ISSN: 2641-1725
Robert Skopec*
Received: October 28, 2019; Published: December 03, 2019
*Corresponding author: Robert Skopec, Researcher-analyst, Dubnik, Slovakia
DOI: 10.32474/LOJMS.2019.04.000180
We study mutual interplay of information and entropy as
quantum field using an information-theoretic (Shannon entropy)
complex-vector analysis to calculate, respectively, the Gibbs free
energy difference and virial mass. We define conjugate hyperbolic
space and entropic momentum co-ordinates to describe these
spiral structures in Minkowski space-time, enabling a consistent
and holographic Hamiltonian-Lagrangian system that is completely
isomorphic and complementary to that of conventional kinematics.
Such double spirals therefore obey a maximum-entropy pathintegral
variational calculus (“the principle of least exertion”. These
simple analytical calculations are quantitative examples of the
application of the Second Law of Thermodynamics as expressed in
entropy terms. They are underpinned by a comprehensive entropic
action (“exertion”) principle based upon Boltzmann’s constant as
the quantum of exertion.
Its overriding significance to us as living beings is its functioning
as an entropy engine. Landauer’s seminal work (following Shannon
and Brillouin teaches us that information has calculable entropy and
obeys physical laws, while the introduction by Jaynes of maximum
entropy (MaxEnt) as the basis of the rules of thermodynamics
(for example, the determination of the partition function) is now
recognised as far-reaching. The associated variational approach to
entropy production first described by Onsager also provides critical
insights into issues of thermodynamic reciprocity and symmetry in
systems far from equilibrium [1-5].
The entropic treatment of information is standard in the
analysis of the efficiency of communications networks in the
presence of noise, and it has become clear that information and
its transfer are associated with discontinuities, implying nonadiabatic:
entropy changing conditions (see the equation p. 7).
Indeed, Brillouin considered information (negative entropy, or
negentropy) to be anti-correlated with entropy, and Bennett
showed elegantly how information erasure has an entropy cost:
note that perfect information copying is excluded by the “nocloning
theorem”. Applying Landauer’s Principle to a computation
involves the transfer of information and therefore also results in a
rise in entropy. Important is the mutual, reciprocity of information
and entropy. In quantum mechanics information and entropy are in
one, same Field, the are not anti-correlated, but correlated MaxEnt
[5-8].
We choose to define the entropy s as the Hodge-dual *h of the
information since this definition can be shown to have the correct
properties; note that Penrose points out that Maxwell’s equations
are self-dual in the orthogonal complement sense of the Hodgedual
operation, with σm = *σn = Iσn:
s=kBln(xm)Iσmm∈{1,2,3};summationconventions=kBln(xm)
Iσmm∈{1,2,3};summationconvention
Thus, we amplify Brillouin’s assertion of the close relation of
information with entropy by treating entropy mathematically as an
orthogonal complement of information.
We choose entropic structures exhibiting a transverse helical
geometry, that is, s3 = h3 = 0, with a “trajectory” axis (plane waves
travelling) in the γ3direction. Then, given that s and h are conjugate
(that is, the orthogonal complements of each other), the entropy
eigenvector can be written as
s=kB(iln(x1)Iσ1−ln(x2)Iσ2)s=kB(iln(x1)Iσ1−ln(x2)Iσ2)
and it’s (conjugate) information term similarly written as
h=kB(ln(x1)σ2−iln(x2)σ1)h=kB(ln(x1)σ2−iln(x2)σ1)
Courant & Hilbert point out that the Maxwell equations are
a hyperbolic version of the Cauchy-Riemann equations, and
Salingaros points out that the vacuum electromagnetic (EM) field is
holomorphic [5,8,9]. To form a holo-morphic info-entropy function
we combine together the expressions in Eqs. for information and
entropy in the same way (and for the same reason) that is done in
the Riemann-Silberstein complex-vector (holomorphic) description
of the EM field:
F––=(E––+icB––)γ0F_=(E_+icB_)γ0
where E and B are the 1-vector electric and magnetic fields; F
is a bivector (see Penrose), hence the need for γ0. The equivalent
complex-vector for the bivector info-entropy case is:
f=s+Ih, f=s+Ih,
so that we have, from Eq.:
f=kBln(x1/x2)I[iσ1+σ2]f=kBln(x1/x2)I[iσ1+σ2]
Note that the argument of the logarithm is now dimensionless,
as is conventional. Note also that meromorphic functions are only
piecewise holomorphic, so they can transmit information.
Just as Maxwell’s equations have a complementary (dual, in a
strong sense) helical structure of the electric and magnetic fields,
we continue to choose a similar double-helical structure to the infoentropic
geometry, such that the loci of the x1 and x2 co-ordinates
of the info-entropic trajectory are related to each other by a pair of
coupled differential equations:
x′1=−κ0x2x′1=−κ0x2
x′2=κ0x1x′2=κ0x1
(5b)
where the coupling parameter is given by κ0 ≡ 2π/λ0 with λ0
being the helical pitch along the γ3-axis (that is, the x3 direction)
and the prime indicating the differential with respect to x3 (the
trajectory axis) xn′ ≡ dx_{n}/dx_{3} as usual [10,11].
In the entropic domain the x_{3} co-ordinate plays a role analogous
to that normally played by time t in conventional kinematics: to
amplify this point, note that x0 ≡ ct and x_{3} are also commensurate
conjugates in the Pauli algebra. Considering only the functional part
of the complex-vector, Eq. allows us to write the ‘local’ geometric
entropy for a double-helical structure as (Eq.):
s=kBln(x′nκ0xn)≡kBlnWn∈{1,2}; summationconventions=k-
Bln(x′nκ0xn)≡kBlnWn∈{1,2};summationconvention
which is functionally equivalent to Boltzmann’s equation for
entropy; where the quantity Wn ≡ xn′/κ0 xn therefore represents
the number of states available for the nth plane wave.
We now consider the case of the double helix in more detail,
and in particular as exhibited by the structure. Without loss of
generality, we define the locus in space l1 of the first informationbearing
with its axis aligned to the γ_{3}direction:
l1(x3)= γ1R0cosκ0x3+γ2R0sinκ0x3l1(x3)=γ1R0cosκ0x3+γ2R
0sinκ0x3
Where R0, κ0 and x3 represent respectively the radius, pitch,
and axial co-ordinate of the helix. The second helix l2, with its
complementary base-pairing and anti-parallel (C2 space group)
symmetry contains the same entropic information content as l1, but
π/2 phase-shifted and propagating in the opposite (i.e. negative) γ_{3}
direction:
l2(x3)=γ1R0sinκ0x3−γ2R0cosκ0x3l2(x3)=γ1R0sinκ0x3−
γ2R0cosκ0x3
These expressions are mathematically equivalent to those for
the electric and magnetic fields of an EM wave, with l1 and l2 being
complementary. Equivalent to Eqs., we now express the complexvector
Σ = l1 + il2 to describe a single holomorphic trajectory in
Euclidean coordinates with spatial basis vectors γn (n ∈ {1, 2}):
Σ=γ1R0eiκ0x3−γ2iR0eiκ0x3Σ=γ1R0eiκ0x3−γ2iR0eiκ0x3
We therefore see in Eq. the functionals represented by x1 = R0
exp(iκ0x3) and x2 = −iR0 exp(iκ0x3), from Eq., where the phase and
sign difference between x1 and x2 are typical for a pair of coupled
mode equations, and which together form a holomorphic function
[12].
The conjugate quantity for position q is the momentum p, so
that moving towards a Lagrangian formalism, we therefore also
define the “entropic momentum” pn vectors in terms of an “entropic
mass” mS and the velocity qn′, where as before qn′ ≡ dqn/dx3. Note
that qn′ is dimensionless, so that either qn′ or its inverse 1/qn′ can
be used as a “velocity” (this ambiguity is a feature of hyperbolic
velocities). It turns out that the inverse definition is more fruitful:
e n t r o p i c m o m e n t u m : p n ≡ m S / q ′ n n ∈ { 1 , 2 }
entropicmomentum:pn≡mS/q′nn∈{1,2}
where the entropic mass mS is defined as:
entropicmass:mS≡iκ0kBentropicmass:mS≡iκ0kB.
We will use Eqs. as the basis for a set of Hamiltonian and
Lagrangian equations. We consider first the entropic equivalent
to kinetic energy, i.e. ‘kinetic entropy’ (KE) TS, based upon the
conventional definition of kinetic energy:
TS(q′)=−∫pdq′=−mSlnq′TS(q′)=−∫pdq′=−mSlnq′
where the additional negative sign accounts for the inverse
velocity. For the three spatial directions, we therefore have:
T S = Σ n − m S l n q ′ n = − 1 / 2 m S l n ( q ′ n q ′ n )
s u m m a t i o n c o n v e n t i o n , n ∈ { 1 , 2 , 3 } T S = Σ n −
mSlnq′n=−1/2mSln(q′nq′n)summationconvention,n∈{1,2,3}
We also define an entropic potential field VS(q) as a function
of hyperbolic position q (the ‘potential entropy’). However, for
the present case of a double helix, Eq. clearly represents a pair of
plane waves travelling in space; which is analogous to the kinematic
“free-particle” situation, such that there is therefore no associated
entropic potential field, VS = 0. The entropic Hamiltonian HS(q(x3),
p(x3), x3) is defined as usual as HS = TS + VS, and is also a conserved
quantity in hyperbolic space [13].
Using the canonical Legendre transformation, the entropic
Lagrangian is given by Eq.:
LS==q′npn−HSsummationconvention,n∈{1,2,3}3mS−
HSLS=q′npn−HSsummationconvention,n∈{1,2,3}=3mS−HS
such that the required canonical equations of state are obeyed:
∂LS/∂x3 = −∂HS/∂x3, as well as p′n=∂LS/∂qnp′n=∂LS/∂qn and
q′n=−∂LS/∂pnq′n=−∂LS/∂pn.
Having defined the exertion integral, Eq. we can also now
see that the equivalent space-trajectory integral of the entropic
Hamiltonian HS (see Eq.) yields a quantity directly proportional to
the entropy:
S=∫HSdl=χ∫HS(q,p,x3)dx3S=∫HSdl=χ∫HS(q,p,x3)dx3
Whereas Eq. describes a ‘local’ entropy s, the integrated
quantity S can be considered as the ‘global’ or the overall system
entropy. Eq. indicates that the overall entropy S depends not only
on the centroidal trajectory of the double helix axis as described
by x3, but principally upon the spiralling path described by l with
its radial dependency such that the entropy is a function of the
full spatial extent (in all spatial dimensions) of the double helix
structure. For convenience, we offset the entropic Hamiltonian
HS by the constant term mS ln(κ0 R0) which is an invariant for a
double helical geometry – any Hamiltonian can be offset by a fixed
(constant) amount to enable more convenient manipulation – such
that the entropic Hamiltonian for a double helix can therefore be
given as HS = πκ0kB; that is, each KE component (n = 1, 2) of the
double helix contributes ½πκ0kB. We can also exploit the Fourier
(periodic) nature of Salong the double helix as characterized by the
parameter iκ0 to write the Fourier differential operator as:
ddx3≡iκ0ddx3≡iκ0
Since the Lagrangian and Hamiltonian are inversely related
(through the Legendre transformation) and the exertion integral
X Eq. is at an extremum (Eq.), δX = 0, then the closely connected
Hamiltonian trajectory integral Eq. (that is, the entropy S) must
also be at an extremum, δS = 0. Given that this represents a highly
stable structure we infer from the Second Law that the entropy S is
at a maximum; ergo the exertion X is at a minimum and topology
represents a MaxEnt (most likely) trajectory in space. In summary,
the overall entropy S is given by:
S=√(1+κ20R20)πκ0LkBS=√(1+κ02R02)πκ0LkB
It is clear that the entropy S is proportional to the length L.
However, in the case of a photon its proper length is actually zero
relativistically, since it travels at the speed of light: L = 0, therefore
S = 0.
TBH=ℏc3/8πGMBHkB=1.5×10−14KTBH=ℏc3/8πGMBHk-
B=1.5×10−14K
MBH is given by Gillessen et al. as 4.3 ± 0.4 million solar masses
M_{◉}, where this 10% uncertainty is entirely due to the uncertainty
in the galactic position of the Sun: the measurement actually has
a precision better than 2% (the mass of the Sun is known very
accurately, to about 10^{-4}: M_{◉} = 1.989 × 10^{30} kg). Applying this
temperature to SMW to obtain the energy (given by the product of
entropy and temperature expressed as a mass through E = mc^{2}) we
naturally recover MBH.
All quantities clearly revert to their respective double-helical
quantities when the logarithmic spiral parameter Λ = 0. We find
that a logarithmic spiral is associated with an entropic potential
field VS ≠ 0 causing a hyperbolic acceleration; indeed, as the
entropic analogy to Newton’s second law of kinematics (F = mẍ),
we solve the Euler-Lagrange equations (defined in hyperbolic
space qn) dpn/dx3=−mSqn′′/qn′=−∂VS/∂qn2dpn/dx3=−mSqn′′/
qn′=−∂VS/∂qn2, where the final term in the equation (the entropic
potential gradient) is therefore equivalent to the entropic force FS.
The associated entropic acceleration is given by Γn=−qn′′/qn′2Γn=−
qn′′/qn′2, the minus sign being due to the inverse velocity nature of
q′. The proof that the double-armed logarithmic spiral satisfies the
Euler-Lagrange equations in hyperbolic space q (that is, obeys the
principle of least exertion).
In Euclidean (x) space, we find that the entropic potential field
VS for the logarithmic double spiral is expressed as:
VS(x)=imSK0eiκGx31−Λx3(x1+ix2x1x2)−mSK3eΛx3R3(1−
Λx3)VS(x)=imSK0eiκGx31−Λx3(x1+ix2x1x2)−mSK3eΛx3R3(1−
Λx3)
It is indeed interesting to note the existence of an inversesquare
law (in Euclidean space) for the γ1 and γ2 directions at the
heart of this entropic potential field; the entropic force varies as
F S , n = − ∂ V S ∂ x n = − m S K 0 e i κ G x 3 x 2 n ( 1 − Λ x 3 )
n=1,2FS,n=−∂VS∂xn=−mSK0eiκGx3xn2(1−Λx3)n=1,2
that is, FS,n ∝ xn−2, with FS also being proportional to the
entropic mass mSassumed located at the centre of the system
and to be the cause of the entropic potential field. We emphasise,
however, that although Eqs and express the entropic field in a more
intuitive Euclidean form, the entropic Hamiltonian and Lagrangian
equations are only correctly applied in hyperbolic space [13-15].
Gauge/gravity duality posits an exact equivalence between
certain conformal field theories (CFT’s) with many degrees of
freedom and higher dimensional theories with gravity. We try
to understand how bulk spacetime geometry and gravitational
dynamics emerge from a non-gravitational theory [16,17]. In recent
years, there have appeared hints that quantum entanglement
entropy a key role. One important development in this direction
was the proposal that the entanglement entropy between spatial
domain D of CFT and its complement is equal to the area of the bulk
extremal surface. Using this showed the emergence of linearized
gravity from entanglement physics of the CFT, we continue this
program. Moreover, we show that bulk stress-energy density in this
region can be reconstructed point-by-point from entanglement on
the boundary [18].
Relative entropy is a measure of distinguishability between
two quantum state in the Hilbert space. The relative entropy of two
density matrices and is defined as
S(/)=tr( log )-tr( log ).
When and are reduced density matrices on a spatial domain D
for two states of a quantum field theory (QFT), which is the case
which implies that S(/) increases with the size of D.
Defining the modular Hamiltonial of implicitly through =
It is easy to see that above is equivalent to
S(/)= , where is the change in the expectation value of the
operator and is the change in entanglement entropy across D as one
goes between the states [19-21].
In general, the modular Hamiltonian associated to a given
density matrix is nonlocal. There are a few simple cases where
it is explicitly known. When is the reduced density matrix of the
vacuum state of a CFT on a disk of radius R which (without loss of
generality) we take to be centered at Χ_{0} = 0,
where is the energy density of the CFT [22].
The Interface between quantum gravity and information
science
1. Theory of quantum gravitation
- Lee Smolin showed an intriguing link between general ideas
in quantum gravity
and the fundamental non-locality of quantum physics,
-We must replace the non-local behavior of quantum mechanics
with the non-local behavior of quantum gravity [22].
2. Quantum entanglement entropy
- Ooguri and Marcolli’s work shows that this quantum
entanglement generates the extra dimensions of the gravitational
theory,
- entangled particles have also complementary properties,
- entangled quantum particles cannot be seen individually, they
form a single quantum object-field, even if they are located far apart,
- If two particles are entangled they have complementary
wavefunction properties and measuring one places meaningful
constraints on the properties of the other.
3. Quantum information
-- The interface between quantum gravity and information
science is becoming increasingly important for both fields [21,22].
- Based on Lee Smolin’s calling for continuing in Einstein’s
Unfinished Revolution, I propose the ultimate quest to supersede
our two current (Mutually Corelated) descriptions of reality:
General Relativity and Quantum Gravity.
- General Relativity and Quantum Gravity including Quantum
Entanglement
- Entropy means that The Twofaced New Main Law of Nature
may lead to a New
Scientific Revolution.
It is a fundamental fact, that everything we do creates a
corresponding energy that comes back to us in some form or
another. From a scientific perspective it is not known enough
that the energy you expend taking some action comes back to
you or someone else. This means that every action become more
or less entangled and produces further complications because of
mutual interplay between information and entropy cooperation at
quantum level. In other words, this quantum entanglement entropy
(QEE) is the key to the human activities. Our above mathematical
formula of changes in QEE is the essence how the Universe works.
The QEE is a Karma of the Universe. Universe doesn’t immediately
respond to your actions with good Karma. It can take time before
Universe repays your intentional actions with more actions that
help you progress toward your goal. The Universe requires energy
to be expended. You might find yourself generating days, if not
weeks or months, of output before you can see the effects of your
efforts. Sometimes, the efforts come in trickle, and other times, they
can come in deluge. The trick is to keep your focus on generating
actions that help also others in the direction you seek to go yourself.
This is only the tip of the iceberg of the metaphysical dynamics of
the Universe directed by the QEE. The more you put forth energies
without expectation of personal gain, the more you’ll be surprised
at how the Universe will open the door to the goal you seek to
attain. Make no mistake, nothing takes the place of commited Allin
action every day. Never, Ever, Give Up on your dreams and soon
you’ll discover that Universe will not give up on you.
It has been shown that information and “entropy” – a
measure of the disorder of a system – are linked together to “infoentropy”
in a way exactly analogous to electric and magnetic
fields (“electromagnetism”). Electric currents produce magnetic
fields, while changing magnetic fields produce electric currents.
Information and entropy influence each other in the same way.
Entropy is a fundamental concept in physics. For example,
because entropy can never decrease (disorder always increases)
you can turn an egg into scrambled eggs but not the other way
around. If you move information around you must also increase
entropy – a phone call has an entropy cost.
Light wave with electric (E) and magnetic (B) fields.
It has been showed that entropy and information can be treated
as a field and that they are related to geometry. Think of the two
strands of the DNA double helix winding around each other. Light
waves have the same structure, where the two strands are the
electric and magnetic fields. We showed mathematically that the
relationship between information and entropy can be visualised
using just the same geometry.
If we want to see if our theory could predict things in the real
world, and decided to try and calculate how much energy you’d
need to convert one form to another form of information and
entropy as one quantum field. For example, the proton’s structure can be modeled along with its attendant fields, showing how even
though it’s made out of point-like quarks and gluons, which has a
finite, substantial size arising from the interplay of the quantum
forces and fields inside it. In the quantum mechanics, the principle
of locality is violated all the time. Locality may be nothing but a
persistent illusion. Quantum gravity tries to combine Einstein’s
general theory of relativity with quantum mechanics. We typically
view objects that are close to one another as capable of exerting
forces on one another, but that might be an illusion. For example,
Schrödinger’s cat: the cat will be either alive or dead, depending
on whether a radioactive particle decayded or not. If the cat
were a true quantum system, the cat would be neither alive nor
dead, but in a superposition of both states until observed. There
are many properties that a particle can have – such as its spin or
polarization – that are fundamentally indeterminate until you make
a measurement. Prior to observing particle, or interacting with it in
such a way that ti’s forced to be in either one state or the other, it’s
actually in a superposition of all possibble outcomes. You can also
take two quantum particles and entangle them, so that these very
same quantum properties are linked between the two entangled
particles. Whenever you interact with one member of the entangled
pair you not only gain information about which particular state
it’s in, but also information about its entangled partner, including
entropy.
By creating two entangled photons from pre-existing system
and separating them by great distances, we can teleport information
about the stateof one by measuring other, even from extraordinary
different locations, including entropy. You’ll find the member you
measure in a particular state and instantly know some information
also about the other entangled member, including entropy. Even
though no information was transmitted faster than the speed of
light, the measurement describes a troubling truth about quantum
physics: it is fundamentally a non-local theory.
Measuring the state of your particle doesn’t tell us the exact
state of its entangled pair, just probabilistic information about its
partner. You can only use this non-locality to predict a statistical
average of entangled particle properties. If two particles are
entangled, they have complementary wavefunction properties and
measuring one places meaningful constraints on the properties
of the other. There is an intriguing link between general ideas in
quantum gravity and the fundamental non-locality of quantum
physics. It means two current mutually compatible, two-faced
descriptions of reality: General Relativity and Quantum Mechanics.
Important is the mutual reciprocity of information and entropy.
In quantum mechanics information and entropy are in one, same
Field, the are not anti-correlated, but correlated MaxEnt [21,22].
Entropy can never decreases, disorder always increases. If you
move information around you must also increase entropy. The
energy is simply product of entropy and temperature. It’s because
info-entropy fields give rise to forces like other fields. Our World is
choreographed by an entropic forces to maximise entropy.
Knowledge of recent neuorobiology is proving our thesis that
Charles Darwin was wrong when formulated his theorem „Survival
of the fittest“. It was the biggest false myth of the modern Western
Science. As we have demonstrated in our above study, the careerist
is psychopat and not „the fittest“. From this reason we must to
correct Charles Darwin to „Survival of the careerist“. Reality in
21 century is showing that Survival of the careerist based on the
Quantum Entanglement Entropy (QEE) is more valid Law of Social
Dynamics in our days because it is under the Universe‘s Law of
Maximising Entropy. Careeristic Competition is the main cause of
the QEE leading to icreased complications through Coincidenses of
Social Dynamics.
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