The Unification of Probabilistic and Geometric …vtpapers.ru/Papers/OntologyAndPLApresentation.pdfThe Unification of Probabilistic and Geometric Languages in Description of Nature

International Conference “Ontological studies in modern Russia” 18-19 November 2010, St. Petersburg, Russia

The Unification of Probabilistic and Geometric

Languages in Description of Nature

the Principle of Least Action, the Path Integral Method, and the

Transition from Possible to Actual Existence

Vladislav Terekhovich

Department of Philosophy of Science and Technology

Saint-Petersburg State University, Russia

INTRODUCTION

In XX century physicists have tried to construct a system of laws of being without aid of modern philosophers. The physicists refer to Pythagoras, Plato, Aristotle, Leibniz, and Kant. It is time for the modern philosophers to join this work.

G. Leibniz (about God): The most economical way He ordered place, space, and time; using the simplest means He made the greatest action ("Principles of Nature and Grace, Based on Reason")

I. Kant: The private laws of nature are subordinate to more general laws, and saving of principles is not only the principle of frugality of mind, but also the internal law of nature ("Critique of Pure Reason")

A. Einstein: In our time, a physicist has to deal with philosophical problems to a much greater extent than previous generations of physicists had to do it ("The Evolution of Physics")

W. Heisenberg: Modern physics is advancing on the same way, in which Plato and the Pythagoreans went ("At the Origin of quantum Theory")

K. Popper: Theory always precedes observations ("Objective Knowledge")

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ONTOLOGICAL PROBLEMS OF PHYSICS

I. The same phenomena of MACRO- and MEGA-worlds can be

described by several geometric languages, which are based

on contradictory axioms

II. The geometric languages of MACRO- and MEGA-worlds

contradict the probabilistic language of MICRO-world

Perhaps, a solution exists!

Simultaneously ... it sheds light on another

ontological problem ...

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THE GOAL IS …

… a describing the model uniting the principle of least action and Feynman’s path integral method, and the way to explain how:

• the probabilistic language of MICRO-world generalizes geometric languages of MACRO-world and MEGA-worlds (classical mechanics and general relativity are special cases of quantum mechanics)

• quantum mechanics shows a general ontological process, under which existence of any object changes from its possible mode to its actual mode

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THE CENTRAL IDEA

The ontological problem

possible existence → actual existence

has a particular physical and mathematical solution

based on the principle of least action and

the quantum path integral method (Richard Feynman, Nobel Prize in Physics, 1965)

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OUTLINE

1. Four ways to describe a flying apple

2. Approaches to quantum mechanics

3. Feynman’s approach to quantum mechanics

4. Feynman’s approach and the flying apple

5. Unification of the descriptions of the flying apple

6. A new view of reality: possible existence turns to actual existence

7. Philosophical foundations

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THE PRINCIPLE OF LEAST ACTION (PLA)

P. Maupertuis formulated PLA in 1744

L. Euler translated PLA into a mathematical form

J. Lagrange, J. D’Alembert, W. Hamilton, C. Gauss, H. Helmholtz and others improved PLA for different areas of physics

A. Einstein: the whole general relativity can be developed based on PLA

M. Planck: PLA dominates above all reversible phenomena of physics (including the Laws of Conservation of Energy and Momentum )

A. Eddington: there are two great generalizations of nature, PLA and the Second Law of Thermodynamics

T. Moore: PLA lies at the core of much of contemporary theoretical physics (Macmillan Encyclopedia of Physics, 1996, Vol. 2, p. 840)

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FLYING APPLE. Description-1

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• The apple has a mysterious internal “tendency” to move straight with constant speed (inertia). If the apple “feels” the effect of external forces, it is accelerated or changes its motion (the vector of velocity) for each small time interval

• A force effects not locally, its value depends on the distance from the center of the Earth and the weight of the apple

Newtonian mechanics

forces act at a distance and

body has inertia

y

x

Fig. 1

Force = Mass x Acceleration

Initial position

Final position

Velocity

FLYING APPLE. Description-2

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• The value of the force is proportional to the speed of the field potential decrease. The vector of force is antigradient of potential energy

• The apple “probes” space around itself and rushes along the path where the potential of gravitational field is minimum

Field theory

potentials in space

and inertia

Fig. 2

Force ≈ max speed of decrease field potential of the Earth

Field potential-1

Field potential-2

Field potential-3

Field potential-3 less than 2 and 1

Initial position

Fnal position

FLYING APPLE. Description-3a

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• Along the actual path the difference between average kinetic energy (KE) and potential energy (PE) reaches a minimum compared with all possible paths

Lagrangian mechanics

the principle of

least or stationary action

Fig. 3

The action along the actual path = integral of the difference (KE – PE) = minimum

Initial position or time

Final position or time

The actual path

The virtual path

FLYING APPLE. Description-3b

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• We do not need to know what is happening in neighboring moment of time or points of space, we need to know only the initial and final position in space and time

• PLA arose from the optical-mechanical analogy with Fermat’s principle, according to which a ray of light moves along the path takes the least time

• Differential equations of body’s motion in a gravitational and other fields (Euler-Lagrange equations) are derived from PLA

• The way predicted by PLA exactly coincides with calculations of the description-1 and description-2

Lagrangian mechanics

the principle of

least or stationary action

FLYING APPLE. Description-4a

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• There are not forces or the field of gravity, there is only a geometric 4D space-time curved under the influence of Earth’s mass

• The apple moves by inertia along the shortest world-line or geodesic

General relativity theory

the principle of maximum proper time

or motion along a

geodesic

Fig. 4

FLYING APPLE. Description-4b

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• "Wristwatches" of twins, moving along different world-lines, show various times

• The actual world-line between two events differs from all possible ones that a value of body’s proper time is maximum or stationary

• For weak field of gravity and low velocities it is reduced to the principle of least action (see Description-3a)

Fig. 5

The virtual path

Proper time along actual path = maximum

Final event

Initial event

General relativity theory

the principle of maximum proper time

or motion along a

geodesic

The actual path

THIS WEIRD PHYSICS

The same result with a similar accuracy can be obtained in four ways …

Each oh them based on different understanding of reality-on its own ontology.

Which one is true?

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CRITERIA FOR SELECTION OF THE THEORIES

1. More unification, fewer axioms and assumptions. The theory includes other as a special case.

2. More explanatory and predictive power.

3. More correspondence with experience (verification/falsification).

4. Simplicity of definition and mathematical notation, internal consistency and harmony.

5. Conformity existing scientific and philosophical paradigm, including its basic principles.

And what happens in the micro world?

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THE VIEWS ON QUANTUM BEHAVIOR

N. Bohr: the observing creates a reality

V. Heisenberg: the potential semi reality goes to observed reality

A. Einstein: the reality does not depend on measurement , there are some hidden parameters

D. Blokhintsev: the reality is a statistical ensemble of states independent of the observer

D. Bohm: the interaction extracts objects from uncertain state in the whole Universum

V. Fock: objective potential possibilities are realized after the measurement, the probable description is primary, the deterministic description is secondary

H. Everett: every measurement divides particles into all possible copies, which actually exists in their own parallel world or branch

K. Popper: probability or propensity is a real property of object

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FEYNMAN’S VIEW

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Quantum mechanics

Feynman’s

path integral method

• Two slit experiment: a particle moves simultaneously along all possible

paths (in coherent superposition), the maximum probability corresponds to the actual path

• The quantum amplitudes near the actual path are nearly in one phase, they reinforce each other and generate significant effect observable as real. Other ways exist too, but they are not observed, more precisely, their probability are very small to be observed

Fig. 6

FEYNMAN’S VIEW

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Quantum mechanics

Feynman’s

path integral method

• Each photon "flew" on the Moon, but we observe the only path, which probability is maximum, it is the geodesic line

• R. Feynman: the particle "explores" or "smells" all neighboring path and chooses one, along which the action is minimum ("Feynman’s Lectures on Physics“)

• Result coincides with the method of Heisenberg or Schrödinger

Fig. 7

The probability of the actual path = maximum Initial state

Final state

The possible path

The actual path

foton

Moon

FLYING APPLE. Description-5

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• The apple, as well as a photon "flew“ simultaneously along all possible paths or world-lines between initial and final states, including the Moon, but we observe the only path, which probability is maximum, it is the geodesic line

• The narrow bundle apple’s paths (world-lines) near the actual path coincides with the path predicted by Classical Mechanics and General Relativity

Quantum mechanics

Feynman’s

path integral method

Рис. 8

The probability of the actual path = maximum

The action of the actual path = minimum

Initial state

Final state

The possible path

The actual path

Moon

UNIFICATION OF DESCRIPTIONS-a

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Nature’s command for the apple:

• Follow the path of least action!

• Follow the path of maximum aging (or maximum proper time)!

• Explore all paths! (Taylor E.F. A call to action. American Journal of Physics. 71 (5), 2003)

Principle of

Maximal Aging

Principle of

Least Action

Principle

“Explore all paths”

RELATIVITY

(Follow world-line of maximal aging)

NEWTONIAN

MECHANICS

(Follow world-line

of Least Action)

QUANTUM

MECHANICS (Follows all world-lines.)

As particle mass increases,

the pencil of effective world-

lines narrows to the world-line

of minimum action

Low speed

Weak gravity

Fig. 9

UNIFICATION OF DESCRIPTIONS-b

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If we propose that the principle of maximum aging is the limiting case of Feynman’s principle “Explore all paths” for strong gravity (and if superstring theory is true), the basic physical theories can be represented as approximations and limiting cases of the quantum laws

Principle of

Maximal Aging

Principle of

Least Action

Principle

“Explore all paths”

RELATIVITY

(Follow world-line of maximal aging)

NEWTONIAN

MECHANICS

(Follow world-line

of Least Action)

QUANTUM

MECHANICS (Follow all world-lines.)

As particle mass increases,

the pencil of effective world-

lines narrows to the world-

line of minimum action

Low speed

Weak gravity

Fig. 10

Under the influence

of massive object the

pencil of effective

world-lines narrows

and curves to the

world-line of

maximum aging

A NEW VIEW ON REALITY

Possible paths of quantum particles and

possible trajectories of classical and relativistic objects are

nor imaginary, nor fictitious.

They are real!

They differ from actual or observed paths because the probability of their detection is not the maximal!

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ADVANTAGES OF ASSOCIATION THE PRINCIPLE OF LEAST ACTION AND FEYNMAN’S VIEW

• New language suitable to the description of micro-, macro-, and mega-objects, any speed, space of any dimension, and fields of any strength.

• It connects the probabilistic and deterministic causality.

• The concepts of “force” and “inertia” are replaced by the phases of quantum amplitudes.

• It uses two powerful mathematical formalisms-the calculus of variations and path integral method.

There is some disadvantages…

• first of all, our common sense protests against the explanation where any classical objects “explore” all possible paths like quantum particles.

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PHILOSOPHY

The new language includes some philosophical foundations.

• All objects in the Universe obey the same simple laws.

• Mathematics describes the actual processes in the Universe.

• There are two modes of existence that turn each other: the potential and actual. The realm of possibility is not less real than the realm of actuality.

• Probabilistic behavior is more fundamental than deterministic one.

• Space is the result of interaction of objects.

• Every object has an internal activity.

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CENTRAL IDEA

The ontological problem

possible existence → actual existence

has a particular physical and mathematical solution

based on the principle of least action and

the quantum path integral method (Richard Feynman, Nobel Prize in Physics, 1965)

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THANK YOU!

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www.vtpapers.ru

ResearcherID: O-1251-2013

Published in “Ontological Studies-I. Ontological studies in modern Russia. 18-20

November 2010. St. Petersburg, Russia”. Saint-Petersburg National Research University of Information Technologies, Mechanics and Optics. p. 123-132.

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