# Antiparticles in Non-Relativistic Quantum Mechanics
## 1 Introduction and Summary
Using the differential geometric approach leads to the assertion that the density of conserved charges for positive- and negative-frequency complex solutions of the Klein-Gordon equation:  
\[\varrho^{\mathrm{KG}}_{\pm}=\frac{\mathrm{i}\hbar}{2 mc^{2}}\left(\psi_{\pm}^{* }\psi_{\pm}-\psi_{\pm}^{*}\psi_{\pm}\right)\ \stackrel{{ c\to\infty}}{{ \longrightarrow}}\ \pm\psi_{\pm}^{*}\psi_{\pm}\,\quad\psi:=\frac{\partial\psi}{ \partial t}\, \tag{3}\]  
representing the density \(\varrho^{\mathrm{Sch}}_{\pm}=\pm\psi_{\pm}^{*}\psi_{\pm}\) of charges \(q_{\
u}=\pm 1\) associated with the bundles \(L^{\pm}_{\mathbb{C}}\) and the continuity equations for the conjugated Schrodinger equations. And the interpretation of functions \(|\psi_{\pm}|^{2}=\psi_{\pm}^{*}\psi_{\pm}\) as probability densities is secondary. This can be illustrated by the example of a one-dimensional harmonic oscillator, considered in this paper. Namely, the function \(\psi_{+}^{*}\psi_{+}\) is the quantum charge density in both the coordinate and holomorphic Segal-Bargmann representations. However, in the holomorphic representation, the function \(\psi_{+}^{*}\psi_{+}\) cannot be interpreted as a probability density (there is no \(\sigma\)-additivity). The inadequacy of introducing the concept of a particle and antiparticle through the sign of the eigenvalues of the operator \(\mathrm{i}\partial_{t}\) is obvious when considering quantum fields in curved space-time, which is discussed in detail in [5]. In general, Killing vectors, with the help of which one could define positive-frequency solutions, do not exist at all, and this leads to the vagueness of the concept of a particle [5]. The definition in terms of bundles \(L^{+}_{\mathbb{C}}\) and \(L^{-}_{\mathbb{C}}\) removes this problem since being a particle or antiparticle means having a conserved quantum charge \(q_{\
u}=+1\) or \(q_{\
u}=-1\). Even in Minkowski space, the definition via the operator \(\mathrm{i}\partial_{t}\) does not work for a non-free particle, for example, for a relativistic oscillator. Its covariant phase space is a homogeneous Kahler space PU(3,1)/U(3), which is not fibered over either coordinate space or the momentum space, and the solutions do not contain terms with \(\exp(\pm\mathrm{i}Et)\). Instead, the oscillating particles and antiparticles are sections of the holomorphic \(L^{+}_{\mathbb{C}}\) and, respectively, antiholomorphic \(L^{-}_{\mathbb{C}}\) bundles over the manifold PU(3,1)/U(3) [6]. The differential-geometric view of quantum mechanics requires another conceptual change. The \"wavefunctions\" \(\psi_{\pm}\) of particles and antiparticles are not functions, but sections of _vector_ bundles \(L^{\pm}_{\mathbb{C}}\). Therefore, they should be added not as scalars, but as vectors:  
\[\Psi=\psi_{+}v_{+}+\psi_{-}v_{-}\in L^{+}_{\mathbb{C}}\oplus L^{-}_{\mathbb{C }}\, \tag{4}\]  
where \(v_{\pm}\) are bases in the fibers of vector bundles \(L^{\pm}_{\mathbb{C}}\) of rank one. Also, in relativistic equations, it is necessary to separate fields with \(q_{\
u}=1\) and \(q_{\
u}=-1\) as in (4), which allows one to avoid the problem of negative energies. In particular, if \(\psi_{\pm}\) in (4) are Dirac spinors, then the sesquilinear form on spinors is given by  
\[\langle\Psi,\Psi\rangle=h_{++}\bar{\psi}_{+}\psi_{+}+h_{--}\bar{\psi}_{-}\psi_ {-}\, \tag{5}\]  
where \(\bar{\psi}=\psi^{\dagger}\gamma^{0}\) and \(h_{++}\) and \(h_{--}\) are components of the Hermitian metric on the bundle \(L^{+}_{\mathbb{C}}\oplus L^{-}_{\mathbb{C}}\). By choosing \(h_{++}=1\) and \(h_{--}=-1\), we obtain a positive definite metric on the space of Dirac spinors and the corresponding Fock space. In the paper [7], a similar change in the inner product was proposed to be obtained by changing the sign of the complex structure on the negative-frequency spinors. This corresponds to the choice \(h_{++}=1\) and \(h_{--}=-1\) for opposite signs of the quantum charges \(q_{\
u}=1\) and \(q_{\
u}=-1\).