A simple model of fundamental physics
By J.A.J. van Leunen
I
http://www.e-physics.eu
A simple model of fundamental physics
By J.A.J. van Leunen
II
http://www.e-physics.eu
A simple model of fundamental physics
By J.A.J. van Leunen
III
http://www.e-physics.eu
Physical Reality
 In no way a model can give a precise description of
physical reality.
 At the utmost it presents a correct view on physical
reality.
 But, such a view is always an abstraction.
 Mathematical structures might fit onto observed
physical reality because their relational structure is
isomorphic to the relational structure of these
observations.
4
Rules Restrict Complexity
 Physical reality applies rules for relational
structures that it accepts
 These rules intent to reduce the complexity
of these relational structures
5
Complexity
 Physical reality is very complicated
 It seems to belie Occam’s razor.
 However, views on reality that apply
sufficient abstraction can be rather simple
 It is astonishing that such simple
abstractions exist
6
What is complexity?
 Complexity is caused by the number and the
diversity of the relations that exist between
objects that play a role
 A simple model has a small diversity of its
relations.
7
Rules and relational Structures
Logic
 The part of mathematics that treats relational structures is
lattice theory.
 Logic systems are particular applications of lattice theory.
 Classical logic has a simple relational structure.
 However since the paper of Birkhoff and von Neumann in
1936, we know that physical reality cheats classical logic.
 Since then we think that nature obeys quantum logic.
 Quantum logic has a much more complicated relational
structure.
8
Physical Reality & Mathematics
 Physical reality is not based on mathematics.
 Instead it happens to feature relational structures that
are similar to the relational structure that some
mathematical constructs have.
 That is why mathematics fits so well in the
formulation of physical laws.
 Physical laws formulate repetitive relational structure
and behavior of observed aspects of nature.
9
Logic systems
 Classical logic and quantum logic only describe the
relational structure of sets of propositions
 The content of these proposition is not part of the
specification of their axioms
 The logic systems only control static relations
 Their specification does not cover dynamics
10
Fundament
 The Hilbert Book Model (HBM) is
strictly based on traditional quantum
logic.
 This foundation is lattice isomorphic
with the set of closed subspaces of an
infinite dimensional separable
Hilbert space.
11
First Model
About 25 axioms
Classical
Logic
Separable Hilbert
Space
Weaker
modularity
isomorphism
Traditional
Quantum
Logic
Particle
location
operator
Countable
Eigenspace
Only
static
status quo
&
No fields
Three alarming facts
The first level model does not support continuums
1.

HS operators have countable eigenspaces
2. The first level model does not support dynamics
 Can only represent static status quo
3. The Hilbert space contains deeper details than
quantum logic does


QL ⟹ propositions ↭ HS ⟹ sub-spaces
HL ⟹ refined propositions ↭ HS ⟹ vectors
13
Threefold hierarchy
Relational
structure
Quantum
Logic
Hilbert
logic
Isomorphisms
Quantum
Hilbert
Logic
space
Atomic
Subspace
quantum
logic
proposition
Atomic
Base vector
Hilbert
Logic
proposition
Set
of
particles
Particle
is swarm
of step
stones
Step stone
Possible interpretation of isomorphisms
14
Physical model
 The isomorphism introduces a set of
particles, where each particle is represented
by a swarm of step stones.
 Particles are represented by atomic
quantum logical propositions.
 Step stones are represented by Hilbert space
vectors that are eigenvectors of operators of
the Hilbert space.
15
Static Representation
Quantum logic
Hilbert space
}
 No full isomorphism
 Cannot represent
continuums
Solution:
• Refine to Hilbert logic
• Add Gelfand triple
16
Discrete sets and continuums
A Hilbert space features operators
that have countable eigenspaces
A Gelfand triple features operators
that have continuous eigenspaces
17
Static Status Quo of the Universe
Classical
Logic
Separable Hilbert
Separable Hilbert
Space
SpaceTriple
Gelfand
Subspaces
Separable Hilbert
Space
Traditional
Quantum
Logic
isomorphisms
Isomorphism’s
Particle
location
location
Continuum
Eigenspace
embedding
Hilbert
Logic
vectors
Countable
Eigenspace
The sub-models can only implement
a static status quo
Representation
Quantum logic
Hilbert logic
Hilbert space
}
 Cannot represent dynamics
 Can only implement a
static status quo
Solution:
An ordered sequence of sub-models
The model looks like a book where the sub-models are the pages.
20
Sequence
· · · |-|-|-|-|-|-|-|-|-|-|-|-| · · · · · · · · · · |-|-|-|-|-|-|-|-|-| · · ·
Prehistory
Reference sub-model has
densest packaging
current
future
Reference Hilbert space delivers via its
enumeration operator the
“flat” Rational Quaternionic Enumerators
Gelfand triple of reference Hilbert space
delivers via its enumeration operator the
reference continuum
HBM has no Big Bang!
21
The Hilbert Book Model
 Sequence ⇔ book ⇔ HBM
 Sub-models ⇔ sequence members ⇔ pages
 Sequence number ⇔page number ⇔ progression parameter
 This results in a
paginated space-progression model
22
Paginated
space-progression model
 Steps through sequence of static sub-
models
 Uses a model-wide clock
 In the HBM the speed of information
transfer is a model-wide constant
 The step size is a smooth function of
progression
 Space expands/contracts in a smooth way
23
Progression step
 The dynamic model proceeds with universe
wide progression steps
 The progression steps have a rather fixed
size
 The progression step size corresponds to an
super-high frequency (SHF)
 The SHF is the highest frequency that can
occur in the HBM
24
Recreation
The whole universe is recreated
at every progression step
If no other measures are taken,
the model will represent
dynamical chaos
25
Dynamic coherence 1
An external correlation
mechanism must take care such
that sufficient coherence
between subsequent pages exist
26
Dynamic coherence 2
The coherence must not be too
stiff, otherwise no dynamics
occurs
27
Storage
The eigenspaces
of operators
can act as storage places
28
Storage details
 Storage places of information that changes
with progression
 The countable eigenspaces of Hilbert space operators
 The continuum eigenspaces of the Gelfand triple
 The information concerns the contents of
logic propositions
 The eigenvectors store the corresponding
relations.
29
Correlation Vehicle
 Supports recreation of the universe at
every progression step
 Must install sufficient cohesion between
the subsequent sub-models
 Otherwise the model will result in
dynamic chaos.
 Coherence must not be too stiff,
otherwise no dynamics occurs
30
Correlation Vehicle Details
 Establishes
 Embedding of particles in continuum
 Causes
 Singularities at the location of the embedding
 Supported by:
 Hilbert space (supports operators)
 Gelfand triple (supports operators)
 Huygens principle (controls information transport)
 Implemented by:
 Enumeration operators
 Blurred allocation function
 Requires identification of atoms / base vectors
31
Correlation vehicle requirements
 Requires ID’s for atomic propositions
 ID generator
 Dedicated enumeration operator

Eigenvalues ⇒ rational quaternions ⇒ enumerators
 Blurred allocation function
 Maps parameter enumerators onto embedding continuum
 Requires a reference continuum
RQE =
Rational
Quaternionic
Enumerator
32
Enumeration
 Hilbert space & Hilbert logic
 Enumerator operator
 Eigenvalues
 Rational quaternionic
enumerators
(RQE’s)
33
Allocation
 Hilbert space & Hilbert logic
 Enumerator operator
 Eigenvalues
 Rational quaternionic
enumerators
(RQE’s)
 Model
 Allocation function 𝒫
 Parameters
 RQE’s
 Image
 Qtargets
34
Enumeration & Allocation
 Hilbert space & Hilbert logic
 Enumerator operator
 Eigenvalues
 Rational quaternionic
enumerators
(RQE’s)
 Model
 Enumeration function
 Parameters
 RQE’s
 Image
 Qtargets
 Function 𝒫 = ℘ ∘ 𝒮
 Blurred 𝒫
 Sharp ℘
 Spread function 𝒮
 Blur 𝜓
35
Enumeration & Allocation & Blur
 Hilbert space & Hilbert logic
 Enumerator operator
 Eigenvalues
 Rational quaternionic
enumerators
(RQE’s)
 Model
 Enumeration function
 Parameters
 RQE’s
 Image
 Qtargets
Swarm
 Function 𝒫 = ℘ ∘ 𝒮
 Blurred 𝒫
 Sharp ℘
 Spread function 𝒮
 Blur 𝜓
36
Blurred allocation function 𝒫
Convolution
 Function 𝒫 = ℘ ∘ 𝒮
 Blurred 𝒫
 Sharp ℘
 Spread function 𝒮
 QPDD
 Quaternionic
Probability
Density
Distribution
⇒ Produces swarm ⇒ Qtarget
⇒ Produces planned Qpatch
⇒ Produces Qpattern ⇒ Swarm
⇓
QPDD
Described by the QPDD
Swarm
37
Blurred allocation function 𝒫
Convolution
 Function 𝒫 = ℘ ∘ 𝒮
 Blurred 𝒫
 Sharp ℘
 Spread function 𝒮
 QPDD 𝜓
 Quaternionic
Probability
Density
Distribution
⇒ Produces swarm ⇒ Qtarget
⇒ Produces planned Qpatch
⇒ Produces Qpattern
Only exists at
current instance
QPDD 𝜓
38
Blurred allocation function 𝒫
 Function 𝒫 = ℘ ∘ 𝒮
 Blurred 𝒫
 Sharp ℘
 Spread function 𝒮
 QPDD 𝜓
 Quaternionic
Probability
Density
Distribution
Curved
space
⇒ Produces swarm ⇒ Qtarget
⇒ Produces planned Qpatch
⇒ Produces Qpattern
Only exists at
current instance
QPDD 𝜓
39
Blurred allocation function 𝒫
 Function 𝒫 = ℘ ∘ 𝒮
 Blurred 𝒫
 Sharp ℘
 Spread function 𝒮
 QPDD 𝜓
 Quaternionic
Probability
Density
Distribution
Curved
space
⇒ Produces swarm ⇒ Qtarget
⇒ Produces planned Qpatch
⇒ Produces Qpattern
Only exists at
current instance
QPDD 𝜓
40
Blurred allocation function 𝒫
 Function 𝒫 = ℘ ∘ 𝒮
 Blurred 𝒫
 Sharp ℘
 Spread function 𝒮
 QPDD 𝜓
 Quaternionic
Probability
Density
Distribution
Curved
space
⇒ Produces QPDD ⇒ Qtarget
⇒ Produces planned Qpatch
⇒ Produces Qpattern
Allocation
function
Swarm 𝜓
41
Hilbert space choices
 The Hilbert space and its Gelfand triple can be defined
using
 Real numbers
 Complex numbers
 Quaternions
 The choice of the number system determines whether
blurring is straight forward
42
Swarming conditions 1, 2 and 3
 In order to ensure sufficient coherence the
correlation mechanism implements
swarming conditions
1. A swarm is a coherent set of step stones
2. A swarm can be described by a
continuous object density distribution
3. That density distribution can be
interpreted as a probability density
distribution
43
Swarming condition 4
 A swarm moves as one unit
 In first approximation this movement can be
described by a linear displacement generator
 This corresponds to the fact that the
probability density distribution has a Fourier
transform
 The swarming conditions result in the
capability of the swarm to behave as part of
interference patterns
44
Swarming conditions
The swarming conditions
distinguish this type of swarm
from normal swarms
45
Mapping Quality Characteristic
 The Fourier transform of the density distribution that
describes the planned swarm can be considered as a
mapping quality characteristic of the correlation
mechanism
 This corresponds to the Optical Transfer Function that
acts as quality characteristic of linear imaging
equipment
 It also corresponds to the frequency characteristic of
linear operating communication equipment
46
Quality characteristic
 Optics versus quantum physics
 In the same way that the Optical Transfer Function is
the Fourier transform of the Point Spread Function
 Is the Mapping Quality Characteristic the Fourier
transform of the QPDD, which describes the planned
swarm. (The Qpattern)
 This view integrates over the set of progression steps
that the embedding process takes to consume the full
Qpattern, such that it must be regenerated
47
Target space
 The quality of the picture that is formed by an optical
imaging system is not only determined by the Optical
Transfer Function, it also depends on the local
curvature of the imaging plane
 The quality of the map produced by quantum physics
not only depends on the Mapping Quality
Characteristic, it also depends on the local curvature
of the embedding continuum
48
Coupling
 For swarms the coupling equation holds
 Φ = 𝛻𝜓 = 𝑚 𝜑
 By requiring that the two sides of the quaternionic differential
equation contain normalized functions, this equation turns into a
coupling equation.
 𝜓 and 𝜑 are normalized quaternionic functions
 They describe quaternionic probability density distributions
 𝛻 is the quaternionic nabla
 Factor 𝑚 is the coupling strength
 P𝜓 = 𝑚 𝜑
 P is the displacement generator
49
Swarms 1
 The correlation mechanism generates swarms of step
stones in a cyclic fashion
 The swarm is prepared in advance of its usage
 This planned swarm is a set of placeholders that is called
Qpattern
 A Qpattern is a coherent set of placeholders
 The step stones are used one by one
 In each static sub-model only one step stone is used per
swarm
 This step stone is called Qtarget
 When all step stones are used, then a new Qpattern is
prepared
50
Planned and actual swarm
Reference
continuum
Swarm of
step stones
Placeholder
generator 𝒮
Embedding
continuum
𝒫 =℘∘𝒮
Qtarget
Set of
placeholders
Qpattern
Continuous
allocation
function ℘
Random
selection
51
Swarms 2
 At each progression step, an image of the planned




swarm (Qpattern) is mapped by a continuous
allocation function onto the embedding continuum
At each progression step, via random selection a single
step stone is selected, whose image becomes the
Qtarget
In fermions that step stone is not used again
A swarm has a “center position”, called Qpatch that can
be interpreted as the expectation value of location of
the swarm
The Qtargets form a stochastic micro-path
52
Placeholders and Step stones
 Together with the allocation function a placeholder
defines where a selected particle can be
 That location is a step stone
 A coherent collection of these placeholders represent
the Qpattern
 The placeholders are generated by the stochastic spatial
spread function 𝒮
 At each progression step a different step stone
becomes the Qtarget location of the particle
53
Generation of placeholders and
step stones
 Per progression step only ONE Qtarget is
generated per Qpattern
 Generation of the whole Qpattern takes a large
and fixed amount of progression steps
 When the Qpatch moves, then the pattern spreads
out along the movement path
 When an event (creation, annihilation, sudden
energy change) occurs, then the enumeration
generation changes its mode
54
Qpattern generation example
(no preferred directions)
 Random enumerator generation at lowest scales
 Let Poisson process produce smallest scale enumerator
 Combine this Poisson process with a binomial process
 This is installed by a 3D spread function
 Generates a 3d “Gaussian” distribution (is example)
The distribution represents an isotropic potential of the form
𝐸𝑟𝑓(𝑟)
𝑟
This quickly reduces to 1/𝑟 (form of gravitational potential)
 The result is a Qpattern
55
Blurred allocation function 𝒫
Convolution
 Blurred function 𝒫 = ℘ ∘ 𝒮
 Sharp ℘
 Spread 𝒮
maps RQE
maps Qpatch
⇒ Qpatch
⇒ Qtarget
 Function 𝒫
 Produces QPDD 𝜓
 Stochastic spatial spread function 𝒮
 Produces Qpattern
 Produces gravitation (1/𝑟)
 Sharp ℘
 Describes space curvature
 Delivers local metric d ℘
56
Micro-path
 The Qpatterns contain a fixed number of step stones
 The step stones that belong to a swarm form a micro-




path
Even at rest, the Qtarget walks along its micro-path
This walk takes a fixed number of progression steps
When the swarm moves or oscillates, then the micropath is stretched along the path of the swarm
This stretching is controlled by the third swarming
condition
57
Wave fronts
 At every arrival of the particle at a new step stone the




embedding continuum emits a wave front
The subsequent wave fronts are emitted from slightly
different locations
Together, these wave fronts form super-high frequency
waves
The propagation of the wave fronts is controlled by
Huygens principle
Their amplitude decreases with the inverse of the
distance to their source
58
Wave front
 Depending on dedicated Green’s functions,
the integral over the wave fronts constitutes
a series of potentials.
 The Green’s function describes the
contribution of a wave front to a
corresponding potential
 Gravitation potentials and electrostatic
potentials have different Green’s functions
59
Potentials & wave fronts
 The wave fronts and the potentials are traces of the
particle and its used step stones.
 The superposition of the singularities smoothens the
effect of these singularities.
 Neither the emitted wave fronts, nor the potentials
affect the particle that emitted the wave front
 Wave fronts interfere
 The wave fronts modulate a field
60
Palestra
 Curved embedding continuum
 Represents universe
Embedded in
continuum
𝑄𝑝𝑎𝑡𝑐ℎ
Collection of
Qpatches
The Palestra is the place where everything happens
61
Mapping
𝒫 =℘∘𝒮
Space curvature
GR
Quantum physics
Quaternionic
metric
𝑑𝒫
16 partial
derivatives
No tensor
needed
Quantum fluid
dynamics
• Continuity equation
𝛻𝜓 = 𝜙
• Dirac equation
𝛻0 𝜓 + 𝛁𝛂 𝜓
• In quaternion format
𝛻𝜓 = 𝑚𝜓 ∗
62
Lattices,
classical logic and
quantum logic
63
Logic – Lattice structure
 A lattice is a set of elements 𝑎, 𝑏, 𝑐, …that is closed for
the connections ∩ and ∪. These connections obey:

 The set is partially ordered. With each pair of elements
𝑎, 𝑏 belongs an element 𝑐, such that 𝑎 ⊂ 𝑐 and 𝑏 ⊂ 𝑐.
 The set is a ∩ half lattice if with each pair of elements
𝑎, 𝑏 an element 𝑐 exists, such that 𝑐 = 𝑎 ∩ 𝑏.
 The set is a ∪ half lattice if with each pair of elements
𝑎, 𝑏 an element 𝑐 exists, such that 𝑐 = 𝑎 ∪ 𝑏.
 The set is a lattice if it is both a ∩ half lattice and a ∪ half
lattice.
64
Partially ordered set
 The following relations hold in a lattice:
𝑎 ∩ 𝑏 = 𝑏 ∩ 𝑎
(𝑎 ∩ 𝑏) ∩ 𝑐
= 𝑎 ∩ (𝑏 ∩ 𝑐)
𝑎 ∩ (𝑎 ∪ 𝑏) = 𝑎
𝑎 ∪ 𝑏 = 𝑏 ∪ 𝑎
(𝑎 ∪ 𝑏) ∪ 𝑐
= 𝑎 ∪ (𝑏 ∪ 𝑐)
𝑎 ∪ (𝑎 ∩ 𝑏) = 𝑎
• has a partial order inclusion ⊂:
a⊂b⇔a⊂b=a
• A complementary lattice
contains two elements 𝑛 and 𝑒
with each element a an
complementary element a’
𝑎 ∩ 𝑎’ = 𝑛 𝑎 ∩ 𝑛 = 𝑛
𝑎 ∩ 𝑒 = 𝑎 𝑎 ∪ 𝑎’ = 𝑒
𝑎 ∪ 𝑒 = 𝑒 𝑎 ∪ 𝑛 = 𝑎
65
Orthocomplemented lattice
 Contains with each element 𝑎 an element 𝑎” such that:
𝑎 ∪ 𝑎” = 𝑒
𝑎 ∩ 𝑎” = 𝑛
(𝑎”)” = 𝑎
𝑎 ⊂ 𝑏 ⟺ 𝑏” ⊂ 𝑎”
Distributive lattice
𝑎 ∩ (𝑏 ∪ 𝑐)
= (𝑎 ∩ 𝑏) ∪ ( 𝑎 ∩ 𝑐)
𝑎 ∪ (𝑏 ∩ 𝑐)
= (𝑎 ∪ 𝑏) ∩ (𝑎 ∪ 𝑐)
Modular lattice
(𝑎 ∩ 𝑏) ∪ (𝑎 ∩ 𝑐) = 𝑎 ∩ (𝑏 ∪ (𝑎 ∩ 𝑐))
Classical logic is an orthocomplemented modular lattice
66
Weak modular lattice
 There exists an element 𝑑 such that
𝑎 ⊂ 𝑐 ⇔ 𝑎 ∪ 𝑏 ∩ 𝑐
= 𝑎 ∪ (𝑏 ∩ 𝑐) ∪ (𝑑 ∩ 𝑐)
 where 𝑑 obeys:
(𝑎 ∪ 𝑏) ∩ 𝑑 = 𝑑
𝑎 ∩ 𝑑 = 𝑛
𝑏 ∩ 𝑑 = 𝑛
[(𝑎 ⊂ 𝑔) and (𝑏 ⊂ 𝑔) ⇔ 𝑑 ⊂ 𝑔
Quantum logic obeys the weak modular law
67
Atoms
 In an atomic lattice
∃𝑝 𝜖 𝐿 ∀𝑥 𝜖 𝐿 {𝑥 ⊂ 𝑝 ⇒ 𝑥 = 𝑛}
∀𝑎 𝜖 𝐿 ∀𝑥 𝜖 𝐿 {(𝑎 < 𝑥 < 𝑎 ∩ 𝑝)
⇒ (𝑥 = 𝑎 𝑜𝑟 𝑥 = 𝑎 ∩ 𝑝)}
𝑝 is an atom
68
Logics
 Classical logic has the structure of an
orthocomplemented distributive
modular and atomic lattice.
 Quantum logic has the structure of an
orthocomplented weakly modular and
atomic lattice.
 Also called orthomodular lattice.
69
Hilbert Space
The set of closed subspaces of an
infinite dimensional separable
Hilbert space forms an
orthomodular lattice
Is lattice isomorphic to quantum
logic
70
Hilbert Logic
 Add linear propositions
 Linear combinations of atomic propositions
 Add relational coupling measure
 Equivalent to inner product of Hilbert space
 Close subsets with respect to relational coupling measure
 Propositions ⇔ subspaces
 Linear propositions ⇔ Hilbert vectors
71
Superposition principle
Linear combinations of linear
propositions are again linear
propositions that belong to the same
Hilbert logic system
72
Isomorphism
Lattice isomorhic
Propositions ⇔ closed subspaces
Topological isomorphic
Linear atoms ⇔ Hilbert base
vectors
73