This post is part of a series on few-body dynamics in the forward-JWKB approximation.
In this post I would like to introduce the toy model I will use to study the forward-JWKB approximation. This model only involves scalar quanta. There are two massive matter fields, \(\Phi_{1}\) and \(\Phi_{2}\), which are complex. There are also two intermedium fields, \(A\) and \(Y\), which are real. One of the intermedium fields is massive and the other is massless. The (free) kinetic operator for these fields are
$$ K_{1}(x, y) \equiv i \delta(x - y) \left[- \frac{1}{2} \hbar^{2} \left\Vert \partial \right\Vert^{2} + \frac{1}{2}m_{1}^{2} \right], $$
$$ K_{2}(x, y) \equiv i \delta(x - y) \left[- \frac{1}{2} \hbar^{2} \left\Vert \partial \right\Vert^{2} + \frac{1}{2}m_{2}^{2} \right], $$
$$ K_{A}(x, y) \equiv i \delta(x - y) \left[- \frac{1}{2} \left\Vert \partial \right\Vert^{2} \right], $$
$$ K_{Y}(x, y) \equiv i \delta(x - y) \left[- \frac{1}{2} \left\Vert \partial \right\Vert^{2} + \frac{1}{2 \lambda^{2}} \right]. $$
That is, both matter fields and both intermedium fields propagate. Note the explicit factors of \(\hbar\) in \(K_{1}\) and \(K_{2}\). These indicate that, unlike the intermedium fields, the matter fields describe quantum mechanical matter (i.e. particles). The intermedium fields describe waves. The (free) Green function for these fields satisfy the relations
$$ \int \mathrm{d}z \left[K_{1}(x, z) G_{1}(z, y)\right] = \delta(x - y), $$
$$ \int \mathrm{d}z \left[K_{2}(x, z) G_{2}(z, y)\right] = \delta(x - y), $$
$$ \int \mathrm{d}z \left[K_{A}(x, z) G_{A}(z, y)\right] = \delta(x - y), $$
$$ \int \mathrm{d}z \left[K_{Y}(x, z) G_{Y}(z, y)\right] = \delta(x - y). $$
These Green functions are given by
$$ G_{1}(x, y) = \left( \frac{1}{\hbar} \right)^{D} \int \mathrm{d}q \left( -\frac{2 i}{ \left\Vert q \right\Vert^{2} + m_{1}^{2} - i \epsilon_{1}} \exp{\left[ - \frac{i}{\hbar} q \cdot \left( x - y \right) \right]} \right), $$
$$ G_{2}(x, y) = \left( \frac{1}{\hbar} \right)^{D} \int \mathrm{d}q \left( -\frac{2 i}{ \left\Vert q \right\Vert^{2} + m_{2}^{2} - i \epsilon_{2}} \exp{\left[ - \frac{i}{\hbar} q \cdot \left( x - y \right) \right]} \right), $$
$$ G_{A}(x, y) = \int \mathrm{d}k \left( -\frac{2 i}{ \left\Vert k \right\Vert^{2} - i \epsilon_{A}} \exp{\left[ - i k \cdot \left( x - y \right) \right]} \right), $$
$$ G_{Y}(x, y) = \int \mathrm{d}k \left( -\frac{2 i \lambda^{2} }{ \lambda^{2} \left\Vert k \right\Vert^{2} + 1 - i \lambda^{2} \epsilon_{Y}} \exp{\left[ - i k \cdot \left( x - y \right) \right]} \right). $$
Note the \(i \epsilon\) prescription. The kinetic part of the action functional is
$$ S_{\text{kin}}\left[ \Phi_{1}, \Phi_{2}, A, Y \right] \equiv S_{0}\left[ \Phi_{1} \right] + S_{0}\left[ \Phi_{2} \right] + S_{0}\left[ A \right] + S_{0}\left[ Y \right], $$
with
$$ S_{0}\left[ \Phi_{1} \right] \equiv \int \mathrm{d}x \mathrm{d}y \left[ \Phi_{1}^{*}(x) K_{1}(x, y) \Phi_{1}(y) \right], $$
$$ S_{0}\left[ \Phi_{2} \right] \equiv \int \mathrm{d}x \mathrm{d}y \left[ \Phi_{2}^{*}(x) K_{2}(x, y) \Phi_{2}(y) \right], $$
$$ S_{0}\left[ A \right] \equiv \frac{1}{2 g^{2}} \int \mathrm{d}x \mathrm{d}y \left[ A(x) K_{A}(x, y) A(y) \right], $$
$$ S_{0}\left[ Y \right] \equiv \frac{1}{2 h^{2}} \int \mathrm{d}x \mathrm{d}y \left[ Y(x) K_{Y}(x, y) Y(y) \right]. $$
Here \(g\) and \(h\) are dimensionful coupling strengths. The interaction part of the action functional is
$$ S_{\text{int}}\left[ \Phi_{1}, \Phi_{2}, A, Y \right] \equiv S_{g}\left[ \Phi_{1}, \Phi_{2}, A \right] + S_{h}\left[ \Phi_{1}, \Phi_{2}, Y \right], $$
with
$$ S_{g}\left[ \Phi_{1}, \Phi_{2}, A \right] \equiv \int \mathrm{d}x \left[ g_{1} \Phi_{1}^{*} A \Phi_{1} + g_{2} \Phi_{2}^{*} A \Phi_{2} \right], $$
$$ S_{h}\left[ \Phi_{1}, \Phi_{2}, Y \right] \equiv \int \mathrm{d}x \left[ h_{1} \Phi_{1}^{*} Y \Phi_{1} + h_{2} \Phi_{2}^{*} Y \Phi_{2} \right]. $$
Here \(g_{1}\), \(g_{2}\), \(h_{1}\), and \(h_{2}\) are dimensionless charges. Both matter fields are coupled to each intermedium field in a cubic way.
Dimensional Analysis
It is always good to be aware of the units of the different quantities that have been introduced so far. I will work with units such that \(c = 1\) but \(\hbar\) is kept explicit and dimensionful. Since \(c = 1\), you have
$$ \left[ \text{time} \right] = \left[ \text{length} \right], $$
$$ \left[ \text{energy} \right] = \left[ \text{momentum} \right] = \left[ \text{mass} \right]. $$
Since \(\hbar\) is dimensionful, you have
$$ \left[ \text{length} \right] = \left[ \hbar \right] - \left[ \text{mass} \right]. $$
Action functionals have units of \(\hbar\). From the kinetic term of \(\Phi_{1}\) and \(\Phi_{2}\) you find the units of the matter fields:
$$ \left[ \Phi_{1} \right] = \left[ \Phi_{2} \right] = \left( \frac{1 - D}{2} \right) \left[ \hbar \right] + \left( \frac{D - 2}{2} \right) \left[ \text{mass} \right]. $$
Here \(D\) is the number of dimensions of spacetime. From the interaction terms you find the units of the intermedium fields:
$$ \left[ A \right] = \left[ Y \right] = 2 \left[ \text{mass} \right]. $$
Obviously, you must have
$$ \left[ \lambda \right] = \left[ \text{length} \right] = \left[ \hbar \right] - \left[ \text{mass} \right]. $$
Finally, from the kinetic terms of \(A\) and \(Y\) you find the units of the coupling strengths:
$$ \left[ g \right] = \left[ h \right] = \left( \frac{D - 3}{2} \right) \left[ \hbar \right] + \left( \frac{6 - D}{2} \right) \left[ \text{mass} \right]. $$
Note that for both coupling strengths the mass-dimension vanishes in \(D = 6\), and the \(\hbar\)-dimension vanishes in \(D = 3\). Also, the \(\hbar\)-dimension is positive for \(D > 3\), and the mass-dimension is positive for \(D < 6\). You can construct dimensionless coupling strengths via:
$$ \alpha_{g} \equiv \hbar^{(3 - D)} \mu^{(D - 6)} g^{2}, $$
$$ \alpha_{h} \equiv \hbar^{(3 - D)} \mu^{(D - 6)} h^{2}. $$
Here \(\mu\) is a constant with units of mass. When \(D > 3\), the JWKB approximation \(\hbar \rightarrow 0\) corresponds to \(\alpha_{g} \rightarrow \infty\) and \(\alpha_{h} \rightarrow \infty\) (i.e. strong-coupling). In this regime you also have
$$ \frac{\hbar}{m_{1}} \rightarrow 0, $$
$$ \frac{\hbar}{m_{2}} \rightarrow 0; $$
which leads to small Compton lenghts, or equivalently, to large matter masses. More details about the semiclassical approximation and dimensional analysis here.
