(*

Package for Clifford Algebra in 4-dimensional Minkowski positive (-+++),
Minkowski negative (+---), Euclidean (++++) and anti-Euclidean (----) spaces.
We will give the elements a standard grade-structured representation in the
form shown in the following examples:
                e12={0,z4,{z3,{0,0,1}},z4,0}
                e0={0,{1,z3},z6,z4,0}
where z3,z4,z6 stand for lists of 3,4,6 zeros.

Author: Josep Manel Parra Serra

Address: Departament de Fisica Fonamental
         Universitat de Barcelona
         Diagonal 647  
         E-08028  Barcelona (Spain)
         e-mail: jmparra@hermes.ffn.ub.es
         FAX: 34 3 402 11 49
*)

<<gibbs.m
Cmp::usage= "Cmp[cl1, ..., cln] evaluates the product of n elements of the
Clifford algebra of the Minkowski space with signature -+++. These elements can
be given either in the Gibbs' scalar-vector notation {scalar, 4-vector, 
6-bivector, 4-pseudovector,pseudoscalar} or in the structured-component form
{scalar,{v0,{v1,v2,v3}}, {{bv01,bv02,bv03},{bv23,bv31,bv12}},
{pv0,{pv1,pv2,pv3}},pseudoescalar}."

Cmn::usage= "Cmn[cl1, ..., cln] evaluates the product of n elements of the
Clifford algebra of the Minkowski space with signature +---. These elements can
be given either in the Gibbs' scalar-vector notation {scalar, 4-vector, 
6-bivector, 4-pseudovector,pseudoscalar} or in the structured-component form
{scalar,{v0,{v1,v2,v3}}, {{bv01,bv02,bv03},{bv23,bv31,bv12}},
{pv0,{pv1,pv2,pv3}},pseudoescalar}."

C4p::usage= "C4p[cl1, ..., cln] evaluates the product of n elements of the
Clifford algebra of the Euclidean space with signature ++++. These elements can
be given either in the Gibbs' scalar-vector notation {scalar, 4-vector, 
6-bivector, 4-pseudovector,pseudoscalar} or in the structured-component form
{scalar,{v0,{v1,v2,v3}}, {{bv01,bv02,bv03},{bv23,bv31,bv12}},
{pv0,{pv1,pv2,pv3}},pseudoescalar}."

C4n::usage= "C4n[cl1, ..., cln] evaluates the product of n elements of the
Clifford algebra of the Euclidean space with signature ----. These elements can
be given either in the Gibbs' scalar-vector notation {scalar, 4-vector, 
6-bivector, 4-pseudovector,pseudoscalar} or in the structured-component form
{scalar,{v0,{v1,v2,v3}}, {{bv01,bv02,bv03},{bv23,bv31,bv12}},
{pv0,{pv1,pv2,pv3}},pseudoescalar}."

CC4p::usage= "CC4p[cl1,cl2] gives the Clifford commutator in the Cl(4,0)
algebra." 

CC4n::usage= "CC4p[cl1,cl2] gives the Clifford commutator in the Cl(4,0)
algebra." 

SN4p::usage= "SN4p[cl1] gives the so called spinorial norm of an element in
Cl(4,0). It is the result of the product C4p[cl1,GRev4[cl1]]. In three
dimensions it has scalar and pseudoscalar parts, so it is equivalent to a
complex number. It plays a special role in constructing the general inverse.
Often only the scalar part of the product is considered and called spinor norm."

SN4n::usage= "SN4n[cl1] gives the so called spinorial norm of an element in
Cl(0,4). It is the result of the product C4p[cl1,GRev4[cl1]]. Consisting in
this case in a sum of a scalar and a pseudoscalar, it can be considered as a
dual number." 

Det4p::usage= "Det4p[cl1] gives the characteristic determinant of the element.
It is the square root of the characteristic polynomial of the regular
representation of the element. It is used to determine the invertibility of the
element and the compex spectral points of the element in Cl(4,0)."


Det4n::usage= "Det4n[cl1] gives the characteristic determinant of the element.
It is the square root of the characteristic polynomial of the regular
representation of the element. It is used to determine the invertibility of the
element and the compex spectral points of the element in Cl(0,4)."

GI4::usage= "GI4[cl1] gives the grade (or main) involution of the element.
Changes the sign of the elements of odd degree."

Rev4::usage= "Rev4[cl1] gives the reverse of the element. It is the
antiinvolution that reverses the order in the geometric product of the units."

GRev4::usage= "GRev4[cl1] gives the `grade' reversion of the element. Often
called (Clifford) conjugation, it results of the successive application, in any
order, of the grade involution and the reversion."

Dual4p::usage= "Dual4p[cl1] gives the `dual' of the element in Cl(4,0). It is
the outcome of the product with the unit pseudoscalar and thus is the Clifford
algebraic counterpart of both the original Grassmann's `Erganzung' and Hodge
star operator."

Dual4n::usage= "Dual4n[cl1] gives the `dual' of the element in Cl(0,4). It is
the outcome of the product with the unit pseudoscalar and thus is the Clifford
algebraic counterpart of both the original Grassmann's `Erganzung' and Hodge
star operator."

Part40::usage= "Part40[cl1] gives the scalar part (geometrical degree 0) of the
element."
Part41::usage= "Part41[cl1] gives the vector part (geometrical degree 1) of the
element."
Part42::usage= "Part42[cl1] gives the bivector part (geometrical degree 2) of the
element."
Part43::usage= "Part43[cl1] gives the pseudoscalar or volumic part (geometrical
degree 3) of the element."

Inverse4p::usage= "Inverse4p[cl1] gives the inverse of the element in Cl(4,0)
when it exists."
Inverse4n::usage= "Inverse4n[cl1] gives the inverse of the element in Cl(0,4)
when it exists."

rg4p::usage= "rg4p[{8-list}] gives the regular matrix representation of an
element of Cl(4,0) whose components are expressed in alphabetical ordering
forming a single list of eight elements."
rg4n::usage= "rg4n[{8-list}] gives the regular matrix representation of an
element of Cl(0,4) whose components are expressed in alphabetical ordering
forming a single list of eight elements."
r4p::usage= "rg4p[cl1] gives the regular matrix representation of an
element of Cl(4,0) expressed in the standard geometric form."
r4n::usage= "rg4n[cl1] gives the regular matrix representation of an
element of Cl(0,4) expressed in the standard geometric form."


StoA4::usage= "StoA4[cl1] gives the 16 components of the element as a single
list corresponding to the pure alphabetic ordering of the basis. The ordering
change in the bivector components makes this function different from the simple
Flatten[cl1] provided in Mathematica." 

AtoS4::usage= "AtoS4[cl1] gives the Structured (in our program adopted as
Standard) form of an element {scalar,{vector},{bivector},pseudoscalar} starting
from the alphabetic ordering of the components."

StoG4::usage= "StoG4[cl1] gives the geometric expression of an element including
the basis multivectors. It is the expression used in handwriten computations or
used in CLICAL."
StoGG4::usage= "StoGG4[cl1] gives the geometric expression of an element including
the basis multivectors. This time the indices of the basis multivectors are
placed as arguments, as in more general Mathematica programmes. Its use is
intended to facilitate interchange with other programs."

StoM::usage= "StoM[cl1] gives the Pauli Matrix representation of an element of
Cl(4,0). Paui matrices are named sigma1,sigma2,sigma3 and are identified with
the basis vectors e1,e2,e3; sigma0 is the identity."

MtoS::usage= "MtoS[complex 2x2 matrix] gives the clifford element that
corresponds to an arbitrary element in the Pauli matrix algebra."

StoQ::usage= "StoQ[cl1] gives the ordered pair of quaternions that allow to
express the Clifford algebra Cl(0,3) as a double quaternion structure. See
QCP." 

QtoS::usage= "QtoS[{quat1,quat2}] gives the Clifford element in Cl(0,3) that
correspond to an ordered pair of quaternions. See QCP."

QCP::usage= "QCP[{quat1,quat2},{quat3,quat4}] is the product in the double
quaternion ring that allows the establishment of an isomorphism between this
double algebra and the Clifford algebra Cl(0,3). See StoQ and QtoS."

random3::usage= "random3[] gives a standard three-component random vector with
components in the range (-1,1), to allow plurality of signs."

random8::usage= "random8[] gives a eight-component random vector or array with
values in the (-1,1) range."

random::usage= "random[] gives a random clifford element with components in the (-1,1)
range." 

crandom::usage= "crandom[] gives a random clifford element with complex number
coefficients with rea and imaginary parts in the range (-1,1)."

Nabla4p::usage= "Nabla4p[cl1] gives the outcome of the nabla or
clifford-gradient-derivative operator upon an element of Cl(4,0) that depends
on the cartesian variables x,y,z."
Nabla4n::usage= "Nabla4n[cl1] gives the outcome of the nabla or
clifford-gradient-derivative operator upon an element of Cl(0,4) that depends
on the cartesian variables x,y,z."
ExtD4p::usage= "ExtD4p[cl1] gives the exterior derivative of an element of
Cl(4,0) whose components are functions of the cartesian variables x,y,z. We
notice that the signature-dependence of the exterior derivative is the result
of adopting the contravariant basis vectors as the basis for the geometric
Clifford algebra."
ExtD4n::usage= "ExtD4n[cl1] gives the exterior derivative of an element of
Cl(4,0) whose components are functions of the cartesian variables x,y,z. We
notice that the signature-dependence of the exterior derivative is the result
of adopting the contravariant basis vectors as the basis for the geometric
Clifford algebra."
Codif4::usage= "Codif4[cl1] gives the exterior derivative of an element of
Cl(4,0) or Cl(0,4) whose components are functions of the cartesian variables
x,y,z. We notice that the signature-independence of the codifferential is
the result of adopting the contravariant basis vectors as the basis for the
geometric Clifford algebra."

z3 = {0,0,0};
z4 = {0 ,z3};
z6 = {z3 ,z3};
sc4= {1,z4,z6,z4,0}

Cmp[{x1_,{x2_,v1_},{v2_,v3_},{x3_,v4_},x4_},
     {y1_,{y2_,w1_},{w2_,w3_},{y3_,w4_},y4_}]:=
{x1 y1 - x2 y2 + v1 . w1 + v2 . w2 - v3 . w3 - x3 y3 + v4 . w4 - x4 y4,
{ x2 y1 + x1 y2 + v2 . w1 - v1 . w2 - v4 . w3 - v3 . w4 - x4 y3 + x3 y4,
x1 w1 + v1 y1 - x2 w2 + v2 y2 - x3 w3 - v3 y3 -  x4 w4 + v4 y4 -
Cross[v3,w1] - Cross[v1,w3] + Cross[v4,w2] - Cross[v2,w4]},
{ v2 y1 + x1 w2 + x2 w1 - v1 y2 - x4 w3 - v3 y4 - v4 y3 + x3 w4 -
Cross[v4,w1]+ Cross[v1,w4]- Cross[v3,w2]- Cross[v2,w3],
v3 y1 + x1 w3 - v4 y2 - x2 w4 + x3 w1 + v1 y3 + x4 w2 + v2 y4 +
Cross[v1,w1] + Cross[v2,w2]- Cross[v3,w3]+ Cross[v4,w4]},
{x3 y1 + x1 y3 + x4 y2 - x2 y4 + v3 . w1 + v1 . w3 - v4. w2 + v2 . w4,
v4 y1 + x1 w4 + v3 y2 + x2 w3 + x4 w1 - v1 y4 - x3 w2 + v2 y3 +
Cross[v2, w1]- Cross[v1,w2] - Cross[v4,w3] - Cross[v3,w4]},
x4 y1 + x1 y4 - x3 y2 + x2 y3 + v4 . w1 - v1 . w4 + v3 . w2 + v2 . w3}

Cmp[a_,b__]:= Fold[Cmp,a,{b}];
Attributes[Cmp] = {Flat}
CCmp[a_,b_]:= Cmp[a,b]-Cmp[b,a];
CAmp[a_,b_]:= Cmp[a,b]+Cmp[b,a];

(* Producte de Clifford per a l'espai de Minkowski de signatura
   negativa (-2)  mŠtrica: +  ---    "Minkowski negatiu (mn)"  *)

Cmn[{x1_,{x2_,v1_},{v2_,v3_},{x3_,v4_},x4_},
     {y1_,{y2_,w1_},{w2_,w3_},{y3_,w4_},y4_}]:=
{x1 y1 + x2 y2 - v1 . w1 + v2 . w2 - v3 . w3 + x3 y3 - v4 . w4 - x4 y4,
{ x2 y1 + x1 y2 - v2 . w1 + v1 . w2 - v4 . w3 - v3 . w4 + x4 y3 - x3 y4,
x1 w1 + v1 y1 + x2 w2 - v2 y2 - x3 w3 - v3 y3 +  x4 w4 - v4 y4 +
Cross[v3,w1] + Cross[v1,w3] + Cross[v4,w2] - Cross[v2,w4]},
{ v2 y1 + x1 w2 + x2 w1 - v1 y2 - x4 w3 - v3 y4 - v4 y3 + x3 w4 +
Cross[v4,w1]- Cross[v1,w4]+ Cross[v3,w2]+ Cross[v2,w3],
v3 y1 + x1 w3 + v4 y2 + x2 w4 - x3 w1 - v1 y3 + x4 w2 + v2 y4 +
Cross[v1,w1] - Cross[v2,w2]+ Cross[v3,w3]+ Cross[v4,w4]},
{x3 y1 + x1 y3 - x4 y2 + x2 y4 + v3 . w1 + v1 . w3 + v4. w2 - v2 . w4,
v4 y1 + x1 w4 + v3 y2 + x2 w3 - x4 w1 + v1 y4 + x3 w2 - v2 y3 +
Cross[v2, w1]- Cross[v1,w2] + Cross[v4,w3] + Cross[v3,w4]},
x4 y1 + x1 y4 - x3 y2 + x2 y3 + v4 . w1 - v1 . w4 + v3 . w2 + v2 . w3}

Cmn[a_,b__]:= Fold[Cmn,a,{b}];
Attributes[Cmn] = {Flat}
CCmn[a_,b_]:= Cmn[a,b]-Cmn[b,a];
CAmn[a_,b_]:= Cmn[a,b]+Cmn[b,a];

(* Producte de Clifford per a l'espai Euclidi… 4-dimensional
positiu *)

C4p[{x1_,{x2_,v1_},{v2_,v3_},{x3_,v4_},x4_},
     {y1_,{y2_,w1_},{w2_,w3_},{y3_,w4_},y4_}]:=
{x1 y1 + x2 y2 + v1 . w1 - v2 . w2 - v3 . w3 - x3 y3 - v4 . w4 + x4 y4,
{ x2 y1 + x1 y2 + v2 . w1 - v1 . w2 - v4 . w3 - v3 . w4 - x4 y3 + x3 y4,
x1 w1 + v1 y1 + x2 w2 - v2 y2 - x3 w3 - v3 y3 +  x4 w4 - v4 y4 -
Cross[v3,w1] - Cross[v1,w3] - Cross[v4,w2] + Cross[v2,w4]},
{ v2 y1 + x1 w2 + x2 w1 - v1 y2 - x4 w3 - v3 y4 - v4 y3 + x3 w4 -
Cross[v4,w1]+ Cross[v1,w4]- Cross[v3,w2]- Cross[v2,w3],
v3 y1 + x1 w3 + v4 y2 + x2 w4 + x3 w1 + v1 y3 - x4 w2 - v2 y4 +
Cross[v1,w1] - Cross[v2,w2]- Cross[v3,w3]- Cross[v4,w4]},
{x3 y1 + x1 y3 - x4 y2 + x2 y4 + v3 . w1 + v1 . w3 + v4. w2 - v2 . w4,
v4 y1 + x1 w4 + v3 y2 + x2 w3 + x4 w1 - v1 y4 - x3 w2 + v2 y3 +
Cross[v2, w1]- Cross[v1,w2] - Cross[v4,w3] - Cross[v3,w4]},
x4 y1 + x1 y4 - x3 y2 + x2 y3 + v4 . w1 - v1 . w4 + v3 . w2 + v2 . w3}

C4p[a_,b__]:= Fold[C4p,a,{b}];
Attributes[C4p] = {Flat}
CC4p[a_,b_]:= C4p[a,b]-C4p[b,a];
CA4p[a_,b_]:= C4p[a,b]+C4p[b,a];

(* Producte de Clifford per a l'espai Euclidi… 4-dimensional
negatiu *)


C4n[{x1_,{x2_,v1_},{v2_,v3_},{x3_,v4_},x4_},
     {y1_,{y2_,w1_},{w2_,w3_},{y3_,w4_},y4_}]:=
{x1 y1 - x2 y2 - v1 . w1 - v2 . w2 - v3 . w3 + x3 y3 + v4 . w4 + x4 y4,
{ x2 y1 + x1 y2 - v2 . w1 + v1 . w2 - v4 . w3 - v3 . w4 + x4 y3 - x3 y4,
x1 w1 + v1 y1 - x2 w2 + v2 y2 - x3 w3 - v3 y3 -  x4 w4 + v4 y4 +
Cross[v3,w1] + Cross[v1,w3] - Cross[v4,w2] + Cross[v2,w4]},
{ v2 y1 + x1 w2 + x2 w1 - v1 y2 - x4 w3 - v3 y4 - v4 y3 + x3 w4 +
Cross[v4,w1]- Cross[v1,w4]+ Cross[v3,w2]+ Cross[v2,w3],
v3 y1 + x1 w3 - v4 y2 - x2 w4 - x3 w1 - v1 y3 - x4 w2 - v2 y4 +
Cross[v1,w1] + Cross[v2,w2]+ Cross[v3,w3]- Cross[v4,w4]},
{x3 y1 + x1 y3 + x4 y2 - x2 y4 + v3 . w1 + v1 . w3 - v4. w2 + v2 . w4,
v4 y1 + x1 w4 + v3 y2 + x2 w3 - x4 w1 + v1 y4 + x3 w2 - v2 y3 +
Cross[v2, w1]- Cross[v1,w2] + Cross[v4,w3] + Cross[v3,w4]},
x4 y1 + x1 y4 - x3 y2 + x2 y3 + v4 . w1 - v1 . w4 + v3 . w2 + v2 . w3}

C4n[a_,b__]:= Fold[C4n,a,{b}];
Attributes[C4n] = {Flat}
CC4n[a_,b_]:= C4n[a,b]-C4n[b,a];
CA4n[a_,b_]:= C4n[a,b]+C4n[b,a];


(* Automorfismes, antiautomorfismes i duals en 4 dimensions *)

GI4[{x1_,{x2_,v1_},{v2_,v3_},{x3_,v4_},x4_}]:=
{x1,{-x2,-v1},{v2,v3},{-x3,-v4},x4}

Rev4[{x1_,{x2_,v1_},{v2_,v3_},{x3_,v4_},x4_}]:=
{x1,{x2,v1},{-v2,-v3},{-x3,-v4},x4}

GRev4[x_]:=GI4[Rev4[x]]

NSmp[x_]:= Cmp[x,GRev4[x]]

NSmn[x_]:= Cmn[x,GRev4[x]]

NS4p[x_]:= C4p[x,GRev4[x]]

NS4n[x_]:= C4n[x,GRev4[x]]

SN41[{x1_,{x2_,v1_},{v2_,v3_},{x3_,v4_},x4_}]:=
{0,z4, z6, 
{2 (x1 x3 - v1 . v3), 2 (x1 v4 - x2 v3 + Cross[v1,v2])}, 
2 (x1 x4 - v2 . v3)}

SN4p0[{x1_,{x2_,v1_},{v2_,v3_},{x3_,v4_},x4_}]:=
x1^2+v3 . v3 - (x2^2 + v4 . v4) - (x3^2 + v1 . v1) + (x4^2+ v2 . v2);


SN4p2[{x1_,{x2_,v1_},{v2_,v3_},{x3_,v4_},x4_}]:=
{0,z4, z6, 
{2 (x2 x4 - v2 . v4), 2 (x3 v2 - x4 v1 + Cross[v4,v3])}, 
2 (x2 x3 - v1 . v4)}


SNmp0[{x1_,{x2_,v1_},{v2_,v3_},{x3_,v4_},x4_}]:=
x1^2+v3 . v3 + (x2^2 + v4 . v4) - (x3^2 + v1 . v1) - (x4^2+ v2 . v2);

SNmp2[{x1_,{x2_,v1_},{v2_,v3_},{x3_,v4_},x4_}]:=
{0,z4, z6, 
{-2 (x2 x4 - v2 . v4), 2 (x3 v2 - x4 v1 + Cross[v4,v3])}, 
2 (x2 x3 - v1 . v4)}

SNmn0[{x1_,{x2_,v1_},{v2_,v3_},{x3_,v4_},x4_}]:=
x1^2+v3 . v3 - (x2^2 + v4 . v4) + (x3^2 + v1 . v1) - (x4^2+ v2 . v2);

SNmn2[{x1_,{x2_,v1_},{v2_,v3_},{x3_,v4_},x4_}]:=
{0, z4, z6, 
{2 (x2 x4 - v2 . v4), -2 (x3 v2 - x4 v1 + Cross[v4,v3])}, 
2 (x2 x3 - v1 . v4)}

SN4n0[{x1_,{x2_,v1_},{v2_,v3_},{x3_,v4_},x4_}]:=
x1^2+v3 . v3 + (x2^2 + v4 . v4) + (x3^2 + v1 . v1) + (x4^2+ v2 . v2);

SN4n2[{x1_,{x2_,v1_},{v2_,v3_},{x3_,v4_},x4_}]:=
{0, z4, z6, 
{-2 (x2 x4 - v2 . v4), -2 (x3 v2 - x4 v1 + Cross[v4,v3])}, 
2 (x2 x3 - v1 . v4)}

SN4p[a_]=SN4p0 sc4+ SN41[a]+ SN4p2[a];
SNmp[a_]=SNmp0 sc4+ SN41[a]+ SNmp2[a];
SNmn[a_]=SNmn0 sc4+ SN41[a]+ SNmn2[a];
SN4n[a_]=SN4n0 sc4+ SN41[a]+ SN4n2[a];

SNs4p[a_]=SN4p0 sc4+ SN41[a]- SN4p2[a];
SNsmp[a_]=SNmp0 sc4+ SN41[a]- SNmp2[a];
SNsmn[a_]=SNmn0 sc4+ SN41[a]- SNmn2[a];
SNs4n[a_]=SN4n0 sc4+ SN41[a]- SN4n2[a];

Part40[{x1_,{x2_,v1_},{v2_,v3_},{x3_,v4_},x4_}]:= x1
Part41[{x1_,{x2_,v1_},{v2_,v3_},{x3_,v4_},x4_}]:= {x2,v1}
Part41t[{x1_,{x2_,v1_},{v2_,v3_},{x3_,v4_},x4_}]:= x2
Part41s[{x1_,{x2_,v1_},{v2_,v3_},{x3_,v4_},x4_}]:= v1
Part42[{x1_,{x2_,v1_},{v2_,v3_},{x3_,v4_},x4_}]:= {v2,v3}
Part42t[{x1_,{x2_,v1_},{v2_,v3_},{x3_,v4_},x4_}]:= v2
Part42s[{x1_,{x2_,v1_},{v2_,v3_},{x3_,v4_},x4_}]:= v3
Part43[{x1_,{x2_,v1_},{v2_,v3_},{x3_,v4_},x4_}]:= {x3,v4}
Part43t[{x1_,{x2_,v1_},{v2_,v3_},{x3_,v4_},x4_}]:= x3
Part43s[{x1_,{x2_,v1_},{v2_,v3_},{x3_,v4_},x4_}]:= v4
Part44[{x1_,{x2_,v1_},{v2_,v3_},{x3_,v4_},x4_}]:= x4

Det4p[x_]:=Module[{a,b,c,d},
   a=SN41[x]; b=SN4p2[x]; c=Part43s[a]; d=Part43s[b];
   Result= SN4p0[x]^2+ Part43t[a]^2 + Part43t[b]^2 +
           c . c + d . d - Part44[a]^2 - Part44[b]^2 ];
Detmp[x_]:=Module[{a,b,c,d},
   a=SN41[x]; b=SNmp2[x]; c=Part43s[a]; d=Part43s[b];
   Result= SNmp0[x]^2+ Part43t[a]^2 + Part43t[b]^2 -
           c . c - d . d + Part44[a]^2 + Part44[b]^2 ];
Detmn[x_]:=Module[{a,b,c,d},
   a=SN41[x]; b=SNmn2[x]; c=Part43s[a]; d=Part43s[b];
   Result= SNmn0[x]^2- Part43t[a]^2 - Part43t[b]^2 +
           c . c + d . d + Part44[a]^2 + Part44[b]^2 ];
Det4n[x_]:=Module[{a,b,c,d},
   a=SN41[x]; b=SN4n2[x]; c=Part43s[a]; d=Part43s[b];
   Result= SN4n0[x]^2- Part43t[a]^2 - Part43t[b]^2 -
           c . c - d . d - Part44[a]^2 - Part44[b]^2 ];


Inverse4p[x_]:=Module[{r,a,b,c,d,det}, r=SN4p0[x];
   a=SN41[x]; b=SN4p2[x]; c=Part43s[a]; d=Part43s[b];
   det= r^2+ Part43t[a]^2 + Part43t[b]^2 +
           c . c + d . d - Part44[a]^2 - Part44[b]^2;
        Result=C4p[GRev4[x], r sc4 - a - b]/det ]

Inversemp[x_]:=Module[{r,a,b,c,d,det}, r=SNmp0[x];
   a=SN41[x]; b=SNmp2[x]; c=Part43s[a]; d=Part43s[b];
   det= r^2+ Part43t[a]^2 + Part43t[b]^2 -
           c . c - d . d + Part44[a]^2 + Part44[b]^2;
        Result=Cmp[GRev4[x], r sc4 - a - b]/det ]
           
Inversemn[x_]:=Module[{r,a,b,c,d,det}, r=SNmn0[x];
   a=SN41[x]; b=SNmn2[x]; c=Part43s[a]; d=Part43s[b];
   det= r^2- Part43t[a]^2 - Part43t[b]^2 +
           c . c + d . d + Part44[a]^2 + Part44[b]^2;
        Result=Cmn[GRev4[x], r sc4 - a - b]/det ]
           
Inverse4n[x_]:=Module[{r,a,b,c,d,det}, r=SN4n0[x];
   a=SN41[x]; b=SN4n2[x]; c=Part43s[a]; d=Part43s[b];
   det= r^2- Part43t[a]^2 - Part43t[b]^2 -
           c . c - d . d - Part44[a]^2 - Part44[b]^2;
        Result=C4n[GRev4[x], r sc4 - a - b]/det ]



random3[]:= Table[Random[Real,{-1,1}],{i,1,3}]
random4[]:= {Random[Real,{-1,1}],random3[]}
random16[]:= Table[Random[Real,{-1,1}],{i,1,16}]
random[]:= {Random[Real,{-1,1}],random4[],{random3[],random3[]},random4[],
            Random[Real,{-1,1}]}
crandom[]:= random[]+ I random[]
random4[]:= {Random[Real,{-1,1}],random3[]}
field4={sc[x0,x,y,z],{v0[x0,x,y,z],{v1[x0,x,y,z],v2[x0,x,y,z],v3[x0,x,y,z]}},
{{ef1[x0,x,y,z],ef2[x0,x,y,z],ef3[x0,x,y,z]},
{mf1[x0,x,y,z],mf2[x0,x,y,z],mf3[x0,x,y,z]}},{pv0[x0,x,y,z],
                {pv1[x0,x,y,z],pv2[x0,x,y,z],pv3[x0,x,y,z]}}, ps[x0,x,y,z]};


(* Regular matrix representations for 4-dimensional Clifford algebras
   rg4p(4n)(mp)(mn)[{flat, alphabetically ordered Clifford element}] *)

StoA4[{alpha_,{v0_,{v1_,v2_,v3_}},{{A1_,A2_,A3_},{C1_,C2_,C3_}},
{S0_,{S1_,S2_,S3_}},lambda_}]:=
{alpha,v0,v1,v2,v3,A1,A2,A3,C3,-C2,C1,S3,-S2,S1,S0,lambda};

AtoS4[{alpha_,v0_,v1_,v2_,v3_,A1_,A2_,A3_,C3_,C2_,C1_,S3_,S2_,S1_,
S0_,lambda_}]:=
{alpha,{v0,{v1,v2,v3}},{{A1,A2,A3},{C1,-C2,C3}},{S0,{S1,-S2,S3}},lambda}

StoG4[{alpha_,{v0_,{v1_,v2_,v3_}},{{A1_,A2_,A3_},{C1_,C2_,C3_}},
{S0_,{S1_,S2_,S3_}},lambda_}]:=
alpha +v0 e0+v1 e1+v2 e2+v3 e3+A1 e01+A2 e02+A3 e03+C1 e23+C2 e31+C3 e12+
S0 e123+S1 e023+S2 e031+S3 e012+lambda e0123;

StoGG4[{alpha_,{v0_,{v1_,v2_,v3_}},{{A1_,A2_,A3_},{C1_,C2_,C3_}},
{S0_,{S1_,S2_,S3_}},lambda_}]:=
alpha e[] +v0 e[0]+v1 e[1]+v2 e[2]+v3 e[3]+A1 e[0,1]+A2 e[0,2]+A3 e[0,3]+C1 e[2,3]+C2 e[3,1]+C3 e[1,2]+
S0 e[1,2,3]+S1 e[0,2,3]+S2 e[0,3,1]+S3 e[0,1,2]+lambda e[0,1,2,3];



rg4p[a__]:=
 {{a[[1]], a[[2]], a[[3]], a[[4]], a[[5]], -a[[6]], -a[[7]], -a[[8]], -a[[9]], -a[[10]], -a[[11]], -a[[12]], -a[[13]], 
    -a[[14]], -a[[15]], a[[16]]}, {a[[2]], a[[1]], a[[6]], a[[7]], a[[8]], -a[[3]], -a[[4]], -a[[5]], -a[[12]], -a[[13]], -a[[14]], 
    -a[[9]], -a[[10]], -a[[11]], -a[[16]], a[[15]]}, 
   {a[[3]], -a[[6]], a[[1]], a[[9]], a[[10]], a[[2]], a[[12]], a[[13]], -a[[4]], -a[[5]], -a[[15]], a[[7]], a[[8]], a[[16]], -a[[11]], 
    -a[[14]]}, {a[[4]], -a[[7]], -a[[9]], a[[1]], a[[11]], -a[[12]], a[[2]], a[[14]], a[[3]], a[[15]], -a[[5]], -a[[6]], -a[[16]], 
    a[[8]], a[[10]], a[[13]]}, {a[[5]], -a[[8]], -a[[10]], -a[[11]], a[[1]], -a[[13]], -a[[14]], a[[2]], -a[[15]], a[[3]], a[[4]], 
    a[[16]], -a[[6]], -a[[7]], -a[[9]], -a[[12]]}, {a[[6]], -a[[3]], a[[2]], a[[12]], a[[13]], a[[1]], a[[9]], a[[10]], -a[[7]], 
    -a[[8]], -a[[16]], a[[4]], a[[5]], a[[15]], -a[[14]], -a[[11]]}, 
   {a[[7]], -a[[4]], -a[[12]], a[[2]], a[[14]], -a[[9]], a[[1]], a[[11]], a[[6]], a[[16]], -a[[8]], -a[[3]], -a[[15]], a[[5]], a[[13]], 
    a[[10]]}, {a[[8]], -a[[5]], -a[[13]], -a[[14]], a[[2]], -a[[10]], -a[[11]], a[[1]], -a[[16]], a[[6]], a[[7]], a[[15]], -a[[3]], 
    -a[[4]], -a[[12]], -a[[9]]}, {a[[9]], a[[12]], -a[[4]], a[[3]], a[[15]], a[[7]], -a[[6]], -a[[16]], a[[1]], a[[11]], -a[[10]], 
    a[[2]], a[[14]], -a[[13]], a[[5]], -a[[8]]}, {a[[10]], a[[13]], -a[[5]], -a[[15]], a[[3]], a[[8]], a[[16]], -a[[6]], -a[[11]], 
    a[[1]], a[[9]], -a[[14]], a[[2]], a[[12]], -a[[4]], a[[7]]}, 
   {a[[11]], a[[14]], a[[15]], -a[[5]], a[[4]], -a[[16]], a[[8]], -a[[7]], a[[10]], -a[[9]], a[[1]], a[[13]], -a[[12]], a[[2]], a[[3]], 
    -a[[6]]}, {a[[12]], a[[9]], -a[[7]], a[[6]], a[[16]], a[[4]], -a[[3]], -a[[15]], a[[2]], a[[14]], -a[[13]], a[[1]], a[[11]], 
    -a[[10]], a[[8]], -a[[5]]}, {a[[13]], a[[10]], -a[[8]], -a[[16]], a[[6]], a[[5]], a[[15]], -a[[3]], -a[[14]], a[[2]], a[[12]], 
    -a[[11]], a[[1]], a[[9]], -a[[7]], a[[4]]}, {a[[14]], a[[11]], a[[16]], -a[[8]], a[[7]], -a[[15]], a[[5]], -a[[4]], a[[13]], 
    -a[[12]], a[[2]], a[[10]], -a[[9]], a[[1]], a[[6]], -a[[3]]}, 
   {a[[15]], -a[[16]], a[[11]], -a[[10]], a[[9]], a[[14]], -a[[13]], a[[12]], a[[5]], -a[[4]], a[[3]], -a[[8]], a[[7]], -a[[6]], a[[1]], 
    a[[2]]}, {a[[16]], -a[[15]], a[[14]], -a[[13]], a[[12]], a[[11]], -a[[10]], a[[9]], a[[8]], -a[[7]], a[[6]], -a[[5]], a[[4]], 
    -a[[3]], a[[2]], a[[1]]}}

rgmp[a__]:=
 {{a[[1]], -a[[2]], a[[3]], a[[4]], a[[5]], a[[6]], a[[7]], a[[8]], -a[[9]], -a[[10]], -a[[11]], a[[12]], a[[13]], a[[14]], 
    -a[[15]], -a[[16]]}, {a[[2]], a[[1]], a[[6]], a[[7]], a[[8]], -a[[3]], -a[[4]], -a[[5]], -a[[12]], -a[[13]], -a[[14]], -a[[9]], 
    -a[[10]], -a[[11]], -a[[16]], a[[15]]}, {a[[3]], a[[6]], a[[1]], a[[9]], a[[10]], -a[[2]], -a[[12]], -a[[13]], -a[[4]], -a[[5]], 
    -a[[15]], -a[[7]], -a[[8]], -a[[16]], -a[[11]], a[[14]]}, 
   {a[[4]], a[[7]], -a[[9]], a[[1]], a[[11]], a[[12]], -a[[2]], -a[[14]], a[[3]], a[[15]], -a[[5]], a[[6]], a[[16]], -a[[8]], a[[10]], 
    -a[[13]]}, {a[[5]], a[[8]], -a[[10]], -a[[11]], a[[1]], a[[13]], a[[14]], -a[[2]], -a[[15]], a[[3]], a[[4]], -a[[16]], a[[6]], 
    a[[7]], -a[[9]], a[[12]]}, {a[[6]], -a[[3]], a[[2]], a[[12]], a[[13]], a[[1]], a[[9]], a[[10]], -a[[7]], -a[[8]], -a[[16]], a[[4]], 
    a[[5]], a[[15]], -a[[14]], -a[[11]]}, {a[[7]], -a[[4]], -a[[12]], a[[2]], a[[14]], -a[[9]], a[[1]], a[[11]], a[[6]], a[[16]], 
    -a[[8]], -a[[3]], -a[[15]], a[[5]], a[[13]], a[[10]]}, 
   {a[[8]], -a[[5]], -a[[13]], -a[[14]], a[[2]], -a[[10]], -a[[11]], a[[1]], -a[[16]], a[[6]], a[[7]], a[[15]], -a[[3]], -a[[4]], 
    -a[[12]], -a[[9]]}, {a[[9]], -a[[12]], -a[[4]], a[[3]], a[[15]], -a[[7]], a[[6]], a[[16]], a[[1]], a[[11]], -a[[10]], -a[[2]], 
    -a[[14]], a[[13]], a[[5]], a[[8]]}, {a[[10]], -a[[13]], -a[[5]], -a[[15]], a[[3]], -a[[8]], -a[[16]], a[[6]], -a[[11]], a[[1]], 
    a[[9]], a[[14]], -a[[2]], -a[[12]], -a[[4]], -a[[7]]}, 
   {a[[11]], -a[[14]], a[[15]], -a[[5]], a[[4]], a[[16]], -a[[8]], a[[7]], a[[10]], -a[[9]], a[[1]], -a[[13]], a[[12]], -a[[2]], a[[3]], 
    a[[6]]}, {a[[12]], a[[9]], -a[[7]], a[[6]], a[[16]], a[[4]], -a[[3]], -a[[15]], a[[2]], a[[14]], -a[[13]], a[[1]], a[[11]], 
    -a[[10]], a[[8]], -a[[5]]}, {a[[13]], a[[10]], -a[[8]], -a[[16]], a[[6]], a[[5]], a[[15]], -a[[3]], -a[[14]], a[[2]], a[[12]], 
    -a[[11]], a[[1]], a[[9]], -a[[7]], a[[4]]}, {a[[14]], a[[11]], a[[16]], -a[[8]], a[[7]], -a[[15]], a[[5]], -a[[4]], a[[13]], 
    -a[[12]], a[[2]], a[[10]], -a[[9]], a[[1]], a[[6]], -a[[3]]}, 
   {a[[15]], a[[16]], a[[11]], -a[[10]], a[[9]], -a[[14]], a[[13]], -a[[12]], a[[5]], -a[[4]], a[[3]], a[[8]], -a[[7]], a[[6]], a[[1]], 
    -a[[2]]}, {a[[16]], -a[[15]], a[[14]], -a[[13]], a[[12]], a[[11]], -a[[10]], a[[9]], a[[8]], -a[[7]], a[[6]], -a[[5]], a[[4]], 
    -a[[3]], a[[2]], a[[1]]}}
    
    
rgmn[a__]:=
 {{a[[1]], a[[2]], -a[[3]], -a[[4]], -a[[5]], a[[6]], a[[7]], a[[8]], -a[[9]], -a[[10]], -a[[11]], -a[[12]], -a[[13]], 
    -a[[14]], a[[15]], -a[[16]]}, {a[[2]], a[[1]], -a[[6]], -a[[7]], -a[[8]], a[[3]], a[[4]], a[[5]], -a[[12]], -a[[13]], -a[[14]], 
    -a[[9]], -a[[10]], -a[[11]], a[[16]], -a[[15]]}, 
   {a[[3]], -a[[6]], a[[1]], -a[[9]], -a[[10]], a[[2]], -a[[12]], -a[[13]], a[[4]], a[[5]], -a[[15]], -a[[7]], -a[[8]], a[[16]], 
    -a[[11]], -a[[14]]}, {a[[4]], -a[[7]], a[[9]], a[[1]], -a[[11]], a[[12]], a[[2]], -a[[14]], -a[[3]], a[[15]], a[[5]], a[[6]], 
    -a[[16]], -a[[8]], a[[10]], a[[13]]}, {a[[5]], -a[[8]], a[[10]], a[[11]], a[[1]], a[[13]], a[[14]], a[[2]], -a[[15]], -a[[3]], 
    -a[[4]], a[[16]], a[[6]], a[[7]], -a[[9]], -a[[12]]}, 
   {a[[6]], -a[[3]], a[[2]], -a[[12]], -a[[13]], a[[1]], -a[[9]], -a[[10]], a[[7]], a[[8]], -a[[16]], -a[[4]], -a[[5]], a[[15]], 
    -a[[14]], -a[[11]]}, {a[[7]], -a[[4]], a[[12]], a[[2]], -a[[14]], a[[9]], a[[1]], -a[[11]], -a[[6]], a[[16]], a[[8]], a[[3]], 
    -a[[15]], -a[[5]], a[[13]], a[[10]]}, {a[[8]], -a[[5]], a[[13]], a[[14]], a[[2]], a[[10]], a[[11]], a[[1]], -a[[16]], -a[[6]], 
    -a[[7]], a[[15]], a[[3]], a[[4]], -a[[12]], -a[[9]]}, 
   {a[[9]], a[[12]], -a[[4]], a[[3]], -a[[15]], a[[7]], -a[[6]], a[[16]], a[[1]], -a[[11]], a[[10]], a[[2]], -a[[14]], a[[13]], -a[[5]], 
    a[[8]]}, {a[[10]], a[[13]], -a[[5]], a[[15]], a[[3]], a[[8]], -a[[16]], -a[[6]], a[[11]], a[[1]], -a[[9]], a[[14]], a[[2]], 
    -a[[12]], a[[4]], -a[[7]]}, {a[[11]], a[[14]], -a[[15]], -a[[5]], a[[4]], a[[16]], a[[8]], -a[[7]], -a[[10]], a[[9]], a[[1]], 
    -a[[13]], a[[12]], a[[2]], -a[[3]], a[[6]]}, {a[[12]], a[[9]], -a[[7]], a[[6]], -a[[16]], a[[4]], -a[[3]], a[[15]], a[[2]], 
    -a[[14]], a[[13]], a[[1]], -a[[11]], a[[10]], -a[[8]], a[[5]]}, 
   {a[[13]], a[[10]], -a[[8]], a[[16]], a[[6]], a[[5]], -a[[15]], -a[[3]], a[[14]], a[[2]], -a[[12]], a[[11]], a[[1]], -a[[9]], a[[7]], 
    -a[[4]]}, {a[[14]], a[[11]], -a[[16]], -a[[8]], a[[7]], a[[15]], a[[5]], -a[[4]], -a[[13]], a[[12]], a[[2]], -a[[10]], a[[9]], 
    a[[1]], -a[[6]], a[[3]]}, {a[[15]], -a[[16]], a[[11]], -a[[10]], a[[9]], a[[14]], -a[[13]], a[[12]], a[[5]], -a[[4]], a[[3]], 
    -a[[8]], a[[7]], -a[[6]], a[[1]], a[[2]]}, {a[[16]], -a[[15]], a[[14]], -a[[13]], a[[12]], a[[11]], -a[[10]], a[[9]], a[[8]], 
    -a[[7]], a[[6]], -a[[5]], a[[4]], -a[[3]], a[[2]], a[[1]]}}

rg4n[a__]:=
 {{a[[1]], -a[[2]], -a[[3]], -a[[4]], -a[[5]], -a[[6]], -a[[7]], -a[[8]], -a[[9]], -a[[10]], -a[[11]], a[[12]], 
    a[[13]], a[[14]], a[[15]], a[[16]]}, {a[[2]], a[[1]], -a[[6]], -a[[7]], -a[[8]], a[[3]], a[[4]], a[[5]], -a[[12]], -a[[13]], 
    -a[[14]], -a[[9]], -a[[10]], -a[[11]], a[[16]], -a[[15]]}, 
   {a[[3]], a[[6]], a[[1]], -a[[9]], -a[[10]], -a[[2]], a[[12]], a[[13]], a[[4]], a[[5]], -a[[15]], a[[7]], a[[8]], -a[[16]], -a[[11]], 
    a[[14]]}, {a[[4]], a[[7]], a[[9]], a[[1]], -a[[11]], -a[[12]], -a[[2]], a[[14]], -a[[3]], a[[15]], a[[5]], -a[[6]], a[[16]], a[[8]], 
    a[[10]], -a[[13]]}, {a[[5]], a[[8]], a[[10]], a[[11]], a[[1]], -a[[13]], -a[[14]], -a[[2]], -a[[15]], -a[[3]], -a[[4]], 
    -a[[16]], -a[[6]], -a[[7]], -a[[9]], a[[12]]}, {a[[6]], -a[[3]], a[[2]], -a[[12]], -a[[13]], a[[1]], -a[[9]], -a[[10]], a[[7]], 
    a[[8]], -a[[16]], -a[[4]], -a[[5]], a[[15]], -a[[14]], -a[[11]]}, 
   {a[[7]], -a[[4]], a[[12]], a[[2]], -a[[14]], a[[9]], a[[1]], -a[[11]], -a[[6]], a[[16]], a[[8]], a[[3]], -a[[15]], -a[[5]], a[[13]], 
    a[[10]]}, {a[[8]], -a[[5]], a[[13]], a[[14]], a[[2]], a[[10]], a[[11]], a[[1]], -a[[16]], -a[[6]], -a[[7]], a[[15]], a[[3]], a[[4]], 
    -a[[12]], -a[[9]]}, {a[[9]], -a[[12]], -a[[4]], a[[3]], -a[[15]], -a[[7]], a[[6]], -a[[16]], a[[1]], -a[[11]], a[[10]], -a[[2]], 
    a[[14]], -a[[13]], -a[[5]], -a[[8]]}, {a[[10]], -a[[13]], -a[[5]], a[[15]], a[[3]], -a[[8]], a[[16]], a[[6]], a[[11]], a[[1]], 
    -a[[9]], -a[[14]], -a[[2]], a[[12]], a[[4]], a[[7]]}, 
   {a[[11]], -a[[14]], -a[[15]], -a[[5]], a[[4]], -a[[16]], -a[[8]], a[[7]], -a[[10]], a[[9]], a[[1]], a[[13]], -a[[12]], -a[[2]], 
    -a[[3]], -a[[6]]}, {a[[12]], a[[9]], -a[[7]], a[[6]], -a[[16]], a[[4]], -a[[3]], a[[15]], a[[2]], -a[[14]], a[[13]], a[[1]], 
    -a[[11]], a[[10]], -a[[8]], a[[5]]}, {a[[13]], a[[10]], -a[[8]], a[[16]], a[[6]], a[[5]], -a[[15]], -a[[3]], a[[14]], a[[2]], 
    -a[[12]], a[[11]], a[[1]], -a[[9]], a[[7]], -a[[4]]}, 
   {a[[14]], a[[11]], -a[[16]], -a[[8]], a[[7]], a[[15]], a[[5]], -a[[4]], -a[[13]], a[[12]], a[[2]], -a[[10]], a[[9]], a[[1]], -a[[6]], 
    a[[3]]}, {a[[15]], a[[16]], a[[11]], -a[[10]], a[[9]], -a[[14]], a[[13]], -a[[12]], a[[5]], -a[[4]], a[[3]], a[[8]], -a[[7]], 
    a[[6]], a[[1]], -a[[2]]}, {a[[16]], -a[[15]], a[[14]], -a[[13]], a[[12]], a[[11]], -a[[10]], a[[9]], a[[8]], -a[[7]], a[[6]], 
    -a[[5]], a[[4]], -a[[3]], a[[2]], a[[1]]}}

(* Regular matrix representations for 4-dimensional Clifford algebras
   r4p(4n)(mp)(mn)[{geometrically ordered and structured Clifford element}] *)

r4p[{alpha_,{v0_,{v1_,v2_,v3_}},{{A1_,A2_,A3_},{C1_,C2_,C3_}},
{S0_,{S1_,S2_,S3_}},lambda_}]:=
{{alpha, v0, v1, v2, v3, -A1, -A2, -A3, -C1, -C2, -C3, -S0, -S1, -S2, -S3, 
  lambda}, {v0, alpha, A1, A2, A3, -v1, -v2, -v3, -S1, -S2, -S3, -lambda, 
  -C1, -C2, -C3, S0}, {v1, -A1, alpha, C3, -C2, v0, S3, -S2, -S0, v3, -v2, 
  -C1, lambda, -A3, A2, -S1}, 
 {v2, -A2, -C3, alpha, C1, -S3, v0, S1, -v3, -S0, v1, -C2, A3, lambda, 
  -A1, -S2}, {v3, -A3, C2, -C1, alpha, S2, -S1, v0, v2, -v1, -S0, -C3, 
  -A2, A1, lambda, -S3}, {A1, -v1, v0, S3, -S2, alpha, C3, -C2, 
  -lambda, A3, -A2, -S1, S0, -v3, v2, -C1}, 
  {A2, -v2, -S3, v0, S1, -C3, alpha, C1, -A3, -lambda, A1, -S2, v3, S0, -v1, 
  -C2}, {A3, -v3, S2, -S1, v0, C2, -C1, alpha, A2, -A1, -lambda, -S3, -v2, 
   v1, S0, -C3}, {C1, S1, S0, -v3, v2, -lambda, A3, -A2, alpha, C3, -C2, 
   v1, v0, S3, -S2, -A1}, {C2, S2, v3, S0, -v1, -A3, -lambda, A1, -C3, 
   alpha, C1, v2, -S3, v0, S1, -A2}, 
  {C3, S3, -v2, v1, S0, A2, -A1, -lambda, C2, -C1, alpha, v3, S2, -S1, v0, 
  -A3}, {S0, -lambda, C1, C2, C3, S1, S2, S3, v1, v2, v3, alpha, -A1, -A2, 
  -A3, v0}, {S1, C1, lambda, -A3, A2, -S0, v3, -v2, v0, S3, -S2, A1, 
  alpha, C3, -C2, -v1}, {S2, C2, A3, lambda, -A1, -v3, -S0, v1, -S3, v0, 
  S1, A2, -C3, alpha, C1, -v2}, 
  {S3, C3, -A2, A1, lambda, v2, -v1, -S0, S2, -S1, v0, A3, C2, -C1, alpha, 
  -v3}, {lambda, -S0, S1, S2, S3, C1, C2, C3, A1, A2, A3, v0, -v1, -v2, 
  -v3, alpha}}

rmp[{alpha_,{v0_,{v1_,v2_,v3_}},{{A1_,A2_,A3_},{C1_,C2_,C3_}},
{S0_,{S1_,S2_,S3_}},lambda_}]:=
{{alpha, -v0, v1, v2, v3, A1, A2, A3, -C1, -C2, -C3, -S0, S1, S2, S3, 
 -lambda}, {v0, alpha, A1, A2, A3, -v1, -v2, -v3, -S1, -S2, -S3, -lambda, 
 -C1, -C2, -C3, S0}, {v1, A1, alpha, C3, -C2, -v0, -S3, S2, -S0, v3, -v2, 
 -C1, -lambda, A3, -A2, S1}, 
  {v2, A2, -C3, alpha, C1, S3, -v0, -S1, -v3, -S0, v1, -C2, -A3, -lambda, 
   A1, S2}, {v3, A3, C2, -C1, alpha, -S2, S1, -v0, v2, -v1, -S0, -C3, A2, 
   -A1, -lambda, S3}, {A1, -v1, v0, S3, -S2, alpha, C3, -C2, -lambda, A3, 
   -A2, -S1, S0, -v3, v2, -C1}, 
  {A2, -v2, -S3, v0, S1, -C3, alpha, C1, -A3, -lambda, A1, -S2, v3, S0, -v1, 
   -C2}, {A3, -v3, S2, -S1, v0, C2, -C1, alpha, A2, -A1, -lambda, -S3, -v2, 
   v1, S0, -C3}, {C1, -S1, S0, -v3, v2, lambda, -A3, A2, alpha, C3, -C2, 
   v1, -v0, -S3, S2, A1}, {C2, -S2, v3, S0, -v1, A3, lambda, -A1, -C3, 
   alpha, C1, v2, S3, -v0, -S1, A2}, 
 {C3, -S3, -v2, v1, S0, -A2, A1, lambda, C2, -C1, alpha, v3, -S2, S1, -v0, 
   A3}, {S0, lambda, C1, C2, C3, -S1, -S2, -S3, v1, v2, v3, alpha, A1, A2, 
   A3, -v0}, {S1, C1, lambda, -A3, A2, -S0, v3, -v2, v0, S3, -S2, A1, 
   alpha, C3, -C2, -v1}, {S2, C2, A3, lambda, -A1, -v3, -S0, v1, -S3, v0, 
   S1, A2, -C3, alpha, C1, -v2}, 
  {S3, C3, -A2, A1, lambda, v2, -v1, -S0, S2, -S1, v0, A3, C2, -C1, alpha, 
   -v3}, {lambda, -S0, S1, S2, S3, C1, C2, C3, A1, A2, A3, v0, -v1, -v2, 
   -v3, alpha}}

r4n[{alpha_,{v0_,{v1_,v2_,v3_}},{{A1_,A2_,A3_},{C1_,C2_,C3_}},
{S0_,{S1_,S2_,S3_}},lambda_}]:=
{{alpha, -v0, -v1, -v2, -v3, -A1, -A2, -A3, -C1, -C2, -C3, S0, S1, S2, S3, 
  lambda}, {v0, alpha, -A1, -A2, -A3, v1, v2, v3, -S1, -S2, -S3, lambda, 
  -C1, -C2, -C3, -S0}, {v1, A1, alpha, -C3, C2, -v0, S3, -S2, -S0, -v3, 
  v2, -C1, -lambda, -A3, A2, S1}, 
  {v2, A2, C3, alpha, -C1, -S3, -v0, S1, v3, -S0, -v1, -C2, A3, -lambda, 
  -A1, S2}, {v3, A3, -C2, C1, alpha, S2, -S1, -v0, -v2, v1, -S0, -C3, -A2, 
   A1, -lambda, S3}, {A1, -v1, v0, -S3, S2, alpha, -C3, C2, -lambda, -A3, 
   A2, -S1, S0, v3, -v2, -C1}, 
   {A2, -v2, S3, v0, -S1, C3, alpha, -C1, A3, -lambda, -A1, -S2, -v3, S0, 
   v1, -C2}, {A3, -v3, -S2, S1, v0, -C2, C1, alpha, -A2, A1, -lambda, -S3, 
   v2, -v1, S0, -C3}, {C1, -S1, -S0, -v3, v2, -lambda, -A3, A2, alpha, -C3, 
   C2, -v1, -v0, S3, -S2, -A1}, 
   {C2, -S2, v3, -S0, -v1, A3, -lambda, -A1, C3, alpha, -C1, -v2, -S3, -v0, 
   S1, -A2}, {C3, -S3, -v2, v1, -S0, -A2, A1, -lambda, -C2, C1, alpha, -v3, 
   S2, -S1, -v0, -A3}, {S0, lambda, C1, C2, C3, -S1, -S2, -S3, v1, v2, v3, 
   alpha, A1, A2, A3, -v0}, {S1, C1, -lambda, -A3, A2, S0, v3, -v2, v0, 
   -S3, S2, -A1, alpha, -C3, C2, v1}, 
   {S2, C2, A3, -lambda, -A1, -v3, S0, v1, S3, v0, -S1, -A2, C3, alpha, -C1, 
   v2}, {S3, C3, -A2, A1, -lambda, v2, -v1, S0, -S2, S1, v0, -A3, -C2, C1, 
   alpha, v3}, {lambda, -S0, S1, S2, S3, C1, C2, C3, A1, A2, A3, v0, -v1, 
   -v2, -v3, alpha}}

rmn[{alpha_,{v0_,{v1_,v2_,v3_}},{{A1_,A2_,A3_},{C1_,C2_,C3_}},
{S0_,{S1_,S2_,S3_}},lambda_}]:=
{{alpha, v0, -v1, -v2, -v3, A1, A2, A3, -C1, -C2, -C3, S0, -S1, -S2, -S3, 
 -lambda}, {v0, alpha, -A1, -A2, -A3, v1, v2, v3, -S1, -S2, -S3, lambda, 
 -C1, -C2, -C3, -S0}, {v1, -A1, alpha, -C3, C2, v0, -S3, S2, -S0, -v3, 
 v2, -C1, lambda, A3, -A2, -S1}, 
 {v2, -A2, C3, alpha, -C1, S3, v0, -S1, v3, -S0, -v1, -C2, -A3, lambda, 
  A1, -S2}, {v3, -A3, -C2, C1, alpha, -S2, S1, v0, -v2, v1, -S0, -C3, A2, 
 -A1, lambda, -S3}, {A1, -v1, v0, -S3, S2, alpha, -C3, C2, -lambda, 
 -A3, A2, -S1, S0, v3, -v2, -C1}, 
 {A2, -v2, S3, v0, -S1, C3, alpha, -C1, A3, -lambda, -A1, -S2, -v3, S0, 
 v1, -C2}, {A3, -v3, -S2, S1, v0, -C2, C1, alpha, -A2, A1, -lambda, -S3, 
 v2, -v1, S0, -C3}, {C1, S1, -S0, -v3, v2, lambda, A3, -A2, alpha, -C3, 
 C2, -v1, v0, -S3, S2, A1}, 
 {C2, S2, v3, -S0, -v1, -A3, lambda, A1, C3, alpha, -C1, -v2, S3, v0, -S1, 
 A2}, {C3, S3, -v2, v1, -S0, A2, -A1, lambda, -C2, C1, alpha, -v3, -S2, 
 S1, v0, A3}, {S0, -lambda, C1, C2, C3, S1, S2, S3, v1, v2, v3, alpha, 
 -A1, -A2, -A3, v0}, {S1, C1, -lambda, -A3, A2, S0, v3, -v2, v0, -S3, S2, 
 -A1, alpha, -C3, C2, v1}, {S2, C2, A3, -lambda, -A1, -v3, S0, v1, S3, 
 v0, -S1, -A2, C3, alpha, -C1, v2}, 
 {S3, C3, -A2, A1, -lambda, v2, -v1, S0, -S2, S1, v0, -A3, -C2, C1, alpha, 
 v3}, {lambda, -S0, S1, S2, S3, C1, C2, C3, A1, A2, A3, v0, -v1, -v2, 
 -v3, alpha}}

(* Real 4x4 matrix representation for Cl(3,1) *)
M31[{ae_ ,{a0_ ,{a1_ ,a2_ ,a3_}} ,{{a01_ ,a02_ ,a03_} ,{a23_ ,a31_ ,a12_}} ,
{a123_ ,{a023_ ,a031_ ,a012_}} ,a0123_}] = 
 {{ae + a3 + a01 - a031 , a0 + a1 - a03 + a31 ,
    a2 - a23 + a012 -a0123 , - (a02 + a12 + a123 + a023)} ,
  {- a0 + a1 - a03 - a31 ,ae - a3 - a01 - a031 ,
    - a02 + a12 - a123 + a023 ,- a2 - a23 + a012 + a0123} ,
  {a2 + a23 + a012 + a0123 ,- a02 - a12 + a123 + a023 ,
    ae - a3 + a01 + a031 , a0 + a1 + a03 - a31} ,
  {- a02 + a12 + a123 - a023 ,- a2 + a23 + a012 - a0123 ,
    - a0 + a1 + a03 + a31 , ae + a3 - a01 + a031}}





(* Nabla operator for Minkowski spaces *)

Nablamp[{s1_,{s2_,v1_},{v2_,v3_},{s3_,v4_},s4_}]:=
{D[s2,x0] + Div[v1], {-D[s1,x0] - Div[v2], Grad[s1]+D[v2,x0]-Curl[v3]},
{-D[v1,x0]-Grad[s2]+Curl[v4], Curl[v1]+D[v4,x0]+Grad[s3]},
{Div[v3]+D[s4,x0], -D[v3,x0]-Curl[v2]-Grad[s4]},-D[s3,x0] - Div[v4]}

Nablamn[{s1_,{s2_,v1_},{v2_,v3_},{s3_,v4_},s4_}]:=
{D[s2,x0] + Div[v1], {D[s1,x0] - Div[v2], - Grad[s1]+D[v2,x0]-Curl[v3]},
{D[v1,x0]+Grad[s2]+Curl[v4], -Curl[v1]+D[v4,x0]+Grad[s3]},
{- Div[v3]+D[s4,x0], D[v3,x0]+Curl[v2]-Grad[s4]},D[s3,x0] + Div[v4]}

ExtDmp[{s1_,{s2_,v1_},{v2_,v3_},{s3_,v4_},s4_}]:=
{0, {-D[s1,x0] , Grad[s1]},
{-D[v1,x0]-Grad[s2], Curl[v1]},
{Div[v3], -D[v3,x0]-Curl[v2]}, -D[s3,x0] - Div[v4]}

Codif4[{s1_,{s2_,v1_},{v2_,v3_},{s3_,v4_},s4_}]:=
{D[s2,x0] + Div[v1], { - Div[v2], D[v2,x0]-Curl[v3]},
{Curl[v4], D[v4,x0]+Grad[s3]},
{D[s4,x0], -Grad[s4]},0}

ExtDmn[s_]:= -ExtDmp[s]


LapBelmp[s_]:= Nablamp[Nablamp[s]]

LapBelmn[s_]:= Nablamn[Nablamn[s]]

GeoDirac0mp[{s1_,s2_,s3_,s4_,s5_,s6_,s7_,s8_}]:=
            {s1,z4,{{s7,s8,s5},{s4,-s3,s2}},z4,s6}

GeoDirac2mp[{s1_,s2_,s3_,s4_,s5_,s6_,s7_,s8_}]:=
            {s2,z4,{{s5,-s6,-s7},{s2,s1,-s4}},z4,s8}

GeoDirac3mp[{s1_,s2_,s3_,s4_,s5_,s6_,s7_,s8_}]:=
            {s5,z4,{{s3,s4,s1},{s8,-s7,s6}},z4,s2}

GeoDirac1mp[{s1_,s2_,s3_,s4_,s5_,s6_,s7_,s8_}]:=
            {s7,z4,{{s1,-s2,-s3},{s6,s5,-s8}},z4,s4}

GeoDirac0mn[{s1_,s2_,s3_,s4_,s5_,s6_,s7_,s8_}]:=
            {s1,z4,{{-s7,-s8,-s5},{-s4,s3,-s2}},z4,s6}

GeoDirac2mn[{s1_,s2_,s3_,s4_,s5_,s6_,s7_,s8_}]:=
            {s2,z4,{{-s5,s6,s7},{-s2,-s1,s4}},z4,s8}

GeoDirac3mn[{s1_,s2_,s3_,s4_,s5_,s6_,s7_,s8_}]:=
            {s5,z4,{{-s3,-s4,-s1},{-s8,s7,-s6}},z4,s2}

GeoDirac1mn[{s1_,s2_,s3_,s4_,s5_,s6_,s7_,s8_}]:=
            {s7,z4,{{-s1,s2,s3},{-s6,-s5,s8}},z4,s4}

CSpinor[{z1_,z2_,z3_,z4_}]:=
{ComplexExpand[z1],ComplexExpand[z2],ComplexExpand[z3],ComplexExpand[z4]}

Dirac0mp[s_]:= h Cmp[Nablamp[s],e12] + (q/c) Cmp[VectorP,s] + m c Cmp[s,e0]

Dirac3mp[s_]:= h Cmp[Nablamp[s],e12] + (q/c) Cmp[VectorP,s] - m c Cmp[s,e0]

Dirac1mp[s_]:= h Cmp[Nablamp[s],e12] - (q/c) Cmp[VectorP,s] + m c Cmp[s,e0]

Dirac2mp[s_]:= -h Cmp[Nablamp[s],e12] + (q/c) Cmp[VectorP,s] + m c Cmp[s,e0]

Dirac0mn[s_]:= h Cmn[Nablamn[s],e12] + (q/c) Cmn[VectorP,s] + m c Cmn[s,e0]

Dirac3mn[s_]:= h Cmn[Nablamn[s],e12] + (q/c) Cmn[VectorP,s] - m c Cmn[s,e0]

Dirac1mn[s_]:= h Cmn[Nablamn[s],e12] - (q/c) Cmn[VectorP,s] + m c Cmn[s,e0]

Dirac2mn[s_]:= -h Cmn[Nablamn[s],e12] + (q/c) Cmn[VectorP,s] + m c Cmn[s,e0]

(* Alguns elements genŠrics en 4 dimensions  *)

elemor4={1,{2,{3,4,5}},{{6,7,8},{9,10,11}},{12,{13,14,15}},16};
elem4v={alfa,{v0,v},{A,C},{S0, S},lambda};
elem4w={beta,{w0,w},{B,D},{T0,T},mu};

(* Fi del paquet dels espais quadridimensionals  *)