\alpha (1-\alpha )}{2}\frac{G^{\prime }\left(u\right)X^{\prime }\left(u\right)+G\left(u\right)X^{″}\left(u\right)}{X^{\prime }(u{)}^2}G\left(u\right)X^{″}\left(u\right)\mathrm{\Delta }t\end{array}$which corrects Equation (88) of Ref. [39] which had a calculation mistake.

12 We recall that the relation between the scalar invariant propagator $\mathbb{K}$ and the actual propagator $\mathbb{P}$ of the process is $\mathbb{K}({\mathbf{x}}_{\mathrm{f}},t_{\mathrm{f}}|{\mathbf{x}}_0,t_0)=\mathbb{P}({\mathbf{x}}_{\mathrm{f}},t_{\mathrm{f}}|{\mathbf{x}}_0,t_0)/\sqrt{\omega \left(\mathbf{x}\right)}$, see Equation (110(110) $\mathbb{K}({\mathbf{x}}_{\mathrm{f}},t_{\mathrm{f}}|{\mathbf{x}}_0,t_0)=\frac{\mathbb{P}({\mathbf{x}}_{\mathrm{f}},t_{\mathrm{f}}|{\mathbf{x}}_0,t_0)}{\sqrt{\omega \left({\mathbf{x}}_{\mathrm{f}}\right)}},$(110) ).

13 We recall that $G^{\mu i}=\frac{\mathrm{\partial }{U}^{\mu }}{\mathrm{\partial }{x}^{\nu }}{g}^{\nu i}$ is read as $G^{\mu i}\left(\mathbf{u}\right)=\frac{\mathrm{\partial }{U}^{\mu }}{\mathrm{\partial }{x}^{\nu }}\left({\mathbf{U}}^{-1}\right(\mathbf{u}\left)\right)g^{\nu i}\left({\mathbf{U}}^{-1}\right(\mathbf{u}\left)\right)$, see e.g. Equation (64(64) $F=\left(f{U}^{\prime }\right)\circ U^{-1},G=\left(g{U}^{\prime }\right)\circ U^{-1},$(64) ) in one dimension.

14 For an operator $\mathbb{O}$, the fraction $\frac{{\mathrm{e}}^{\mathbb{O}}-1}{\mathbb{O}}$ appearing e.g. in (279(279) ${\mathbb{T}}_{\mathbf{f},\mathbf{g}}h=\frac{\exp \left\{(f^{\mu }\mathrm{\Delta }t+g^{\mu i}\mathrm{\Delta }{\eta }_i){\mathrm{\partial }}_{\mu }\right\}-1}{(f^{\mu }\mathrm{\Delta }t+g^{\mu i}\mathrm{\Delta }{\eta }_i){\mathrm{\partial }}_{\mu }}h,$(279) ) is understood as a series $\sum _{k\ge 1}\frac{1}{n!}{\mathbb{O}}^{n-1}$.

15 The quantity $\delta \mathcal{L}$ of Equation (328(328) $\delta \mathcal{L}\left[\mathbf{u}\right]=\frac{1}{2}\frac{\mathrm{\partial }{\omega }_{\mu \nu }}{\mathrm{\partial }{x}^{\eta }}\frac{{\mathrm{\partial }}^2{X}^{\eta }}{\mathrm{\partial }{u}^{\alpha }\mathrm{\partial }{u}^{\text{β}}}\frac{\mathrm{\partial }{X}^{\mu }}{\mathrm{\partial }{u}^{\rho }}\frac{\mathrm{\partial }{X}^{\nu }}{\mathrm{\partial }{u}^{\sigma }}{\mathrm{\Omega }}^{\alpha \rho }{\mathrm{\Omega }}^{\text{β}\sigma }-\frac{1}{2}{\omega }^{\alpha \text{β}}\frac{\mathrm{\partial }{U}^{\nu }}{\mathrm{\partial }{x}^{\text{β}}}\frac{{\mathrm{\partial }}^2{X}^{\mu }}{\mathrm{\partial }{u}^{\varphi }\mathrm{\partial }{u}^{\nu }}\frac{{\mathrm{\partial }}^2{U}^{\varphi }}{\mathrm{\partial }{x}^{\alpha }\mathrm{\partial }{x}^{\mu }}.$(328) ) corresponds, in the arXiv v1 Ref. [117], to the integrand of the last line of Equation (E.22) multiplied by $-D$ (with $D=\frac{1}{2}$ in our settings), which gives $-g^2(u^{″}/u^{\prime }{)}^2/2-g{g}^{\prime }{u}^{″}/u^{\prime }$. This is the same result above but when changing variables from u to x.

Additional information

Funding

LFC & FvW acknowledge financial support from the ANR-20-CE30-0031 grant THEMA and VL from the ANR-18-CE30-0028-01 grant LABS.