quantum mechanics - Boundary conditions in holomorphic path integral

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").

Answer

OP's question is essentially pondering (in the context of the holomorphic/coherent state path integral) if a pair of variables is a complex conjugate pair or$^1$ truly independent variables.

Notation in this answer: In this answer, let $z,z^{\ast}\in \mathbb{C}$ denote two independent complex numbers. Let $\overline{z}$ denote the complex conjugate of $z$. Also Planck's constant $\hbar=1$ is put equal to one.

Recall that the coherent ket state is

$$\tag{1} |z \rangle~:=~e^{z\hat{a}^{\dagger}}|0 \rangle, \qquad \hat{a}|z \rangle~=~z|z \rangle .$$

It is customary$^2$ to define the coherent bra state

$$\tag{2} \langle z | ~:=~ |\bar{z} \rangle^{\dagger} ~\stackrel{(1)}{=}~\langle 0 |e^{z\hat{a}}$$

in terms of the coherent ket state (1) by including a complex conjugation, cf. e.g. Ref. 1. In other words, we have the convenient rule that

$$\tag{3} \langle z^{\ast} | ~\stackrel{(2)}{=}~ \langle 0 |e^{z^{\ast}\hat{a}}, \qquad \langle z^{\ast} |\hat{a}^{\dagger} ~=~z^{\ast} \langle z^{\ast} | .$$

With this convention (2), the completeness relation reads

$$\tag{4} \int_{\mathbb{C}} \frac{dz~d\bar{z}}{2 \pi i} e^{-\bar{z}z} |z \rangle\langle \bar{z} |~=~{\bf 1}.$$

It is important to realize that the coherent states are an overcomplete set of states

$$\tag{5} \langle z^{\ast}|z \rangle~=~e^{z^{\ast} z}$$

with non-orthogonal overlaps. The coherent state path integral reads

$$\tag{6} \langle z_f^{\ast}, t_f | z_i, t_i \rangle ~=~ \int_{z(t_i)=z_i}^{\bar{z}(t_f)=z^{\ast}_f} \! {\cal D}z~{\cal D}\bar{z} ~e^{iS[z,\bar{z}]}, \qquad {\cal D}z~{\cal D}\bar{z}~:=~ \prod_{n=1}^N \frac{dz_n~d\bar{z}_n}{2 \pi i} ,\qquad$$

$$iS[z,z^{\ast}]~:=~ (1-\lambda)z^{\ast}(t_f)~z(t_f)+ \lambda z^{\ast}(t_i) z(t_i)\qquad \qquad$$ $$\tag{7}\qquad+ \int_{t_i}^{t_f}\! dt \left[\lambda \dot{z}^{\ast} z -(1-\lambda) z^{\ast} \dot{z}- iH_N(z^{\ast},z) \right],$$

where $\lambda\in \mathbb{R}$ is a real constant which the action (7) does not actually depend on, due to the fundamental theorem of calculus. The Hamiltonian function

$$\tag{8} H_N(z^{\ast},z)~:=~\frac{\langle z^{\ast}|\hat{H}(a^{\dagger},a)|z \rangle}{\langle z^{\ast}|z \rangle}$$

is the normal/Wick-ordered function/symbol corresponding to the quantum Hamiltonian operator $\hat{H}(a^{\dagger},a)$. Concerning operator ordering in the path integral, see also e.g. this Phys.SE post.

In the standard Feynman path integral there are 2 real boundary conditions (BCs), typically Dirichlet BCs

$$\tag{9} q(t_i)~=~q_i \quad\text{and}\quad q(t_f)~=~q_f.$$

The position $\hat{q}$ and the momentum $\hat{p}$ are related to

$$\tag{10} {\rm Re}(\hat{a})~:=~\frac{\hat{a}+\hat{a}^{\dagger}}{2} \quad\text{and}\quad{\rm Im}(\hat{a})~:=~\frac{\hat{a}-\hat{a}^{\dagger}}{2i},$$

respectively. In the coherent state path integral (6), there are 2 complex (= 4 real) BCs

$$\tag{11} z(t_i)~=~z_i \quad\text{and}\quad \bar{z}(t_f)~=~z^{\ast}_f.$$

In other words, we specify both initial position and initial momentum, naively violating HUP. Similar for the final state. This is related to the overcompleteness (5) of the coherent states.

The overcomplete BCs (11) means that there typically is not an underlying physical real classical path that fulfills all the BCs (11) and the Euler-Lagrange equations simultaneously unless we tune the BCs (11) appropriately, cf. e.g. Ref. 1. The precise tuning depends on the theory at hand.

References:

1. L.S. Brown, QFT; Section 1.8.

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$^1$ For more on complex conjugation and independence of variables, see also e.g. this Phys.SE post.

$^2$ Notabene: Some authors do not include a complex conjugation in definition (2), cf. e.g. Wikipedia!