Current quantum physical models treat Hilbert spaces, function theory and differential calculus and integral calculus as separate entities. In the past nothing existed that directly relates these ingredients, which together constitute the quantum physical model. Thus, a need exists for a methodology that intimately binds these ingredients into a consistent description of the structure and the phenomena that occur in the model.

Quaternions can store progression and spatial location in a single data element. Quaternionic Hilbert spaces are no more and no less than structured storage media. Function theory relates these data. Differential calculus describes the change of data and integral calculus collects common characteristics of data.

A need exists to be able to treat sets of discrete dynamic data and related fields independent of the equations that describe their behavior. This is possible by exploiting the fact that Hilbert spaces can store discrete quaternions and quaternionic continuums in the eigenspaces of operators that reside in Hilbert spaces. A method that applies Paul Dirac’s bra-ket notation can create natural parameter spaces from quaternionic number systems and can relate the combination of a mostly continuous function and its parameter space to the eigenspace and the eigenvectors of a corresponding operator that resides in a non-separable Hilbert space. This also works for separable Hilbert spaces.

In addition the defining functions relate the separable Hilbert space with a unique non-separable companion. This enables the view that the separable Hilbert space is embedded inside its non-separable companion.

This new method is called the reverse bra-ket method because it uses the Dirac bra's and ket’s in the reverse way. It uses quaternions because these numbers represent the most elaborate numbers that Hilbert spaces can handle. In the separable Hilbert space, the method applies the rational members of a quaternionic number system and an orthonormal set of Hilbert vectors in order to construct what we will call a reference operator by attaching the rational numbers as eigenvalues and use the orthonormal base vectors as eigenvectors. In this way the reference operator defines its eigenspace as a discrete parameter space. Next the method uses the same eigenvectors and continuous functions of the rational parameter values in order to define a new operator that uses the function values, which belong to the parameter values as eigenvalues of the new operator. This operator construction method works for a special class of operators. It does not work for stochastic operators that get their eigenvalues from mechanisms, which use stochastic processes in order to generate those values. However, if the mechanism generates a coherent location swarm that corresponds to a hopping path, then an operator can be constructed that applies the location density distribution, which describes the location swarm as its defining function.

Every infinite dimensional separable Hilbert space owns a non-separable companion Hilbert space. This can be achieved by taking the continuum which is represented by a quaternionic number system as eigenspace of a reference operator that resides in the non-separable Hilbert space. Now a similar trick is performed with the same continuous function, but now the continuous parameter space is applied. This delivers a new operator in the non-separable Hilbert space, which is the companion operator of the operator that was defined in the separable Hilbert space. The procedure intimately binds the infinite dimensional separable Hilbert space to its non-separable companion.

Quaternionic number systems exist in several versions that differ in the ordering that determines their symmetry flavor. Thus, in Hilbert spaces several different versions of parameter spaces can coexist. It is possible that the members of a category of parameter spaces float over another parameter space. This can be used by modelling elementary objects, whose platforms float over a background space.

Let {qᵢ} represent the set of rational quaternions that completely covers a version of a quaternionic number system. Now let {|qᵢ〉} represent a set of orthonormal Hilbert vectors that form a base of an infinite dimensional separable Hilbert space. Each of these base vectors is enumerated with a rational quaternion that is taken from the set {qᵢ}. Now the reference operator ℛ can be defined by the flat quaternionic function ℛ(q)≝q :

ℛ≝|qᵢ〉qᵢ〈qᵢ|=|qᵢ〉ℛ(qᵢ)〈qᵢ|

A new operator ℱ can be defined by:

ℱ≝|qᵢ〉ℱ(qᵢ)〈qᵢ|

Both equations are shorthand definitions that in the full definition uses a sum over all elements of the sets {qᵢ} and {|qᵢ〉}. For all bra’s 〈x| and all ket's |y〉 hold:

〈x|ℱ|y〉≝∑ᵢ [〈x|qᵢ〉ℱ(qᵢ)〈qᵢ|y〉]

In the non-separabel Hilbert space, the sum is replaced by an integral. In the formula we skip the enumerator i, because the sets are no longer countable.

〈x|ℱ|y〉≝∫∭ [〈x|q〉ℱ(q)〈q|y〉] dq

The last formula and its shorthand equivalent are only valid in domains where the defining function ℱ(q) is sufficiently continuous. That is where the differential dℱ(q)/dq exists. Subdomains where this requirement is not met must be excluded. This becomes important where function ℱ(q) stands for a differential of another function or for a convolution of a function with a blurring function. It is also possible that subdomains exist where ℱ(q) is not defined. These subdomains must be circumvented. However, these subdomains may possess a border in which something can be said about the behavior of the eigenspace at this border. Often something can be said about the eigenspace of the companion operator in the separable Hilbert space, while the behavior of the corresponding operator in the non-separable Hilbert space is not covered by current mathematical methodology. These examples show the importance of the reverse bra-ket method in the investigation of models that use quaternionic Hilbert spaces.

An important application of the quaternionic Hilbert spaces is the split of the Hilbert space in multiple domains. The necessity for separating domains where the analyzed defining function shows discontinuities is already mentioned. A very interesting split concerns the separation of the real part of the domain of reference operator ℛ into a past part and a future part. At the rim between these parts exists a static status quo that is characterized by a fixed progression value. Two different views are possible for this split. One view sees the combination of the separable and its companion non-separable Hilbert space as a repository that contains all eigenvalues of all existing operators. In this view a vane moves over the real part of the eigenspace of ℛ and over the corresponding eigenvectors. During this travel the vane encounters what exists in this eigenspace and in the eigenspaces of related operators. The other view takes the position of an observer that travels with the split. The past is already precisely stored in the eigenspaces of the existing operators, but the future is not known. Also information about what exists at a distance of the observer must still travel through the spatial part of future static status quo's in order to reach the observer. The situation becomes complicated when this observer lives inside a continuum eigenspace of one of the operators that reside in the non–separable Hilbert space. This last version of the second view seems to agree with the description that current physical theories tend to produce about how they see physical reality.

These examples show how the reverse bra-ket method can help in investigating dynamic models that apply quaternionic Hilbert spaces. The method relates Hilbert space operators, functions, fields and operators that act on functions. It offers deep insight in multidimensional integration technology where multiple parameter spaces are used in parallel. Hilbert spaces allow the parallel existence of multiple parameter spaces that differ in their symmetry flavor. Mathematics cannot yet properly handle these situations.

## Comments