''p''-adic quantum mechanics is a collection of related research efforts in
quantum physics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
that replace
real numbers
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every re ...
with
''p''-adic numbers. Historically, this research was inspired by the discovery that the
Veneziano amplitude
In theoretical physics, the Veneziano amplitude refers to the discovery made in 1968 by Italian theoretical physicist Gabriele Veneziano that the Euler beta function, when interpreted as a scattering amplitude, has many of the features needed to ...
of the open
bosonic string, which is calculated using an
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
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
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
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 linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
,
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.
See also
*
P-adic analysis § P-adic quantum mechanics
References
External links
* {{nlab, id=p-adic+physics, title=''p''-adic physics
P-adic numbers
Quantum mechanics
String theory