Update README.md

README.md

@@ -103,7 +103,7 @@ is the pushforward of the tangent vector $\frac{d \alpha}{d\tau}(\tau) \in T_\al
 
 By the time-dependent Schrödinger equation, the state $\ket{\psi(\alpha(t))} \in \mathcal{H}$ at some time $t$ will evolve according to $i \frac{d}{dt} \ket{\psi(\alpha(t))} = H \ket{\psi(\alpha(t))}$, which is satisfied (up to a constant) by the propagator $U(t_2 - t_1) = \exp[-i(t_2-t_1)H]$ given the Hamiltonian $H$. Here, the Hamiltonian $H$ is the infinitesimal generator of the one-parameter unitary group of time translations whose elements are the unitary transformations $U(t_2 - t_1):\mathcal{H} \to \mathcal{H}$ on the state space $\mathcal{H}$ for any $t_1, t_2 \in \mathbb{R}$. By performing a Wick rotation $\tau = it$, the state $\ket{\psi(\alpha(\tau))} \in \mathcal{H}$ at some imaginary time $\tau$ will evolve according to the imaginary-time Schrödinger equation $-\frac{d}{d\tau} \ket{\psi(\alpha(\tau))} = H \ket{\psi(\alpha(\tau))}$, which is satisfied (up to a constant) by the non-unitary propagator $U(\tau_2 - \tau_1) = \exp[-(\tau_2-\tau_1)H]$. Taking $\tau_1 = \tau$ and $\tau_2 = \tau + \delta\tau$, we propagate the state $\ket{\psi(\alpha(\tau))} \in \mathcal{H}$ by
  $$\ket{\psi(\alpha(\tau + \delta \tau))} = U(\delta\tau) \ket{\psi(\alpha(\tau))} = \ket{\psi(\alpha(\tau))} - \delta \tau H \ket{\psi(\alpha(\tau))} + \frac{(-\delta\tau)^2}{2}H^2 \ket{\psi(\alpha(\tau))} + \cdots \approx \ket{\psi(\alpha(\tau))} - \delta \tau H \ket{\psi(\alpha(\tau))} \in \mathcal{H}$$
-approximated to first order over the interval $[\tau, \tau + \delta \tau]$, which becomes exact in the limit $\delta\tau \to 0$. Enforcing a different normalization, we may also evolve the state $\ket{\psi(\alpha(\tau))} \in \mathcal{H}$ according to the imaginary-time Schrödinger equation $- \Delta\frac{d}{d\tau} \ket{\psi(\alpha(\tau))} = \Delta H \ket{\psi(\alpha(\tau))}$ involving the deviations $\Delta \frac{d}{d\tau} = \frac{d}{d\tau} - \left\langle \frac{d}{d\tau} \right\rangle_{\psi(\alpha)}$ and $\Delta H = H - \langle H \rangle_{\psi(\alpha)}$, which is often more advantageous in a stochastic framework.
+approximated to first order over the interval $[\tau, \tau + \delta \tau]$, which becomes exact in the limit $\delta\tau \to 0$. Enforcing a different normalization for the propagator, we may also evolve the state $\ket{\psi(\alpha(\tau))} \in \mathcal{H}$ according to $- \Delta\frac{d}{d\tau} \ket{\psi(\alpha(\tau))} = \Delta H \ket{\psi(\alpha(\tau))}$ involving the deviations $\Delta \frac{d}{d\tau} = \frac{d}{d\tau} - \left\langle \frac{d}{d\tau} \right\rangle_{\psi(\alpha)}$ and $\Delta H = H - \langle H \rangle_{\psi(\alpha)}$, which is often more advantageous in a stochastic framework.
 
 To determine the actual form of the tangent vector $\frac{d \alpha}{d\tau}(\tau) \in T_\alpha \mathcal{M}$ at time $\tau$, we impose the constraint that the projection
  $$\left\langle \frac{d}{d\tau} \psi(\alpha(\tau)), \bigg[ \Delta \frac{d}{d\tau} + \Delta H \bigg] \psi(\alpha(\tau)) \right\rangle = 0$$