Quantum mechanics andp-adic numbers

Quadratic equations over p-adic fields and their applications in statistical mechanics

“…The set ∆ ⊂ p × p is called a solvability domain of the quadratic equation (6). Since p is a disordered field, we could not describe the solvability domain ∆ in the picture.…”

“…5, a p-adic counterpart of statistical mechanics is also studied in the context of the p-adic theory of probability and stochastic processes. More recently, numerous applications of p-adic numbers have shown up in theoretical physics and quantum mechanics [6][7][8][9] .…”

“…In this case, the two solutions of the quadratic equation (6) are formally given by x ± = 1 2 (−a± D). Based on the last formula, it is quite difficult to verify whether the solution of the quadratic equation (6) belongs to the classical sets *…”

“…Based on the last formula, it is quite difficult to verify whether the solution of the quadratic equation (6) belongs to the classical sets *…”

Solvability of Cubic Equations over Q3

“…p + x 2 . p 2 + …) where x 0 ∈ {1,2,… p -1} and x i ∈ {0,1,2,… p -1}, i ≥ 1, (Borevich & Shafarevich 1966;Koblitz 1984) More recently, numerous applications of p-adic numbers have shown up in theoretical physics and quantum mechanics (Beltrametti & Cassinelli 1972;Khrennikov 1994Khrennikov , 1991Volovich 1987). Unlike the field  of real numbers, in general, the cubic equation ax 3 + bx 2 + cx + d = 0 is not necessary to have a solution in  p , where a,b,c,d ∈  p with a ≠ 0.…”