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.
