# Antiparticles in Non-Relativistic Quantum Mechanics
## 1 Introduction and Summary
Antiparticles are introduced after the transition to relativistic QM with phase space \(T^{*}\mathbb{R}^{3,1}=\mathbb{R}^{3,1}\times\mathbb{R}^{3,1}\), where positive-frequency solutions of free wave equations are associated with particles and negative-frequency solutions are associated with antiparticles. This interpretation is carried over to quantum field theory (QFT), where the coefficients in the expansion in positive- and negative-frequency basis solutions are replaced by operators. In the non-relativistic limit \(c\to\infty\), positive-frequency solutions \(\psi_{+}\) of relativistic equations are identified with the wave functions of free non-relativistic particles with energy \(E=p^{2}/2m\), and the analogous limit of negative-frequency solutions is declared non-physical. However, antiparticles do exist at low velocities \(v<<c\). That is why attempts have been made repeatedly to find a transition from QFT to non-relativistic QM while preserving the concept of antiparticles (see e.g., [3; 4] and references therein), but all these attempts were not considered successful. Returning to Formulae (1) and (2), we note that differential geometry asserts that, if a particle is associated with sections \(\psi_{+}\) of the bundle \(L^{+}_{\mathbb{C}}\), then sections of the dual bundle must be associated with antiparticles (opposite charges \(q_{\
u}=+1\) and \(q_{\
u}=-1\)). It is well known that, if \(\mathcal{E}\) is a rank \(n=1,2,\ldots\) complex vector bundle, then there are three more complex vector bundles: the complex conjugate bundle \(\bar{\mathcal{E}}\), the dual bundle \(\mathcal{E}^{\
u}\) and the dual of the complex conjugate bundle \(\mathcal{E}^{\
u}\). For example, if quarks are described by a complex vector bundle \(\mathcal{E}\) of rank 3 as \(\mathbb{C}^{3}\)-columns, then antiquarks are described by the dual bundle \(\mathcal{E}^{\
u}\) as \(\mathbb{C}^{3}\)-rows. If \(\mathcal{E}\) is a Hermitian complex vector bundle, then bundles \(\mathcal{E}\) and \(\mathcal{E}^{\
u}\) are isomorphic, as well as bundles \(\mathcal{E}\) and \(\mathcal{E}^{\
u}\). Bundles \(\mathcal{E}\) and \(\mathcal{E}\) are not isomorphic and, in particular, the \(k\)-th Chern class of \(\bar{\mathcal{E}}\) is given by \(c_{k}(\bar{\mathcal{E}})=(-1)^{k}c_{k}(\mathcal{E})\) so that \(c_{1}(\bar{\mathcal{E}})=-c_{1}(\mathcal{E})\). The bundle \(L^{+}_{\mathbb{C}}\) introduced in non-relativistic QM is a Hermitian complex line bundle [1; 2]. Therefore, we can and must introduce the complex conjugate bundle \(L^{-}_{\mathbb{C}}\) and its sections \(\psi_{-}\), which describe antiparticles when \(\psi_{-}\
eq\psi_{+}^{*}\). In this case, the energy \(E\) of particles and antiparticles is always the same since, in the Schrodinger equations for dual objects, \(\mathrm{i}\partial_{t}\) is replaced by \(-\mathrm{i}\partial_{t}\). Differential geometry requires distinguishing particles and antiparticles by the quantum charge \(q_{\
u}=\pm 1\) introduced above, and not by the sign of energy, and the definition in terms of bundles \(L^{\pm}_{\mathbb{C}}\) is preserved for particles in any external fields and in the relativistic case. In particular, the energy of a free non-relativistic electron is positive, but, for an electron in the field of central forces generated by a proton, it is negative. Moreover, if the hydrogen atom is described by the function \(\psi_{+}\in\Gamma(L^{+}_{\mathbb{C}})\), then the positron in the field of forces generated by the antiproton is described by the function \(\psi_{-}\in\Gamma(L^{-}_{\mathbb{C}})\), and the experimentally observed energy levels of hydrogen and antihydrogen atoms are the same.