Transformation Having Applications in Quantum Mechanics

Abstract

By properly ordering functions of noncommuting operators, a one-to-one transformation between operator functions and corresponding functions of commuting algebraic variables can be made. With this transformation, boson operator equations such as the Schrödinger equation can be converted to differential equations for the transformed functions, the resulting equations containing solely commuting variables. Once the solution to the transformed equation is obtained, the inverse transformation may be applied to yield the solution to the original operator equation. The method is extended to include angular momentum operators.

Detectors for probing relativistic quantum physics beyond perturbation theory

“…The exact form of these transformations can be obtained nonperturbatively by employing a technique introduced by Heffner and Louisell [28]. For the cases of interests (for the family of Hamiltonians (20)), and as it is thoroughly detailed in appendix A, due to the algebraic nature of the interaction, the evolution can be reduced to…”

“…The covariance matrix σ of the detector+field state is defined as in the previous section. Once we have solved for the squeezing and rotation operators, in the sense that we have solved for z(t) and φ(t), it is then straightforward to compute the evolved covariance matrix [28]. If we split the covariance matrix into the block form…”

“…The exact form of these transformations can be obtained nonperturbatively by employing a technique introduced by Heffner and Louisell [28]. We will utilize the polar decomposition of z into a product of a hermitian and a unitary matrix, which can always be achieved.…”

Vacuum-Matter Interaction through Hyper-Dimensional Time-Space Shifting

“…The correlation puts the invariant time components on the reference planes, and as in a synoptic system, we can only rotate space-time plane and the observation point on the timeline placed at the base of the scheme. Spacetime conjugation is defined as by Lorentz 12 [12] transforms and does not introduce variables and differentiable functions such that they can describe or represent a hyperplane of time reference, which is "uncurvable" or shiftable compared to space-time plane.…”

“…(The topology of Minkowski space; E C. Zeeman). 12 The Lorentz transformation (or transformations) are coordinate transformations between two coordinate frames that move at constant velocity relative to each other. The Lorentz transformation is a linear transformation.…”