Dear Donald

In modern physics, space( 3 real spatial dimension) could not be separated from

"Time" as the fourth coordinate (or dimension) in such a way that talking about the coordinates of each physical events should be defined on the space-time continuum which is a conjoined of 3 real dimension with a "complex Time" dimension in a brief

expression: (x,y,z,ict) where"c" is celerity. So" ict " with a complex value substitutes the fourth dimension and space-time is a partially complex manifold where infinitesimal interval remains invariant under Lorentz transformation as mentioned in special Relativity. However the totally complex manifold reveals some application in String theory, supersymmetry and etc.

Regards

Space is curved and its curvature changes with progression.

In this sense it has all aspects of a field and that field has all aspects of a quaternionic function.

So, my question is partially mathematical, regarding the character of the mathematical model used to represent Space. Today, mathematics realizes a Real continuum and our models of space incorporate this Real continuum.

However, we appear to need models with complex quantities involved - even for all space axes. Are axes, with 'only' Real values, sufficient for an adequate model of space or do we need a complex continuum to properly model space?

If we need 'more than' Real values to adequately model space, then maybe our mathematical tools need updating in order to handle what we are perceiving as space.  A particular mathematical tool needing updating could be the continuum, with the need for a complex continuum shown by the need for complex values in models of space.

If space can be represented by a field and a field can be represented by a multidimensional function of a multidimensional parameter space, the particles can be represented by singularities of that function.

Hans

Will these multidimensional functions be Real functions or Complex valued functions?

If complex, are the field lines complex? Which would indicate a complex continuum is needed, but we only have a Real continuum.  So we need new mathematical tools.

Donald,

Hilbert spaces can cope with members of a division ring. Only three suitable division rings exist: real numbers, complex numbers and quaternions.

Separable Hilbert spaces cannot cope with continuums, but every infinite dimensional separable Hilbert space owns a Gelfand triple, also known as rigged Hilbert space. It is a non-separable Hilbert space and it features operators that have continuums as eigenspaces.

A Physical theory that is based on Hilbert spaces can use an eigenspace of a quaternionic non-separable Hilbert space in order to represent our living space.

That continuum can also be seen as the target of a continuous quaternionic function that has a flat parameter space. This parameter space is then spanned by a quaternionic number system. In that parameter space the real part of the quaternion plays the role of progression.

Due to their four dimensions quaternionic number systems exist in 16 versions that differ in their discrete symmetries. This also hold for continuous quaternionic functions.

Nature seems to use this fact in order to create a large diversity of elementary particles.

In order to support my own insights I have developed a completely deduced mathematical model of reality: http://vixra.org/abs/1409.0050

@Hans:

I think the first thing to do is to define what a Continuum is.  I will suggest that we are using a single instance of a continuum today, and that a more general definition should be used that does not limit a continuum to the Real Numbers.  It is this implicit, and somewhat arbitrary, constraint that we have placed on the Continuum that causes us to think that the Continuum Hypothesis might or might not be true according to infinite sets strictly set to that of Real Numbers.

Why not a Complex continuum, defined as a linear set of complex (or quaternion) numbers?  We know of Real extensions to Rational Numbers - so if q(x) is a Rational Number, then q(1) + q(2) sqrt(2)) is an extension of the Rationals, extended by a single Real value.  What if Complex Numbers, defined as r(1) + i r(2) are simply an extension of Reals into the Complex hyperfield?  This would suggest that quaternions are the true 'Complex Numbers' and that what we call Complex Numbers today are only an extension of Reals into the full Complex Numbers (which do not actually represent the full set of quanterion-level complex numbers; just as q(1) + sqrt(2) q(2) does not represent the full set of Reals.

I believe there are still many levels of mathematics we have not yet touched - and hence should be a bit humbler about what we think we know (and about how we apply this knowledge).

Doanld

@Donald

Cantor solved the distinction between countable sets and continuums.

The rational numbers (whether they are real, complex or quaternionic) are allways forming  countable sets. Continuums include all limits of converging series.

Hans:

i'm not sure how you consider Real, Complex, or Quaternions to be equivalent to Rational numbers.  Cantor provided a proof that Reals are not countable and Complx and Quanternios are of at least the same order as Reals (uncountable) - since these both contain the Reals as a subset.

The limits of converging series are the Reals.

so I am not understanding your last post.

Complex numbers are assembled from two real numbers. When both members are rationals, then the complex number is a rational complex number. The  set of rational complex numbers is countable.

Similarly, quaternions are assembled from four real numbers. When all members are rationals, then the quaternion is a rational quaternion. The set of rational quaternions is countable.

When members are real, then the sets no longer need to be countable and then they may form continuums. It means that in that case all series of numbers that are approaching to a limit will have this limit included in the set.

The normal operators of separable Hilbert spaces can only accept rational numbers as their eigenvalues. This can be rationals that belong to a real number set, rationals that belong to a complex number set or rationals that belong to a quaternionic number set.

The normal operators of non-separable Hilbert spaces can offer real valued, complex valued or quaternion valued continuums as their eigenspaces.

The subspaces of a separable Hilbert space have a dimension. This is not ensured for the subspaces of a non-separable Hilbert space! 

It is possible to consider the separable Hilbert space as being embedded inside the non-separable Hilbert space. This embedding leaves freedom. That freedom is used by nature in order to implement quantum physics.

Thank you, Hans

Now I understand your distinction. The set of 'Rational Complex numbers' is a (proper) sub-set of Complex numbers, using only rational values to construct the Complex values. Complex numbers, as a full set, are uncountable.

From a mathematical standpoint, Rationals are a (countable) sub-set of the Real numbers, which are an uncountable sub-set of Complex numbers.

The limitation of real and imaginary parts of a complex number to only rational values pertains to a specific application of complex numbers (eg. For separable Hilbert spaces).

i would not agree that nature has anything to do with Hilbert spaces or quantum mechanics - just that we humans find these concepts useful to describe and model nature.

@Donald

In contrast to your point of view I think that the real secret of quantum physics is borrowed in the relation between separable and non-separable Hilbert spaces.

I think that reality impersonates (my native language is Dutch and I do not know a better word for it) a small set of chains of mathematical structures, where each chain starts with a reliable and simple foundation and subsequent members of the chain are model extensions of the previous member. For example one of the chains starts with an orthomodular lattice. A subsequent member is a separable Hilbert space, which is then followed by a non-separable Hilbert space.

The chains are interrelated. For example the Hilbert spaces relate to the chain that contains number systems and starts with countable sets.

The Hilbert spaces restrict the number systems that it can handle to members of a division ring.

This indicates that the next steps in model development also introduce mutual restrictions. With other words the model has built-in limitations that prevent the model from becoming more complicated in an uncontrolled way.

This indicates that the lower development levels of the model are rather simple. You need not be a genius in order to comprehend the lower levels of the model.

Thus models that are deducible from simple foundations are characterized by a hierarchy in which complexity grows rather slowly with the model development levels.

Dear Donald

There is a new theory that relates complex space time to real space time as assumed by special theory of relativity. The complex terms are only appeared when relating elementary particles to classical objects otherwise if the transformations are undergone between classical objects the imaginary terms will be cancelled from both sides to get the real counterparts:

please see the following link for the mentioned theory

https://www.researchgate.net/publication/259996539_Solving_the_instantaneous_response_paradox_of_entangled_particles_using_the_time_of_events_theory

Article Solving the instantaneous response paradox of entangled part...

Maxwell-Minkowski based approach versus Hamilton-Euclidean based approach

The difference between the Maxwell-Minkowski based approach and the quaternionic Hamilton-Euclidean based approach will become clear when the difference between the coordinate time t and the proper time τ is investigated. This becomes difficult when space is curved, but for infinitesimal steps space can be considered flat. In that situation holds:

Coordinate time step vector = proper time step vector + spatial step vector

Or in Pythagoras format:

(∆t)² = (∆τ)² + (∆x)² + (∆y)² + (∆z)²

This influence is easily recognizable in the corresponding wave equations:

In Maxell-Minkowski format the wave equation uses coordinate time t. It runs as:

∂²ψ/∂t²−∂²ψ/∂x²−∂²ψ/∂y²−∂²ψ/∂z²=0

Papers on Huygens principle work with this formula or it uses the version with polar coordinates.

For 3D the general solution runs:

ψ =f(r−ct)/r, where c=±1; f is real

For 1D the general solution runs:

ψ =f(x−ct), where c=±1; f is real

For the Hamilton-Euclidean version, which uses proper time τ, we use the quaternionic nabla ∇:

∇={∂/∂τ, ∂/∂x, ∂/∂y, ∂/∂z}=∇₀+▽; ∇*=∇₀−▽

∇ψ = ∇₀ ψ₀ – (▽,Ψ) + ∇₀ Ψ + ▽ ψ₀ ± ▽ × Ψ

The ± sign reflects the choice between right handed and left handed quaternions.

In this way the Hamilton-Euclidean format of the wave equation runs:

∇*∇ψ = ∇₀∇₀ψ +(▽,▽)ψ =0

∂²ψ/∂τ²+∂²ψ/∂x²+∂²ψ/∂y²+∂²ψ/∂z²=0

Where ψ= ψ₀+Ψ

For the general solution holds: f= f₀+F

For the real part ψ₀ of ψ:

ψ₀ =f₀ (î r−c τ)/r, where c=±1 and î is an imaginary base vector in radial direction

For the imaginary part Ψ of ψ:

Ψ = F(î z−c τ), where c=±1 and î= î(z) is an imaginary base vector in the x,y plane

The orientation θ(z) of î(z) in the x,y plane determines the polarization of the 1D wave front.

Quaternionic coordinate data, which are based on proper time τ, fit as eigenvalues in Hilbert spaces. This is not the case for the spacetime coordinates that are based on coordinate time t. Hilbert spaces require that their eigenvalues are members of a division ring. Only three suitable division rings exist: real numbers, complex numbers and quaternions.

The formula:

(∆t)² = (∆τ)² + (∆x)² + (∆y)² + (∆z)²

indicates that the coordinate time step corresponds to the step of a full quaternion, which is a superposition of a proper time step and a spatial step.

An infinitesimal spacetime step ∆s is usually presented as an infinitesimal proper time step ∆τ.

(∆s)² = (∆t)² - (∆x)² - (∆y)² - (∆z)², with signature + - - - .

Above it is indicated that the coordinate time step ∆t corresponds to a quaternionic step. It mixes progression and 3D space. Proper time corresponds to pure progression.

If one considers the field of complex numbers and uses variables to name its elements, one cannot define either real numbers nor a linear ordering inside. Subsets in C which are definable are only finite or cofinite. Subsets in C^n have always an even number real dimension. But in Physics we need ordering and also real line movement, for trajectories, etc. So I find that it is not a very good idea to start with the complex numbers. Better we start with the real numbers and we spometimes use the imaginary unit i or quaternions units i, j, k or even more complicated terms, but on a basis which has a definable ordering. (the real numbers!)

A good reason for selection of number systems is formulated in the link below.

http://arxiv.org/abs/1101.5690

    Generalization of the kinetic theory, using a complex representation of the phase space, has been proposed in [Physical Review E 94(2), 023316, (2016)], where the phase space variables (velocity, position, and time) and the probability density function are all represented by complex valued quantities. Please see the following link for more information:

https://www.researchgate.net/publication/308116942_Generalizing_the_Boltzmann_equation_in_complex_phase_space

or

http://journals.aps.org/pre/abstract/10.1103/PhysRevE.94.023316

Article Generalizing the Boltzmann equation in complex phase space

@Donald

Very interesting question endeed. This article elaborates on the case of imaginary

spatial coordinates that co-exsist with the real ones in Minkowsky spacetime

Article Time vector defined in imaginary space of spatial coordinate

Before answering with standard 4D please recall spinores and they use in QM, also be sure to read https://arxiv.org/pdf/0910.0352.pdf