In Wigner's classification of particles Wigner, E. (1939). On Unitary Representations of the Inhomogeneous Lorentz Group. Annals of Mathematics, 40(1), 149–204. http://www.jstor.org/stable/1968551 the Hilbert space of a single spinless particles of mass $m$ is given by functions $\tilde{\varphi}$ of 4-momentum $p$ which are supported on the mass shell $p^2=m^2$ and have inner product $$\langle\tilde{\varphi}_1,\tilde{\varphi}_2\rangle=\int \frac{d^3\mathbf{p}}{2E_\mathbf{p}}\tilde{\varphi}_1^*(p)\tilde{\varphi}_2(p)|_{p^0=E_\mathbf{p}}.$$ In particular, in terms of the position space $$\varphi(x)=\frac{1}{(2\pi)^{3/2}}\int\frac{d^3\mathbf{p}}{2E_\mathbf{p}}e^{-i p\cdot x}\tilde{\varphi}(p)|_{p^0=E_\mathbf{p}},$$ the inner product is $$\langle \varphi_1,\varphi_2\rangle=i\int d^3\mathbf{x}\left(\varphi_1^*\frac{\partial\varphi_2}{\partial t}-\frac{\partial\varphi_1^*}{\partial t}\varphi_2\right)$$ at any time slice. These position space functions satisfy the Klein-Gordon equation since the momentum space ones where concentrated on the mass shell. It is usually claimed that this cannot be interpreted as a probability amplitude for position given that this inner product is not positive definite. Moreover, in this scheme there doesn't seem to be any plausible notion of position operator (see Ticciati, R. (1999). Quantum Field Theory for Mathematicians. Cambridge University Press section 1.6). A nice summary of these facts is in the first chapter of Haag, R. (1996). Local Quantum Physics (2nd ed.). Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-642-61458-3.

On the other hand, in section 4.2 of Polchinski, J. (1998). String Theory. In Cambridge Monographs on Mathematical Physics. Cambridge University Press. https://doi.org/10.1017/CBO9780511816079, there is another approach to the quantum theory of a single spinless particle. Instead of being based on a classification of the representations of the Lorentz group, it is given by quantization of the classical theory of the particle $$S=\int d\tau\left(\frac{1}{2e}\dot{x}^\mu\dot{x}_\mu+\frac{1}{2}em^2\right).$$ The BRST procedure here yields the bosonic operators $x^\mu$ and $p^\mu$ satisfying $[x^\mu,p^\nu]=i\eta^{\mu\nu}$ fermionic operators $b$ and $c$ satisfying ${b,c}=1$, and a BRST operator $Q=cH$, where the Hamiltonian $H=\frac{1}{2}(p^2+m^2)$. Then the Stone - von Neumann theorem says that the Hilbert space of the theory is given by $\mathbb{C}^2$ valued functions $\varphi$ of position $x$ with inner product $$\langle\varphi_1,\varphi_2\rangle=\int d^4x\varphi_1^*(x)\varphi_2(x).$$ After taking the BRST cohomology, one can essentially forget that they are $\mathbb{C}^2$ valued and one obtains the additional restriction that they must satisfy the Klein-Gordon equation. However, unlike the previous case, in this perspective there is a position operator and one can interpret $\varphi$ as the spatial (spacetime) probability amplitude of the particle.

I am very confused by the fact that these two approaches give different answers. What is the relationship between them?

A comment:

The Klein-Gordon QFT yields a Hilbert space which is the second quantization of the one described by Wigner. On the other hand, the Feynman diagram expansion of the theory can be expressed in the worldline formalism in terms of the quantum theory described in Polchinski's book.