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 / RWhat does one need to discover the Ohm’s law?
Ohm’s law
Battery(voltage U)
Resistor(resistance R)
Current I = U / RWhat does one need to discover the Ohm’s law?
1. many different batteries
Ohm’s law
Battery(voltage U)
Resistor(resistance R)
Current I = U / RWhat does one need to discover the Ohm’s law?
1. many different batteries
2. many different resistors
Ohm’s law
Battery(voltage U)
Resistor(resistance R)
Current I = U / RWhat does one need to discover the Ohm’s law?
1. many different batteries
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
How to proceed to discover the Ohm’s law?
1. select some “reference” battery and “reference” resistor
2. get the voltage U for each battery by connecting it to the “reference” resistor and measuring the current
A
ref.
ref.
G
How to proceed to discover the Ohm’s law?
1. select some “reference” battery and “reference” resistor
2. get the voltage U for each battery by connecting it to the “reference” resistor and measuring the current
3. get the conductance G for each resistor by connecting it to the “reference” battery and measuring the current
AI
U
G
I
How to proceed to discover the Ohm’s law?
1. select some “reference” battery and “reference” resistor
2. get the voltage U for each battery by connecting it to the “reference” resistor and measuring the current
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
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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2
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Bat
teri
esResistors
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1 2 3 4 5
1
2
3
4
5
Bat
teri
esResistors
11 12 21 22, , ,
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1 2 3 4 5
1
2
3
4
5
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teri
esResistors
11 13 21 23, , ,
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1 2 3 4 5
1
2
3
4
5
Bat
teri
esResistors
11 14 21 24, , ,
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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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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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1 2 3 4 5
1
2
3
4
5
Bat
teri
esResistors
, , , , , , 0i i j ji j
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
3D hyper-surface in 4D space
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
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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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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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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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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
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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
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
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)
Once again about Ohm’s law: physical structure of rank (2,3)
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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
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
Once again about Ohm’s law: physical structure of rank (2,3)
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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
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).
Once again about Ohm’s law: 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
Once again about Ohm’s law: 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
1 1 1
1 2
1 1 1
i i i
j j j
I I I
M M I I I
Once again about Ohm’s law: 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
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
Once again about Ohm’s law: 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
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
six quantities are not independent of each other
Once again about Ohm’s law: 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
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)).