Now that the physical content of time and its distinction from space have been withdrawn, the approach of quantum dynamics is pure more geometrical than it can already be in the classical. The starting point is a Riemannian metric:

 

(1)               ds² = g_ab(x)dx^adx^b = dx_adx^a

 

in curved quantum D-space X. A MOTION through X is a quantum curve x^a(s). VELOCITY is introduced as:

 

(2)               v^a(s) = cdx^a(s)/ds = cu^a(s)

 

with c, the CLASSICAL velocity of light. Indeed, we don’t need c to turn complex-valued: both s and x^a are complex-valued and c is, after all, a mere COEFFICIENT. As usual in Riemannian geometry, u(s) is a unit vector in T_sX, the space tangent to X at s:

 

(3)               u^a(s)u_a(s) = 1          for all s

 

The projections of u^a(s) are:

 

(4)               [u^a(s)]_A = s_A+2^BC(d/ds^B)x_C^a(s₁,s₂) = s_A+2^BCu_BC^a(s₁,s₂)

 

they give two VELOCITY SURFACES. The rest is similar to the geometrical approach of classical dynamics. Such an approach has the merits of treating all D position variables x^a on an equal footing, thus giving a UNIFIED vision of the dynamics involved, whereas the dynamical parameter s, being invariant, is the only ABSOLUTE quantity (the same at all points and in all reference frames). The only thing that matters is the RIEMANN AXIOM: for X to be Riemannian, there MUST exist a coordinate system at all its regular points where the metrical tensor field g_ab(x) can be made diagonal (and locally constant). In such a coordinate system:

 

(5)               ds² = S_a=1^D g_aa(x’)(dx’^a)²  ,  Det[g_aa(x’)] <> 0

 

at each regular point of X. Singularities on X are revealed by Det[g_ab(x)] = 0. At such points, the transformation of coordinates:

 

(6)               x^a = f^a(x’)               (a = 1,…,D)

 

is no longer invertible and one has to deal with sub-spaces of X.

 

Given a quantum mass m = m₁ + im₂, we have an invariant elementary action:

 

(7)               dS = mcds = Lds/c

 

generating a Lagrange functional,

 

(8)               L[x(s),v(s),s] = ½ mv²(s) + W[x(s),s]  ,  v²(s) = v^a(s)v_a(s)

 

The work function making the potential contribution to that functional can always be elementarily defined as a 1-form:

 

(9)               dW[x(s),s] = F_a[x(s),s]dx^a(s) = F_a[x(s),s]u^a(s)ds

 

over the MOBILE space X(s). F[x(s),s] is a vector-valued FORCE FIELD applied onto the body. Momenta are:

 

(10)           p_a(s) = dL/dv^a(s) = mv_a(s) = mcu_a(s)

 

Following (3), they verify the invariant identity:

 

(11)           p_a(s)p^a(s) = m²c² = E₀²/c²

 

ENERGIES have now to be understood first as:

 

-         the ENERGY AT REST,

 

(12)           E₀ = mc²  ,  E_0A = m_Ac²      (A = 1,2)

 

-         the D DIRECTIONAL ENERGIES,

 

(13)           E_a(s) = p_a(s)c = E₀u_a(s)

 

-         and the HAMILTON FUNCTIONAL,

 

(14)           H[x(s),p(s),s] = p_a(s)v^a(s) - L[x(s),v(s),s] = ½ mv²(s) - W[x(s),s]

     = p²(s)/2m - W[x(s),s] = E²(s)/2E₀ - W[x(s),s]

 

now realized as an INVARIANT SQUARE OF DIRECTIONAL ENERGIES (reported to the body’s energy at rest). It shows that:

 

The concept of energy is maintained in the quantum and DOES NOT REQUIRE TIME.

 

It doesn’t either in the classical, but we’re so used to reason with a time parameter that the Jacobi formalism established a duality between time and energy on one side and space and momenta on the other side. NO SUCH SCHISM SURVIVES IN THE QUANTUM. The Jacobi formalism takes the following unified form. The differential of the “Jacobi function” (or “Hamilton’s principal function”) is:

 

(15)           dS[x(s),s] = p_a(s)dx^a(s) + H[x(s),p(s),s]ds/c

 

It follows that:

 

(16)           p_a(s) = dS/dx^a(s)

 

and,

 

(17)           E_a(s) = cdS/dx^a(s)

(18)           H[x(s),p(s),s] = (2m)^-1[dS/dx^a(s)][dS/dx_a(s)] - W[x(s),s] = cdS/ds

 

according to (13) and (14). This gives back the Jacobi equation in (D + 1)-dimensional space for the functional S[x(s),s]. Newton’s equations of motion read:

 

(19)           cdp_a(s)/ds = dE_a(s)/ds = dW/dx^a(s) = F_a[x(s),s]

 

If one encounters a boundary problem, the integral-differential formulation may also be interesting:

 

(20)           E_a(s) = S F_a[x(s),s]ds = mc²dx_a(s)/ds

 

where the boundaries of the integral need be specified.

 

Time (…) to resume a bit. The PHYSICAL frame is a 3- maybe 4-dimensional quantum space and the CONFIGURATION space is a D-dimensional quantum space X. The MOBILE space related to X is X(s). It’s an infinite-dimensional space of quantum curves x(s). The PHASE space of X(s) is T_s*X(s), the fiber bundle made of all spaces cotangent to X(s) at s, with local coordinates [x^a(s),p_a(s)]. For EACH s, it’s a 2D-dimensional space over the “fold” X(s), endowed with a natural SYMPLECTIC structure (we’re going to explicit below). Finally, the CARTAN or STATE SPACE is the Cartesian product T_s*X(s) x R (or a finite interval of R) with local coordinates [x^a(s),p_a(s),s]. At each s, it’s a (2D+1)-dimensional space over the fold X(s) made of T_s*X(s) and the point {s}.

 

The CURVILINEAR DERIVATIVE on T_s*X(s) is built as:

 

(21)           D_s = d/ds + u^a(s)d/dx^a(s) + [dp_a(s)/ds]d/dp_a(s)

 

Applied to the Hamilton functional (14), it gives:

 

D_sH = dH/ds + u^a(s)dH/dx^a(s) + [dp_a(s)/ds]dH/dp_a(s)

       = dH/ds - u^a(s)dL/dx^a(s) + cu^a(s)dp_a(s)/ds

 

that is,

 

(22)           D_sH = dH/ds

 

while the Lagrange equations,

 

(23)           c(d/ds)dL/dv^a(s) = (d/ds)dL/du^a(s) = dL/dx^a(s)

 

lead to the Hamilton system

 

(24)           dH/dx^a(s) = -cdp_a(s)/ds  ,  dH/dp_a(s) = cdx^a(s)/ds

 

Inserted into the definition of the curvilinear derivative (21), these last equations give the Poisson bracket:

 

(25)           D_s = d/ds + (1/c){[dH/dp_a(s)]d/dx^a(s) - [dH/dx^a(s)]d/dp_a(s)}

     = d/ds + (1/c){H,.}

 

The Poisson bracket {H,.} gives the symplectic structure of the phase space, since it’s skew-symmetric. We retrieve the fact that {H,H} = 0, implying (22). Hamilton’s energy H will therefore remain a constant all along the trajectory iff H does not explicitly depend on s:

 

(26)           D_sH = 0  <=>  dH/ds = 0  <=>  H = Cte              for all s

 

This is how CONSERVATIVE systems should be understood in the quantum.

 

Setting the D directional energies E_a(s) to constant would be far too restrictive, as p(s) would need to be a constant vector, which is only verified by uniform motions x^a(s) = u^a(s - s₀) + x^a(s₀), u^a = ctes. Instead, relations (26) are MUCH WEAKER (since H is a scalar), therefore allowing a lot more possible motions.

 

According to what was said in B 151 and IN BLATANT CONTRADICTION WITH THE CLASSICAL, the one and only requirement is:

 

(27)           g_aa(x’) <> 0  ,  g_ab(x’) = 0              (a <> b)

 

in a coordinate system at each point of X. Then, the choice of the g_aas, even in the plane situation where they’re globally defined, is TOTALLY IRRELEVANT: as all the g_aas are complex-valued, ALL THEIR PROJECTIONS HAVE EQUAL RELEVANCE. In a coordinate transform as (6), the invariance of the ds² implies that:

 

(28)           g_ab(x’) = g_cd(x)(df^c/dx’^a)(df^d/dx’^b)

 

and, conversely,

 

(29)           g_cd(x) = g_ab(x’)(dx’^a/df^c)(dx’^b/df^d) = g_ab(x’)(df’^a/dx^c)(df’^b/dx^d)

 

by the inverse transformation

 

(30)           x’^a = f’^a(x)

 

at any regular point of X. So, from the time that we can INDIFFERENTLY go from a REPRESENTATION to another, we can find a coordinate system where g_aa = +1, another one where g_aa = -1, still another where g_aa = +1 for a = 1,…,D₁ and g_aa = -1 for a = D₁ + 1,…,D; we can find coordinate systems where some g_aas only or all of them are complex-valued,… we can find ANYTHING and our metrical coefficients will remain globally constant in the plane situation under ALL LINEAR TRANSFORMS. So, no particular choice is better than another, as was in the classical, because we were dealing there with REAL-VALUED quantities: the reality of ds² imposed that ds² < 0 was excluded from the physical frame. Here, our ds² is complex-valued.

 

IN THE QUANTUM, NO PHYSICAL SECTOR IS OUT OF OBSERVATION REACH.

 

I’d like to come back more specifically on the question of the geometry of D-dimensional quantum space, because the calculi given at the end of B 150 aren’t explicit enough.

 

Let’s first notice that, because of the three fundamental relations:

 

(1)               cos(ipsi₂) = ch(psi₂)  ,  sin(ipsi₂) = ish(psi₂)  ,  psi₂ in [0,2pi[

(2)               ch²(.) - sh²(.) = 1

(3)               cos²(.) + sin²(.) = 1

 

relation (3) extends to COMPLEX-VALUED angles psi = psi₁ + ipsi₂. This is important because one might thought at first glance that going from classical D-space to quantum one would change the rotation group from SO(D₁,D₂) into SU(D₁,D₂), D₁ + D₂ = D. This is true for the HERMITIAN spaces of supersymmetry, but not for the RIEMANNIAN spaces we are to use here. It’s indeed well-known that SU(D₁,D₂) preserves the REAL-VALUED quadratic form S_a=1^D1 |x^a|² - S_a=D1+1^D2 |x^a|², not the STILL COMPLEX-VALUED form S_a=1^D1 (x^a)² - S_a=D1+1^D2 (x^a)². Furthermore, we’re going to show that, at all regular point of X, we can set D₂ = 0 and remain in a FULLY RIEMANNIAN SPACE.

 

We begin with reminding the simplest case of a plane rotation in 2D real space:

 

(4)               x’¹ = x¹cos(psi₁) + x²sin(psi₁)  ,  x’² = -x¹sin(psi₁) + x²cos(psi₁)

 

with psi₁ real. This transformation has determinant +1 [relation (3)], thus describing a DIRECT rotation. It preserves the quadratic form (x¹)² + (x²)², since:

 

(5)               (x’¹)² + (x’²)² = (x¹)² + (x²)²

 

This is SO_R(2), the rotation group of 2D real space. It has a single rotation angle psi₁ so that the dimension OF THE GROUP is dim_R[SO_R(2)] = 1. One can easily show that (4) is formally equivalent to:

 

(6)               x’ = exp(-ipsi₁)x  ,  x = x₁ + ix₂  ,  x’ = x’₁ + ix’₂

 

so that there’s a group isomorphism between SO_R(2) and U(1), the rotation group of 1D COMPLEX space. However, (6) preserves the HERMITIAN form |x|² = (x¹)² + (x²)², since the only angle-free relation is |x’|² = |x|². So, if we complexify (4) replacing all x-variables as well as the angle with complex-valued quantities, the extended transformation:

 

(7)               x’¹ = x¹cos(psi) + x²sin(psi)  ,  x’² = -x¹sin(psi) + x²cos(psi)

(8)               x’^a = x₁^a + ix₂^a        (a = 1,2)  ,  psi = psi₁ + ipsi₂

 

remains of determinant +1 thanks to (3) still holding, but it now preserves the complex-valued quadratic form (x¹)² + (x²)², which is exactly what we need so as to keep the Bose-Einstein nature of the quantum vacuum. It follows that the rotation group leaving this form invariant is not U(1), but SO_C(2), the COMPLEXIFIED rotation group of 2D COMPLEX space. As we can see, this group has TWO real-valued angles psi₁ and psi₂ or a SINGLE COMPLEX-valued angle psi. Its COMPLEX dimension is therefore unchanged, it’s one, but its REAL dimension is 2. More generally, for D-dimensional quantum space X, we will find SO_C(D) has the rotation group leaving the quadratic form S_a=1^D (x^a)² invariant, with:

 

(9)               dim_C[SO_C(D)] = D(D - 1)/2  =>  SO_C(D) = SO_R(D) x SO_R(D)

 

(the equality being understood as a group isomorphism). Translations, scalings and inversions follow the same principle so that the larger invariance groups are respectively: the Poincaré group P_C(D) of displacements and the conformal group C_C(D), with complex dimensions:

 

(10)           dim_C[P_C(D)] = D(D + 1)/2  =>  P_C(D) = SO_C(D + 1)

(11)           dim_C[C_C(D)] = (D+1)(D+2)/2  =>  C_C(D) = P_C(D+1) = SO_C(D + 2)

 

Let’s now show why X can (locally) remain EUCLIDIAN. Let’s reconsider the local COORDINATES transformation:

 

(12)           x^a = f^a(x’)

 

from an “old” coordinate system x^a to a “new” one x’^a in the vicinity of any point of X. Let’s remind that such a transformation is a mere CHANGE OF REPRESENTATION, it induces NO PHYSICAL MOTION. One goes, for instance, from a planar representation to a polar or axisymmetric one. We know that the induced transformation on the METRICAL TENSOR of X is:

 

(13)           g’_cd = g_ab(df^a/dx’^c)(df^b/dx’^d)

 

Well, it suffices to set D = 1 and consider a linear transform x = ux’ with u constant to see that the metrical tensor in the NEW reference frame:

 

(14)           g’ = gu²

 

will have state projections,

 

(15)           g’₁ = u₁² - u₂²  ,  g’₂ = 2u₁u₂

 

with NO DEFINITE SIGNS, EVEN IF WE SET g = 1 in the old reference frame. It means two things:

 

1)      that starting from a FULLY EUCLIDIAN and REAL-VALUED metrical tensor in a given coordinate system, it’s always possible to obtain a COMPLEX-VALUED metrical tensor in another coordinate system, and

2)      at any REGULAR point of X, where transformation (12) is invertible, it’s always possible to find a coordinate system into which A COMPLEX-VALUED metrical tensor REDUCES TO A FULLY EUCLIDIAN REAL-VALUED ONE.

 

In physical terms, this reads:

 

A coordinate transform in the vicinity of any regular point of X is a PURELY INERTIAL EFFECT, a mere CHANGE OF REPRESENTATION WITH NO PHYSICAL CONTENT. And the ability to get back to a fully Euclidian metric means that TIME-LIKE DIMENSIONS ARE PROJECTIVE “ILLUSIONS” THAT CAN ALWAYS BE ELIMINATED.

 

Indeed, if we set u₂ = 0 in (15), we obtain g’₁ = u₁² > 0 (“space-like”) and g’₂ = 0 (on state 2, the geometry SEEMS to collapse into a mere point) and, if we set u₁ = 0, we obtain g’₁ = -u₂² < 0 (“time-like”) and g’₂ = 0. But, as we can go back to the previous coordinate system, in that system, we have g = 1, that is, g₁ = 1 and g₂ = 0 and there’s no time-like direction anymore. This is of course due to the fact that complex multiplication induces PROJECTIVE HYPERBOLIC SQUARES, whereas real multiplication only induces ELLIPTIC squares (squares with a DEFINITE sign). In other words:

 

TIME IS A CLASSICAL NOTION THAT DOES NOT “SURVIVE” IN THE QUANTUM.

 

And this has very important impacts on mechanics as well as thermodynamics, because time is dual to the concept of energy. So, eliminating time for another space dimension means “eliminating” energy to the benefit of another momentum, but also reviewing the concept of TEMPERATURE, since there’s a direct link between thermal energy and temperature. In the quantum, they all appear as “inertial effects”. And, indeed, there was no solid reason why, in special relativity, “space-time” would only be a FICTITIOUS MATH FRAME more suitable than the 3D one to describe phenomena due to the finite value of c and would “suddenly” turn a PHYSICAL frame in general relativity, especially if gravity was to be considered an INERTIAL EFFECT… K Something didn’t match, which was exactly the bottom of the painstaking question about the SIGNATURE of “space-time”, namely: “why space-like dimensions and time-like ones and why, more specifically, 3 space-like ones for only a single time-like one?”.

 

Well, we now have an answer: there’s ZERO “time-like dimension” in the quantum and time appear as a PROJECTIVE ILLUSION when reported to classical or semi-classical things. The only thing that should remain relevant is THE INVARIANCE OF THE METRIC OF X: starting from a (local) coordinate system where

 

ds² = S_a=1^D (dx^a)²  =>  g_ab = real-valued Kronecker delta

 

and such a system can always be found around any regular point (the Riemann axiom itself!), we can always end up with a metric

 

ds² = g’_ab(x’)dx’^adx’^b

 

through a (NON)-LINEAR transform x^a = f^a(x’) in a coordinate system x’^a and back, with

 

ds² = g’_ab(x’)dx’^adx’^b = S_a=1^D (dx^a)²

 

Signed metrical coefficients are PROJECTIVE EFFECTS on states 1 and 2. The quantum reality is unsigned.

 

Following this series of results, it’s always possible to keep on taking a MINKOWSKI metric if, for some purposes, we don’t want “shifts” to emerge (see next bidouille) in our familiar notions. It suffices to take, in (13):

 

(16)           f^a(x) = ix^a  (a = 1,2,3) , f⁴(x) = x⁴

 

to obtain a “time-like” metrical tensor, or

 

(17)           f^a(x) = x^a  (a = 1,2,3) , f⁴(x) = ix⁴

 

to obtain a “space-like” one. It WON’T restaure a physical nature to time for as much, since both transformations are invertible and we can always be back to the initial Euclidian metric. Thus, preferring the Minkowski metric to the Euclidian one would only be a matter of “keeping the good old habits”.

 

But, actually, all these “classically preconceived choices” won’t be necessary in the quantum, as we will show it in the next bidouille.

 

Pas besoin de chercher bien loin: il n'y a, à ce jour, QU'UN SEUL pays du continent Américain qui soit dirigé par un membre tribal. Afin d'éviter le genre de désagrément trouvé, par exemple, dans:
https://newfoodeconomy.org/tanka-bar-general-mills-epic-provisions-bison-bars/?fbclid=IwAR1_jFjSZwSVOXrrp9aAdFHrFX0IbJpEcW8c8_R1W1fahIes9h1f2whfyT0

voici mes suggestions:
- vous avez un produit INNOVANT à lancer, basé sur une science tribale;
- vous DELOCALISEZ en Bolivie et vous ENREGISTREZ là;
- vous PROTEGEZ IMMEDIATEMENT le produit aux Instituts de la Propriété Industrielle, afin de déposer marques et brevets;
- vous démarrez l'activité, phase de recherche si préalable, puis commercialisation;
- vous focalisez les ventes AU SEIN DES TRIBUS AMERICAINES ET A L'ETRANGER (export);
- vous payez vos taxes et impôts société et individuels au pays hôte.

De ce fait, vous aurez BEAUCOUP MOINS DE RISQUES d'être copiés "à la sauvage" et de vous voir détourner votre chiffre d'affaires. Le dépôt de marques et concepts vous ouvrira la PROTECTION JURIDIQUE et SECURISERA VOTRE MARCHé, gardant en tête que les situations de monopole restent quand même intenables à moyen terme. Il faudra donc, par la suite, faire EVOLUER VOS PRODUITS, REDEPOSER DE NOUVEAUX BREVETS ET MARQUES,...

C'est comme ça qu'on procède, en Occident... :) ça vous évite de plonger et d'aller au tribunal pour rien...

Bonjour à tous. A compter d'aujourd'hui, des posts en français sur le même blog que mes travaux théoriques, mais qui n'ont rien à voir avec ceux-ci et portent surtout sur DIPLOMATIE ET FINANCE. Aujourd'hui, à votre connaissance:

https://newsmaven.io/indiancountrytoday/news/cheyenne-river-citizens-plead-against-water-permit-for-pe-sla-gold-prospecting-H0oGwlzEa0K_92ujJW_LFQ/
https://www.navajotimes.com/wires-wp/index.php?id=1825030680&kid=IZXbj2m0mvvKAgbc#apfeed

(normalement, les liens sont actifs). Voici le pb. Le sort des tribus locales peut ne pas entrez dans vos intérêts c'est une toute autre question. EN REVANCHE, si les affirmations de la Cie canadienne rapportée dans l'article de ICT s'avère correctes, tenant compte de toutes les AUTRES zones d'extraction minière en plein (re)développement, et dans la perspective logique que le minerai suive les lois du marché, alors il faut s'attendre à ce que les cours de l'or S'EFFONDRENT DANS UN AVENIR TOUT PROCHE. Or, les devises "papier-monnaie" restent indexées sur le cours de l'or. Une dépréciation considérable de celui-ci, lié à une surabondance brutale sur le marché des devises, entrainerait des dépréciations équivalentes des monnaies.
POUR CETTE RAISON, je pense que la question devrait être examinée, indépendamment du sort réservé aux tribus locales. Mais, le soutien à leur opposition pourrait faire l'objet d'une activité DIPLOMATIQUE complémentaire.

Vous devriez être informés par ce site des nouveaux posts à venir. Alors, il y aura sans doute un mélange d'articles techniques et non-techniques, mais j'ai préféré conserver le même blog et même fournisseur de contenu.

Mes amitiés à tous et mon bon souvenir.

Quantum physics is essentially replacing the real-valued classical frame with a complex-valued one. It may look simplistic, but that’s what the wave-corpuscle duality says, nothing more. So, we give ourselves a physical frame X with COMPLEX dimension D, that’s REAL dimension 2D, and we realize X as the direct sum X = X₁ + iX₂ of two “projective spaces”, real, with same dimension D. This is the planar representation: each point x of X with coordinates x^a (a = 1,…,D) has “projections” x₁ in X₁ with coordinates x₁^a and x₂ in X₂ with coordinates x₂^a. Saying that a quantum object is “located at point x in quantum space X” becomes formally equivalent to saying that the PROJECTIONS of this location are x₁ in X₁ and x₂ in X₂. And it goes the same with every other physical quantity: parameters, variables, functions. There’s therefore absolutely no mathematical difficulty whatsoever, nothing new, the main difficulty stands on how to INTERPRET RESULTS, their PHYSICAL CONTENT.

 

As we will work with projective states all along, we will introduce the s-MATRICES we will widely use in our calculations. These are the real-valued pendings of the sigma-matrices of spin theory. Basically, each time we have a PRODUCT of complex-valued quantities, the real-valued components of the result can be expressed in terms of these matrices, which are all INVERTIBLE. Let’s then consider two complex quantities x = x₁ + ix₂ and y = y₁ + iy₂. We have:

 

(1)               u = xy = x₁y₁ - x₂y₂ + i(x₁y₂ + x₂y₁) = (s₁^AB + is₂^AB)x_Ay_B

(2)               v = x*y = x₁y₁ + x₂y₂ + i(x₁y₂ - x₂y₁) = (s₃^AB + is₄^AB)x_Ay_B

 

Capital Latin indices run from 1 to 2 and label projective states. Thus:

 

(3)               u_C = s_C^ABx_Ay_B  ,  v_C = s_C+2^ABx_Ay_B

 

These s-matrices have components:

 

(4)               s₁¹¹ = -s₁²² = +1  ,  s₁¹² = s₁²¹ = 0

(5)               s₂¹¹ = s₂²² = 0  ,  s₂¹² = s₂²¹ = +1

(6)               s₃¹¹ = s₃²² = +1  ,  s₃¹² = s₃²¹ = 0  =>  s₃ = Id

(7)               s₄¹¹ = s₄²² = 0  ,  s₄¹² = -s₄²¹ = +1 =>  s₄ = J

 

The “identity matrix” is s₃. s₁ and s₃ are diagonal; s₂ and s₄, “anti-” or “off-diagonal”. s₁, s₂ and s₃ are symmetric matrices, s₄ is skew-symmetric. Their genuine properties are:

 

(8)               Tr(s₁,s₂,s₄) = 0  ,  Tr(s₃) = 2

(9)               Det(s₁,s₂) = -1  ,  Det(s₃,s₄) = +1

(10)           (s₁)² = (s₂)² = (s₃)² = -(s₄)² = s₃

(11)           (s₁)^-1 = s₁  ,  (s₂)^-1 = s₂  ,  (s₃)^-1 = s₃  ,  (s₄)^-1 = -s₄

(12)           s₁s₂ = -s₂s₁ = s₄  ,  s₁s₄ = -s₄s₁ = s₂  ,  s₄s₂ = -s₂s₄ = s₁

 

while s₁, s₂ and s₄ obviously commute with s₃. These properties linearize the products of s-matrices. For instance,

 

s₁s₂s₄ = s₄² = -s₁² = -s₃

 

etc., so that we find the CYCLIC properties of the integer powers of i (i² = -1, i³ = -i, i⁴ = +1) REPORTED to the real-valued s-matrices, which make a BASE of M₂(R).

 

Let’s now consider a quantum point x^a = x₁^a + ix₂^a, a = 1,…,D. Both projections x₁^a and x₂^a are a priori SIGNED quantities. If X₁ and X₂ are both Euclidian, they both have SO(D) as EXTERNAL rotation group; if they are pseudo-Euclidian with signature (D₁,D₂), D₁ + D₂ = D, they both have SO(D₁,D₂) as external rotation group. This group defines ORIENTATION in both projective spaces. What about X = X₁ + iX₂? It inherits the EXTERNAL symmetries of its projections, plus an INTERNAL symmetry introduced by the presence of the imaginary unit i. This is a U(1) symmetry group. It defines ANOTHER ORIENTATION, now dealing with PHYSICAL STATES. If we extend (1) and (2) to the tensor product of two quantum vectors x and y of X, we find:

 

(13)           u^ab = x^ay^b = x₁^ay₁^b - x₂^ay₂^b + i(x₁^ay₂^b + x₂^ay₁^b) = (s₁^AB + is₂^AB)x_A^ay_B^b

(14)           v^ab = x*^ay^b = x₁^ay₁^b + x₂^ay₂^b + i(x₁^ay₂^b - x₂^ay₁^b) = (s₃^AB + is₄^AB)x_A^ay_B^b

(15)           u_C^ab = s_C^ABx_A^ay_B^b  ,  v_C^ab = s_C+2^ABx_A^ay_B^b

 

It is clear the small Latin indices representing the EXTERNAL components aren’t affected at all by the INTERNAL transformation involving only capital Latin indices. What we find instead is that the state-1 projection of u^ab is u₁^ab = x₁^ay₁^b - x₂^ay₂^b, its state-2 projection is u₂^ab = x₁^ay₂^b + x₂^ay₁^b; the state-1 projection of v^ab is v₁^ab = x₁^ay₁^b + x₂^ay₂^b and its state-2 projection, v₂^ab = x₁^ay₂^b - x₂^ay₁^b. Each time, it involves ALL FOUR INITIAL VECTORS, x₁, y₁, x₂ and y₂ and this is what the INTERNAL SCALAR PRODUCTS express in (15). Notice that:

 

x₁^ay₁^b = ½ (u₁ + v₁) , x₁^ay₂^b = ½ (u₂ - v₂)

x₂^ay₁^b = ½ (u₂ + v₂) , x₂^ay₂^b = -½ (u₁ - v₁)

 

so that,

 

(16)           x_A^ay_B^b = ½ (u_Cs^C_AB + v_Cs^C+2_AB)

 

In the special case y = x,

 

(17)           u^ab = x^ax^b = x₁^ax₁^b - x₂^ax₂^b + i(x₁^ax₂^b + x₂^ax₁^b) = (s₁^AB + is₂^AB)x_A^ax_B^b

(18)           v^ab = x*^ax^b = x₁^ax₁^b + x₂^ax₂^b + i(x₁^ax₂^b - x₂^ax₁^b) = (s₃^AB + is₄^AB)x_A^ax_B^b

(19)           u_C^ab = s_C^ABx_A^ax_B^b  ,  v_C^ab = s_C+2^ABx_A^ax_B^b

(20)           x_A^ax_B^b = ½ (u_Cs^C_AB + v_Cs^C+2_AB)

 

and external traces give:

 

(21)           u = S_b=a=1^D u^ab = S_a=1^D [(x₁^a)² - (x₂^a)² + 2ix₁^ax₂^a]

(22)           v = S_b=a=1^D v^ab = S_a=1^D [(x₁^a)² + (x₂^a)²]

 

While v is always non-negative and real-valued in an Euclidian geometry, both u₁ and u₂ can either be positive, negative or zero, EVEN IN AN EUCLIDIAN GEOMETRY. Here is the effect of the INTERNAL SYMMETRY: the EXTERNAL geometry remain unchanged, but PROJECTIVE SQUARES are affected by the INTERNAL geometry (in circle). This is the great difference with the classical situation. Here, we have:

 

(23)           u₁ = S_a=1^D [(x₁^a)² - (x₂^a)²]

(24)           u₂ = 2S_a=1^D x₁^ax₂^a

 

Both are HYPERBOLIC squares, whereas v is an ELLIPTIC square. It follows that, independent of the external geometry, we have to define INTERNAL GENUS. If u₁ is found > 0, the state-1 contribution is HIGHER than the state-2 contribution and we’ll say that u₁ is “(state)1-like”. If u₁ < 0, it will be “(state)2-like” and if u₁ = 0, we’ll keep the name “isotropic” used in the mathematics of space-time relativity. It means that, if the Euclidian areas measured in both states are EQUAL, than the resulting area, OBSERVED IN STATE 1, will be ZERO… otherwise said, it will reduce to a POINT. If u₂ now is equal to zero, the two projected vectors x₁ and x₂ appear ORTHOGONAL to a state-2 observer. In the POLAR representation now:

 

(25)           x₁^a = r^acos(ksi) , x₂^a = r^asin(ksi)

(26)           u₁ = [S_a=1^D (r^a)²]cos(2ksi)

(27)           u₂ = [S_a=1^D (r^a)²]sin(2ksi)

 

and, whether S_a=1^D (r^a)² = 0 in which case x₁ = x₂ = 0 and  u₁ = u₂ = 0, or S_a=1^D (r^a)² <> 0 and then u₁ = 0 for ksi = (2k+1)pi/4, k in Z, giving u₂ = (-1)^kS_a=1^D (r^a)² or u₂ = 0 for ksi = kpi/2, giving u₁ = (-1)^kS_a=1^D (r^a)². We can see that, in both situations, we alternate between a positively-counted and a negatively-counted area and that, more importantly:

 

A NEGATIVELY-COUNTED area is a POSITIVELY-COUNTED one in PHASE OPPOSITION.

 

This enables us to talk of “negatively-counted distances”. What happens is that we have a state-1⁺ and a state-2⁺, where distances are positively counted, plus a state-1^- and a state-2^-, where distances are negatively counted, because 1^- is opposite in phase to 1⁺ and 2^- to 2⁺.

 

We don’t encounter this in the classical, since ksi is set to zero there, giving x₁^a = x^a, x₂^a = 0 and u₁ = S_a=1^D (r^a)² >= 0 while u₂ = 0: the component X₂ is reduced to a point (as to know, {0}) and X₁ identifies with classical D-space.

 

Notice that there’s no contradiction of principle in finding negatively-counted areas, because these are PROJECTIVE EFFECTS: the INVARIANT area remains S_a=1^D (r^a)² IN ALL CASES [however, each r^a in (25) is SIGNED, since it remains subjected to EXTERNAL orientation].

 

Much more generally, when X is CURVED but RIEMANNIAN, its elementary area is given by the second quadratic form:

 

(28)           dl² = g_abdx^adx^b

 

where ALL quantities are complex-valued. Developing in real-valued components, it’s not difficult to show that:

 

(29)           dl² = (dl²)₁ + i(dl²)₂ = (s₁^AB + is₂^AB)dl_AB²

(30)           (dl²)_C = s_C^ABdl_AB²

(31)           dl_AB² = g_Aabs_B^CDdx_C^adx_D^b

 

Explicitly:

 

(32)           (dl²)₁ = g_1ab(dx₁^adx₁^b - dx₂^adx₂^b) - g_2ab(dx₁^adx₂^b + dx₁^bdx₂^a)

(33)           (dl²)₂ = g_1ab(dx₁^adx₂^b + dx₁^bdx₂^a) + g_2ab(dx₁^adx₁^b - dx₂^adx₂^b)

 

Also notice the linear form:

 

(34)           dl = u_adx^a  =>  dl_C = s_C^ABu_Adx_B^a

 

If we square dl = dl₁ + idl₂, we find dl² = dl₁² - dl₂² + 2idl₁dl₂. Comparing with (29) gives:

 

(35)           (dl²)₁ = dl₁² - dl₂² = s₁^ABdl_Adl_B

(36)           (dl²)₂ = 2dl₁dl₂ = s₂^ABdl_Adl_B

 

So, we should not confuse (dl²)_C, which are the projections of dl², with (dl_C)², which are the squares of the projections of dl.