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Particles and Wavesin Conceptual Time Systems
Karl Erich Wolff
Mathematics and Science Faculty
University of Applied Sciences Darmstadt
Ernst Schröder Center for Conceptual Knowledge Processing
Research Group Concept Analysis at
Darmstadt University of Technology
1. Introduction: Problems in Physics
2. Temporal Concept Analysis
3. Examples of Conceptual Time Systems
4. Time Dimension and Branching Life Tracks
5. Object Identification: Conceptual Representation of a Tale
6. Particles and Waves
Outline
1. Reality – Observation - Theory
2. Formal Representations: Numbers, Scales, and Systems
3. Continuity and Discreteness
4. Granularity, States, and Situations
5. Transitions and Life Tracks
6. Objects, Particles, and Waves
Problems in Physics
Theories of Space and Time
•Aristoteles: continuum, point of time, duration, time as a category
•Classical Physics: example: x“(t)=a; x(t)=x(0) + v t + ½at²
•Special and General Theory of Relativity: Space-Time (curved)
•Quantum Theory: Time-dependent Schrödinger function ψ(t)
•Automata theory: States, Transitions (without an explicit time description)
•Mathematical System Theory: State? System?
•Quantum Gravity: Looking for a theory with an appropriate time description
Einstein’s Granularity Remark
Albert Einstein:
„Zur Elektrodynamik bewegter Körper“
Annalen der Physik 17 (1905): 891-921
Footnote on page 893:
„Die Ungenauigkeit, welche in dem Begriff der Gleichzeitigkeit zweier Ereignisse an (annähernd) demselben Orte steckt und gleichfalls durch eine Abstraktion überbrückt werden muß, soll hier nicht erörtert werden.“
A General Theory for Objects in Space and Time ...
• should cover discrete and continuous descriptions
• needs a theory of granularity
• needs a suitable notion of ‚system‘ and a notion of ‚states‘ depending on the granularity
• transitions: time-dependent changes between states
• objects as subsystems (and not as „atoms“)
• should construct a temporal logic based on these temporal systems
Conceptual Time Systems with Objects
and a Time Relation (CTSOT)
g
h
Object 2
Object 1
i
j
Time part T Event part C
Time scales Event scales
v w
K(C)
time states states
situationsobject concepts:
derived context K(T) |
Harmonic Oscillator: State Space
y x
Harmonic Oscillator: Situation Space
Stars, Morning Star, Evening Star, Venus
morning eveningMonday
east west
Stars, Morning Star, Evening Star, Venus
morning eveningTuesday
east west
Morning Star Evening Star
morning evening
Monday
east west
MondayTuesday
Tuesday
Venus
morning evening
Monday
east west
MondayTuesday
Tuesday
Venus: Morning Star and Evening Star
luminous
west
eveningmorning
east
TuesdayMonday
4321
Mo Tu mo ev ea we lu
1
2
3
4
Life Tracks: Morning Star, Evening Star
luminous
west
eveningmorning
east
TuesdayMonday
4321
Example: A Journey of John and Mary
Using the order dimension we define:
• The time dimension is the order dimension of the concept lattice of the time part
• The time scale dimension is the order dimension of the time scale
• These dimensions may be greater than one!
• Are life tracks 1-dimensional ?
Time Dimension = 1 ???
Branching Life Tracks?Unique-State-Lemma:A conceptual time system is at each time granule in exactly one situation, in exactly one state, and in exactly one time state.
But branching of life tracks is possible!Example: We send an „abstract letter“ in two copies.
Time Relation
Day Town
(letter,0) 0 A
(letter,1) 1 B
(letter,2) 2 C
(letter,3) 3 D
A
B
C
D
Two Gymnasts: Roll and Jump
The Waves of Rolls and Jumps
Wave of a Population
Rigid Body ?
Translation
Rotation
Reflection
Indistinguishable Objects ?
An observer of a temporal system who describes that system by aCTSOT has to decide how to choose the objects of the CTSOT.
Like tennis balls they may be indistinguishable with repectto some parts of the chosen description.
In the following German taleDer Wettlauf zwischen dem Hasen und dem Igel(The Race Between the Hare and the Hedgehog)the hare cannot distinguish the hedgehog and the hedgehog‘s wife.
Hare and Hedgehog: partial data
actual objects object timehare-object
place running
hedgehog0 hedgehog 0 hedgehog house no
hh-wife0 hh-wife 0 hedgehog house no
hedgehog1 hedgehog 1 hedgehog field no
Hare and Hedgehog: State Space
hedgehog
hare
hh-wife
Hare and Hedgehog: Situation Space
The Hare‘s View: Situation Space
Thank you!
www.fbmn.fh-darmstadt.de/home/wolff/index.htm
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