Download The Hilbert Book Model

A simple model of fundamental physics
By J.A.J. van Leunen
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.
2
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
3
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.
4
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 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.
5
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.
6
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
 They only control static relations
 Their specification does not cover dynamics
7
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.
8
Correspondences
 ≈1930 Garret Birkhoff and John von Neumann discovered
the lattice isomorphy:
 Infinite, but
countable
number of
atoms /
base vectors
Quantum logic
Propositions:
𝑎, 𝑏
atoms
𝑐, 𝑑
Relational complexity:
𝐶𝑜𝑚𝑝𝑙𝑒𝑥𝑖𝑡𝑦 𝑎 ∩ 𝑏
Inclusion:
𝑎 ∪ 𝑏
For atoms 𝑐𝑖 :
Hilbert space
Vectors:
|𝑎⟩, |𝑏⟩
Base vectors:
|𝑐⟩, |𝑑⟩
Inner product:
𝑎|𝑏
Sum:
|𝑎⟩ + |𝑏⟩
Subspace
𝑐𝑖
𝒊
𝛼𝑖 |𝑐𝑖 ⟩
𝑖
∀ 𝛼𝑖
9
Atoms & base vectors
 Atom
 Contents not important
 Set is unordered
 Many sets possible
 Logic
 Lattice
 Only relations
important
10
Atoms & base vectors
 Atom
 Contents not important
 Set is unordered
 Many sets possible
 Base vector
 Set is unordered
 Many sets possible
 Can be eigenvector

 Logic
 Lattice
 Only relations
important
Eigenvalue
 Real
 Complex
 Quaternionic
11
Atoms & base vectors
 Atom
 Contents not important
 Set is unordered
 Many sets possible
 Base vector
 Set is unordered
 Many sets possible
 Can be eigenvector

Eigenvalue
 Real
 Complex
 Quaternionic
 Hilbert space
 Inner product
 Real
 Complex
 Quaternionic
Constantin Piron:
Inner product 𝑥|𝑦 must be
real, complex or quaternionic
𝑎|𝑃𝑎 = 𝑎|𝑝𝑎 = 𝑎|𝑎 𝑝
The eigenvalues are the same type
of numbers as the inner products
12
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
Representation
Quantum logic
Hilbert space
}
 No full isomorphism
 Cannot represent
continuums
Solution:
• Refine to Hilbert logic
• Add Gelfand triple
14
Discrete sets and continuums
A Hilbert space features operators
that have countable eigenspaces
A Gelfand triple features operators
that have continuous eigenspaces
15
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 form a threefold hierarchy
Three structures, three levels
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
Swarm
of step
stones
Step stone
18
No support for dynamics
None of the three structures has
a built-in mechanism for
supporting dynamics
19
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.
21
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!
22
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
23
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
24
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
25
Recreation
The whole universe is recreated
at every progression step
If no other measures are taken,
the model will represent
dynamical chaos
26
Dynamic coherence 1
An external correlation
mechanism must take care such
that sufficient coherence
between subsequent pages exist
27
Dynamic coherence 2
The coherence must not be too
stiff, otherwise no dynamics
occurs
28
Storage
The eigenspaces
of operators
can act as storage places
29
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.
30
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
31
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
32
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
33
Reference continuum
 Select a reference Hilbert space
 Has countable number of dimensions/base vectors
 Criterion is densest packaging of enumerators*.
 Take its Gelfand triple (rigged “Hilbert space”)
 Has over-countable number of dimensions/base vectors
 Has operators with continuum eigenspaces
 Select equivalent of enumeration operator in Hilbert space
 Use its eigenspace as reference continuum
(*Cyclic: Densest with respect to reference continuum)
34
Atoms & base vectors
 Atom
 Set is unordered
 Many sets possible
 Contents not important
 Base vector
 Set is unordered
 Many sets possible
 Can be eigenvector

Eigenvalue
 Real
 Complex
 Quaternionic
35
Atoms & base vectors
 Atom
 Set is unordered
 Many sets possible
 Contents not important
 Base vector
 Set is unordered
 Many sets possible
 Can be eigenvector

Eigenvalue
 Real
 Complex
 Quaternionic
 Hilbert space & Hilbert logic
 Inner product
 Real
 Complex
 Quaternionic
 Enumerator operator
 Eigenvalues
 Rational
quaternionic
enumerators
(RQE’s)
 Enumerates atoms
36
Enumeration
 Hilbert space & Hilbert logic
 Enumerator operator
 Eigenvalues
 Rational quaternionic
enumerators
(RQE’s)
37
Enumeration
 Hilbert space & Hilbert logic
 Enumerator operator
 Eigenvalues
 Rational quaternionic
enumerators
(RQE’s)
 Model
 Allocation function 𝒫
 Parameters
 RQE’s
 Image
 Qtargets
38
Enumeration
 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
39
Enumeration
 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
40
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
41
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
42
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
43
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
44
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
45
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
46
Real Hilbert space model
 Progression separated
 Use rational eigenvalues in Hilbert space
 Use real eigenvalues in Gelfand triple
 Cohesion not too stiff (otherwise no dynamics!)
 Keep sufficient interspacing ⇛ 1D blur ?
 Define lowest rational

May introduce scaling as function of progression
 Rather fixed progression steps
47
Complex Hilbert space model
 Progression at real axis
 Use rational complex numbers in Hilbert space
 Use real complex numbers in Gelfand triple
 Cohesion not too stiff (otherwise no dynamics!)
 Keep sufficient interspacing ⇛ 1D blur ?
 Lowest rational at both axes (separately)
 May introduce scaling as function of progression
 No scaling or blur at progression axis
48
Quaternionic
Hilbert space model
 Progression at real axis
 Use rational quaternions in Hilbert space and in
Gelfand triple
 Cohesion not too stiff (otherwise no dynamics!)
 Keep sufficient interspacing⇛ 3D blur
 Lowest rational at all axes (same for imaginary
axes)
 May introduce scaling as function of progression

No scaling and no blur at progression axis
 Blur installed by correlation vehicle
49
Why blurred ?
 Pre-enumerated objects
(atoms, base vectors) are not ordered
 No origin

Affine-like space
 Enumeration must not introduce
extra properties
 No preferred directions
50
Swarming conditions 1
 In order to ensure sufficient coherence the
correlation mechanism implements
swarming conditions
 A swarm is a coherent set of step stones
 A swarm can be described by a continuous
object density distribution
 That density distribution can be
interpreted as a probability density
distribution
51
Swarming conditions 2
 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
52
Swarming conditions
The swarming conditions
distinguish this type of swarm
from normal swarms
53
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
54
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
55
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
56
Coupling
 For swarms the coupling equation holds
 Φ = 𝛻𝜓 = 𝑚 𝜑
 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
57
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
58
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
59
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
 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
60
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
61
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
62
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
63
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 ℘
64
Micro-path
 The Qpatterns contain a fixed number of step stones
 The step stones that belong to a Qpattern form a




micro-path
Even at rest, the Qpattern walks along its micro-path
This walk takes a fixed number of progression steps
When the Qpattern moves or oscillates, then the
micro-path is stretched along the path of the Qpattern
This stretching is controlled by the third swarming
condition
65
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
66
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
67
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
68
Photon & gluon emission
 A sudden decrease in the energy of the emitting




particle causes a modulation of the amplitude of the
emitted wave fronts
The creation of this modulation lasts a full micro-walk
The modulation of the SHF carrier wave becomes
observable as a photon or a gluon
The modulation represents an energy quantum
E=ℏ∙ν
The energy is shown in the modulation frequency ν
69
Embedding continuum
 A curved continuum embeds the elementary
particles
 The continuum is constituted by a
background field
70
Photon & gluon absorption
 A modulation of the embedding continuum
can be absorbed by an elementary particle
 The modulation frequency determines the
absorbable energy quantum
 The modulation must last during a full
micro-walk
71
Photons and gluons
 Photons and gluons are energy quanta
 Photons and gluons are NOT electro-
magnetic waves!
 Photons and gluons are NOT particles
72
Palestra
 Curved embedding continuum
 Represents universe
Embedded in
continuum
𝑄𝑝𝑎𝑡𝑐ℎ
Collection of
Qpatches
The Palestra is the place where everything happens
73
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 ∗
74
How to use
Quaternionic Distributions
and
Quaternionic Probability Density Distributions
The HBM is a quaternionic model
 The HBM concerns quaternionic physics rather than
complex physics.
 The peculiarities of the quaternionic Hilbert model are
supposed to bubble down to the complex Hilbert space
model and to the real Hilbert space model
 The complex Hilbert space model is considered as an
abstraction of the quaternionic Hilbert space model
 This can only be done properly in the right
circumstances
76
Continuous
Quaternionic Distributions
 Quaternions
𝑎 = 𝑎0 + 𝒂
c = 𝑎𝑏 = 𝑎0 𝑏0 − 𝒂, 𝒃 +
𝑎0 𝒃 + 𝑏0 𝒂 + 𝒂 × 𝒃
 Quaternionic distributions

Two
equations
Differential equation
g = 𝛻𝑓 = 𝛻0 𝑓0 − 𝛁, 𝒇 +
𝛻0 𝒇 + 𝛁𝑏0 + 𝛁 × 𝒃
Three
kinds
𝜙 = 𝛻𝜓 = 𝑚 𝜑
{
𝑔0 = 𝛻0 𝑓0 − 𝛁, 𝒇
𝐠 = 𝛻0 𝒇 + 𝛁𝑏0 + 𝛁 × 𝒃
Differential
Coupling
Continuity
}
equation
77
Field equations
 𝜙 = 𝛻𝜓
 𝜙0 = 𝛻0 𝜓0 − 𝛁, 𝜓
 𝝓 = 𝛻0 𝜓 + 𝛁𝜓0 + 𝛁 × 𝜓
Spin of a field:
𝜮𝑓𝑖𝑒𝑙𝑑 =
𝕰 × 𝝍 𝑑𝑉
𝑉
 𝕰 ≡ 𝛻0 𝝍 + 𝜵𝜓0
 𝕭≡𝜵×𝝍
 𝝓=𝕰+𝕭
𝐸≡ 𝜙 =
𝜙0 𝜙0 + 𝝓, 𝝓
= 𝜙0 𝜙0 + 𝕰, 𝕰 + 𝕭, 𝕭 + 𝟐 𝕰, 𝕭
Is zero
?
78
QPDD’s
 Quaternionic distribution
 𝑓 = 𝑓0 + 𝒇
Scalar
field
Vector
field
 Quaternionic Probability Density Distribution
 𝜓 = 𝜓0 + 𝝍 = 𝜌0 + 𝜌0 𝒗
Density
distribution
Current density
distribution
79
Coupling equation
 Differential
𝜓 and φ
are normalized
𝜙 = 𝛻𝜓 = 𝑚𝜑
𝜓 = 𝜑
 Integral
𝜓
2
𝑉
𝜑
𝑉
𝜙
𝑉
𝑑𝑉 =
2
2
𝑑𝑉 = 1
𝑚 = total energy
= rest mass +
kinetic energy
𝑑𝑉 = 𝑚2
Flat space
80
Coupling in Fourier space
𝛻𝜓 = 𝜙 = 𝑚 𝜑
ℳ𝜓 = 𝜙 = 𝑚 𝜑
𝜓|ℳ 𝜓 = 𝑚 𝜓|𝜑
ℳ = ℳ0 + 𝞛
ℳ0 𝜓0 − 𝞛, 𝝍 = 𝑚 𝜑0
ℳ0 𝝍 + 𝞛𝜓0 + 𝞛 × 𝝍 = 𝑚 𝝋
2
𝜙 𝑑𝑉 =
𝑉
ℳ𝜓
𝑉
2
𝑑 𝑉 = 𝑚2
In general 𝜓 is not an
eigenfunction of operator
ℳ.
That is only true when 𝜓
and 𝜑 are equal.
For elementary particles
they are equal
apart from their difference
in discrete symmetry.
81
Dirac equation
Approximately
flat space
𝛻0 𝜓 + 𝛁𝛂 𝜓 = 𝑚𝛽 𝜓
 Spinor 𝜓
 Dirac matrices 𝛂, 𝛽
• 𝛻0 𝜓𝑅 + 𝛁𝜓𝑅 = 𝑚𝜓𝐿
• 𝛻0 𝜓𝐿 − 𝛁𝜓𝐿 = 𝑚𝜓𝑅
 In quaternion format
• 𝛻𝜓 = 𝑚𝜓 ∗
• 𝛻 ∗ 𝜓 ∗ = 𝑚𝜓
𝜓𝑅 = 𝜓𝐿∗ = 𝜓0 + 𝝍
Qpattern
QPDD
82
Entanglement
83
Entanglement
 The correlation mechanism manages entanglement
 At every progression instant the quantum state




function of an entangled system equals the
superposition of the quantum state functions of its
components
Entangled systems obey the swarming conditions
For entangled systems the coupling equation holds
Φ = 𝛻𝜓 = 𝑚 𝜑
𝜓 and 𝜑 are normalized
Entanglement acts as a binding mechanism
84
Binding
 The fact that superposition coefficients define internal
movements can best be explained by reformulating the
definition of entangled systems.
 Composites that are equipped with a quantum state
function whose Fourier transform at any progression
step equals the superposition of the Fourier
transforms of the quantum state functions of its
components form an entangled system.
 Now the superposition coefficients can define internal
displacements. As a function of progression they define
internal oscillations.
85
Geoditches
 In an entangled system the micro-paths of the constituting




elementary particles are folded along the internal
oscillation paths.
Each of the corresponding step stones causes a local pitch
that describes the temporary (singular) curvature of the
embedding continuum.
These pitches quickly combine in a ditch that like the
micro-path folds along the oscillation path.
These ditches form special kinds of geodesics that we call
“Geoditches”.
The geoditches explain the binding effect of entanglement.
86
Pauli principle
 If two components of an entangled (sub)system that have the
same quantum state function are exchanged, then we can take
the system location at the center of the location of the two
components. Now the exchange means for bosons that the
(sub)system quantum state function is not affected:

 For all α and β{αφ(-x)+βφ(x)=αφ(x)+βφ(-x)}⇒φ(-x)=φ(x)

 and for fermions that the corresponding part of the (sub)system
quantum state function changes sign.

 For all α and β{αφ(-x)+βφ(x)=-αφ(x)-βφ(-x)}⇒φ(-x)=-φ(x)
 This conforms to the Pauli principle.
87
Non-locality
 Action at a distance cannot be caused via information




transfer
Non-locality already plays a role inside the realm of
separate elementary particles.
Hopping along the step stones occurs much faster than
the information carrying waves can follow.
Similar features occur inside entangled systems.
Due to the exclusion principle, observing the state of a
sub-module has direct (instantaneous) consequences
for the state of other sub-modules.
88
Focus
 If in an entangled system the focus is on the system,
then the whole system acts as a swarm and the
correlation mechanism causes hopping along ALL step
stones that are involved in the system
 When the focus shifts to one or more of the
constituents, then the entanglement get at least partly
broken
 The separated particles and the resulting entangled
system act as separate swarms
89
Binding
90
Binding mechanism
 When it is used, each step stone that is involved in an
entangled system produces a singularity. The influence
of that singularity spreads over the embedding
continuum in the form of a wave front that folds and
thus curves this continuum
 The traces of these Qtargets mark paths where the
wave fronts dig pitches into the continuum that
combine into channels that act as geodesics.
91
The effect of modularization
92
Modularization
 Modularization is a very powerful influencer.
 Together with the corresponding encapsulation it
reduces the relational complexity of the ensemble of
objects on which modularization works.
 The encapsulation keeps most relations internal to the
module.
 When relations between modules are reduced to a few
types , then the module becomes reusable.
 If modules can be configured from lower order
modules, then efficiency grows exponentially.
93
Modularization
 Elementary particles can be considered as the lowest
level of modules. All composites are higher level
modules.
 Modularization uses resources efficiently.
 When sufficient resources in the form of reusable
modules are present, then modularization can reach
enormous heights.
 On earth it was capable to generate intelligent
species.
94
Complexity
 Potential complexity of a set of objects is a measure
that is defined by the number of potential relations
that exist between the members of that set.
 If there are n elements in the set,
then there exist n·(n-1) potential relations.
 Actual complexity of a set of objects is a measure that
is defined by the number of relevant relations that
exist between the members of the set.
 Relational complexity is the ratio of the number of
actual relations divided by the number of potential
relations.
95
Relations and interfaces
 Modules connect via interfaces.
 Relations that act within modules are lost to the
outside world of the module.
 Interfaces are collections of relations that are used by
interactions.
 Physics is based on relations. Quantum logic is a set of
axioms that restrict the relations that exist between
quantum logical propositions.
96
Types of physical interfaces
 Interactions run via (relevant) relations.
 Inbound interactions come from the past.
 Outbound interactions go to the future.
 Two-sided interactions are cyclic.
 They take multiple progression steps.
 They are either oscillations or rotations of the interactor.
 Cyclic interactions bind the corresponding modules
together.
97
Modular systems
 Modular (sub)systems consist of connected modules.
 They need not be modules.
 They become modules when they are encapsulated
and offer standard interfaces that makes the
encapsulated system a reusable object.
 All composites are modular systems
98
Binding in sub-systems
 Let 𝜓 represent the renormalized superposition of the
involved distributions.
 𝛻𝜓 = 𝜙 = 𝑚 𝜑
 𝑉 𝜓 2 𝑑𝑉 = 𝑉 𝜑 2 𝑑𝑉 = 1
 𝑉 𝜙 2 𝑑𝑉 = 𝑚2
 𝑚 is the total energy of the sub-system
 The binding factor is the total energy of the sub-
system minus the sum of the total energies of the
separate constituents.
99
Random versus intelligent design
 At lower levels of modularization nature designs
modular structures in a stochastic way.
 This renders the modularization process rather slow.
 It takes a huge amount of progression steps in order to
achieve a relatively complicated structure.
 Still the complexity of that structure can be orders of
magnitude less than the complexity of an equivalent
monolith.
 As soon as more intelligent sub-systems arrive, then
these systems can design and construct modular
systems in a more intelligent way.
 They use resources efficiently.
 This speeds the modularization process in an enormous way.
100
The noise of low dose imaging
Low dose X-ray imaging
Film of cold cathode emission
101
Shot noise
Low dose X-ray image of the moon
102
Shot noise
103
Large scale fluid dynamics
104
Physical fields-1
 SHF wave modulations
 Photon
𝛻𝜓 = 0
 Gluon
𝛻2𝜓 = 0
}

harmonic
𝛻𝜓 = 𝑚𝜑
Energy quanta
𝑛𝑖 𝑒𝑖 𝜓𝑖
𝑖
𝑒𝑖 = ±𝑒
 SHF wave potentials
 Electromagnetic field
 Gravitation field
𝑛𝑖 𝑚𝑖 𝜑 𝑖
𝑖
105
Physical fields-2
 Fields from step stone distributions
 Quaternionic quantum state function
 QPDD


Quaternionic distributions
Charges are preserved
𝛻𝜓 = 𝑚𝜑
106
Inertia-1
 Inertia is implemented via the embedding
continuum
 The embedding continuum is formed by a
curved background field that forms our
living space
107
Inertia-2
Potential fields of
distant particles
 Φ0 =
𝑉
𝜓 dV
In a uniform background:
𝜓 = 𝜌0 𝑟 ; 𝜌0 is constant
Everywhere present
background field
𝜌0
 Φ0 =
𝐺=
𝑉
−𝑐 2 Φ
𝜌0 𝒗
𝑟
dV = 𝜌0
1
𝑉
𝑟
dV = 2π𝑅2 𝜌0
(Dennis Sciama)
; 𝚽 = Φ0 𝒗
𝑐
 𝕰 = 𝛻𝟎 𝚽 + 𝛁Φ0 = 𝚽 + 𝛁Φ0 = Φ0 𝒗
𝑐
𝚽=
𝑉
𝑐𝑟
dV = Φ 𝒗
𝑐
+ 𝛁Φ0
108
Inertia-3
 Φ0 is a scalar background field
 𝜱 is a vector background field
 𝐺 is gravitational constant
 𝕰 = Φ0 𝒗 𝑐 + 𝛁Φ0
 𝕰 ≈ Φ0 𝒗
= 𝐺𝒗
 Acceleration goes together with an extra field 𝕰
 This field counteracts the acceleration
𝑐
109
Inertia-4
 Starting from coupling equation
 𝛻𝜓 = 𝑚𝜑
 𝜓 = χ + χ0 𝒗
 χ represents particle at rest
 𝜓0 = χ0
Small
 𝝍 = χ + χ0 𝒗
 𝛻0 𝝍 = χ0 𝒗 = 𝑚𝝋 − 𝜵𝜓0 − 𝜵× 𝝍
 𝕰 ≡ 𝛻0 𝝍 + 𝜵𝜓0
Represents influence
of distant particles
110
Continuity equation
 Balance equation
 Total change within V
= flow into V + production inside V


𝑑
𝑑𝜏 𝑉
𝜌0 𝑑𝑉 =
𝛻𝜌
𝑉 0 0
𝑑𝑉 =
𝒗
𝒏𝜌0
𝑆
𝑐
𝑉
𝑑𝑆 +
𝛁, 𝝆 𝑑𝑉 +
𝑠
𝑉 0
𝑠
𝑉 0
𝑑𝑉
𝑑𝑉
Gauss
 𝝆 = 𝜌0 𝒗/𝑐
 𝜌 = 𝜌0 + 𝝆
 𝑠 = 𝛻𝜌
 𝑠0 = 2𝛻0 𝜌0 − 𝒗 𝑞 , 𝛁𝜌0 − 𝛁, 𝒗 𝜌0
 𝒔 = 𝛻0 𝒗 + 𝛁𝜌0 +𝜌0 𝛁 × 𝒗 − 𝒗 × 𝛁𝜌0
111
Inversion surfaces


𝑑
𝑑𝜏 𝑉
𝑉
𝜌 𝑑𝑉 +
𝛻 𝜌 𝑑𝑉 =
 The criterion
𝑆
𝑉
𝒏𝜌 𝑑𝑆 =
𝑉
𝑠 𝑑𝑉
𝑠 𝑑𝑉
𝑆
𝒏𝜌 𝑑𝑆=0 divides universe in
compartments
Inversion surface
112
Compartments
universe
Huge
BH
Black holes
Huge BH ⇔ s tart of new episode
BH ⇔ densest packaging
Merge
Compartments
Never ending story
113
History of Cosmology
 Black hole represents natal state of compartment
 Black holes suck all mass from their compartment
 A passivized huge black hole represents start of new
episode of its compartment
 Driving force is enormous mass present outside
compartment ⇒ expansion
 Whole universe is affine space
 Result is never ending story
114
Gravitation
 The Palestra is a curved space
 𝒫𝑏𝑙𝑢𝑟𝑟𝑒𝑑 = ℘𝑠ℎ𝑎𝑟𝑝 ∘ 𝒮𝑠𝑝𝑟𝑒𝑎𝑑
𝜈
 𝑑𝑠 𝑥 = 𝑑𝑠 𝑥 𝑒𝜈 = 𝑑℘ =
 𝑞 𝜇 is quaternion
c dτ
dr
𝜕℘
𝑑𝑥𝜇 = 𝑞 𝜇 𝑥 𝑑𝑥𝜇
𝜇=0…3 𝜕𝑥𝜇
16 partial derivatives
 𝑐 2 𝑑𝑡 2 = 𝑑𝑠 𝑑𝑠 ∗ = 𝑑𝑥02 + 𝑑𝑥12 +𝑑𝑥22 +𝑑𝑥32
 𝑑𝑥02 = 𝑑𝜏 2 = 𝑐 2 𝑑𝑡 2 − 𝑑𝑥12 −𝑑𝑥22 −𝑑𝑥32
 ∆𝑠𝑓𝑙𝑎𝑡 = ∆𝑥0 + 𝒊 ∆𝑥1 + 𝒋 ∆𝑥2 + 𝒌 ∆𝑥3
 ∆𝑠℘ = 𝑞 0 ∆𝑥0 + 𝑞1 ∆𝑥1 + 𝑞 2 ∆𝑥2 + 𝑞 3 ∆𝑥3
Pythagoras
Minkowski
Flat space
Curved space
115
Metric
 𝑑℘ is a quaternionic metric
 It is a linear combination of 16 partial derivatives
 𝑑℘ =
𝜕℘
𝑑𝑥𝜇
𝜕𝑥
𝜇=0…3 𝜇
=
𝜈=0,…3
= 𝑞 𝜇 𝑥 𝑑𝑥𝜇
𝜕℘𝜈
𝑒𝜈
𝑑𝑥𝜇 =
𝜕𝑥𝜇
𝜇=0…3
𝜇
𝑒𝜈 𝑞𝜈 𝑑𝑥𝜇
𝜈=0,…3
𝜇=0…3
 Avoids the need for tensors
116
The primary building blocks
117
Elementary particles
 Coupling equation
 𝛻𝜓 𝑥 = 𝑚 𝜓 𝑦
 𝛻𝜓 𝑥 ∗ = 𝑚 𝜓 𝑦 ∗
 Coupling occurs between
pairs
 {𝜓 𝑥 , 𝜓 𝑦 }
 Colors x, y
 N, R, G, B, R, G, B, W
 Right and left handedness
 R,L
Sign flavors
𝝍⓪ 𝑁 𝐑
𝝍① 𝑅 𝐋
Imaginary
part
𝝍② 𝐺 𝐋
𝝍③ 𝐵 𝐋
𝝍④ 𝐵 𝐑
𝝍⑤ 𝐺 𝐑
𝝍⑥ 𝑅 𝐑
𝝍⑦ 𝑁 L
𝝍⓪
is the
Reference
QPDD
Discrete
symmetries
118
Spin
 HYPOTHESIS : Spin relates to the fact whether the
coupled QPDD is the reference Qpattern 𝝍⓪ .
 Each generation has its own reference QPDD.
 Fermions couple to the reference QPDD 𝝍⓪ .
 Fermions have half integer spin.
 Bosons have integer spin.
 The spin of a composite equals the sum of the spins of
its components.
119
Sign of spin
 The micro-path can be walked in two directions
 This determines the sign of spin
120
Electric charge
 HYPOTHESIS : Electric charge depends on the
number of dimensions in which the discrete symmetry
of Qpattern elements differ from the discrete
symmetry of the embedding field.
 Each sign difference stands for one third of a full
electric charge.
 Further it depends on the fact whether the handedness
differs.
 If the handedness differs then the sign of the count is
changed as well.
121
Color charge
 HYPOTHESIS : Color charge is related to the direction of the




anisotropy of the considered QPDD with respect to the reference
QPDD.
The anisotropy lays in the discrete symmetry of the imaginary parts.
The color charge of the reference QPDD is white.
The corresponding anti-color is black.
The color charge of the coupled pair is determined by the colors of its
members.
 All composite particles are black or white.
 The neutral colors black and white correspond to isotropic QPPDs.
 Currently, color charge cannot be measured.
 In the Standard Model the existence of color charge is derived via the
Pauli principle.
122
Total energy
 Mass is related to the coupling factor of the involved




QPPDs.
It is directly related to the square root of the volume
integral of the square of the local field energy 𝐸.
Any internal kinetic energy is included in 𝐸.
The same mass rule holds for composite particles.
The fields of the composite particles are dynamic
superpositions of the fields of their components.
123
Leptons
Pair
s-type
e-charge c-charge
{𝜓 ⑦ , 𝜓 ⓪ }
fermion
-1
{𝜓 ⓪ , 𝜓 ⑦ }
Antifermion
+1
Handed
ness
SM Name
N
LR
electron
W
RL
positron
124
Quarks
Pair
s-type
e-charge
c-charge
Handedness
SM Name
{𝜓 ① , 𝜓 ⓪ }
fermion
-1/3
R
LR
down-quark
{𝜓 ⑥ , 𝜓 ⑦ }
Anti-fermion
+1/3
R
RL
Anti-down-quark
{𝜓 ② , 𝜓 ⓪ }
fermion
-1/3
G
LR
down-quark
{𝜓 ⑤ , 𝜓 ⑦ }
Anti-fermion
+1/3
G
RL
Anti-down-quark
{𝜓 ③ , 𝜓 ⓪ }
fermion
-1/3
B
LR
down-quark
{𝜓 ④ , 𝜓 ⑦ }
Anti-fermion
+1/3
B
RL
Anti-down-quark
{𝜓 ④ , 𝜓 ⓪ }
fermion
+2/3
B
RR
up-quark
{𝜓 ③ , 𝜓 ⑦ }
Anti-fermion
-2/3
B
LL
Anti-up-quark
{𝜓 ⑤ , 𝜓 ⓪ }
fermion
+2/3
G
RR
up-quark
{𝜓 ② , 𝜓 ⑦ }
Anti-fermion
-2/3
G
LL
Anti-up-quark
{𝜓 ⑥ , 𝜓 ⓪ }
fermion
+2/3
R
RR
up-quark
{𝜓 ① , 𝜓 ⑦ }
Anti-fermion
-2/3
R
LL
Anti-up-quark
125
Reverse quarks
Pair
s-type
e-charge
c-charge
Handedness
SM Name
{𝜓 ⓪ , 𝜓 ① }
fermion
+1/3
R
RL
down-r-quark
{𝜓 ⑦ , 𝜓 ⑥ }
Anti-fermion
-1/3
R
LR
Anti-down-r-quark
{𝜓 ⓪ , 𝜓 ② }
fermion
+1/3
G
RL
down-r-quark
{𝜓 ⑦ , 𝜓 ⑤ }
Anti-fermion
-1/3
G
LR
Anti-down-r-quark
{𝜓 ⓪ , 𝜓 ③ }
fermion
+1/3
B
RL
down-r-quark
{𝜓 ⑦ , 𝜓 ④ }
Anti-fermion
-1/3
B
LR
Anti-down-r_quark
{𝜓 ⓪ , 𝜓 ④ }
fermion
-2/3
B
RR
up-r-quark
{𝜓 ⑦ , 𝜓 ③ }
Anti-fermion
+2/3
B
LL
Anti-up-r-quark
{𝜓 ⓪ , 𝜓 ⑤ }
fermion
-2/3
G
RR
up-r-quark
{𝜓 ⑦ , 𝜓 ② }
Anti-fermion
+2/3
G
LL
Anti-up-r-quark
{𝜓 ⓪ , 𝜓 ⑥ }
fermion
-2/3
R
RR
up-r-quark
{𝜓 ⑦ , 𝜓 ① }
Anti-fermion
+2/3
R
LL
Anti-up-r-quark
126
W-particles
{𝜓 ⑥ , 𝜓 ① }
boson
-1
RR
RL
𝑊−
{𝜓 ① , 𝜓 ⑥ }
Anti-boson
+1
RR
LR
𝑊+
{𝜓 ⑥ , 𝜓 ② }
boson
-1
RG
RL
𝑊−
{𝜓 ② , 𝜓 ⑥ }
Anti-boson
+1
GR
LR
𝑊+
{𝜓 ⑥ , 𝜓 ③ }
boson
-1
RB
RL
𝑊−
{𝜓 ③ , 𝜓 ⑥ }
Anti-boson
+1
BR
LR
𝑊+
{𝜓 ⑤ , 𝜓 ① }
boson
-1
GG
RL
𝑊−
{𝜓 ① , 𝜓 ⑤ }
Anti-boson
+1
GG
LR
𝑊+
{𝜓 ⑤ , 𝜓 ② }
boson
-1
GG
RL
𝑊−
{𝜓 ② , 𝜓 ⑤ }
Anti-boson
+1
GG
LR
𝑊+
{𝜓⑤ , 𝜓③ }
boson
-1
GB
RL
𝑊−
{𝜓 ③ , 𝜓 ⑤ }
Anti-boson
+1
BG
LR
𝑊+
{𝜓 ④ , 𝜓 ① }
boson
-1
BR
RL
𝑊−
{𝜓 ① , 𝜓 ④ }
Anti-boson
+1
RB
LR
𝑊+
{𝜓 ④ , 𝜓 ② }
boson
-1
BG
RL
𝑊−
{𝜓 ② , 𝜓 ④ }
Anti-boson
+1
GB
LR
𝑊+
{𝜓 ④ , 𝜓 ③ }
boson
-1
BB
RL
𝑊−
{𝜓 ③ , 𝜓 ④ }
Anti-boson
+1
BB
LR
𝑊+
127
Z-particles
Pair
s-type
e-charge
c-charge
Handedness
SM Name
{𝜓 ② , 𝜓 ① }
boson
0
GR
LL
Z
{𝜓 ⑤ , 𝜓 ⑥ }
Anti-boson
0
GR
RR
Z
{𝜓 ③ , 𝜓 ① }
boson
0
BR
LL
Z
{𝜓 ④ , 𝜓 ⑥ }
Anti-boson
0
RB
RR
Z
{𝜓 ③ , 𝜓 ② }
boson
0
BR
LL
Z
{𝜓 ④ , 𝜓 ⑤ }
Anti-boson
0
RB
RR
Z
{𝜓 ① , 𝜓 ② }
boson
0
RG
LL
Z
{𝜓 ⑥ , 𝜓 ⑤ }
Anti-boson
0
RG
RR
Z
{𝜓 ① , 𝜓 ③ }
boson
0
RB
LL
Z
{𝜓 ⑥ , 𝜓 ④ }
Anti-boson
0
RB
RR
Z
{𝜓 ② , 𝜓 ③ }
boson
0
RB
LL
Z
{𝜓 ⑤ , 𝜓 ④ }
Anti-boson
0
RB
RR
Z
128
Neutrinos
type
s-type
e-charge
c-charge
Handedness
SM Name
{𝜓 ⑦ , 𝜓 ⑦ }
fermion
0
NN
RR
neutrino
{𝜓 ⓪ , 𝜓 ⓪ }
Anti-fermion
0
WW
LL
neutrino
{𝜓 ⑥ , 𝜓 ⑥ }
boson?
0
RR
RR
neutrino
{𝜓 ① , 𝜓 ① }
Anti- boson?
0
RR
LL
neutrino
{𝜓 ⑤ , 𝜓 ⑤ }
boson?
0
GG
RR
neutrino
{𝜓 ② , 𝜓 ② }
Anti- boson?
0
GG
LL
neutrino
{𝜓 ④ , 𝜓 ④ }
boson?
0
BB
RR
neutrino
{𝜓 ③ , 𝜓 ③ }
Anti- boson?
0
BB
LL
neutrino
129
Color confinement
The color confinement rule forbids the
generation of individual particles that
have non-neutral color charge
130
Color confinement
 Color confinement forbids the generation of
individual quarks
 Quarks can appear in hadrons
 Color confinement blocks observation of
gluons
131
Photons & gluons
type
s-type
e-charge
c-charge
Handedness
SM Name
{𝜓 ⑦ }
boson
0
N
R
photon
{𝜓 ⓪ }
boson
0
W
L
photon
{𝜓 ⑥ }
boson
0
R
R
gluon
{𝜓 ① }
boson
0
R
L
gluon
{𝜓 ⑤ }
boson
0
G
R
gluon
{𝜓 ② }
boson
0
G
L
gluon
{𝜓 ④ }
boson
0
B
R
gluon
{𝜓 ③ }
boson
0
B
L
gluon
132
Photons & gluons
 Photons and gluons are NOT particles
 Ultra-high frequency waves are constituted
by wave fronts that at every progression step
are emitted by elementary particles
 Photons and gluons are modulations of
ultra-high frequency carrier waves.
133
Fundamental particles
 Due to color confinement some elementary
particles cannot be created as individuals
 Quarks can only be created combined in
hadrons
 Fundamental particles form a category of
particles that are created in one integral action
 The color charge of fundamental particles is
neutral
134
135
Dual space distributions
 A subset of the (quaternionic) distributions have the




same shape in configuration space and in the linear
canonical conjugated space.
We call them dual space distributions
These are functions that are invariant under Fourier
transformation.
The Qpatterns and the harmonic and spherical
oscillations belong to this class.
Fourier-invariant functions show iso-resolution, that
is, ∆p = ∆q in the Heisenberg’s uncertainty relation.
136
Why has nature a preference?
 Nature seems to have a preference for this class of
quaternionic distributions.
 A possible explanation is the two-step generation
process, where the first step is realized in
configuration space and the second step is realized in
canonical conjugated space.
 The whole pattern is generated two-step by two-step.
 The only way to keep coherence between a distribution
and its Fourier transform that are both generated step
by step is to generate them in pairs.
137
Conclusion
 Fundament
 Quantum logic
 Book model
 Correlation vehicle
 Main features
 Fundamentally countable ⇛ Quanta
 Embedded in continuum ⇛ Fields
 Fundamentally stochastic ⇛ Quantum Physics
 Palestra is curved
⇛ Quaternionic “GR”
 Quaternionic metric
}
138
Conclusion
 Contemporary physics works (QED, QCD)
 But cannot explain fundamental features
 Origin of dynamics
 Space curvature
 Inertia
 Existence of Quantum Physics
 What photons are
139
End
 Physics made its greatest misstep in the
thirties when it turned away from the
fundamental work of Garret Birkhoff and
John von Neumann.
 This deviation did not prohibit pragmatic
use of the new methodology.
 However, it did prevent deep understanding
of that technology because the
methodology is ill founded.
140
Lattices,
classical logic and
quantum logic
141
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.
142
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
143
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
144
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
145
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
146
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.
147
Hilbert space
The set of closed subspaces of an
infinite dimensional separable
Hilbert space forms an
orthomodular lattice
Is lattice isomorphic to quantum
logic
148
Hilbert logic
Back
 Add linear propositions
 Linear combinations of atomic propositions
 Add relational coupling measure
 Equivalent to inner product of Hilbert space
 Close subsets with respect to realational coupling
measure
 Propositions ⇔ subspaces
 Linear propositions ⇔ Hilbert vectors
149
Superposition principle
Linear combinations of linear
propositions are again linear
propositions that belong to the same
Hilbert logic system
150
Isomorphism
Lattice isomorhic
Propositions ⇔ closed subspaces
Topological isomorphic
Linear atoms ⇔ Hilbert base
vectors
151