The results of Chapter 3 may be generalized to a multi-category system by noting that the sum of a set of natural numbers is itself a natural number, so the sum of natural number operators is just a natural number operator. So, let 4.1 https://www.w3.org/1998/Math/MathML" display="block"> N ^ = ∑ i = 1 S N ^ i , 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/math4_1.tif "/>
In this section we begin with the multiple-category system of S independent categories that was introduced in Chapter 4 , but then we will apply a unitary transformation to the annihilation operators in order to model interactions between the categories. This constitutes a generalization of the unitary transformation that was introduced in Chapter 6 to deal with the two-category case. That required a https://www.w3.org/1998/Math/MathML" display="inline"> 2 × 2 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/math9_1.tif "/> matrix. Here we will require an https://www.w3.org/1998/Math/MathML" display="inline"> S × 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/math9_2.tif "/> matrix transformation.
In the previous chapter we identified a component of angular momentum that was orthogonal to the two-dimensional configuration space that had emerged from a degenerate two-category bosonic system. This implied that we needed a third dimension of configuration space to complete our configuration space picture. This suggests we explore a bosonic three-category system. So, we now let https://www.w3.org/1998/Math/MathML" display="inline"> N ^ 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/math8_1.tif "/> be 8.1 https://www.w3.org/1998/Math/MathML" display="block"> N ^ = N ^ 1 + N ^ 2 + N ^ 3 , 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/math8_2.tif "/>
In this chapter we are going to explicitly identify the parameter, t , from Chapter 4 as a continuous variable that corresponds to time in a physical system. This will enable us to show how natural number dynamics can be converted into Hamiltonian form and thus develop dynamical equations that can be applied to physical systems, that form the basis for standard quantum mechanics. It is instructive to begin with a single category system with n items. We will take the system to be bosonic so that n can be any natural number. We then simply parameterize the variable ξ with t , as we did for the multi-category case, by writing https://www.w3.org/1998/Math/MathML" display="inline"> ξ = ω 0 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/math5_1.tif "/> , where ω 0 , is a constant angular frequency. Eq. (3.14) can then be transformed into a time evolution S-type equation applied to an eigenfunction, https://www.w3.org/1998/Math/MathML" display="inline"> Φ n 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/math5_2.tif "/> of the operator https://www.w3.org/1998/Math/MathML" display="inline"> N ^ 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/math5_3.tif "/> . Comparing the result with Eq. (4.9) , one then gets 5.1 https://www.w3.org/1998/Math/MathML" display="block"> i d Φ n d t = Ω ^ Φ n = ω 0 N ^ Φ n , 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/math5_4.tif "/>
The properties of a quantum mechanical system composed of many identical particles are most conveniently described in terms of the second‐quantized, Heisenberg representation, particle‐creation, and annihilation operators. The creation operator ψ ^ † ( r , t ) $ \hat{\psi }^{\dag } ({\mathbf{r}}, t) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math2_1.tif"/> , when acting to the right on a state of the system, adds a particle to the state at the space‐time point r , $ {\mathbf{r}}, $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math2_2.tif"/> t ; the annihilation operator ψ ^ ( r , t ) $ \hat{\psi }({\mathbf{r}}, t) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math2_3.tif"/> , the adjoint of the creation operator acting to the right, removes a particle from the state at the point r , t .
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 "/>