Journal of Applied Mathematics and Physics, 2013, 1, 25-44
Published Online November 2013 (http://www.scirp.org/journal/jamp)
http://dx.doi.org/10.4236/jamp.2013.15005
Open Access JAMP
Mathematical Nanotechnology: Quantum
Field Intentionality
Francisco Bulnes
Research Department in Mathematics and Engineering, Technological Institute of High Studies of Chalco,
Federal Highway Mexico-Cuautla s/n Tlapala “La Candelaria” Chalco, Mexico City, Mexico
Received July 29, 2013; revised August 29, 2013; accepted September 15, 2013
Copyright © 2013 Francisco Bulnes. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
Considering the finite actions of a field on the matter and the space which used to infiltrate their quantum reality at level
particle, methods are developed to serve to base the concept of “intentional action” of a field and their ordered and
supported effects (synergy) that must be realized for the “organized transformation” of the space and matter. Using path
integrals, these transformations are decoded and their quantum principles are shown.
Keywords: Intentional Action; Field Infiltration; Hyper-Reality; Synergic Action; Events Synchronization; Synergic
Operators; Cyber-Quantum Algebra; Perception Integral Transforms; Feynman-Bulnes Integrals; Newman
Dimension
1. Introduction
The nanotechnology will bring new paradigms of the
scientific thought and will change our limited vision of
the world since the real causes of any phenomenon initi-
ate from an atomic scale, the fifth part of the one hun-
dred millionth part of a meter (≈ (1/5)10−9 meters) (and
even subatomic if we want to understand the communi-
cation and transference of the states of energy between
atoms). Only in these small dimensions, the real reality
exists, since everything what our senses can receive there
are ordered holograms that obey a cosmic perception that
has been developed according to a field conscience that
spills their creative intentions from a quantum zone.
We consider a set of particles in the space , under
certain law of movement defined by their Lagrangian ,
we have that the action defined by a field that acts with
this movement law and that causes it, as is defined by the
map [1]:
E
L
:T ,ER (1)
with rule of correspondence









Flux ,
x
sLxs xs (2)
We can establish that the global action in a particle
system with instantaneous action can be re-interpreted
locally as a permanent action of the field considering the
synergy of the instantaneous temporary actions under this
permanent action of the field. This passes to the follow-
ing principle:
Principle 1. The temporary or instantaneous action on
a global scale can be measured like a local permanent
action
The previous principle together with certain laws of
synchronicity of events in the space time will shape one
of the governing principles of the nanotechnology, why?
Because, at microscopic level, the permanence of a
field is constant, in proportion to the interminable state
and permanence of energy that exist in the atoms. As a
result of this, a nano-technological process will be di-
rected to the manipulation of the microstructures of the
components of the matter using this principle of “inten-
tional action”. The time at quantum level is a time meas-
ured in ns (nano-seconds) which is compared to the time
measured in the scales that measure the visible effects in
any physical process; they turn out to be almost nonexis-
tent or void. Nevertheless, this nullity characterizes the
flatness of the space time to these quantum scales (the
time stops existing like an entity not separated from the
space), and is where the field finds permanence shaped
by a quantum sea of light of bosons. Then any macro-
scopic temporary action is measured by an action that is
born from a permanent field at local level, and every-
where, since it belongs to this quantum light sea.

F. BULNES 26
Let be the Lie group of actions defined by their
automorphisms
G
G


X

, such that these actions from ,
are defined in their algebra g, through the diffeomor-
phism , [2]. If we consider the alge-
bra e, that is to say, find their local application in
their algebra , then we can access to the quantum zone
of the nullity where the temporal effects are annulled.
Then in ,
G

ptX
g
g
ex
g
g
g
e

0,Xx


xM [3].
Then supposing that the field X can control under fi-
nite actions like the described for , and under the es-
tablished principle, we can execute an action on a micro-
structure always and when the sum of the actions of all
the particles is major than their algebraic sum (to give an
order to only one particle so that the others continue it).

How to obtain this combined effect of all the particles
that move and that is wanted to realize a coordinated
action (of tidy effect) and simultaneously (synchronicity),
with the only effect?
Inside the universe of minimal trajectories that satis-
fies the variation functional [1], we can choose a


t
  , such that
   
2
1
Exed ,
t
t
p
t
p
Lxs xs



  


 (3)
which is not arbitrary, since we can define any action on
γt, like
  
2
1
,,d
t
p
p
Lxs xs ss




,
(4)
that is to say, there is an intention defined by the field
action that infiltrates into the whole space of the particles
influencing or “infecting” the temporary or instantaneous
actions making that the particles arrange themselves all
and with added actions not in the algebraic sense, but in
the holistic sense. This action is the “conscience” that has
the field to exercise their action in “intelligent” form that
is to say, in organized form through their path integrals
like the described by the classical Feynman integrals.
Then extending the above mentioned integral to the
whole space 

, we have the synergic principle of
the total field X,








TOTALd
t
j
j
,
x
sxs


 (5)
the length and breadth of .
E
The order conscience is described by the operator of
execution of a finite action of a field X, on a target (re-
gion of space that must be infiltrated by the action of the
field which is that for which we realize our re-walked


).
Likewise if we choose a geodesic t

, of the field X,
on which an action will be applied , measurable in the
Lagrange ambience, we have that their execution is as
given in (3). The action must be realized in form sup-
ported along the object to which is required apply the
transformation due to X.

How to measure this transference of conscience of
transformation due to the field X, on an object defined by
a portion of the space



? Which is the limit of this
supported action or transference of conscience so that it
feeds the effect in the portion of the space



, and
the temporary or instantaneous actions for every particle
xi, are founded on only one global synergic action on

?
We measure this transference of conscience (or inten-
tion) of X, on a particle


x
s, through the value of the
integral of the spilled intelligence (path integral) given
as [4]:








11
,
,
Xxs x
xxxx xx


 


 




(6)
We left at level conjecture and based on our researches
of nanotechnology and advanced quantum mechanics
that a sensor for the quantum sensitization of any particle
that receives an instruction given by a field X, must sat-
isfy the inequality of Hilbert type [5], for this transfer-
ence of conscience is defined in (6) on the region



,
to know [4]:





,
loglog ,2
ab
Xxs t
xxcona

 

 b


(7)
The conscience transference demands a synchroniza-
tion of events that obeys to an effect of simultaneity and
coordination of temporary actions on a set of particles
that must behave under the same intensity that could be
programmed through “revisited” path integrals [6], pro-
ducing a joint effect called synergy [7].
In the above mentioned process, the time and the space
are interchangeable in the quantum zone as we can ob-
serve it in the integrals (3), and where a particle will be
and when it will be there, are aspects that go together.
Therefore not only the energy must be quantized but also
the magnitude “distance”, whose discrete length quantity
must perform a low order to 12−13 cm (spin relation be-
tween two connected virtual particles [8]). The energy is
to form a particle for an observer who lies in a space -
time . M
The above mentioned particle will disappear to the
view of the same observer if this one lives in one of the
coordinated axes of the space , if the observer is not
in any of these axes. On the other hand, for an observer
in the region
M



, this one will see the particles as tra-
jectories that fill the space as instantons with time [9].
Out of these ambiences the alone observer perceives a
cloud of energy that is permanent.
Any transformation that is wanted to realize a space,
Open Access JAMP

F. BULNES 27
takes as constant the same energy that comes from the
permanent field of the matter and which is determined by
the quantum field of the constituents particles


x
s, of
the space and matter. If we want to define a conscience
in the above mentioned field, that is to say, an action that
involves an intention is necessary to establish this inten-
tion inside the argument of the action.
Likewise, if


xs, and



x
s, is their action
due to a field of particles X, and is a spilled intention
defined by (3) showing the length and breadth of the
space , such that to satisfy (5), for all the possible
trajectories that fill , we have
M

   
TOTAL d,d
t
E
j
j
E
xs xs




 






 




(8)
where the total action is an intentional action (for all the
infinity of paths t

, that define to ) 

   

  
TOTAL
()
d
d,
c
Oxt xs
EE xsdxst




 



 







 





t















(9)
where the energy factor , represents the energy
needed by the always present force to realize the action
and c, is the conscience operator who defines the value
or record of the field X, (direction), on every particle of
the space
EE


O


, which along their set of trajectories

,
realizes the action of permanent field , which is
c
O

    

d
c
Oc
X
,
x
sO xsxs

M
(10)
where the operator c
O, invests an energy quasi infinite,
encapsulated in a microscopic region of the space (quan-
tum space ), and with applications and influence in
an unlimited space of the sub-particles (boson space).
Likewise a photon of certain class

M

x
s, will be gener-
ated by the quantum field (if it manages to change its
field spin) and will be moved for the intention on a tra-
jectory , by the path integral

 

   

d,
c
Oc
XC
xsO xsxs









I 


(11)
Interesting applications of the formula (9) to nano-
sciences will happen at the end of the present chapter.
Also it will be demonstrated that (9) is a quantum inte-
gral transform of bundles or distortions of energy in the
space-time if it involves a special kernel. The bundle
stops existing if certain intention is applied (path inte-
gral transform). The operator c, involves a connec-
tion of the tangent bundle of the space of trajectories
O


.
The integral (11) will determine on certain hypotheses
the interdependence between the material, quantum and
virtual realities in , (see formula (44)). M
Then under this perspective we can define the quan-
tum zone of the space-time as susceptible to the appli-
cation of intentions which establish statistical weights in
the elected trajectories of



 to do it. These statisti-
cal weights are born of considering the existence of dif-
feomorphisms between elements of the topological group
, that is subjacent in the space (structure of the
operators of the actions of the field X, in the space )
that wants to transform and corresponding algebra g
(Lie algebra) of this group that records the operators of
the actions of the field locally (space time of every par-
ticle).
GM
M
Definition 1.1. [12]. The space-time in the quantum
zone , is the spatial region of the flux of permanent
field (region of application of the field) where are the
diffeomorphic correspondences which are subjacent to
the relations between the macroscopic effects, and the
quantum phenomena of . The diffeomorphisms come
given for a uni-parametric group
M
M



i
etX
 , of the
quantum field of the space , [10,11] (generator of
quantum isometries) which defines the Lagrangian spe-
cial c, of a conscience operator defined for an integral
of action [12], as given by (10).
M
O
The macroscopic effects, that is to say, the effects
generated in the space , come from causes that take
place from the set of automorphisms of the group G
(topological group) in which the cause—effect laws are
established between elements of G, evaluated in every
particle.
M
Considering a particle system , in a space
time
12
,,pp
4

MR. Let


3,
t
x
tI

 R
x
being a tra-
jectory which predetermines a position for all
time
3,R
.
t
tI

A field
X
that infiltrates their action to
the whole space of points predetermined by all the tra-
jectories






,,,t xt

,
123 is the field that
pre-determines the points
xt x




,
ii
x
t 
which are fields
whose determination is given by the action of the field
X
, and evaluated in the position of every particle. Every
point has a defined force by the action , of 
X
, along
the geodesic ,
t

and determined direction by their tan-
gent bundle given for



,
1TX




1,X 
that is to say, the field provides direction to every point
,
i

because their tangent bundle has a spinor bundle
[13], where the field
S
X
comes given

,,
ii
i
ix
i
X




 1
,,, , 

X on every
particle




,1,2,
ii
pxti. Then to direct an inten-
tion we have the map or connection:







1
:*TT T

 MX,
(12)
with rule of correspondence
Open Access JAMP

F. BULNES 28



,,
iii
it
xx

,

  (13)
which produces the spinor field i thi-

[14], where the
action , of the field 
X
, infiltrates and transmits from
particle to particle in the whole space




, using a
configuration given by their Lagrangian L (conscience
operator), along all the trajectories of

. Then from
a sum of trajectories




,
F
Dxt
 one has the sum



d

,
x



, 
on all the possible field configurations n, m C.
Extending these intentions to the whole space
on all the elections of possible paths whose
statistical weight corresponds to the determined by the
intention of the field, and realizing the integration in
paths for an infinity of particles—fields in
M


,T it is
had that


 

 
1
0
,
1
1d d
lim
ed
n
ii
i
T
Ns
ii
i
xx
BB B
,
x
s







 
  













I

(14)
where
12
,
2is 





m
B is the amplitude of their
propagator and in the second integral of (14), we have
expressed the Feynman integral using the form of vol-
ume



,

x


T
of the space of all the paths that are
added in to obtain the real path of the particle
(where we have chosen quantized trajectories, that is to
say,

,



d

,
x





x
. Remember that the sum of all these
paths is the interference amplitude between paths that are
established under an action whose Lagrangian is

d,
x
x 


M

where, if is a complex
with , the space-time, and is a complex or
configuration space on , (interfered paths in the ex-
periment given by multiple split [11,15]), endowed with a
pairing

M

M

C
M




: *C 
MM,R

(15)
where is some dual complex (“forms on con-
figuration spaces”), i. e. such that “Stokes theorem”
holds:

*M
,d,
C




 (16)
then the integrals given by (14) can be written as (to
border points and inner points as: m- n-
  



1
1
1
12
T
1
1
()
d
dd
dd
n
m
tt
n
m
tt t
m m
qn
mm
n
xx 



 
 











 
 

This is an infiltration in the space-time by the direct
action

[4,6] that happens in the space ,C

 to
each component of the space through the ex-
pressed Lagrangian in this case by

,

, de (10) in (17),
the integration of the space is realized with the infiltra-
tion of the time, integrating only spinor elements of the
field.
Definition 1.2. (intentional action of
X
) [12]. Let
X
be a field acting on the particles






123
,,,,xt xt xtM
M
,
, and let be their action
on the mentioned particles above, under an operator who
recognizes the “target” in , (conscience operator). We
say that
,

is an conscientious intentional action (or
simply intention) of the field
X
, if and only if
1) ,

is the determination of the field
X
, to realise
or execute (their force);
2) ,

recognizes well their target, knowing what to
do of the field
X
(their direction follows a con-
figuration patron)
,
The definition 1. 2., is summed establishing the prin-
ciple:
Principle 2. [12,13]
Intention = Determination of the field to realize a ac-
tion + Knowledge of their objective
The knowledge of their target is the knowledge of their
effect. This brings with it the cause—effect knowledge,
which anticipates the action and determines their appli-
cation.
But how to construct a conscience that integrates the
time and space through the proper energy of the matter
that constitutes it, and that could transform the matter
constituted by these particles x(s), what the intentional
action wants?
The action must expand their intention, that is to say,
must be transmitted from particle to particle “infecting”
all space that it wants transform taking the same intention
given by the action, using the relation of distance that is
subjacent between connected virtual particles that take
this intention (since these that are photons (bosons, fer-
mions, gravitons, baryons, etc) shape the exchange of
energy information between atoms and weave shaping
the network of specific messages between the atoms).
,




(17)
Likewise, if is a space-time constituted by matter
defined by the particles and is desir-
able to realize a transformation of this matter transmit-
ting an intention of an action , given by a field
M
 
12
,,xt xt

X
,
through the states ,,,



 such that it satisfies (17)
with the angular relation between two connected states
given by their angular momentum (and this way succes-
sively to all the remaining states) then the intention is
transmitted like the quantum wave

 of the state ,
i

replaced with the state in the infinite
homomorphism [12,16] (which is of the type (14), later
in the Section 4, we will talk more about it)

,1,2,3,
ii


Open Access JAMP

F. BULNES
Open Access JAMP
29
 

 
11223344,tttt
 
 

 (18) In the quantum zone, the quantum particles field is
permanent and interminable, since matter and energy are
equivalent and the atoms are interminable (they do not
suffer wear), and what gets worn there are the linkages
between atoms which can weaken or get lost for absence
of a transmission of the states of suitable energy (routes
given by path integrals). Nevertheless, infiltrating the
intention on every path

, and under the condition of
permanent field given by the operators the trans-
mission of the states will be able to turn revitalized by
every node, transmitting the same information about
every box. We will call this characteristic a
intentionality [12]. But every particle with regard to oth-
ers takes their corresponding position, since they all have
the same infiltrated intention, by which the synergic ac-
tion is realized.
,
c
O
0000 - 
where the transmission of the quantum wave is realized
on the spinor space of ,t

where ,t

is the nanotech-
nology that is wanted to create in the class

. An inter-
esting re-interpretation of (18) is when this quantum
wave is a quantum code of intelligence spilled in the
memory of a microprocessor for the achievement of an
action of correction, re-composition, alignment, or resto-
ration of a field of particles [7]. Here the distinguishable
particles states are already bosons or fermions, they have
a job or arrangement that eliminates an infinity of the
states that by their sum of spins are annulled, leaving
alone those who realize an effective action. They annul
the shaken states or realize scattering [7].
Considering a space of configuration equiva-
lent to the complex given for composed for
hypercubes defined by 00 boxes, we can
define a network of paths that will be able to establish
routes of organized transformations on diagrams of
Feynman type (with path integrals with actions given by
and path integrals as given in (11)) which will es-
tablish the ideal route of the intention, considering the
action in every node of the network. Then these ar-
rangements can happen in the networks designed on a
field of particles that can be arranged in boxes
[16], where the action can be calculated in a point (node
of the crystalline network of a field [17]) corresponding
to the n-states of energy
,
n, m
C
000 0 

,CM
00 -n-
,
c
O
U,
-
i



1, 2, 3i, ,,
Principle 3. The nanotechnology is a science of the
organization of nano-components of the matter, space or
energy with quantum intention.
Example 1.1. The nano-components of different ob-
jects are the same, the only thing that changes is the ar-
rangement of their atoms to form particles or molecules
to create the different objects and realities in the
space-time (see Figure 1). M
n super-
posed near the node for


  
12
12
U
000 0-boxes
d
n
n
nXWXWXW
x
z
 

 








,


(19)
Example 1.2. The intention of an action takes implicit
the atomic visualization of every nano-component.
Likewise, if 12
,,, ,
n
pp p

M are particles with
an finite arrangement of the atoms 1 1NANO
n
,,aa , ,
m
a


i
S
m NA S
n
given by a struct,,
nm
C
 (configuration
spa in M), which considers configurations froNO ,
up to the particles of the material reality M, the in-
tention froNO , generates the organised transfor-
mation in the space. Likewise, we can say that the
nano-space is the zone where there initiates an organized
transformation of nano type (see Figure 2).
ure

CM
NA
S
M

ce
m
(a) (b)
Figure 1. (a) Nano-components; (b) Fi eld. (a) The preservation of the measure of the energy process in the nano-components
is given for their invariance. The ensemble of the nano-compone nts is realized without energy invariance. This demonstrates
that the supported action in all nanotechnological process is the showed by the figure (I) remember that the object obtains
their finished transformation in an established limit (Figure 3). (II) The action of the field is expanded to the whole space
achieving that the showed configuration in (I), is transmitted to the whole space, object of the transformation in continuous
form. This continuous transformation involves a homomorphism between the space of configuration and the space-time that
characterizes the continuous map to the na notechnological transformation. In the given simulating in (I), has been considered
a field


VectorPlot32, 42,XDxx

, xwhose expansion in is the incise (II). The expansion preserve volume of matter,
that is to say the volume of particles set that is transformed remains invariant.
3,R

F. BULNES 30
Figure 2. The intention infiltrated by the conscience given for establishes that the differential of the action
c
O,


hd


(using the energy (amplitude ) that their propagator contributes
F
O) can be visualised i nside the config uration space through
their boarder points (“targets” of the intention of the field X, and that happen in M

), being also the interior points of the
space , , are the proper sources of the field (particles of the space , that generate the field MintM M
X
). Then the intention
of the field
X
, is the total action


Tin
.
MM
 
t
We consider the following postulate:
Postulate 1. Infinitely big is contained in infinitely
small. It allows comprising the cause of any observable
effect.
Due to the postulate 1, the measurement of a process
of nanotechnology is realized according to their capacity
of ensemble of the nano-components (particles) under an
energy condition, that is to say, for every time
s
, of the
particles

x
s, a map exists
0
:
t
TH H, (20)
with rule of correspondence






,
s
x
Tx   (21)
that for a sustainable period of time and considering
their operator it had that the supported action satis-
fies
,T
,
c
O

0
1
limd ,
ct
T
OTc s
TOT s
T



 

 (22)
We will see eventually that a ,

is always associated
with a measure ,


1,
on energy region space
with so that
,
E
JH

E
J 






d
E
cc
J
OOs ,



 (23)
Thus the limit given in (22) exists for each fixed ,

and is independent of ,
E
J

then there is a measure
,

on ,
E
J
so that
 
0
1
limdd ,
E
T
cs c
TJ
OT sO
T



 


 (24)
The measure ,

has a very important property. Let
,

be fixed and suppose ,
E
J
the characteristic func-
tion of a measurable set ,
E
F
J then



1
1
0
0
0
1d
1d
1d,
u v
u
T
TF
T
TF v
T
Fuv
s
T
Ts
T
TT s
T


















so if the Texists, then lim ,

1
(T ),
u
F
F
 
,T
that is,
the measure is invariant. Then we say that is measure
preserving (see Figure 1).
The volume conservation of particles is the conserva-
tion of the flow of particles ,
c
F
luxO in then we
have the following invariability principle of the meas-
urement in any nanotechnology process:
,H
Principle 4. In any nanotechnological process there is
preserved the measurement of volume of the set of parti-
cles led by c of the region of space-time that is trans-
formed. There must not be losses of volume.
,O
Example 1.3. In the Hamiltonian formulation of clas-
sical mechanics, the position and momentum of a system
are described by the points of the cotangent bundle
. Likewise, this is a symplectic manifold in a natu-
ral way. The physics aspects of the system are described
by a so-called Hamiltonian function and
the evolution of the system is given by the flow of the
Hamiltonian vector field
T*M
:T*,HMR
,
H
X
associated to ,
H
satis-
fying the Lioville integration theorem [1,18].
In the results of this paper, this property will be dis-
cussed as fundamental in the nanotechnological chore.
2. Intentionality
We consider t
3,
I

MR the space-time of certain
particles


,
x
s in movement, and let be an opera-
tor that explains certain law of movement that governs
,L
Open Access JAMP

F. BULNES 31
the movement of the set of particles in , of such way
that the energy conservation law is applied for the total
action of each one of their particles. The movement of all
the particles of the space , is given geometrically for
their tangent vector bundle . Then the action due to
on , is defined like [1]:
M

,
M
T
:TM
M
,LM
,
L
R (25)
with rule of correspondence




x
s xsFlux
E
L

xs (26)
and whose energy due to the movement is
,L

(27)
But this energy is given from their Lagrangian
defined like [19]

T, ,LC
MR


















,,T ,,,Lxxs sss xs s

,xs Vx,s xs

(28)
If we want to calculate the action defined in (13) and
(14), along a given path

,
x
s

we have that the
action is





,,Lxs s



d,ss
M
x
T*
(29)
If this action involves an intention (that is to say, is an
intentional action) then the action is translated in all the
possible field configurations, considering all the varia-
tions of the action along the fiber derivative defined by
the Lagrangian L. Of this way, the conscience operator is
the map
:T ,
c
OM (30)
with corresponding rule
 
0, 
d
ds
cs
OvwtwLv (31)
That is, , is the derivative of , along the
fiber in direction . In the case of

c
Ovw
w
L


,
x
s

  and


,sqx



1,EVVq   ,,q2  ,qLM


we see that

,
c
Ov
T * M
w
,

M
w , so we recover the usual
map (with b, Euclidean in ) as-
sociated with the bilinear form
s: T
b3
R
,. Is here where the
spin structure subjacent appears in the momentum of the
particle x(s).
As we can see, , carries a canonical symplectic
form, which we call
T*M

. Using we obtain a closed
two-form
,
c
O
,
L

on , by setting [12] TM

*
Lc
O,

  (32)
Considering the local coordinates

,,
ii 

 to
L

,
modeling the space-time , through spaces, we
have that (32) is
MH
22
dd
ij dd, i j
ij ij
LL


L

 
 



(33)
Likewise, the variation of the action from the operator




d,
c
OL

d,

  is translated in the differ-
ential
  


d
d,
d
LL
hss
t



 




d,
hss (34)
where


:Ths M, and is such that ,h



M
and




12
0,hx hx

 to extreme points of ,



1,
x
sq

and


2.
x
sq

The total differential (34) is
the symplectic form ,
L

that constructs the application
of the field intention expanding coordinates in (20).
The space
2n




1,

X is the space of differentiable
vector fields on


,

 and is the manifold of
trajectories (space-time of curves) that satisfies the varia-
tion principle given by the Lagrange equation that ex-

,
presses the force



F,
j
xs generated

1, 2,,j

n
by a field that creates one “conscience” of order given by
their Lagrangian (to see the Figure 1(a)).
The operator is an operator that involves the La-
grangian but directing this Lagrangian in one specific
fiber (direction) that is to say, prefixing that Lagrangian
action in one direction. We remember that said map:
have rule of correspondence
,
c
O
, M:T T*
c
OM


c
wOvw , where


,wLv with the classic
Lagrangian. This defines the quantum conscience. If we
locally restrict to that is to say, on the tangent
space
,L
,
c
O
T,T
xx

MM



,x

M we have that [12]


locally
:TTTT ,
cxx
O M M M M* (35)
with rule of correspondence


, ,
c
vwOvw


,
c
Ovw generalises the means of




,T
c
Lv vM,
x
Ovv


.x Likewise, if
:T ,

MR with rule of correspondence








LvO v
,
c, then the total action along
the trajectory
Lv v

will be





,
c
Ovv Lv

 
 

36)
that is the integral given in (16).
In the forms language, the conscience operator comes
given by the map :TT* ,
L

MM with rule of corre-
spondence given by (32). The quantum conscience shape
a continuous flux of energy with an intention, involving a
smooth map ,

(defined in the example 1). Then the
conscience operator is related with the action ,

and
the trajectories ,
s

through of the following diagram
[12]:
C
t
O






MM
RM
TT

*
(37)
Open Access JAMP

F. BULNES 32
Proposition 2.1. The diagram (37) is commutative.
Proof. By construction and nature of the operator .
c
O
Consider the following basic properties of the opera-
tors Let .
c
O

x
x'O
,H
[20] defined in the footnote 1,
and where the space H, is the set of
points

x  
 


T*xmm   MH1 [3], (38)
Points of phase space are called states of the particle
system acting in the cotangent space of . Thus, to
give the state of a system, one must specify their con-
figuration and momentum.
M
Example 2.1 [12]. Let :T* ,

MM be (like given
by commutative diagram) and , :T
nm
C,

R then
,

 R M describes the curve in the configuration
space, which also describes the sequence of configure-
tions through which the particles system passes to dif-
ferent strata of co-dimension one (see Figure 1). Every
strata correspond to a phase space of , particles that
are moved by curve
m
,

and directed from their energy
states


d
x
  by ,

to , particles n


x
 
This defines our intentional conscience. Then are true
the following properties:
1) 2

,x x'xxx'xxx'   MO
2)








,
c
Ox'xsx x'xx' 
MO and
,
s
t
3)
 
d;
c
cO
Os  
H
   
dd

c
Oc
s
Oxs  
, H
 in the unlimited space
4) if and only if

,
c
Os s 

 
 
x
sxs's s' s t  , then






F ,
x
sxs
5)
 
()
c

x
x'Oxs
 O =



F
x
x'x x'  
and
,xx'M, 
s
t
6)


 

d d
c
Osxx xsxs 

 

HO.
,
All properties are demonstrated in [12], the reader cans
to find all details in this reference.
Now we consider the space-time , like space where M
,3
d
t
Id

R
M
is the macroscopic component of the
space-time and we called , the microscopic component
of the space-time of ratio (length of a string
[21]). For previously described the quantum zone of the
space-time , is connected with , which we will
called virtual zone of the space-time (zone of the
space-time where the process and transformation of the
virtual particles happens) are connected by possibilities
causal space generated by certain class of photons and by
the material particles interacting in the material space
time, with permanent energy and the material particles
re-combining their states they become in waves on hav-
ing moved in
F
10 33cm

N
,
d
t
I

R on any Feynman path. Likewise
we can define the space of this double fibration of quan-
tum processing as [12]:
 

 

2
2
2
2
,,
0,
d
ct
c
OxttCI
t
Oxtt











 

RL
(39)
with the states

, of quantum field are in the quantum
zone . Let , the ambi-space (set of connection
and field) defined as:
M N
 


, 0,XX


xy
xy
NL` (40)
where ,

is the connection of virtual field ,
X
with
the quantum field and is the field whose action
is always present to create perceptions in the quantum
zone connected with
,Y,

(2-form) [22]. Then we can
create the correspondence given by the double fibration
[16]:
(virtual zone of
the space-time)N
M(quantum zone
of the space-time)
(space of processing of quantum particles)
L
πθ
(41)
This double fibration conforms the interrelation be-
tween , and . M N


xtM, give beginning to a
ubmanifold (that represents the spaces where
are the quantum hologram) that includes all these quan-
tum images given by quantum holograms, why? Because
these complex submanifolds, considering the causal
structure given in the space-time by the light cones [23],
of all trajectories that follows a particle in the space-time
[16], they can write using (41) as:
complex s
1The corresponding cotangent space to vector fields is:


 


11



MMXHX X
1
M
Here


*
,
T.
nm
mC
2In the general sense the functional derivative


 
,
an
ba
b
yyx
x
 


implies
ba
x
 
d,
n
ba
a
yyxx  

but does not imply






.
n
bbaa
y
yx x   

1,
x
x






 (42)
of , such that which by space-time
N11
, PP
x
Open Access JAMP

F. BULNES 33
properties to quantum level represents the space of all
light rays that transit through ,
x
conforming a hyper-
surface (projective surface) that is a light surface. This
surface is called the sky
x
[24]. A sky in this context
represents the set of light rays through
x
(bosons) that
comes of the virtual field.
If then
4,CM,
x
QMMis the complete uni-
verse (include the super-symmetries [25]). But, what is
there of our quantum universe with regard to our real
universe (included the material part given by the atoms)?
The answer is the same, we have an universe of ten
dimensions and where the quantum rep-
resentation of the object
,MNM


,
x
s is the quantum
space-time 3
t
I
R
,
M
2
(which is the space-time as the
Einstein perception) then the images of the virtual parti-
cles are
x
QC [11], then the execution operator
that proceeds to connect virtual particles through the
paths which have path integrals on a double fibration,
establishing the material-quan tum-virtual connection
required to a total reality:
,I
N
M
L
π θ
M C
`
σ
ρ

(43)
where is the material part connected with the quan-
tum zone of the space-time (space taken by atoms) .
The corresponding path integral that connects virtual
particles in the whole fibration is the integral of line type
(5) defining feedback connection:
,C
M
 







11 ,
xcs
xs Ox

 




IQ
(44)
always with the space


,xs
x
MN to the
permanent field actions. Then the reality state is the ob-
tained through the integral of perception (44), consider-
ing the fibre of the corresponding reality in the argument
of the operator of the integrating from (44).
,
c
O
Proposition 2.2. All transformation of a reality in-
cludes bosons of the field. N
Proof. To demonstrate this, is necessary to prove that
the cohomological group on , is the same coho-
mological group modulus a seated class in , of the
material reality and the corresponding for the quantum
reality. For it result useful the stacks concept in physics,
where is possible to tack bosons to construct superior
physical spaces using bosons branes (see corollary of
[26]). On the other hand, any open numerable covering
of bosons in , contains a finite subcovering in ,
which is guaranteed by the sky .
N
N
N M
x
Q But a sky in this
context represents the set of light rays through x (bosons)
that comes of the virtual field.
In a plane of reality of the space N
a commutative diagram similar to the given in (37), con-
sidering fibers of the topological space sky
, we can establish
x
Qhat at
all times do that the integral submanifolds in L
(through of the double fibrations of (59)) connect both
realities determined in , and , along these sub-
manifolds:
, t
N M


T*
g






LPT*M
MM

(45)
where






10,
  

L with


T1O



L PM* ,1, where is a ho-

1, 1,O
mogeneous bundle of lines due to that the sky
since the normal bundle
11
,
x
QPP
N
,
x
Q in
every sky ;
x
Q is isomorphic to


11,1 .JO In particular
there is an exact sucession


1
01,1N1, 1OO 0,
(46)
which allows to have a composition of the reality in ,
through fields that come of Then the quantum -
virtual composition of both realities is given by the
moduli stack:
M
.N
3. Intention Transmission and Introspection
Considering the quantizations of our Lagrangian system
describe in (17), (31) and (32) on coordi- ,2
nnR,
nated by


,
j
x
we describe terms of a graded commu-
tative


C

Malgebra ,
H
with generating elements


12 12
,, ,,,
k
aa a
xxx x
 
  
a,
(47)
and the bi-graded differential algebra of differen-
tial forms (the Chevalley-Eilenberg differential calculus
[4]) over
*,H
0,
H
as an -algebra [4,7,27]. One can think
R
Open Access JAMP

F. BULNES 34
of generating elements (44) of ,
H
as being sui generis
coordinates of even and odd fields and their partial de-
rivatives. The graded commutative -algebra R0,
H
is
provided with the even graded derivations (called total
derivatives)
1
0
,k
a
a
dddd





 

  (48)
where

1,, k and
k  


1
,
k,

   
are symmetric multi-indices. One can think of even ele-
ments


0
, d,x
dd
(1) dd
an
an
a
an
q
Lxx
L xx
d ,
x
Lx








 
 

L
E


(49)
where we observe that ,L

is the 2-form given by ,
L

in the formula (33) with and
2,n12 .

 
Now we consider the dual part of the space




,H , that is to say, the space

*, .

H
L

n
R
We con-
sider quantize this Lagrangian system in the framework
of perturbative Euclidean QFT. We suppose that is a
Lagrangian of Euclidean fields on The key
point is the algebra of Euclidean quantum fields
,L
B
.
,

given as 0,
H
the graded commutative. It is generated
by elements



,xx
 .
a For any



x

 ,
there is a homomorphism belonging to the space


T, ,
H
HomH D


T, ,
(with homomorphisms
H
HomHD given for -algebra of cy-
cles):
DG
aa


  
11
11
11 1
111
:
,C
rr
rr
aa
xaa
aa x
xx
x




 







I

r
r


rr
rr
aa
x
II
(50)
of the algebra 0,
H
of classical fields to the algebra
which sends the basic elements
,B
0a,
x
H

 to the
elements a and replaces coefficient functions
of elements of
,B



,I0,
H
with their values


x
I (exe-
cutions) at a point
x
. Then a state , of is given
by symbolic functional integrals
,B


11
11
1
exp d
kk
kk
aa
xx
an
Ox

d ,
aa
xx
a
cxx
x
N







 




H




(51)
where this is an integral of type





,


c
Oxsd
H
as
was given by the properties. Then when the intention
expands to the whole space, infiltrating their information
on the tangent spaces images (map (35)) of the cotangent
bundle (given by the imagen of T* ,M


x


under
) then their intentionality will be the property of the
dc
O

field to spill or infiltrate their intention from a nano level
of strings inside the quantum particles. Then from the
energy states of the particles, and considering the inten-
tion spilled in them given by


,
c
O

we have the
homomorphism (50) that establishes the action from




,

M to ,

M for their transformation through
the action T,

defined in (28) to any derivation given
through their conscience operator (fiber (31)), like the
graded derivation ,

 (considering the derivatives




cc
Ox O*,

 d
F
D


):


 


:, ,
,
c
cc
xx Ox
Ox O











x
a
xL
aa
(52)
of the algebra of quantum fields . With an odd pa-
rameter α, let us consider the automorphism
,B


ˆexp ,UId



 

of the algebra ,B

This automorphism yields a new
state ,, of ,B

given by the equality
 



11
11
1ˆˆ
ˆˆ
expdd,
kk
kk
aa aa
xxx x
an a
cx x
x
UU
N
OUx U
 
















(
H


(53)
where the energy state has survived, since


ˆ
d
a
d
a
x
x
U

 . That because the intention is the same.
The intention has not changed.
What happens towards the interior of every particle?
What is the field intention mechanism inside every parti-
cle?
To answer these questions we have to internalise the
actions of field
X
, on the particles of the space , and
consider their spin. But for it, it is necessary to do the
immersion of the Lagrangian
M

, defined as the map
2, n
  L M3
with rule of correspondence


ii
Z
Ζ
 
where the image of the 1-form

, that the Lagrangian
defines,


i


, is a symplectic form [1], and the vari-
able i
Ζ

is constructed through the algebraic equations


a
Wi Ζ

 [28]. They describe the dimensional
hypersurfaces denoted by such that where
-k
S  S H
3Having chosen is to consider the two components of any point
2, n
M
in the space (that we are considering isomorphic to the ambient
space of any quantum particle
,
N
C


x
s, in the space-time) to have the
two components that characterise any quantum particle


x
s, that is
their spin (direction) and their energy state (density of energy or “liv-
ing force of the particle”). , is the corresponding Lagrangian sub-
manifold of the symplectic structure given by
L

2, n

M.
Open Access JAMP

F. BULNES 35
H
a
W
, is the phase space defined in the Section 2. The in-
dex runs over the number of polynomials
i
1,, ,a

,
q
Ζ
in the variables i
Ζ
 and , runs over the
dimension of the ambient manifold which is assumed to
be
i
.
N
C If the space is a complete intersection, the con-
straints

a
i
W

,
Ζ
(there is exact solution to


a
W

i), are linearly independent and the differen-
tial form
Ζ

n

k

,WΖ

T
k

,ANkq



Nk

1
d
a
1
n
a

1d
N-K
Nk
a
aa WW

 

k
a
WW
,q




(54)
is not vanishing. In this case, , and the dimen-
sion of the surface is easily determined. For example, if
the hypersurface is described by a single algebraic equa-
tion the form (60) is given by
On the other hand, if the hypersurface is not a complete
intersection, then there exists a differential form
qN

1dW.


1
,
dd
NK
Nk
a


 
ANkq Aa



,
(55)
where is a set of forms defined
such that is non-vanishing on the constraints
Nk

a
W

i
Ζ

,ANkq


, and , is a numerical tensor which

1Nk
Aaa  
T

is anti-symmetric in the indices 1 The construc-
tion of depends upon the precise form of the
algebraic manifold (variety of the equations
.
q
aa



a
Wi). In some cases a general form can be given,
but in general it is not easy to find it and we did not find
a general procedure for that computation.
Ζ
To construct a global form on the space

S one can
use a modification of the Griffiths residue method [28],
by observing that given the global holomorphic form on
the ambient space
1N
ii
 N1N iidd,
Z
Z

's
we can decompose the
i, into a set of coordinates

Ζ


Yaa W
Ζ
 and the
rest. By using the contraction with respect to vectors ,q

,
a
i
Ζ the top form for can be written as  S



,
a
a
k
Nk











 


N

(56)
which is independent from


,
a
i
Ζ as can be easily
proved by using the constraints


a
i
WΖ . Notice that
this form is nowhere-vanishing and non singular only to
the case of space (Calabi-Yau manifold). The Cal-
abi-Yau manifold is a spin manifold and their existence
in our space like product of this construction is
the first evidence that a spin manifold is the spin of our
space-time due to their holomorphicity [29]. The vectors
Y-C
2, n
M

,
a
i
Ζ play the role of gauge fixing parameters needed
to choose a polarisation of the space into the ambi-
ent space.
 S
For example, in the case of pure spinor we have: the
ambient form

16 16
16 dd
i 
,
i

  
 and

5ddddd
mmm
mnp







  . From these data,
we can get the holomorphic top form , introducing
five independent parameters

11

,

and using the formula
(56).
The latter is independent from the choice of parame-
ters ,

(however, some care has to be devoted to the
choice of the contour of integration and of the integrand:
in the minimal formalism, the presence of delta function


,

 might introduce some singularities which pre-
vent from proving the independence from ,

as was
pointed out in [30,31]). Using one can
compute the correlation functions by integrating globally
defined functions. When the space is Calabi-Yau, also
exists a globally-defined nowhere vanishing holomorphic
form

k


,
k


0, k
hol
 such that
 
0
0,
k
k
holhol is pro-
portional to


.
kk
 The ratio of the two top forms
is a globally defined function on the space. In the
case of the holomorphic measure
Y-C

0, k
hol
 the integra-
tion of holomorphic functions is related to the definition
of a contour ,


S in the complex space

 ,0k

,
iA i A
AA
Z
,pZ p
 


0
S
OO (57)
where


iA
Z
,pO, are the vertex operators of the theory
localized at the points pA, of the Riemann surface and


,
iA
Z
p

,0 . k

0, is the zero-mode component of the vertex
operators. Newly our conscience operator come given by
the form
O
Example 3.1. All Calabi-Yau manifolds are spin . In
hypothetical quantum process (from point of QFT view),
to obtain a Calabi-Yau manifold is necessary add (or sum)
strings in all directions. In the inverse imaginary process,
all these strings define a direction or spin. The strings
themselves are Lagrangian submanifolds whose Lagran-
gian action is a path integral.
Let be a complex orientable Riemannian mani-
fold (manifold that we consider as model of the space of
the material and quantum reality) and the topologi-
cal subjacent group to the manifold. Let be a
conic and regular involutive submanifold.
,M
,G
T* ,SM
The causality of all particle in the space ,
is determined by the causality in the cotangent space
to know
,M
pS
T* ,M



*
T, p
ppp
CC MMC (see Figure 3)
where ,
p
C is the corresponding light cone of the parti-
cle p, where all universe line is tp Cp

S, with
, is a microscopic space of . Given that this micro-
scopic space have a structure of cause-effect induced for
their proper causal structure , then by the postulate 1,
we will refer to the above mentioned space like
nano-space.
SM
C
In a nano-space there is a structure of Hilbert space,
since this has a pre-hilbert structure represented as a
Open Access JAMP

F. BULNES 36
Figure 3. Causal structure and tree of the strata.
structure of interior product “(,)” which generates a Hil-
bert space through the quadratic form of their endomor-
phisms [32]. The space spin or spinor is a fibre of a bun-
dle spinor in the orientable complex Riemanniana mani-
fold [33]. M
We write


SE, for the space defined as
 

2,,SEId E
  , (58)
If


dim ,nE is odd then let

, denote one of
.

 Recall that their corresponding algebra
 

End,,,,,EXSXwXwwE  so
(59)
If ,wE

, are such that


,w  0,
,
then let

,E

nd
X
wV  be defined by



,,,,,
X
uuwuwuV    (60)
Then
 
,.
X
u  so
.V
E

,
Let be an or-
thonormal basis of Set ij
1,,, n
ee

,
i j
X
Xee for .ij

Then ij
X
, , is a basis for We define a
linear map
ij

.Eso

, of


,Eso

into by


End SE

,







12 ,
iji j
X
ee   (61)
A direct calculation shows that
  


,,,,,
X
YXYXYE    
 so (62)
This implies that




,SE  , defines a module for
Dicho modulo es el espacio

.Eso .
N
ANO
S
Proposition 3.1. If is odd then up to equivalence ,n




,SE  , is independent of the choice of ,


and




,SE 

, is irreducible. If is even then we set ,n

,W
ev
SV
 Then

SV .
 odd W


,SV
 is
invariant under

, and each defines an irreducible rep-
resentation of

.Vso
,n
Proof. If is odd, set




0

,0. VV
Then our construction implies that




.SVSV

 If
V


, set



0
.

 

.


,
Then

2
I

   ,S
 Let be a non-zero subspace of


,SV of minimal dimension that is invariant under


V 

. Then ,S

satisfies the definition of spinor
space. If is odd then up to isomorphism there are
exactly two spaces of spinors and they are each of di-
mension
,n
2
2.
n In the case of even dimension, only there
is one space of spinor under isomorphism of the dimen-
sion also 2.2
n Then Hence

dim dim.SSV



.VSS
 So

, is irreducible in this case.
We now relabeling the orthonormal basis that we are
using. If 2nk1,

 let 00 2
,, ,,
ik
eee


2,nk
212
,
be an or-
thonormal basis of . If then take
We assume that
V12 ,, .
k
ee
j
jj
ie

we


,
(63)
with 1jk

 and that .
j
WCw
Set
21 2j
,1 .jk
jj
hXi
e

 Then ,
j
Rh t

.Vso
is a
maximal Abelian sub-algebra of Let ,
j


t
be defined by


.
j
jk
h



 A direct calculation yields






1121
2,
kk
iw whw wk      
(64)
and







,
jj j
ihw h  hw 
.
(65)
where h

t This implies that
() the weights of the Lie algebra on ,t


,SV are
precisely the linear functional



12
12

1
2,
1,
p
kii i
p
ii k
   
 


i
,
and each occurs with multiplicity 1.
Notice that this last is independent of our choice of


when is odd. If then ,n,n


,Vso is
semi-simple and is the Lie algebra of the compact Lie
group
 


,,,,,LVggwww V   OVgG
(66)
which is


,SO n in the spinor endomorphisms version
en Weyl’s theorem implies that the connected,
simply connected Lie group with Lie algebra,
.V,G


,Vso is
compact. Also exp ,T

t is a maximal torus in
and
,G
,

integrates to a representation of . Then (),
implies that the character of
G
,

is independent of our
choice of .


So the assertion of the proposition 3. 2. 1.,
has been proved in the when is odd. ,n
We therefore confine our attention to the case when
is even. It is clear that the spaces are in-
variant. Set
,n

,SV



.
dim 2,
Vgso
V
We assume that (in
the case that
dimV2

the result that we are proving is
an easy exercise). We set






1
1,
rs
rs
irsk
irsk


  


 (67)
Open Access JAMP

F. BULNES
Open Access JAMP
37
Then is a system of positive roots for ,




, CC gt .
The only dominant weight in () are
their Lagrangian of effective energy are applicable [20]),
there must be a co-action between the conscience of the
field that applies the intention and the conscience of the
field of the object that is wanted to transform. Likewise
this coercion must be realized in synchrony and tuning,
with the same intensity for both fields. This way, the
proposition 3.2.2, establishes that the nano-space is the
zone where initiates an organized transformation of nano
type.


12 2, k
i    and

121 2. kk
i  

 Then assertion for
even now follows from the theorem of the highest
weight.
,n
Proposition 3.2.2 (F. Bulnes). The set of hypersur-
faces conforms under holomorphicity and polariza-
tion of the fields inside each hypersurfaces our spin space
, S
N
ANO
S (see Figure 4(c)). 4. Results
Proof. The corresponding holomorphic Chern-Simons
theory establish the action
3,00,1 0,1
12
,
23
hCS
M
tr XXXXX

 





(68)
In mathematics, an isotropic manifold is a manifold in
which the geometry doesn’t depend on directions. A sim-
ple example is the surface of a sphere. This directional
independence grants us freedom to generate a quantum
dimension process, since it does not import what direc-
tion falls ill through a string, the space is the same way
affected and is presented the same aspect in any direction
that is observed creating this way their isotropy.
originated from the field
0, 20, 10 ,1
FXXX

 (69)
which need of Calabi-Yau manifolds. Being a Cal-
abi-Yau manifold a spin manifold, as is mentioned above,
then the related strings with the fields and their polariza-
tion (orbits of the space-time (see Figure 4(a)) need of
dimensional manifolds embeddings into space-time.
In every one of these manifolds subjacent Lagrangian
submanifolds , that is to say, these shape special em-
beddings in a manifold [10,28,33]. These are the
required holomorphic hypersurfaces in a manifold.
As in this decomposition of dimensional submani-
folds to appear inevitably
2n
L
CY-
Y-C
2n
D

branes (see Figure 4(b)),
in these, subjacent Hilbert spaces like spaces of bundle
sections where , and , are
holomorphic bundle of lines of corresponding gauge
fields [34]. The tensor products have as
square elements of their endomorphic elements de

0, 1
T S
The importance of this isotropy property in our spin
manifold, helps us to establish that the transformations
applied to the space that are directed to use (awakening)
their nano-structure are did through an organized trans-
formation that introduces the time as isotropic variable,
creating a momentary timelessness in the space where
the above mentioned transformation is created. Then the
intentionality like an organized transformation is a
co-action compose by field that act to realise the trans-
formation of space and the field of the proper space that
is transformed. Then the symplectic structure subjacent
in haves sense. ,M
,
ab
E

Ea E
,
ab
E
b
E

E
N
ANO
S.
Likewise, if , is a transformation on the space
whose subjacent group , have endomorphisms
1
T,M
G
,, ,
n

  such that are iso-
tropic then the infinite tensor product of isotropic sub-
manifolds is a isotropic manifold, and is a organized
transformation equivalent to tensor product of spin rep-
resentations
 
,, ,
n

MTT
 M




,   MTT
n
M 
Here the space of bundle sections
is one of the mentioned in the
image of the algebra to the two components in the
strings context of the field.
 0, 1
T
a 
EE S
t
,
b
,
Proposition (Bulnes) 4.1. The length of the quantum
path that represents the angular relation between two
“connected” spin manifolds (spins) is of the inferior or-
[6,14].
So that the intention becomes effective (that is to say,
(a) (b) (c)
Figure 4. (a) Waking up to the space with a guided action. (b) Elementary quantum structure being content for the arrival of
trings that are added. (c)


-21Ndimensional spin space and the dimensional phas e space. -2Ns

F. BULNES 38
der to 12−13 cm.
Proof. Consider hypersurface (defined from (53)
in, such, where
S
that
th
,
to (57)), defined 2n M
ype
dd
ii
H S
es giv
H


N
, is the phase space defined in e structure (38). One
element of this hrsurface comen by

1
1dd
NK
Nk
na a
aa WW


 

k but the global
holomorphic form on the ambient space is
1
1
N
N
ii
Z
Z
ic scale. B
 

er the atom
which represent the in-
finitesimal volume of the microscopic space-time, if we
ut in this scale the before
volume have the value 10−27 m3. But this number is rep-
resented approximately by the AMU (Atomic Mass Unit),
that is to say 1.6604  10−27 kg, [10] which is induced
from the intention given by the action ,
c
O
 (Figure
4(b)) until the curve that measures the linking energy for
nucleon in function of the mass number (re 4(b)).
Then the infinitesimal volume element

consid
Figu
N
, to every
particle that have received this intention is approximately
the Max Planck length except in 10−3 nm t is a little
volume of the pure intention transmitted in nanoseconds.
Also the same argument to the hypersurface in the space
H (considering the little surface 10−2 nm) that is to say,
the length 10−20 m. Then by the (62) considering the in-
ance hypothesis to the considered measures we have

, tha
vari




33 27
13
10cm10cm10 cm,
a
N





 


 

k
22 18
10 cm 10cm
a
Nk







(70)
Then the path that connect the two spin manifolds
have the length
But 12−13 cm = 9.3463  10−15 cm = 9.346
0.0009346.




13
10m 9.3463,

  k (71)
3  10−13 m =
Theorem 4.1 (Bulnes). The BRST-cohomology


0
, ,HGGM0
l
and with
Proof. We app the BRSTmology to the
space-timewith the following decomposition
0,m ,ml
dim . M
ly -coho
,M


0
,

MGG (72)
where is the Fock space at ze
mome and
0,G
ntum
ro center-of-mass
,

G are Verma modules whose highest
rato
ite cohomology also breaks up as
energy that
does possible all transformations from particles to parti-
cles (energy support). Here space , is the virtual
zone of the space-time (Higgs fields). From now on and
we focus on
weight vectors are obtained by repeated application of
the creation opers in the full spectrum generating
algebra.
According to the decomposition of ,M the
semi-infin

0
, ,.HH
 
GGNN SS We ignore
because it represents the permanent quantum
0,G
N


,HGN S
, where d ,G is a fixe.

G
c Consider the plex cverelative subcomonsisting of tors
H


Fock space satisfying 0
00. c
bO   The
induced differential in this subcomplex is easily seen to
be the operator Q, (extended BRST-oerator) who in-
clude the more and deep action of ,
c
O reflected in (32).
The cohomoloplex is ning but the rela-
tive semi-infinite cohomology

p
othgy of this com

0
,,,HGNN SS
0,N
S is the subalgebra genecThen
there exists a filtration of this complex giving rise to a
spectral sequence converging to
where rated by 0.O


0
,,,HGNN SS
whose 1 E, t obeys 1 E0,
m
erm

for 0.m

Therefore

0
,,0,
m
H

GNN SS for
0.m

What happen with the cnsidase 0m ? We coer
the “Po duality” theorem [10], which sayincaré s

 
0
;, ;,,
mm
HH


GGNN NN SS SS (73)
ude tha

0,, m
0
allows us to conclt

0,
0
H

GNN SS
m.l


Lemma 4. 1 (Bulnes). Let M ,
t
M
I the unlimited
space of the quantum space (Fock space [24]). A particle


,
x
t
energy
that is focalized by a given
load function
bad evolutionfor the


,,wts comes given for
,
  
dd
x
sxxt xttsxt t 





M
Then to time ,ts
(74)

begin the singularity.
Proof. [11,12].
If we consider a ce operator with sinonscienc gularity


,
c
O

for the presence of an energy load


,.wst
Then the elimination of the singularity


,
s
 comes
given by the theorem [12]:












1
1
dim d
jj
j
C
correc restoringc tion, d
,
XC
x
tOs
wtst
t



(75)
where
nF n x
A







  







dim ,

 is the Neumann dimension corre-
sponding to the Weyl camera of the roots ,
j

[12,23]
used in theocess to elim
created by the singularity.
Proof. [11,12].
Arguments for algebras and homotopies, and
mwe can dem
ves
pair of quantum integral
tra
rotation prinate the deviation [8],
their
eetings onstrate the nature of sum “correc-
tion + restoring” established in (75) that ha their re-
alization by the path integral established in (10) y (11).
Then we can enunciate the
nsforms as [12]







 


 

,d d
cc
XC XC
xtOxtwtssOxt




Q
(76)
O
pen Access JAMP

F. BULNES 39



 




1dim, d
c
XC
tOswtst 
 


Q
(77)
 

d,
c
XC
Os
x

Let ,
have t
oal or
sp
be the potential that the integral transform
(76) o realize (that is to say, there is a realiz
(g target)) a quantum transformation of a region
ace-time from the field
ation
R
of the ,C .
X
The inte
derived tough a bundle ofes O(1, 1)
innce there is a topological space
grals
, are of line
“sky”
that
projectivized
hr
,C si
lin
,
x
Q such that x QPP Likewise one inte-
gral of line (of the type described in (44)), is the realiza-
tn
 
 
Rxs  . Then the exact succession
(46) is translated for thotential of this quantum reali-
zation in the commutative diagram [6, 11]:
11
.
io
e p
(78)
where ,,

 and ,

are differentiable maps in all the
space
Consider to a set of transformation potentials of
the space that
.C
,
such ,M
 




,
,andT*,
,CD
D

F 




 



R
M
oion (Bulnes) 4.2. Let
(79)
Prop sit,

 be the set of po-
tentials in nano. ,

 is potentiality if only if ,

is a
fiber throuwhich , gh

is a potential.
Proof. Let
N
ANO
S
o, the f

.
, the nano of the space-time In-
side of nanorce of potential is given by
M.
F
x
x'

 
ty iv,

(by the proper
But by the defi
Then their conscience oper
given in the Section 2. 1)
s
at
is said for
or c O
ce.
nition of our conscience operator c O, is a
fiber of the corresponding Lagrangian operator that in-
volves states. In term of the space T*M, this is a Ham-
iltonian which satisfies the
g
rad

, in question. But
,

  is the potential that have the zone

, in the
space-time. In special, along of anyone trajectory ,

this camera ,

is a potentiality. The reciprocate, if
,

 is potentiality then ,

is a fiber through which
,

is a potential, is deduced immediately applying the
corresponding path integral whose potential is .

Then ,

is the sum over all bastes correspond-
ing to a fixed choice of 0,
ic sta

(',
s

and '
s

), of the
ponding “amplitude” (to be defined s:
all basic statesof
,
corres )hortly

 



(80) 
Note. The true amplitude of the process would involve
a sum over all states, when the values of 0,

varies on
thena intervertices while the state of the boundary , l

is fixed. The sum over values on the edges amounts to a
contraction process (traces etc.).
Now ,


(,


in (86) or also ,in of the se

ontribu
c-
tion IVduct over the c) is a protions


,



over the vertices of ,

,n internal
ty
and ,m boundary
pe.
For an internal vertex

, we have:
 
 
 Re
vv v
v
ve,,d,
ei n
x




 




ev out
rtex
(81)
where in(v) (or out(v)) denotes the set of incoming (out-
going) edges of the ve

, and the shorthand notation

1,

  was used since 0,

is fi
te-sum.
xed within this
Towars an “propagation amplitude” interpretation (as
have established in the space
sta


ct (,
) replace the evaluation
pairing with the inner produ), such that the ab
basis
ove


,1,,
iiin ,be ortal. Also collec
Defin sic
honormt the
“in” and “out” products, introducing the following ter-
minology.
ition 4.1. For any ba state ,

of the graph

:



inout ev
e(v)
e, e,
out
in
 






 (82)
are called the in and out states of the scattering process at
the vertex


.
Very process that gives beginning to the shape singu-
rity are related with the scattering amitude obtained la pl
from the internal vertex inside space, body or object that
is wanted to realize a transformation by nano (see
5). Of fact, some invasive methods in nanote
(like the use of nanobots, follows this idea) require that
Figure
chnology
the transformation of the space ,
M
initiate inside of the
space, objet ocf the transformtion. Nevertheless, this
id
a
ea is implicitly given with the intentionality and the
operators c
O, since require of the two actions, the action
from field that is applied and the action of the proper
field of the space, object or body. Being this important
we have:
Proposition 4.3.


 


out in
v
vv,adv,

   (83)
O
pen Access JAMP

F. BULNES
Open Access JAMP
40
(a) (b) (c)
Figure 5. The inte ntion is appl ie d and begins an inne r transfor mation of the space (the singulari ty is born). The acti on of
the field (from observer) is transmitted to the field of the proper space that we wan transform. Using direct transform, the
singularity is implanted in the space by the intention of the field (a)-(c). For the of the inverse transform we annul the
transmission of the conscience in the sinularity
M
t to
case
, and focus to the space M means of the operator , by




c
Oxt


c
O

g (to
revise the interjections of the figures in the inverse order).
is the scattering amplitude





out in
v
v,adv ,
v


(84)
of the elementary process at internal vertex

:

in(v)
1
ou
t
(v) out(v)
1
in(v)
k(v)
k(v)

v



 
  
(85)
Proof. Here




ad, ,
X
YXY
erential operators.
is the commu
bracket on diff Therefore if
tation
,
X
and
,Y commute, then




,.
X
YX  
component, use the a
Y
bov
Now ttrieve
propriatee inner uct:
o re
prodthe ap
 

,, ,
out  
 

in in 

  

   
 (86)
This can be put in the form of propagation amplitude
establishing the above claim.
The integral transforms apply inverse methods of
quantum re-compositions that produce quantum
formations “appropriate” to establish an answer of the
st
blishing an image of the
fie
trans-
ates of energy of the part M, affected by singularity,
and re-compose the space esta
ld

,XM adapted in same resonance, quantum
composition of the space points

,
X
t of the whole
space M. This last for the synergic principle of the ac-
tions of the field (postulate in [17]), and due to the
field intention that has the proper matter of the space M,
with tto an auto-ordering in the nano-level.
The answers between densities alized in accor-
dance w th the correlation densities established in certain
3,
endency
re rea
i
re the co
sponding Feynman diagram on a logic algebra
commutative diagrams that can be shaped by spaces 2 L,
on the space-time of the particles [35].
Coding thisgion of transition states ofrre-


, ,

A (like full states or empty of electrons like
particle/wave, is to say,


00   (is not the particle
electron, but is like wave)


11   (is not the wave
electron, but is like particle) and their compleme
w
nts),
gi
h a
here the given actions in [8], are applied and
re-interpreting the region of the space-time of the parti-
cles like a electronic complex of a hypothetical loc
nano-floodgate (that is to say, like a space 2 L, wit
logic given by


, , ,

A with values in 0, 1 M
[8,17], on their transition states), we can define the Feyn-
ma integrals, [6,7,17,23], as those that establish
the transition amplitude of ourstems of particles
through of a binary code thas the action of cor-
rection and restoration of the field established in [8].
Likewise a Feynman-Bulnes integral [17,36,37], is a path
integral of digital spectra with the composition of the
fast Fourier transform of densities of states of the corre-
sponding Feynman diagrams. Thus, if 12 3 ,,
n-Bulnes
sy
t realize

 and 4,

are four transitive states corresponding to a Feyn
diagram top holes of the field

,XM then the path
integral of Feynman-Bulnes is [36]:
   
123 4
1234
0001101001 ,
nnn n
man
I
Fn FnFnFn


 






The integrals of Feynman-Bulnes, establish the ampli-
tude of transition to that the input of a sith signal ystem w


,
x
t can be moved through of a synergic action of
electronic charges ,

doing thro pre-determined ugh of
s functions by




,Lxt wave and encoded in a binary

F. BULNES 41
algebra (pre-defined by states


0, 


0,  , and


1

),
(in the kernel of the space -
tion
of solutions of the wave equa

AA AA
xF
x
x' 

 □
of a circuit
16) of a point to
into
other
j
. Their int
of paths or re-walked in
egral
cluded i
it is ext
nto of
e
space the re
La
nd to all
gion of
grangian action
kk

j
jj



nals in


 


of sig
 with a topology

2,,L [4, 6]. If we want corrective
actions for stretch
j
, of
string
cohomology of strings [6
ln
a path , these we can realize
them using diagrams of of corrective action using
the direct codification of pathrth staf
emission-reception of electrons (by means of one sym-
bolic ,25]). Then the evaluation
of the Feynman-Bues integrals it reduces to the evalua-
tion of the integrals:

s
integals wites o




0
C
,I   
 ,where ,

is the orientation of ng

0,C
h used to co
X
, is the correspondi
model of graprrect after of identify the sin-
gularity of the field, that distorts it. For example, ob-
serve that is can to null the corrective action of erroneous
encoding through of a sub-graph:






 
0
110001 1
01111110,


 

 

The correspondingohomologof equation
gs is [6,38]:
in the cy
strin
The quantum transformations due
d by
cyberqu
), base
pace
this
the
antum alg
d in
inverse action
logic
in eb
path inte
belo
in this level of deep stay establishe
0, 1
M (acquaintance as
pplications in cybernetic
nging to the cohomological s
defined
for
grals
ra
their a


1,,
H
Ok
PM and whose action organize and cor-
relating all movements of
that devolve of initial shape of the m
Proposition (Bulnes) 4.4.
tions as given in

particles x(
at
s), in an
ter.
Using organized transfor
sym
-
ma
phony



, we
n
 MMTT
all particles in set, is their

can to establish that the state of
 
corresponding Fock image [39].
Proof. Is necessary to apply the group of permutations
j,

on at least ,n components of the space
1
HH H,
n
 with 1 HH.
n


Inside of the Fock space begins a re
ention, since the Fock pure state in-
volves all the states of particles of the space, object of the
transformation.
alization
pote
of the
ntial of the int
We consider ,
F
ISMAT
E the space of the all physical
applications to make wl-
edges in mathematics and physics: “Remember that the
intention must know their goal”)
ack
[6]. We will define as
ho
space
nonanotechnolog y, (set of
momorphism of certain class to the application on the
of knowledge applications ,
F
ISMAT
Eand with val-


, whues in FISMAT
advance useful in the nanotechnology creation in the
class
E

ich we ify as the set
of
will ident

[40,41].
If we consider an action like the product of diverse ac-
tions 1,
n

 ,t

determined in a technology the
inorrtegral operator esponding execution will come
determined for
of c












 
1
1
12
1
1
:
,
n
n
n
n
EE
n
EE
x
s
xs
xs











exe exeexe
(87)
In t case the m


his e
lar
tric evaluation onanotechnol-
og
f the
y is carried out in an n ndimensional space. If in
particu ,
F
IS
EE
MAT

then ,

 with ,

a ho-
meomorphism (neologism of technologies), then









,
FISMAT
E Et

 

  
 
exe exe 
(88)




,
FISMAT
E

 and xs ,


the c
stituted wealth
logical p
logy [6,40,
orresponding
measulre on the techno
responding cu
to the evolutio
ogical con this, for
the corrves of technorofiles ac-
cordn of techno41]. ing
Proposition (Bulnes) 4.5. The image of homomorph-
ism (88) is a transmission of intention from

, to ,

with connected

.
r c O
sult is risked.
have considere
devices, co
Proof. By the proposition 4.2, with the properties of
ope (Section 2.1), and roposition 4.3, the re-
Wed to thho
mts a
rato
entities of
p
e technologies like
nd
listic
ponen implements of any
physical nature that under laws of the quantum mechan-
ics and field theory contained in ,
F
ISMAT
E they act under
an operator for the design of experiments it has more
th
o
ng such traby means of
fu
rehearsalsnd re-tests of
te
an enough minimum trajectories of evolution of the
process of these experiments and their evaluation f
these technolog aieslo
us
of
jectories
or tests a
hnologi
nctional corresponding.
Basing on this thesis, it is need to characterize a
space topologic
chnologies whose evaluation should be carried out
through instrumentation that throws measures and values
in topological spaces 2 L [35]. These measures should
spread to a limit of nano-teccal application
(bench mark [12]) which is a measure inside the space of
functions


2GL [40,42].
The relation between the space SNANO, and the Banach
space ,
F
ISMAT
E is endow of Hilbert structure to the
space ,
F
ISMAT
E doing that thorphism let
m
eir endom
easurable in the space


2
LG. The instruments are in
O
pen Access JAMP

F. BULNES 42
the space of measures

2GL.
Let us consider to ,
L
AB
X like the space of rehearsals
or tests of technological process. Let us endow ,
L
AB
X
of a structure of Hilbespace also of the Hamiltonian
structure that have due at their ergodicity (see Ergodic
theory of dynamical systems (principle 3), [39,43]), then
the space of t
rt
rverified in ue propositions ,
L
AB
X satisfies
 

Implies tautologically,
aLAB
X
Xf   (89)
fGL


2,


,,
SYSTE t 
   (the transfer
technologies operator), ere belonging to a class 
(that is to say, applica
M
and
ble to
wh
a certain class of technolo-
gies) they are isometries from ,
F
ISMAT
E to ,
L
AB
X hav-
ing that for a variety of technological application
with the relevancy of considering widespread measures
of the technological applications, we can consider

s and


t
 

11
,
G
tttt t
,Xf t


 








 


 (90)
which bears according the topology of G (as topo-
logical group), the approach of following inequality:

to


log ab Xftt t
 
 
(91)
with values in 2.ab The controls to
log,


,
X
ft


are given for


log ,t

 and

log ,t

 [35,40].
e context ofBut the
, it bears to that in the space of technologic
under the technology transfers are satisfie
Theorem (Bulnes) 4.3. 1)
one that are isometries in th

2GL
plications
al ap-
d:
,
F
ISMAT
E is isometric to
,
L
AB
X
2)




.
FISMAT LAB
EX

 
Proof. Consider the space



0,
FISMAT LAB
EX

 
equal intransfer o
o that
(92)
That which is the sense of the f
technologies tSYSTE,
M 

 


,, 0,tt tt
 
  (93)
,tt [4]. Likewise, in the context of the spac

Ges

H, Hn, the relationship of pres symmetry that Li
in the space
of
fin
viou
is transcd in the environment
ng to on the in-
,
SYSTEM
 ribe
theing integral operators belo
ite straight line R, as

HL,








dxsf sFstxs

 
0,s
(94)

This integral is thexecu integral of etion of the actions
of each component o particle given in gen
dinates
eralized coor-


1,,, n
qq
nam
that they specify the
of us dy
[4]. We co
action on these
,
configuration
ical system in certain ndimensional space
nsider Hamilton’s principle of the minimal
generalized coordinates then

d0
i
i
t
TQq t 
2
1
t


 (95)
By Ergodic theory is not difficult prove that the m
ure on any curve in
eas-
6N
M

R
easure
(,N
33
dd
NN
qp
is the numb
particles) with the m (well-known to
be
triction to
er of
invariant under the Hamiltonian flow (Lioville’s theo-
rem)), is a measure in )(
2GL [40,42]. This measure
has a res
E

(domain of energy of
M
) given
formally by






33
,dd,
NN
E
F
F
Hpq Epq  
 (96)
F
where , is a measurable set of
E
, and


,,
H
pq is
the Hamnian. ilto
But (94) for other side, is the Parseval’s theorem for
the isometries group in


HL
ce
. This induces us a sym-
metrical structure in the spa


,
FISMAT that ex-
tended to the infinitude of the space ,
E


F
ISMAT
E is ob-
tained




,
SY Et
 
,,
STEM FISMAT
Lin H HLinHCBH H

  ,
(97)
where


H, H,B is the space of enclosed operators,
with bench marks

2
log ,Nt

  white-
gr

ch are in
ables mayoralties in the ways or Hermitian forms


,,tt
 
 then ,
L
AB
Xisometric to , is
F
ISMAT
Eand
it follows (a), sentence. For other side,




,,
SYSYEM
 is the integrals space with
FISMAT
Et

coefficients in


L
AB
X

, and isomorphic to space




,tE

 the which is isomorphic to space


,
SMAT FI
E

hen the integral is evaluated in points of w
erefore the Banach space. Th





0,
LAB FISMAT
G
XE t
 





 (98)
This integapplicaral is true for all technology tion
t


G, for dual spaces in spectrum of technologies.
ore it follows (2).
s
The future ion of the
organized tran
o. In the foreseen exposi-
tion along of the Sections 1, 2 and 3, it is demons
that nanotechnology processes are the same with that
in
ity. Th
ich realizes the transformation
Theref
5. Conclusion
of the nanotechnology is the creat
sformations on space-time, matter and en-
ergy using codes that are born of quantum intention of
the field that we want to apply t
trated
tention processes where in this case we call intentional-
is property is formulated with integration of two
elements the field of wh
O
pen Access JAMP

F. BULNES 43
and the object which is applied in this transformation.
ie Groups,” Princeton Univer-
.
This defines the organized transformation in nanotech-
nology and the actions involved which shape the inten-
tion.
But also this intention realizes an integration of the
field of the proper object to transform with the field that
realizes this transformation. Cause and effect shape only
one unit in this process. In the nanotechnology theory, it
is necessary to realize that one third unification of the
time and space considering a shortest of uncertainly in all
the quantum process to define the synchronicity must be
born in more studies and mathematical research of the
motivic cohomology, K-theory, Lie infinite theory and
motives theory applied to the QFT. In the nanotechnol-
ogy process it will be fundamentally legitimized through
the quantum electrodynamics and their new version that
is necessary to obtain this synchronicity, given the prin-
ciples to the advanced process that requires the teleporta-
tion, quantum communication and other phenomena that
will be the next technologies to interstellar step and sur-
vivor of the humanity. In this work the maximum prin-
ciple is included that must govern all the process in
nanotechnology which is the intentionality.
6. Acknowledgements
I am grateful for the financing support offered by De-
metrio Moreno Árcega, M. in L. and Principal of the Te-
chnological Institute of High Studies of Chalco (TESCHA).
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