Quant-Math Group is an independent research group under the Theoretical Physics Group of the National Institute of Physics, University of the Philippines Diliman. The group focuses on research in quantum foundations and mathematical physics, exploring the conceptual and formal structures underlying quantum theory and related areas of theoretical physics.
Research Contributions
Time in quantum mechanics
The group is recognized for its fundamental contributions to the quantum time problem, one of the central conceptual issues in quantum mechanics. This problem concerns whether time is a legitimate quantum observable represented by an operator and, if so, under what precise mathematical conditions such a representation is possible. Through a rigorous clarification of Pauli’s no-go theorem, the group demonstrated that the commonly cited impossibility of a self-adjoint time operator is not absolute but instead depends critically on spectral properties and domain considerations.
By carefully identifying appropriate canonical domains, the group constructed self-adjoint and essentially self-adjoint time operators in both finite- and infinite-dimensional Hilbert spaces. These developments led to a consistent operator-theoretic formulation of quantum time of arrival, where eigenfunctions correspond to states with definite arrival times. The resulting framework yields direct and unambiguous predictions for dynamical phenomena such as quantum tunneling, offering explanation to the conflicting experimental results.
Finite-part integration
Recently, the group developed finite-part integration, a systematic analytic method for extracting meaningful results from well-defined problems whose intermediate formulations involve divergent integrals. This framework extends and reinterprets the classical finite-part integral introduced by Jacques Hadamard. Whereas the classical approach treats the finite part as the endpoint of a regularization procedure, finite-part integration elevates finite parts to structural components within the analytic process as a means of obtaining the solution.
Within this framework, divergent integrals are decomposed, reorganized, and recombined in a manner that preserves exact analytic information which is not transparent in conventional regularization schemes. The method has been applied to derive new identities among special functions of mathematical physics, rigorously obtain asymptotic expansions involving divergent integrals, offer a new Stieltjes resummation scheme in quantum mechanics and quantum field theory, and clarify hidden analytic cancellations inside formally divergent expressions.
Current Research Directions
Fundamental Limits of Quantum Dynamics
Building on its foundational work on time–energy relations, the group investigates the fundamental limits on the speed of quantum processes. By translating time–energy constraints into the geometry of state transitions in Hilbert space, the group analyzes optimal dynamical trajectories that minimize temporal cost under spectral constraints. This program is expected to yield generalized formulations of quantum speed limits that extend known bounds and provides principled criteria for engineering quantum evolutions that implement target transformations in minimal time.
Quantum Decoherence Mitigation
The group develops active strategies for protecting quantum coherence and entanglement against environmental decoherence. By embedding a target subsystem within a larger interacting “cage” of auxiliary system, we modify the collective wavefunction in a way that reshapes system–environment interaction. Instead of allowing correlations to dissipate irreversibly, part of the quantum information is redistributed within the enlarged system. In appropriate limits, coherence and entanglement is preserved. This systematically allows engineering open quantum systemss to extend their coherence times.
Extensions of Finite-part Integration in Higher Dimensions
(Under construction.)
Applications of Analytic Divergence
(Under construction.)
Principal Investigator
- Eric A. Galapon (egalapon@nip.upd.edu.ph)
PhD Students
- Philip Jordan D. Blancas
- Nathalie Liezel R. Rojas (nrrojas@up.edu.ph)
- John Jaykel P. Magadan
- Ralf E. Faralles
- Paul Allen Chavit
- Rexcel de Peralta
MS Students
- Alyann Mhar Rosero
- Carl Anthony Arguelles
- Amiel Chua
- Oscar Arellano Jr.
BS Students
- Karl Henry Daho
- Jerald D. Magcalas
- Sophia Nhor Quigao
- Narl Jhon Rico
- Toffee Galolo
- Samuel Zurriell Lamayo