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 "/>
In the previous chapter, we did not examine the change of a state with time https://www.w3.org/1998/Math/MathML"> t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_1.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . It has been assumed that we have been examining a state at a given moment https://www.w3.org/1998/Math/MathML"> t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_2.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . In the general case, the state of a system changes with time https://www.w3.org/1998/Math/MathML"> t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_3.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . For example, say the motion of an electron in a potential field, its wave function depends on time https://www.w3.org/1998/Math/MathML"> t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_4.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> : https://www.w3.org/1998/Math/MathML"> Ψ = Ψ q , t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_5.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . Such a dependence has to be described by a differential equation in which we may have https://www.w3.org/1998/Math/MathML"> ∂ Ψ ∂ t , ∂ 2 Ψ ∂ t 2 , ⋯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_6.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> .
The quantum revolution in the beginning of the 20th century changed our concepts of dynamics. The dynamic variables such as position 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-math1_1.tif"/> and momentum p $ {\mathbf{p}} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math1_2.tif"/> are replaced by the corresponding position and momentum operators r ^ $ {\hat{\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-math1_3.tif"/> and p ^ $ {\hat{\mathbf{p}}} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math1_4.tif"/> , respectively When one would like to observe a dynamical variable, say ω , of a particle, one has to introduce a wavefunction ψ , which contains all the dynamical information of particles, and to apply the corresponding dynamical operator Ω ^ $ \hat{\Omega } $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math1_5.tif"/> to the wavefunction ψ . Then one may obtain the desired dynamical value of the particle as the eigenvalue of the operator, 1.1 Ω ^ ψ = ω ψ . Ω^ψ=ωψ. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/math1_1.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 .
Components of a field function are functions of coordinates and time and satisfy some definite differential equations, which are called field equations. Field equations can be given a very general variational or Lagrangian form, if one accepts, that a field as the dynamic system must be characterized by a definite Lagrangian function. For a mechanical system this function is written down as the sum over all the material points of the system. For a continuous system like a wave field, Lagrangian function is expressed by the space integral from Lagrangian function density: L(t) = ∫ Ld3x. (4.1) However, the variational principle deals not with the Lagrangian function but with an action S obtained from it by means of integration over x0 = ct S = c ∫ L(t)dt. (4.2) Therefore, noncovariant expression (4.1) in Lagrangian formalism is in fact intermediate and it is sufficient to consider the Lagrangian function density L, which depends on the four space-time coordinates x in an implicit way. Hereinafter, we shall simply call L by the Lagrangian. Analogously, the Hamiltonian function density H will be named the Hamiltonian. The Lagrangian depends on components of a field wavefunction ψk and on their first derivatives with respect to the space-time coordinates ∂ψk/∂x ν . In so doing we assign the meaning of generalized field coordinates to ψk, and the meaning of generalized field velocities to ∂ψk/∂x ν . The demand of lacking derivatives with respect to field functions above the first order in the Lagrangian is caused by the following reason. Dynamics of all known quantum fields is described either by equations of the first or the second order. On the one hand, these equations follow from the principle of a least action: δS = 0, (4.3) and under action variation the order of derivatives, entering L, is raised by one. The Lagrangian must be a relativistically covariant quantity, namely, a relativistic scalar L′(x′) = L(x). Thanks to the invariance under the space-time translations, we conclude that the Lagrangian must not manifestly contain any space-time coordinates x. Since dynamical observables (four-dimensional vector of the energy-momentum, the moment of momentum tensor, and so on) are real quantities, then the Lagrangian must be also a real quantity. We shall also impose the locality condition on the Lagrangian, that is, we assume that the Lagrangian depends only on the state of fields in the infinitely small vicinity of the point Particles, Fields, and Quantum x. So: L(x) = L(ψk, ∂ψk ∂xν ) = L (ψk(x), ψk;ν (x)) . General variation of an action, related with varying both field wave functions and of integration domain boundaries, is equal to: δS = δ ∫ L(x)dx = ∫ L(x′)dx′ − ∫ L(x)dx, (4.4) where dx = dx0dr. For L′(x′) we have the expression: L′(x′) = L (ψ′k(x′), ψ′k;ν(x′)) = L(x) + δL(x) = L(x) + δL(x) + dLdxλ δxλ, (4.5) In Eq. (4.5) we denote a form variation of corresponding quantities by the symbol δ. This way δL(x) represents varying the Lagrangian only on the account of the variation of field functions and field function derivatives: δL(x) = ∂L ∂ψk δψk + ∂L ∂ψk;ν δψk;ν , (4.6) and δψk(x) does a variation of a field function form: δψk(x) = ψ ′ k(x) − ψk(x).
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 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.