Alexey Nenashev Institute of Semiconductor Physics ...

Theory of physical structures:

hidden symmetries in some basic laws of general physics

Alexey Nenashev

Institute of Semiconductor Physics

Novosibirsk, Russia

Theory of Physical Structures

is an unusual point of view on

“usual” laws of physics and

geometry, such as

• Newton’s second law: F = ma ,

• Ohm’s law: U = IR ,

• distance between two points:

rab2 = (xa-xb)2 + (ya-yb)2 + (za-zb)2,

etc.

Yuri Kulakovsuggested this theory in 1960s

Ohm’s law

Battery(voltage U)

Resistor(resistance R)

Current I = U / R

Ohm’s law

Battery(voltage U)


Current I = U / RWhat does one need to discover the Ohm’s law?

Ohm’s law

Battery(voltage U)



1. many different batteries

Ohm’s law

Battery(voltage U)




2. many different resistors

Ohm’s law

Battery(voltage U)




2. many different resistors

3. ammeter(well-calibrated)

How to proceed to discover the Ohm’s law?

1. select some “reference” battery and “reference” resistor

I

U



2. get the voltage U for each battery by connecting it to the “reference” resistor and measuring the current

A

ref.

ref.

G




3. get the conductance G for each resistor by connecting it to the “reference” battery and measuring the current

AI

U

G

I




4. connect different batteries to different resistors, and discover that the current I is proportional to U and to G I = U G.

3. get the conductance G for each resistor by connecting it to the “reference” battery and measuring the current

A

U

I I

G

NWhat if the ammeter is not calibrated(and we don’t know how to calibrate it)?

N

What if the ammeter is not calibrated(and we don’t know how to calibrate it)?

N

“conductance”

“voltage”

“cu

rren

t”“c

urr

ent”

What if the ammeter is not calibrated(and we don’t know how to calibrate it)?

Nonlinear dependencies not clear whether there is a linear law behind them

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1 2 3 4 5

1

2

3

4

5

Bat

teri

esResistors

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21 22 23 24 25

31 32 33 34 35

41 42 43 44 45

51 52 53 54 55

1 2 3 4 5

1

2

3

4

5

Bat

teri

esResistors

11 12 21 22, , ,

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21 22 23 24 25

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41 42 43 44 45

51 52 53 54 55

1 2 3 4 5

1

2

3

4

5

Bat

teri

esResistors

11 13 21 23, , ,

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41 42 43 44 45

51 52 53 54 55

1 2 3 4 5

1

2

3

4

5

Bat

teri

esResistors

11 14 21 24, , ,

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21 22 23 24 25

31 32 33 34 35

41 42 43 44 45

51 52 53 54 55

1 2 3 4 5

1

2

3

4

5

Bat

teri

esResistors

11 12 31 32, , ,

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51 52 53 54 55

1 2 3 4 5

1

2

3

4

5

Bat

teri

esResistors

, , ,i i j j

, , ,i i j j

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41 42 43 44 45

51 52 53 54 55

1 2 3 4 5

1

2

3

4

5

Bat

teri

esResistors

3D hyper-surface in 4D space

, , ,i i j j

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51 52 53 54 55

1 2 3 4 5

1

2

3

4

5

Bat

teri

esResistors

, , , , , , 0i i j ji j


, , ,i i j j

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21 22 23 24 25

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1 2 3 4 5

1

2

3

4

5

Bat

teri

esResistors

, , , , , , 0i i j ji j


Two batteries and two resistors → a functional dependence between 2×2 measured quantities. This is a physical structures of rank (2,2).

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1 2 3 4 5

1

2

3

4

5

I I I I I

I I I I I

I I I I I

I I I I I

I I I I I

Bat

teri

esResistors

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21 22 23 24 25

31 32 33 34 35

41 42 43 44 45

51 52 53 54 55

1 2 3 4 5

1

2

3

4

5

I I I I I

I I I I I

I I I I I

I I I I I

I I I I I

Bat

teri

esResistors

0i i i i

j j j j

I I U G U G

I I U G U G

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41 42 43 44 45

51 52 53 54 55

1 2 3 4 5

1

2

3

4

5

I I I I I

I I I I I

I I I I I

I I I I I

I I I I I

Bat

teri

esResistors

0i i i i

j j j j

I I U G U G

I I U G U G

i if U G

1 1

1 1

( ) ( ), , ,

( ) ( )

f a f ba b c d

f c f d

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21 22 23 24 25

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41 42 43 44 45

51 52 53 54 55

1 2 3 4 5

1

2

3

4

5

I I I I I

I I I I I

I I I I I

I I I I I

I I I I I

Bat

teri

esResistors

0i i i i

j j j j

I I U G U G

I I U G U G

i if U G

1 1

1 1

( ) ( ), , ,

( ) ( )

f a f ba b c d

f c f d

, , , , , , 0i i j ji j

“conductance” g

“voltage” u

0

0u

0g

,u g

1

“conductance” g

“voltage” u

0

0u

0g

,u g

1

1u 1u 2u2u

1g

2g

1g2g3g

“conductance” g

“voltage” u

0

1

1u 1u 2u0u2u

0g

1g

2g

1g2g

,u g

3g

“conductance” g

“voltage” u

0

1

1u 1u 2u0u2u

0g

1g

2g

1g2g

,u g

3g

2

“conductance” g

“voltage” u

0

1

1u 1u 2u0u2u

0g

1g

2g

1g2g

,u g

2 3

3g

“conductance” g

“voltage” u

0

1

1u 1u 2u0u2u

0g

1g

2g

1g2g

,u g

2 3

,k n k nu g

3g

“conductance” g

“voltage” u

0

1

1u 1u 2u0u2u

0g

1g

2g

1g2g

,u g

2 3

,k n k nu g

k

kU u e

n

nG g e

m

mI e

3gLet us define U, G and I as

“conductance” g

“voltage” u

0

1

1u 1u 2u0u2u

0g

1g

2g

1g2g

,u g

2 3

,k n k nu g

k

kU u e

n

nG g e

m

mI e

I U G

3gLet us define U, G and I as

F = ma: law or definition?

i iF m a usual formulation of Newton’s second law

F = ma: law or definition?

i iF m a

0i i

j j

a a

a a

usual formulation of Newton’s second law

Newton’s second law as a physical structure of rank (2,2)

Battery(e.m.f. Ui and inner

resistance ri)

Resistor(resistance Rα)

i

ii

i

UI

r R

Once again about Ohm’s law: physical structure of rank (2,3)


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1

2

3

4

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Bat

teri

es

Resistors

, , ,i i j j

points are everywhere in 4D space no physical structure of rank (2,2)

ii i

i

Uf I f

r R


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1

2

3

4

5

Bat

teri

es

Resistors

, , , , ,i i i j j j

ii i

i

Uf I f

r R

All points are in a 5D hyper-surface of 6D space (why?). This is a physical structure of rank (2,3).


1

1

1

0

0

0 0 1

i i i

j j j

U r U

M U r U

2

1 1 1

1 1 1

M R R R

ii

i

UI

r R


1

1

1

0

0

0 0 1

i i i

j j j

U r U

M U r U

2

1 1 1

1 1 1

M R R R

ii

i

UI

r R

1 1 1

1 1 1

1 2

1 1 1

i i i

j j j

I I I

M M I I I


1

1

1

0

0

0 0 1

i i i

j j j

U r U

M U r U

2

1 1 1

1 1 1

M R R R

ii

i

UI

r R

1 1 1

1 1 1

1 2

1 1 1

i i i

j j j

I I I

M M I I I

1 1 1

1 1 1

1 2det det 0 0

1 1 1

i i i

j j j

I I I

M M I I I

six quantities are not independent of each other


1

1

1

0

0

0 0 1

i i i

j j j

U r U

M U r U

2

1 1 1

1 1 1

M R R R

ii

i

UI

r R

1 1 1

1 1 1

1 2

1 1 1

i i i

j j j

I I I

M M I I I

1 1 1

1 1 1

1 2det det 0 0

1 1 1

i i i

j j j

I I I

M M I I I

ii

i

Uf

r R

, , , , ,i i i j j j

Examples of functional equations

equation solution

( ) ( ) ( )f x y f x f y ( )f x C x

( ) ( ) ( )f x y f x f y ( ) expf x C x

( ) ( ) ( ) ( ) ( )

0 ( ) ( ) for 0 1

f x y f x f y g x g y

x f x g x x x

( ) cos , ( ) sinf x x g x x

1 1 1 2

2 1 2 2

( , ), ( , ),

0( , ), ( , )

x x

x x

( , )

or, equivalently,

( , )

x f x

x f x

Functional equations of the Theory of physical structures

equation solution

, ,

, 0

i i

j j

or, equivalently,

i i

i i

f x

f x

, , ,

, , 0

i i i

j j j

i i if x y

, , , ,

, , , 0

i i i i

j j j j

or

i i i

i i i

f x y

f x y

i ii

i

x yf

z

rank

(2,2)

(2,3)

(2,4)

(3,3)

, , ,

, , ,

, , 0

i i i

j j j

k k k

Possible ranksand solutions

(found by G. Mikhailichenko in 1970s)

(2,2) (2,3) (2,4)

(3,2) (3,3) (3,4)

(4,2) (4,3) (4,4) (4,5)

(5,6)(5,4) (5,5)

(6,6) (6,7)(6,5)

(7,7)(7,6)

i if x i i if x y

i i i

i i i

f x y

f x y

i ii

i

x yf

z

Some works on theory of physical structures

Yu. I. Kulakov, Theory of physical structures, Journal of Soviet Mathematics, v. 27, pp. 2616-2646 (1984) (review of results and possible applications).

G. G. Mikhailichenko, A problem in the theory of physical structures, Siberian Mathematical Journal, v. 18, pp. 951-961 (1977) (proofs for ranks (2,2), (2,3), (2,4)).

G. G. Mikhailichenko, On a functional equation with two-index variables, Ukrainian Mathematical Journal, v. 25, pp. 490-497 (1973) (proofs for ranks (3,3), (3,4)).

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