_{1}

^{*}

It’s created a canonical Lie algebra in electrodynamics with all the “nice” algebraic and geometrical properties of an universal enveloping algebra with the goal of can to obtain generalizations in quantum electrodynamics theory of the TQFT, and the Universe based in lines and twistor bundles to the obtaining of irreducible unitary representations of the Lie groups SO(4) andO(3,1) , based in admissible representations of U(1) , and SU(n) . The obtained object haves the advantages to be an algebraic or geometrical space at the same time. This same space of ￡-modules can explain and model different electromagnetic phenomena in superconductor and quantum processes where is necessary an organized transformation of the electromagnetic nature of the space- time and obtain nanotechnologies of the space-time and their elements.

Let

Let the Lorentz group

where^{1}

Is the pseudo Riemannian metric of the manifold

Induced for the orientation of

Consider the electromagnetic field or Maxwell field defined as the differential 2-form of the forms space

Which can be described in the endomorphism space of

where

We want to obtain a useful form to define the actions of the group

Likewise, the electromagnetic field is the 2-form given by (6) with the property of the transformation

In

In the context of the gauge theories (that is to say, in the context of bundles with connection as the principal ^{2}

Consider the

And let the spaces

And

where^{3},

where^{4}, is the dual electromagnetic tensor of

In the absence of sources, the Maxwell equations are symmetric under a duality transformation, which interchanges electric and magnetic fields.

Proposition 2.1. (F. Bulnes) [

Proof. Using the definition of^{5} defined as the map

With rule of correspondence

where the images of

Then to a new coordinate system

where

Let

And such that the

which is completely equivalent to (8). But is enunciated in this moment because it legitimizes the Maxwell tensor from the scalar and vector potentials and we have (12).

We consider the space of electromagnetic power where we will define the domain of electromagnetic space transformation,^{6} that is to say, the cross product of

where^{7} having by pro- perties of tensor product of free modules that is:

which is universal in the following sense:

For every Abelian group

there is a unique group homomorphism

such that

for all ^{8} where in this case

We want describe energy flux in liquid and elastic media in a completely generalized diffusion of electromagnetic energy from the source view (particles of the space-time), which must be much seemed as a multi-ra- diative tensor insights space or a electromagnetic insights tensor space. This will permits us to express and model the flux of electromagnetic energy and any their characteristics.

The rate of energy transfer (per unit volume) from a region of space equals the rate of work done on a charge distribution plus the energy flux leaving that region.

Of fact these are elements

Then a source inside the electromagnetic multi-radiative space is obtained with the divergence, to know:

where

Proposition (F. Bulnes) 3.1. The electrodynamical space

Proof. [

Due to that we are using a torsion-free connection (e.g. the Levi Civita connection), then the partial derivative

Proposition (F. Bulnes) 3.2. The closed algebra

Proof.

Then the other properties of Lie algebra are trivially satisfied. Thus

Proposition (F. Bulnes) 3.3. The closed algebra

Proof. Since as Lie algebra, the space

Thus

vectors

Related

Theorem (F. Bulnes) 3.1. The electro-anti-gravitational effects produced from superconductivity have that to be governed by the actions of the superconducting Lie-QED-algebra

Proof. [

We want establish the electromagnetic principle that produce levitation or anti-gravity from the electro-anti- gravitational source that include the proper movements in the space-time that are connected with the actions of the group

These proper movements are determined through elements of

action of their Maxwell fields

calibrate the gravitational elements through electromagnetic elements such that these last can change the gravitational effects changing the spin characteristic of the affected region by these superconductor electromagnetic fields.

The initial ideas to this respect are replace the Abelian group

We want these identifications because our superconductivity theory establish the principles to risk the electro- anti-gravitational flight of an object as a sidereal object in the space-time, such that a galaxy or a star. In these sidereal objects, there are electromagnetic transformations explained MHD^{9}, where the superconducting phenomena go given form the accretion rings, and their rotation (see the

Use through the model that consists of a complex scalar field^{10} minimally coupled to a gauge field given by 1-forms (

where

If the potential is such that their minimum occurs at non-zero value of

of ^{11}) and appears as a topological charge associated with the topological current [

Developing these topological electromagnetic elements using the tensor

precisely is our tensor algebra given in proposition 3.1., with their conserved Lie structure.

The essential difference between both versions consists in the coupling to a charged

Considering the supercurrent

where

Considering to an electron field, a representation

And let^{12} the two-sided ideal in the tensor algebra defined in Section 2.1,

Proposition 2.2.1. There is a natural one-to-one correspondence between the set of all representations of

Proof. [

Def. 2.2.1. [

Before of this, we pass to the fundamental lemma to characterize the algebra

Lemma (F. Bulnes) [

Proof. [

Theorem (F. Bulnes) 5.1. [

The demonstrations was realized in [

And the field is transformed as

where explicitly the image

Which belongs to the charge-conjugated particle. The anti-particle is obtained of accord to the contragradient

There are not charge-conjugated in gravity, since if the gauge group is Lorentz group

But we need affect the immediate space-time at least locally through of these

We define the field

Under a general diffeomorphism

Then the principal equivalence requires that the fields on our manifold locally transform be as in special relativity, that is to say, if, is an element of the Lorentz group^{13} the fields are transformed like Lorentz-vectors. Of fact this property is extended to all electro-physical modules

However, the generalization to a general diffeomorphism is not unique. We could have chosen the field

But as

It will be useful to clarify the emerging picture of space-time properties by having a close look at a contravariant vector field

Then for completeness, let us also define the combined mappings through the relations:

Newly introducing the fields ^{14}. Then we can have (after of involve the relations of

where

which is a new connection. Then the Maxwell-anti-gravity Lagrangian (that is to say, for anti-gravitational pendants

Staying a Lagrangian of the type

Different microscopic aspects of electromagnetic nature are analyzed through the construction of an anti-com- mutative algebra of

# | QED-Lie-Algebra | ||
---|---|---|---|

Electrodynamics Object | Phenomena | ||

1 | Poynting Vector | Electromagnetic Power Density | |

2 | 4-Tensor Of Stress Energy | Electromagnetic Stress Energy | |

3 | Bi-Sided Ideal | Magnetic Flux Carried by the Fluxoids | |

4 | Product of Tensors | Super-Currents | |

5 | Product of Spins | Photon Spin | |

6 | ^{15} | Fermionic Fock Space in Superconducting | Electro-Anti-Gravitational Effect Produced from Superconductivity |

In this table are resumed all the applications mentioned in the sections of this work.

I am grateful for the invitation realized by the SCET-2015, organizers to participate with a talk in applied mathematics and physics.

Francisco Bulnes, (2015) QED-Lie Algebra and Their £ -Modules in Superconductivity. Journal of Applied Mathematics and Physics,03,417-427. doi: 10.4236/jamp.2015.34053