1. Relativistic QM for spinless particles is not a physically consistent theory. Let the scalar wave function \(\phi\) satisfy the KG eq. \(\partial_a \partial^a\phi+m^2\phi=0\). One proves directly from the KG eq. that the currents \(j_a\propto \phi^*\partial_a\phi-\phi\partial_a\phi^*\) are conserved, i.e., its covariant divergence vanishes: \(\partial^aj_a=0\). In a relativistic version of *single-particle* QM (read: first quantized theory), one would expect the time-like component of the KG current, \(j_0\propto\phi^*\partial_t\phi-\propto\phi\partial_t\phi^*\), to be interpreted as a probability current density. Differently from the non-relativistic QM, where the wave-function \(\psi\) obeying a Schrodinger eq. \(i\partial_t\psi=H\psi\) have as probability density the positive quantity \(\rho\propto\phi^*\phi\), the KG current \(j_0\) allows both positive as well as negative values, spoiling its probabilistic interpretation. This was one of the reasons that lead Dirac to find a new relativistic wave equation, which indeed cures the non-positiveness of the time-like component of the current.

**Remark. **Even if one insists with carrying on a relativistic QM theory for a spinless particle described by KG, ignoring therefore the problem mentioned above regarding its probabilistic interpretation as well the one which I will mention below (about the negative-energy solutions), the expression for \(j_0\) given above shows that it vanishes for real-valued \(\phi\). The situation now is then even worse, that now its not only that negative values are allowed for \(j_0\), but what should be regarded as a "probability" current vanishes identically in the whole space-time, and then there is no particle left at all.

2. The KG eq. have as plane wave solutions both positive as well as negative frequency solutions, namely, \(\phi_\boldsymbol{k}^{(+)}(t,\boldsymbol{r})=e^{i(\omega_k t - k\cdot r)}/N\) and \(\phi_\boldsymbol{k}^{(-)}(t,\boldsymbol{r})=e^{-i(\omega_k t + k\cdot r)}/N\) (\(N\) is the normalization constant), where \(\omega_k=+\sqrt{\boldsymbol{k}^2+m^2}\) for all possible momenta \(\boldsymbol{k} \in \mathbb{R}^3\). In a single particle theory, both positive and negative frequency yields both positive as well as negative energy states, since relativity only imposes that \(E_k^2=\omega_k^2=\boldsymbol{k}^2+m^2\), giving a physically unstable vacuum for the theory (the particle could radiate its energy away down to \(-\infty\)). This situation reappears in the solutions to Dirac equation, and the latter tried to solve this issue by introducing the hypothesis that all negative-energy states are already filled, so that, his particle having spin one-half and therefore obeying Pauli exclusion principle, no transition to negative-energy states are allowed. This theory, historically known as the Dirac hole theory, is a manifestation that relativistic QM is only consistent with many-particle states.

3. This still left open the problem of defining a Lorentz invariant quantum theory for spinless particles. The hole theory interpretation is not possible since these particles do not obey the exclusion principle. Pauli (in what he dubbed his "anti-Dirac" paper) was one of the first to indicate that the right solution is to introduce *quantum fields* (which has been known from the 1930s as *second* quantization, since one treats the wave-function for a single-particle state as a classical field and then quantize again, but now the field; this is outdated terminology, however.) As opposed to the single-particle theory discussed above, we begin by interpreting \(\phi\) as a classical field. Its currents \((j_a)\) can be seen as the Nother charges derived from the KG Lagrangian \(\mathcal{L}=(1/2)\partial_a\phi\partial^a\phi+(1/2)m^2\phi^2\), so that real-valued \(\phi\) for which \(j_a=0\) should describe a *classical* non-charged field, while complex-valued \(\phi\) are to be identified with classical fields carrying charges. Quantum theory is obtained either by introducing the quantum commutation relations on the field variables, for example, the equal-time commutation relations on (the now field-operators) \(\phi\) and \(\pi=\partial\mathcal {L}/\partial{(\partial_t\phi)}\), or going to the path-integral formalism by working out the partition function \(Z_0[J]=\int\mathcal{D}\phi e^{i\int \mathcal{L}+J\phi}\), whose functional derivatives generates the theory Green's functions. In any case, one then obtains a consistent, Lorentz invariant, theory for spinless particles, as a quantum theory of the *field *\(\phi\). (*Note. *For Fermi-Dirac fields, one replaces commutation relations by anticommutation ones, or in the path-integral approach, one introduce Grassman variables).

**Remark. **The Fourier decomposition of the field operator \(\phi\) in terms of the (positive and negative frequency) plane-wave solutions \(\phi_\boldsymbol{k}^{(+)}\) and \(\phi_\boldsymbol{k}^{(-)}\), namely, \(\phi \propto\int (d^3\boldsymbol{k}/\sqrt{\omega_\boldsymbol{k}}) \big( \phi_\boldsymbol{k}^{(+)} a_\boldsymbol{k}+\phi_\boldsymbol{k}^{(-)} a_\boldsymbol{k}^{+}\big)\), shows at once that: (i) only positive-energy solutions exists, but (ii) the theory must be contain many-particle states, where the coefficients \(a_k\) and \(a_k^+\) acts as annihilation and creation operators.

4. Finally, two further remarks. (i) About the the role of the term \(e^{imt}\) in the derivation of the Sch. eq. from the KG, the author of the question should give more details about what reference he is using to study this. Nevertheless, I guess it must came from the expression of the time-evolution operator using the relativistic Hamiltonian, \(U(t)=e^{i\sqrt{p^2+m^2}t}\). In this case, the non-relativistic limit is \(\propto e^{imt+i\boldsymbol{p}^2t/2m}\), and so the first term can be ignored since it only gives an overall phase to the quantum state. (ii) As a further inquire, suppose now you begin with the Dirac equation, and then decide to take its non-relativistic limit. How one obtains the *spinless* Schrodinger eq. from it? (Tip: the state space of nonrelativistic QM will be recovered from the nonrelativistic limit of the Dirac eq. as an eigenspace of the spin operator, namely, the nonrelativistic wave-functions are going to arise as spin eigenfunctions.)

5. I think that the best way to relate the KG current for a complex scalar field to electric charge is by discussing the gauge-invariance of the theory. Begin with the KG Lagrangian for a complex field:

\[\mathcal{L}=\partial_a\phi\partial^a\phi^*+m^2\phi\phi^*.\]

Under the transformation \((\phi,\phi^*)\mapsto(\phi e^{-i\Lambda},\phi^* e^{i\Lambda})\) for a constant \(\Lambda\), the complex KG Lagrangian is invariant. Using Noether theorem, you can show that under \(\delta \phi=-i\Lambda\phi\), \(\delta \phi^*=i\Lambda\phi^*\), the Noether current

\[\mathcal{J_a}=\frac{\delta \mathcal{L}}{\delta (\partial^a \phi)}\delta\phi+\frac{\delta \mathcal{L}}{\delta (\partial^a \phi^*)}\delta\phi^*
=-i(\phi \partial_a \phi^*-\phi^* \partial_a \phi)
\]

is proportional to the KG current which I introduced above (namely, \(\propto j_a\)). One might worry that a proportionality constant, in particular the electric charge \(e\), is missing. This happens because we have been considering a *global* gauge transformation, where the "generator" of the gauge \(\Lambda\) was assumed to be constant. If we allow it to vary in space-time, so that we begin to consider *local* transformations, the infinitesimal variations becomes \(\delta \phi=-i\Lambda(x_\mu)\phi\), \(\delta \phi^*=i\Lambda(x_\mu)\phi^*\). But now, the field derivatives transforms as

\[\delta (\partial_a \phi)=-i\Lambda \partial_a\phi-i\phi\partial_a\Lambda, \\
\delta (\partial_a \phi^*)=i\Lambda \partial_a^*\phi+i\phi^*\partial_a\Lambda.\]

KG Lagrangian is not invariant under these variations. One is then lead to introduce the following term to the Lagrangian density,

\[\mathcal{L}'=-e\mathcal{J}_aA^a+e^2A_aA^a\phi\phi^*,\]

which couples with coupling strength \(e\) the KG current \(\mathcal{J}_a\) to a new vector field \(A^a\), which is supposed to transform as covariant gradient \(e^{-1}\partial_a\Lambda\). This is nothing but the gauge transformation that we have meet in classical EM. In order to give dynamical content to the field \(A^a\) (in the sense of making it contribute itself to the Lagrangian), we add the Maxwell term to the total Lagrangian, \(\mathcal{L}''=-(1/4)F_{ab}F^{ab}\) for the covariant curl \(F^{ab}=\partial^{[a}A^{b]}\), and we end up with the theory of electrodynamics coupled to a charged complex scalar field.

From this perspective then, we obtain the EM field (and the charge \(e\) coupling the scalar field to the EM one) as a result of requiring gauge invariance. I believe this point of view is more modern (certainly more sophisticated) than the traditional approach: relativistic QM through wave equations \(\mapsto\) second quantization.