Negative Time in Vacuum to Vacuum Transition Amplitude In the path integral formulation of quantum mechanics, the transition amplitude for going from $|\phi_i, t_i \rangle$ to $|\phi_f, t_f \rangle$ is given by
$$\langle \phi_f, t_f |\phi_i, t_i \rangle = \int \frac{\mathcal{D}q\mathcal{D}p}{h} \, \exp\left[\frac{i}{\hbar} \int_{t_i}^{t_f} \, dt \, \left( p(t)\dot{q}(t) - H(p, q) \right) \right], $$
and, the vacuum to vacuum transition amplitude in the presence of a source $J(t)$ is given by
$$\langle 0, \infty |0, -\infty \rangle^J \propto Z[J] = \int \mathcal{D}q \, \exp\left[\frac{i}{\hbar} \int_{-\infty}^{\infty} dt \left( L + \hbar J q + \frac{i}{2} \epsilon q^2 \right) \right], $$
where the symbols have their usual meanings. 
Question
While deriving the expression for $\langle 0, \infty |0, -\infty \rangle$, we need to take a limit: $t \to -\infty$. But how can the time be negative? In other words, what is the physical significance of the negative time? 
 A: I don't think this has anything to do with negative time, just time intervals. 
This is something that is generally done whenever we work with scattering amplitudes in Quantum Field Theory. This is the way I see it: let's choose some arbitrary origin for your time axis $t=0$ during the interaction process. Using this definition, everything 'left' of our origin, i.e. on the negative time axis occurs in the "past" and everything to the 'right' occurs in the "future".
The typical scattering experiment starts off with particles so far apart at some time (say $t = t_\text{in}$) in the past that they are not yet interacting, and ends with particles again so far apart in the future that they have ceased interacting at some time (say $t = t_\text{out}$). These are now the limits of the time integral within the path integral. We now consider these two points to be considerably in the past and the future respectively, in other words both the time intervals can be taken to be infinite$^1$.
Thus, the integral from $t_\text{in}$ to $t_\text{out}$ effectively reduces to an integral from $-\infty$ to $+\infty$, the entire 'history' of the particle.
$^1$ Since time is a quantity with dimension, the question of course is "infinite with respect to what?" Well, it's with respect to the duration of the interaction. We assume that the time scale associated with the interaction is significantly small compared to the time taken by the total scattering process.
