In this thesis we look at how it is possible to construct models in quantum mechanics by using p-adic numbers. First we look closely at different quantum mechanical models using the real numbers, as it is necessary to understand them well before moving on to p-adic numbers. The most promising model, where Weyl systems are used, is studied in detail. Here time translation is not generated by the Hamiltonian, but constructed directly as an operator possessing some fundamental structure in relation to the classical dynamics. Then we develop the relevant theory of the field of p-adic numbers Qp , with a focus on the properties of Qp as a locally compact abelian group. Here we present alternative proofs to those found in the literature. In particular, we give an independent proof of the selfduality of Qp. In the last chapters we look at some models using Qp . We generalize the idea of Weyl systems from real to p-adic numbers, and we discuss the physical implications. When using Weyl systems, time is p-adic. We also produce MatLab algorithms for numerical computations in connection with approximations of p-adic models by finite models.
In this thesis we look at how it is possible to construct models in quantum mechanics by using p-adic numbers. First we look closely at different quantum mechanical models using the real numbers, as it is necessary to understand them well before moving on to p-adic numbers. The most promising model, where Weyl systems are used, is studied in detail. Here time translation is not generated by the Hamiltonian, but constructed directly as an operator possessing some fundamental structure in relation to the classical dynamics. Then we develop the relevant theory of the field of p-adic numbers Qp , with a focus on the properties of Qp as a locally compact abelian group. Here we present alternative proofs to those found in the literature. In particular, we give an independent proof of the selfduality of Qp. In the last chapters we look at some models using Qp . We generalize the idea of Weyl systems from real to p-adic numbers, and we discuss the physical implications. When using Weyl systems, time is p-adic. We also produce MatLab algorithms for numerical computations in connection with approximations of p-adic models by finite models.
A formulation of quantum mechanics on p-adic number fields is presented. Quantum amplitudes are taken as complex functions of p-adic variables and it is shown how the Weyl approach to quantum mechanics can be generalized to the p-adic case. The p-adic analogs of simple one-dimensional systems (free particle, compact and noncompact oscillators) are defined by a ‘‘group of motion,’’ which is an Abelian subgroup of SL (2,Qp). In each case the evolution operator is a unitary representation of the appropriate group. Its spectrum is given by characters and its eigenstates are calculated.
The Heisenberg-Weyl group
HW(d) related to a
d-dimensional Hilbert space
H(d), is enlarged into the Heisenberg-Weyl-parity group
HWP(d) that incorporates parity transformations. It consists of
2d3 elements, of which
d3 elements belong to the
HW(d) subgroup, and extra
d3 elements which are related through a Fourier transform with the former ones. It is shown that
HWP(d) is a generalised version of the dihedral group. The properties of operators that combine displacements and parity, are discussed.
HWP(d) is shown to be a solvable group, and commutators of its elements perform displacement and parity transformations of quantum states, along loops in the discrete phase space.
2d2 coherent states related to the
HWP(d) group are introduced, which consist of
d2 coherent states related to the
HW(d) subgroup, and extra
d2 coherent states which are related through a Fourier transform with the former ones. In noisy cases, expansion of an arbitrary state in terms of the
2d2 coherent states with Bargmann coefficients, is advantageous in comparison to expansion in terms of the
d2 coherent states related to
HW(d). One of the consequences of the
HWP(d) group, is a natural unification of the Wigner and Weyl functions. The properties of the unified Wigner-Weyl function are discussed.