Dirac introduced the bra-ket notation that strongly popularized the usage of Hilbert spaces. The bra’s and ket’s ease the handling of Hilbert vectors and the usage of the inner products of these vectors.
In quaternionic Hilbert spaces the values of inner products, the values of coefficients in linear combinations of Hilbert vectors and the eigenvalues of operators are quaternions. The factors of the product of two quaternions do in general not commute. Thus, the bra’s and ket’s must be handled carefully.
Like the quaternions themselves, the product of two quaternions can be split in a real scalar and an imaginary vector.
a = a₀+A
a ̽ = a₀−A
c = (c₀+C) = a b = (a₀+A) (b₀+B) = a₀ b₀ − 〈a,b〉 + a₀ B+b₀ A ± A×B
c₀ = a₀ b₀−〈a,b〉
C = a₀ B + b₀ A ± A×B
The ± sign reflects the choice between a right handed and a left handed external vector product.
〈x| is a bra vector. |y〉 is a ket vector. α is a quaternion.
〈x|y〉=〈y|x〉 ̽
〈x+y|z〉=〈x|z〉+〈y|z〉
〈αx|y〉=α 〈x|y〉
〈x|αy〉=〈x|y〉 α ̽
The reverse bra-ket notation can be used to relate a category of quaternionic operators to quaternionic functions.
Let {aᵢ} be a set of quaternions and let {|aᵢ〉} be a set of ket vectors that are normalized and mutually orthogonal.
〈aᵢ|aᵢ〉 = 1 and 〈aᵤ|aᵥ〉 = δᵤᵥ
Now in the separable Hilbert space ℋ the operator ℴ is defined by the reverse bra-ket method
ℴ ≡ |aᵢ〉aᵢ〈aᵢ|
The set {aᵢ} forms the eigenspace of operator ℴ. The set {|aᵢ〉} are the eigenvectors of normal operator ℴ. They span a subspace of the separable Hilbert space ℋ.
If the set {qᵢ} comprises all rational quaternions (quaternions with rational numbers as components), then operator ℛ is defined by
ℛ ≡ |qᵢ〉qᵢ〈qᵢ|
and represents a reference operator, whose eigenspace represents a parameter space.
Quaternionic number systems exist in several versions that differ in the way that they are ordered.
The set of orthonormal ket vectors {|qᵢ〉} represents a base of a complete infinite dimensional separable Hilbert space ℋ. We will use the symbol ℛ also for the eigenspace of operator ℛ.
The operator F defined by
F ≡ |qᵢ〉F(qᵢ)〈qᵢ|
relates quaternionic function F(q) with operator F and with parameter space ℛ.
For all bras 〈x| and all kets |y〉 holds:
〈x|F|y〉 ≡ ∑ᵢ 〈x|qᵢ〉F(qᵢ)〈qᵢ|y〉
We can perform the same trick in the Gelfand triple, which is a non-separable Hilbert space. Every infinite dimensional separable Hilbert space owns a companion Gelfand triple. These two Hilbert spaces can be related via the defining functions of their operators. The eigenspaces of operators that reside in the separable Hilbert space are per definition countable. The Gelfand triple supports operators that have continuums as eigenspaces.
The reference operators in the Gelfand triple are defined by:
ℛ ≡ |q〉q〈q|
Here we left the subscript ᵢ. The eigenvalues {q} form a continuum. Similarly, for the continuous function F(q):
F ≡ |q〉F(q)〈q|
〈x|F|y〉 ≡ ∫〈x|q〉F(q)〈q|y〉 dq
The summation is replaced by an integration over all {q}, thus over all quaternions that have real components.
We use the same symbol F for the operator F, the function F(q) and the continuum eigenspace. It is important to note that the operator F and the continuum F exist independent from the function F(q)! Selecting another parameter space {q}, results in a different function F(q).
The reverse bra-ket method relates the parameter spaces
ℛ ≡ |qᵢ〉qᵢ〈qᵢ| and ℛ ≡ |q〉q〈q|.
Further, it relates the operators F ≡ |qᵢ〉F(qᵢ)〈qᵢ| and F ≡ |q〉F(q)〈q|.
The method shows that the same continuum F can be related to several pairs of functions and parameter spaces.
Warning:
The reverse bra-ket method goes a step further than the outer bra-ket product between a ket |p〉 and a bra 〈q|.
|p〉〈q|
The definition:
F ≡ |q〉F(q)〈q|
implicitly involves a complete base of eigenvectors {|q〉} and eigenvalues {F(q)}
In fact it implies a sum:
〈x|F|y〉 ≡ ∑ᵢ 〈x|qᵢ〉F(qᵢ)〈qᵢ|y〉
or an integral
〈x|F|y〉 ≡ ∫〈x|q〉F(q)〈q|y〉 dq
Charles
The tensor product of two quaternionic Hilbert spaces is a real number based Hilbert space. Thus a theory based on quaternionic Hilbert spaces cannot use Fock spaces.
See the link.
http://arxiv.org/abs/1101.5690
@Hans This is really a question within your question: Can quaternionic Hilbert spaces handle description of non-Hermitian quantum dynamics more efficiently? As the product of two quaternions can be split into real scalar and an imaginary vector components, it seem to suggest that one can easily tract evolutions of open systems where non-Hermitian factors come into play. What is your view on this?
If the question : Why is the powerful reverse bra-ket method not used more intensively? as far it concerne the use of the bra-ket, the answer is peole still use this notation but in the same time you've different approach to describe quantum phenomon think about the path integral where still you have the braket but it's dominate but the integration approach, which give you the feeling that the brakets were not used. It's not the case. there is the algebraic approach and notations and functional approach. Recently people start use differential forms (which are very efficient for the quaternions approach if you're intersted by the quaternion approach). Geometric approach you use the fiber bundle structure, and Hilbert space is a functional space defined on the T^*M on the cotangent fiber bundle and the states are elements of this functional space. Here we use the geometric quantization approach.
So it depend in the way you want to resolve your problem or to present the phenomenon.
Best,
Joseph
One shouldn't confuse notions with notation. bra-ket notation is, just, that-a notation. It describes notions, such as scalar products, that's all.
to Stam Nicolis
Not only, scalar product need a structure of metric, this is the big problem in Hilbert space, you should define the metric before, they are a vectors (and covector) elements of a linear vectorial space we called Hilbert space (and it's dual).
I think it's more naturally to think the braket in terms of differential forms, something physicists didn't use often before before 20 years and so unfortunatly . In this framework of the forms, duality give you the scalar product and you don't need additional structure as the metric.
remark: In the (complexe) projectif case Hilb/U(1) (hopf bundle) the product of the brakets give you the density of probability and in the same time it's a distance in this projectif space and you have two structures, the metric and the symplectic form
Charles,
It is quite sensible to represent the whole universe with the combination of an infinite dimensional quaternionic separable Hilbert space and its companion Gelfand triple. Fock spaces are used to represent binding between particles. That can be performed by other means. Each elementary particle can be represented by a subspace of the separable Hilbert space. Continuums (the physical fields) can be represented by eigenspaces of operators in the Gelfand triple. This idea is exploited in the link below.
http://vixra.org/abs/1511.0074
Alagu,
What most scientists do not know is that quaternionic number systems exist in many versions that differ in the ordering of their elements. As a result quaternions with a right handed and quaternions with a left handed external vector product exist. As is explained in the link below, this is the reason of existence of electric charges.
Further, it is hardly known by scientists that Hilbert spaces can only cope with number systems, which are division rings. This limits these number systems to real numbers, complex numbers and/or quaternions.
Quaternions can be seen as a combination of a real scalar and a three dimensional vector. These parts can be handled separately as is done in Maxwell equations, but Maxwell equations uses the equivalent of quaternionic distance as the real scalar in their parameter space. That is why the corresponding spacetime model is characterized by a Minkowski signature. Quaternionic parameter spaces are characterized by a Euclidean signature.
These facts introduce much confusion.
Personally I appreciate the compactness and clarity of the quaternionic differential equations. This is due to the fact that the quaternionic differential operator acts as a multiplier. Maxwell equations must split their differentials in a scalar oriented part and a vector oriented part, because their parameter space does not represent a number system.
In a separable quaternionic Hilbert space, normal operators have sets of quaternions as their eigenspaces and the quaternionic Gelfand triple, which is a non-separable Hilbert space, offers operators that have quaternionic continuums as eigenspaces. These quaternionic continuums can be represented by pairs of quaternionic functions and quaternionic parameter spaces. The quaternionic parameter spaces are formed by quaternionic number systems. As mentioned earlier, these number systems may be ordered in different ways. Thus the continuums (the fields) exist independent of the representing functions and parameter spaces!
The parameter spaces that are used by Maxwell equations must first be split in real components before they can be handled by Hilbert spaces. This also holds for quaternionic Hilbert spaces!
The most important message is that the quaternionic differential equations and the Maxwell based differential equations can treat the same continuum (the same field)
Another important message is that fields are not distinguished by the set of differential equations that treats them. Instead the fields differ due their start and boundary conditions. These conditions are set by collections of artifacts that trigger vibrations or local discontinuities in these fields. These artifacts are generated by mechanisms, which reside outside of the Hilbert spaces! Still their presence can be archived in the separable Hilbert space.
http://vixra.org/abs/1511.0007
Joseph,
The Hilbert spaces are no more and no less than structured storage places. They can administer what happens in universe, but they do not trigger what happens. The real actors are mechanisms that reside external to the Hilbert spaces.
The reverse bra-ket method shows how a category of operators can be related to pairs of functions and parameter spaces. Also these functions only act as descriptors and not as activators. These functions relate continuums with the differential equations that describe the behavior of the fields. Again, also the differential equations are descriptors and not activators. However, they support the presentation of the fields as manifolds and the toolbox of tensor calculus, fiber bundles and more of that stuff. None of these objects or tools act as activator. Only the mentioned mechanisms that generate the artifacts, which affect the fields are the actual activators. Obviously these mechanisms apply stochastic processes in order to generate these artifacts. The presence and history of the artifacts are stored in the separable Hilbert space. The corresponding data form eigenspaces of a category of stochastic operators. That category differs from the category of operators that get their eigenspaces via pairs of functions and parameter spaces. The reverse bra-ket method defines this relation.
I have not yet seen a physical theory that describes the mechanisms, which supply the stochastic operators with their data.
This classifies all physical theories as description tools and not as resources where an explanation of the origins of the phenomena can be found.
Stam,
The bra-ket notation is a way to describe Hilbert vectors and their inner products.
The reverse bra-ket method is a way to relate pairs of functions and parameter spaces with a corresponding normal operator. The method involves summation or integration.
In contrast to the outer bra-ket product |p〉〈q| it is far more than a notation.
F ≡ |q〉F(q)〈q| is a definition that relates the function F(q), the parameter space {q} and the orthonormal set of Hilbert vectors {|q〉}, with operator F via the integral:
〈x|F|y〉 ≡ ∫〈x|q〉F(q)〈q|y〉 dq
Which holds for all vectors |x〉 and |y〉 in the non-separable Hilbert space.
The method also works for sets {aᵢ} of discrete quaternions and an orthonormal set of Hilbert vectors {|aᵢ〉}, which span a subspace of the separable Hilbert space.
〈x|A|y〉 ≡ ∑ᵢ 〈x|aᵢ〉aᵢ〈aᵢ|y〉
Charles
I use the name stochastic operator for a category of operators that get their eigenspaces filled by the actions of mechanisms, which use stochastic processes in order to generate the corresponding data. For example the process can be a combination of a Poisson process and a binomial process. The binomial process locally reduces the efficiency of the Poisson process. It is implemented by a spread function, which distributes the output of the Poisson process over a region of space, such that a coherent location swarm results, which can be characterized by a spatial distribution such as a Gaussian distribution. The coherent swarm can be described by a location density distribution. The squared modulus of the wave function of an elementary particle conforms to something like this Gaussian location distribution.
The continuous location density distribution can be used to define an operator via the reverse bra-ket method. That operator will then represent the object that is generated by the corresponding mechanism. The stochastic operator resides in the separable Hilbert space and registers the generation process in a step-wise fashion. The operator that represents the continuous location density distribution resides in the Gelfand triple that belongs to the separable Hilbert space.
The problem with creation and annihilation operators is that they conflict with conservation laws. They create from nothing and they annihilate into nothing.
Their action appears to have no cause. This degrades these operators to descriptors of what happened, rather than to the actual actors.
So, what stimulates this operator to start an interaction? What is the cause of dynamics in your model?
The Hilbert spaces contain nothing that activates the generation of objects. Its operators only describe what happens by administering the corresponding data in their eigenspaces. The eigenspaces of operators in Hilbert space store present data and historic data together with a time stamp. They can only store these combinations as quaternions or as quaternionic continuums. The operators in the separable Hilbert space do not store future data that can act as artifacts, which affect the continuums in the Gelfand triple. They have no built-in mechanism that drives them. The mechanisms that feed the operators with present data reside external to the Hilbert spaces. Without these mechanisms nothing occurs.
This model means that the separable Hilbert space is recreated at each step of a model wide clock. In that recreation the history stays as it was. Only the present data are added. Thus. the data generating mechanisms act as a function of the progression value of the model wide clock. The discrete part of universe steps with this universe wide clock. The separable Hilbert space is embedded into its companion Gelfand triple. In this way the continuums that are stored in the eigenspaces of operators, which reside in the Gelfand triple follow what happens in the separable Hilbert space. In this way progression steps in the separable Hilbert space and flows in the Gelfand triple.
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