''p''-adic quantum mechanics is a collection of related research efforts in
quantum physics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
that replace
real numbers
In mathematics, a real number is a number that can be used to measurement, measure a continuous variable, continuous one-dimensional quantity such as a time, duration or temperature. Here, ''continuous'' means that pairs of values can have arbi ...
with
''p''-adic numbers. Historically, this research was inspired by the discovery that the
Veneziano amplitude of the open
bosonic string, which is calculated using an
integral
In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
over the real numbers, can be generalized to the ''p''-adic numbers. This observation initiated the study of ''p''-adic string theory. Another approach considers particles in a ''p''-adic potential well, with the goal of finding solutions with smoothly varying complex-valued
wave function
In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
s. Alternatively, one can consider particles in ''p''-adic potential wells and seek ''p''-adic valued wave functions, in which case the problem of the probabilistic interpretation of the ''p''-adic valued wave function arises. As there does not exist a suitable ''p''-adic
Schrödinger equation
The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
,
path integrals are employed instead. Some one-dimensional systems have been studied by means of the path integral formulation, including the
free particle, the particle in a constant field, and the
harmonic oscillator
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force ''F'' proportional to the displacement ''x'':
\vec F = -k \vec x,
where ''k'' is a positive const ...
.
See also
*
''p''-adic analysis
References
External links
* {{nlab, id=p-adic+physics, title=''p''-adic physics
P-adic numbers
Quantum mechanics
String theory