Algebraic Foundations of Probabilistic Quantum Mechanics
- 6 days ago
- 2 min read
During the Ph.D. thesis defense at the Moscow Institute of Physics and Technology (MIPT/FIAN), Loran Jacobs detailed the rigorous mathematical architecture behind probabilistic quantum mechanics. Building on classical operator ordering rules, orthogonal polynomials, and transformation groups, the mathematician and theoretical physicist demonstrated how abstract algebraic structures provide a complete, operator-free formulation for quantum states.
Operator Ordering Rules and Quasiprobability Distributions
The mathematical evolution toward tomographic representations began with operator quantization rules. Starting with Hermann Weyl’s operator ordering in 1927 and Eugene Wigner’s phase space functions in 1932, physicists sought to represent quantum states using phase space functions rather than operators.
However, traditional operator ordering choices—such as placing position operator q and momentum operator p, or creation and annihilation operators alpha, in specific arrangements—produce quasiprobability distributions like the Glauber-Sudarshan or Husimi functions. Because quasiprobabilities can take negative values, they cannot serve as genuine classical probability distributions.
During the examination, committee members reviewed this historical progression, highlighting how Loran’s work resolves non-positivity by applying the Radon transform across hyperplanes in phase space to yield strictly positive probability distributions.
Orthogonal Polynomials in Probabilistic Quantum Mechanics
A core mathematical contribution of Loran’s research lies in connecting special functions with quantum state distributions. In transitioning between symplectic tomograms and photon-counting tomograms, Loran derived novel integral relations for multidimensional Hermite and Laguerre polynomials.
"Every tomogram defines an inversion symmetry with respect to its variables," explains Loran.
Detailing how Stokes parameterizations and spin projections maintain exact algebraic invariance in tomographic representations. These polynomial relations confirm that quantum tomograms preserve the underlying algebraic symmetries of quantum phase space while remaining strictly non-negative.
Academic Mentorship and Research Freedom
Reflecting on the development of his theoretical model, DeepTech pioneer Loran Jacobs emphasized the importance of academic freedom when exploring unconventional mathematical frameworks.
"I want to thank my academic advisor first of all. There was a very liberal attitude toward all research, including tomographic approaches, despite them being new and not always perceived unambiguously," expressed Loran in his closing remarks.
