In this chapter we look at what happens when we apply the important symmetry condition of degeneracy to a two-category system. This involves imposing the condition that the frequencies in the Hamiltonian of the two categories are identical, i.e., we let the frequencies in the Hamiltonians in Eqs. (6.2) and (6.23) be equal, so https://www.w3.org/1998/Math/MathML" display="inline"> ω a = ω b = ω 0 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/math7_1.tif "/> . We are first going to examine bosonic systems, but in the last section of the chapter we will look at an important fermionic case.
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 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.
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 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 "/>
In the previous chapters, the focus was on systems of particles following the law of classical mechanics. Of course, this was mainly done within the framework of classical mechanics but a quantum description of individual (discretised) states was also used, while assuming distinguishability of the particles (for instance, particles located at the nodes of a lattice (spin system, quantum harmonic oscillators,) ). The present chapter as well as Chapters 7 and 8 are dedicated to the study of statistical physics in a fully quantum framework. The first obstacle encountered by a quantum approach is the impossibility to define a quantum phase space (in which the ergodic hypothesis would hold for instance). This problem is inherent to quantum mechanics: because of Heisenberg’s uncertainty principle, one cannot define both the momentum and the position of a particle. Besides, statistical physics considers systems made of a large number of particles, a priori in mutual interaction. In classical physics, model situations for which the interactions between the constituents of the system are neglected can be studied. In quantum mechanics, the indistinguishability of identical particles makes the study of the problem in the absence of interaction already difficult: even for non-interacting particles, the symmetrisation (or antisymmetrisation) of the wave function of 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/9781003272427/5ca4d37a-9786-43a3-8b38-8fa9eb3d1a19/content/math_25.tif "/> identical particles induces correlations which have non-trivial thermodynamical effects such as the existence of a “Fermi sea” for fermions (see Section 8.1 ) or the Bose–Einstein condensation phenomenon for bosons (see Section 7.4 ). The present chapter addresses these two points consecutively.
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 "/>