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