Abstract This paper reviews Ladenburg’s development of the phenomenological theory of radiative transitions between the stationary states of an atom put forward by Einstein in 1917. The historical background as well as the far reaching outcomes of his work are considered and discussed; among them the Kramers–Heisenberg quantum dispersion theory that paved the way to Heisenberg’s formulation of matrix mechanics and the quantum-mechanical calculation of the spectral line profiles.

European Journal of PhysicsJun 30, 20160 citations

Continuous matrix product operators for quantum fields

Erickson Tjoa, J. Ignacio Cirac

quant-pharXiv:2511.04545Physical Review ResearchNov 7, 20250 citations

Elsevier eBooks

Construction of quantum theory II: matrix mechanics

Mario S. Reis

Published Sep 27, 2025• 0 citations

Publisher

DOI: 10.1016/b978-0-44-332826-8.00012-9

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In this work we introduce an ansatz for continuous matrix product operators for quantum field theory. We show that (i) they admit a closed-form expression in terms of finite number of matrix-valued functions without reference to any lattice parameter; (ii) they are obtained as a suitable continuum limit of matrix product operators; (iii) they preserve the entanglement area law directly in the continuum, and in particular they map continuous matrix product states (cMPS) to another cMPS. As an application, we use this ansatz to construct several families of continuous matrix product unitaries beyond quantum cellular automata.In this work we introduce an ansatz for continuous matrix product operators for quantum field theory. We show that (i) they admit a closed-form expression in terms of finite number of matrix-valued functions without reference to any lattice parameter; (ii) they are obtained as a suitable continuum limit of matrix product operators; (iii) they preserve the entanglement area law directly in the continuum, and in particular they map continuous matrix product states (cMPS) to another cMPS. As an application, we use this ansatz to construct several families of continuous matrix product unitaries beyond quantum cellular automata.

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Functional Integral Approach to

C^*

-algebraic Quantum Mechanics II: Symplectic Quantum Mechanics

We propose

Sp\,(8,\R)

and its Langlands dual

SO(9,\R)

as dynamical groups for closed quantum systems. Restricting here to the non-compact group

Sp\,(8,\R)

, the quantum theory is constructed and investigated. The functional Mellin transform plays a prominent role in defining the quantum theory. It provides a bridge between the quantum algebra of observables and the algebra of operators on Hilbert spaces furnishing unitary representations that are induced from a distinguished parabolic subgroup of $Sp\,(8,\C)$. As well, the parabolic subgroup renders a fiber bundle construction that models what can be described as a matrix quantum gauge theory. The formulation is strictly quantum mechanics: no \emph{a priori} space-time is assumed and the only geometrical input comes indirectly from the group manifold. But what appears on the surface to be a fairly simple-minded model turns out to have a capacious structure suggesting some compelling physical interpretations regarding space-time and fundamental interactions.