Consider the holomorphic representation of the path integral (for a single degree of freedom):

$$ U(a^{*}, a, t'', t') = \int e^{\alpha^{*}(t'') \alpha(t'')} \exp\left\{\intop_{t'}^{t''} dt \left( -a^{*} \dot{a} - i h(a, a^{*}) \right) \right\} \prod_t \frac{da^{*}(t) da(t)}{2\pi i}. $$

The proper boundary conditions are of form

$$ a(t') = a; \quad a^{*}(t'') = a^{*}. $$

My question is: how are $a$ and $a^{*}$ related and why?

One observation is that we treat them as independent variables in the path integral, so they can't be complex-conjugate ad hoc.

Another observation is that we should impose the reality condition on the boundary (which is analogous to the $\text{Im} \left(x(t',t'')\right) = 0$ condition in the coordinate representation). But how (and why) should we relate $a(t')$ to $a^{*}(t'')$ which are taken at different instants of time?

UPDATE: my original idea is that they are not related at all. We simply constrain our description to holomorphic wavefunctions $\Psi(a^{*}(t''))$ and $\Phi(a(t'))$ which is analogous to constraining it to the real-variable wavefunctions in the coordinate basis. But my professor keeps insisting otherwise (he doesn't actually care to give a convincing argument though, just keeps saying "no").

Consider the holomorphic representation of the path integral (for a single degree of freedom):

$$ U(a^{*}, a, t'', t') = \int e^{\alpha^{*}(t'') \alpha(t'')} \exp\left\{\intop_{t'}^{t''} dt \left( -a^{*} \dot{a} - i h(a, a^{*}) \right) \right\} \prod_t \frac{da^{*}(t) da(t)}{2\pi i}. $$

The proper boundary conditions are of form

$$ a(t') = a; \quad a^{*}(t'') = a^{*}. $$

My question is: how are $a$ and $a^{*}$ related and why?

One observation is that we treat them as independent variables in the path integral, so they can't be complex-conjugate ad hoc.

Another observation is that we should impose the reality condition on the boundary (which is analogous to the $\text{Im} \left(x(t',t'')\right) = 0$ condition in the coordinate representation). But how (and why) should we relate $a(t')$ to $a^{*}(t'')$ which are taken at different instants of time?

UPDATE: my original idea was that they are not related at all. We simply constrain our description to holomorphic wavefunctions $\Psi(a^{*}(t''))$ and $\Phi(a(t'))$ which is analogous to constraining it to the real-variable wavefunctions in the coordinate basis. But my professor keeps insisting otherwise (he doesn't actually care to give a convincing argument though, just keeps saying "no").