Quantum mechanics on ${\mathbb Q}/{\mathbb Z}$Q/Z

Abstract

Quantum mechanics with positions in Q/Z and momenta in b Z is considered. Displacement operators and coherent states, parity operators, Wigner and Weyl functions, and time evolution, are discussed. The restriction of the formalism to certain finite subspaces, is equivalent to Good's factorization of quantum mechanics on Z(q). I. INTRODUCTIONQuantum mechanics and quantum field theory on the field Q p of p-adic numbers have been studied by various authors (e.g., [1][2][3][4][5][6][7][8][9][10][11][12]). Applications to condensed matter physics have been discussed in [13][14][15][16]. Mathematical work related to these problems has been presented in [17][18][19][20][21]. General references on p-adic numbers are [22,23], and on the related topic of profinite groups [24,25].Recently we have studied quantum mechanics with positions in Z p (p-adic integers) and momenta in Q p /Z p , with emphasis on both the physical aspects [26], and the mathematical aspects (using inverse limits) [27]. Physical applications include the physics at the Planck scale, condensed matter, etc. Our interest is in quantum engineering of devices with p-adic arithmetic [26]. Such devices might have applications in the general area of information processing (e.g., [28]).In the present paper we extend this work and study quantum mechanics with positions in Q/Z and momenta in the Pontryagin dual group Z (where Q are the rational numbers, Z are the integers and Z is discussed below). We study the Heisenberg-Weyl group and discuss various properties of the displacement and parity operators, Wigner and Weyl functions and coherent states. Hamiltonians in this context, and the corresponding time evolution, are also studied. We show that a finite number of coupled finite quantum systems can be embedded into quantum mechanics on Q/Z and in this sense there are many realistic physical systems, where the formalism is applicable.There has been a lot of work recently (for reviews see [29][30][31][32][33]) on various aspects of finite quantum systems with positions and momenta in Z(d) (the integers modulo d). When d = p n (with p a fixed prime number), the large d limit of these systems is precisely the system with phase space Z p × (Q p /Z p ) (studied in [26,27]). When d takes all integer values, the large d limit of these systems is the system with phase space Z × (Q/Z) (studied here). This is shown in table I. The mathematical tools that bring these finite systems to the 'edge' are the inverse limit and direct limit. Using the inverse limit with the groups corresponding to momenta and the direct limit with the groups corresponding to positions, ensures the Pontryagin duality between the two groups for momenta and positions. The inverse limit of Z(p n ) is the profinite group Z p and the direct limit of Z(p n ) is Q p /Z p [27]. The inverse limit of Z(d) is the profinite group Z, and the direct limit of Z(d) is Q/Z. In this paper we only mention the inverse limits very briefly (with reference to the literature) because the emphasis is on the physical aspect...

The complete Heyting algebra of subsystems and contextuality

“…This set contains the quantum system Σ[C(p ∞ ), C(p ∞ )] = Σ(Q p /Z p , Z p ) where the position takes values in Q p /Z p and the momentum takes values in Z p , which has been studied as a subject in its own right in refs [10,11]. It also contains the quantum system Σ[C(Ω), C(Ω)] = Σ(Q/Z, Z), where the position takes values in Q/Z and the momentum takes values in Z, which has been studied as a subject in its own right in refs [12]. Table 1 presents a summary of these systems.…”

“…In the former case the corresponding quantum system is Σ(Q p /Z p , Z p ) and has been studied in [10,11] (Q p denotes p-adic numbers). In the latter case the corresponding quantum system is Σ(Q/Z, Z) and has been studied in [12] (Q denotes rational numbers, Z denotes integers, and Z is defined below). This work can be regarded as a study of 'large finite quantum systems' and it factorizes them (using the Chinese remainder theorem) as tensor products of 'mathematical component systems' with dimension p e (where p is a prime number).…”

Quantum mechanics on profinite groups and partial order

“…This topology is the same as the topology endowed by the p-adic metric in Eq. (34). It is easily seen that…”

Partial order and a T0-topology in a set of finite quantum systems

“…In analogous way we extend the H and R which are not dcpo, into H 1 and R 1 correspondingly, which are dcpo. For example, H 1 will contain the space of the system Σ(τ ) (which is described in detail in [27]).…”

“…We have studied such a system from both a mathematical and a physical point of view in in ref [26,27].…”