T. R. Robinson
The phase, ξ , that we introduced as part of natural number dynamics in Chapter 3 , plays an absolutely key role in developing the dynamical theory that is the central theme of this book. We saw in Chapter 4 , where we introduced multi-category systems, how the phase, ξ i , on which the raising and lowering operators of each of the categories of a system functionally depend, needs to be parameterized in order to be able to both distinguish between the categories, but also to allow a universal scalar parameter to apply to the system as a whole and thus enable us to construct a single equation of motion for the whole system. Initially we chose the write https://www.w3.org/1998/Math/MathML" display="inline"> ξ i = ω i t https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377504/0865e0f9-b6f9-4666-9620-3b3ea10ce510/content/math10_1.tif "/> , where ω i serves as a label for the https://www.w3.org/1998/Math/MathML" display="inline"> i t h https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377504/0865e0f9-b6f9-4666-9620-3b3ea10ce510/content/math10_2.tif "/> category and t is a universal parameter that applies to the system as a whole. We have seen that we can associate t with time and then ω i behaves as an angular frequency that also serves as a scaled energy parameter. However, there is nothing to stop us parameterizing the phase with more than one universal parameter, and adding other labels for the categories. So we could, for example, generalize the phase, ξ i to 10.1 https://www.w3.org/1998/Math/MathML" display="block"> ξ i = ω i t + κ i s , https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377504/0865e0f9-b6f9-4666-9620-3b3ea10ce510/content/math10_3.tif "/>
CRC Press eBooks•Mar 21, 2023•0 citations