How to path-integrate over the half-line? Consider the path-integral over a scalar field $\varphi$:
$$
Z=\int_{\mathcal S}\ \mathrm e^{iS[\varphi]}\mathrm d\varphi
$$
where $\mathcal S$ is some function space (say, Schwartz or its dual).
How can we implement the condition $\varphi\ge0$? what effect does this restriction have, both at the perturbative level and at the non-perturbative level?
Note that this is not merely an "out of curiosity" question. The situation above occurs in practice (in its simplest manifestation, in the Higgs mechanism where we decompose $H(x)=\rho(x)\ \mathrm e^{i\sigma(x)}$; here the integral over $\rho$ is only over the half-line $\rho\ge0$).
 A: For a free particle on the half line, the requirement that the Hamiltonian:
$$H = \frac{p^2}{2m}$$
is self adjoint
$$ (\psi ,H \phi) =  (H\psi , \phi) = \phi(0) \frac{d\psi}{dx}(0)-\psi(0) \frac{d\phi}{dx}(0)$$
(this condition means that there is no probability current crossing $x=0$), leads to a family of self adjoint extensions characterized by  the boundary conditions:
$$\psi(0) = \gamma  \frac{d\psi}{dx}(0)$$
The propagator of the general solution of the Schrodinger equation with this boundary condition is given by:
$$G(x_1, x_2) = G_0(x_2-x_1)+G_0(x_2+x_1) - 2 \gamma \int_0^{\infty}d\lambda e^{-\gamma \lambda} G_0(x_2+x_1+\lambda)$$
($G_0$ is the free unconstrained propagator; Please, see Gamboa)
Both Clark,Menikoff, and Sharp  and  Fahri and Gutmann
showed that this propagator can be obtained from a path integral quantization of the free action with a delta potential at the origin for paths extending from $-\infty$ to $+\infty$.
$$L = \frac{m \dot{x}^2}{2} + \gamma \delta(x)$$
However, this is not the unique way to quantize the motion on the half line:
Isham , defined a quantization scheme based on the canonical transformation:
$$(x, p) \rightarrow (e^x, e^{-x}p)$$
Another quantization scheme is based on the quatization of $\mathbb{R}^+$ as a quotient space:Please see Tanimura
$$\mathbb{R}^+ = \mathbb{R}^2/SO(2),$$
This method leads to another family of Hamiltonians:
I am not aware of any path integral formulation of the last two methods. Clearly, the last method leads to inequivalent quantizations with respect to the first one. I am quite confident that both methods can be merged to a unified method including both families of quatizations. 
