Equivalence and Antiequivalence of Irreducible Sets of Operators. I. Finite Dimensional Spaces

Journal Article · Tue Jan 01 04:00:00 EST 1963 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.1703970· OSTI ID:4718045

Lomont, J. S.; Moses, H. E.; Yu, Kwang

Since an elementary dynamical system is specified by giving an irreducible set of dynamical variables, one is led to investigate the following problem: Given two irreducible sets of operators with a one-to-one correspondence between them, find the algebraic properties of the two sets that make it possible to infer the existence of a unitary or antiunitary operator relating them. A series of theorems is obtained from such considerations for finite dimensional spaces. It is shown that if the second set of operators contains some of the algebraic properties of the first set, the two sets are related by a similarity transformation. By altering the requirements, this transformation is a unitary transformation. Indications are also given to show how the theorems can be extended to Hilbert spaces. Finally, in the Appendix there is given a definition of invariance of elementary quantum-mechanical systems based on the above theorems, giving the same results as Wigner's definition in terms of transition probabilities.

• Research Organization: Polytechnic Inst. of Brooklyn
• Sponsoring Organization: USDOE
• NSA Number: NSA-17-022321
• OSTI ID: 4718045
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 3 Vol. 4; ISSN JMAPAQ; ISSN 0022-2488
• Publisher: American Institute of Physics (AIP)
• Country of Publication: Country unknown/Code not available
• Language: English

Related Subjects

FIELD THEORY
GROUP THEORY
HILBERT SPACE
MATHEMATICS
OPERATORS
PHYSICS
QUANTUM MECHANICS
TRANSIENTS
WIGNER THEORY