# Unitary theory

## Definitions

• Webster's Revised Unabridged Dictionary
• Unitary theory (Chem) the modern theory that the molecules of all complete compounds are units, whose parts are bound together in definite structure, with mutual and reciprocal influence on each other, and are not mere aggregations of more or less complex groups; -- distinguished from the dualistic theory.
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## Usage

### In science:

It is well known in Random Matrix Theory as the distribution of the eigenvalues in the Gaussian Unitary Ensemble (G.U.E.).
Determinantal random point fields
Therefore, theories with spacetime noncommutativity do not have a unitary S-matrix.
Space-Time Noncommutative Field Theories And Unitarity
Gaussian unitary ensemble in the random matrix theory (see e.g.
Riemann-Hilbert problems for last passage percolation
These are connected with Hermitian Brownian motion and the Gaussian Unitary Ensemble of random matrix theory.
A path-transformation for random walks and the Robinson-Schensted correspondence
Voiculescu, Circular and semicircular systems and free product factors, “Operator Algebras, Unitary representations, Algebras, and Invariant Theory”, Progress in Math.
Random matrices, free probability and the invariant subspace problem relative to a von Neumann Algebra
In random matrix theory, unitary ensembles of N × N matrices {H } play a central role .
Products and Ratios of Characteristic Polynomials of Random Hermitian Matrices
Shevitz, Unitary-Matrix Models as Exactly Solvable String Theories, Phys.
Toeplitz determinants, random growth and determinantal processes
For example, there do not exist vector particles with helicity = 1, which is a consequence of the theory of unitary representations of the Poincar´e group, as was shown by J. Lopusza´nski .
Generally covariant Quantum Mechanics
Mackey, Unitary Group Representations in Physics, Probability, and Number Theory.
Generally covariant Quantum Mechanics
Voiculescu, Circular and Semicircular Systems and Free Product Factors, ”Operator Algebras, Unitary Representations, Algebras, and Invariant Theory”, Progress in Math.
Non-commutative Polynomials of Independent Gaussian Random Matrices. The Real and Symplectic Cases
Let us recall that the relations (4.13) give a canonical transformation classically, and a unitary transformation in the quantum theory.
Perturbative dynamics of matrix string for the membrane
In fact, using Hodge theory for unitary local systems (as in [Tim87]), we may replace R by C.
The deformation theory of representations of the fundamental group of a smooth variety
Mixed Hodge theory for unitary local systems. J.
The deformation theory of representations of the fundamental group of a smooth variety
Two important consequences of our general theory are: (i) we obtain a natural generalization of a theorem of Diaconis and Shahshahani to the case of several independent unitary matrices; (ii) we can show that global ﬂuctuations in unitarily invariant multi-matrix models are not universal.
Second Order Freeness and Fluctuations of Random Matrices: II. Unitary Random Matrices
Mackey, G.W.: Unitary Group representations in Physics, Probability, and Number Theory.
Mackey Theory for \$p\$-adic Lie groups
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