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Cyclic mechanics
The principle of cyclicity
Vasil Penchev
Associate Professor, Doctor of Science,
Bulgarian Academy of Science
[email protected]
http://www.scribd.com/vasil7penchev
http://vsil7penchev.wordpress.com
The mutual transformation between mass,
energy, time, and quantum information
Notations:
Quantities:
Q − quantum
information
S − entropy
E − energy
t − time
m − mass
x − distance
quantum
information
−𝟏
[°𝑲 ]
𝒕
S
k
𝒉
𝑬
𝒄
𝒎
Constants:
h − Planck
G
c − light speed
𝟑
𝑸
=
𝒙
= 𝒄𝒐𝒏𝒔𝒕 =
G − gravitational
= 𝟏 𝒒𝒖𝒃𝒊𝒕
k − Boltzmann
Quantum information in terms of quantum
temperature and the Bekenstein bound
𝑺𝟐
𝟏
𝟐π𝒌
𝑸𝟐 =
=
≤
𝒙𝟐
𝑬𝟐 𝑻𝟐
ħ𝒄
𝑺𝟏 𝒕𝟏 𝑺𝟏 𝟐π𝒌
𝟐π𝒌
𝑸𝟏 =
=
≤
𝒙𝟏 =
𝒕𝟏
ħ
𝑬𝟏
ħ𝒄
ħ
Here 𝑥1 , 𝑥2 are the corresponding radiuses of spheres,
which can place (2) the energy-momentum and
(1) the space-time of the system in question
𝑸 = 𝑸𝟏 + 𝒊𝑸𝟐 ; 𝑸 =
(𝑸𝟏 )𝟐 +(𝑸𝟐 )𝟐
The transformation in terms of quantum
measure
Notations:
Quantities:
Q − quantum
information
E − energy
t − time
m − mass
x − distance
quantum
information
Q
𝑸𝟐 𝑸𝟏
𝒄
G
Constants:
𝑬 𝒎 𝒕
h − Planck
𝒉
c − light speed
𝟑
𝑸
=
𝒙
= 𝒄𝒐𝒏𝒔𝒕 =
G − gravitational
= 𝟏 𝒒𝒖𝒃𝒊𝒕
k − Boltzmann
the axiom of choice,
“0” “1”
Yes
A qubit
No,
the Kochen-Specker
theorem
Y
I
N
Y
A
N
G
A bit
The universe as an infinite cocoon
of light = one qubit
Minkowski space
The Kochen-Specker
theorem stars as Yin
The axiom of choice
stars as Yang
Light cone
Energymomentum
Space
-time
All the universe can arise trying to divide
one single qubit into two distinctive parts,
i.e. by means of quantum invariance
Mass at rest as another “Janus”
between the forces in nature
Gravity
PseudoRiemannian
space
?
Banach space
Entanglement
? The Higgs mechanism
Weak
interaction
Electromagnetism
Minkowski
space
Groups
represented
in Hilbert
space
?
Mass at
rest
Strong
interaction
The “Standard
Model”
How the mass at rest can arise by
a mathematical mechanism
The mass at rest
Spaceis a definite mass
localized in
time
The Kochena definite space EnergySpecker
theorem
momendomain
Entanglement=
tum
Quantum
m
invariance
= The mass at rest
The axiom
of choice
The universe as
a cocoon of light
Mass at rest in relativity
and wave-particle duality
𝒕
The qubit
𝒗
𝒑
Any
qubit
in
corresponding
Hilbert
space
in its dual
space
𝑟𝐸𝑀
𝒑
𝒎~
𝒗
𝑟𝑆𝑇
The light
cone
Minkowski space
Relativity
𝑟𝑆𝑇
𝒓𝑬𝑴
𝒎~
𝒓𝑺𝑻
dual space
space
Hilbert space
Wave-particle duality
Wave function as gravitational field
and gravitational field as wave function
Infinity
Wave
function
Gravitational
field
Wholeness
How to compare qubits, or a
quantum definition of mass at rest
The qubit Any qubit in
𝒓𝒂𝒏𝒚 𝒒𝒖𝒃𝒊𝒕 = 𝟏 ⇒
corresponding
⇒ 𝒓𝑬𝑴 = 𝒓𝑺𝑻 ⇔ 𝒎𝟎 = 𝟎 in its dual Hilbert space
space
𝒓 > 𝒓 ⇔ (𝒓 = 𝟏) &
𝑬𝑴
𝑺𝑻
𝑺𝑻
& (𝒓𝑬𝑴 > 𝟏) ⇒ 𝒎𝟎 > 𝟎
𝒓𝑸≡𝒂𝒏𝒚 𝒒𝒖𝒃𝒊𝒕 ≠ 𝟏 ⇒
⇒ 𝑸 ⊬ 𝑸𝟎 (𝒓𝑸𝟎 = 𝟏)
𝑸𝟎 ≡ 𝑨𝒆𝒊𝒏𝝎𝟎 +𝝋𝟎 ⇒
⇒ 𝑸 = 𝑨𝒆𝒊𝒏𝜶𝝎𝟎 +𝝋𝟎
𝜶 ≠ 𝟏 ≡ 𝒆𝒏𝒕𝒂𝒏𝒈𝒍𝒆𝒎𝒆𝒏𝒕
Mass at rest means
entanglement
𝑟𝐸𝑀
𝑟𝑆𝑇
𝑟𝑆𝑇
𝒓𝑬𝑴
𝒎~
𝒓𝑺𝑻
dual space
space
Hilbert space
Wave-particle duality
How the mass at rest can arise by
a mathematical mechanism
Mass at rest
SpaceEnergyarises if a bigger
time
EM qubit (domain) momentum
The Kochenmust be inserted
Specker
theorem
in a smaller ST
Entanglement=
qubit (domain)
Quantum
m
𝑬𝑴
invariance
= The mass at rest
The axiom
of choice
The universe as
a cocoon of light
Mass at rest and quantum uncertainty:
a resistless conflict
“At rest” means: (∆𝒙 ≡ 𝟎) & (∆𝒗 ≡ 𝟎) ⇒
⇒ (∆𝒑 = 𝟎) & (∆𝒙. ∆𝒑 = 𝟎 < 𝒉)
Consequently, the true notions of “rest” and
“quantum uncertainty” are inconsistent
Observers
Generalized
Internal
External
probability
speed
Mass at rest and quantum uncertainty:
a vincible conflict
𝒎
quantity
𝒕
𝑷
𝒄𝟐
The
= is a power. According to general
relativity this is the power of gravitational energy,
and to quantum mechanics an additional degree of
freedom or uncertainty:
Gravitational
field with
the power p(t)
in any point:
𝒕𝒊
𝑷𝒊
𝑷𝒊 (𝒕𝒊 )
Quantum mechanics
General relativity
The Bekenstein bound as a thermodynamic law for the upper limit of entropy
𝑺≤
𝟐𝝅𝒌𝑹𝑬
ħ𝒄
⇔𝑺≤
𝟐𝝅𝒌𝒄𝒕𝑬
ħ𝒄
⇔𝑺≤
𝟐
⇔ 𝑺 ≤ 𝟒𝝅
𝑬
𝒌
𝑬
⇔
𝟐
𝟒𝝅 𝒌
The necessary and sufficient condition for the above
equivalence: 𝑹 = 𝒄𝒕 & 𝑬 = 𝒉ν (ν−frequency). This
means that the upper bound is reached for radiation,
and any mass at rest decreases the entropy
proportionally to the difference to the upper limit:
𝒎𝟎 ~∆𝑺 = 𝟒𝝅𝟐 𝒌 − 𝑺𝟎
∴ Mass at rest represents negentropy ≡ information
The Bekenstein bound as a function of
two conjugate quantities (e.g. t and E)
𝟐𝝅𝒌𝒕𝑬
𝑺≤
= 𝟒𝝅𝟐 𝒌𝒏 = 𝟒𝝅𝟐 𝒌 𝟏 + Я ,
ħ
where 𝒏 ≥ 𝟏, 𝒂𝒏𝒅 Я ≥ 𝟎
𝑺𝟎 = 𝟒𝝅𝟐 𝒌 − 𝒎𝟎
𝑰𝒇 𝑺 = 𝑺 𝟎 , 𝒕𝒉𝒆𝒏:
𝒎𝟎 = −𝟒𝝅𝟐 𝒌Я
That is the quantum uncertainty (Я)
as a rest mass (𝒎𝟎 )
About the “new” invariance
to the generalized observer
Any internal
observer
System
System
Any external
observer
The generalized observer
as any “point” or any
relation (or even ratio)
between any internal and
any external observer
An(y) external observer
An(y) internal
Reference frame
observer
Special & general relativity
Quantum mechanics All classical mechanics
and science
System
Cyclicity from the “generalized observer”
The generalized
observer
Any external
Any internal
observer
:
observer
The generalized observer
System
is (or the process of) the cyclic
return of any internal observer
into itself as an external
observer
All physical
Any internal
Any external laws should
be invariant
observer
observer
to that
cyclicity,
The generalized observer
or to “the
The universe
generalized
observer”
=
General relativity as the superluminal
generalization of special relativity
The curvature in “
“ can be represenred as a second speed in “
“. Then the
former is to the usual, external observer,
and the latter is to an internal one
Minkowski space where:
“
“ means its imaginary region, and “
“ its
real one. The two ones are isomorphic, and as a pair
are isomorphic to two dual Hilbert spaces.
Gravitational energy 𝑬𝒈 by the energy to an external
observer 𝑬𝒆 or to an internal one 𝑬𝒊 :
𝑺𝒆 = 𝑺𝒊 ⇒ 𝑬𝒈 = 𝑬𝒊 𝟏 − 𝜷 ⇔ 𝑬𝒈 =
𝟏−𝜷
𝑬𝒆
𝜷
Cyclicity as a condition of gravity
A space-time
cycle
𝑺𝒉 = 𝑺𝒕 ⇒
𝑷𝒉 + 𝑷𝒈 = 𝑷𝒕 ⇒
𝑬𝒈 =
𝑷𝒕 − 𝑷𝒉 𝒅𝒕
h – homebody
t – traveller S – action
g - gravity P – power
E – energy
Cyclicity as the foundation of
conservation of action
The universe
The Newton
absolute time
and space
Simultaneity of
all points
𝑺𝒊 = 𝑺𝒆
𝒑𝒆𝒓 𝒂 𝒖𝒏𝒊𝒕 𝒐𝒇 𝒆𝒏𝒆𝒓𝒈𝒚
Apparatus
Entanglement
Simultaneity
of quantum
entities
𝒕𝒊 = 𝒕𝒆
Mathematical and physical uncertainty
Certainty Uncertainty Independence
Any
Set theory
Any set
Disjunctive
element of
sets
any set (the
axiom of
choice)
Bound
Free
Independent
Logic
variable
variable
variables
Physics
(relativity)
Quantum
mechanics
Force
Degree of
freedom
Independent
quantities
The
measured
value of a
conjugate
Any two
conjugates
Independent
quantities
(not
conjugates)
General relativity is entirely a
thermodynamic theory!
The laws of thermodynamics
The Bekenstein bound
⇒
General
Relativity
Since the Bekenstein bound is a thermodynamic law,
too, a quantum one for the use of 𝑬 = 𝒉ν, this implies
that the true general relativity is entirely a thermodynamic theory! However if this is so, then which is the
statistic ensemble, to which it refers?
To any quantum whole, and first of all,
to the universe, represented as a statistic ensemble!
Cycling and motion
Cycle 1 = Phase 1
The universe
Cycle 3
Mechanical motion
of a mass point in it
Cycle 2 = Phase 2
General relativity is entirely a
thermodynamic theory!
A quantum thermodynamic law
A quantum whole
unorderable in
principle
⇒
⇒
General
Relativity
⇒
The laws of classical
thermodynamics
The Bekenstein bound
A relevant
well-ordered,
statistical
ensemble:
SPACE-TIME
The statistic ensemble of general relativity
A quantum
whole
The KochenSpecker
theorem
The axiom
of choice
Quantum information =
= Action =
SPACE-TIME
different
energy –
momentum
and rest mass
in any point
in general
Energy (Mass)⨂Space-Time (Wave Length)
𝟐
Einstein’s emblem: 𝑬 = 𝒎𝒄
The question is:
What is the common fundament of energy and mass?
Energy conservation defines the energy as such: The
rest mass of a particle can vanish (e.g. transforming
into photons), but its energy never! Any other fundament would admit as its violation as another physical
entity equivalent to energy and thus to mass?!
However that entity has offered a long time ago, and
that by Einstein himself and another his famous
formula, 𝑬 = 𝒉𝝂, Nobel prized
The statistic ensemble of general relativity
The Bekenstein bound
A body with
A domain of
nonzero mass as
space-time as
an “ideal gas”
The particular informational
of space-time case if 𝒕 = 𝒄𝒐𝒏𝒔𝒕
“coagulate”
points
𝟐
𝑬 = 𝒎𝒄
Information
Information
as a nonzero
as pure
rest mass
energy
(photons) = 𝒕 𝑬 = 𝒕 𝒎𝒄𝟐 − 𝒉𝑰 (a body) <
𝟐
max entropy
max entropy 𝟏
𝟏
𝒕𝟐
𝒅𝒕𝟐
The general case: = or
- speed of body time,
𝜷
𝒕𝟏
𝒅𝒕𝟏
which is 1 in the particular case above
Reflections on the information equation:
𝟐
𝒕𝟏 𝑬 = 𝒕𝟐 𝒎𝒄 − 𝒉𝑰
The information equation for the Bekenstein bound:
𝟐
𝑬
𝒎𝒄
𝟐𝝅𝒌𝑬𝟎
=
−
𝒗𝟏
𝒗𝟐
𝒄
The information equation for the “light time”:
𝟏
𝟐
𝑬 = 𝒎𝒄 − 𝑬𝟎
𝜷
The distinction between energy
and rest mass
If one follows a space-time trajectory (world line),
then energy corresponds to any moment of time,
and rest mass means its (either minimal or average)
constant component in time
Energy (mass)
𝑬𝟎
𝒎𝟎
𝒕𝟎
𝑬𝒏
𝒎𝟎
𝑬𝟏
𝒕𝟏
... ... ... ...
𝒕𝒏
𝒎𝟎
Time
Gravitational field as a limit, to which tends
the statistical ensemble of an ideal gas
Gravitational field
Differential
representation
The laws of classical
thermodynamics
An infinitely
A back
small volume transformation
of an ideal gas to the differenThe Bekenstein
bound (a quantum law)
tials of mechanical quantities
The rehabilitated aether, or:
Gravitational field as aether
The laws of classical
thermodynamics
Space-time of
general
relativity
A finite volume
of an ideal gas
A point under infinitely
large magnification
momentum
pressure
The gas
into the point
energy
temperature
The back transformation
The Bekenstein
bound (a quantum law)
An additional step consistent with the
“thermodynamic” general relativity
The infinity of
ideal field
A finite volume
of ideal field
The universe
as a whole
A point in it
A cyclical structure
The cyclicity of the universe by the
cyclicality of gravitational field
D
u
a
l
H
i
l
b
e
r
t
s
p
a
c
e
H
i
l
b
e
r
t
𝒕𝒏𝟏 , 𝒕𝒏𝟐 two successive cycles
“Light”
“Light”
𝒕𝒏𝟏 The universe 𝒕𝒏𝟐
Two “successive”
points in it
𝒏𝟏
𝒏𝟐
As to the universe,
as to any point in it
by means of
the axiom of choice and
the Kochen – Specker theorem
⇔
The cyclicity of gravitational and of
quantum field as the same cyclicity
Quantum
mechanics
The universe
The Standard
General
Model
relativity
Strong,
Gravity
electromagnetic, and weak
A point in it
interaction
Gravitational and quantum field as an ideal
gas and an ideal “anti-gas” accordingly
D
u
a
l
s
H p
i a
l c
b e
e
r
t
The universe
H
i
l
b
e
r
t
All the space-time
A volume of
ideal gas or
ideal field
PseudoRiemannian
space
Gravitational
A point in it
field
Quantum field
Conjugate B
Specific gravity as a ratio of qubits
𝒓𝟐
is
uncertain
𝒓𝟏
Quantum uncertainty
Gravity as if determines
the quantum uncertainty
being a ratio of conjugates
Conjugate A
An “ideal gas”
composed of 𝒓
mass points ( 𝟐 𝒓𝟏 ~ 𝑬 𝒕)
Qubits
Quantum mechanics
General relativity
The gas constant R of space-time
The axiom of choice needs suitable fundamental
constants to act physically:
The Boltzmann
Avogadro’s number 𝑵𝑨
constant 𝑲𝑩
?
⇔
How much to (or per) how many?
⇔
In Paradise: No choice
Paradise on earth!
An ideal gas (aether) of
𝑲𝑩
space-time points:
𝑵𝑨
𝑹 = 𝑲𝑩 . 𝑵𝑨
On earth: Choices, choices ...
Quantum mechanics
General relativity
Time as entropy: “relic” radiation as
a fundamental constant or as a variable
+Energy (S) flow(S) +Energy (D) flow(D)
𝑫𝒆𝒄𝒆𝒍𝒆𝒓𝒂𝒕𝒊𝒐𝒏𝒕𝒊𝒎𝒆𝟏−𝟐
𝑺𝒑𝒆𝒆𝒅𝒕𝒊𝒎𝒆𝟏
𝑺𝒑𝒆𝒆𝒅𝒕𝒊𝒎𝒆𝟐
𝒕
𝒉
𝟏
𝒉
𝑺𝒑𝒆𝒆𝒅𝒕𝒊𝒎𝒆 = 𝑺𝒕 =
=
. =
𝒕𝟎 𝑪𝑴𝑩 𝒕𝟎
𝑪𝑴𝑩 𝒕 =𝟏
𝟎
Seen “inside”:
Seen “outside”:
Our immense and
A black hole
expanding universe
among many ones
determined by
determined by
the fundamental
its physical parameters
constants
like
mass,
energy,
etc.
Horizon
How much
should the deceleration of time be?
The ideal gas equation is:
𝒑𝑽 = 𝒏𝑹𝑻
The universe
Any separate
point in it
The “Supreme Pole”
(the Chinese Taiji 太極)
𝐒 = 𝒑𝒙 = (𝑵𝑲𝑩 /𝑲𝒖 )𝑬𝒕
𝑺𝒖 = 𝑵𝒉
𝑲𝑩
𝑲𝒖 =
𝑵
𝑵𝑨
𝟏
𝑪𝑴𝑩
= 𝑲𝑩
𝒕
𝒉
𝟏
𝑪𝑴𝑩
𝝎
= 𝑲𝑩
=𝝂=
𝒕
𝒉
𝟐𝝅
The Einstein and Schrödinger equation:
the new cyclic mechanics
The universe
Any and all points in it
The Great Pole Cyclic mechanics:
Conservation of information
d(Information) =
d(Energy of gravity)
Pseudo-Riemannian
space-time ≠ 0 info
The Einstein equation
Space
d(Info)= & Time
d(Energy) = “0” Info
Schrödinger’s equation