Creation of contours around of points of application
Combination of quantum factors of the field X.
Submitted: April 13th, 2011 Published: February 24th, 2012
From the Edited Volume
Edited by Mohammad Reza Pahlavani
In the universe all the phenomena of physical, energetic and mental nature coexist of functional and harmonic management, since they are interdependent ones; for example to quantum level a particle have a harmonic relation with other or others, that is to say, each particle has a correlation energy defined by their energy density
There is an infinite number of paths of this kind Γ, in the space-time of the phenomena to quantum scale, that permits the transition or impermanence of the particles, that is to say, these can change from wave to particle and vice versa, or suffer energetic transmutations due to the existing relation between matter and energy, and of themselves in their infinity of the states of energy. Of this form we can realize calculations, which take us to the determination of amplitudes of transition inside a range of temporary equilibrium of the particles, that is to say, under the constant action of a field, which in this regime, remains invariant under proper movements in the space-time. Then exist a
Nevertheless, this temporal equilibrium due to the space-time between particles can turn aside, and even get lost (suffer scattering) in the expansion of the space-time when the trajectory that joins the transition states gets lost, this due to the absence of correlation of the particles, or of an adequate correlation, whose transition states do not turn out to be related, or turn out to be related in incorrect form. From a deep point of view of the knowledge of the matter, this succeeds when the chemical links between the atoms weaken and break, or get lost for lack of an electronic exchange adapted between these (
To eliminate this distortion of the field is necessary to remember the paths and to continue them of a systematised form through of
In quantum mechanics, the spectral and vibration knowledge of the field of particles in the space, facilitates the application of corrective and restorer actions on the same field using their space of energy states through of the meaning of their electromagnetic potentials studied in quantum theory (
In particular, if we want the evaluation of this action to along of certain elected trajectory (path), inside of the field of minimal trajectories that governs the principle of minimal action established by
If we consider to the trajectory Γj, in terms of their deviation θ
Observe that the term ζ(θ
Now well, the relation between field and matter is realised through a quantum jump and only to this level succeeds. In the quantum mechanics, all the particles like pockets of energy works like points of transformation (states defined by energy densities). The field in the matter of a space-time of particles is evident like answer between these energy states, as it is explained in the
The integrals of Feynman-Bulnes, establish the amplitude of transition to that the input of a system with signal
The total correction of a field requires the action to a deep level as the established in (2), and developing in (4), and only this action can be defined by a logic that organises and correlates all and each one of the movements of the particles
We consider a space of quantum particles under a regime of permanent energy defined by an operator of conservation called the Lagrangian, which establishes a field action on any trajectory of constant type. A particle has energy of interrelation defined by their energy density which relates the states of energy of the particle over to along of time considering the path or trajectory that joins both states in the space - time of their trajectories. Thus an infinite number of possible paths exist in the space - time that can take the particle to define their transition or impermanence in the space - time, the above mentioned due to the constant action of the field in all the possible trajectories of their space - time. In fact, the particle transits in simultaneous form all the possible trajectories that define their movement. Likewise, if Ω(Γ) ⊂ R3 × I
In the quantum conception the perspective different from the movement of a particle in the space - time answers the previous question enunciating:
We consider M≈ R3 × I
with rule of correspondence
and whose energy due to the movement is
But this energy is due to their Lagrangian
For a classic trajectory, it is observed that the action is an extreme (minimum), namely,
Thus there are obtained the famous equations of Euler-Lagrange equivalent to the movement equations of Newton,
That is to say, we have obtained a differential equation of the second order in the time for the freedom grade
we define the Hamiltonian or energy operator to the ith-momentum
and Hamilton's equations are obtained
Nevertheless, it is not there clear justification of this extreme principle that happens in the classic systems, since any of the infinite trajectories that fulfill the minimal variation principle, the particle can transit, investing the same energy. Nevertheless the Feynman exposition establishes that it is possible to determine the specific trajectory that the particle has elected as the most propitious for their movement to go from
The concrete Feynman proposal is that the trajectory or real path of movement continued by a particle to go from one point to another in the space-time is the amplitude of interference of all the possible paths that fulfill the condition of extreme happened in (12) (to see figure 1 a)). Now then, this proposal is based on the probability amplitude that comes from a sum of all the possible actions due to the infinite possible trajectories that set off initially in
Using the duality principle of the quantum mechanics we find that the particle as wave satisfies for this superposition
where the term
where to two arbitrary points in the space-time (
Then we consider to this classic trajectory:
Of this manner, the action on this covered comes given for
It reduces us to calculate then the standardization term
Therefore, the exact expression is had in the probability amplitude
This type of exact results from the Feynman expression can also be obtained for potentials of the form:
But the condition given in (12), establishes that the paths that minimize the action are those who fulfill with the sum of paths given in terms of a functional integral, that is to say, those paths on the space Ω(Γ) ⊂ R3 × I
An interesting option that we can bear in mind here is to discrete the time (figure 2). Thus if the number of temporary intervals from
Thus, on a possible path γ, it is had:
Thus it is observed that the propagator
Realizing the change we re-write:
where Developing the first integral of (31), we have
Then integrating for
Then a recurrence has in the integrals of such form that we can express the general term as Therefore it is had for the propagator (30), which
identifying in this case:
it is had that the integration in paths is given for:
where in the first member of (36) we have expressed the Feynman integral using the form of volume ω(
where Ω*(M), is some dual complex (“forms on configuration spaces”), that is to say. such that “Stokes theorem” holds:
This is due to the infiltration in the space-time by the direct action ζ, that happens in the space Ω ×
Two versions of (36), that use the evolution operator and their unitarity are their differential version and numerical version of Trotter (numerical version of (36)). The first version is re-obtain the Schröndinger equation from the Feynman path integral. In this case the wave function involves the corresponding electronic propagator given in (30) with a temporal step
Realising the integral we obtain the differential version of the Feynman integral (36).
Since it has been argued in (Ionescu, 2004), this
Likewise, an action on G (“ζint”), is a character ζint :
A QFT (
Example 1. In the compute of path integrals on the graph configuration space C(Ω), The graphs Γ∈Gn, will be used in the string schemes given by BRST-quantization on gauge theory. For example, the BRST-quantization is always nilpotent around a vertex: . The Kontsevich class not has loops everywhere (figure 4 a)). The Feynman diagrams (figure 4 b)) conforms a subclass in the Kontsevich class, that is to say, restricted in the deformation quantization in respective micro-local structure of the Riemannian manifold (Kontsevich, 2003).
While the concept of subgraph of γ, is clear (will be modeled after that of a subcategory), we will define the quotient of Γ, by the subgraph γ as the graph Γ’, obtained by collapsing γ, (vertices and internal edges) to a vertex of the quotient. Then it satisfies the graph class succession under
We enunciate the following basic properties of the classic Feynman integrals. Let γ, γ’∈Γ, where Γ∈G, and ω(Γ), their corresponding Lagrangian with the property like in (38). We consider the path integral IΓ, like a map given in (37). Let
If ω(Γ), is a Lagrangian on (41) with Γ/γ = γ’, then
ɄΓ∈G (Feynman graph),
As consequence of the integral (44), we have the composition formulas
Feynman integrals over codimension one strata corresponding to non-normal subgraphs vanish. A graph Γ∈G, is normal if the corresponding quotient Γ/γ, belongs to the same class of Feynman graphs G.
The remaining terms corresponding to normal proper subgraphs meeting the boundary [
where the proper normal subgraph γ, meets non-trivially the boundary of Γ.
If the Lagrangian ω(Γ), is a closed form then the corresponding Feynman integral ζ is a cocycle. Then
Consider the space of hypercomplex coordinates (coordinates in the
This space is the corresponding to the group of Feynman
This functional belongs to the integral operator cohomology on homogeneous bundles of lines H1(PT, Ο(-2-2)), where PT = PT + ∪ PT − for example, for
The elements FU, can be expressed in a low unique way the map in the complex manifold Pm, like
with rule of correspondence
that allows us to identify ΦD, with L(P
which bears to the isomorphism among the cohomological spaces
which would be a quaternion version of these integrals? It would be the one given for integral of type Cauchy of functions of H-modules (Shapiro & Kravchenko, 1996), on opened D, that turn out to be Liapunov domains in R
But this cohomology of diagrams of contour integrals is applicable to 1-functions for P(C), in PT ±, that which is not chance, since it is a consequence of the
If we consider that the for-according complex manifolds have a pseudo-Hermitian complex structure not symmetrical and induced by the sheaf of quadratic forms
The Feynman integrals are invariants in R3, under rotations of Wick, that is to say
to a coordinates system in E4,
then Ω(Γ), represents a region W(
where the potential energy
Considers a microelectronic device that is fundamented in the functional space The integrals of Feynman-Bulnes give solution to the functional equation of a automatic micro-device to control (micro-processor) F(XZ+, YZ-) = 0 (Bulnes, 2006c). The informatics theory assign a cybernetic complex to C, (Gorbatov, 1986) and each cube in this cybernetic net establish a path on the which exist a vector of input XZ+, and a vector of output YZ-, signed with a time of transition τ, to carry a information given in XZ+, on a curve γj, (path) to a state YZ-, through logic certain (conscience), that include all the circuit C (Bulnes, 2006c). In the case of C, the logic is the real conscience of interpretation of C, (criteria of C). As C, has a real conscience of recognition; into of their corrective action and reexpert, elect the adequate path to the application of the corrective action. For it, the integrals of Feynman-Bulnes can be explained on the electable model , (path, see figure 1 a)) as:
The integrals of Feynman-Bulnes give solution to the functional equation of a automatic micro-device to control (micro-processor) F(XZ+, YZ-) = 0 (Bulnes, 2006c). The informatics theory assign a cybernetic complex to C, (Gorbatov, 1986) and each cube in this cybernetic net establish a path on the which exist a vector of input XZ+, and a vector of output YZ-, signed with a time of transition τ, to carry a information given in XZ+, on a curve γj, (path) to a state YZ-, through logic certain (conscience), that include all the circuit C (Bulnes, 2006c). In the case of C, the logic is the real conscience of interpretation of C, (criteria of C). As C, has a real conscience of recognition; into of their corrective action and reexpert, elect the adequate path to the application of the corrective action. For it, the integrals of Feynman-Bulnes can be explained on the electable model , (path, see figure 1 a)) as:
Def. 1 (Path Integrals of Feynman-Bulnes). A integral of Feynman-Bulnes is a path integral of digital spectra with composition with Fast Transform of densities of state of Feynman diagrams.
Since a duality exists between wave and particle, a duality also exists between field and matter in the natural sense (Schwinger, 1998). Both dualities are isomorphic in the sense of the exchange of states of quantum particles and the interaction of a field. Indeed in this quantum exchange of information of the particles, that happen in the space-time Ω(Γ) region, the pertinent transformations are due to realise to correct, restore, align or re-compose (put together) a field
|Elements of fiel Nano-metric application Effect obtained on field|
|0-lines localization of anomalous points encoding nodes to application|
|1-lines application of electronic propagator alignment of lines of field|
|-1-lines inversion of actions* reflections of restoration|
Any anomalous declaration in a quantum field shows like a distortion, deviation, non-definition or not existence of the field in the space-time where this must exist like physical declaration of the matter (existence of quantum particles in the space). The quantum particles are transition states of the material particles. We remember that from the point of mathematical view, a singularity of the space X(M), is a section of TM, in a mathematical sense (Marsden et al., 1983).
X(M), is a section of TM, in a mathematical sense (Marsden et al., 1983).
Def. 2.  If
A anomaly in a trajectory and thus in M, will be a singular point which can be a knot (multiform points), a discontinuity (a hole (source or fall hole)) in M)) or a indeterminate point (without information of the field in whose point or region in M). But we require their electromagnetic mean into the context of
If we consider the space C0
Γ(ψ) (Watanabe, 2007), as space of configuration associate with sub-graph (Γ, ψ), where ψ, is the corresponding smooth embedding to
In this study the path integrals and their applications in the re-composition, alignment, correction and restoration of fields due to their particles realise using certain rules of fundamental electronic state and their sub-graphs, through considering the identification. We define as correction of a field
A restoration is a re-establishment of the field, strengthening their force lines (properties of the Dirac and Heaviside function on particles: with
All anomalies in the space-time produce scattering effects that can be measured by the proper states using the following rules, considering these anomalies like a process of scattering risked by the particle with negative potential effect of energy:
The negative actions in one perturbation created by an anomaly in the quantum field X, acts deviating and decreasing the action of the “healthy” quantum energy states
Example 2. The energy in this Feynman diagram is the given by Eoutput = W− = −
One result that explains and generalises all actions of correction and restoring of a quantum field including the electromagnetic effects that observes with vector tomography is:
Theorem 1 (F. Bulnes) (Bulnes et al., 2010). Be M =
The effect on the field is re-construct and re-establish their lines of field (channels of enery) by synergic action (see figure 8).
The fundamental consequences are great, and they have to do with the reinterpretation of the anomalies of the field in an electromagnetic spectra (Schwinger, 1998), (see the figure 7), which we can measure across detectors of electromagnetic radiation, detectors and meters of current, voltage or amperage calibrated in micro or nano-units (Bulnes et al., 2011).
An important result (that can be a consequence in a sense of the previous one (for example in integral geometry and gauge theory)) that applies the vector tomography to electromagnetic fields used to measure fields of another nature and classify the anomalies by their electromagnetic resonance (at least in the first approach) is given by:
Theorem 2 (Bulnes, F) (Bulnes, 2006b). If the Radon transform (tomography on
In the demonstration of the theorem 1, the Stokes theorem guarantees the invariance of the value of the integrals of path under the application of an electromagnetic field (Landau & Lifshitz, 1987), like gauge of a quantum field, since the value of these integrals does not depend on the contour measured for the detection of a field anomaly (Bulnes et al., 2011).
In nanomedicine the applications of the corrective actions and restorers of a field are essential and they are provided by the called integral medicine, which bases their methods on the regeneration of the codes of cellular energy across the conduits of energy of the vital field that keeps healthy the human body, the above mentioned for the duality principle of mind-body. But the transformations are realised in the quantum area of the mind of the body, that is to say the electronic memory of the healthy body. The mono-pharmacists of integral medicine contain codes of electronic memory at atomic level that return the information that the organs have lost for an atomic collateral damage.
Diagram of strings belonging to the cohomology of strings equivalent to the code of electronic memory spilled to a patient sick with the duodenum (Bulnes et al., 2011) (see figure 9 a)). In nanomedicine, the path integrals are intelligence codes of corrective and restoration actions to cure all sicknesses. In the (see figure 9 b), W, is the topological group of the necessary reflections to the recognition of the object space of the cure (Bulnes et al., 2010). This recompose the amplitude of the wave defined in the spectra Iαβ
The study of the resultant energy due to the meta-stables conditions that it is obtains in the quasi-relaxation phenomena establishes clearly their plastic nature for the suffered deformations on the specimen. Nevertheless their study can to require the evaluation of the field of plastic deformation on determined sections to a detailed study on the liberated energy in the produced dislocations when the field of plastic deformation acts. Thus, it is doing necessary the introduction of certain evaluations of the actions of the field to along of the dislocation trajectories in mono-crystals of the metals with properties of asymptotic relaxation. Then we consider like specimens, mono-crystals of
By the theorem of Bulnes-Yermishkin (Bulnes, 2008), all functional of stress-deformation to along of the time must satisfy for hereditary integrals in the quasi-relaxation phenomena that have considered the foreseen actions inside of trajectory of quasi-relaxation like path integrals measuring field actions on crystal particles of metals:
The square bracket in (60), is the one differential form ω(Γ
Since it has been mentioned previously if we consider a set of particles in the space E, under certain law of movement defined by their Lagrangian
with rule of correspondence as given in (8), we can establish that the global action in a particles 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 following principle:
Principle. The temporary or instantaneous action on a global scale can be measured like a local permanent action.
The previous proposition together with certain laws of
Inside the universe of minimal trajectories that satisfies the variation functional (12) we can choose a γt∈Ω(Γ), such that
which is not arbitrary, since we can define any action on γ
that is to say, there exists an intention defined by the field action that infiltrates into the whole space of the particles influencing or "infecting" the temporary or instantaneous actions doing 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 across his path integrals like the already described ones. Then extending the above mentioned integral to the whole space Ω(Γ), we have the synergic principle of the whole field
the length and breadth of E. The order conscience is described by the operator of execution of a finite action of a field
How to measure this transference of conscience of transformation due to the field
We measure this transference of conscience (or intention) of
We let at level conjecture and based on our investigations of nanotechnology and advanced quantum mechanics, that a sensor for the quantum sensitisation of any particle that receives an instruction given by a field
Example 3. A force is spilled F(
Finally and based on the development that the quantum mechanics has had along their history, we can affirm that the classic quantum mechanics evolves to the advanced quantum mechanics (created by Feynman) and known like quantum electrodynamics reducing the uncertainty of Heissenberg of the frame of the classic quantum mechanics, on having established and having determined a path or trajectory of the region of space-time where a particle transits. Therefore the following step will demand the evolution of the quantum mechanics of Feynman to a synchronous quantum mechanics that should establish rules of path integrals that they bear 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 across their "revisited" path integrals, producing a joint effect called synergy. The time and the space they are interchangeable in the quantum area as we can observe it in the integrals (61). Where a particle will be and when it will be there, are aspects that go together. This way the energy is not separated from the space-time and forms with them only one piece in the mosaic of the universe.
I am grateful with Carlos Sotero Zamora, Eng., and with Juan Maya Castellanos, Msc., for the help offered for the digital process of the images that were included in this chapter.
Submitted: April 13th, 2011 Published: February 24th, 2012
© 2012 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.