We solve the particle-antiparticle and cosmological constant problems proceeding from quantum theory, which postulates that: various states of the system under consideration are elements of a Hilbert space$cal{H}$ with a positive definite metric; each physical quantity is defined by a self-adjoint operator in $cal{H}$;symmetry at the quantum level is defined by a representation of a real Lie algebra $A$ in $cal{H}$ such that the representation operator of any basis element of $A$ is self-adjoint. These conditions guarantee the probabilistic interpretation of quantum theory. We explain that in the approaches to solving these problems that are described in the literature, not all of these conditions have been met. We argue that fundamental objects in particle theory are not elementary particles and antiparticles but objects described by irreducible representations (IRs) of the de Sitter (dS) algebra.One might ask why, then, experimental data give the impression that particles and antiparticles are fundamental and there are conserved additive quantum numbers (electric charge, baryon quantum number and others). The matter is that, at the present stage of the universe, the contraction parameter $R$ from the dS to the Poincare algebra is very large and, in the formal limit $Rtoinfty$, one IR of the dS algebra splits into two IRs of the Poincare algebra corresponding to a particle and its antiparticle with the same masses. The problem why the quantities $(c,hbar,R)$ are as are does not arise because they are contraction parameters for transitions from more general Lie algebras to less general ones. Then the baryon asymmetry of the universe problem does not arise. At the present stage of the universe, the phenomenon of cosmological acceleration (PCA) is described without uncertainties as an inevitable {it kinematical} consequence of quantum theory in semiclassical approximation. In particular, it is not necessary to involve dark energy the physical meaning of which is a mystery. In our approach, background space and its geometry are not used and $R$ has nothing to do with the radius of dS space. In semiclassical approximation, the results for PCA are the same as in General Relativity if $Lambda=3/R^2$, i.e., $Lambda>0$ and there isno freedom in choosing the value of $Lambda$.