Journal of Applied Mathematics and Physics, 2013, 1, 2544 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 MexicoCuautla s/n Tlapala “La Candelaria” Chalco, Mexico City, Mexico Email: francisco.bulnes@tesch.edu.mx 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; HyperReality; Synergic Action; Events Synchronization; Synergic Operators; CyberQuantum Algebra; Perception Integral Transforms; FeynmanBulnes 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 , sLxs xs (2) We can establish that the global action in a particle system with instantaneous action can be reinterpreted 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 nanotechnological 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 (nanoseconds) 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 xM [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 , 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 rewalked ). 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 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 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 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 , 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 subparticles (boson space). Likewise a photon of certain class M 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 spacetime 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 spacetime 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 spacetime 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 uniparametric 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 tI R x being a tra jectory which predetermines a position for all time 3,R . t tI A field that infiltrates their action to the whole space of points predetermined by all the tra jectories ,,,t xt , 123 is the field that predetermines the points xt x , ii t which are fields whose determination is given by the action of the field , and evaluated in the position of every particle. Every point has a defined force by the action , of , along the geodesic , t and determined direction by their tan gent bundle given for , 1TX 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 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 , 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 , , 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 , 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 , 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 . 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 M where, if is a complex with , the spacetime, 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 spacetime 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 ) [12]. Let be a field acting on the particles 123 ,,,,xt xt xtM 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 , if and only if 1) , is the determination of the field , to realise or execute (their force); 2) , recognizes well their target, knowing what to do of the field (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 spacetime 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 , 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 reinterpretation 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, recomposition, 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 nstates 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 nanocomponents of the matter, space or energy with quantum intention. Example 1.1. The nanocomponents 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 spacetime (see Figure 1). M n super posed near the node for 12 12 U 000 0boxes d n n nXWXWXW x z , (19) Example 1.2. The intention of an action takes implicit the atomic visualization of every nanocomponent. 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 nanospace 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) Nanocomponents; (b) Fi eld. (a) The preservation of the measure of the energy process in the nanocomponents is given for their invariance. The ensemble of the nanocompone 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 spacetime 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 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 ). Then the intention of the field , 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 nanocomponents (particles) under an energy condition, that is to say, for every time , of the particles s, a map exists 0 : t TH H, (20) with rule of correspondence , s 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 JH E J d E cc J OOs , (23) Thus the limit given in (22) exists for each fixed , and is independent of , then there is a measure , on , 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 , the characteristic func tion of a measurable set , J then 1 1 0 0 0 1d 1d 1d, u v u T TF T TF v T Fuv T Ts T TT s T so if the Texists, then lim , 1 (T ), u 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 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 spacetime 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 socalled Hamiltonian function and the evolution of the system is given by the flow of the Hamiltonian vector field T*M :T*,HMR , associated to , 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, MR the spacetime of certain particles , 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 s xsFlux 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 , 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 , s and ,sqx 1,EVVq ,,q2 ,qLM 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 twoform , c O , on , by setting [12] TM * Lc O, (32) Considering the local coordinates ,, ii to , modeling the spacetime , through spaces, we have that (32) is MH 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, sq and 2. sq The total differential (34) is the symplectic form , 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 (spacetime 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 OM 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 vM, 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 , 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'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 codimension 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 by , to , particles n 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 , t 3) d; c cO Os H dd c Oc Oxs , H in the unlimited space 4) if and only if , c Os s sxs's s' s t , then F , sxs 5) () c x'Oxs O = F x'x x' and ,xx'M, 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 spacetime , like space where M ,3 d t Id R M is the macroscopic component of the spacetime and we called , the microscopic component of the spacetime of ratio (length of a string [21]). For previously described the quantum zone of the spacetime , is connected with , which we will called virtual zone of the spacetime (zone of the spacetime 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 recombining their states they become in waves on hav ing moved in F 10 33cm N , d t 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 ambispace (set of connection and field) defined as: M N , 0,XX xy xy NL` (40) where , is the connection of virtual field , with the quantum field and is the field whose action is always present to create perceptions in the quantum zone connected with ,Y, (2form) [22]. Then we can create the correspondence given by the double fibration [16]: (virtual zone of the spacetime)N M(quantum zone of the spacetime) (space of processing of quantum particles) L πθ (41) This double fibration conforms the interrelation be tween , and . M N xtM, 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 spacetime by the light cones [23], of all trajectories that follows a particle in the spacetime [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 yx x 1, x (42) of , such that which by spacetime N11 , PP x Open Access JAMP
F. BULNES 33 properties to quantum level represents the space of all light rays that transit through , conforming a hyper surface (projective surface) that is a light surface. This surface is called the sky [24]. A sky in this context represents the set of light rays through (bosons) that comes of the virtual field. If then 4,CM, QMMis the complete uni verse (include the supersymmetries [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 , s is the quantum spacetime 3 t R , M 2 (which is the spacetime as the Einstein perception) then the images of the virtual parti cles are QC [11], then the execution operator that proceeds to connect virtual particles through the paths which have path integrals on a double fibration, establishing the materialquan tumvirtual connection required to a total reality: ,I M L π θ M C ` σ ρ (43) where is the material part connected with the quan tum zone of the spacetime (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 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 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* 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 QPP , Q in every sky ; 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 nnR, nated by , j we describe terms of a graded commu tative C Malgebra , with generating elements 12 12 ,, ,,, k aa a xxx x a, (47) and the bigraded differential algebra of differen tial forms (the ChevalleyEilenberg differential calculus [4]) over *,H 0, as an algebra [4,7,27]. One can think R Open Access JAMP
F. BULNES 34 of generating elements (44) of , as being sui generis coordinates of even and odd fields and their partial de rivatives. The graded commutative algebra R0, 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 multiindices. One can think of even ele ments 0 , d,x dd (1) dd an an a an q Lxx L xx d , Lx L E (49) where we observe that ,L is the 2form given by , in the formula (33) with and 2,n12 . Now we consider the dual part of the space ,H , that is to say, the space *, . 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, the graded commutative. It is generated by elements ,xx . a For any x , there is a homomorphism belonging to the space T, , HomH D T, , (with homomorphisms HomHD 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, of classical fields to the algebra which sends the basic elements ,B 0a, H to the elements a and replaces coefficient functions of elements of ,B ,I0, with their values I (exe cutions) at a point . 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 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 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 , 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 where the image of the 1form , that the Lagrangian defines, i , is a symplectic form [1], and the vari able i 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 s, in the spacetime) to have the two components that characterise any quantum particle 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 in the variables i and , runs over the dimension of the ambient manifold which is assumed to be i . C If the space is a complete intersection, the con straints a i W , (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 NK 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 nonvanishing on the constraints Nk a W i Ζ ,ANkq , and , is a numerical tensor which 1Nk Aaa T is antisymmetric 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 's we can decompose the i, into a set of coordinates Ζ Yaa W 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 nowherevanishing and non singular only to the case of space (CalabiYau manifold). The Cal abiYau 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 spacetime due to their holomorphicity [29]. The vectors YC 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 CalabiYau, also exists a globallydefined 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 YC 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 ,pZ p 0 S OO (57) where iA ,pO, are the vertex operators of the theory localized at the points pA, of the Riemann surface and , iA p ,0 . k 0, is the zeromode component of the vertex operators. Newly our conscience operator come given by the form O Example 3.1. All CalabiYau manifolds are spin . In hypothetical quantum process (from point of QFT view), to obtain a CalabiYau 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 , 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 causeeffect induced for their proper causal structure , then by the postulate 1, we will refer to the above mentioned space like nanospace. SM C In a nanospace there is a structure of Hilbert space, since this has a prehilbert 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 wV be defined by ,,,,, uuwuwuV (60) Then ,. u so .V E , Let be an or thonormal basis of Set ij 1,,, n ee , i j Xee for .ij Then ij , , is a basis for We define a linear map ij .Eso , of ,Eso into by End SE , 12 , iji j ee (61) A direct calculation shows that ,,,,, YXYXYE so (62) This implies that ,SE , defines a module for Dicho modulo es el espacio .Eso . 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 ,S Let be a nonzero 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 jj ie we , (63) with 1jk and that . WCw Set 21 2j ,1 .jk jj hXi e Then , Rh t .Vso is a maximal Abelian subalgebra of Let , j t be defined by . jk h A direct calculation yields 1121 2, kk iw whw wk (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 semisimple 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 coaction 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 nanospace 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 ANO S (see Figure 4(c)). 4. Results Proof. The corresponding holomorphic ChernSimons 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 CalabiYau manifolds. Being a Cal abiYau manifold a spin manifold, as is mentioned above, then the related strings with the fields and their polariza tion (orbits of the spacetime (see Figure 4(a)) need of dimensional manifolds embeddings into spacetime. 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 YC 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 nanostructure 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 coaction 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 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) 21Ndimensional 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 ic scale. B er the atom which represent the in finitesimal volume of the microscopic spacetime, 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 , 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 BRSTcohomology 0 , ,HGGM0 l and with Proof. We app the BRSTmology to the spacetimewith 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 centerofmass , 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 spacetime (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 semiinfin 0 , ,.HH GGNN SS We ignore because it represents the permanent quantum 0,G N ,HGN S , where d ,G is a fixe. G c Consider the plex cverelative subcomonsisting of tors Fock space satisfying 0 00. c bO The induced differential in this subcomplex is easily seen to be the operator Q, (extended BRSToerator) who in clude the more and deep action of , c O reflected in (32). The cohomoloplex is ning but the rela tive semiinfinite cohomology p othgy of this com 0 ,,,HGNN 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 ,,,HGNN 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 I the unlimited space of the quantum space (Fock space [24]). A particle , t energy that is focalized by a given load function bad evolutionfor the ,,wts comes given for , dd 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 , comes given by the theorem [12]: 1 1 dim d jj j C correc restoringc tion, d , XC tOs wtst t (75) where nF n x A dim , is the Neumann dimension corre sponding to the Weyl camera of the roots , [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 acetime from the field ation R of the ,C . 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 , Q such that x QPP 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 ANO S o, the f . , the nano of the spacetime In side of nanorce of potential is given by M. F 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 rad , in question. But , is the potential that have the zone , in the spacetime. 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 (', and ' ), 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 tesum. 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 , 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 (v) out(v) 1 in(v) k(v) k(v) v (85) Proof. Here ad, , YXY erential operators. is the commu bracket on diff Therefore if tation , and ,Y commute, then ,. 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 recompositions 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 recompose the space esta ld ,XM adapted in same resonance, quantum composition of the space points , 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 autoordering in the nanolevel. 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 spacetime 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 reinterpreting the region of the spacetime of the parti cles like a electronic complex of a hypothetical loc nanofloodgate (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 FeynmanBulnes 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 ,, nBulnes 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 FeynmanBulnes is [36]: 123 4 1234 0001101001 , nnn n man Fn FnFnFn The integrals of FeynmanBulnes, establish the ampli tude of transition to that the input of a sith signal ystem w , t can be moved through of a synergic action of electronic charges , doing thro predetermined ugh of s functions by ,Lxt wave and encoded in a binary
F. BULNES 41 algebra (predefined by states 0, 0, , and 1 ), (in the kernel of the space  tion of solutions of the wave equa AA AA xF x' □ of a circuit 16) of a point to into other . Their int of paths or rewalked 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 , 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 emissionreception of electrons (by means of one sym bolic ,25]). Then the evaluation of the FeynmanBues 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 subgraph: 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,, 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 , 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 , 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 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 ndimensional space. If in particu , 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 , ISMAT E they act under an operator for the design of experiments it has more th o ng such traby means of fu rehearsalsnd retests 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 nanoteccal 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 , ISMAT E is endow of Hilbert structure to the space , 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 , AB X like the space of rehearsals or tests of technological process. Let us endow , 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 , AB X satisfies Implies tautologically, aLAB Xf (89) fGL 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 , ISMAT E to , 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, , 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: , ISMAT E is isometric to , 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 ndimensional 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 R easure (,N 33 dd NN qp is the numb particles) with the m (wellknown 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 (domain of energy of ) given formally by 33 ,dd, NN E F Hpq Epq (96) where , is a measurable set of , and ,, 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 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 , AB Xisometric to , is ISMAT Eand it follows (a), sentence. For other side, ,, SYSYEM is the integrals space with FISMAT Et coefficients in 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 spacetime, 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, Ktheory, 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). REFERENCES [1] J. E. Marsden and R. Abraham, “Manifolds, Tensor Ana lysis and Applications,” AddisonWesley, Massachusetts, 1993. [2] C. Chevalley, “Theory of L sity Press, Princeton, 1946 [3] F. E. Burstall and J. H. 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Shapiro, “On General Theory of Inte gral Operators to Analysis and Geometry (Monograph in Mathematics),” 2007. F. Bulnes, H. F. Bulnes, E. Hernandez and J. Maya, “In[17] tegral Medicine: New Methods of OrganRegeneration by Cellular Encoding through Path Integrals ap Quantum Medicine,” Journal of Nanotechnology in En gineering and Medicine, Vol. 1, No. 7, 2010, Article ID: 030019. [18] D. McDuff and D. Salamon, “Introduction to Symplectic Topology,” Oxford University Press, Oxford, 1998. [19] I. S. Sokolnikof tions,” Wiley and Sons, New York, 1951. [20] B. R. Holstein, “Topics in Advanced Quantum Mechan ics,” AddisonWesley Publ 1992. [21] A. Kapustin, M. Kreuser and K. G. Schlesinger, “Homo logical Mirror Symmetry: New Developments and Per spectives,” Springer, Berlin, 2009. O pen Access JAMP
F. BULNES Open Access JAMP 44 tentials in Quantum Theory,” Physical Review [22] Y. Aharonov and D. Bohm, “Significance of Electroma gnetic Po, Vol. 115, No. 3, 1959, pp. 485491. http://dx.doi.org/10.1103/PhysRev.115.485 [23] F. Bulnes, “Doctoral Course of Mathematical Electrody namics,” In: National Polytechnique Institute, Ed., Ap pliedmath 3: Advanced Courses, Proceedings of the Ap plied Mathematics International Congress, 2529 October 2006, México. [24] E. R. LeBrun, “Twistors, Ambitwistors and Conformal Gravity,” In: T. N. Bailey and R. J. Baston, Eds., Twistors in Mathematics and Physics, Cambridge University, Cam bridge, 1990, pp. 7186. [25] L. P. Hughston and W. T. Shaw, “Classical Strings in Ten Dimensions,” Proceedings of the Royal Society of London Series A: Mathematical and Physical Scien No. 1847, 1987, pp. 423431. ces, Vol. 414. [26] F. Bulnes, “Penrose Transform on DModules, Moduli Spaces and Field Theory,” Advances in Pure Mathematics, Vol. 2, No. 6, 2012, pp. 379390. http://dx.doi.org/10.4236/apm.2012.26057 [27] R. Sobreiro, “Quantum Gravity,” InTech, Rijeka, 2012. http://www.intechopen.com/books/quantumgravity [28] P. Griffiths and J. Harris, “Principles of Algebraic Ge ometry,” WileyInterscience, New York, 1994. http://dx.doi.org/10.1002/9781118032527 [29] M. Gross, D. Huybrechts and D. Joyce, “CalabiYau Ma nifolds and Related Geometries,” Springer, Norway, 2001. [30] J. Hoogeveen and K. Skenderis, “Decoupling of Unph . Skenderis, “Decoup l Pure S han “Riemannian Geometry and Geometric Analysis,” Press, Cambridge, 2002. y sical States in the Minimal Pure Spinor Formalism I,” 2010. [31] N. Berkovits, J. Hoogeveen and K ling of Unphysical States in the Minimapinor Formalism II,” 2009. [32] T. W. B. Kibbe, “Geometrization of Quantum Mec ics,” Springer Online Journal Archives, Vol. 65, No. 2, 1979, pp. 189201. [33] J. Jost, 4th Edition, SpringerVerlag, Berlin, 2005. [34] Y. Makeenko, “Methods of Contemporary Gauge Theory,” Cambridge University http://dx.doi.org/10.1017/CBO9780511535147 [35] F. Bulnes, “Treatise of Advanced Mathematics: System and Signals Analysis,” 1998. noPharma l. 1, No. [36] F. Bulnes and H. F. Bulnes, “Quantum Medicine Actions: Programming Path Integrals on Integral Mo cists for Strengthening and Arranging of the Mind on Body,” Journal of Frontiers of Public Health, Vo 3, 2012, pp. 5765. http://dx.doi.org/10.5963/PHF0103001 [37] F. Bulnes, “Analysis and Design of Algorithms to the ethods for Phys , 1972. ns, osophical Meth esearch,” R. A. Topin, “The Classical Field Theo Master in Applied Informatics,” Research Course, Uni versity of Informatics Sciences (UCI), Habana, 2007. [38] F. Bulnes, “Cohomology of Cycles and Integral Topol ogy,” 2008. www.Appliedmath4.ipn.mx [39] B. Simon and M. Reed, “Mathematical M ics,” Vol. 1, Academic Press, New York [40] F. Bulnes, “Conferences of Mathematics: Seminar of Re presentation Theory of Reductive Lie Groups,” Compila tion of Institute of Mathematics, UNAM Publicatio Mexico, 2000. [41] F. Bulnes, “Handle of Scientific and Phil ods of the Research: Guide of Prospective and Quality of the International Scientific and Technological R 2011. [42] J. Dieudonne, “Treatise on Analysis,” Vol. 6, Academic Press, New York, 1976. [43] C. Truesdell and ries (in Encyclopaedia of Physics, Vol. III/1),” Springer Verlag, Berlin, 1960.
