The (E) theory: A new attempt to unify EM and gravity

[member 11137]

13 April 2006
Let us consider the stress energy tensor for an EM field (Lichnerowicz; Masson and Co; 1955; Théories relativistes de la gravitation et de l’électromagnétisme):
$$T_{ab}$$ =¼. $$g_{ab}.F_{cd}.F^{cd}$$ – $$F_{ar}.F_{b}^{r}$$

Let us consider the quadratic form
F = ½. $$F_{ab}.dx^{a}$$ x $$dx^{b}$$

1°) Let us then remark that:
½. $$F_{cd}.F^{cd}$$ = <F. F>
is the scalar product of F with itself [and can be sometimes interpreted as |E|² - |B|² where E is the electric field and B the magnetic field].

2°) Also note that my own hypothesis, the existence of a decomposition of the Maxwell EM field tensor so that (see etfgb03.doc above; equation 10.21):
$$F^{ab}$$ = scalar. $$[g_{bc}.A_{ea}^{c}$$ – $$A_{eb}^{c}.g_{ca}].v^{e}$$

has the necessary consequence that the term:
$$T_{ab}$$ = … – $$F_{ar}.F_{b}^{r}$$

can be written:
$$T_{ab}$$ = … – $$D_{ef}.v^{e} v^{f}$$

At the end, note that the stress energy tensor for any EM field inside my theory can be written:
$$T_{ab}$$ = ½. $$g_{ab}.<F. F>$$ – $$D_{ef}.v^{e}.v^{f}$$

and make a comparison with the stress energy tensor proposed for a perfect fluid… within the generalized theory of relativity, … one could propose to identify:
1°) – ½. <F. F> and p the pressure of this fluid;
2°) the eigenvalues of $$D_{ef}$$ should give us the possible values for (density + pressure).

Thus, to make short, if logical with itself, my theory is investigating the possibility to interpret any EM field with a fluid and sometimes with a perfect fluid. This would change a little bit the usual analysis made in Lichnerowicz where (although the author notes himself – page 18 of the same reference- the provisory character of the form proposed for) the stress energy tensor is the sum of the general form of a stress energy tensor in harmony with the requirements of the generalized theory plus the stress energy tensor for any EM field arising from the considerations made within the restricted (special) theory of relativity.

In my approach, these two components are the same. EM fields can be understood as being fluids with eigen-fluctuations. Rationalistic (= in relation with some experiments) or totally crazy (= only a mathematical toy theory)?

 

[member 11137]

2°) Also note that my own hypothesis, the existence of a decomposition of the Maxwell EM field tensor so that (see etfgb03.doc above; equation 10.21):
$$F^{ab}$$ = scalar. $$[g_{bc}.A_{ea}^{c}$$ – $$A_{eb}^{c}.g_{ca}].v^{e}$$

has the necessary consequence that the term:
$$T_{ab}$$ = … – $$F_{ar}.F_{b}^{r}$$

can be written:
$$T_{ab}$$ = … – $$D_{ef}.v^{e} v^{f}$$

At the end, ...

First autocritic: $$D_{ef}.v^{e} v^{f}$$ is unfortunately not $$D_{ab}.v^{a} v^{b}$$. Thus an identification appears to be possible for a limited number of circumstances. But not impossible, a priori.

 

[member 11137]

Perhaps you did not realize that my new homepage adress is now:
http://www.vacuum-world-net.eu
The discussion is going further here; thanks for your attention.

 

[member 11137]

The above discussion concernin the relationship between extended vector product and parallel transport makes sense and allows a conform re-formulation of the FAraday Maxwell tensor if one applies it to the parallel transport of the EM potential 4-vector A. This is supposing a parallel transport of this (in fact) gauge vector by respect for the trajectory of what (a wave, a particle, or whatsoever) is moving. Does it makes sense: that's another question. Independently of the answer to this fundamental question, the new mathematical formalism allways is (local decomposition of the tensor in any 4D space):
F = (metric tensor x trivial matrix) - (transposed of the trivial matrix x metric tensor) + complementary term related to the (first order) variations of the metric.

 

[member 11137]

demonstrating my affirmations

The above discussion concernin the relationship between extended vector product and parallel transport makes sense and allows a conform re-formulation of the FAraday Maxwell tensor if one applies it to the parallel transport of the EM potential 4-vector A. This is supposing a parallel transport of this (in fact) gauge vector by respect for the trajectory of what (a wave, a particle, or whatsoever) is moving. Does it makes sense: that's another question. Independently of the answer to this fundamental question, the new mathematical formalism allways is (local decomposition of the tensor in any 4D space):
F = (metric tensor x trivial matrix) - (transposed of the trivial matrix x metric tensor) + complementary term related to the (first order) variations of the metric.

The correct proposition:
You didn’t hear of me since a long time. Theoretical research is a challenge for professionals and only a “Neben-produkt” for amateurs. The progression is depending on how many free time one gets to do it …
In between, you certainly realized that I have made a lot of errors concerning the description of the (covariant) components of the Faraday-Maxwell tensor (henceforth called “the tensor”). I should have written for the trivial proposition, or better said, for the proposition involving a trivial matrix in the split of the extended vector product (I note it here with the symbol “x” because I don’t have a better possibility to translate my own symbol with Tex)$$^{(4)}u$$ “x” $$^{(4)}w$$ supposed to be associated with this representation of the tensor, in extenso: F = scalar one. (metric tensor time trivial matrix) + scalar two. (transposed of the trivial matrix time metric tensor), the following relation:
$$F_{ab}$$ = ($$s_{1}. g_{ac}. A_{eb}^{c}$$ + $$s_{2}. A_{ea}^{c}. g_{cb}). u^{e}$$ (1)

The test:
If we consider that EM physical phenomenon are occurring when following conditions are realized:
a) The local cube defining the extended vector product supposed to be involved in the discussion (the E Theory hypothesis) contains 64 scalars corresponding to a local connection;
b) The fundamental extended vector product under consideration is in fact the extended vector product of the "EM-potential 4-vector" by the local position 4-vector;
c) The "EM-potential 4-vector" is parallel transported with respect to the local position 4-vector.

Then:
1) starting from the historical definition of the tensor (involving neither the total derivates D nor complementary terms, e.g. Yang Mills, but only the partial derivation that I shall exceptionally note here d –problem with tex):
$$F_{ab}$$ = $$d_{a} A_{b}$$ – $$d_{b} A_{a}$$ (2)
2) and supposing that usual universal rules of the differential calculus are locally valid, it is straightforward to demonstrate that hypothesis a), b) and c) above lead to:
$$F_{ab}$$ = $$[g_{ac}. A_{eb}^{c}$$ - $$A_{ea}^{c}. g_{cb}]. A^{e}$$ + $$[d_{a} g_{eb}$$ – $$d_{b} g_{ea}]. A^{e}$$ (3)
where it is easy to recognize the equation (1) for $$s_{1}$$ = $$- s_{2}$$ = 1 in the first part of (3).

Conclusion:
Assuming the hypothesis a) b) and c) above allow to write:
$$F$$ = $$(G. T$$ – $$T^{t}. G)$$ + [… $$(d_{a} g_{eb}$$ – $$d_{b} g_{ea}). u^{e}$$ … ] (4)
where the $$u^{e} …$$ = $$A^{e} …$$ are now the contra-variant components of the EM potential four vector A, G is the matrix representation of the local metric tensor whilst T is those of the trivial matrix and $$T^{t}$$ of its transposed.

For negligible variations of the local metric, the second term in (4) vanishes and we stay with:
$$F$$ = $$(G. T$$ – $$T^{t}. G)$$
which is the expression corresponding to our intuitive representation of the tensor. Next steps will analyze the possibility to generalize the proposition (1) to others circumstances than those described by a) b) and c).

 

[member 11137]

Some remarks

A) The above demonstration starting from (2) did not verify if the proposed formalism satisfies some any basic properties of the tensor. In fact it is not necessary since it lies on an accepted definition of the tensor and since it only needs a minimum of very simple and acceptable circumstances to be valid. These verifications must only be done if we conversely start from (1). For example, the proposition (1) must satisfy elementary constraints:
$$F_{aa}$$ = 0 (5)
This is equivalent to:
$$F_{aa}$$ = ($$s_{1}. g_{ac}. A_{ea}^{c}$$ + $$s_{2}. A_{ea}^{c}. g_{ca}). u^{e}$$ (6)
Since a GR compatible metric should be symmetric, this is yielding:
$$F_{aa}$$ = ($$s_{1}. g_{ac}. A_{ea}^{c}$$ + $$s_{2}. A_{ea}^{c}. g_{ac}). u^{e}$$ (7)
At this moment, we theoretically get two possibilities:
a) the body K on which the theory is built is commutative and we get:
0 = ($$s_{1}$$ + $$s_{2}$$). $$g_{ac}. A_{ea}^{c}. u^{e}$$ (8)
1) For this proposition to be valid independently of the contra-variant components $$u^{e} …$$ we must write:
0 = $$s_{1}$$ + $$s_{2}$$ (9)
This is corresponding to the special case encountered in the demonstration made above and starting from (2).
2) For this proposition to be valid independently of the two scalars we must write:
0 = $$g_{ac}. A_{ea}^{c}. u^{e}$$ (10)

b) the body K on which the theory is built is not commutative and we get complications that we leave for later.

B) The demonstration is apparently valid for any local metric. The formalism:
$$F$$ = $$(G. T$$ – $$T^{t}. G)$$ (11)
as still explained in former interventions on this forum allows a discussion within an A.D.M. time-slicing procedure.

 

[member 11137]

If the connection is metric compatible

Now there is something else that we did not examine in the demonstration above [starting from (2)]: the nature of the connection represented by the cube A. And we also didn’t precise if it is a connection compatible with the $$g_{ab}$$ metric. For the coherence of our approach we shall now suppose that yes. Yes the cube A is coinciding with a Levi-Civita connection metric compatible and we can write the well-known formula for it:
$$A_{ea}^{c}$$ = ½. $$g^{cp}$$. ($$d_{e}g_{ap}$$ + $$d_{b}g_{ep}$$ - $$d_{p}g_{ae}$$) (12)

Then (although it is a little bit tedious) we can demonstrate that:
$$(d_{a}g_{eb}$$ – $$d_{b}g_{ea}). A^{e}$$ = - 2. $$[g_{ac}. A_{eb}^{c}$$ - $$A_{ea}^{c}. g_{cb}]. A^{e}$$ (13)

Thus, starting from (4), this is yielding:
$$F$$ = - $$(G. T$$ – $$T^{t}. G)$$ (14)
So we see that it gives the great advantage of a very simple non zero formulation for the tensor and this formulation, up to a minus sign, is in coincidence with our first intuition. But the most interesting here lies in the fact that the second term of (4) in someway vanishes, as if it was absorbed by the special property of the connection to be metric compatible. And we finally stay with the desiderated formalism independently of the eventual local variations of the local metric.

If we adopt the Yang Mills formulation for the tensor, this means that the tensor is:
F = ($$T^{t}. G$$ - $$G. T$$) + […[$$A_{a}$$, $$A_{b}$$] …] (15)

 

[member 11137]

Generalization

Non trivial solutions of the splits
It must be clear now that we deal with $$^{(4)}A$$ “x” $$^{(4)}X$$ extended vector products where $$^{(4)}X$$ is the local four vector giving the position. These products can split into P. $$^{(4)}X$$ + $$^{(4)}Z$$ where P is a (4-4) matrix built with elements of K and where $$^{(4)}Z$$ is called the rest of the split.

Since it can be shown that non trivial solutions P exist that own the following formalism:
G. (P – T) = - Q (16)
Where Q is a Hessian matrix depending on a h($$^{(4)}X$$) function it is straightforward to show that:
G. P = G. T - Q
$$P^{t}. G$$ = $$T^{t}. G$$ - $$Q^{t}$$
($$T^{t}. G$$ - $$G. T$$) = ($$P^{t}. G$$ - $$G. P$$) + ($$Q^{t}$$ - $$Q$$)
and:
F = ($$P^{t}. G$$ - $$G. P$$) + ($$Q^{t}$$ - $$Q$$) + […[$$A_{a}$$, $$A_{b}$$] …] (17)

Thus, the tensor can own the intuitive formalism:
F = ($$P^{t}. G$$ - $$G. P$$) (18)
If:
$$^{(4)}A$$ “x” $$^{(4)}X$$ = P. $$^{(4)}X$$ + $$^{(4)}Z$$ (19)
and if there exists a h($$^{(4)}X$$) function so that:
[0] = ($$Q^{t}$$ - $$Q$$) + […[$$A_{a}$$, $$A_{b}$$] …] (20)
The special property of the Q matrix is so that if h($$^{(4)}X$$) has no discontinuity, then:
[0] = ($$Q^{t}$$ - $$Q$$) (21)
and consequently :
[0] = […[$$A_{a}$$, $$A_{b}$$] …] (22)
The non Abelian term vanishes.

This suggests that the formalism (18) can also hold when a discontinuity of h($$^{(4)}X$$) exists. In this case one must have (20). The difficulty is to find the solutions of this equation.

 

[member 11137]

Why all these things?

Good question isn't it? The demonstration above lies on (16) for which I have a general demonstration which is too long to be exposed here. The result of my approach is that we can connect the non abelian term of the Yang Mils formulation of the tensor with a discontinuity of a vector(ial) fonction h(X). As we know through other chapters of the physics that propagation of waves can be related with the propagation of discontinuities (Hadamard), we are pushed to believe that the h(X) function has something to do with the propagation of the EM phenomenon under consideration in this theory... A door for a solution concerning the (mass) gap in the Yang Mils Theory? I hope it. I would, at this step enjoy critics and discussions because it is hard to work alone. Thanks. Blackforest.

 

[member 11137]

Apologize

Then (although it is a little bit tedious) with (12) we can demonstrate that:
0 = $$[g_{ac}. A_{eb}^{c}$$ - $$A_{ea}^{c}. g_{cb}]. A^{e}$$ (13)

So, I must apologize for this stupid error; ... sorry. I slowly think that either I should do something else or look into the direction of non abelian mathematical structure. Otherwise: what does it mean: "a parallel transported EM potential 4-vector"? Since this potential is understood as a gauge field, it owns an arbitrary signification more for the use of the mathematician than for the use of the physics; even if it can help to explain phenomenon (Aharonov-Bohm shift). The point of view of the demonstration starting from (2) pre-supposes the existence of such a gauge field every where and at each moment that would be effectively encountered by the EM phenomenon under consideration... It says nothing on the procedure involved in the eventual motion from a point to the other.

At the end, concerning our approach in the case of a metric compatible connection, the tensor is reduced to the second term on the right hand of (4) post 40. Plus the non abelian terms eventually in a quite more general approach.

 

[member 11137]

In a metric compatible connection (02)

By the wonder of the calculus, this embarrassing situation (13) post 45 is equivalent to:
$$T^{t}. G$$ - $$G. T$$ = [0] (23)
and to the residual result :
$$F_{ab}$$ = $$(d_{a}g_{eb}$$ – $$d_{b}g_{ea}). A^{e}$$ (24)
But if the connection (cube) A is metric compatible, this is also leading to:
$$F_{ab}$$ = (2. $$A_{cba}$$ – $$d_{c}g_{ab}$$. $$A^{c}$$ (25)
and to:
$$F_{ab}$$ = (2. $$g_{be}. A_{ca}^{e}$$ – $$d_{c}g_{ab}$$. $$A^{c}$$ (26)
We recognize:
F = 2. $$T^{t}. G$$ – […$$d_{c}g_{ab}$$. $$A^{c}$$…] (27)
Thus, because of (23), we could write:
F = ($$G. T$$ + $$T^{t}. G$$) – […$$d_{c}g_{ab}$$. $$A^{c}$$…] (28)
… This is the correct expression for the tensor in a metric compatible connection; instead of (14) post (42).

Why all these things (bis). Not only for the challenge but for the pleasure to learn and to research. Essentially to connect with the problematic of the polarizations; here in a 4-D space. The important variables of this theory are the metric tensor G and the matrix T (or more generally P) resulting from the split of the fundamental extended vector product under consideration. This approach allows also a theoretical connection with the problematic of the Berry's phases (shift) because it is intimely connecting the expression of the Faraday Maxwell tensor with the local metric. A lot of work must now be done to develop these foundations, I agree.

 

[member 11137]

Non commutative structures

The former steps could convince ourself that the intuitive formalism proposed for the tensor in a 4D space was not totally stupid. Let us now demonstrate that the esthetic of the things is not a sufficient argument to valid this approach. At least one more important condition is needed.

Since we try to connect our own approach with the historical one which is related to the existence of a gauge field $$^{4}A$$ which is a 4-vector of $$E_{4}$$, a vector space built on what we call a “body” K, and since a gauge field offers some freedom, even if this freedom will have later a price (i.e. we shall have to verify if the choice made for the gauge yields at the end a coherent theory), we decide to fix the gauge with the following relation:
{For any subscripts a and b = 0, 1, 2 or 3: $$d_{c}g_{ab}$$. $$A^{c}$$ = 0} (29)
This choice leaves us with:
F = ($$G. T$$ + $$T^{t}. G$$) (30)
We know that the $$T_{ab}$$ are the 16 components of T and that the component in position (a, b) of $$T^{t}$$ writes $$T_{ba}$$. So that (30), in a coordinates language is:
$$F_{ab}$$ = $$g_{ae}. T_{eb}$$ + $$T_{ea}. g_{eb}$$ (31)

For this proposition to really describe an EM field one must have:
$$F_{ab}$$ + $$F_{ba}$$ = 0 (C.1)
$$F_{aa}$$ = 0 (C.2)
Let us study the implications of these condition, and recall that GR compatible metrics are symmetric (H.1).

For the first condition (C.1) above, let us make an inversion of the subscripts a and b in (31):
$$F_{ba}$$ = $$g_{be}. T_{ea}$$ + $$T_{eb}. g_{ea}$$ (32)
Let us make use of the symmetry of the metric:
$$F_{ba}$$ = $$g_{eb}. T_{ea}$$ + $$T_{eb}. g_{ae}$$ (33)
Let us make the important hypothesis that K owns an anti commutative multiplication:
{For any element a and b of K: a. b + b. a = 0} (H.2)
Let us make use of this hypothesis (H.2) in (33):
$$F_{ba}$$ = - ($$T_{ea}. g_{eb}$$ + $$g_{ae}. T_{eb}$$) (34)
If the addition of two scalars of K is “usual”, then:
$$F_{ba}$$ = - ($$g_{ae}. T_{eb}$$ + $$T_{ea}. g_{eb}$$) (35)
and finally, we get (C.1):
$$F_{ba}$$ = - $$F_{ab}$$

For the second condition (C.2) above, let us write, starting from (31):
$$F_{aa}$$ = $$g_{ae}. T_{ea}$$ + $$T_{ea}. g_{ea}$$ (36)
Let us make use of (H.1):
$$F_{aa}$$ = $$g_{ae}. T_{ea}$$ + $$T_{ea}. g_{ae}$$ (37)
It is obvious that (C.2) holds if K is an anti commutative body.

Conclusion at this moment:
Our proposition of a locally parallel transported gauge field (by respect for a local position 4-vector) in a metric compatible connection fixed by the relation (29) above is acceptable ( = satisfies two elementary conditions C.1 and C.2) if the theory extends on a vector space $$E_{4}$$ built on an anti commutative “body” K.

Next step:
We shall have to verify an important condition written:
$$e^{abcd}. d_{b}F_{cd}$$ = 0 (C.3)

 

[member 11137]

Maxwell's homogeneous law; first part

We reach now the next important step for this theory. We shall have to verify:
$$e^{abcd}. d_{b}F_{cd}$$ = 0 (C.3)

Let us recall that $$e^{abcd}$$ is a so-called « pseudo-tensor » of which the only non zero components are those for which all indices are different. Let us consider a given value for a (e.g. a = 0). It remains only 3 possible values for the b, c and d. Accordingly to the fact that there are 2 x 3 = 6 permutations for three indices without repetition, each sum described by (C.3) contains 6 terms. In extenso and for example, we have the following possibilities for the component 0: {0123} {0231} {0312} {0321} {0213} {0132}.

Let us write the generic term $$d_{b}F_{cd}$$ in taking care of (31):
$$d_{b}F_{cd}$$ = $$d_{b}g_{ce}. T_{ed}$$ + $$d_{b}T_{ec}. g_{ed}$$ + $$g_{ce}. d_{b}T_{ed}$$ + $$T_{ec}. d_{b}g_{ed}$$ (38)

And let us write the sum related to a = 0 in extenso. Terms for which the permutation is clockwise have a +1 signature; the others have a signature – 1. This gives:
$$d_{1}F_{23}$$ = $$d_{1}g_{2e}. T_{e3}$$ + $$d_{1}T_{e2}. g_{e3}$$ + $$g_{2e}. d_{1}T_{e3}$$ + $$T_{e2}. d_{1}g_{e3}$$ (38.1)

$$d_{3}F_{12}$$ = $$d_{3}g_{1e}. T_{e2}$$ + $$d_{3}T_{e1}. g_{e2}$$ + $$g_{1e}. d_{3}T_{e2}$$ + $$T_{e1}. d_{3}g_{e2}$$ (38.2)

$$d_{2}F_{31}$$ = $$d_{2}g_{3e}. T_{e1}$$ + $$d_{2}T_{e3}. g_{e1}$$ + $$g_{3e}. d_{2}T_{e1}$$ + $$T_{e3}. d_{2}g_{e1}$$ (38.3)

$$d_{3}F_{21}$$ = $$d_{3}g_{2e}. T_{e1}$$ + $$d_{3}T_{e2}. g_{e1}$$ + $$g_{2e}. d_{3}T_{e1}$$ + $$T_{e2}. d_{3}g_{e1}$$ (38.4)

$$d_{2}F_{13}$$ = $$d_{2}g_{c1}. T_{e3}$$ + $$d_{2}T_{e1}. g_{e3}$$ + $$g_{1e}. d_{2}T_{e3}$$ + $$T_{e1}. d_{2}g_{e3}$$ (38.5)

$$d_{1}F_{32}$$ = $$d_{1}g_{3e}. T_{e2}$$ + $$d_{1}T_{e3}. g_{e2}$$ + $$g_{3e}. d_{1}T_{e2}$$ + $$T_{e3}. d_{1}g_{e2}$$ (38.6)

Let us organize all these 24 terms better. We first consider all terms with a component of the metric and regroup them in taking care of the (H.2) hypothesis (K is anti commutative). Let us begin with terms of this kind in (38.1, 2 and 3):
$$d_{1}T_{e2}. g_{e3}$$ + $$g_{2e}. d_{1}T_{e3}$$ + $$d_{3}T_{e1}. g_{e2}$$ + $$g_{1e}. d_{3}T_{e2}$$ + $$d_{2}T_{e3}. g_{e1}$$ + $$g_{3e}. d_{2}T_{e1}$$ =

- $$g_{e3}. d_{1}T_{e2}$$ + $$g_{2e}. d_{1}T_{e3}$$ - $$g_{e2}. d_{3}T_{e1}$$ + $$g_{1e}. d_{3}T_{e2}$$ - $$g_{e1}. d_{2}T_{e3}$$ + $$g_{3e}. d_{2}T_{e1}$$ =
Let us make the (H.1) hypothesis (symmetric metric):
$$g_{1e}$$. ($$d_{3}T_{e2} - d_{2}T_{e3}$$) + $$g_{2e}$$. ($$d_{1}T_{e3} - d_{3}T_{e1}$$) + $$g_{3e}$$. ($$d_{2}T_{e1} - d_{1}T_{e2}$$) (g+)

Let us continue with terms of this kind in (38.4, 5 and 6):
$$d_{3}T_{e2}. g_{e1}$$ + $$g_{1e}. d_{2}T_{e3}$$ + $$d_{1}T_{e3}. g_{e2}$$ + $$g_{2e}. d_{3}T_{e1}$$ + $$g_{3e}. d_{1}T_{e2}$$ + $$d_{2}T_{e1}. g_{e3}$$ =

- $$g_{e1}. d_{3}T_{e2}$$ + $$g_{1e}. d_{2}T_{e3}$$ - $$g_{e2}. d_{1}T_{e3}$$ + $$g_{2e}. d_{3}T_{e1}$$ + $$g_{3e}. d_{1}T_{e2}$$ - $$g_{e3}. d_{2}T_{e1}$$ =

Let us make the (H.1) hypothesis (symmetric metric):
$$g_{1e}$$. ($$d_{2}T_{e3} - d_{3}T_{e2}$$) + $$g_{2e}$$. ($$d_{3}T_{e1} - d_{1}T_{e3}$$) + $$g_{3e}$$. ($$d_{1}T_{e2} - d_{2}T_{e1}$$) (g-)

Let us compare (g+) and (g-) and not forget that (g+) has the signature + 1 whilst (g-) has the signature -1. It is thus obvious that (H.1 and 2) are not sufficient hypothesis to prove that in fact: (g+) - (g-) = 0. We must add a new important hypothesis (H.3), namely that the trivial matrix must be a symplectic one:
T + T$$^{t}$$ = [0] (H.3)

Since T is a trivial matrix for the extended vector product under consideration and since the gauge field is parallel transported, this is yielding:
$$T_{eb}$$ = - $$d_{e}A^{b}$$ (39)
Such that if T is a symplectic matrix, then:
$$T_{eb}$$ = - $$T_{be}$$ = - $$d_{b}A^{e}$$ (40)
This is a crucial statement because each expression of the following formalism then vanishes:
($$d_{a}T_{eb} - d_{b}T_{ea}$$)
= ($$d_{a}d_{b}A^{e} - d_{b}d_{a}A^{e}$$)
= 0 (41)
This is thus now clearly eliminating each term of (g+) and of (g-) separately; at the end we have eliminated 12 terms of the component 0 with the three hypothesis (H.1, 2 and 3). Remark that we did loose a part of the generality and that because of (H.3) the tensor must now be written in a reduced form:
F = G. T – T. G (42)

At this moment we only did have made the half of the way that we have to do to get (C.3)

 

[member 11137]

Maxwell's homogeneous law; second part

We actually are studying 6 terms with a + 1 signature:
$$d_{1}g_{2e}. T_{e3}$$ + $$T_{e2}. d_{1}g_{e3}$$ (38.1)
$$d_{3}g_{1e}. T_{e2}$$ + $$T_{e1}. d_{3}g_{e2}$$ (38.2)
$$d_{2}g_{3e}. T_{e1}$$ + $$T_{e3}. d_{2}g_{e1}$$ (38.3)
and 6 other terms with a (– 1) signature:
$$d_{3}g_{2e}. T_{e1}$$ + $$T_{e2}. d_{3}g_{e1}$$ (38.4)
$$d_{2}g_{c1}. T_{e3}$$ + $$T_{e1}. d_{2}g_{e3}$$ (38.5)
$$d_{1}g_{3e}. T_{e2}$$ + $$T_{e3}. d_{1}g_{e2}$$ (38.6)
If they would not be the question with the signature, it would be evidently zero for a theory working with an anti commutative body K.

Since we have to take care of the signature, the result of this calculation is not obvious. It looks like if we could regroup the terms so that:
$$T_{e1}$$. ($$d_{3}g_{e2} - d_{2}g_{e3}$$) +
($$d_{2}g_{3e} – d_{3}g_{2e}$$). $$T_{e1}$$ =
Since K satisfies (C.2):
$$T_{e1}$$. ($$d_{3}g_{e2} - d_{2}g_{e3}$$) -
$$T_{e1}$$. ($$d_{2}g_{3e} – d_{3}g_{2e}$$) =
and since the metric satisfies (C.1):
2. $$T_{e1}$$. ($$d_{3}g_{e2} - d_{2}g_{e3}$$) =
Since the connection is metric compatible:
2. $$T_{e1}$$. (2. $$g_{3p}. A_{2e}^{p}$$ – $$d_{e}g_{32}$$) = (42)
In the generalized theory of the relativity, we have the following relation (Lichnerowicz; page 266; 79-8):
$$d_{e}g_{ab}$$ = $$g_{pb}. A_{ae}^{p}$$ + $$g_{ap}. A_{eb}^{p}$$
This yields here for a = 3 and b = 2:
$$d_{e}g_{32}$$ = $$g_{p2}. A_{3e}^{p}$$ + $$g_{3p}. A_{e2}^{p}$$
Making use of this relation in (42):
2. $$T_{e1}$$. [2. $$g_{3p}. A_{2e}^{p}$$ – ($$g_{p2}. A_{3e}^{p}$$ + $$g_{3p}. A_{e2}^{p}$$)] = (42)
2. $$T_{e1}$$. ($$g_{3p}. A_{2e}^{p}$$ – ($$g_{p2}. A_{3e}^{p}$$) = (42)
We can expect to get similar terms by permutation:
2. $$T_{e2}$$. ($$g_{1p}. A_{3e}^{p}$$ – ($$g_{p3}. A_{1e}^{p}$$)
2. $$T_{e3}$$. ($$g_{2p}. A_{1e}^{p}$$ – ($$g_{p1}. A_{2e}^{p}$$)
We can now regroup terms with the same metric component and obtain the generic factor:
$$T_{e2}$$. $$A_{3e}^{p}$$ - $$T_{e3}$$. $$A_{2e}^{p}$$
Since T is a trivial matrix, this yields:
$$A_{q2}^{e}. A^{q}$$. $$A_{3e}^{p}$$ - $$A_{q3}^{e}. A^{q}$$. $$A_{2e}^{p}$$
and finally, we get:
($$A_{q2}^{e}$$. $$A_{3e}^{p}$$ - $$A_{q3}^{e}$$. $$A_{2e}^{p}$$). $$A^{q}$$
If the connection defines an “associative” extended vector product (see other sections of this theory), then this term is zero in any gauge field $$^{4}A$$ and the second part of the demonstration valids the homogeneous Maxwell’s law.

Conclusion of this part:
Except an error (who does not make some?) concerning the delicate procedure of a term ordering (that would have been done for example here above at the step where the relation connecting the Christoffel’s symbols of second kind and the components of the metric has been introduced), our approach seems to allow a coherent new definition of the Faraday Maxwell tensor within a theory extending in a space vector E$$_{4}$$ built on an anti commutative body K, with a metric compatible connection A defining an associative extended vector product corresponding to the picture of a parallel transported gauge field respecting the fixation rule (29): For any subscripts a and b = 0, 1, 2 or 3: $$d_{c}g_{ab}$$. $$A^{c}$$ = 0. At the end we have the simple symmetric relation: F = G. T – T. G. It will be the starting point of a further development.

I hope you did enjoy this presentation. And I hope I could give some interesting proposition for professional searchers in physics; specially people working about non commutative geometry. Best regards.

 

[member 11137]

I shall now verify my calculus; it is a tedious thing as you could see. Independently of the result of this verification I think that the presentation above contains important ideas and informations about the nature of some EM fields. The formulation that has been obtained, i.e. F = G.T - T. G allows a very special interpretation of F if G, the metric tensor, can be interpreted as a spinor. Fundamentaly speaking, spinors always own a tensorial character. We actually know some litterature proving that this interpretation is possible in a Minkowskian space. The representation of F, above, allows to understand the EM field as a "parallel transport" inside a theory based on arguments due to Cartan. Bye. I shall come later again. Be patient.

 

[member 11137]

Why all these things (03)?

As still mentioned somewhere else on these forums about some other discussion (important or not), I should ask myself if the approach under study owns any significant importance before doing so much calculations and, perhaps, loose so many time. If you did loose your time because of my research, I apologize. I had the pleasure to learn more about a fascinating domain: physics. Since gluons fields are described via gauge fields satisfying the usual equations within a Yang Mills formulation (Quantum Chromo Dyn.) of the EM fields, it looks like if my theory could describe gluons interacting with a gravitational field; any one: the own gravitational field or an exterior one. The next question is: where do we encounter gluons interacting with a gravitational field in the nature? If gluons can gravitationaly interact with themself, the answer is: everywhere where gluons exist. This is giving a new consistence to this essay... It seems to be very actual and in someway the boarder of the science. This is why I am not sure to be able to go further. It's over my head (intellectual level) and it's a country-land for professionals only, I suppose. Have a nice day. Blackforest

 

[member 11137]

No revolutionary informations. I did verify my calculation above. Everything seems to be ok with the mathematics except that the trivial matrix does not need to be a symplectic one to valid the homogeneous Maxwell's law. (It makes this approach one step more general). Concerning the physics, I am now learning about interactions between elementary particles (difficult I must say) to try to discover if my scenario makes sense. You can now read all these things in clear text in english on my home page. Thanks for attention.
p.s. Don't forget to write me directly if you think that it was a relevant "essay" or if you have critics. I do appreciate some human communication and feel sometimes like "a lonesome cow-boy far from home... (lucky lucky, a wellknown French cartoon)". Bye

 

[member 11137]

Hey people, I think I have get it ! Shake the vacuum in a certain manner and you get the ligth ... More seriously: it can be proved that some geometric deformations (= some gravitational fields) generate EM fields and conversely. Relatively to the gravitational fields, EM fields are perhaps what the water is relatively to the molecules of water ... (just a pictorial illustration).

 

[eljose]

-But..isn,t supposed that the Kaluza-Klein model unified EM and Gravity?...by adding a fifth dimension?...and getting a similar Einstein Lagrangian?..i don,t know what,s the purpose of this post...:?:?

 

[member 11137]

-But..isn,t supposed that the Kaluza-Klein model unified EM and Gravity?...by adding a fifth dimension?...and getting a similar Einstein Lagrangian?..i don,t know what,s the purpose of this post...:?:?

If the Klein-Kaluza model would be the more useful one, this would be known. Since we don't have the certitude to live in a world with 5 dimensions, I think that any attempt to unify gravitation and EM in our usual 4-D world certainly represents a progress. If the problem is not interesting, then I ask why so many clicks on this thread or on the other one. I ask why so many theoretical efforts to unify these two separate fields of our knowledge, why - for example - people are working so hard to understand the quantified Hall effect or any other effect where not only the EM side of the reality is involved in (but also the topology), a.s.a...

In fact I don't understand that you don't understand : why this post.

 

[member 11137]

-But..isn,t supposed that the Kaluza-Klein model unified EM and Gravity?...by adding a fifth dimension?...and getting a similar Einstein Lagrangian?..i don,t know what,s the purpose of this post...:?:?

Sorry for having been a little bit direct with you in my answer to you.

But I insist. Look at post 35 of this ("discussion") quasi-monologue. And consider my argumentation attentively. All others attempts (e.g. Mortimer here on this subforum; Klein - Kaluza at the beginning of the 20th century) follow the historic example of the construction of the relativity. I.e.: the progress at the end of the 19th century was to be able to go from the 3D+1 world to a full 4-D space. Every researcher is now thinking that "increasing the number of dimensions of the theoretical discussion" is a good way to follow because it did work once. This way is equivalent to my example with the basket full of apples and of oranges. Of course, they are both fruits but, for example, it doesn't explain to me if they have any common points in their genetic code that would allow me to classify them in the same familly via this deeper argument.

The claim of the (E) approach is to find realistic calculations and physical situations for which a reasonable relationship between gravitation fields (represented by the connections; Christoffel's symbols) and EM fields are possible in a 4D background. This is why I think it is an original approach and, I hope it, a progress. Some lectures that I could recently do encourage me in this direction.

Best regards

 

[member 11137]

Thank you for the patience. This is absolutely not easy to put the chaotic development of my thoughts into a well organized work. If I try to take some distance with my own research, I slowly get the sensation that all my efforts are concentrated into one direction: to demonstrate the existence of some physical circumstances inside a usual 4-D world where gravitational and EM phenomenon are strongly connected together. Since EM phenomenon are quantized, a success story in this try would immediately imply a quantized version for the gravitation; which is the “holy Graal” of all modern research.
The reduction of any tensor within any group theory giving a representation of it (my example: the Faraday Maxwell tensor) does not represent a theoretical scoop; I agree. The particularity of my representation lies in two facts, I believe: i) it takes place in M(4 x 4, K) where K is any “corps” on which the space vector E is built; eventually an anti-commutative one; and ii) the reductions that I propose could allow a comparison with some other usual one involving spinors. For this proposition to be acceptable is requiring that spinors also own a representation in any ad hoc sub set of M(4 x 4, K). If this proposition – interpretation holds, then we propose to interpret some of the possible reductions as prototype representations of the Lorentz-Einstein Law.
This motivates my answers concerning the Lagrangian and the necessary limited informations that I can actually propose concerning it. I think that it is still too soon to give a definitive answer and a correct interpretation of the Lagrangian naturally implied by the reductions of the tensor. It should be more convenient to first get a serious link with a measure theory before doing any prediction concerning the energy contained into the EM field.

 

[member 11137]

If you did follow the last developments of my theory, you certainly understand that I did find an interesting course on non commutative geometry and that it gives me the link to a measure theory. My actual efforst are made to prove that some ad hoc trilinear form (see etf71.pdf) is a cyclic cocycle of dimension 2; things are turning out so that they suggest that the relativistic invariant $$0 = (ds)^2 = g_{\alpha \beta} dx^\alpha d x^\beta$$ could also be seen as an invariant logicaly arising from the co-homology theory... if some criterium are fullfilled (This is the matter of the present research but I am stopped by some stupid difficulties). If this proposition is true it will have enormous consequences for the physics.
The reduction of the Faraday Maxwell tensor that I propose in my (E) theory can be also interpreted inside this co-homologie theory*. In extenso F = [G, P] could be understood as F = dP and for me this is the door for a natural quantization inside *.
Best regards.

 

[member 11137]

The Lagrangian

At the beginning of this thread, I said that I wanted to propose an other formulation for the Lagrangian of the EM field.

This section of the (E) Theory develops the consequences of the proposed reductions of the Faraday Maxwell tensor F for the Lagrangian of the EM field. As a matter of facts we could demonstrate the perfect coherence with the Maxwell’s laws of the F = G. P + P*. G (1) reduction if the cube A defines an associative extended product on (E4, K) when
i) K is anti-commutative
ii) G is the local representation of the metric tensor (symmetric) and
iii) P is a trivial matrix (and P* is its transposed here; sorry for the notation but it is a little bit long with tex) relatively to this product.

See pleasae etgb100.pdf on my homepage or the discussion on this subforum. Note that reductions in (E4, K) with K abelian also exist; of course.

Since, in language matrix F = G. F'. G, (where F' is exceptionally here the dual of F) we get F' = G$$^-1$$. F. G$$^-1$$.
For the simplicity let us exceptionally work on basis where G² = I.
This yields a simplification in the calculations:
F' = G. F. G
and consequently:
F. F'
= (G. P + P*. G). [G. (G. P + P* G). G]
= (G. P + P*. G). [G. (G. P. G + P*)]
= (G. P + P*. G). [P. G + G. P*]
= G. P². G + (G. P). (G. P*) + (P*. G). (P. G) + (P*. G). (G. P*)
= G. P². G + (G. P). (G. P*) + (P*. G). (P. G) + (P*)²
This Lagrangian contains 4 terms and one of them has no direct relation with the metric tensor. The same kind of conclusion would hold for:
F'. F
= [G. (G. P + P* G). G]. (G. P + P*. G)
= (P. G + G. P*). (G. P + P*. G)
= (P. G). (G. P) + (P. G). (P*. G) + (G. P*). (G. P) + (G. P*). (P*. G)
= P² + (P. G). (P*. G) + (G. P*). (G. P) + G. (P*)². G
In one case or in the other we do have a priori four different components and the “pure” term not directly depending on the metric is related to this extended product that we decided to introduce.

If we refer to some classical lecture concerning the Lagrangian of the EM field, we should incorporate a complementary expression and so we have theoretically in fact 4 supplementary terms but we can suspect that they own the same structure than the four that we still got. Perhaps something like:
G. F. F' = P². G + P. (G. P*) + G. (P*. G). (P. G) + G. (P*)²
This means that if we propose:
T = G. F. F' + F. F'
then we get:
T = P². G + P. (G. P*) + G. (P*. G). (P. G) + G. (P*)² + P² + (P. G). (P*. G) + (G. P*). (G. P) + G. (P*)². G
If the multiplication would also be anti-commutative for the matrices, then we would have:
(P. G). (P*. G) + (G. P*). (G. P)
= (P. G). (P*. G) + (-P*. G). (-P. G)
= (P. G). (P*. G) + (P*. G). (P. G)
= [0]
and the reduced form:
T » P². [I + G] + G. (P*)² + P. (G. P*) + G. (P*)². G + G. (P*. G). (P. G)
In the Minkowskian limit of the metric (Here we note it h; Note that this metric satisfies h² = I) :
T » P². [I + h] + h. (P*)² + F. (h. P*) + h. (P*)². h + h. (P*. h). (P.h)
Where we must remark that the first term yields T(0,0) = [P²](0,0) and T(k,beta) = 0 because [I + h] is a matrix with only one component which is non zero : the one in position (0,0).

That is: the first term introduces no impulsion and no deformation but only the mass (or the energetic density) and, in my approach, it seems to directly depend on the square of the trivial matrix P.

Continuing...

 

[member 11137]

60 th and last intervention

Hye, dear members of this forum; here we are: it is the 60th intervention and accordingly to old rules here the last one. It was the possibility for me to catch your attention and try to convince you of the interest of my approach.

So let us dream a while and think that instead of one unique representation F related to a given and momentary value of an EM field under consideration, we could consider a set of such matrices; i.e. the F$$_i$$ matrices for i = 1, 2, ..., N and now F = sum of F$$_i$$.
The Lagrangian obtained before (and with it a kind of correspondance between the energetic density and the trivial matrices) is suggesting that, if each event in the world is a mixture of EM fields, we could propose a kind of probabilistic interpretation for the components of the P² matrices (and for the P² matrices themselves) involved in a representation F of a given mixture.
I am not sure that I am really clear with my bad English language but I hope that you understand my idea.
So, at the end, we get a way to connect a relativistic approach and a statistic one, that is a possible link between relativity and a probabilistic approach.
If, to this vision one adds the fact that a symplectic collection of trivial matrices would give us the possibility to write any reduction of the F$$_i$$ under the bracket representation F$$_i$$ = [G, P$$_i$$] that we could interpret within the context of a "co-homological" theory introducing the quantized calculus with this bracket notation (which is supposing that G² = I and that G is a self-adjoint operator in an Hilbert space), I think we get an interesting link between the relativistic approach (involving the metric tensor and its variations) and the quantum approach...

The strange thing of my approach is that it is suggesting that, accordingly to the equivalence principle, EM fields around an atom could be so strong that they are locally curving space-time as if they would define geodesics where we would have such or such probability to find the electrons ...

So, it was just a dream... a vision to connect two sides of our theories. There is certainly still a lot of work to do to precise and confirm this vision, I agree. No idea if I did really success and bring some progress. Hope you enjoyed my proposition and wish you all a very good and long life. If it was good enough and if you need my help, please tell me.
Blackforest

 

[member 11137]

Contribution to another great discussion

Dear members, "never say never more"
commentaries in my mouth are perhaps not so relevant as commentaries coming from a professional staff. But, if you give me the permission, I shall try to give point of view as “amateur”.

Steven Carlip in his book: “Quantum Gravity in 2 +1 dimensions” mentions a lot of arguments (page 2) sketching the essential differences between the Quantum mechanical approach and the relativistic one. They agree with the analyze made by Lee Smolin at the beginning of the article arXiv: quant-ph/0609109v1 14 September 2006; example: the question of non-locality. (see other sub-forum: beyond the standard model)

For me, the essential question-answer asked by L. Smolin is (page 2): “Is it possible to solve the measurement problem with a realistic ontology that is not doubled, as Boehm’s is? The idea that quantum mechanic is an approximation to a non-local, cosmological theory offers new possibility for doing this because the missing information, which makes quantum theory statistical, would be found, not in a more detailed description of the sub-system, as it is in the Bohmian mechanics, but in hidden variables which describe relationship between the subsystem and the rest of the universe.”

I would like to say thank you to Mr. Smolin because he is just saying with words what I was saying with my calculations in my investigations devoted to the following question: “Is there any possibility to analyze the Lorentz-Einstein Law as a PDE of second order?”

Even if it does not appear clearly at the beginning of the investigation, the later owns a deep relationship with the way of thinking proposed by L. Smolin. Why? The answer to my modest question is: yes, it is quasi-unique and it is compatible with a space time that would be for a short while without curvature. What do we learn with this? In fact that we can begin a Sturm Liouville analysis of the Lorentz-Einstein Law that is a Law describing locally a long distance effect of some masses and (or and) electrical charges repartitions in the universe. And this analysis can be done in a temporary flat space time which is exactly the frame where the quantum mechanics seems to take place.

For me, the sub-system where things are quantized is the slice of time where we live in; at each instant of our life. And naturally, we get informations inside of our sub-system from the rest of the universe around us via the natural time evolution which is only a temporary and local translation of the laws describing how the nature of the things are changing every where at any time. In other words, the quantum mechanic is a tool developed in accordance with our extremely strong locality. It reports on the apparent flatness of every slice of universe as soon as the slice is tiny enough. And our life is a very tiny thing.

I hope I could help to increase the understanding of the position developed by a) Smolin and b) by my self.

 

[member 11137]

Dear members, this is really my last intervention since nobody wants to speak directly with me about the subject that I want to develop. But as "thank you" for the time you gave me here on this "podium" and to demonstrate to Mortimer or to Sweet that I did care of their critics, I propose this small work in the attachment. Every new developmemt is on my home page: http:www.vacuum-world-net.eu/4579/[/URL]

Best regards

 

[member 11137]

Dear members, this is really my last intervention since nobody wants to speak directly with me about the subject that I want to develop. But as "thank you" for the time you gave me here on this "podium" and to demonstrate to Mortimer or to Sweet that I did care of their critics, I propose this small work in the attachment. Every new developmemt is on my home page: http:www.vacuum-world-net.eu/4579/[/URL]

Best regards[/QUOTE]

I wish you all an happy Christmas. Some commentaries on the etgb76.pdf document are available on my homepage and here if you want. My research is progressing slowly but I did not give up.

 

[member 11137]

News

Recent developments of my approach.

 

[member 11137]

Time, thesis, derivation, angular momentum

Life is hard; are we free to discuss about these things? I don't know. Nevermind I continue my intellectual progression and I hope you can enjoy it.

Here is a small work to tell you about the recent progresses and I think that it is a good example to illustrate my approach : the "extended angular momentum" (introduction)

 

[member 11137]

Commentaries

This demonstrates how the introduction of the notion of extended product modifies our way to calculate the variations (derivations) of a vector. This also illustrates the fact that we have to separate the role of the geometric connection (the Christoffel’s cube) from those of the cube defining the extended product locally; even if both cubes can be sometimes in coincidence.

Since the Christoffel’s cube vanishes in any inertial frame: we state that the “derive” of L(M, t), [the first term of the LHT in (6)], disappears in any inertial frame. The derive of the angular momentum, as we call it, is the variation of the angular momentum which is induced by a variation of the cube defining the initial very classical cross product. As expected, there is no derive of this definition in any inertial frame. Otherwise, we would be informed of that!

This is exactly the point of view that we want to develop with our principle of elasticity:
“… There must be (or there should be) a relationship between a) the "manner how" we intuitively and historically decided to define the different mathematical operations we are customized to calculate with (e.g. scalar and cross product of two vectors, and so forth) and b) our Euclidian geometry. With other words: the geometry acts on our brain in a way of which we do not necessarily have the consciousness but determinates our strategy to calculate. If our world would have been curved (like it certainly is for the sub-atomic particles), we would certainly make use of the same operations (scalar and cross products,... ), but in another way, with modified definitions... "

The relation (6) is the necessary condition to annihilate the effects of an eventual variation of the definition of the cross product induced by a variation of the geometry. As long as this relation holds, we cannot be informed of these variations, … if they exist really. We can also imagine a validity of this relation (6) … in average only. The result would be exactly the same four our instruments: an incapacity to detect the “underground” variations.

The second LHT of (6) is examined in the updated version of etgb94.pdf (see my homepage). At the end my theory predicts a natural “derive” of the definition of the cross product. Since the angular momentum is quantized, this derive must be quantized too.

 

[member 11137]

Well I can now define involutive algebras with the extended product and I begin the construction of Fredholm modules... We will see if it gives us something interesting or not.

 

[member 11137]

Coming back to the reductions of the EM tensor

Here is a discussion concerning the reductions of the EM tensor proposed in the post 40-59.

 

[member 11137]

Extended products and loops

Natural introduction of the notion of extended product

 

[member 11137]

important commentaries

As claimed a lot of months before, an ocean of complicated calculations owning no serious motivation do not bring any light into a discussion. Thus, I have to motivate my construction; at least to try to do it.

In the following text, formula are not correctly written (I am not a specialist of text); see attachment for this.

The formula we are referring, i.e. ∫daa = ∫Gabg. ag. dxb (1) [01; page 88] is directly parented with the notion of parallel transport of the vector a = aq. eq along a path xb = xb(s) in a curved space (Here a 4D space). To convince us self of this, let us compare (1) with the formula in [02; Appendix C; page 243]: dar + Gbnr. dxb. an = 0 (2). There is no real difficulty to state the similitude (perhaps up to a minus sign) between (1) and (2); one only needs to consider the covariant formulation of a and makes the hypothesis of the absence of torsion for the connection defined by the so-called Christoffel’s cube. Thus, either with (1) or with (2), we should arrive to the same conclusion than in etgb96.pdf if the hypothesis concerning the infinitesimal elements of surface holds true (These elements can be referred to two main directions). By side, what was not said in reference [01], but is very clearly exposed in [02], is that formula (1) or (2) are the starting point for the calculation of the parallel transport matrix and of the holonomy of the curve along which a is parallel transported. If the Stockes’ theorem is valid [02; Appendix C; (C.10) page 245]: Urn = drn - òå Rrnbd . dfbd (3) where å is the surface delimited by the path. So, we get for small enough elements of surface with main directions (db, dc): ∫daa = -2. a. ⌂(▼R), a (db, dc). This is situating the discussion on the notion of extended products directly into the domain of application of the loops.

In an approach based on the loops, like LQG, the vector a has to be an element of a Lie algebra. And it can be proved that the space vector (E4, R)can be equipped with a structure of Lie algebra via the extended products.

Now, referring to this citation (THIS FORUM; INTRODUCTION TO LQG): “… Well, let’s steal some ideas from particle physics... In QFT we have fermionic matter-fields and bosonic force-fields. The quanta of these force-fields or the so called force-carrier-particles that mediate forces between matter-particles. Sometimes force-carriers can also interact with each other, like strong-force-mediating gluons for example. These force carriers also have wavelike properties and in this view they are looked as excitations of the bosonic-force fields. For example some line in a field can start to vibrate (think of a guitar-string) and in QFT one then says that this vibration is a particle. This may sound strange but what is really meant is that the vibration has the properties of some particle with energy, speed, and so on, corresponding to that of the vibration. These lines are also known as Faraday’s lines of force. Photons are "generated" this way in QFT, where they are excitations of the EM-field. Normally these lines go from one matter-particle to another and in the absence of particles or charges they form closed lines, loops. Loop Quantum Gravity is the mathematical description of quantum gravity in terms of loops on a manifold. We have already shown how we can work with loops on a manifold and still be assured of background-independence and gauge-invariance for QFT. So we want to quantize the gravitational field by expressing it in terms of loops. These loops are quantum excitations of the Faraday-lines that live in the field and who represent the gravitational force. Gravitons or closed loops that arise as low-energy-excitations of the gravitational field and these particles mediate the gravitational force between objects.
It is important to realize that these loops do not live on some space-time-continuum, they are space-time ! The loops arise as excitations of the gravitational field, which on itself constitutes “space”. Now the problem is how to incorporate the concept of space or to put it more accurately : “how do we define all these different geometries in order to be able to work with a wave function … ?”,

I introduce the notion of split. It is for me the way to connect a set of second order differential operator to the set of the different expressions of the Lorentz Einstein Law (LEL). Indeed, supposing that $ ([P]a, ka) Î M4(Â) x (E4, Â, ⌂(▼R), a) | ⌂(▼R), a. Extpr(db, dc) = [P]. dc + k yields ∫daa = -2. a. {[P]a. dc + ka}. For the special cases where a = dc, this is nothing but ∫da = -2. a. {[P]. a + k} and obviously a quadratic equations depending on a. With a few complementary hypothesis (see etgb96.pdf), this is exactly this kind of relation that can connect any second order differential operator with a precise expression of the LEL provided I interpret a as being the position vector in the Riemannian space and ∫daa as being the ath coordinate of the “projection” in a flat space of aa , and indirectly, because of that, of a. Let us note it: xa(a). The advantage of this procedure is that the second order differential operator is acting on x and that a Sturm-Liouville treatment can give him the Schrödinger equation formalism. Inside this procedure, we didn’t make any precise hypothesis concerning the nature of x. This is suggesting the possibility to consider it as a spinor.

This procedure must be analyzed: is it coherent? Does it make sense? If it holds true, than I can say that I have developed a new tool to connect QM and GTR. This was the foundations of this (E) attempt.

Bibliography:
[01] ART ;Torsten Fließbach ; 4. Auflage; Spektrum der Wissenschaft; 2003
[02] Quantum Gravity in 2 + 1 Dimensions; Steven Carlip; Cambridge Monographs on Mathematical Physics; 2003;