research article minimal length and the existence of some...

Research ArticleMinimal Length and the Existence of Some InfinitesimalQuantities in Quantum Theory and Gravity

A E Shalyt-Margolin

National Centre of Particles and High Energy Physics Pervomayskaya Street 18 220088 Minsk Belarus

Correspondence should be addressed to A E Shalyt-Margolin ashalytmailru

Received 23 September 2014 Accepted 25 November 2014 Published 10 December 2014

Academic Editor George Siopsis

Copyright copy 2014 A E Shalyt-Margolin This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited The publication of this article was funded by SCOAP3

It is demonstrated that provided a theory involves a minimal length this theory must be free from such infinitesimal quantities asinfinitely small variations in surface of the holographic screen its volume and entropy The corresponding infinitesimal quantitiesin this case must be replaced by the ldquominimal variations possiblerdquomdashfinite quantities dependent on the existent energies As a resultthe initial low-energy theory (quantum theory or general relativity) inevitably must be replaced by a minimal length theory thatgives very close results but operates with absolutely other mathematical apparatus

At the present time all high-energy generalizations (limits)of the basic components in fundamental physics (quantumtheory [1] and gravity [2]) of necessity lead to a minimallength on the order of the Planck length 119897min prop 119897

119875 This

follows from string theory [3ndash6] loop quantum gravity [7]and other approaches [8ndash22]

But it is clear that provided a minimal length exists itis existent at all the energy scales and not at high (Planckrsquos)scales only

What is inferred on this basis for real physics At least itis suggested that the use of infinitesimal quantities 119889119909

120583in a

mathematical apparatus of both quantum theory and gravityis incorrect despite the fact that both these theories give theresults correlating well with the experiment (eg [23])

Indeed in all cases the infinitesimal quantities 119889119909120583bring

about an infinitely small length 119889119904 [2]

1198891199042= 119892120583]119889119909120583119889119909] (1)

that is inexistent because of 119897minThe same is true for any function Υ dependent only on

different parameters 119871119894whose dimensions of length of the

exponents are equal to or greater than 1 ]119894ge 1

Υ equiv Υ (119871]119894119894) (2)

Obviously the infinitely small variation 119889Υ of Υ is senselessas according to (2) we have

119889Υ equiv 119889Υ (]119894119871]119894minus1119894

119889119871119894) (3)

But because of 119897min the infinitesimal quantities 119889119871119894make no

sense and hence 119889Υmakes no sense tooInstead of these infinitesimal quantities it seems reason-

able to denote them as ldquominimal variations possiblerdquo Δmin ofthe quantity 119871 having the dimension of length that is thequantity

Δmin119871 = 119897min (4)

And then

ΔminΥ equiv ΔminΥ (]119894119871]119894minus1119894

Δmin119871 119894) = ΔminΥ (]119894119871]119894minus1119894

119897min)

(5)

However the ldquominimal variations possiblerdquo of any quan-tity having the dimensions of length (4) which are equal to119897min prop 119897

119875require according to the Heisenberg Uncertainty

Principle (HUP) [24] maximal momentum 119901max prop 119875119875119897

and energy 119864max prop 119864119875 Here 119897

119875 119875119875119897 119864119875are Planckrsquos length

momentum and energy respectivelyBut at low energies (far from the Planck energy) there are

no such quantities and hence in essence Δmin119871 = 119897min prop

119897119875(4) corresponds to the high-energy (Planckrsquos) case only

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2014 Article ID 195157 8 pageshttpdxdoiorg1011552014195157

2 Advances in High Energy Physics

For the energies lower than Planckrsquos energy the ldquominimalvariations possiblerdquo Δmin119871 of the quantity 119871 having thedimensions of lengthmust be greater than 119897min and dependenton the present 119864

Δmin equiv Δmin119864 Δmin119864119871 gt 119897min (6)

Besides as we have a minimal length unit 119897min it isclear that any quantity having the dimensions of length isldquoquantizedrdquo that is its valuemeasured in the units 119897min equalsan integer number and we have

119871 = 119873119871119897min (7)

where119873119871is positive integer number

Theproblem is that how the ldquominimal variations possiblerdquoΔmin119864 (6) are dependent on the energy or similarly on thescales of the measured lengths

To solve the above-mentioned problem initially we canuse the space-time quantum fluctuations (STQF) with regardto quantum theory and gravity

The definition (STQF) is closely associated with thenotion of ldquospace-time foamrdquo The notion ldquospace-time foamrdquointroduced byWheeler about 60 years ago for the descriptionand investigation of physics at Planckrsquos scales (EarlyUniverse)[25 26] is fairly settled Despite the fact that in the last decadenumerous works have been devoted to physics at Planckrsquosscales within the scope of this notion for example [27ndash46]by this time still their is no clear understanding of the ldquospace-time foamrdquo as it is

On the other hand it is undoubtful that a quantum theoryof the Early Universe should be a deformation of the well-known quantum theory

In my works with colleagues [47ndash56] I have put forwardone of the possible approaches to resolution of a quantumtheory at Planckrsquos scales on the basis of the density matrixdeformation

In accordance with the modern concepts the space-timefoam [26] notion forms the basis for space-time at Planckrsquosscales (Big Bang) This object is associated with the quantumfluctuations generated by uncertainties in measurements ofthe fundamental quantities inducing uncertainties in anydistance measurement A precise description of the space-time foam is still lacking along with an adequate quantumgravity theory But for the description of quantum fluctua-tions we have a number of interesting methods (eg [36ndash46 57 58])

In what follows we use the terms and symbols from [38]Then for the fluctuations 120575119897 of the distance 119897 we have thefollowing estimate

(120575119897)120574≳ 119897120574

1198751198971minus120574

= 119897119875(

119897

119897119875

)

1minus120574

= 119897 (119897119875

119897)

120574

= 119897120582120574

119897 (8)

or that same one

100381610038161003816100381610038161003816(120575119897)120574

100381610038161003816100381610038161003816min= 120573119897120574

1198751198971minus120574

= 120573119897119875(

119897

119897119875

)

1minus120574

= 120573119897120582120574

119897 (9)

where 0 lt 120574 le 1 coefficient 120573 is of order 1 and 120582119897equiv 119897119875119897

From (8) and (9) we can derive the quantum fluctuationsfor all the primary characteristics specifically for the time(120575119905)120574 energy (120575119864)

120574 and metrics (120575119892

120583])120574 In particular for(120575119892120583])120574 we can use formula (10) in [38]

(120575119892120583])120574≳ 120582120574 (10)

Further in the text it is assumed that the theory involvesa minimal length on the order of Planckrsquos length

119897min prop 119897119875 (11)

or similarly

119897min = 120585119897119875 (12)

where the coefficient 120585 is on the order of unity tooIn this case the origin of the minimal length is not

important For simplicity we assume that it comes from thegeneralized uncertainty principle (GUP) that is an extensionof HUP for Planckrsquos energies where gravity must be takeninto consideration [3ndash22]

Δ119909 geℎ

Δ119901+ 12057210158401198972

119875

Δ119901

ℎ (13)

Here1205721015840 is themodel-dependent dimensionless numericalfactor

Inequality (13) leads to the minimal length 119897min = 120585119897119875

=

2radic1205721015840119897119875

Therefore in this case replacement of Planckrsquos length bythe minimal length in all the above formulae is absolutelycorrect and is used without detriment to the generality [59]

119897119875997888rarr 119897min (14)

Thus 120582119897equiv 119897min119897 and then (8)ndash(10) upon the replacement

of (14) remain unchangedAs noted in the overview [38] the value 120574 = 23 derived

in [57 58] is totally consistent with the Holographic Principle[60ndash63]

The following points of importance should be noted [59](11) It is clear that at Planckrsquos scales that is at theminimal

length scales

119897 997888rarr 119897min (15)

models for different values of the parameter 120574 are coincident(12) As noted specifically in (7) provided some quantity

has a minimal measuring unit values of this quantity aremultiples of this unit

Naturally any quantity having a minimal measuring unitis uniformly discrete

The latter property is not met in particular by the energy119864

As 119864 sim 1119897 where 119897 is measurable scale the energy 119864 is adiscrete but nonuniformquantity It is clear that the differencebetween the adjacent values of 119864 is the less the lower 119864 Inother words for 119897 ≫ 119897min that is

119864 ≪ 119864119875 (16)

119864 becomes a practically continuous quantity

Advances in High Energy Physics 3

(13) In fact the parameter 120582119897was introduced earlier

in [47ndash56] as a deformation parameter on going from thecanonical quantum mechanics to the quantum mechanics atPlanckrsquos scales (Early Universe) that are considered to be thequantum mechanics with the fundamental length (QMFL)

0 lt 120572119909=

1198972

min1199092

le1

4 (17)

where 119909 is the measuring scale 119897min sim 119897119901

The deformation is understood as an extension of aparticular theory by inclusion of one or several additionalparameters in such a way that the initial theory appears inthe limiting transition [64]

Obviously everywhere apart from the limiting point120582119909= 1 or 119909 = 119897min we have

120582119909= radic120572119909 (18)

From (17) it is seen that at the limiting point 119909 = 119897minthe parameter 120572

119909is not defined due to the appearance of

singularity [47ndash56] But at this point its definition may beextended (regularized)

The parameter 120572119897has the following clear physical mean-

ing

120572minus1

119897sim 119878119861119867

(19)

where

119878119861119867

=119860

41198972119901

(20)

is the well-known Bekenstein-Hawking formula for the blackhole entropy in the semiclassical approximation [65 66] forthe black hole event horizon surface 119860 with the character-istics linear dimension (ldquoradiusrdquo) 119877 = 119897 This is especiallyobvious in the spherically symmetric case

In what follows we use both parameters 120582119909and 120572

119909

Turning back to the introductory section of this work andto the definition Δmin119864119871 we assume the following

1003816100381610038161003816Δmin1198641198711003816100381610038161003816 =

100381610038161003816100381610038161003816(120575119871)120574

100381610038161003816100381610038161003816min (21)

where |(120575119871)120574|min is from formula (9) 120574 is fixed parameter

from formulae (8) and (9) and 119864 = 119888ℎ119871In physics and in thermodynamics in particular the

extensive quantities or parameters are those proportionalto the mass of a system or to its volume Proceeding fromdefinition (2) of the function Υ(119871

]119894119894) one can generalize this

notion taking as a generalized extensive quantity (GEQ) ofsome spatial system Ω the function dependent only on thelinear dimensions of this system with the exponents no lessthan 1

The function Υ(119871]119894119894) ]119894ge 1 (2) is GEQ of the system Ω

with the characteristic linear dimensions 119871119894 119894 = 1 119899 or

identically a sumof the systemsΩ119894 119894 = 1 119899 each ofwhich

has its individual characteristic linear dimension 119871119894

Then from the initial formulae (2)ndash(6) it directly followsthat provided the minimal length 119897min is existent there are noinfinitesimal variations of GEQ

In the first place this is true for such simplest objects asthe 119899-dimensional sphere 119861

119899 119899 ge 2 whose surface area (area

of the corresponding hypersphere 119878119899) and volume 119881

119899repre-

sent GEQs and are equal to the following

119878119899= 119899119862119899119877119899minus1

119881119899= 119862119899119877119899 (22)

where 119877mdashradius of a sphere the length of which is a charac-teristic linear size 119862

119899= 1205871198992

Γ((1198992) + 1) and Γ(119909) is agamma-function

Of course the same is true for the 119899-dimensional cube (orhypercube)119860

119899 its surface area and its volume are GEQs and

a length of its edge is a characteristic linear dimensionProvided 119897min exists there are no infinitesimal increments

for both the surface area and volume of 119860119899or 119861119899 only

minimal variations possible for these quantities are the caseIn what follows we consider only the spatial systems

whose surface areas and volumes are GEQsLet us consider a simple but very important example of

gravity in horizon spacesGravity and thermodynamics of horizon spaces and their

interrelations are currently most actively studied [67ndash79] Letus consider a relatively simple illustrationmdashthe case of a staticspherically symmetric horizon in space-time the horizonbeing described by the metric

1198891199042= minus119891 (119903) 119888

21198891199052+ 119891minus1

(119903) 1198891199032+ 1199032119889Ω2 (23)

The horizon location will be given by a simple zero of thefunction 119891(119903) at the radius 119903 = 119886

This case is studied in detail by Padmanabhan in hisworks[67 78] and by the author of this paper in [80] We usethe notation system of [78] Let for simplicity the space bedenoted byH

It is known that for horizon spaces one can introduce thetemperature that can be identified with an analytic continua-tion to imaginary time In the case under consideration ([78]Equation (116))

119896119861119879 =

ℎ1198881198911015840(119886)

4120587 (24)

Therewith the conditions 119891(119886) = 0 and 1198911015840(119886) = 0 must

be fulfilledThen at the horizon 119903 = 119886 Einsteinrsquos equations have the

form

1198884

119866[1

21198911015840(119886) 119886 minus

1

2] = 4120587119875119886

2 (25)

where 119875 = 119879119903

119903is the trace of the momentum-energy tensor

and radial pressureNow we proceed to the variables ldquo120572rdquo from formula (17)

to consider (25) in a new notation expressing 119886 in terms ofthe corresponding deformation parameter 120572 In what followswe omit the subscript in formula (17) of 120572

119909 where the context

implies which index is the case In particular here we use 120572

instead of 120572119886 Then we have

119886 = 119897min120572minus12

(26)


Therefore

1198911015840(119886) = minus2119897

minus1

min12057232

1198911015840(120572) (27)

Substituting this into (25) we obtain in the consideredcase of Einsteinrsquos equations in the ldquo120572-representationrdquo asfollows [80]

1198884

119866(minus120572119891

1015840(120572) minus

1

2) = 4120587119875120572

minus11198972

min (28)

Multiplying the left- and right-hand sides of the lastequation by 120572 we get

1198884

119866(minus1198911015840(120572) 1205722minus

1

2120572) = 4120587119875119897

2

min (29)

Lhs of (29) is dependent on 120572 Because of this rhs of(29) must be dependent on 120572 as well that is 119875 = 119875(120572) thatis

1198884

119866(minus1198911015840(120572) 1205722minus

1

2120572) = 4120587119875 (120572) 119897

2

min (30)

Note that in this specific case the parameter 120572 withinconstant factors is coincident with the Gaussian curvature119870

119886

[81] corresponding to 119886

1198972

min1198862

= 1198972

min119870119886 (31)

Substituting rhs of (31) into (30) we obtain the Einsteinequation on horizon in this case in terms of the Gaussiancurvature

1198884

119866(minus1198911015840(119870119886)1198702

119886minus

1

2119870119886) = 4120587119875 (119870

119886) (32)

This means that up to the constants

minus1198911015840(119870119886)1198702

119886minus

1

2119870119886= 119875 (119870

119886) (33)

that is the Gaussian curvature 119870119886is a solution of Einstein

equations in this caseThen we examine different cases of thesolution (33) with due regard for considerations of formula(21)

(21) First let us assume that 119886 ≫ 119897min As according to(7) the radius 119886 is quantized we have 119886 = 119873

119886119897min with the

natural number 119873119886≫ 1 Then it is clear that the Gaussian

curvature119870119886= 1119886

2asymp 0 takes a (nonuniform) discrete series

of values close to zero and within the factor 11198972min this seriesrepresents inverse squares of natural numbers

(119870119886) = (

1

1198732119886

1

(119873119886plusmn 1)2

1

(119873119886plusmn 2)2 ) (34)

Let us return to formulas (9) and (21) for 119897 = 119886 consider1003816100381610038161003816100381610038161003816((120575119886)

120574)min

1003816100381610038161003816100381610038161003816= 120573119873119886119897min119873

minus120574

119886= 120573119873

1minus120574

119886119897min (35)

where 120573 in this case contains the proportionality factor thatrelates 119897min and 119897

119875

Then according to (21) 119886plusmn1

is a measurable value of theradius 119903 following after 119886 and we have

(119886plusmn1)120574equiv 119886 plusmn ((120575119886)

120574)min

= 119886 plusmn 1205731198731minus120574

119886119897min

= 119873119886(1 plusmn 120573119873

minus120574

119886) 119897min

(36)

But as 119873119886≫ 1 for sufficiently large 119873

119886and fixed 120574 the

bracketed expression in rhs (36) is close to 1

1 plusmn 120573119873minus120574

119886asymp 1 (37)

Obviously we get

lim119873119886rarrinfin

(1 plusmn 120573119873minus120574

119886) 997888rarr 1 (38)

As a result the Gaussian curvature119870119886plusmn1

corresponding to119903 = 119886plusmn1

119870119886plusmn1

=1

1198862plusmn1

prop1

1198732119886(1 plusmn 120573119873

minus120574

119886 )2 (39)

in the case under study is only slightly different from119870119886

And this is the case for sufficiently large values of 119873119886

for any value of the parameter 120574 and for 120574 = 1 as wellcorresponding to the absolute minimum of fluctuations asymp

119897min or more precisely to 120573119897min However as all the quantitiesof the length dimension are quantized and the factor 120573 is onthe order of 1 actually we have 120573 = 1

Because of this provided the minimal length is involved119897min (9) is read as

10038161003816100381610038161003816(120575119897)1

10038161003816100381610038161003816min= 119897min (40)

But according to (12) 119897min = 120585119897119875is on the order of

Planckrsquos length and it is clear that the fluctuation |(120575119897)1|min

corresponds to Planckrsquos energies and Planckrsquos scales TheGaussian curvature119870

119886 due to its smallness (119870

119886≪ 1 up to the

constant factor 119897minus2min) and smooth variations independent of 120574(formulas (36)ndash(39)) is insensitive to the differences betweenvarious values of 120574

Consequently for sufficiently small Gaussian curvature119870119886 we can take any parameter from the interval 0 lt 120574 le 1 as

120574It is obvious that the case 120574 = 1 that is |(120575119897)

1|min = 119897min

is associated with infinitely small variations 119889119886 of the radius119903 = 119886 in the Riemannian geometry

Since then119870119886is varying practically continuously in terms

of 119870119886up to the constant factor we can obtain the following

expression

119889 [119871 (119870119886)] = 119889 [119875 (119870

119886)] (41)

where we have

119871 (119870119886) = minus119891

1015840(119870119886)1198702

119886minus

1

2119870119886 (42)

that is lhs of (32) (or (33))


But in fact as in this case the energies are low it is morecorrect to consider

119871 ((119870119886plusmn1

)120574)

minus 119871 (119870119886) = [119875 (119870

119886plusmn1)120574] minus [119875 (119870

119886)] equiv 119865

120574[119875 (119870

119886)]

(43)

where 120574 lt 1 rather than (41)In view of the foregoing arguments (21) the difference

between (43) and (41) is insignificant and it is perfectlycorrect to use (41) instead of (43)

(22) Now we consider the opposite case or the transitionto the ultraviolet limit

119886 997888rarr 119897min = 120581119897min (44)

That is

119886 = 120581119897min (45)

Here 120581 is on the order of 1Taking into consideration point (11) stating that in this

case models for different values of the parameter 120574 arecoincident by formula (40) for any 120574 we have

100381610038161003816100381610038161003816(120575119897)120574

100381610038161003816100381610038161003816min=10038161003816100381610038161003816(120575119897)1

10038161003816100381610038161003816min= 119897min (46)

But in this case the Gaussian curvature119870119886is not a ldquosmall

valuerdquo continuously dependent on 119886 taking according to(39) a discrete series of values119870

119886 119870119886plusmn1

119870119886plusmn2

Yet (25) similar to (32) (33) is valid in the semiclassical

approximation only that is at low energiesThen in accordance with the above arguments the

limiting transition to high energies (44) gives a discrete chainof equations or a single equation with a discrete set ofsolutions as follows

minus1198911015840(119870119886)1198702

119886minus

1

2119870119886= Θ (119870

119886)

minus1198911015840(119870119886plusmn1

)1198702

119886plusmn1minus

1

2119870119886plusmn1

= Θ (119870119886plusmn1

)

(47)

and so on Here Θ(119870119886) is some function that in the limiting

transition to low energies must reproduce the low-energyresult to a high degree of accuracy that is 119875(119870

119886) appears for

119886 ≫ 119897min from formula (33)

lim119870119886rarr0

Θ(119870119886) = 119875 (119870

119886) (48)

In general Θ(119870119886) may lack coincidence with the high-

energy limit of the momentum-energy tensor trace (if any)

lim119886rarr 119897min

119875 (119870119886) (49)

At the same timewhenwenaturally assume that the staticspherically-symmetric horizon space-time with the radiusof several Planckrsquos units (45) is nothing else but a microblack hole then the high-energy limit (49) is existing andthe replacement of Θ(119870

119886) by 119875(119870

119886) in rhs of the foregoing

equations is possible to give a hypothetical gravitationalequation for the event horizon micro black hole But aquestion arises for which values of the parameter 119886 (45) (or119870119886) is this valid and what is a minimal value of the parameter

120574 = 120574(119886) in this caseIn all the cases under study (21) and (22) the deforma-

tion parameter 120572119886(17) (120582

119886(18)) is within the constant factor

coincident with the Gaussian curvature 119870119886(resp radic119870

119886) that

is in essence continuous in the low-energy case and discretein the high-energy case

In this way the above-mentioned example shows thatdespite the absence of infinitesimal spatial-temporal incre-ments owing to the existence of 119897min and the essentialldquodiscretenessrdquo of a theory this discreteness at low energies isnot felt the theory being actually continuous The indicateddiscreteness is significant only in the case of high (Planckrsquos)energies

In [78] it is shown that the Einstein equation for horizonspaces in the differential formmay be written as a thermody-namic identity (the first principle of thermodynamics) ([78]formula (119))

ℎ1198881198911015840(119886)

4120587⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896119861119879

1198883

119866ℎ119889(

1

441205871198862)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119889119878

minus1

2

1198884119889119886

119866⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

minus119889119864

= 119875119889(4120587

31198863)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119875119889119881

(50)

where as noted above 119879 is temperature of the horizonsurface 119878 is corresponding entropy 119864 is internal energy and119881 is space volume

Note that because of the existing 119897min practically allquantities in (50) (except of 119879) represent GEQ Apparentlythe radius of a sphere 119903 = 119886 its volume 119881 and entropyrepresent such quantities

119878 =41205871198862

41198972119875

=1205871198862

1198972119875

(51)

within the constant factor 141198972

119875equal to a sphere with the

radius 119886Because of this there are no infinitesimal increments

of these quantities that is 119889119886 119889119881 119889119878 And provided 119897min isinvolved the Einstein equation for the above-mentioned casein the differential form (50) makes no sense and is uselessIf 119889119886 may be purely formerly replaced by 119897min then as thequantity 119897min is fixed it is obvious that ldquo119889119878rdquo and ldquo119889119881rdquo in (50)will be growing as 119886 and 119886

2 respectively And at low energiesthat is for large values of 119886 ≫ 119897min this naturally leads toinfinitely large rather than infinitesimal values

In a similar way it is easily seen that the ldquoEntropicApproach to Gravityrdquo [82] in the present formalism is invalidwithin the scope of the minimal length theory In fact theldquomain instrumentrdquo in [82] is a formula for the infinitesimalvariation 119889119873 in the bit numbers119873 on the holographic screenSwith the radius119877 andwith the surface area119860 ([82] formula(418))

119889119873 =1198883

119866ℎ119889119860 =

119889119860

1198972119875

(52)

As 119873 = 1198601198972

119875and 119860 represents GEQ it is clear that 119873 is

also GEQ and hence neither 119889119860 nor 119889119873makes sense


It is obvious that infinitesimal variations of the screensurface area 119889119860 are possible only in a continuous theoryinvolving no 119897min

When 119897min prop 119897119875is involved the minimal variation Δ119860 is

evidently associated with a minimal variation in the radius 119877

119877 997888rarr 119877 plusmn 119897min (53)

is dependent on 119877 and growing as 119877 for 119877 ≫ 119897min

Δplusmn119860 (119877) = (119860 (119877 plusmn 119897min) minus 119860 (119877)) prop (

plusmn2119877

119897min+ 1)

= plusmn2119873119877+ 1

(54)

where119873119877= 119877119897min as indicated above

But as noted above a minimal increment of the radius 119877equal to |Δmin119877| = 119897min prop 119897

119875corresponds only to the case

of maximal (Planckrsquos) energies or similarly to the parameter120574 = 1 in formula (21) However in [82] the considered lowenergies are far from the Planck energies and hence in thiscase in (21) 120574 lt 1 (53) and (54) are respectively replacedby

119877 997888rarr 119877 plusmn1198731minus120574

119877119897min (55)

Δplusmn119860 (119877) = (119860 (119877 plusmn 119873

1minus120574

119877119897min) minus 119860 (119877)) prop plusmn119873

2minus120574

119877+ 1198732minus2120574

119877

= 1198732minus2120574

119877(plusmn119873120574

119877+ 1)

(56)

An increase of rhs in (56) with the growth of 119877 (oridentically of119873

119877) for 119877 ≫ 119897min is obvious

So if 119897min is involved formula (418) from [82] makes nosense similar to other formulae derived on its basis (419)(420) (422) (532)ndash(534) in [82] and similar to thederivation method for Einsteinrsquos equations proposed in thiswork

Proceeding from the principal parameters of this work120572119897(or 120582119897) the fact is obvious and is supported by formula (19)

given in this paper meaning that

120572minus1

119877sim 119860 (57)

That is small variations of 120572119877(low energies) result in large

variations of 120572minus1119877 as indicated by formula (54)

In fact we have no-go theoremsThe last statements concerning 119889119878 119889119873 may be explicitly

interpreted using the language of a quantum informationtheory as follows due to the existence of the minimal length119897min the minimal area 1198972min and volume 1198973min are also involvedand that means ldquoquantizationrdquo of the areas and volumes Asup to the known constants the ldquobit numberrdquo119873 from (52) andthe entropy 119878 from (51) are nothing else but

119878 =119860

41198972min 119873 =

119860

1198972min (58)

it is obvious that there is a ldquominimalmeasurerdquo for the ldquoamountof datardquo that may be referred to as ldquoone bitrdquo (or ldquoone qubitrdquo)

The statement that there is no such quantity as 119889119873 (andresp 119889119878) is equivalent to claiming the absence of 025 bit0001 bit and so on

This inference completely conforms to the Hooft-Susskind Holographic Principle (HP) [60ndash63] that includestwo main statements as follows

(a) All information contained in a particular spatialdomain is concentrated at the boundary of thisdomain

(b) A theory for the boundary of the spatial domainunder study should contain maximally one degree offreedom per Planckrsquos area 1198972

119875

In fact (but not explicitly) HP implicates the existence of119897min = 119897

119875 The existence of 119897min prop 119897

119875totally conforms to HP

providing its generalization Specifically without the loss ofgenerality 1198972

119875in point (b) may be replaced by 119897

2

minSo the principal inference of this work is as follows

provided the minimal length 119897min is involved its existencemust be taken into consideration not only at high but alsoat low energies both in a quantum theory and in gravityThis becomes apparent by rejection of the infinitesimalquantities associated with the spatial-temporal variations119889119909120583 In other words with the involvement of 119897min the

general relativity (GR) must be replaced by a (still unframed)minimal length gravitation theory that may be denoted asGrav119897min In their results GR and Grav119897min should be very closebut as regards their mathematical apparatus (instruments)these theories are absolutely different

Besides Grav119897min should offer a rather natural transitionfrom high to low energies

[119873119871asymp 1] 997888rarr [119873

119871≫ 1] (59)

and vice versa

[119873119871≫ 1] 997888rarr [119873

119871asymp 1] (60)

where 119873119871

is integer from formula (7) determining thecharacteristics scale of the lengths 119871 (energies 119864 sim 1119871 prop

1119873119871)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] M E Peskin and D V Schroeder An Introduction to QuantumField Theory Addison-Wesley Reading Mass USA 1995

[2] R M Wald General Relativity University of Chicago Press1984

[3] G Veneziano ldquoA stringy nature needs just two constantsrdquoEurophysics Letters vol 2 no 3 p 199 1986

[4] D Amati M Ciafaloni and G Veneziano ldquoCan spacetime beprobed below the string sizerdquo Physics Letters B vol 216 no 1-2pp 41ndash47 1989


[5] E Witten ldquoReflections on the fate of spacetimerdquo Physics Todayvol 49 no 4 pp 24ndash30 1996

[6] J Polchinski String Theory vol 1 amp 2 Cambridge UniversityPress 1998

[7] C Rovelli ldquoLoop quantum gravity the first 25 yearsrdquo Classicaland Quantum Gravity vol 28 no 15 Article ID 153002 2011

[8] G Amelino-Camelia ldquoQuantum spacetime phenomenologyrdquoLiving Reviews in Relativity vol 16 p 5 2013

[9] L Garay ldquoQuantum gravity and minimum lengthrdquo Interna-tional Journal of Modern Physics A vol 10 no 2 pp 145ndash1651995

[10] G Amelino-Camelia and L Smolin ldquoProspects for constrainingquantum gravity dispersion with near term observationsrdquoPhysical Review D vol 80 no 8 Article ID 084017 2009

[11] G Gubitosi L Pagano G Amelino-Camelia A Melchiorriand A Cooray ldquoA constraint on Planck-scale modificationsto electrodynamics with CMB polarization datardquo Journal ofCosmology and Astroparticle Physics vol 2009 no 8 article 212009

[12] G Amelino-Camelia ldquoBuilding a case for a PLANCK-scale-deformed boost action the Planck-scale particle-localizationlimitrdquo International Journal of Modern Physics D vol 14 no 12pp 2167ndash2180 2005

[13] S HossenfelderM Bleicher S Hofmann J Ruppert S SchererandH Stocker ldquoSignatures in the Planck regimerdquoPhysics LettersB vol 575 no 1-2 pp 85ndash99 2003

[14] S Hossenfelder ldquoRunning coupling with minimal lengthrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 70 no 10 Article ID 105003 2004

[15] S Hossenfelder ldquoA note on theories with a minimal lengthrdquoClassical and Quantum Gravity vol 23 no 5 pp 1815ndash18212006

[16] S Hossenfelder ldquoMinimal length scale scenarios for quantumgravityrdquo Living Reviews in Relativity vol 16 p 2 2013

[17] R J Adler and D I Santiago ldquoOn gravity and the uncertaintyprinciplerdquoModern Physics Letters A vol 14 p 1371 1999

[18] D V Ahluwalia ldquoWave-particle duality at the Planck scalefreezing of neutrino oscillationsrdquo Physics Letters A vol 275 no1 pp 31ndash35 2000

[19] D V Ahluwalia ldquoInterface of gravitational and quantumrealmsrdquoModern Physics Letters A vol 17 no 15ndash17 p 1135 2002

[20] MMaggiore ldquoThe algebraic structure of the generalized uncer-tainty principlerdquo Physics Letters B vol 319 no 1ndash3 pp 83ndash861993

[21] A Kempf G Mangano and R B Mann ldquoHilbert space repre-sentation of the minimal length uncertainty relationrdquo PhysicalReview D vol 52 no 2 pp 1108ndash1118 1995

[22] K Nozari and A Etemadi ldquoMinimal length maximal momen-tum and Hilbert space representation of quantum mechanicsrdquoPhysical Review D vol 85 Article ID 104029 2012

[23] R Penrose ldquoQuantum theory and space-timerdquo in The Natureof Space and Time S Hawking and R Penrose Eds FourthLecture Prinseton University Press 1996

[24] W Heisenberg ldquoUber den anschaulichen Inhalt der quanten-theoretischen Kinematik und Mechanikrdquo Zeitschrift fur Physikvol 43 pp 172ndash198 1927 English translation J A Wheeler andH Zurek QuantumTheory and Measurement Princeton UnivPress pp 62ndash84 1983

[25] J A Wheeler Geometrodynamics Academic Press New YorkNY USA 1962

[26] C W Misner K S Thorne and J A Wheeler Gravitation WH Freeman San Francisco Calif USA 1973

[27] R Garattini ldquoA spacetime foam approach to the cosmologicalconstant and entropyrdquo International Journal of Modern PhysicsD vol 11 no 4 pp 635ndash651 2002

[28] R Garattini ldquoA spacetime foam approach to the Schwarzschild-de Sitter entropyrdquo Entropy vol 2 no 1 pp 26ndash38 2000

[29] R Garattini ldquoEntropy and the cosmological constant aspacetime-foam approachrdquo Nuclear Physics B vol 88 pp 297ndash300 2000

[30] RGarattini ldquoEntropy from the foamrdquoPhysics Letters B vol 459no 4 pp 461ndash467 1999

[31] F Scardigli ldquoBlack hole entropy a spacetime foam approachrdquoClassical andQuantumGravity vol 14 no 7 pp 1781ndash1793 1997

[32] F Scardigli ldquoGeneralized uncertainty principle in quantumgravity from micro-black hole gedanken experimentrdquo PhysicsLetters B vol 452 no 1-2 pp 39ndash44 1999

[33] F Scardigli ldquoGravity coupling frommicro-black holesrdquoNuclearPhysics BmdashProceedings Supplements vol 88 pp 291ndash294 2000

[34] L J Garay ldquoThermal properties of spacetime foamrdquo PhysicalReview D vol 58 Article ID 124015 1998

[35] L J Garay ldquoSpacetime foam as a quantum thermal bathrdquoPhysical Review Letters vol 80 no 12 pp 2508ndash2511 1998

[36] Y J Ng and H van Dam ldquoMeasuring the foaminess of space-timewith gravity-wave interferometersrdquo Foundations of Physicsvol 30 no 5 pp 795ndash805 2000

[37] Y J Ng ldquoSpacetime foamrdquo International Journal of ModernPhysics D vol 11 p 1585 2002

[38] Y J Ng ldquoSelected topics in planck-scale physicsrdquo ModernPhysics Letters A vol 18 no 16 p 1073 2003

[39] Y J Ng ldquoQuantum foamrdquo in Proceedings of the 10th MarcelGrossmannMeeting pp 2150ndash2164 Rio de Janeiro Brazil 2003

[40] Y J Ng and H van Dam ldquoSpacetime foam holographicprinciple and black hole quantum computersrdquo InternationalJournal of Modern Physics A vol 20 no 6 pp 1328ndash1335 2005

[41] W A Christiansen Y J Ng and H van Dam ldquoProbingspacetime foam with extragalactic sourcesrdquo Physical ReviewLetters vol 96 no 5 Article ID 051301 2006

[42] Y J Ng ldquoHolographic foam dark energy and infinite statisticsrdquoPhysics Letters B vol 657 no 1ndash3 pp 10ndash14 2007

[43] Y Jack Ng ldquoSpacetime foam and dark energyrdquo AIP ConferenceProceedings vol 1115 p 74 2009

[44] W A Christiansen Y J Ng D J E Floyd and E S PerlmanldquoLimits on spacetime foamrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 83 no 8 Article ID 0840032011

[45] G Amelino-Camelia ldquoGravity-wave interferometers as quan-tum-gravity detectorsrdquo Nature vol 398 no 6724 pp 216ndash2181999

[46] L Diosi and B Lukacs ldquoOn theminimum uncertainty of space-time geodesicsrdquo Physics Letters A vol 142 no 6-7 pp 331ndash3341989

[47] A E Shalyt-Margolin and J G Suarez ldquoQuantummechanics ofthe early universe and its limiting transitionrdquo httparxivorgabsgr-qc0302119

[48] A E Shalyt-Margolin and J G Suarez ldquoQuantum mechanicsat Planckrsquos scale and density matrixrdquo International Journal ofModern Physics Part D Gravitation Astrophysics Cosmologyvol 12 no 7 pp 1265ndash1278 2003


[49] A E Shalyt-Margolin and A Y Tregubovich ldquoDeformeddensity matrix and generalized uncertainty relation in thermo-dynamicsrdquo Modern Physics Letters A vol 19 no 1 pp 71ndash812004

[50] A E Shalyt-Margolin ldquoNon-unitary and unitary transitionsin generalized quantum mechanics new small parameter andinformation problem solvingrdquoModern Physics Letters A Parti-cles and Fields Gravitation Cosmology Nuclear Physics vol 19no 5 pp 391ndash403 2004

[51] A E Shalyt-Margolin ldquoPure states mixed states and Hawkingproblem in generalized quantum mechanicsrdquo Modern PhysicsLetters A vol 19 no 27 pp 2037ndash2045 2004

[52] A E Shalyt-Margolin ldquoThe universe as a nonuniform latticein finite-volume hypercube I Fundamental definitions andparticular featuresrdquo International Journal of Modern Physics Dvol 13 no 5 pp 853ndash863 2004

[53] A E Shalyt-Margolin ldquoTheUniverse as a nonuniform lattice inthe finite-dimensional hypercube II Simple cases of symmetrybreakdown and restorationrdquo International Journal of ModernPhysics A vol 20 no 20-21 pp 4951ndash4964 2005

[54] A E Shalyt-Margolin and V I Strazhev ldquoThe density matrixdeformation in quantum and statistical mechanics of the earlyuniverserdquo in Proceedings of the 6th International Symposium-Frontiers of Fundamental and Computational Physics B GSidharth Ed pp 131ndash134 2006

[55] A Shalyt-Margolin ldquoEntropy in the present and early universenew small parameters and dark energy problemrdquo Entropy vol12 no 4 pp 932ndash952 2010

[56] A E Shalyt-Margolin Quantum Cosmology Research Trendsvol 246 ofHorizons inWorld Physics Nova ScienceHauppaugeNY USA 2005

[57] E PWigner ldquoRelativistic invariance and quantumphenomenardquoReviews of Modern Physics vol 29 pp 255ndash268 1957

[58] H Salecker and E P Wigner ldquoQuantum limitations of themeasurement of space-time distancesrdquo vol 109 pp 571ndash5771958

[59] A E Shalyt-Margolin ldquoSpace-time fluctuations quantum fieldtheory with UV-cutoff and einstein equationsrdquo Nonlinear Phe-nomena in Complex Systems vol 17 no 2 pp 138ndash146 2014

[60] G rsquot Hooft ldquoDimensional reduction in quantum gravityrdquo Essaydedicated to Abdus Salam httparxivorgabsgr-qc9310026

[61] G Hooft ldquoThe holographic principlerdquo httparxivorgabshep-th0003004

[62] L Susskind ldquoTheworld as a hologramrdquo Journal ofMathematicalPhysics vol 36 no 11 pp 6377ndash6396 1995

[63] R Bousso ldquoThe holographic principlerdquo Reviews of ModernPhysics vol 74 no 3 pp 825ndash874 2002

[64] LD Faddeev ldquoAmathematical viewof the evolution of physicsrdquoPriroda no 5 pp 11ndash16 1989

[65] J D Bekenstein ldquoBlack holes and entropyrdquo Physical Review Dvol 7 pp 2333ndash2345 1973

[66] S W Hawking ldquoBlack holes and thermodynamicsrdquo PhysicalReview D vol 13 no 2 pp 191ndash204 1976

[67] T Padmanabhan ldquoClassical and quantum thermodynamics ofhorizons in spherically symmetric spacetimesrdquo Classical andQuantum Gravity vol 19 no 21 pp 5387ndash5408 2002

[68] T Padmanabhan ldquoA new perspective on gravity and dynamicsof space-timerdquo International Journal of Modern Physics D vol14 no 12 p 2263 2005

[69] T Padmanabhan ldquoThe holography of gravity encoded in arelation between entropy horizon area and action for gravityrdquoGeneral Relativity and Gravitation vol 34 no 12 pp 2029ndash2035 2002

[70] T Padmanabhan ldquoHolographic gravity and the surface term inthe Einstein-Hilbert actionrdquo Brazilian Journal of Physics vol 35no 2 p 362 2005

[71] T Padmanabhan ldquoGravity a new holographic perspectiverdquoInternational Journal of Modern Physics D vol 15 p 1659 2006

[72] G A Mukhopadhyay and T Padmanabhan ldquoHolography ofgravitational action functionalsrdquo Physical Review D vol 74 no12 Article ID 124023 15 pages 2006

[73] T Padmanabhan ldquoDark energy and gravityrdquo General Relativityand Gravitation vol 40 no 2-3 pp 529ndash564 2008

[74] T Padmanabhan and A Paranjape ldquoEntropy of null surfacesand dynamics of spacetimerdquo Physical ReviewD Particles FieldsGravitation and Cosmology vol 75 no 6 Article ID 0640042007

[75] T Padmanabhan ldquoGravity as an emergent phenomenon aconceptual descriptionrdquo AIP Conference Proceedings vol 939p 114 2007

[76] T Padmanabhan ldquoGravity and the thermodynamics of hori-zonsrdquo Physics Reports vol 406 no 2 pp 49ndash125 2005

[77] A Paranjape S Sarkar and T Padmanabhan ldquoThermodynamicroute to field equations in Lanczos-LOVelock gravityrdquo PhysicalReview D vol 74 no 10 Article ID 104015 2006

[78] T Padmanabhan ldquoThermodynamical aspects of gravity newinsightsrdquo Reports on Progress in Physics vol 73 no 4 ArticleID 046901 2010

[79] T Padmanabhan ldquoEquipartition of energy in the horizondegrees of freedom and the emergence of gravityrdquo ModernPhysics Letters A Particles and Fields Gravitation CosmologyNuclear Physics vol 25 no 14 pp 1129ndash1136 2010

[80] A E Shalyt-Margolin ldquoQuantum theory at planck scalelimiting values deformed gravity and dark energy problemrdquoInternational Journal of Modern Physics D vol 21 no 2 ArticleID 1250013 2012

[81] S Kobayashi and K Nomozu Foundations of DifferentialGeometry VII Interscience Publishers New York NY USA1969

[82] E Verlinde ldquoOn the origin of gravity and the laws of NewtonrdquoJournal of High Energy Physics vol 2011 no 4 article 29 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


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Statistical MechanicsInternational Journal of


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Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



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AerodynamicsJournal of

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Biophysics


ThermodynamicsJournal of


For the energies lower than Planckrsquos energy the ldquominimalvariations possiblerdquo Δmin119871 of the quantity 119871 having thedimensions of lengthmust be greater than 119897min and dependenton the present 119864

Δmin equiv Δmin119864 Δmin119864119871 gt 119897min (6)

Besides as we have a minimal length unit 119897min it isclear that any quantity having the dimensions of length isldquoquantizedrdquo that is its valuemeasured in the units 119897min equalsan integer number and we have

119871 = 119873119871119897min (7)

where119873119871is positive integer number

Theproblem is that how the ldquominimal variations possiblerdquoΔmin119864 (6) are dependent on the energy or similarly on thescales of the measured lengths

To solve the above-mentioned problem initially we canuse the space-time quantum fluctuations (STQF) with regardto quantum theory and gravity

The definition (STQF) is closely associated with thenotion of ldquospace-time foamrdquo The notion ldquospace-time foamrdquointroduced byWheeler about 60 years ago for the descriptionand investigation of physics at Planckrsquos scales (EarlyUniverse)[25 26] is fairly settled Despite the fact that in the last decadenumerous works have been devoted to physics at Planckrsquosscales within the scope of this notion for example [27ndash46]by this time still their is no clear understanding of the ldquospace-time foamrdquo as it is

On the other hand it is undoubtful that a quantum theoryof the Early Universe should be a deformation of the well-known quantum theory

In my works with colleagues [47ndash56] I have put forwardone of the possible approaches to resolution of a quantumtheory at Planckrsquos scales on the basis of the density matrixdeformation

In accordance with the modern concepts the space-timefoam [26] notion forms the basis for space-time at Planckrsquosscales (Big Bang) This object is associated with the quantumfluctuations generated by uncertainties in measurements ofthe fundamental quantities inducing uncertainties in anydistance measurement A precise description of the space-time foam is still lacking along with an adequate quantumgravity theory But for the description of quantum fluctua-tions we have a number of interesting methods (eg [36ndash46 57 58])

In what follows we use the terms and symbols from [38]Then for the fluctuations 120575119897 of the distance 119897 we have thefollowing estimate

(120575119897)120574≳ 119897120574

1198751198971minus120574

= 119897119875(

119897

119897119875

)

1minus120574

= 119897 (119897119875

119897)

120574

= 119897120582120574

119897 (8)

or that same one

100381610038161003816100381610038161003816(120575119897)120574

100381610038161003816100381610038161003816min= 120573119897120574

1198751198971minus120574

= 120573119897119875(

119897

119897119875

)

1minus120574

= 120573119897120582120574

119897 (9)

where 0 lt 120574 le 1 coefficient 120573 is of order 1 and 120582119897equiv 119897119875119897

From (8) and (9) we can derive the quantum fluctuationsfor all the primary characteristics specifically for the time(120575119905)120574 energy (120575119864)

120574 and metrics (120575119892

120583])120574 In particular for(120575119892120583])120574 we can use formula (10) in [38]

(120575119892120583])120574≳ 120582120574 (10)

Further in the text it is assumed that the theory involvesa minimal length on the order of Planckrsquos length

119897min prop 119897119875 (11)

or similarly

119897min = 120585119897119875 (12)

where the coefficient 120585 is on the order of unity tooIn this case the origin of the minimal length is not

important For simplicity we assume that it comes from thegeneralized uncertainty principle (GUP) that is an extensionof HUP for Planckrsquos energies where gravity must be takeninto consideration [3ndash22]

Δ119909 geℎ

Δ119901+ 12057210158401198972

119875

Δ119901

ℎ (13)

Here1205721015840 is themodel-dependent dimensionless numericalfactor

Inequality (13) leads to the minimal length 119897min = 120585119897119875

=

2radic1205721015840119897119875

Therefore in this case replacement of Planckrsquos length bythe minimal length in all the above formulae is absolutelycorrect and is used without detriment to the generality [59]

119897119875997888rarr 119897min (14)

Thus 120582119897equiv 119897min119897 and then (8)ndash(10) upon the replacement

of (14) remain unchangedAs noted in the overview [38] the value 120574 = 23 derived

in [57 58] is totally consistent with the Holographic Principle[60ndash63]

The following points of importance should be noted [59](11) It is clear that at Planckrsquos scales that is at theminimal

length scales

119897 997888rarr 119897min (15)

models for different values of the parameter 120574 are coincident(12) As noted specifically in (7) provided some quantity

has a minimal measuring unit values of this quantity aremultiples of this unit

Naturally any quantity having a minimal measuring unitis uniformly discrete

The latter property is not met in particular by the energy119864

As 119864 sim 1119897 where 119897 is measurable scale the energy 119864 is adiscrete but nonuniformquantity It is clear that the differencebetween the adjacent values of 119864 is the less the lower 119864 Inother words for 119897 ≫ 119897min that is

119864 ≪ 119864119875 (16)

119864 becomes a practically continuous quantity




0 lt 120572119909=

1198972

min1199092

le1

4 (17)




120582119909= radic120572119909 (18)





ing

120572minus1

119897sim 119878119861119867

(19)

where

119878119861119867

=119860

41198972119901

(20)



119909


1003816100381610038161003816Δmin1198641198711003816100381610038161003816 =

100381610038161003816100381610038161003816(120575119871)120574

100381610038161003816100381610038161003816min (21)














119899repre-


119878119899= 119899119862119899119877119899minus1

119881119899= 119862119899119877119899 (22)


119899= 1205871198992










1198891199042= minus119891 (119903) 119888

21198891199052+ 119891minus1

(119903) 1198891199032+ 1199032119889Ω2 (23)




119896119861119879 =

ℎ1198881198911015840(119886)

4120587 (24)



form

1198884

119866[1

21198911015840(119886) 119886 minus

1

2] = 4120587119875119886

2 (25)

where 119875 = 119879119903







119886 = 119897min120572minus12

(26)


Therefore

1198911015840(119886) = minus2119897

minus1

min12057232

1198911015840(120572) (27)


1198884

119866(minus120572119891

1015840(120572) minus

1

2) = 4120587119875120572

minus11198972

min (28)


1198884

119866(minus1198911015840(120572) 1205722minus

1

2120572) = 4120587119875119897

2

min (29)


1198884

119866(minus1198911015840(120572) 1205722minus

1

2120572) = 4120587119875 (120572) 119897

2

min (30)


119886


1198972

min1198862

= 1198972

min119870119886 (31)


1198884

119866(minus1198911015840(119870119886)1198702

119886minus

1

2119870119886) = 4120587119875 (119870

119886) (32)


minus1198911015840(119870119886)1198702

119886minus

1

2119870119886= 119875 (119870

119886) (33)






curvature119870119886= 1119886



(119870119886) = (

1

1198732119886

1

(119873119886plusmn 1)2

1

(119873119886plusmn 2)2 ) (34)


120574)min

1003816100381610038161003816100381610038161003816= 120573119873119886119897min119873

minus120574

119886= 120573119873

1minus120574

119886119897min (35)


119875




120574)min

= 119886 plusmn 1205731198731minus120574

119886119897min

= 119873119886(1 plusmn 120573119873

minus120574

119886) 119897min

(36)




1 plusmn 120573119873minus120574

119886asymp 1 (37)

Obviously we get


(1 plusmn 120573119873minus120574

119886) 997888rarr 1 (38)



119870119886plusmn1

=1

1198862plusmn1

prop1

1198732119886(1 plusmn 120573119873

minus120574

119886 )2 (39)






10038161003816100381610038161003816(120575119897)1

10038161003816100381610038161003816min= 119897min (40)





119886≪ 1 up to the




1|min = 119897min




expression

119889 [119871 (119870119886)] = 119889 [119875 (119870

119886)] (41)

where we have

119871 (119870119886) = minus119891

1015840(119870119886)1198702

119886minus

1

2119870119886 (42)




119871 ((119870119886plusmn1

)120574)

minus 119871 (119870119886) = [119875 (119870

119886plusmn1)120574] minus [119875 (119870

119886)] equiv 119865

120574[119875 (119870

119886)]

(43)




119886 997888rarr 119897min = 120581119897min (44)

That is

119886 = 120581119897min (45)



100381610038161003816100381610038161003816(120575119897)120574

100381610038161003816100381610038161003816min=10038161003816100381610038161003816(120575119897)1

10038161003816100381610038161003816min= 119897min (46)



119886 119870119886plusmn1

119870119886plusmn2




minus1198911015840(119870119886)1198702

119886minus

1

2119870119886= Θ (119870

119886)

minus1198911015840(119870119886plusmn1

)1198702

119886plusmn1minus

1

2119870119886plusmn1

= Θ (119870119886plusmn1

)

(47)



119886) appears for

119886 ≫ 119897min from formula (33)

lim119870119886rarr0

Θ(119870119886) = 119875 (119870

119886) (48)




119875 (119870119886) (49)


119886) by 119875(119870




tion parameter 120572119886(17) (120582



119886) that




ℎ1198881198911015840(119886)

4120587⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896119861119879

1198883

119866ℎ119889(

1

441205871198862)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119889119878

minus1

2

1198884119889119886

119866⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

minus119889119864

= 119875119889(4120587

31198863)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119875119889119881

(50)



119878 =41205871198862

41198972119875

=1205871198862

1198972119875

(51)







119889119873 =1198883

119866ℎ119889119860 =

119889119860

1198972119875

(52)

As 119873 = 1198601198972







119877 997888rarr 119877 plusmn 119897min (53)



plusmn2119877

119897min+ 1)

= plusmn2119873119877+ 1

(54)






119877119897min (55)

Δplusmn119860 (119877) = (119860 (119877 plusmn 119873

1minus120574


2minus120574

119877+ 1198732minus2120574

119877

= 1198732minus2120574

119877(plusmn119873120574

119877+ 1)

(56)






120572minus1

119877sim 119860 (57)





119878 =119860

41198972min 119873 =

119860

1198972min (58)






119875






2





[119873119871asymp 1] 997888rarr [119873

119871≫ 1] (59)

and vice versa

[119873119871≫ 1] 997888rarr [119873

119871asymp 1] (60)

where 119873119871


1119873119871)



References


























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






0 lt 120572119909=

1198972

min1199092

le1

4 (17)




120582119909= radic120572119909 (18)





ing

120572minus1

119897sim 119878119861119867

(19)

where

119878119861119867

=119860

41198972119901

(20)



119909


1003816100381610038161003816Δmin1198641198711003816100381610038161003816 =

100381610038161003816100381610038161003816(120575119871)120574

100381610038161003816100381610038161003816min (21)














119899repre-


119878119899= 119899119862119899119877119899minus1

119881119899= 119862119899119877119899 (22)


119899= 1205871198992










1198891199042= minus119891 (119903) 119888

21198891199052+ 119891minus1

(119903) 1198891199032+ 1199032119889Ω2 (23)




119896119861119879 =

ℎ1198881198911015840(119886)

4120587 (24)



form

1198884

119866[1

21198911015840(119886) 119886 minus

1

2] = 4120587119875119886

2 (25)

where 119875 = 119879119903







119886 = 119897min120572minus12

(26)


Therefore

1198911015840(119886) = minus2119897

minus1

min12057232

1198911015840(120572) (27)


1198884

119866(minus120572119891

1015840(120572) minus

1

2) = 4120587119875120572

minus11198972

min (28)


1198884

119866(minus1198911015840(120572) 1205722minus

1

2120572) = 4120587119875119897

2

min (29)


1198884

119866(minus1198911015840(120572) 1205722minus

1

2120572) = 4120587119875 (120572) 119897

2

min (30)


119886


1198972

min1198862

= 1198972

min119870119886 (31)


1198884

119866(minus1198911015840(119870119886)1198702

119886minus

1

2119870119886) = 4120587119875 (119870

119886) (32)


minus1198911015840(119870119886)1198702

119886minus

1

2119870119886= 119875 (119870

119886) (33)






curvature119870119886= 1119886



(119870119886) = (

1

1198732119886

1

(119873119886plusmn 1)2

1

(119873119886plusmn 2)2 ) (34)


120574)min

1003816100381610038161003816100381610038161003816= 120573119873119886119897min119873

minus120574

119886= 120573119873

1minus120574

119886119897min (35)


119875




120574)min

= 119886 plusmn 1205731198731minus120574

119886119897min

= 119873119886(1 plusmn 120573119873

minus120574

119886) 119897min

(36)




1 plusmn 120573119873minus120574

119886asymp 1 (37)

Obviously we get


(1 plusmn 120573119873minus120574

119886) 997888rarr 1 (38)



119870119886plusmn1

=1

1198862plusmn1

prop1

1198732119886(1 plusmn 120573119873

minus120574

119886 )2 (39)






10038161003816100381610038161003816(120575119897)1

10038161003816100381610038161003816min= 119897min (40)





119886≪ 1 up to the




1|min = 119897min




expression

119889 [119871 (119870119886)] = 119889 [119875 (119870

119886)] (41)

where we have

119871 (119870119886) = minus119891

1015840(119870119886)1198702

119886minus

1

2119870119886 (42)




119871 ((119870119886plusmn1

)120574)

minus 119871 (119870119886) = [119875 (119870

119886plusmn1)120574] minus [119875 (119870

119886)] equiv 119865

120574[119875 (119870

119886)]

(43)




119886 997888rarr 119897min = 120581119897min (44)

That is

119886 = 120581119897min (45)



100381610038161003816100381610038161003816(120575119897)120574

100381610038161003816100381610038161003816min=10038161003816100381610038161003816(120575119897)1

10038161003816100381610038161003816min= 119897min (46)



119886 119870119886plusmn1

119870119886plusmn2




minus1198911015840(119870119886)1198702

119886minus

1

2119870119886= Θ (119870

119886)

minus1198911015840(119870119886plusmn1

)1198702

119886plusmn1minus

1

2119870119886plusmn1

= Θ (119870119886plusmn1

)

(47)



119886) appears for

119886 ≫ 119897min from formula (33)

lim119870119886rarr0

Θ(119870119886) = 119875 (119870

119886) (48)




119875 (119870119886) (49)


119886) by 119875(119870




tion parameter 120572119886(17) (120582



119886) that




ℎ1198881198911015840(119886)

4120587⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896119861119879

1198883

119866ℎ119889(

1

441205871198862)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119889119878

minus1

2

1198884119889119886

119866⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

minus119889119864

= 119875119889(4120587

31198863)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119875119889119881

(50)



119878 =41205871198862

41198972119875

=1205871198862

1198972119875

(51)







119889119873 =1198883

119866ℎ119889119860 =

119889119860

1198972119875

(52)

As 119873 = 1198601198972







119877 997888rarr 119877 plusmn 119897min (53)



plusmn2119877

119897min+ 1)

= plusmn2119873119877+ 1

(54)






119877119897min (55)

Δplusmn119860 (119877) = (119860 (119877 plusmn 119873

1minus120574


2minus120574

119877+ 1198732minus2120574

119877

= 1198732minus2120574

119877(plusmn119873120574

119877+ 1)

(56)






120572minus1

119877sim 119860 (57)





119878 =119860

41198972min 119873 =

119860

1198972min (58)






119875






2





[119873119871asymp 1] 997888rarr [119873

119871≫ 1] (59)

and vice versa

[119873119871≫ 1] 997888rarr [119873

119871asymp 1] (60)

where 119873119871


1119873119871)



References


























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Therefore

1198911015840(119886) = minus2119897

minus1

min12057232

1198911015840(120572) (27)


1198884

119866(minus120572119891

1015840(120572) minus

1

2) = 4120587119875120572

minus11198972

min (28)


1198884

119866(minus1198911015840(120572) 1205722minus

1

2120572) = 4120587119875119897

2

min (29)


1198884

119866(minus1198911015840(120572) 1205722minus

1

2120572) = 4120587119875 (120572) 119897

2

min (30)


119886


1198972

min1198862

= 1198972

min119870119886 (31)


1198884

119866(minus1198911015840(119870119886)1198702

119886minus

1

2119870119886) = 4120587119875 (119870

119886) (32)


minus1198911015840(119870119886)1198702

119886minus

1

2119870119886= 119875 (119870

119886) (33)






curvature119870119886= 1119886



(119870119886) = (

1

1198732119886

1

(119873119886plusmn 1)2

1

(119873119886plusmn 2)2 ) (34)


120574)min

1003816100381610038161003816100381610038161003816= 120573119873119886119897min119873

minus120574

119886= 120573119873

1minus120574

119886119897min (35)


119875




120574)min

= 119886 plusmn 1205731198731minus120574

119886119897min

= 119873119886(1 plusmn 120573119873

minus120574

119886) 119897min

(36)




1 plusmn 120573119873minus120574

119886asymp 1 (37)

Obviously we get


(1 plusmn 120573119873minus120574

119886) 997888rarr 1 (38)



119870119886plusmn1

=1

1198862plusmn1

prop1

1198732119886(1 plusmn 120573119873

minus120574

119886 )2 (39)






10038161003816100381610038161003816(120575119897)1

10038161003816100381610038161003816min= 119897min (40)





119886≪ 1 up to the




1|min = 119897min




expression

119889 [119871 (119870119886)] = 119889 [119875 (119870

119886)] (41)

where we have

119871 (119870119886) = minus119891

1015840(119870119886)1198702

119886minus

1

2119870119886 (42)




119871 ((119870119886plusmn1

)120574)

minus 119871 (119870119886) = [119875 (119870

119886plusmn1)120574] minus [119875 (119870

119886)] equiv 119865

120574[119875 (119870

119886)]

(43)




119886 997888rarr 119897min = 120581119897min (44)

That is

119886 = 120581119897min (45)



100381610038161003816100381610038161003816(120575119897)120574

100381610038161003816100381610038161003816min=10038161003816100381610038161003816(120575119897)1

10038161003816100381610038161003816min= 119897min (46)



119886 119870119886plusmn1

119870119886plusmn2




minus1198911015840(119870119886)1198702

119886minus

1

2119870119886= Θ (119870

119886)

minus1198911015840(119870119886plusmn1

)1198702

119886plusmn1minus

1

2119870119886plusmn1

= Θ (119870119886plusmn1

)

(47)



119886) appears for

119886 ≫ 119897min from formula (33)

lim119870119886rarr0

Θ(119870119886) = 119875 (119870

119886) (48)




119875 (119870119886) (49)


119886) by 119875(119870




tion parameter 120572119886(17) (120582



119886) that




ℎ1198881198911015840(119886)

4120587⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896119861119879

1198883

119866ℎ119889(

1

441205871198862)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119889119878

minus1

2

1198884119889119886

119866⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

minus119889119864

= 119875119889(4120587

31198863)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119875119889119881

(50)



119878 =41205871198862

41198972119875

=1205871198862

1198972119875

(51)







119889119873 =1198883

119866ℎ119889119860 =

119889119860

1198972119875

(52)

As 119873 = 1198601198972







119877 997888rarr 119877 plusmn 119897min (53)



plusmn2119877

119897min+ 1)

= plusmn2119873119877+ 1

(54)






119877119897min (55)

Δplusmn119860 (119877) = (119860 (119877 plusmn 119873

1minus120574


2minus120574

119877+ 1198732minus2120574

119877

= 1198732minus2120574

119877(plusmn119873120574

119877+ 1)

(56)






120572minus1

119877sim 119860 (57)





119878 =119860

41198972min 119873 =

119860

1198972min (58)






119875






2





[119873119871asymp 1] 997888rarr [119873

119871≫ 1] (59)

and vice versa

[119873119871≫ 1] 997888rarr [119873

119871asymp 1] (60)

where 119873119871


1119873119871)



References


























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119871 ((119870119886plusmn1

)120574)

minus 119871 (119870119886) = [119875 (119870

119886plusmn1)120574] minus [119875 (119870

119886)] equiv 119865

120574[119875 (119870

119886)]

(43)




119886 997888rarr 119897min = 120581119897min (44)

That is

119886 = 120581119897min (45)



100381610038161003816100381610038161003816(120575119897)120574

100381610038161003816100381610038161003816min=10038161003816100381610038161003816(120575119897)1

10038161003816100381610038161003816min= 119897min (46)



119886 119870119886plusmn1

119870119886plusmn2




minus1198911015840(119870119886)1198702

119886minus

1

2119870119886= Θ (119870

119886)

minus1198911015840(119870119886plusmn1

)1198702

119886plusmn1minus

1

2119870119886plusmn1

= Θ (119870119886plusmn1

)

(47)



119886) appears for

119886 ≫ 119897min from formula (33)

lim119870119886rarr0

Θ(119870119886) = 119875 (119870

119886) (48)




119875 (119870119886) (49)


119886) by 119875(119870




tion parameter 120572119886(17) (120582



119886) that




ℎ1198881198911015840(119886)

4120587⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896119861119879

1198883

119866ℎ119889(

1

441205871198862)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119889119878

minus1

2

1198884119889119886

119866⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

minus119889119864

= 119875119889(4120587

31198863)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119875119889119881

(50)



119878 =41205871198862

41198972119875

=1205871198862

1198972119875

(51)







119889119873 =1198883

119866ℎ119889119860 =

119889119860

1198972119875

(52)

As 119873 = 1198601198972







119877 997888rarr 119877 plusmn 119897min (53)



plusmn2119877

119897min+ 1)

= plusmn2119873119877+ 1

(54)






119877119897min (55)

Δplusmn119860 (119877) = (119860 (119877 plusmn 119873

1minus120574


2minus120574

119877+ 1198732minus2120574

119877

= 1198732minus2120574

119877(plusmn119873120574

119877+ 1)

(56)






120572minus1

119877sim 119860 (57)





119878 =119860

41198972min 119873 =

119860

1198972min (58)






119875






2





[119873119871asymp 1] 997888rarr [119873

119871≫ 1] (59)

and vice versa

[119873119871≫ 1] 997888rarr [119873

119871asymp 1] (60)

where 119873119871


1119873119871)



References


























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







119877 997888rarr 119877 plusmn 119897min (53)



plusmn2119877

119897min+ 1)

= plusmn2119873119877+ 1

(54)






119877119897min (55)

Δplusmn119860 (119877) = (119860 (119877 plusmn 119873

1minus120574


2minus120574

119877+ 1198732minus2120574

119877

= 1198732minus2120574

119877(plusmn119873120574

119877+ 1)

(56)






120572minus1

119877sim 119860 (57)





119878 =119860

41198972min 119873 =

119860

1198972min (58)






119875






2





[119873119871asymp 1] 997888rarr [119873

119871≫ 1] (59)

and vice versa

[119873119871≫ 1] 997888rarr [119873

119871asymp 1] (60)

where 119873119871


1119873119871)



References


























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics
























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



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