On Parametrization of the Linear GL(4,C) and Unitary SU(4) Groups in Terms of Dirac Matrices
Parametrization of 4 × 4-matrices G of the complex linear group GL(4,C) in terms of four complex 4-vector parameters (k,m,n,l) is investigated. Additional restrictions separating some subgroups of GL(4,C) are given explicitly. In the given parametrization, the problem of inverting any 4 × 4 matrix G...
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irk-123456789-1489912019-02-20T01:25:29Z On Parametrization of the Linear GL(4,C) and Unitary SU(4) Groups in Terms of Dirac Matrices Red'kov, V.M. Bogush, A.A. Tokarevskaya, N.G. Parametrization of 4 × 4-matrices G of the complex linear group GL(4,C) in terms of four complex 4-vector parameters (k,m,n,l) is investigated. Additional restrictions separating some subgroups of GL(4,C) are given explicitly. In the given parametrization, the problem of inverting any 4 × 4 matrix G is solved. Expression for determinant of any matrix G is found: det G = F(k,m,n,l). Unitarity conditions G⁺ = G⁻¹ have been formulated in the form of non-linear cubic algebraic equations including complex conjugation. Several simplest solutions of these unitarity equations have been found: three 2-parametric subgroups G₁, G₂, G₃ - each of subgroups consists of two commuting Abelian unitary groups; 4-parametric unitary subgroup consisting of a product of a 3-parametric group isomorphic SU(2) and 1-parametric Abelian group. The Dirac basis of generators Λk, being of Gell-Mann type, substantially differs from the basis λi used in the literature on SU(4) group, formulas relating them are found - they permit to separate SU(3) subgroup in SU(4). Special way to list 15 Dirac generators of GL(4,C) can be used {Λk} = {αiÅβjÅ(αiVβj = KÅL ÅM )}, which permit to factorize SU(4) transformations according to S = eiaα eibβeikKeilLeimM, where two first factors commute with each other and are isomorphic to SU(2) group, the three last ones are 3-parametric groups, each of them consisting of three Abelian commuting unitary subgroups. Besides, the structure of fifteen Dirac matrices Λk permits to separate twenty 3-parametric subgroups in SU(4) isomorphic to SU(2); those subgroups might be used as bigger elementary blocks in constructing of a general transformation SU(4). It is shown how one can specify the present approach for the pseudounitary group SU(2,2) and SU(3,1). 2008 Article On Parametrization of the Linear GL(4,C) and Unitary SU(4) Groups in Terms of Dirac Matrices / V.M. Red'kov, A.A. Bogush, N.G. Tokarevskaya // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 75 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 20C35; 20G45; 22E70; 81R05 http://dspace.nbuv.gov.ua/handle/123456789/148991 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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Parametrization of 4 × 4-matrices G of the complex linear group GL(4,C) in terms of four complex 4-vector parameters (k,m,n,l) is investigated. Additional restrictions separating some subgroups of GL(4,C) are given explicitly. In the given parametrization, the problem of inverting any 4 × 4 matrix G is solved. Expression for determinant of any matrix G is found: det G = F(k,m,n,l). Unitarity conditions G⁺ = G⁻¹ have been formulated in the form of non-linear cubic algebraic equations including complex conjugation. Several simplest solutions of these unitarity equations have been found: three 2-parametric subgroups G₁, G₂, G₃ - each of subgroups consists of two commuting Abelian unitary groups; 4-parametric unitary subgroup consisting of a product of a 3-parametric group isomorphic SU(2) and 1-parametric Abelian group. The Dirac basis of generators Λk, being of Gell-Mann type, substantially differs from the basis λi used in the literature on SU(4) group, formulas relating them are found - they permit to separate SU(3) subgroup in SU(4). Special way to list 15 Dirac generators of GL(4,C) can be used {Λk} = {αiÅβjÅ(αiVβj = KÅL ÅM )}, which permit to factorize SU(4) transformations according to S = eiaα eibβeikKeilLeimM, where two first factors commute with each other and are isomorphic to SU(2) group, the three last ones are 3-parametric groups, each of them consisting of three Abelian commuting unitary subgroups. Besides, the structure of fifteen Dirac matrices Λk permits to separate twenty 3-parametric subgroups in SU(4) isomorphic to SU(2); those subgroups might be used as bigger elementary blocks in constructing of a general transformation SU(4). It is shown how one can specify the present approach for the pseudounitary group SU(2,2) and SU(3,1). |
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Red'kov, V.M. Bogush, A.A. Tokarevskaya, N.G. |
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Red'kov, V.M. Bogush, A.A. Tokarevskaya, N.G. On Parametrization of the Linear GL(4,C) and Unitary SU(4) Groups in Terms of Dirac Matrices Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Red'kov, V.M. Bogush, A.A. Tokarevskaya, N.G. |
| author_sort |
Red'kov, V.M. |
| title |
On Parametrization of the Linear GL(4,C) and Unitary SU(4) Groups in Terms of Dirac Matrices |
| title_short |
On Parametrization of the Linear GL(4,C) and Unitary SU(4) Groups in Terms of Dirac Matrices |
| title_full |
On Parametrization of the Linear GL(4,C) and Unitary SU(4) Groups in Terms of Dirac Matrices |
| title_fullStr |
On Parametrization of the Linear GL(4,C) and Unitary SU(4) Groups in Terms of Dirac Matrices |
| title_full_unstemmed |
On Parametrization of the Linear GL(4,C) and Unitary SU(4) Groups in Terms of Dirac Matrices |
| title_sort |
on parametrization of the linear gl(4,c) and unitary su(4) groups in terms of dirac matrices |
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Інститут математики НАН України |
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2008 |
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http://dspace.nbuv.gov.ua/handle/123456789/148991 |
| citation_txt |
On Parametrization of the Linear GL(4,C) and Unitary SU(4) Groups in Terms of Dirac Matrices / V.M. Red'kov, A.A. Bogush, N.G. Tokarevskaya // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 75 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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2025-07-12T20:49:53Z |
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| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 4 (2008), 021, 46 pages
On Parametrization of the Linear GL(4, C) and
Unitary SU(4) Groups in Terms of Dirac Matrices?
Victor M. RED’KOV, Andrei A. BOGUSH and Natalia G. TOKAREVSKAYA
B.I. Stepanov Institute of Physics, National Academy of Sciences of Belarus, Minsk, Belarus
E-mail: redkov@dragon.bas-net.by, bogush@dragon.bas-net.by, tokarev@dragon.bas-net.by
Received September 19, 2007, in final form January 24, 2008; Published online February 19, 2008
Original article is available at http://www.emis.de/journals/SIGMA/2008/021/
Abstract. Parametrization of 4 × 4-matrices G of the complex linear group GL(4, C) in
terms of four complex 4-vector parameters (k, m, n, l) is investigated. Additional restrictions
separating some subgroups of GL(4, C) are given explicitly. In the given parametrization, the
problem of inverting any 4×4 matrix G is solved. Expression for determinant of any matrix G
is found: det G = F (k, m, n, l). Unitarity conditions G+ = G−1 have been formulated in the
form of non-linear cubic algebraic equations including complex conjugation. Several simplest
solutions of these unitarity equations have been found: three 2-parametric subgroups G1,
G2, G3 – each of subgroups consists of two commuting Abelian unitary groups; 4-parametric
unitary subgroup consisting of a product of a 3-parametric group isomorphic SU(2) and 1-
parametric Abelian group. The Dirac basis of generators Λk, being of Gell-Mann type,
substantially differs from the basis λi used in the literature on SU(4) group, formulas
relating them are found – they permit to separate SU(3) subgroup in SU(4). Special way to
list 15 Dirac generators of GL(4, C) can be used {Λk} = {αi⊕βj⊕ (αiV βj = K⊕L⊕M)},
which permit to factorize SU(4) transformations according to S = ei~a~αei~b~βeikKeilLeimM ,
where two first factors commute with each other and are isomorphic to SU(2) group, the
three last ones are 3-parametric groups, each of them consisting of three Abelian commuting
unitary subgroups. Besides, the structure of fifteen Dirac matrices Λk permits to separate
twenty 3-parametric subgroups in SU(4) isomorphic to SU(2); those subgroups might be
used as bigger elementary blocks in constructing of a general transformation SU(4). It
is shown how one can specify the present approach for the pseudounitary group SU(2, 2)
and SU(3, 1).
Key words: Dirac matrices; linear group; unitary group; Gell-Mann basis; parametrization
2000 Mathematics Subject Classification: 20C35; 20G45; 22E70; 81R05
1 Introduction
The unitary groups play an important role in numerous research areas: quantum theory of
massless particles, cosmology models, quantum systems with dynamical symmetry, nano-scale
physics, numerical calculations concerning entanglement and other quantum information pa-
rameters, high-energy particle theory – let us just specify these several points:
• SU(2, 2) and conformal symmetry, massless particles [7, 19, 20, 39, 72];
• classical Yang–Mills equations and gauge fields [64];
• quantum computation and control, density matrices for entangled states [2, 31, 65];
• geometric phases and invariants for multi-level quantum systems [55];
?This paper is a contribution to the Proceedings of the Seventh International Conference “Symmetry in
Nonlinear Mathematical Physics” (June 24–30, 2007, Kyiv, Ukraine). The full collection is available at
http://www.emis.de/journals/SIGMA/symmetry2007.html
mailto:redkov@dragon.bas-net.by
mailto:bogush@dragon.bas-net.by
mailto:tokarev@dragon.bas-net.by
http://www.emis.de/journals/SIGMA/2008/021/
http://www.emis.de/journals/SIGMA/symmetry2007.html
2 V.M. Red’kov, A.A. Bogush and N.G. Tokarevskaya
• high-temperature superconductivity and antiferromagnets [36, 52];
• composite structure of quarks and leptons [67, 68, 51];
• SU(4) gauge models [73, 29];
• classification of hadrons and their interactions [30, 34, 28].
Because of so many applications in physics, various parametrizations for the group elements
of unitary group SU(4) and related to it deserve special attention. Our efforts will be given to
extending some classical technical approaches proving their effectiveness in simple cases of the
linear and unitary groups SL(2, C) and SU(2), so that we will work with objects known by every
physicist, such as Pauli and Dirac matrices. This paper, written for physicists, is self-contained
in that it does not require any previous knowledge of the subject nor any advanced mathematics.
Let us start with the known example of spinor covering for complex Lorentz group: consider
the 8-parametric 4× 4 matrices in the quasi diagonal form [18, 32, 45]
G =
∣∣∣∣∣ k0 + k~σ 0
0 m0 −m~σ
∣∣∣∣∣ .
The composition rules for parameters k = (k0,k) and m = (m0,m) are
k′′0 = k′0k0 + k′k, k′′ = k′0k + k′k0 + ik′ × k,
m′′
0 = m′
0m0 + m′m, m′′ = m′
0m + m′m0 − im′ ×m.
With two additional constraints on 8 quantities k2
0 − k2 = +1, m2
0−m2 = +1, we will arrive at
a definite way to parameterize a double (spinor) covering for complex Lorentz group SO(4, C).
At this, the problem of inverting of the G matrices with unit determinant det G is solved straight-
forwardly: G = G(k0,k,m0,m), G−1 = G(k0,−k,m0,−m). Transition from covering 4-spinor
transformations to 4-vector ones is performed through the known relationship GγaG−1 = γcL a
c
which determine 2 =⇒ 1 map from ±G to L.
There exists a direct connection between the above 4-dimensional vector parametrization
of the spinor group G(ka,ma) and the Fedorov’s parametrization [32] of the group of complex
orthogonal Lorentz transformations in terms of 3-dimensional vectors Q = k/k0, M = m/m0,
with the simple composition rules for vector parameters
Q′′ =
Q + Q′ + iQ′ ×Q
1 + Q′Q
, M ′′ =
M + M ′ − iM ′ ×M
1 + M ′M
.
Evidently, the pair (Q,M) provides us with possibility to parameterize correctly orthogonal
matrices only. Instead, the (ka,ma) represent correct parameters for the spinor covering group.
When we are interested only in local properties of the spinor representations, no substantial
differences between orthogonal groups and their spinor coverings exist. However, in opposite
cases global difference between orthogonal and spinor groups may be very substantial as well as
correct parametrization of them.
Restrictions specifying the spinor coverings for orthogonal subgroups are well known [32].
In particular, restriction to real Lorentz group O(3, 1) is achieved by imposing one condition
(including complex conjugation) (k, m) =⇒ (k, k∗). The case of real orthogonal group O(4) is
achieved by a formal change (transition to real parameters) (k0,k) =⇒ (k0, ik), (m0,m) =⇒
(m0, im), and the real orthogonal group O(2, 2) corresponds to transition to real parameters
according to (k0, k1, k2, k3) =⇒ (k0, k1, k2, ik3), (m0,m1,m2,m3) =⇒ (m0,m1,m2, im3).
To parameterize 4-spinor and 4-vector transformations of the complex Lorentz group one
may use curvilinear coordinates. The simplest and widely used ones are Euler’s complex angles
On Parametrization of GL(4, C) and SU(4) 3
(see [32] and references in [18]). In general, on the basis of the analysis given by Olevskiy [58]
about coordinates in the real Lobachevski space, one can propose 34 different complex coordinate
systems appropriate to parameterize the complex Lorentz group and its double covering.
A particular, Euler angle parametrization is closely connected with cylindrical coordinates
on the complex 3-sphere, one of 34 possible coordinates. Such complex cylindrical coordinates
can be introduced by the following relations [18]:
k0 = cos ρ cos z, k3 = i cos ρ sin z, k1 = i sin ρ cos φ, k2 = i sin ρ sinφ,
m0 = cos R cos Z, m3 = i cos R sinZ, m1 = i sinRΦ, m2 = i sinR sinΦ.
Here 6 complex variables are independent, (ρ, z, φ), (R,Z, Φ), additional restrictions are satisfied
identically by definition. Instead of cylindrical coordinates in (ρ, z, φ) and (R,Z, Φ) one can
introduce Euler’s complex variables (α, β, γ) and (A,B, Γ) through the simple linear formulas:
α = φ + z, β = 2ρ, γ = φ− z, A = Φ + Z, B = 2R, Γ = Φ− Z.
Euler’s angles (α, β, γ) and (A,B, Γ) are referred to ka,ma-parameters by the formulas (see
in [32])
cos β = k2
0 − k2
3 + k2
1 + k2
2, sin β = 2
√
k2
0 − k2
3
√
−k2
1 − k2
2,
cos α =
−ik0k1 + k2k3√
k2
0 − k2
3
√
−k2
1 − k2
2
, sinα =
−ik0k2 − k1k3√
k2
0 − k2
3
√
−k2
1 − k2
2
,
cos γ =
−ik0k1 − k2k3√
k2
0 − k2
3
√
−k2
1 − k2
2
, sin γ =
−ik0k2 + k1k3√
k2
0 − k2
3
√
−k2
1 − k2
2
,
cos B = m2
0 −m2
3 + m2
1 + m2
2, sinB = 2
√
m2
0 −m2
3
√
−m2
1 −m2
2,
cos A =
+im0m1 + m2m3√
m2
0 −m2
3
√
−m2
1 −m2
2
, sinA =
+im0m2 −m1m3√
m2
0 −m2
3
√
−m2
1 −m2
2
,
cos Γ =
+im0m1 −m2m3√
m2
0 −m2
3
√
−m2
1 −m2
2
, sin Γ =
+im0m2 + m1m3√
m2
0 −m2
3
√
−m2
1 −m2
2
.
Complex Euler’s angles as parameters for complex Lorentz group SO(4, C) have a distin-
guished feature: 2-spinor constituents are factorized into three elementary Euler’s transforms
(σi stands for the known Pauli matrices):
B(k) = e−iσ3α/2eiσ1β/2e+iσ3γ/2 ∈ SL(2, C),
B(m̄) = e−iσ3Γ/2eiσ1B/2e+iσ3A/2 ∈ SL(2, C)′.
The main question is how to extend possible parameterizations of small orthogonal group
SO(4, C) and its double covering to bigger orthogonal and unitary groups1. To be concrete we
are going to focus attention mainly on the group SU(4) and its counterparts SU(2, 2), SU(3, 1).
There exist many publications on the subject, a great deal of facts are known – in the following
we will be turning to them. A good classification of different approaches in parameterizing finite
transformations of SU(4) was done in the recent paper by A. Gsponer [35]. Recalling it, we will
try to cite publications in appropriate places though many of them should be placed in several
different subclasses – it is natural because all approaches are closely connected to each other.
1In this subject, especially concerned with generalized Euler angles, we have found out much from Murnaghan’s
book [56].
4 V.M. Red’kov, A.A. Bogush and N.G. Tokarevskaya
• Canonical form [53, 54, 56, 57, 60, 61, 75]. They use explicitly the full set of the
Lie generators2 so that the group element is expressed as the exponential of the linear
combination
G = expn[i(a1λ1 + · · ·+ anλN )]
the infinite series of terms implied be exp – symbol is usually very difficult to be summed
in closed form – though there exists many interesting examples of those:
• Non-canonical forms [3, 4, 8, 9, 11, 35, 38, 40, 42, 57, 59, 61, 63, 74]. As a consequence
of the Baker–Campbell–Hausdorff theorem [1, 26, 37] it is possible to break-down the
canonical form into a product
G = expn(1) × · · · × expn(k) , n(1) + · · ·+ n(k) = N.
with the hope that expn(i) could be summed in closed form and also that these factors have
simple properties. This possibility for the groups SU(4) and SU(2, 2) will be discussed in
more detail in sections below.
• Product form [12, 13, 14, 16, 27, 38, 56, 57]. An extreme non-canonical form is to
factorize the general exponential into a product of n simplest 1-parametric exponentials
G = exp[ia1λ1]× · · · × exp[iaNλN ].
• Basic elements (the main approach in the present treatment) [8, 9, 35, 40, 44, 45, 46,
47, 74]. This way is to expand the elements of the group (matrices or quaternions) into
a sum over basis elements and to work with a linear decomposition of the matrices over
basic ones:
G′ = x′nλm, G = xnλm, λ0 = I, k ∈ {0, 1, . . . , N},
G′′ = G′G, x′′kλk = x′mλmxnλn = x′mxnλmλn, (1.1)
as by definition the relationships λmλn = emnkλk must hold, the group multiplication rule
for parameters xk looks
x′′k = emnkx
′
mxn. (1.2)
The main claim is that the all properties of any matrix group are straightforwardly deter-
mined by the bilinear function, the latter is described by structure constants emnk entering
the multiplication rule λmλn = emnkλk.
• Hamilton–Cayley form [8, 9, 11, 12, 13, 14, 15, 16, 17, 33, 63]. It is possible to expand
the elements of the group into a power series of linear combination of generators:
λ(a) = i(a1λ1 + · · ·+ aNλN ),
because of Hamilton–Cayley theorem this series has three terms for SU(3) and four terms
for SU(4):
SU(3), G(a) = e0(a)I + e1(a)λ(a) + e2(a)λ2(a),
SU(4), G(a) = e0(a)I + e1(a)λ(a) + e2(a)λ2(a) + e3(a)λ3(a).
2In the paper we will designate generators in Dirac basis by Λi whereas another set of generators mainly used
in the literature will be referred as λi.
On Parametrization of GL(4, C) and SU(4) 5
• Euler-angles representations [10, 21, 22, 23, 24, 25, 35, 42, 56, 57, 69]. In Euler-angles
representations only a sub-set {λn} ⊂ {λN} of the Lie generators are sufficient to produce
the whole set (for SU(N) we need only 2(N − 1) generators). In that sense all other way
to obtain the whole set of elements are not minimal.
In our opinion, we should search the most simplicity in mathematical sense while work-
ing with basic elements λk and the structure constants determining the group multiplication
rule (1.1), (1.2).
The material of this paper is arranged as follows.
In Section 2 an arbitrary 4 × 4 matrix G ∈ GL(4, C) is decomposed into sixteen Dirac
matrices3
G = AI + iBγ5 + iAlγ
l + Blγ
lγ5 + Fmnσmn =
∣∣∣∣∣ k0 + k~σ n0 − n~σ
−l0 − l~σ m0 −m~σ
∣∣∣∣∣ , (1.3)
for definiteness we will use the Weyl spinor basis; four 4-dimensional vectors (k, m, l, n) are
definite linear combinations of A, B, Al, Bl, Fmn – see (2.4). In such parameters (2.3), the
group multiplication law G′′ = G′G is found in explicit form.
Then we turn to the following problem: at given G = G(k, m, n, l) one should find parameters
of the inverse matrix: G−1 = G(k′,m′, n′, l′) – expressions for (k′,m′, n′, l′) have been found
explicitly (for details of calculation see [62]). Also, several equivalent expressions for determinant
det G have been obtained, which is essential when going to special groups SL(4, C) and its
subgroups.
In Section 3, with the help of the expression for the inverse matrix G−1(k′,m′, l′, n′) we
begin to consider the unitary group SU(4). To this end, one should specify the requirement of
unitarity G+ = G−1 to the above vector parametrization – so that unitarity conditions are given
as non-linear cubic algebraic equations for parameters (k, m, l, n) including complex conjugation.
In Section 4 we have constructed three 2-parametric solutions of the produced equations of
unitarity4, these subgroups G1, G2, G3 consist of two commuting Abelian unitary subgroups.
In Section 5 we have constructed a 4-parametric solution5 – it may be factorized into two
commuting unitary factors: G = G0 ⊗ SU(2) – see (5.15).
The task of complete solving of the unitarity conditions seems to be rather complicated. In
remaining part of the present paper we describe some relations of the above treatment to other
considerations of the problem in the literature. We hope that the full general solution of the
unitary equations obtained can be constructed on the way of combining different techniques
used in the theory of the unitary group SU(4) and it will be considered elsewhere.
We turn again to the explicit form of the Dirac basis and note that all 15 matrices are of
Gell-Mann type: they have a zero-trace, they are Hermitian, besides their squares are unite:
SpΛ = 0, (Λ)2 = I, (Λ)+ = Λ, Λ ∈ {Λk, k = 1, . . . , 15}.
Exponential function of any of them equals to
Uj = eiajΛj = cos aj + i sin ajΛj , det eiajΛj = +1, U+
j = U−1
j , ai ∈ R.
Evidently, multiplying such 15 elementary unitary matrices (at real parameters xi) gives again
an unitary matrix
U = eia1Λ1eia2Λ2 · · · eia14Λ14eial5Λl5 , U+ = e−ialΛle−iakΛk · · · e−iajΛje−iaiΛi .
3That Dirac matrices-based approach was widely used in physical context (see [5, 8, 9, 6, 45, 48, 49, 50, 66]
and especially [40]).
4At this, the unitarity equations may be considered as special eigenvalue problems in 2-dimensional space.
5The problem again is reduced to solving of a special eigenvalue problem in 2-dimensional space.
6 V.M. Red’kov, A.A. Bogush and N.G. Tokarevskaya
At this there arises one special possibility to determine extended Euler angles a1, . . . , a15. For
the group SU(4) the Euler parametrization of that type was found in [69]. A method to solve
the problem in [69] was based on the use yet known Euler parametrization for SU(3) – the latter
problem was solved in [25]. Extension to SU(N) group was done in [70, 71]. Evident advantage
of the Euler angles approach is its simplicity, and evident defect consists in the following: we
do not know any simple group multiplication rule for these angles – even the known solution
for SU(2) is too complicated and cannot be used effectively in calculation.
In Section 6 the main question is how in Dirac parametrization one can distinguish SU(3),
the subgroup in SU(4). In this connection, it should be noted that the basis λi used in [25]
substantially differs from the above Dirac basis Λi – this peculiarity is closely connected with
distinguishing the SU(3) in SU(4). In order to have possibility to compare two approaches we
need exact connection between λi and Λi – we have found required formulas6. The separation
of SL(3, C) in SL(4, C) is given explicitly, at this 3× 3 matrix group is described with the help
of 4× 4 matrices7. The group law for parameters of SL(3, C) is specified.
In Section 7 one different way to list 15 generators of GL(4, C) is examined8
α1 = γ0γ2, α2 = iγ0γ5, α3 = γ5γ2, β1 = iγ3γ1, β2 = iγ3, β3 = iγ1,
these two set commute with each others αjβk = βkαj , and their multiplications provides us with
9 remaining basis elements of fifteen:
A1 = α1β1, B1 = α1β2, C1 = α1β3,
A2 = α2β1, B2 = α2β2, C2 = α2β3,
A3 = α3β1, B3 = α3β2, C3 = α3β3.
We turn to the rule of multiplying 15 generators αi, βi, Ai, Bi, Ci and derive its explicit form
(see (7.3)).
Section 8 adds some facts to a factorized structure of SU(4). To this end, between 9 generators
we distinguish three sets of commuting ones
K = {A1, B2, C3}, L = {C1, A2, B3}, M = {B1, C2, A3},
an arbitrary element from GL(4, C) can be factorized as follows9
S = ei~a~αei~b~βeikKeilLeimM , (1.4)
where K, L, M are 3-parametric groups, each of them consists of three Abelian commuting
unitary subgroups10. On the basis of 15 matrices one can easily see 20 ways to separate SU(2)
subgroups, which might be used as bigger elementary blocks in constructing a general transfor-
mation11.
In Sections 9 and 10 we specify our approach for pseudounitary groups SU(2, 2) and SU(3, 1)
respectively. All generators Λ′
k of these groups can readily be constructed on the basis of the
known Dirac generators of SU(4) (see (9.1)).
6This problem evidently is related to the task of distinguishing GL(3, C) in GL(4, C) as well.
7Interesting arguments related to this point but in the quaternion approach are given in [35].
8Such a possibility is well-known – see [40]; our approach looks simpler and more symmetrical because we use
the Weyl basis for Dirac matrices instead of the standard one as in [40].
9These facts were described in main parts in [40].
10Note that existence of three Abelian commuting unitary subgroups was shown in [40] as well.
11This possibility was studied partly in [13, 14] on the basis of the Hamilton–Cayley approach.
On Parametrization of GL(4, C) and SU(4) 7
2 On parameters of inverse transformations G−1
Arbitrary 4×4 matrix G ∈ GL(4, C) can be decomposed in terms of 16 Dirac matrices (such an
approach to the group L(4, C) was discussed and partly developed in [5, 8, 9, 6, 45, 48, 49, 50, 66]
and especially in [40]):
G = AI + iBγ5 + iAlγ
l + Blγ
lγ5 + Fmnσmn, (2.1)
where
γaγb + γbγa = 2gab, γ5 = −iγ0γ1γ2γ3,
σab = 1
4(γaγb − γbγa), gab = diag(+1,−1,−1,−1).
Taking 16 coefficients A, B, Al, Bl, Fmn as parameters in the group G = G(A,B, Al, Bl, Fmn)
one can establish the corresponding multiplication law for these parameters:
G′ = A′I + iB′γ5 + iA′
lγ
l + B′
lγ
lγ5 + F ′
mnσmn,
G = AI + iBγ5 + iAlγ
l + Blγ
lγ5 + Fmnσmn,
G′′ = G′G = A′′I + iB′′γ5 + iA′′
l γ
l + B′′
l γlγ5 + F ′′
mnσmn,
where
A′′ = A′A−B′B −A′
lA
l −B′
lB
l − 1
2F ′
klF
kl,
B′′ = A′B + B′A + A′
lB
l −B′
lA
l + 1
4F ′
mnFcdε
mncd,
A′′
l = A′Al −B′Bl + A′
lA + B′
lB + A′kFkl
+ F ′
lkA
k + 1
2B′
kFmnε kmn
l + 1
2F ′
mnBkε
mnk
l ,
B′′
l = A′Bl + B′Al −A′
lB + B′
lA + B′kFkl
+ F ′
lkB
k + 1
2A′
kFmnεkmn
l + 1
2F ′
mnAkε
mnk
l,
F ′′
mn = A′Fmn + F ′
mnA− (A′
mAn −A′
nAm)− (B′
mBn −B′
nBm) (2.2)
+ A′
lBkε
lkmn −B′
lAkε
lkmn + 1
2B′Fklε
kl
mn + 1
2F ′
klBεkl
mn + (F ′
mkF
k
n − F ′
nkF
k
m).
The latter formulas are correct in any basis for Dirac matrices. Below we will use mainly Weyl
spinor basis:
γa =
∣∣∣∣ 0 σ̄a
σa 0
∣∣∣∣ , σa = (I, σj), σ̄a = (I,−σj), γ5 =
∣∣∣∣ −I 0
0 +I
∣∣∣∣ .
With this choice, let us make 3 + 1-splitting in all the formulas:
G ∈ GL(4, C), G =
∣∣∣∣∣ k0 + k~σ n0 − n~σ
−l0 − l~σ m0 −m~σ
∣∣∣∣∣ , (2.3)
where complex 4-vector parameters (k, l,m, n) are defined by [18]:
k0 = A− iB, kj = aj − ibj , m0 = A + iB, mj = aj + ibj ,
l0 = B0 − iA0, lj = Bj − iAj , n0 = B0 + iA0, nj = Bj + iAj . (2.4)
For such parameters (2.3), the composition rule (2.2) will look as follows:
(k′′,m′′;n′′, l′′) = (k′,m′;n′, l′)(k, m;n, l),
8 V.M. Red’kov, A.A. Bogush and N.G. Tokarevskaya
k′′0 = k′0k0 + k′k − n′0l0 + n′l,
k′′ = k′0k + k′k0 + ik′ × k − n′0l + n′l0 + in′ × l,
m′′
0 = m′
0m0 + m′m− l′0n0 + l′n,
m′′ = m′
0m + m′m0 − im′ ×m− l′0n + l′n0 − il′ × n,
n′′0 = k′0n0 − k′n + n′0m0 + n′m,
n′′ = k′0n− k′n0 + ik′ × n + n′0m + n′m0 − in′ ×m,
l′′0 = l′0k0 + l′k + m′
0l0 −m′l,
l′′ = l′0k + l′k0 + il′ × k + m′
0l−m′l0 − im′ × l. (2.5)
Now let us turn to the following problem: with given G = G(k, m, n, l) one should find
parameters of the inverse matrix: G−1 = G(k′,m′, n′, l′). In other words, starting from
G(k,m, n, l) =
∣∣∣∣∣∣∣∣
+(k0 + k3) +(k1 − ik2) +(n0 − n3) −(n1 − in2)
+(k1 + ik2) +(k0 − k3) −(n1 + in2) +(n0 + n3)
−(l0 + l3) −(l1 − il2) +(m0 −m3) −(m1 − im2)
−(l1 + il2) −(l0 − l3) −(m1 + im2) +(m0 + m3)
∣∣∣∣∣∣∣∣ , (2.6)
one should calculate parameters of the inverse matrix G−1. The problem turns to be rather
complicated12, the final result is (D = detG, (mn) ≡ m0n0 −mn, and so on)
k′0 = D−1[k0(mm) + m0(ln) + l0(nm)− n0(lm) + il(m× n)],
k′ = D−1[−k(mm)−m(ln)− l(nm) + n(lm) + 2l× (n×m)
+ im0(n× l) + il0(n×m) + in0(l×m)],
m′
0 = D−1[k0(ln) + m0(kk)− l0(kn) + n0lk) + in(l× k)],
m′ = D−1[−k(ln)−m(kk) + l(kn)− n(kl) + 2n× (l× k)
+ in0(k × l) + il0(k × n) + ik0(n× l)],
l′0 = D−1[+k0(ml)−m0(kl)− l0(km)− n0(ll) + im(l× k)],
l′ = D−1[+k(ml)−m(kl)− l(km)− n(ll) + 2m× (k × l)
+ im0(l× k) + ik0(l×m) + il0(m× k)],
n′0 = D−1[−k0(nm) + m0(kn)− l0(nn)− n0(km) + ik(m× n)],
n′ = D−1[−k(nm) + m(kn)− l(nn)− n(km) + 2k × (m× n)
+ ik0(m× n) + im0(k × n) + in0(m× k)]. (2.7)
Substituting equations (2.7) into equation G−1G = I one arrives at
D = k′′0 = k′0k0 + k′k − n′0l0 + n′l,
0 = k′′ = k′0k + k′k0 + ik′ × k − n′0l + n′l0 + in′ × l,
D = m′′
0 = m′
0m0 + m′m− l′0n0 + l′n,
0 = m′′ = m′
0m + m′m0 − im′ ×m− l′0n + l′n0 − il′ × n,
0 = n′′0 = k′0n0 − k′n + n′0m0 + n′m,
0 = n′′ = k′0n− k′n0 + ik′ × n + n′0m + n′m0 − in′ ×m,
0 = l′′0 = l′0k0 + l′k + m′
0l0 −m′l,
0 = l′′ = l′0k + l′k0 + il′ × k + m′
0l−m′l0 − im′ × l.
12For more details see [62]; also see a preceding paper [41].
On Parametrization of GL(4, C) and SU(4) 9
After calculation, one can prove these identities and find the determinant:
D = detG(k,m, n, l) = (kk)(mm) + (ll)(nn) + 2(mk)(ln) + 2(lk)(nm)− 2(nk)(lm)
+ 2i[k0l(m× n) + m0k(n× l) + l0k(n×m) + n0l(m× k)]
+ 4(kn)(ml)− 4(km)(nl). (2.8)
Let us specify several more simple subgroups.
Case A
Let 0-components k0, m0, l0, n0 be real-valued, and 3-vectors k, m, l, n be imaginary. Per-
forming in (2.5) the formal change (new vectors are real-valued)
k =⇒ ik, m =⇒ im, l =⇒ il, n =⇒ in,
G =
∣∣∣∣∣ k0 + ik~σ n0 − in~σ
−l0 − il~σ m0 − im~σ
∣∣∣∣∣ , (2.9)
then the multiplication rules (2.5) for sixteen real variables look as follows
k′′0 = k′0k0 − k′k − n′0l0 − n′l,
k′′ = k′0k + k′k0 − k′ × k − n′0l + n′l0 − n′ × l,
m′′
0 = m′
0m0 −m′m− l′0n0 − l′n,
m′′ = m′
0m + m′m0 + m′ ×m− l′0n + l′n0 + l′ × n,
n′′0 = k′0n0 + k′n + n′0m0 − n′m,
n′′ = k′0n− k′n0 − k′ × n + n′0m + n′m0 + n′ ×m,
l′′0 = l′0k0 − l′k + m′
0l0 + m′l,
l′′ = l′0k + l′k0 − l′ × k + m′
0l−m′l0 + m′ × l.
Correspondingly, expression for determinant (2.8) becomes
D = [kk][mm] + [ll][nn] + 2[mk][ln] + 2[lk][nm]− 2[nk][lm]
+ 2[k0l(m× n) + m0k(n× l) + l0k(n×m) + n0l(m× k)]
+ 4(kn)(ml)− 4(km)(nl),
where the notation is used: [ab] = a0b0 + ab.
Case B
Equations (2.5) permit the following restrictions:
ma = k∗a, la = n∗a,
and become
k′′0 = k′0k0 + k′k − n′0n
∗
0 + n′n∗,
k′′ = k′0k + k′k0 + ik′ × k − n′0n
∗ + n′n∗0 + in′ × n∗,
n′′0 = k′0n0 − k′n + n′0k
∗
0 + n′k∗,
n′′ = k′0n− k′n0 + ik′ × n + n′0k
∗ + n′k∗0 − in′ × k∗.
Determinant D is given by
D = (kk)(kk)∗ + (nn)∗(nn) + 2(k∗k)(n∗n) + 2(n∗k)(nk∗)− 2(nk)(nk)∗
+ 2i[k0k
∗(n× n∗)− k∗0k(n∗ × n) + n∗0n(k × k∗)− n0n
∗(k∗ × k)]
+ 4(kn)(k∗n∗)− 4(kk∗)(nn∗).
10 V.M. Red’kov, A.A. Bogush and N.G. Tokarevskaya
Case C
In (2.9) one can impose additional restrictions
m0 = k0, l0 = n0, m = −k, l = −n; (2.10)
at this G(k0,k, n0,n) looks
G =
∣∣∣∣∣ (k0 + ik~σ) (n0 − in~σ)
−(n0 − in~σ) (k0 + ik~σ)
∣∣∣∣∣ ;
and the composition rule is
k′′0 = k′0k0 − k′k − n′0n0 + n′n,
k′′ = k′0k + k′k0 − k′ × k + n′0n + n′n0 + n′ × n,
n′′0 = k′0n0 + k′n + n′0k0 + n′k,
n′′ = k′0n− k′n0 − k′ × n− n′0k + n′k0 − n′ × k.
Determinant equals to
det G = [kk][kk] + [nn][nn] + 2(kk)(nn) + 2(nk)(nk)− 2[nk][nk])
+ 4(kn)(kn)− 4(kk)(nn).
Case D
There exists one other subgroup defined by
na = 0, la = 0, G =
∣∣∣∣∣ (k0 + k~σ) 0
0 (m0 −m~σ)
∣∣∣∣∣ ,
the composition law (2.5) becomes simpler
k′′0 = k′0k0 + k′k, k′′ = k′0k + k′k0 + ik′ × k,
m′′
0 = m′
0m0 + m′m,m′′ = m′
0m + m′m0 − im′ ×m,
as well as the determinant D
det G = (kk)(mm).
If one additionally imposes two requirements (kk) = +1, (mm) = +1, the Case D describes
spinor covering for special complex rotation group SO(4, C); this most simple case was consid-
ered in detail in [18].
It should be noted that the above general expression (2.8) for determinant can be transformed
to a shorter form
det G = (kk)(mm) + (nn)(ll) + 2[kn][ml]
− 2(k0n + n0k − ik × n)(m0l + l0m + im× l),
which for the three Cases A, B, C becomes yet simpler:
(A) : det G = [kk][mm] + [nn][ll] + 2(kn)(ml)
+ 2(k0n + n0k + k × n)(m0l + l0m−m× l),
(B) : det G = (kk)(k∗k∗) + (nn)(n∗n∗) + 2[kn][k∗n∗]
− 2(k0n + n0k − ik × n)(k∗0n
∗ + n∗0k
∗ + ik∗ × n∗),
(C) : det G = [kk]2 + [nn]2 + 2(kn)2 − 2(k0n + n0k + k × n)2.
On Parametrization of GL(4, C) and SU(4) 11
3 Unitarity condition
Now let us turn to consideration of the unitary group SU(4). One should specify the requirement
of unitarity G+ = G−1 to the above vector parametrization. Taking into account the formulas
G+ =
∣∣∣∣∣ k∗0 + k∗~σ −l∗0 − l∗~σ
n∗0 − n∗~σ m∗
0 −m∗~σ
∣∣∣∣∣ , G−1 =
∣∣∣∣∣ k′0 + k′~σ n′0 − n′~σ
−l′0 − l′~σ m′
0 −m′~σ
∣∣∣∣∣ , (3.1)
which can be represented differently
G+ = G(k∗0,k
∗;m∗
0,m
∗;−l∗0, l
∗,−n∗0,n
∗), G−1 = G(k′0,k
′;m′
0,m
′;n′0,n
′, l′0, l
′),
we arrive at
k∗0 = k′0, k∗ = k′, m∗
0 = m′
0, m∗ = m′,
−l∗0 = n′0, l∗ = n′, −n∗0 = l′0, n∗ = l′. (3.2)
With the use of expressions for parameters of the inverse matrix with additional restriction
det G = +1 equations (3.2) can be rewritten as
k∗0 = +k0(mm) + m0(ln) + l0(nm)− n0(lm) + il(m× n),
m∗
0 = +m0(kk) + k0(nl) + n0(lk)− l0(nk)− in(k × l),
k∗ = −k(mm)−m(ln)− l(nm) + n(lm) + 2l× (n×m)
+ im0(n× l) + il0(n×m) + in0(l×m),
m∗ = −m(kk)− k(nl)− n(lk) + l(nk) + 2n× (l× k)
− ik0(l× n)− in0(l× k)− il0(n× k),
l∗0 = +k0(nm)−m0(kn) + l0(nn) + n0(km) + ik(n×m),
n∗0 = +m0(lk)− k0(ml) + n0(ll) + l0(mk)− im(l× k),
l∗ = −k(nm) + m(kn)− l(nn)− n(km) + 2k × (m× n)
+ ik0(m× n) + im0(k × n) + in0(m× k),
n∗ = −m(kl) + k(ml)− n(ll)− l(mk) + 2m× (k × l)− im0(k × l)
− ik0(m× l)− il0(k ×m). (3.3)
Thus, the known form for parameters of the inverse matrix G−1 makes possible to write easily
relations (3.3) representing the unitarity condition for group SU(4). Here there are 16 equations
for 16 variables; evidently, not all of them are independent.
Let us write down several simpler cases.
Case A
With formal change13
k =⇒ ik, m =⇒ im, l =⇒ il, n =⇒ in, (3.4)
equations (3.3) give
k0 = +k0[mm] + m0[ln] + l0[nm]− n0[lm] + l(m× n),
m0 = +m0[kk] + k0[nl] + n0[lk]− l0[nk]− n(k × l),
13Let 0-components k0, m0, l0, n0 be real-valued, and 3-vectors k, m, l, n be imaginary.
12 V.M. Red’kov, A.A. Bogush and N.G. Tokarevskaya
k = k[mm] + m[ln] + l[nm]− n[lm] + 2l× (n×m)
+ m0(n× l) + l0(n×m) + n0(l×m),
m = +m[kk] + k[nl] + n[lk]− l[nk] + 2n× (l× k)
− k0(l× n)− n0(l× k)− l0(n× k),
l0 = +k0[nm]−m0[kn] + l0[nn] + n0[km] + k(n×m),
n0 = +m0[lk]− k0[ml] + n0[ll] + l0[mk]−m(l× k),
l = +k[nm]−m[kn] + l[nn] + n[km] + 2k × (m× n)
+ k0(m× n) + m0(k × n) + n0(m× k),
n = +m[kl]− k[ml] + n[ll] + l[mk] + 2m× (k × l)
−m0(k × l)− k0(m× l)− l0(k ×m).
Here there are 16 equations for 16 real-valued variables.
Case B
Let
m0 = k∗0, m = k∗, l0 = n∗0, l = n∗,
k0 = m∗
0, k = m∗, n0 = l∗0, n = l∗,
or symbolically m = k∗, l = n∗. The unitarity relations become
k∗0 = +k0(k∗k∗) + k∗0(n
∗n) + n∗0(nk∗)− n0(n∗k∗) + in∗(k∗ × n),
k∗ = −k(k∗k∗)− k∗(n∗n)− n∗(nk∗) + n(n∗k∗)
+ 2n∗ × (n× k∗) + ik∗0(n× n∗) + in∗0(n× k∗) + in0(l×m),
n∗0 = +k∗0(n
∗k)− k0(k∗n∗) + n0(n∗n∗) + n∗0(k
∗k)− ik∗(n∗ × k),
n∗ = −k∗(kn∗) + k(k∗n∗)− n(n∗n∗)− n∗(k∗k)
+ 2k∗ × (k × n∗)− ik∗0(k × n∗)− ik0(k∗ × n∗)− in∗0(k × k∗),
and 8 conjugated ones
k0 = +k∗0(kk) + k0(nn∗) + n0(n∗k)− n∗0(nk)− in(k × n∗),
k = −k∗(kk)− k(nn∗)− n(n∗k) + n∗(nk)
+ 2n× (n∗ × k)− ik0(n∗ × n)− in0(n∗ × k)− in∗0(n× k),
n0 = +k0(nk∗)− k∗0(kn) + n∗0(nn) + n0(kk∗) + ik(n× k∗),
n = −k(nk∗) + k∗(kn)− n∗(nn)− n(kk∗)
+ 2k × (k∗ × n) + ik0(k∗ × n) + ik∗0(k × n) + in0(k∗ × k).
It may be noted that latter relations are greatly simplified when n = 0, or when k = 0. Firstly,
let us consider the case n = 0:
k∗0 = +k0(k∗k∗), k∗ = −k(k∗k∗).
Taking in mind the identity
det G = (kk)(kk)∗ = +1 =⇒ (kk) = +1, (kk)∗ = +1,
we arrive at k∗0 = +k0, k∗ = −k. It has sense to introduce the real-valued vector ca:
k∗0 = +k0 = c0, k∗ = −k : k = ic,
On Parametrization of GL(4, C) and SU(4) 13
then matrix G is
G(k, m = k∗, 0, 0) =
∣∣∣∣ c0 + ic~σ 0
0 c0 − ic~σ
∣∣∣∣ ∼ SU(2).
Another possibility is realized when k = 0:
n∗0 = +n0(nn)∗, n∗ = −n(nn)∗.
With the use of identity
det G = (nn)(nn)∗ = +1 =⇒ (nn) = +1, (nn)∗ = +1,
we get
n∗0 = +n0 = c0, n∗ = −n, n ≡ ic,
Corresponding matrices G(0, 0, n, l = n∗) make up a special set of unitary matrices
G =
∣∣∣∣ 0 c0 − ic~σ
−(c0 + ic~σ) 0
∣∣∣∣ , G+ =
∣∣∣∣ 0 −(c0 − ic~σ)
(c0 + ic~σ) 0
∣∣∣∣ . (3.5)
However, it must be noted that these matrices (3.5) do not provide us with any subgroup because
G2 = −I.
Case C
Now in equations (3.4) one should take
m0 = k0, l0 = n0, m = −k, l = −n,
then
k0 = +k0[kk] + k0(nn) + n0(nk)− n0[nk],
k = k[kk]− k(nn)− n(nk)− n[nk] + 2n× (n× k),
n0 = +k0(nk)− k0[kn] + n0[nn] + n0(kk),
n = −k(kn)− k[kn] + n[nn]− n(kk) + 2k × (k × n). (3.6)
4 2-parametric subgroups in SU(4)
To be certain in correctness of the produced equations of unitarity, one should try to solve them
at least in several most simple particular cases. For instance, let us turn to the Case C and
specify equations (3.6) for a subgroup arising when k = (k0, k1, 0, 0) and n = (n0, n1, 0, 0):
k0 = +k0[kk] + k0(nn) + n0(nk)− n0[nk],
k1 = +k1[kk]− k1(nn)− n1(nk)− n1[nk],
n0 = +k0(nk)− k0[kn] + n0[nn] + n0(kk),
n1 = −k1(kn)− k1[kn] + n1[nn]− n1(kk), (4.1)
they are four non-linear equations for four real variables. It may be noted that equations (4.1)
can be regarded as two eigenvalue problems in two dimensional space (with eigenvalue +1):∣∣∣∣ (k2
0 + n2
0)− 1 + (k2
1 − n2
1) −2n1k1
−2n1k1 (k2
0 + n2
0)− 1− (k2
1 − n2
1)
∣∣∣∣ ∣∣∣∣ k0
n0
∣∣∣∣ =
∣∣∣∣ 0
0
∣∣∣∣ ,
14 V.M. Red’kov, A.A. Bogush and N.G. Tokarevskaya∣∣∣∣ (k2
1 + n2
1)− 1 + (k2
0 − n2
0) −2n0k0
−2n0k0 (k2
1 + n2
1)− 1− (k2
0 − n2
0)
∣∣∣∣ ∣∣∣∣ k1
n1
∣∣∣∣ =
∣∣∣∣ 0
0
∣∣∣∣ .
The determinants in both problems must be equated to zero:
[(k2
0 + n2
0)− 1]2 − (k2
1 − n2
1)
2 − 4n2
1k
2
1 = 0,
[(k2
1 + n2
1)− 1]2 − (k2
0 − n2
0)
2 − 4n2
0k
2
0 = 0,
or
[(k2
0 + n2
0)− 1]2 − (k2
1 + n2
1)
2 = 0, [(k2
1 + n2
1)− 1]2 − (k2
0 + n2
0)
2 = 0.
The latter equations may be rewritten in factorized form:
[(k2
0 + n2
0)− 1− (k2
1 + n2
1)][(k
2
0 + n2
0)− 1 + (k2
1 + n2
1)] = 0,
[(k2
1 + n2
1)− 1− (k2
0 + n2
0)][(k
2
1 + n2
1)− 1 + (k2
0 + n2
0)] = 0.
They have the structure: AC = 0, BC = 0. Four different cases arise.
(1) Let C = 0, then
k2
0 + n2
0 + k2
1 + n2
1 = +1. (4.2)
(2) Now, let A = 0, B = 0, but a contradiction arises: A + B = 0, A + B = −2.
(3)–(4) There are two simples cases:
A = 0, C = 0 k2
0 + n2
0 = 1, k1 = 0, n1 = 0, (4.3)
B = 0, C = 0 k2
1 + n2
1 = 1, k0 = 0, n0 = 0. (4.4)
Evidently, (4.3) and (4.4) can be regarded as particular cases of the above variant (4.2). Now,
one should take into account additional relation
det G = [kk][kk] + [nn][nn] + 2(kk)(nn)
+ 2(nk)(nk)− 2[nk][nk]) + 4(kn)(kn)− 4(kk)(nn) = 1,
which can be transformed to
det G = (k2
0 + k2
1 + n2
0 + n2
1)
2 − 4(k1n0 + k0n1)2 = +1. (4.5)
Both equations (4.2) and (4.5) are to be satisfied
(k2
0 + n2
0 + k2
1 + n2
1) = 1, (k2
0 + k2
1 + n2
0 + n2
1)
2 − 1− 4(k1n0 + k0n1)2 = 0,
from where it follows
k1n0 + k0n1 = 0, k2
0 + n2
0 + k2
1 + n2
1 = +1.
They specify a 2-parametric unitary subgroup in SU(4)
G+
1 = G−1
1 , det G1 = +1,
k1n0 + k0n1 = 0, k2
0 + n2
0 + k2
1 + n2
1 = +1,
G1 =
∣∣∣∣∣ k0 + ik1σ
1 n0 − in1σ
1
−n0 + in1σ
1 k0 + ik1σ
1
∣∣∣∣∣ =
∣∣∣∣∣∣∣∣
k0 ik1 n0 −in1
ik1 k0 −in1 n0
−n0 in1 k0 ik1
in1 −n0 ik1 k0
∣∣∣∣∣∣∣∣ . (4.6)
On Parametrization of GL(4, C) and SU(4) 15
Two analogous subgroups are possible:
G+
2 = G−1
2 , det G2 = +1,
k2n0 + k0n2 = 0, k2
0 + n2
0 + k2
2 + n2
2 = +1,
G2 =
∣∣∣∣ k0 + ik2σ
2 n0 − in2σ
2
−n0 + in2σ
2 k0 + ik2σ
2
∣∣∣∣ =
∣∣∣∣∣∣∣∣
k0 k2 n0 −n2
−k2 k0 n2 n0
−n0 n2 k0 k2
−n2 −n0 −k2 k0
∣∣∣∣∣∣∣∣ ; (4.7)
G+
3 = G−1
3 , det G3 = +1,
k3n0 + k0n3 = 0, k2
0 + n2
0 + k2
3 + n2
3 = +1,
G3 =
∣∣∣∣∣∣∣∣
(k0 + ik3) 0 (n0 − in3) 0
0 (k0 − ik3) 0 (n0 + in3)
−(n0 − in3) 0 (k0 + ik3) 0
0 −(n0 + in3) 0 (k0 − ik3)
∣∣∣∣∣∣∣∣ . (4.8)
Let us consider the latter subgroup (4.8) in some detail. The multiplication law for parame-
ters is
k′′0 = k′0k0 − k′3k3 − n′0n0 + n′3n3, k′′3 = k′0k3 + k′3k0 + n′0n3 + n′3n0,
n′′0 = k′0n0 + k′3n3 + n′0k0 + n′3k3, n′′3 = k′0n3 − k′3n0 − n′0k3 + n′3k0.
For two particular cases (see (4.3) and (4.4)), these formulas take the form:
{G′0
3 } : k2
3 + n2
3 = 1, k0 = 0, n0 = 0,
k′′0 = −k′3k3 + n′3n3, k′′3 = 0,
n′′0 = +k′3n3 + n′3k3, n′′3 = 0,
{G0} : k2
0 + n2
0 = 1, k3 = 0, n3 = 0,
k′′0 = k′0k0 − n′0n0,
n′′0 = k′0n0 + n′0k0. (4.9)
Therefore, multiplying of any two elements from G
′0
3 does not lead us to any element from
G
′0
3 , instead belonging to G0: G0′
3 G0
3 ∈ G0. Similar result would be achieved for G1 and G2:
G0′
1 G0
1 ∈ G0, G0′
2 G0
2 ∈ G0. In the subgroup given by (4.9) one can easily see the structure of the
1-parametric Abelian subgroup:
k0 = cos α, n0 = sinα,
G0(α) =
∣∣∣∣∣∣∣∣∣
cos α 0 sinα 0
0 cos α 0 sinα
− sin α 0 cos α 0
0 − sinα 0 cos α
∣∣∣∣∣∣∣∣∣ , α ∈ [0, 2π]. (4.10)
In the same manner, similar curvilinear parametrization can be readily produced for 2-
parametric groups (4.6)–(4.8). For definiteness, for the subgroup G3 such coordinates are
given by
k0 = cos α cos ρ, k3 = cos α sin ρ,
n0 = sinα cos ρ, −n3 = sinα sin ρ, α ∈ [0, 2π],
16 V.M. Red’kov, A.A. Bogush and N.G. Tokarevskaya
and matrix G3 is
G3(ρ, α) =
∣∣∣∣∣∣∣∣
cos αeiρ 0 sinαeiρ 0
0 cos αe−iρ 0 sinαe−iρ
− sinαeiρ 0 cos αeiρ 0
0 − sinαe−iρ 0 cos αe−iρ
∣∣∣∣∣∣∣∣ . (4.11)
One may note that equation (4.11) at ρ = 0 will coincide with G0(α) in (4.10): G3(ρ =
0, α) = G0(α). Similar curvilinear parametrization may be introduced for two other subgroups,
G1 and G2.
One could try to obtain more general result just changing real valued curvilinear coordinates
on complex. However it is easily verified that it is not the case: through that change though
there arise subgroups but they are not unitary. Indeed, let the matrix (4.10) be complex: then
unitarity condition gives
cos α cos α∗ + sinα sinα∗ = 1, − cos α sin α∗ + sinα cos α∗ = 0.
These two equations can be satisfied only by a real valued α. In the same manner, the the
formal change {G1, G2, G3} =⇒ {GC
1 , GC
2 , GC
3 } again provides us with non-unitary subgroups.
It should be noted that each of three 2-parametric subgroup G1, G2, G3, in addition to G0(α),
contains one additional Abelian unitary subgroup:
K1 =
∣∣∣∣ k0 + ik1σ
1 0
0 k0 + ik1σ
1
∣∣∣∣ , K1 ⊂ G1, k2
0 + k2
1 = 1,
K2 =
∣∣∣∣ k0 + ik2σ
2 0
0 k0 + ik2σ
2
∣∣∣∣ , K2 ⊂ G2, k2
0 + k2
2 = 1,
K3 =
∣∣∣∣ k0 + ik3σ
3 0
0 k0 + ik3σ
3
∣∣∣∣ , K3 ⊂ G3, k2
0 + k2
3 = 1.
It may be easily verified that
G1 = G0K1 = K1G0, G2 = G0K2 = K2G0, G3 = G0K3 = K3G0.
Indeed
G0(α)K1 = K1G0(α) =
∣∣∣∣ cos αk0 + i cos αk1σ
1 sinαk0 + i sinαk1
− sinαk0 − i sinαk1 cos αk0 + i cos αk1σ
1
∣∣∣∣ ,
and with notation
cos αk0 = k′0, cos αk1 = k′1, sin αk0 = n′0, sinαk1 = −n′1,
k′0n
′
1 + k′1n
′
0 = 0, k
′2
0 + k
′2
3 + n
′2
0 + n
′2
3 = 1
we arrive at
G0K1 = K1G0 =
∣∣∣∣∣ k′0 + ik′1σ
1 n′0 − in′1
−n′0 + in′1 k′0 + ik′1σ
1
∣∣∣∣∣ ⊂ G1.
5 4-parametric unitary subgroup
Let us turn again to the subgroup in GL(4, C) given by Case C (see (2.10)):
G =
∣∣∣∣ (k0 + k~σ) (n0 − n~σ)
−(n0 − n~σ) (k0 + k~σ)
∣∣∣∣ ,
On Parametrization of GL(4, C) and SU(4) 17
when the unitarity equations look as follows:
k0 = +k0[kk] + k0(nn) + n0(nk)− n0[nk],
n0 = +k0(nk)− k0[kn] + n0[nn] + n0(kk),
k = k[kk]− k(nn)− n(nk)− n[nk] + 2n× (n× k),
n = −k(kn)− k[kn] + n[nn]− n(kk) + 2k × (k × n).
They can be rewritten as four eigenvalue problems:∣∣∣∣ [kk] + (nn) (nk)− [nk]
(nk)− [nk] (kk) + [nn]
∣∣∣∣ ∣∣∣∣ k0
n0
∣∣∣∣ = (+1)
∣∣∣∣ k0
n0
∣∣∣∣ , (5.1)∣∣∣∣ +([kk]− [nn]) −2(nk)
−2(nk) −([kk]− [nn])
∣∣∣∣ ∣∣∣∣ k1
n1
∣∣∣∣ = (+1)
∣∣∣∣ k1
n1
∣∣∣∣ ,∣∣∣∣ +([kk]− [nn]) −2(nk)
−2(nk) −([kk]− [nn])
∣∣∣∣ ∣∣∣∣ k2
n2
∣∣∣∣ = (+1)
∣∣∣∣ k2
n2
∣∣∣∣ ,∣∣∣∣ +([kk]− [nn]) −2(nk)
−2(nk) −([kk]− [nn])
∣∣∣∣ ∣∣∣∣ k3
n3
∣∣∣∣ = (+1)
∣∣∣∣ k3
n3
∣∣∣∣ . (5.2)
These equations have the same structure∣∣∣∣ A C
C B
∣∣∣∣ ∣∣∣∣ Z1
Z2
∣∣∣∣ = λ
∣∣∣∣ Z1
Z2
∣∣∣∣ ,
where λ = +1. Non-trivial solutions may exist only if
det
∣∣∣∣ A− λ C
C B − λ
∣∣∣∣ = 0,
which gives two different eigenvalues
λ1 =
A + B +
√
(A−B)2 + 4C2
2
, λ2 =
A + B +
√
(A−B)2 + 4C2
2
.
In explicit form, equations (5.1) looks as follows:∣∣∣∣ A C
C B
∣∣∣∣ ∣∣∣∣ k0
n0
∣∣∣∣ = λ
∣∣∣∣ k0
n0
∣∣∣∣ ,
A = (k2
0 + n2
0) + (k2 − n2), B = (k2
0 + n2
0)− (k2 − n2), C = −2kn, (5.3)
λ1 = (k2
0 + n2
0) +
√
(k2 − n2)2 + 4(kn)2,
λ2 = (k2
0 + n2
0)−
√
(k2 − n2)2 + 4(kn)2.
The eigenvalue λ = +1 might be constructed by two ways:
λ1 = +1, k2
0 + n2
0 = 1−
√
(k2 − n2)2 + 4(kn)2,
λ2 = +1, k2
0 + n2
0 = 1 +
√
(k2 − n2)2 + 4(kn)2. (5.4)
These two relations (5.4) are equivalent to the following one:
(1− k2
0 − n2
0)
2 = (k2 − n2)2 + 4(kn)2.
18 V.M. Red’kov, A.A. Bogush and N.G. Tokarevskaya
Thus, equations (5.3) have two different types:
Type I
(A− 1)k0 + Cn0 = 0, Ck0 + (B − 1)n0 = 0,
k2
0 + n2
0 = 1−
√
(k2 − n2)2 + 4(kn)2, (5.5)
k2
0 + n2
0 < +1, (k2 − n2)2 + 4(kn)2 < +1;
Type II
(A− 1)k0 + Cn0 = 0, Ck0 + (B − 1)n0 = 0, (5.6)
k2
0 + n2
0 = 1 +
√
(k2 − n2)2 + 4(kn)2, k2
0 + n2
0 > +1.
Now let us turn to equations (5.2). They have the form∣∣∣∣ A C
C −A
∣∣∣∣ ∣∣∣∣ ki
ni
∣∣∣∣ = λ
∣∣∣∣ ki
ni
∣∣∣∣ , i = 1, 2, 3,
A = k2
0 + k2 − n2
0 − n2, B = −A, C = −2(k0n0 − kn),
λ1 = +
√
(k2
0 + k2 − n2
0 − n2)2 + 4(k0n0 − kn)2,
λ2 = −
√
(k2
0 + k2 − n2
0 − n2)2 + 4(k0n0 − kn)2.
As we are interested only in positive eigenvalue λ = +1, we must use only one possibility
λ = +1 = λ1, so that
(A− 1)k + Cn = 0, Ck − (A + 1)n = 0, (5.7)
1 = (k2
0 + k2 − n2
0 − n2)2 + 4(k0n0 − kn)2.
Vector condition in (5.7) says that k and n are (anti)collinear:
k = Ke, n = Ne, e2 = 1, e ∈ S2, (5.8)
so that (5.7) give
(A− 1)K + CN = 0, CK − (A + 1)N = 0,
1 = (k2
0 + K2 − n2
0 −N2)2 + 4(k0n0 −KN)2, (5.9)
A = k2
0 + K2 − n2
0 −N2, C = −2(k0n0 −KN).
With notation (5.8), equations (5.5)–(5.6) take the form:
Type I
(A− 1)k0 + Cn0 = 0, Ck0 + (B − 1)n0 = 0, k2
0 + n2
0 = 1− (K2 + N2),
Type II
(A− 1)k0 + Cn0 = 0, Ck0 + (B − 1)n0 = 0, k2
0 + n2
0 = 1 + (K2 + N2), (5.10)
where
A = (k2
0 + n2
0) + (K2 −N2), B = (k2
0 + n2
0)− (K2 −N2), C = −2KN. (5.11)
On Parametrization of GL(4, C) and SU(4) 19
Therefore, we have 8 variables e, k0, n0, K, N and the set of equations, (5.9)–(5.11) for them.
Its solving turns to be rather involving, so let us formulate only the final result:
k0, k = Ke, n0, n = Ne,
k2
0 + K2 + n2
0 + N2 = +1, k0N + n0K = 0,
G =
∣∣∣∣ (k0 + iKe~σ) (n0 − iNe~σ)
−(n0 − iNe~σ) (k0 + iKe~σ)
∣∣∣∣ . (5.12)
It should be noted that
det G = (k2
0 + K2 + n2
0 + N2)2 = +1.
The unitarity of the matrices (5.12) may be verified by direct calculation. Indeed,
G+ =
∣∣∣∣ (k0 − iKe~σ) −(n0 + iNe~σ)
(n0 + iNe~σ) (k0 − iKe~σ)
∣∣∣∣ ,
and further for GG+ = I we get (by 2× 2 blocks)
(GG+)11 = k2
0 + K2 + n2
0 + N2 = +1, (GG+)12 = −2i(n0K + k0N)(e~σ) = 0,
(GG+)22 = k2
0 + K2 + n2
0 + N2 = +1, (GG+)21 = +2i(n0K + k0N)(e~σ) = 0.
One different way to parameterize (5.12) can be proposed. Indeed, relations (5.12) are
k0, k = Ke, n0, n = Ne,
k2
0(1 +
K2
k2
0
) + n2
0(1 +
N2
n2
0
) = +1,
K
k0
= −N
n0
≡ W,
or
k0, k = k0We, n0, n = −n0We,
(k2
0 + n2
0)(1 + W 2) = +1, K = k0W, N = −n0W, 0 ≤ k2
0 + n2
0 ≤ 1.
Therefore, matrix G can be presented as follows:
G =
∣∣∣∣∣ k0(1 + iWe~σ) n0(1 + iWe~σ)
−n0(1 + iWe~σ) k0(1 + iWe~σ)
∣∣∣∣∣ , (5.13)
(k2
0 + n2
0)(1 + W 2) = +1 =⇒ W = ±
√
1
k2
0 + n2
0
− 1.
Evidently, it suffices to take positive values for W . The constructed subgroup (5.13) depends
upon four parameters k0, n0, e:
0 ≤ k2
0 + n2
0 ≤ 1, e2 = 1, (k2
0 + n2
0)(1 + W 2) = +1.
Let us establish the law of multiplication for four parameters k0, n0, W = We:
G′′ =
∣∣∣∣ k′0(1 + iW ′~σ) n′0(1 + iW ′~σ)
−n′0(1 + iW ′~σ) k′0(1 + iW ′~σ)
∣∣∣∣ ∣∣∣∣ k0(1 + iW~σ) n0(1 + iW~σ)
−n0(1 + iW~σ) k0(1 + iW~σ)
∣∣∣∣
or by 2× 2 blocks
(11) = (k′0k0 − n′0n0)(1 + iW ′~σ)(1 + iW~σ),
20 V.M. Red’kov, A.A. Bogush and N.G. Tokarevskaya
(12) = (k′0n0 + n′0k0)(1 + iW ′~σ)(1 + iW~σ),
(21) = −(k′0n0 + n′0k0)(1 + iW ′~σ)(1 + iW~σ),
(22) = (k′0k0 − n′0n0)(1 + iW ′~σ)(1 + iW~σ).
As (11) = (22), (12) = −(21); further one can consider only two blocks:
(11) = (k′0k0 − n′0n0)(1−W ′W )
(
1 + i
W ′ + W −W ′ ×W
1−W ′W
~σ
)
,
(12) = (k′0n0 + n′0k0)(1−W ′W )
(
1 + i
W ′ + W −W ′ ×W
1−W ′W
~σ
)
.
So the composition rules should be
k′′0 = (k′0k0 − n′0n0)(1−W ′W ), n′′0 = (k′0n0 + n′0k0)(1−W ′W ),
W ′′ =
W ′ + W −W ′ ×W
1−W ′W
.
The later formula coincides with the Gibbs multiplication rule (see in [32]) for 3-dimensional
rotation group SO(3, R). It remains to prove the identity:
(k
′′2
0 + n
′′2
0 )(1 + W
′′2) = +1
which reduces to
(k′0k0 − n′0n0)2 + (k′0n0 + n′0k0)2
[
(1−W ′W )2 + (W ′ + W −W ′ ×W )2
]
. (5.14)
First terms are
(k′0k0 − n′0n0)2 + (k′0n0 + n′0k0)2 = (k
′2
0 + n
′2
0 )(k2
0 + n2
0).
Second term is
(1−W ′W )2 + (W ′ + W −W ′ ×W )2 = (1 + W ′2)(1 + W 2).
Therefore, (5.14) takes the form
(k′ 20 + n′ 20 )(k2
0 + n2
0)(1 + W ′2)(1 + W 2) = 1
which is identity due to equalities
(k′ 20 + n′ 20 )(1 + W ′2) = 1, (k2
0 + n2
0)(1 + W 2) = 1.
It is matter of simple calculation to introduce curvilinear parameters for such an unitary
subgroup:
e = (sin θ cos φ, sin θ sinφ, cos θ),
k0 = cos α cos ρ, K = cos α sin ρ, n0 = sinα cos ρ, N = − sinα sin ρ,
and G looks as follows
G =
∣∣∣∣ ∆ Σ
−Σ ∆
∣∣∣∣ , ∆ =
∣∣∣∣ cos α(cos ρ + i sin ρ cos θ) i cos α sin ρ sin θe−iφ
i cos α sin ρ sin θeiφ cos α(cos ρ− i sin ρ cos θ
∣∣∣∣ ,
Σ =
∣∣∣∣ sinα(cos ρ + i sin ρ cos θ) +i sinα sin ρ sin θe−iφ
i sinα sin ρ sin θeiφ sinα(cos ρ− i sin ρ cos θ)
∣∣∣∣ .
On Parametrization of GL(4, C) and SU(4) 21
It should be noted that one one may factorize 4-parametric element into two unitary factors,
1-parametric and 3-parametric. Indeed, let us consider the product of commuting unitary groups,
isomorphic to Abelian group G0 and SU(2):
G = G0 ⊗ SU(2) = SU(2)⊗G0 =
∣∣∣∣ k′0 n′0
−n′0 k′0
∣∣∣∣ ∣∣∣∣ a0 + ia~σ 0
0 a0 + ia~σ
∣∣∣∣
=
∣∣∣∣ k′0a0 + ik′0a~σ n′0a0 + in′0a~σ
−n′0a0 − in′0a~σ k′0a0 + ik′0a~σ
∣∣∣∣ , k′ 20 + n′ 20 = 1, a2
0 + a2 = 1,
with the notation
k′0a0 = k0, k′0a = k0W , n′0a0 = n0, n′0a = n0W ,
(k2
0 + n2
0)(1 + W 2) = (k′ 20 a2
0 + n′ 20 a2
0)(1 + W 2) = (k′ 20 + n′ 20 )(a2
0 + a2) = 1,
takes the form
G0 ⊗ SU(2) = SU(2)⊗G0 =
∣∣∣∣ k0(1 + iW~σ) n0(1 + iW~σ)
−n0(1 + iW~σ) k0(1 + iW~σ)
∣∣∣∣ = G. (5.15)
Let us summarize the main results of the previous sections:
Parametrization of 4 × 4 matrices G of the complex linear group GL(4, C) in
terms of four complex vector-parameters G = G(k, m, n, l) is developed and the
problem of inverting any 4×4 matrix G is solved. Expression for determinant of any
matrix G is found: detG = F (k, m, n, l). Unitarity conditions have been formulated
in the form of non-linear cubic algebraic equations including complex conjugation.
Several simplest solutions of these unitarity equations have been found: three 2-
parametric subgroups G1, G2, G3 – each of subgroups consists of two commuting
Abelian unitary groups; 4-parametric unitary subgroup consisting of a product of
a 3-parametric group isomorphic SU(2) and 1-parametric Abelian group.
The task of full solving of the unitarity conditions seems to be rather complicated and it will
be considered elsewhere. In the remaining part of the present paper we describe some relations
of the above treatment to other considerations of the problem in the literature. The relations
described give grounds to hope that the full general solution of the unitary equations obtained
can be constructed on the way of combining different techniques used in the theory of the unitary
group SU(4).
6 On subgroups GL(3, C) and SU(3), expressions
for Gell-Mann matrices through the Dirac basis
In this section the main question is how in the Dirac parametrization one can distinguish
GL(3, C), subgroup in GL(4, C). To this end, let us turn to the explicit form of the Dirac
basis (the Weyl representation is used; at some elements the imaginary unit i is added)
γ5 =
∣∣∣∣∣∣∣∣
−1 0 0 0
0 −1 0 0
0 0 1 0
0 0 0 1
∣∣∣∣∣∣∣∣ , γ0 =
∣∣∣∣∣∣∣∣
0 0 1 0
0 0 0 1
1 0 0 0
0 1 0 0
∣∣∣∣∣∣∣∣ , iγ5γ0 =
∣∣∣∣∣∣∣∣
0 0 −i 0
0 0 0 −i
i 0 0 0
0 i 0 0
∣∣∣∣∣∣∣∣ ,
iγ1 =
∣∣∣∣∣∣∣∣
0 0 0 −i
0 0 −i 0
0 i 0 0
i 0 0 0
∣∣∣∣∣∣∣∣ , γ5γ1 =
∣∣∣∣∣∣∣∣
0 0 0 1
0 0 1 0
0 1 0 0
1 0 0 0
∣∣∣∣∣∣∣∣ , iγ2 =
∣∣∣∣∣∣∣∣
0 0 0 −1
0 0 1 0
0 1 0 0
−1 0 0 0
∣∣∣∣∣∣∣∣ , (6.1)
22 V.M. Red’kov, A.A. Bogush and N.G. Tokarevskaya
γ5γ2 =
∣∣∣∣∣∣∣∣
0 0 0 −i
0 0 i 0
0 −i 0 0
i 0 0 0
∣∣∣∣∣∣∣∣ , iγ3 =
∣∣∣∣∣∣∣∣
0 0 −i 0
0 0 0 i
i 0 0 0
0 −i 0 0
∣∣∣∣∣∣∣∣ , γ5γ3 =
∣∣∣∣∣∣∣∣
0 0 1 0
0 0 0 −1
1 0 0 0
0 −1 0 0
∣∣∣∣∣∣∣∣ ,
2σ01 =
∣∣∣∣∣∣∣∣
0 1 0 0
1 0 0 0
0 0 0 −1
0 0 −1 0
∣∣∣∣∣∣∣∣ , 2σ02 =
∣∣∣∣∣∣∣∣
0 −i 0 0
i 0 0 0
0 0 0 i
0 0 −i 0
∣∣∣∣∣∣∣∣ , 2σ03 =
∣∣∣∣∣∣∣∣
1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 1
∣∣∣∣∣∣∣∣ ,
2iσ12 =
∣∣∣∣∣∣∣∣
1 0 0 0
0 −1 0 0
0 0 1 0
0 0 0 −1
∣∣∣∣∣∣∣∣ , 2iσ23 =
∣∣∣∣∣∣∣∣
0 1 0 0
1 0 0 0
0 0 0 1
0 0 1 0
∣∣∣∣∣∣∣∣ , 2iσ31 =
∣∣∣∣∣∣∣∣
0 −i 0 0
i 0 0 0
0 0 0 −i
0 0 i 0
∣∣∣∣∣∣∣∣ .
All these 15 matrices Λi are of Gell-Mann type: they have a zero-trace, they are Hermitian,
besides, their squares are unite:
SpΛ = 0, (Λ)2 = I, (Λ)+ = Λ, Λ ∈ {Λk : k = 1, . . . , 15}.
Exponential function of any of them equals to
U = eiaΛ = cos a + i sin aΛ, det eiaΛ = +1, U+ = U−1, a ∈ R.
Evidently, multiplying of such 15 elementary unitary matrices (at real parameters xi) results in
an unitary matrix
U = eia1Λ1eia2Λ2 · · · eia14Λ14eial5Λl5 .
At this there arise 15 generalized angle-variables a1, . . . , a15. Evident advantage of this ap-
proach is its simplicity, and evident defect consists in the following: we do not know any simple
group multiplication rule for these angles.
It should be noted that the basis λi used in [69] substantially differs from the above Dirac
basis Λi – this peculiarity is closely connected with distinguishing SU(3) in SU(4). This problem
is evidently related to the task of distinguishing GL(3, C) in GL(4, C) as well.
In order to have possibility to compare two approaches we need exact connection between λi
and Λi. In [69] the following Gell-Mann basis for SU(4) were used:
λ1 =
∣∣∣∣∣∣∣∣
0 1 0 0
1 0 0 0
0 0 0 0
0 0 0 0
∣∣∣∣∣∣∣∣ , λ2 =
∣∣∣∣∣∣∣∣
0 −i 0 0
i 0 0 0
0 0 0 0
0 0 0 0
∣∣∣∣∣∣∣∣ , λ3 =
∣∣∣∣∣∣∣∣
1 0 0 0
0 −1 0 0
0 0 0 0
0 0 0 0
∣∣∣∣∣∣∣∣ ,
λ4 =
∣∣∣∣∣∣∣∣
0 0 1 0
0 0 0 0
1 0 0 0
0 0 0 0
∣∣∣∣∣∣∣∣ , λ5 =
∣∣∣∣∣∣∣∣
0 0 −i 0
0 0 0 0
i 0 0 0
0 0 0 0
∣∣∣∣∣∣∣∣ , λ6 =
∣∣∣∣∣∣∣∣
0 0 0 0
0 0 1 0
0 1 0 0
0 0 0 0
∣∣∣∣∣∣∣∣ ,
λ7 =
∣∣∣∣∣∣∣∣
0 0 0 0
0 0 −i 0
0 i 0 0
0 0 0 0
∣∣∣∣∣∣∣∣ , λ8 =
1√
3
∣∣∣∣∣∣∣∣
1 0 0 0
0 1 0 0
0 0 −2 0
0 0 0 0
∣∣∣∣∣∣∣∣ , λ9 =
∣∣∣∣∣∣∣∣
0 0 0 1
0 0 0 0
0 0 0 0
1 0 0 0
∣∣∣∣∣∣∣∣ ,
λ10 =
∣∣∣∣∣∣∣∣
0 0 0 −i
0 0 0 0
0 0 0 0
i 0 0 0
∣∣∣∣∣∣∣∣ , λ11 =
∣∣∣∣∣∣∣∣
0 0 0 0
0 0 0 1
0 0 0 0
0 1 0 0
∣∣∣∣∣∣∣∣ , λ12 =
∣∣∣∣∣∣∣∣
0 0 0 0
0 0 0 −i
0 0 0 0
0 i 0 0
∣∣∣∣∣∣∣∣ ,
On Parametrization of GL(4, C) and SU(4) 23
λ13 =
∣∣∣∣∣∣∣∣
0 0 0 0
0 0 0 0
0 0 0 1
0 0 1 0
∣∣∣∣∣∣∣∣ , λ14 =
∣∣∣∣∣∣∣∣
0 0 0 0
0 0 0 0
0 0 0 −i
0 0 i 0
∣∣∣∣∣∣∣∣ , λ15 = 1√
6
∣∣∣∣∣∣∣∣
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 −3
∣∣∣∣∣∣∣∣ . (6.2)
All the λ excluding λ8, λ15 possess the same property:
λ3
i = +λi, i 6= 8, 15.
The minimal polynomials for λ8, λ15 can be easily found. Indeed,
(λ8)2 = 1
3
∣∣∣∣∣∣∣∣
1 0 0 0
0 1 0 0
0 0 4 0
0 0 0 0
∣∣∣∣∣∣∣∣ , (λ8)3 = 1
3
√
3
∣∣∣∣∣∣∣∣
1 0 0 0
0 1 0 0
0 0 −8 0
0 0 0 0
∣∣∣∣∣∣∣∣ ,
therefore
(λ8)3 = 2
3λ8 − 1√
3
(λ8)2.
Analogously, for λ15 we have
(λ15)2 = 1
6
∣∣∣∣∣∣∣∣
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 9
∣∣∣∣∣∣∣∣ , (λ15)3 = 1
6
√
6
∣∣∣∣∣∣∣∣
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 −27
∣∣∣∣∣∣∣∣ ,
and
(λ15)3 = 1
2λ15 − 2√
6
(λ15)2.
Comparing Λi and λi one can readily derive the linear combinations:
γ0 + γ5γ3 = 2λ4, γ0 − γ5γ3 = 2λ11, iγ5γ0 + iγ3 = 2λ5,
iγ5γ0 − iγ3 = 2λ12, γ5γ1 + iγ2 = 2λ6, γ5γ1 − iγ2 = 2λ9,
iγ1 + γ5γ2 = 2λ10, iγ1 − γ5γ2 = 2λ7, 2σ01 + 2iσ23 = 2λ1,
2σ01 − 2iσ23 = −2λ13, 2σ02 + 2iσ31 = 2λ2, 2σ02 − 2iσ31 = −2λ14,
and additional six combinations
2σ03 + 2iσ12 =
∣∣∣∣∣∣∣∣
2 0 0 0
0 −2 0 0
0 0 0 0
0 0 0 0
∣∣∣∣∣∣∣∣ , 2σ03 − 2iσ12 =
∣∣∣∣∣∣∣∣
0 0 0 0
0 0 0 0
0 0 −2 0
0 0 0 +2
∣∣∣∣∣∣∣∣ ,
γ5 + 2σ03 =
∣∣∣∣∣∣∣∣
0 0 0 0
0 −2 0 0
0 0 0 0
0 0 0 2
∣∣∣∣∣∣∣∣ , γ5 − 2σ03 =
∣∣∣∣∣∣∣∣
−2 0 0 0
0 0 0 0
0 0 2 0
0 0 0 0
∣∣∣∣∣∣∣∣ ,
γ5 + 2iσ12 =
∣∣∣∣∣∣∣∣
0 0 0 0
0 −2 0 0
0 0 2 0
0 0 0 0
∣∣∣∣∣∣∣∣ , γ5 − 2iσ12 =
∣∣∣∣∣∣∣∣
−2 0 0 0
0 0 0 0
0 0 0 0
0 0 0 2
∣∣∣∣∣∣∣∣ ,
24 V.M. Red’kov, A.A. Bogush and N.G. Tokarevskaya
they should contain three linearly independent matrices. Those three linearly independent
matrices might be chosen in different ways. Let us introduce the notation:
a =
∣∣∣∣∣∣∣∣
1 0 0 0
0 −1 0 0
0 0 0 0
0 0 0 0
∣∣∣∣∣∣∣∣ , b =
∣∣∣∣∣∣∣∣
0 0 0 0
0 −1 0 0
0 0 1 0
0 0 0 0
∣∣∣∣∣∣∣∣ , c =
∣∣∣∣∣∣∣∣
−1 0 0 0
0 0 0 0
0 0 1 0
0 0 0 0
∣∣∣∣∣∣∣∣ ,
A =
∣∣∣∣∣∣∣∣
0 0 0 0
0 0 0 0
0 0 −1 0
0 0 0 +1
∣∣∣∣∣∣∣∣ , B =
∣∣∣∣∣∣∣∣
−1 0 0 0
0 0 0 0
0 0 0 0
0 0 0 1
∣∣∣∣∣∣∣∣ , C =
∣∣∣∣∣∣∣∣
0 0 0 0
0 −1 0 0
0 0 0 0
0 0 0 1
∣∣∣∣∣∣∣∣ .
The matrices a, b, c have the 3 × 3 blocks different from zero, so they could be generators
for SU(3) transformations; whereas A, B, C may be generators only of the group SU(4). All
six matrices a, b, c, A, B, C have the same minimal polynomial:
λ3 = λ.
Linear space to which these six matrices a, b, c, A, B, C belong is 3-dimensional. Indeed, one
easily obtains
c = b− a, C −A = b, C −B = a, B −A = c,
basis {a, b, C} =⇒ c = b− a, A = C − b, B = C − a.
These relations can be rewritten differently
a = b− c, C −A = b, C −B = b− c, B −A = c,
basis {b, c, A} =⇒ a = b− c, C = A + b, B = A + c
or
b = a + c, C −A = b, C −B = b− c, B −A = c,
basis {a, c, B} =⇒ b = a + c, C = B + a, A = B − c.
One should note that in the basis λi (6.2) the corresponding three linearly independent
elements λ3, λ8, λ15 are taken as follows:
λ3 = 1
22σ03 + 1
22iσ12, λ8 = − 1√
3
γ5 + 1
2
√
3
2σ03 − 1
2
√
3
2iσ12,
λ15 = − 1√
6
γ5 − 1√
6
2σ03 + 1√
6
2iσ12,
their minimal polynomials look
(λ3)3 = λ3, (λ8)3 = 2
3λ8 − 1√
3
(λ8)2, (λ15)3 = 1
2λ15 − 2√
6
(λ15)2.
It is the matter of simple calculation to find relationships between λ3, λ8, λ15 and the basis
{a, b, C} :
λ3 = a, λ8 = 1√
3
a− 2√
3
b, λ15 = 1√
6
a + 1√
6
b− 3√
6
C. (6.3)
In the following we will use the notation
a = λ3, b = λ′8, C = λ′15,
On Parametrization of GL(4, C) and SU(4) 25
so the previous formulas (6.3) will read ({λ3, λ8, λ15} ⇐⇒ {λ3, λ
′
8, λ
′
15})
λ3 = λ3, λ8 = 1√
3
λ3 − 2√
3
λ′8, λ15 = 1√
6
λ3 + 1√
6
λ′8 − 3√
6
λ′15.
The inverse relations are
λ3 = λ3, λ′8 = 1
2λ3 −
√
3
2 λ8, λ′15 = −
√
6
3 λ15 + 1
2λ3 − 1
2
√
3
λ8. (6.4)
Now, starting with the linear decomposition of G ∈ GL(4, C) the in Dirac basis (2.1):
G = a0I + ib0γ
5 + iA0γ
0 + iAkγ
k + B0γ
0γ5 + Bkγ
kγ5
+ ak2σ0k + b12σ23 + b22σ31 + b32σ12
= a0I + ib0γ
5 + iA0γ
0 + Ak(iγk) + iB0(iγ5γ0)−Bk(γ5γk)
+ ak(2σ0k)− ib1(2iσ23)− ib2(2iσ31)− ib3(2iσ12),
with the help of the formulas
γ0 + γ5γ3 = 2λ4, γ0 − γ5γ3 = 2λ11, iγ5γ0 + iγ3 = 2λ5,
iγ5γ0 − iγ3 = 2λ12, γ5γ1 + iγ2 = 2λ6, γ5γ1 − iγ2 = 2λ9,
iγ1 + γ5γ2 = 2λ10, iγ1 − γ5γ2 = 2λ7, 2σ01 + 2iσ23 = 2λ1,
2σ01 − 2iσ23 = −2λ13, 2σ02 + 2iσ31 = 2λ2, 2σ02 − 2iσ31 = −2λ14,
2σ03 + 2iσ12 = 2λ3, 2σ03 − 2iσ12 = 2(λ′15 − λ′8), γ5 + 2σ03 = 2λ′15,
γ5 − 2σ03 = 2(λ′8 − λ3), γ5 + 2iσ12 = 2λ′8, γ5 − 2iσ12 = 2(λ′15 − λ3)
and inverse ones
γ0 = λ4 + λ11, γ5γ3 = λ4 − λ11, iγ5γ0 = λ5 + λ12, iγ3 = λ5 − λ12,
γ5γ1 = λ6 + λ9, iγ2 = λ6 − λ9, iγ1 = λ10 + λ7, γ5γ2 = λ10 − λ7,
2iσ23 = λ1 + λ13, 2σ01 = λ1 − λ13, 2iσ31 = λ2 + λ14, 2σ02 = λ2 − λ14,
2σ03 = λ3 − (λ′8 − λ′15), 2iσ12 = λ3 + (λ′8 − λ′15), γ5 = −λ3 + (λ′8 + λ′15),
we will arrive at
G = a0I + (a1 − ib1)λ1 + (a2 − ib2)λ2 + (iA0 −B3)λ4 + (A3 + iB0)λ5 + (A2 −B1)λ6
+ (A1 + B2)λ7 + (a3 − ib3 − ib0)λ3 + (−ib3 + ib0 − a3)λ′8 + (ib0 + a3 + ib3)λ′15
+ (−B1 −A2)λ9 + (A1 −B2)λ10 + (iA0 + B3)λ11 + (−A3 + iB0)λ12
+ (−a1 − ib1)λ13 + (−a2 − ib2)λ14.
In variables (k,m, l, n) (see (2.3), (2.4))
B0 − iA0 = l0, Bj − iAj = lj , B0 + iA0 = n0, Bj + iAj = nj ,
a0 − ib0 = k0, aj − ibj = kj , a0 + ib0 = m0, aj + ibj = mj
the previous expansion looks
G = 1
2(k0 + m0)I + k1λ1 + k2λ2+ 1
2 [(n0 − n3)− (l0 + l3)]λ4+ 1
2i [−(n0 − n3)− (l0 + l3)]λ5
+ 1
2 [−(n1 + in2)− (l1 − il2)]λ6 + 1
2i [(n1 + in2)− (l1 − il2)]λ7 +
[
k3 + 1
2(k0 −m0)
]
λ3
+
[
−m3 + 1
2(m0 − k0)
]
λ′8 +
[
m3 + 1
2(m0 − k0)
]
λ′15 + 1
2 [−(n1 − in2)− (l1 + il2)]λ9
+ 1
2i [(n1 − in2)− (l1 + il2)]λ10 + 1
2 [(n0 + n3)− (l0 − l3)]λ11
26 V.M. Red’kov, A.A. Bogush and N.G. Tokarevskaya
+ 1
2i [−(n0 + n3)− (l0 − l3)]λ12 −m1λ13 −m2λ14.
Let coefficients at λ9, λ10, λ11, λ12, λ13, λ14 be equal to zero:
(n1 − in2) + (l1 + il2) = 0, (n1 − in2)− (l1 + il2) = 0, (n0 + n3)− (l0 − l3) = 0,
(n0 + n3) + (l0 − l3) = 0, m1 = 0, m2 = 0. (6.5)
Note that we do not require vanishing of the coefficient at λ′15:
λ′15
(
m3 +
m0 − k0
2
)
6= 0.
As a result we have a subgroup of 4 × 4 matrices defined by 10 complex parameters. At this
four elements are diagonal matrices:
I, λ3, λ′8, λ′15,
all other matrices have on the diagonal only zeros. Equations (6.5) give
in2 = n1, n3 = −n0, il2 = −l1, l3 = l0, m1 = 0, m2 = 0, (6.6)
so that any matrix G(ka, n0, n1, l0, l1,m0,m3) is decomposed according to
G = k1λ1+ k2λ2+ (n0− l0)λ4+ i(n0+ l0)λ5+ (−n1− l1)λ6+ i(−n1+ l1)λ7+ 1
2(k0 + m0)I
+
[
k3 + 1
2(k0 −m0)
]
λ3 +
[
−m3 + 1
2(m0 − k0)
]
λ′8 +
[
m3 + 1
2(m0 − k0)
]
λ′15. (6.7)
Explicit form of the matrices parameterized by (6.7) can be obtained from representation for
arbitrary element of GL(4, C) (2.6)
G(k, m, n, l) =
∣∣∣∣∣∣∣∣
+(k0 + k3) +(k1 − ik2) +(n0 − n3) −(n1 − in2)
+(k1 + ik2) +(k0 − k3) −(n1 + in2) +(n0 + n3)
−(l0 + l3) −(l1 − il2) +(m0 −m3) −(m1 − im2)
−(l1 + il2) −(l0 − l3) −(m1 + im2) +(m0 + m3)
∣∣∣∣∣∣∣∣
with additional restrictions (6.6):
G =
∣∣∣∣∣∣∣∣
k0 + k3 k1 − ik2 +2n0 0
k1 + ik2 k0 − k3 −2n1 0
−2l0 −2l1 m0 −m3 0
0 0 0 m0 + m3
∣∣∣∣∣∣∣∣ . (6.8)
If additionally one requires m0 + m3 = 1, then
G =
∣∣∣∣∣∣∣∣
k0 + k3 k1 − ik2 +2n0 0
k1 + ik2 k0 − k3 −2n1 0
−2l0 −2l1 1− 2m3 0
0 0 0 1
∣∣∣∣∣∣∣∣
with decomposition rule
G = k1λ1 + k2λ2 + (n0 − l0)λ4 + i(n0 + l0)λ5 + (−n1 − l1)λ6 + i(−n1 + l1)λ7
+ 1
2(1 + k0 −m3)I +
[
k3 + 1
2(k0 + m3 − 1)
]
λ3
+
[
−m3 + 1
2(1−m3 − k0)
]
λ′8 + 1
2(1− k0 + m3)λ′15. (6.9)
On Parametrization of GL(4, C) and SU(4) 27
In the diagonal part of (6.9), there are four independent matrices because equation (6.9)
represents 4× 4 matrix with the structure
G ∼
∣∣∣∣ GL(3, C) 0
0 1
∣∣∣∣ .
To deal with the matrices from GL(3, C), in the diagonal part of (6.9) one should separate
only a 3× 3 block:
Diag = 1
2(1 + k0 −m3)I(3) +
[
k3 + 1
2(k0 + m3 − 1)
]
λ
(3)
3
+
[
−m3 + 1
2(1−m3 − k0)
]
λ
′(3)
8 + 1
2(1− k0 + m3)λ
′(3)
15 =
= 1
2(1 + k0 −m3)
∣∣∣∣∣∣
1 0 0
0 1 0
0 0 1
∣∣∣∣∣∣ +
[
k3 + 1
2(k0 + m3 − 1)
] ∣∣∣∣∣∣
1 0 0
0 −1 0
0 0 0
∣∣∣∣∣∣
+
[
−m3 + 1
2(1−m3 − k0)
] ∣∣∣∣∣∣
0 0 0
0 −1 0
0 0 1
∣∣∣∣∣∣ + 1
2(1− k0 + m3)
∣∣∣∣∣∣
0 0 0
0 −1 0
0 0 0
∣∣∣∣∣∣ .
Resolving λ
′(3)
15 in terms of I(3), λ
(3)
3 , λ
′(3)
8 :
λ
′(3)
15 = −1
3I(3) + 1
3λ
(3)
3 + 1
3λ
′(3)
8 ,
we arrive at a 3-term relation:
Diag = 1
3(1 + 2k0 − 2m3)I(3) +
[
k3 + 1
3(k0 + 2m3 − 1)
]
λ
(3)
3 + 1
3(−4m3 − 2k0 + 2)λ′(3)8 .
The group law for parameters of SL(3, C) has the form (the notation M = 1− 2m3 is used)
k′′0 = k′0k0 + k′k + 2(−n′0l0 + n′1l1),
(k′′)1 = (k′0k + k′k0 + ik′ × k)1 + 2(−n′0l1 + n′1l0),
(k′′)2 = (k′0k + k′k0 + ik′ × k)2 + 2(−in′0l1 − in′1l0),
(k′′)3 = (k′0k + k′k0 + ik′ × k)3 + 2(−n′0l0 − n′1l1),
n′′0 = (k′0 + k′3)n0 − (k′1 − ik′2)n1 + n′0M,
n′′1 = (k′0 − k′3)n1 − (k′1 + ik′2)n0 + n′1M,
l′′0 = l′0(k0 + k3) + l′1(k1 + ik2) + M ′l0,
l′′1 = l′0(k1 − ik2) + l′1(k0 − k3) + M ′l1,
M ′′ = M ′M − 4(l′0n0 − l′1n1).
These rules determine multiplication of the matrices
G =
∣∣∣∣∣∣∣∣
k0 + k3 k1 − ik2 +2n0 0
k1 + ik2 k0 − k3 −2n1 0
−2l0 −2l1 M 0
0 0 0 1
∣∣∣∣∣∣∣∣ .
If additionally, in equation (6.8) one requires
n0 = 0, n1 = 0, l0 = 0, l1 = 0, m3 = 0, m0 = 1,
28 V.M. Red’kov, A.A. Bogush and N.G. Tokarevskaya
then
G =
∣∣∣∣∣∣∣∣
k0 + k3 k1 − ik2 0 0
k1 + ik2 +k0 − k3 0 0
0 0 1 0
0 0 0 1
∣∣∣∣∣∣∣∣ ,
with the decomposition rule
G = k1λ1 + k2λ2 + 1
2(1 + k0)I +
[
k3 − 1
2(1− k0)
]
λ3 + 1
2(1− k0)λ′8 + 1
2(1− k0)λ′15.
One can readily verify that the 2× 2 block is given by
G(2)(ka) = k0I
(2) + k1λ
(2)
1 + k2λ
(2)
2 + k3λ
(2)
3 .
7 On the multiplication law for GL(4, C) in Dirac basis
In the Gell-Mann basis λi, an element of GL(4, C) is
G = a0I + (a1 − ib1)λ1 + (a2 − ib2)λ2 + (iA0 −B3)λ4 + (A3 + iB0)λ5 + (A2 −B1)λ6
+ (A1 + B2)λ7 + (a3 − ib3 − ib0)λ3 + (−ib3 + ib0 − a3)λ′8 + (−B1 −A2)λ9
+ (A1 −B2)λ10 + (iA0 + B3)λ11 + (−A3 + iB0)λ12 + (−a1 − ib1)λ13
+ (−a2 − ib2)λ14 + (ib0 + a3 + ib3)λ′15,
or in variables (k, m, l, n):
G = 1
2(k0 + m0)I+ k1λ1 + k2λ2+ 1
2 [(n0 − n3)− (l0 + l3)]λ4+ 1
2i [−(n0 − n3)− (l0 + l3)]λ5
+ 1
2 [−(n1 + in2)− (l1 − il2)]λ6 + 1
2i [(n1 + in2)− (l1 − il2)]λ7 +
[
k3 + 1
2(k0 −m0)
]
λ3
+
[
−m3 + 1
2(m0 − k0)
]
λ′8 + 1
2 [−(n1 − in2)− (l1 + il2)]λ9
+ 1
2i [(n1 − in2)− (l1 + il2)]λ10 + 1
2 [(n0 + n3)− (l0 − l3)]λ11
+ 1
2i [−(n0 + n3)− (l0 − l3)]λ12 −m1λ13 −m2λ14 +
[
m3 + 1
2(m0 − k0)
]
λ′15.
The problem is to establish the multiplication rule G′′ = G′G in λ-basis:
x′′kλk = x′mλmxnλn = x′mxnλmλn.
As by definition the relationships λmλn = emnkλk must hold, the multiplication rule is
x′′k = emnkx
′
mxn. (7.1)
The main claim is that the all properties of the GL(4, C) with all its subgroups are determined
by the bilinear function (7.1), the latter is described by structure constants emnk. It is evident
that these group constants should be simpler in the Dirac basis Λi than in the basis λi. Our
next task is to establish the multiplication law G′′ = G′G in Λ-basis:
ΛmΛn = EmnkΛk, X ′′
k = EmnkX
′
mXn.
Before searching for structural constants Emnk, let us introduce a special way to list the Dirac
basis Λi:
α1 = γ0γ2, α2 = iγ0γ5, α3 = γ5γ2 α2
i = I, α1α2 = iα3, α2α1 = −iα1,
β1 = iγ3γ1, β2 = iγ3, β3 = iγ1, β2
i = I, β1β2 = iβ3, β2β1 = −iβ3,
On Parametrization of GL(4, C) and SU(4) 29
these two set commute with each others αjβk = βkαj , and their multiplications provides us with
9 remaining basis elements of {Λk}:
A1 = α1β1 = −γ5, B1 = α1β2 = γ5γ1, C1 = α1β3 = γ3γ5,
A2 = α2β1 = −iγ2, B2 = α2β2 = −iγ1γ2, C2 = α2β3 = −iγ2γ3,
A3 = α3β1 = γ0, B3 = α3β2 = γ0γ1, C3 = α3β3 = γ0γ3. (7.2)
The multiplication rules for basic elements
α1, α2, α3, β1, β2, β3, A1, A2, A3, B1, B2, B3, C1, C2, C3,
are
α1 α2 α3
α1 I iα3 −iα2
α2 −iα3 I iα1
α3 iα2 −iα1 I
β1 β2 β3
α1 A1 B1 C1
α2 A2 B2 C2
α3 A3 B3 C3
A1 A2 A3
α1 β1 iA3 −iA2
α2 −iA3 β1 iA1
α3 iA2 −iA1 β1
B1 B2 B3
α1 β2 iB3 −iB2
α2 −iB3 β2 iB1
α3 iB2 −iB1 β2
C1 C2 C3
α1 β3 iC3 −iC2
α2 −iC3 β3 iC1
α3 iC2 −iC1 β3
α1 α2 α3
β1 A1 A2 A3
β2 B1 B2 B3
β3 C1 C2 C3
β1 β2 β3
β1 I iβ3 −iβ2
β2 −iβ3 I iβ1
β3 iβ2 −iβ1 I
A1 A2 A3
β1 α1 α2 α3
β2 −iC1 −iC2 −iC3
β3 iB1 iB2 iB3
B1 B2 B3
β1 iC1 iC2 iC3
β2 α1 α2 α3
β3 −iA1 −iA2 −iA3
C1 C2 C3
β1 −iB1 −iB2 −iB3
β2 iA1 iA2 iA3
β3 α1 α2 α3
α1 α2 α3
A1 β1 iA3 −iA2
A2 −iA3 β1 iA1
A3 iA2 −iA1 β1
β1 β2 β3
A1 α1 iC1 −iB1
A2 α2 iC2 −iB2
A3 α3 iC3 −iB3
A1 A2 A3
A1 I iα3 −iα2
A2 −iα3 I iα1
A3 iα2 −iα1 I
B1 B2 B3
A1 iβ3 −C3 C2
A2 C3 iβ3 −C1
A3 −C2 C1 iβ3
C1 C2 C3
A1 −iβ2 B3 −B2
A2 −B3 −iβ2 B1
A3 B2 −B1 −iβ2
α1 α2 α3
B1 β2 iB3 −iB2
B2 −iB3 β2 iB1
B3 iB2 −iB1 β2
β1 β2 β3
B1 −iC1 α1 iA1
B2 −iC2 α2 iA2
B3 −iC3 α3 iA3
A1 A2 A3
B1 −iβ3 C3 −C2
B2 −C3 −iβ3 C1
B3 C2 −C1 −iβ3
B1 B2 B3
B1 I iα3 −iα2
B2 −iα3 I iα1
B3 iα2 −iα1 I
C1 C2 C3
B1 iβ1 −A3 A2
B2 A3 iβ1 −A1
B3 −A2 A1 iβ1
α1 α2 α3
C1 β3 iC3 −iC2
C2 −iC3 β3 iC1
C3 iC2 −iC1 β3
β1 β2 β3
C1 iB1 −iA1 α1
C2 iB2 −iA2 α2
C3 iB3 −iA3 α3
A1 A2 A3
C1 iβ2 −B3 B2
C2 B3 iβ2 −B1
C3 −B2 B1 iβ2
B1 B2 B3
C1 −iβ1 A3 −A2β1
C2 −A3 −iβ1 A1
C3 A2 −A1 −iβ1
30 V.M. Red’kov, A.A. Bogush and N.G. Tokarevskaya
C1 C2 C3
C1 I iα3 −iα2
C2 −iα3 I iα1
C3 iα2 −iα1 I
(7.3)
These relations provide us with simple formulas for fifteen coordinates of the element of GL(4, C)
G = γI + ajαj + bjβj + XjAj + YjBj + ZjCj ,
γ = 1
4SpG, aj = 1
4SpαjG, bj = 1
4SpβjG,
Xj = 1
4SpAjG, Yj = 1
4SpBjG, Zj = 1
4SpCjG.
With the use of relations (7.3) an explicit form of the group law for (15 + 1) parameters can be
found:
(γ′I + a′iαi + b′iβi + X ′
iAi + Y ′
i Bi + Z ′
iCi)(γI + ajαj + bjβj + XjAj + YjBj + ZjCj)
= γ′γI + (γ′aj + a′jγ)αj + (γ′bj + b′jγ)βj + (γ′Xj + X ′
jγ)Aj + (γ′Yj + Y ′
i γ)Bj
+ (γ′Zj + Z ′
jγ)Cj + a′1(a1 + a2iα3 − a3iα2) + a′2(−a1iα3 + a2 + a3iα1)
+ a′3(a1iα2 − a2iα1 + a3) + a′1(b1A1 + b2B1 + b3C1) + a′2(b1A2 + b2B2 + b3C2)
+ a′3(b1A3 + b2B3 + b3C3) + a′1(X1β1 + X2iA3 −X3iA2)
+ a′2(−X1iA3 + X2β1 + X3iA1) + a′3(X1iA2 −X2iA1 + X3β1)
+ a′1(Y1β2 + Y2iB3 − Y3iB2) + a′2(−Y1iB3 + Y2β2 + Y3iB1)
+ a′3(Y1iB2 − Y2iB1 + Y3β2) + a′1(Z1β3 + Z2iC3 − Z3iC2)
+ a′2(−Z1iC3 + Z2β3 + Z3iC1) + a′3(Z1iC2 − Z2iC1 + Z3β3)
+ b′1(a1A1 + a2A2 + a3A3) + b′2(a1B1 + a2B2 + a3B3) + b′3(a1C1 + a2C2 + a3C3)
+ b′1(b1 + b2iβ3 − b3iβ2) + b′2(−b1iβ3 + b2 + b3iβ1) + b′3(b1iβ2 − b2iβ1 + b3)
+ b′1(X1α1 + X2α2 + X3α3) + b′2(−X1iC1 −X2iC2 −X3iC3)
+ b′3(X1iB1 + X2iB2 + X3iB3) + b′1(Y1iC1 + Y2iC2 + Y3iC3)
+ b′2(Y1α1 + Y2α2 + Y3α3) + b′3(−Y1iA1 − Y2iA2 − Y3iA3)
+ b′1(−Z1iB1 − Z2iB2 − Z3iB3) + b′2(Z1iA1 + Z2iA2 + Z3iA3)
+ b′3(Z1α1 + Z2α2 + Z3α3) + X ′
1(a1β1 + a2iA3 − a3iA2)
+ X ′
2(−a1iA3 + a2β1 + a3iA1) + X ′
3(a1iA2 − a2iA1 + a3β1)
+ X ′
1(b1α1 + b2iC1 − b3iB1)+ X ′
2(b1α2 + b2iC2 − b3iB2)+ X ′
3(b1α3 + b2iC3 − b3iB3)
+ X ′
1(X1+ X2iα3−X3iα2)+ X ′
2(−X1iα3+ X2 + X3iα1)+ X ′
3(X1iα2−X2iα1 + X3)
+ X ′
1(Y1iβ3− Y2C3+ Y3C2)+ X ′
2(Y1C3+ Y2iβ3− Y3C1)+ X ′
3(−Y1C2+ Y2C1+ Y3iβ3)
+ X ′
1(−Z1iβ2 + Z2B3 − Z3B2) + X ′
2(−Z1B3 − Z2iβ2 + Z3B1)
+ X ′
3(Z1B2 − Z2B1 − Z3iβ2) + Y ′
1(a1β2 + a2iB3 − a3iB2)
+ Y ′
2(−a1iB3 + a2β2 + a3iB1) + Y ′
3(a1iB2 − a2iB1 + a3β2)
+ Y ′
1(−b1iC1 + b2α1 + b3iA1) + Y ′
2(−b1iC2 + b2α2 + b3iA2)
+ Y ′
3(−b1iC3 + b2α3 + b3iA3) + Y ′
1(−X1iβ3 + X2C3 −X3C2)
+ Y ′
2(−X1C3 −X2iβ3 + X3C1) + Y ′
3(X1C2 −X2C1 −X3iβ3)
+ Y ′
1(Y1 + Y2iα3 − Y3iα2) + Y ′
2(−Y1iα3 + Y2 + Y3iα1) + Y ′
3(Y1iα2 − Y2iα1 + Y3)
+ Y ′
1(Z1iβ1 − Z2A3 + Z3A2) + Y ′
2(Z1A3 + Z2iβ1 − Z3A1)
+ Y ′
3(−Z1A2 + Z2A1 + Z3iβ1) + Z ′
1(a1β3 + a2iC3 − a3iC2)
+ Z ′
2(−a1iC3 + a2β3 + a3iC1) + Z ′
3(a1iC2 − a2iC1 + a3β3)
On Parametrization of GL(4, C) and SU(4) 31
+ Z ′
1(b1iB1 − b2iA1 + b3α1) + Z ′
2(b1iB2 − b2iA2 + b3α2) + Z ′
3(b1iB3 − b2iA3 + b3α3)
+ Z ′
1(X1iβ2 −X2B3 + X3B2) + Z ′
2(X1B3 + X2iβ2 −X3B1)
+ Z ′
3(−X1B2 + X2B1 + X3iβ2) + Z ′
1(−Y1iβ1 + Y2A3 − Y3A2)
+ Z ′
2(−Y1A3 − Y2iβ1 + Y3A1) + Z ′
3(Y1A2 − Y2A1 − Y3iβ1)
+ Z ′
1(Z1 + Z2iα3 − Z3iα2) + Z ′
2(−Z1iα3 + Z2 + Z3iα1) + Z ′
3(Z1iα2 − Z2iα1 + Z3).
From these relations we arrive at the following composition rules:
γ′′ = γ′γ + (a′1a1 + a′2a2 + a′3a3) + (b′1b1 + b′2b2 + b′3b3)
+ (X ′
1X1 + X ′
2X2 + X ′
3X3) + (Y ′
1Y1 + Y ′
2Y2 + Y ′
3Y3) + (Z ′
1Z1 + Z ′
2Z2 + Z ′
3Z3),
a′′1 = (γ′a1 + a′1γ) + (b′1X1 + b′2Y1 + b′3Z1) + (X ′
1b1 + Y ′
1b2 + Z ′
1b3)
+ i(a′2a3 − a′3a2) + i(X ′
2X3 −X ′
3X2) + i(Y ′
2Y3 − Y ′
3Y2) + i(Z ′
2Z3 − Z ′
3Z2),
a′′2 = (γ′a2 + a′2γ) + (b′1X2 + b′2Y2 + b′3Z2) + (X ′
2b1 + Y ′
2b2 + Z ′
2b3)
+ i(a′3a1 − a′1a3) + i(X ′
3X1 −X ′
1X3) + i(Y ′
3Y1 − Y ′
1Y3) + i(Z ′
3Z1 − Z ′
1Z3),
a′′3 = (γ′a3 + a′3γ) + (b′1X3 + b′2Y3 + b′3Z3) + (X ′
3b1 + Y ′
3b2 + Z ′
3b3)
+ i(a′1a2 − a′2a1) + i(X ′
1X2 −X ′
2X1) + i(Y ′
1Y2 − Y ′
2Y1) + i(Z ′
1Z2 − Z ′
2Z1),
b′′1 = γ′b1 + b′1γ + i(b′2b3 − b′3b2) + (a′1X1 + a′2X2 + a′3X3) + (X ′
1a1 + X ′
2a2 + X ′
3a3)
+ i(Y ′
1Z1 + Y ′
2Z2 + Y ′
3Z3)− i(Z ′
1Y1 + Z ′
2Y2 + Z ′
3Y3),
b′′2 = γ′b2 + b′2γ + i(b′3b1 − b′1b3) + (a′1Y1 + a′2Y2 + a′3Y3) + (Y ′
1a1 + Y ′
2a2 + Y ′
3a3)
+ i(Z ′
1X1 + Z ′
2X2 + Z ′
3X3)− i(X ′
1Z1 + X ′
2Z2 + X ′
3Z3),
b′′3 = γ′b3 + γb′3 + i(b′1b2 − b′2b1) + (a′1Z1 + a′2Z2 + a′3Z3) + (Z ′
1a1 + Z ′
2a2 + Z ′
3a3)
+ i(X ′
1Y1 + X ′
2Y2 + X ′
3Y3)− i(Y ′
1X1 + Y ′
2X2 + Y ′
3X3),
X ′′
1 = (γ′X1 + γX ′
1) + (a′1b1 + a1b
′
1) + i(Y ′
1b3 − Y1b
′
3) + i(b′2Z1 − b2Z
′
1)
+ i(a′2X3 − a′3X2)− i(a2X
′
3 − a3X
′
2) + (Z2Y
′
3 − Z3Y
′
2) + (Z ′
2Y3 − Z ′
3Y2),
X ′′
2 = (γ′X2 + γX ′
2) + (a′2b1 + a2b
′
1) + i(Y ′
2b3 − Y2b
′
3) + i(b′2Z2 − b2Z
′
2)
+ i(a′3X1 − a′1X2)− i(a3X
′
1 − a1X
′
3) + (Z3Y
′
1 − Z1Y
′
3) + (Z ′
3Y1 − Z ′
1Y3),
X ′′
3 = (γ′X3 + γX ′
3) + (a′3b1 + a3b
′
1) + i(Y ′
3b3 − Y3b
′
3) + i(b′2Z3 − b2Z
′
3)
+ i(a′1X2 − a′2X1)− i(a1X
′
2 − a2X
′
1) + (Z1Y
′
2 − Z2Y
′
1) + (Z ′
1Y2 − Z ′
2Y1),
Y ′′
1 = (γ′Y1 + γY ′
1) + (a′1b2 + a1b
′
2) + i(Z ′
1b1 − Z1b
′
1) + i(b′3X1 − b3X
′
1)
+ i(a′2Y3 − a′3Y2)− i(a2Y
′
3 − a3Y
′
2) + (X2Z
′
3 −X3Z
′
2) + (X ′
2Z3 −X ′
3Z2),
Y ′′
2 = (γ′Y2 + γY ′
2) + (a′2b2 + a2b
′
2) + i(Z ′
2b1 − Z2b
′
1) + i(b′3X2 − b3X
′
2)
+ i(a′3Y1 − a′1Y3)− i(a3Y
′
1 − a1Y
′
3) + (X3Z
′
1 −X1Z
′
3) + (X ′
3Z1 −X ′
1Z3),
Y ′′
3 = (γ′Y3 + γY ′
3) + (a′3b2 + a3b
′
2) + i(Z ′
3b1 − Z3b
′
1) + i(b′3X3 − b3X
′
3)
+ i(a′1Y2 − a′2Y1)− i(a1Y
′
2 − a2Y
′
1) + (X1Z
′
2 −X2Z
′
1) + (X ′
1Z2 −X ′
2Z1),
Z ′′
1 = (γ′Z1 + γZ ′
1) + (a′1b3 + a1b
′
3) + i(Y1b
′
1 − Y ′
1b1) + i(X ′
1b2 −X1b
′
2)
+ i(a′2Z3 − a′3Z2)− i(a2Z
′
3 − a3Z
′
2) + (Y2X
′
3 − Y3X
′
2) + (Y ′
2X3 − Y ′
3X2),
Z ′′
2 = (γ′Z2 + γZ ′
2) + (a′2b3 + a2b
′
3) + i(Y2b
′
1 − Y ′
2b1) + i(X ′
2b2 −X2b
′
2)
+ i(a′3Z1 − a′1Z3)− i(a3Z
′
1 − a1Z
′
3) + (Y3X
′
1 − Y1X
′
3) + (Y ′
3X1 − Y ′
1X3),
Z ′′
3 = (γ′Z3 + γZ ′
3) + (a′3b3 + a3b
′
3) + i(Y3b
′
1 − Y ′
3b1) + i(X ′
3b2 −X3b
′
2)
+ i(a′1Z2 − a′2Z1)− i(a1Z
′
2 − a2Z
′
1) + (Y1X
′
2 − Y2X
′
1) + (Y ′
1X2 − Y ′
2X1). (7.4)
With the help of the index notation
X = C(1), Y = C(2), Z = C(3),
32 V.M. Red’kov, A.A. Bogush and N.G. Tokarevskaya
it is easy to see a cyclic symmetry in the above relationships:
γ′′ = γ′γ + a′kak + b′kbk + C
(1)′
k C
(1)
k + C
(2)′
k C
(2)
k + C
(3)′
k C
(3)
k ,
a′′k = γ′ak + γa′k + (b′1C
(1)
k + b1C
(1)′
k ) + (b′2C
(2)
k + b2C
(2)′
k ) + (b′3C
(3)
k ) + b3C
(3)′
k )
+ iεklna′lan + iεklnC
(1)′
l C(1)
n + iεklnC
(2)′
l C(2)
n + iεklnC
(3)′
l C(3)
n ,
b′′k = γ′bk + γb′k + iεklnb′lbn + (a′1C
(k)
1 + a1C
(k)′
1 ) + (a′2C
(k)
2 + a2C
(k)′
2 )
+ (a′3C
(k)
3 + a3C
(k)′
3 ) + iεklnC(l)′
m C(n)
m ,
C
(1)
k = γ′C
(1)
k + γC
(1)′
k + (a′kb1 + akb
′
1) + iε(1)ln(C(l)′
k bn − C
(l)
k b′n)
+ iεkln(a′lC
(1)
n − alC
(1)′
n ) + εkln(C(2)′
l C(3)
n + C
(2)
l C(3)′
n ),
C
(2)
k = γ′C
(2)
k + γC
(2)′
k + (a′kb2 + akb
′
2) + iε(2)ln(C(l)′
k bn − C
(l)
k b′n)
+ iεkln(a′lC
(2)
n − alC
(2)′
n ) + εkln(C(3)′
l C(1)
n + C
(3)
l C(1)′
n ),
C
(3)
k = γ′C
(3)
k + γC
(3)′
k + (a′kb3 + akb
′
3) + iε(3)ln(C(l)′
k bn − C
(l)
k b′n)
+ iεkln(a′lC
(3)
n − alC
(3)′
n ) + εkln(C(1)′
l C(2)
n + C
(1)
l C(2)′
n ). (7.5)
It is readily seen that these group multiplication laws (7.4), (7.5) permit 15 two-parametric
subgroups:
(γ, a) ∈ {(γ, a1), (γ, a2), (γ, a3), (γ, b1), (γ, b2), γ, b3), (γ, X1), (γ, X2), (γ, X3),
(γ, Y1), (γ, Y2), (γ, Y3), (γ, Z1), (γ, Z2), (γ, Z3)}
with the same composition law:
γ′′ = γ′γ + a′a, a′′ = γ′a + γa′,
which in variables γ = W cos φ, a = iW sinφ takes the form
W ′′ = W ′W, α′′ = α′ + α.
The variable W is determined by det G(W,α) = W 4, the choice W = 1 guarantees det G = +1.
All 15 basis elements Λ(ρ) ∈ {αk, βk, Ak, Bk, Ck} possess the same properties:
Λ+
(ρ) = Λ(ρ), Λ2
(ρ) = I.
Therefore, one can construct 15 different elementary unitary (at real valued parameters) matrices
by one the same recipe:
U(ρ) = eiφ(ρ)Λ(ρ) = cos φ(ρ) + i sinφ(ρ)Λ(ρ),
U+
(ρ) = U−1
(ρ) = e−iφ(ρ)Λ(ρ) = cos φ(ρ) − i sinφ(ρ)Λ(ρ).
The whole set of unitary matrices SU(4) may be constructed on the basis of a simple factorized
formula:
U = eiφ(1)Λ(1) · · · eiφ(15)Λ(15) .
The order of the factors is important. Every such order leads us to a definite parametrization
for the group SU(4) – all them seem to be equivalent.
On Parametrization of GL(4, C) and SU(4) 33
In the end of the section let us write down the explicit form of these 15 elementary unitary
transformations:
α1 =
∣∣∣∣ σ2 0
0 −σ2
∣∣∣∣ , α2 =
∣∣∣∣ 0 i
−i 0
∣∣∣∣ , α3 =
∣∣∣∣ 0 σ2
σ2 0
∣∣∣∣ ,
Uα
1 =
∣∣∣∣ cos φ + i sinφσ2 0
0 cos φ− i sinφσ2
∣∣∣∣ , Uα
2 =
∣∣∣∣ cos φ − sinφ
sinφ cos φ
∣∣∣∣ ,
Uα
3 =
∣∣∣∣ cos φ i sinφσ2
i sinφσ2 cos φ
∣∣∣∣ , β1 =
∣∣∣∣ σ2 0
0 σ2
∣∣∣∣ , β2 =
∣∣∣∣ 0 −iσ3
iσ3 0
∣∣∣∣ ,
β3 =
∣∣∣∣ 0 −iσ1
iσ1 0
∣∣∣∣ , Uβ
1 =
∣∣∣∣ cos φ + i sinφσ2 0
0 cos φ + i sinφσ2
∣∣∣∣ ,
Uβ
2 =
∣∣∣∣ cos φ sinφσ3
− sinφσ3 cos φ
∣∣∣∣ , Uβ
3 =
∣∣∣∣ cos φ sin φσ1
− sinφσ1 cos φ
∣∣∣∣ , A1 =
∣∣∣∣ I 0
0 −I
∣∣∣∣ ,
A2 =
∣∣∣∣ 0 iσ2
−iσ2 0
∣∣∣∣ , A3 =
∣∣∣∣ 0 I
I 0
∣∣∣∣ , UA
1 =
∣∣∣∣ cos φ + i sinφ 0
0 cos φ− i sinφ
∣∣∣∣ ,
UA
2 =
∣∣∣∣ cos φ − sinφσ2
sinφσ2 cos φ
∣∣∣∣ , UA
3 =
∣∣∣∣ cos φ i sinφ
i sinφ cos φ
∣∣∣∣ , B1 =
∣∣∣∣ 0 σ1
σ1 0
∣∣∣∣ ,
B2 =
∣∣∣∣ −σ3 0
0 −σ3
∣∣∣∣ , B3 =
∣∣∣∣ −σ1 0
0 σ1
∣∣∣∣ , UB
1 =
∣∣∣∣ cos φ i sinφσ1
i sinφσ1 cos φ
∣∣∣∣ ,
UB
2 =
∣∣∣∣ cos φ− i sinφσ3 0
0 cos φ− i sinφσ3
∣∣∣∣ ,
UB
3 =
∣∣∣∣ cos φ− i sinφσ1 0
0 cos φ + i sinφσ1
∣∣∣∣ , C1 =
∣∣∣∣ 0 −σ3
−σ3 0
∣∣∣∣ ,
C2 =
∣∣∣∣ −σ1 0
0 −σ1
∣∣∣∣ , C3 =
∣∣∣∣ σ3 0
0 −σ3
∣∣∣∣ , UC
1 =
∣∣∣∣ cos φ −i sinφσ3
−i sinφσ3 cos φ
∣∣∣∣ ,
UC
2 =
∣∣∣∣ cos φ− i sinφσ1 0
0 cos φ− i sinφσ1
∣∣∣∣ ,
UC
3 =
∣∣∣∣ cos φ + i sinφσ3 0
0 cos φ− i sinφσ3
∣∣∣∣ . (7.6)
Certainly, these relations provide us with 15 elementary solutions of the unitarity equations (3.3).
For instance, the generator α2 gives rise to the above 1-parametric Abelian subgroup G0(α);
whereas the above 4-parametric subgroup G0 × SU(2) (5.15) is generated by (α2;β1, B2, C2).
The question is how one could describe all combinations of the above 15 simple sub-solutions
by a single unifying formula – the latter should evidently exist.
8 On factorization SU(4) and the group fine-structure
On the basis of 9 matrices (7.2) one can construct six 3-dimensional sub-sets:
K = {A1 = α1β1, B2 = α2β2, C3 = α3β3},
L = {C1 = α1β3, A2 = α2β1, B3 = α3β2},
M = {B1 = α1β2, C2 = α2β3, A3 = α3β1},
K ′ = {−C ′
1 = −α1β3, −B′
2 = −α2β2, −C ′
3 = −α3β3},
L′ = {−B′
1 = −α1β2, −A′
2 = −α2β1, −B′
3 = −α3β2},
M ′ = {−A′
1 = −α1β1, −C ′
2 = −α2β3, −B′
3 = −α3β2},
34 V.M. Red’kov, A.A. Bogush and N.G. Tokarevskaya
(one may recall the rule to calculate the determinant of a 3× 3 matrix) with the same commu-
tation relations:
Γ1Γ2 = −Γ3, Γ2Γ1 = −Γ3, Γ1Γ2 − Γ2Γ1 = 0, Γ1Γ2 + Γ2Γ1 = −2Γ3, (8.1)
and analogous by cyclic symmetry. The whole set of the above 9 matrices coincides with
~α, ~β, K, L, M , (8.2)
or
~α, ~β, K ′, L′, M ′.
It suffices to consider one variant, let it be (8.2). It seem reasonable to suppose that arbitrary
element from GL(4, C) can be factorized as follows
S = ei~a~αei~b~βeikKeilLeimM . (8.3)
When all parameters are real-valued, the formula provides us with the rule to construct elements
from SU(4) group14. The order of factors might be different. Let us specify the group law for
these 5 subsets. First are the two groups:
ei~a~α = cos a + i sin a(n1α1 + n2α2 + n3α3), ei~b~β = cos b + i sin b(n1β1 + n2β2 + n3β3).
They are isomorphic, so one can consider only the first one:
ei~a~α = cos a + i sin a(n1α1 + n2α2 + n3α3) = x0 − ix1α1 − ix2α2 − ix3α3.
Multiplying two matrices we arrive at
x′′0 = x′0x0 − x′1x1 − x′2x2 − x′3x3, x′′1 = x′0x1 + x′1x0 + (x′2x3 − x′2x3),
x′′2 = x′0x2 + x′2x0 + (x′3x1 − x′1x3), x′′3 = x′0x3 + x′3x0 + (x′1x2 − x′2x1). (8.4)
Parameters (x0, xi) should obey
x2
0 + x2
1 + x2
2 + x2
3 = 1 ⇐⇒ det ei~a~α = +1.
The inverse matrix looks
(x0,x)−1 = (x0,−x).
With real (x0, xi) we have a group isomorphic to SU(2), spinor covering for SO(3, R):
c =
x
x0
, c′′ =
c′ + c + c′ × c
1− c′c
.
At complex (x0, xi) we have a group isomorphic to GL(2, C), spinor covering for SO(3, C) or
Lorentz group.
Now let us turn to finite transformations from remaining subsets. It is readily verified that
these 1-parametric finite elements
eiy1Γ1 = cos y1 + i sin y1Γ1, eiy2Γ1 = cos y2 + i sin y2Γ2, eiy3Γ3 = cos y3 + i sin y3Γ3,
14Just such a structure was described in [40].
On Parametrization of GL(4, C) and SU(4) 35
commute with each other:
eiy1Γ1eiy2Γ2 = (cos y1 + i sin y1Γ1)(cos y2 + i sin y2Γ2) =
= cos y1 cos y2 + i cos y1 sin y2Γ2 + i cos y2 sin y1Γ1 + sin y1 sin y2Γ3,
eiy2Γ2eiy1Γ1 = (cos y2 + i sin y2Γ2)(cos y1 + i sin y1Γ1) =
= cos y2 cos y1 + i cos y2 sin y1Γ1 + i cos y1 sin y2Γ2 + sin y2 sin y1Γ3,
that is eiy1Γ1eiy2Γ2 = eiy2Γ2eiy1Γ1 , and so on. Evidently, this property correlates with the com-
mutative relations (8.1). Thus, each of tree subgroups can be constructed as multiplying of
elementary 1-parametric commuting transformations. Their explicit forms are:
subgroup K
K = {A1 = α1β1, B2 = α2β2, C3 = α3β3},
A1 =
∣∣∣∣ I 0
0 −I
∣∣∣∣ , B2 =
∣∣∣∣ −σ3 0
0 −σ3
∣∣∣∣ , C3 =
∣∣∣∣ σ3 0
0 −σ3
∣∣∣∣ ,
eik1K1 = cos k1 + i sin k1A1, eik2K1 = cos k2 + i sin k2B2,
eik3K3 = cos k3 + i sin k3C3;
subgroup L
L = {C1 = α1β3, A2 = α2β1, B3 = α3β2},
C1 =
∣∣∣∣ 0 −σ3
−σ3 0
∣∣∣∣ , A2 =
∣∣∣∣ 0 iσ2
−iσ2 0
∣∣∣∣ , B3 =
∣∣∣∣ −σ1 0
0 σ1
∣∣∣∣ ,
eil1L1 = cos l1 + i sin l1C1, eil2L1 = cos l2 + i sin l2A2, eil3L3 = cos l3 + i sin l3B3;
subgroup M
M = {B1 = α1β2, C2 = α2β3, A3 = α3β1},
B1 =
∣∣∣∣ 0 σ1
σ1 0
∣∣∣∣ , C2 =
∣∣∣∣ −σ1 0
0 −σ1
∣∣∣∣ , A3 =
∣∣∣∣ 0 I
I 0
∣∣∣∣ ,
eim1M1 = cos m1 + i sinm1B1, eim2M1 = cos m2 + i sinm2C2,
eim3M3 = cos m3 + i sinm3A3.
One additional note should be made. In the recent paper by A. Gsponer [35] on the quater-
nion approach to the problem of building the finite transformations from SU(3) and SU(4) an
important point was to divide 15 basis 4× 4 matrices into three sets:
set A of antisymmetrical matrices,
set S of symmetrical matrices,
set D of diagonal traceless ones.
It is easily seen that
set A = {αi ⊕ βi} ;
set S = {A2, A3, B1, B3, C1, C2} = {L⊕M};
set D = {A1, B2, C3 = K}.
36 V.M. Red’kov, A.A. Bogush and N.G. Tokarevskaya
Turning again to relationship (8.3), let us rewrite it as follows
S = ei~a~α[eikKeilLeimM ]ei~b~β.
which exactly corresponds to the structure used in [35] in connection with the Lanczos decom-
position theorem [43].
Several last comments should be made. On the basis of 15 matrices
α1, α2, α3, β1, β2, β3,
A1 = α1β1, B1 = α1β2, C1 = α1β3,
A2 = α2β1, B2 = α2β2, C2 = α2β3,
A3 = α3β1, B3 = α3β2, C3 = α3β3
one can easily see the following 20 ways to separate SU(2) subgroups (certainly, those arise at
real-valued parameters; complex-valued parameters give rise to linear subgroups GL(2, C)):
(α1, α2, α3), (β1, β2, β3),
(α1, A2, A3), (A1, α2, A3), (A1, A2, α3),
(α1, B2, B3), (B1, α2, B3), (B1, B2, α3),
(α1, C2, C3), (C1, α2, C3), (C1, C2, α3),
(β1, B1, C1), (β1, B2, C2), (β1, B3, C3),
(A1, β2, C1), (A2, β2, C2), (A3, β2, C3),
(A1, B1, β3), (A2, B2, β3), (A3, B3, β3). (8.5)
Certainly, they provide us with twenty different 3-parametric solutions of the unitarity equa-
tions (3.3). Such 3-subgroups might be used as bigger elementary blocks in constructing of a
general transformation [25, 28].
For instance, for the variant from (8.5): (α1, A2, A3) =⇒ (a1, X2, X3) the general multiplica-
tion law (7.4) gives
γ′′ = γ′γ + a′1a1 + X ′
2X2 + X ′
3X3, a′′1 = γ′a1 + a′1γ + i(X ′
2X3 −X ′
3X2),
a′′2 = 0, a′′3 = 0, b′′1 = 0, b′′2 = 0, b′′3 = 0, X ′′
1 = 0,
X ′′
2 = γ′X2 + γX ′
2 + +i(−a′1X2) + a1X
′
3), X ′′
3 = γ′X3 + γX ′
3 + i(a′1X2 − a1X
′
2),
Y ′′
1 = 0, Y ′′
2 = 0, Y ′′
3 = 0, Z ′′
1 = 0, Z ′′
2 = 0, Z ′′
3 = 0,
that is
γ′′ = γ′γ + a′1a1 + X ′
2X2 + X ′
3X3, a′′1 = γ′a1 + a′1γ + i(X ′
2X3 −X ′
3X2),
X ′′
2 = γ′X2 + γX ′
2 + i(X ′
3a1 − a′1X2), X ′′
3 = γ′X3 + γX ′
3 + i(a′1X2 − a1X
′
2),
which coincides with equation (8.4). The same can be done for any other representative
from (8.5).
9 On pseudo-unitary group SU(2, 2)
As said, the Dirac basis was used previously [9, 40, 48, 50] in studying the exponentials for
SU(2, 2) matrices. Let us show how the above formalism can apply to this pseudo-unitary
group SU(2, 2). Transformations from SU(2, 2) should leave invariant the following form
(z∗1 , z
∗
2 , z
∗
3 , z
∗
4)
∣∣∣∣∣∣∣∣
+1 0 0 0
0 +1 0 0
0 0 −1 0
0 0 0 −1
∣∣∣∣∣∣∣∣
∣∣∣∣∣∣∣∣
z1
z2
z3
z4
∣∣∣∣∣∣∣∣ , z+ηz = z
′+ηz′,
On Parametrization of GL(4, C) and SU(4) 37
which leads to
z′ = Uz, where U+η = ηU−1, η =
∣∣∣∣∣∣∣∣
+1 0 0 0
0 +1 0 0
0 0 −1 0
0 0 0 −1
∣∣∣∣∣∣∣∣ .
Any generator Λ′ of those transformations must obey relation
Uk = eiaΛ′
k , (Λ′
k)
+η = ηΛ′
k, k ∈ {1, . . . , 15}
or allowing for identity η = −γ5
(Λ′
k)
+γ5 = γ5Λ′
k, k ∈ {1, . . . , 15}.
All generators Λ′
k of the group SU(2, 2) can be readily constructed on the basis of the known
generators Λk of SU(4) (see (6.1))
Λ′
1 = Λ1 = γ5, Λ′
1 = iΛ1 = iγ0, Λ′
3 = iΛ3 = −γ5γ0,
Λ′
4 = iΛ4 = −γ1, Λ′
5 = iΛ5 = iγ5γ1, Λ′
6 = iΛ6 = −γ2,
Λ′
7 = iΛ7 = iγ5γ2, Λ′
8 = iΛ8 = −γ3, Λ′
9 = iΛ9 = iγ5γ3,
Λ′
10 = Λ10 = 2σ01, Λ′
11 = Λ11 = 2σ02, Λ′
12 = Λ12 = 2σ03,
Λ′
13 = Λ13 = 2iσ12, Λ′
14 = Λ14 = 2iσ23, Λ′
15 = Λ15 = 2iσ31. (9.1)
Basis elements may be listed as follows:
α′1 = α1 = γ0γ2, α′2 = iα2 = −γ0γ5, α′3 = iα3 = iγ5γ2
(α′1)
2 = I, (α′2)
2 = −I, (α′3)
2 = −I,
α′2α
′
3 = −iα1, α′3α
′
1 = iα′2, α′1α
′
2 = iα′3;
β′1 = β1 = iγ3γ1, β′2 = iβ2 = −γ3, β′3 = iβ3 = −γ1,
(β′1)
2 = I, (β′2)
2 = −I, (β′3)
2 = −I,
β′2β
′
3 = −iβ1, β′3β
′
1 = iβ′2, β′1β
′
2 = iβ′3.
These two sets commute with each other: αjβk = βkαj ; and their multiplications give remaining
9 elements:
A′
1 = A1 = α′1β
′
1 = −γ5, B′
1 = iB1 = α′1β
′
2 = iγ5γ1, C ′
1 = iC1 = α′1β
′
3 = iγ3γ5,
A′
2 = iA2 = α′2β
′
1 = γ2, B′
2 = −B2 = α′2β
′
2 = +iγ1γ2, C ′
2 = −C2 = α′2β
′
3 = +iγ2γ3,
A′
3 = iA3 = α′3β
′
1 = iγ0, B′
3 = −B3 = α′3β
′
2 = −γ0γ1, C ′
3 = −C3 = α′3β
′
3 = −γ0γ3.
Making in relations (7.6) a formal change in accordance with
eiaΛ = cos a + i sin aΛ, Λ′ = iΛ,
cos a + i sin aΛ = eiaΛ =⇒ eiaΛ′
= cos ia + i sin iaΛ = cosh a− sinh aΛ,
we arrive at explicit form of elementary pseudo-unitary SU(2, 2)-transformations:
α′1 = α1 =
∣∣∣∣ σ2 0
0 −σ2
∣∣∣∣ , Uα
1 =
∣∣∣∣ cos φ + i sinφσ2 0
0 cos φ− i sinφσ2
∣∣∣∣ ,
α′2 = iα2 =
∣∣∣∣ 0 −1
1 0
∣∣∣∣ , Uα
2 =
∣∣∣∣ coshχ −i sinhχ
i sinhχ coshχ
∣∣∣∣ ,
38 V.M. Red’kov, A.A. Bogush and N.G. Tokarevskaya
α′3 = iα3 =
∣∣∣∣ 0 iσ2
iσ2 0
∣∣∣∣ , Uα
3 =
∣∣∣∣ coshχ − sinhχσ2
sinhχσ2 coshχ
∣∣∣∣ ,
β′1 = β1 =
∣∣∣∣ σ2 0
0 σ2
∣∣∣∣ , Uβ
1 =
∣∣∣∣ cos φ + i sinφσ2 0
0 cos φ + i sinφσ2
∣∣∣∣ ,
β′2 = iβ2 =
∣∣∣∣ 0 σ3
−σ3 0
∣∣∣∣ , Uβ
2 =
∣∣∣∣ coshχ i sinhχσ3
−i sinhχσ3 coshχ
∣∣∣∣ ,
β′3 = iβ3 =
∣∣∣∣ 0 σ1
−σ1 0
∣∣∣∣ , Uβ
3 =
∣∣∣∣ coshχ i sinhχσ1
−i sinhχσ1 coshχ
∣∣∣∣ ,
A′
1 = A1 =
∣∣∣∣ I 0
0 −I
∣∣∣∣ , UA
1 =
∣∣∣∣ cos φ + i sinφ 0
0 cos φ− i sin φ
∣∣∣∣ ,
A′
2 = iA2 =
∣∣∣∣ 0 −σ2
σ2 0
∣∣∣∣ , UA
2 =
∣∣∣∣ coshχ −i sinhχσ2
i sinhχσ2 coshχ
∣∣∣∣ ,
A′
3 = iA3 =
∣∣∣∣ 0 i
i 0
∣∣∣∣ , UA
3 =
∣∣∣∣ coshχ − sinhχ
− sinhχ coshχ
∣∣∣∣ ,
B′
1 = iB1 =
∣∣∣∣ 0 iσ1
iσ1 0
∣∣∣∣ , UB
1 =
∣∣∣∣ coshχ − sinhχσ1
− sinhχσ1 coshχ
∣∣∣∣ ,
B′
2 = −B2 =
∣∣∣∣ σ3 0
0 σ3
∣∣∣∣ , UB
2 =
∣∣∣∣ cos φ + i sin φσ3 0
0 cos φ + i sinφσ3
∣∣∣∣ ,
B′
3 = −B3 =
∣∣∣∣ σ1 0
0 −σ1
∣∣∣∣ , UB
3 =
∣∣∣∣ cos φ + i sinφσ1 0
0 cos φ− i sinφσ1
∣∣∣∣ ,
C ′
1 = iC1 =
∣∣∣∣ 0 −iσ3
−iσ3 0
∣∣∣∣ , UC
1 =
∣∣∣∣ coshχ sinhχσ3
sinhχσ3 coshχ
∣∣∣∣ ,
C ′
2 = −C2 =
∣∣∣∣ σ1 0
0 σ1
∣∣∣∣ , UC
2 =
∣∣∣∣ cos φi sinφσ1 0
0 cos φ + i sinφσ1
∣∣∣∣ ,
C ′
3 = −C3 =
∣∣∣∣ −σ3 0
0 +σ3
∣∣∣∣ , UC
3 =
∣∣∣∣ cos φ− i sinφσ3 0
0 cos φ + i sinφσ3
∣∣∣∣ . (9.2)
We can easily obtain unitarity equations for SU(2, 2) group, simple solutions to which are
given by (9.2). Indeed, taking into account the formulas (see (3.1)
G+η =
∣∣∣∣∣ k∗0 + k∗~σ −l∗0 − l∗~σ
n∗0 − n∗~σ m∗
0 −m∗~σ
∣∣∣∣∣
∣∣∣∣ −I 0
0 +I
∣∣∣∣ =
∣∣∣∣∣ −k∗0 − k∗~σ −l∗0 − l∗~σ
−n∗0 + n∗~σ m∗
0 −m∗~σ
∣∣∣∣∣ ,
ηG−1 =
∣∣∣∣ −I 0
0 I
∣∣∣∣
∣∣∣∣∣ k′0 + k′~σ n′0 − n′~σ
−l′0 − l′~σ m′
0 −m′~σ
∣∣∣∣∣ =
∣∣∣∣∣ −k′0 − k′~σ −n′0 + n′~σ
−l′0 − l′~σ m′
0 −m′~σ
∣∣∣∣∣ ,
from G+η = ηG−1 we produce
k∗0 = k′0, k∗ = k′, m∗
0 = m′
0, m∗ = m′,
l∗0 = n′0, l∗ = −n′, n∗0 = l′0, n∗ = −l′.
These relations differ from analogous ones (3.2) for SU(4) group only in all signs of the second
line. Therefore, the unitarity conditions for the group SU(2, 2) are (compare with (3.3))
k∗0 = +k0(mm) + m0(ln) + l0(nm)− n0(lm) + il(m× n),
m∗
0 = +m0(kk) + k0(nl) + n0(lk)− l0(nk)− in(k × l),
k∗ = −k(mm)−m(ln)− l(nm) + n(lm) + 2l× (n×m)
On Parametrization of GL(4, C) and SU(4) 39
+ im0(n× l) + il0(n×m) + in0(l×m),
m∗ = −m(kk)− k(nl)− n(lk) + l(nk) + 2n× (l× k)
− ik0(l× n)− in0(l× k)− il0(n× k),
−l∗0 = +k0(nm)−m0(kn) + l0(nn) + n0(km) + ik(n×m),
−n∗0 = +m0(lk)− k0(ml) + n0(ll) + l0(mk)− im(l× k),
−l∗ = −k(nm) + m(kn)− l(nn)− n(km) + 2k × (m× n)
+ ik0(m× n) + im0(k × n) + in0(m× k),
−n∗ = −m(kl) + k(ml)− n(ll)− l(mk) + 2m× (k × l)− im0(k × l)
− ik0(m× l)− il0(k ×m).
Several words on factorization SU(2, 2) = SU(1, 1)× [K × L×M ]× SU(1, 1) and a further
group fine-structure for SU(2, 2). On the basis of 9 matrices
A′
1 = A1 = α′1β
′
1 = −γ5, B′
1 = iB1 = α′1β
′
2 = iγ5γ1, C ′
1 = iC1 = α′1β
′
3 = iγ3γ5,
A′
2 = iA2 = α′2β
′
1 = γ2, B′
2 = −B2 = α′2β
′
2 = +iγ1γ2, C ′
2 = −C2 = α′2β
′
3 = iγ2γ3,
A′
3 = iA3 = α′3β
′
1 = iγ0, B′
3 = −B3 = α′3β
′
2 = −γ0γ1, C ′
3 = −C3 = α′3β
′
3 = −γ0γ3
one can construct three 3-dimensional subsets (omitting three others):
K ′ = {A′
1 = A1 = α′1β
′
1, B′
2 = −B2 = α′2β
′
2, C ′
3 = −C3 = α′3β
′
3},
L′ = {C ′
1 = iC1 = α′1β
′
3, A′
2 = iA2 = α′2β
′
1, B3 = −B3 = α′3β
′
2},
M ′ = {B′
1 = iB1 = α′1β
′
2, C ′
2 = −C2 = α′2β
′
3, A′
3 = iA3 = α′3β
′
1},
with the same commutation relations:
Γ1Γ2 = +Γ3, Γ2Γ1 = +Γ3, Γ1Γ2 − Γ2Γ1 = 0, Γ1Γ2 + Γ2Γ1 = +2Γ3,
and analogous ones by cyclic symmetry. Arbitrary element from SU(2, 2) can be factorized as
follows
S = ei~a~α′
ei~b~β′
eikK′
eilL′
eimM ′
,
all parameters are real-valued. Let us specify the group law for these 5 sub-sets. Two groups
ei~a~α′
, ei~b~β′
are isomorphic so one can consider only the first one:
ei~a~α′
= I + i(a1α
′
1 + a2α
′
2 + a3α
′
3)− 1
2!(a
2
1 − a2
2 − a2
3)
− i 1
3!(a
2
1 − a2
2 − a2
3)(a1α
′
1 + a2α
′
2 + a3α
′
3) + 1
4!(a
2
1 − a2
2 − a2
3)
2 + · · · .
In the variables
ai = ani, n2
1 − n2
2 − n2
3 = 1
we have
ei~a~α′
= I + i(n1α
′
1 + n2α
′
2 + n3α
′
3)a− 1
2!a
2 − i 1
3!(n1α
′
1 + n2α
′
2 + n3α
′
3)a
3
+ 1
4!a
4 + 1
5!(n1α
′
1 + n2α
′
2 + in3α
′
3)a
5 − 1
6!a
6 + · · · ,
that is
ei~a~α′
= cos a + i sin a(n1α
′
1 + n2α
′
2 + n3α
′
3) = x0 − ix1α
′
1 − ix2α
′
2 − ix3α
′
3.
40 V.M. Red’kov, A.A. Bogush and N.G. Tokarevskaya
Multiplying two matrices we arrive at
x′′0 = x′0x0 − x′1x1 + x′2x2 + x′3x3, x′′1 = x′0x1 + x′1x0 − (x′2x3 − x′2x3),
x′′2 = x′0x2 + x′2x0 + (x′3x1 − x′1x3), x′′3 = x′0x3 + x′3x0 + (x′1x2 − x′2x1).
The inverse matrix looks
(x0,x)−1 = (x0,−x).
Parameters (x0, xi) should obey the following condition
x2
0 + x2
1 − x2
2 − x2
3 = 1.
In the variables a, ni it will look
cos2 a + sin2 a(n2
1 − n2
2 − n2
3) = 1, n2
1 − n2
2 − n2
3 = 1.
For three particular cases we will have:
n = (1, 0, 0), cos2 a + sin2 a = 1, eiaα′
1 = cos a + i sin aα′1, a ∈ R;
n = (0, i, 0), cos2 a− sin2 a = 1,
a = ib, cos ib = cosh b, sin ib = i sinh b, b ∈ R,
eiaα′
2 = cos a + i sin an2α
′
2 = cosh b− i sinh bα′2;
n = (0, 0, i), cos2 a− sin2 a = 1,
a = ib, cos ib = cosh b, sin ib = i sinh b, b ∈ R,
eiaα′
3 = cos a + i sin an3α
′
3 = cosh b− i sinh bα′3.
Now let us turn to finite transformations from remaining sub-sets K ′, L′, M ′. Each of tree
subgroups can be constructed as multiplying of elementary 1-parametric commuting transfor-
mations. Their explicit forms are:
subgroup K ′
K ′ = {A′
1 = A1 = α′1β
′
1, B′
2 = −B2 = α′2β
′
2, C ′
3 = −C3 = α′3β
′
3},
A′
1 =
∣∣∣∣ I 0
0 −I
∣∣∣∣ , B′
2 =
∣∣∣∣ σ3 0
0 σ3
∣∣∣∣ , C ′
3 =
∣∣∣∣ −σ3 0
0 σ3
∣∣∣∣ ;
subgroup L′
L′ = {C ′
1 = iC1 = α′1β
′
3, A′
2 = iA2 = α′2β
′
1, B′
3 = −B3 = α′3β
′
2},
C ′
1 =
∣∣∣∣ 0 −iσ3
−iσ3 0
∣∣∣∣ , A′
2 =
∣∣∣∣ 0 −σ2
σ2 0
∣∣∣∣ , B′
3 =
∣∣∣∣ σ1 0
0 −σ1
∣∣∣∣ ;
subgroup M ′
M ′ = {B′
1 = iB1 = α1β2, C ′
2 = −C2 = α2β3, A′
3 = iA3 = α3β1},
B′
1 =
∣∣∣∣ 0 iσ1
iσ1 0
∣∣∣∣ , C ′
2 =
∣∣∣∣ σ1 0
0 σ1
∣∣∣∣ , A′
3 =
∣∣∣∣ 0 i
i 0
∣∣∣∣ .
On the basis of 15 matrices
α′1, α′2, α′3, β′1, β′2, β′3,
On Parametrization of GL(4, C) and SU(4) 41
A′
1 = α′1β
′
1, B′
1 = α′1β
′
2, C ′
1 = α′1β
′
3,
A′
2 = α′2β
′
1, B′
2 = α′2β
′
2, C ′
2 = α′2β
′
3,
A′
3 = α′3β
′
1, B′
3 = α′3β
′
2, C ′
3 = α′3β
′
3
one can easily see the following 20 ways to separate SU(1, 1) subgroups:
(α′1, α
′
2, α
′
3), (β′1, β
′
2, β
′
3),
(α′1, A
′
2, A
′
3), (A′
1, α
′
2, A
′
3), (A′
1, A
′
2, α
′
3),
(α′1, B
′
2, B
′
3), (B′
1, α
′
2, B
′
3), (B′
1, B
′
2, α
′
3),
(α′1, C
′
2, C
′
3), (C ′
1, α
′
2, C
′
3), (C ′
1, C
′
2, α
′
3),
(β′1, B
′
1, C
′
1), (β′1, B
′
2, C
′
2), (β′1, B
′
3, C
′
3),
(A′
1, β
′
2, C
′
1), (A′
2, β
′
2, C
′
2), (A′
3, β
′
2, C
′
3),
(A1, B1, β3), (A2, B2, β3), (A3, B3, β3).
such 3-subgroups might be used as bigger elementary blocks in constructing a general transfor-
mation.
10 On pseudo-unitary group SU(3, 1)
Let us show how the above formalism can apply to the pseudo-unitary group SU(3, 1). Trans-
formations from SU(3, 1) should leave invariant the following form
(z∗1 , z
∗
2 , z
∗
3 , z
∗
4)
∣∣∣∣∣∣∣∣
+1 0 0 0
0 +1 0 0
0 0 +1 0
0 0 0 −1
∣∣∣∣∣∣∣∣
∣∣∣∣∣∣∣∣
z1
z2
z3
z4
∣∣∣∣∣∣∣∣ , z+ηz = z
′+ηz′,
which leads to
z′ = Uz, where U+η = ηU−1, η =
∣∣∣∣∣∣∣∣
+1 0 0 0
0 +1 0 0
0 0 +1 0
0 0 0 −1
∣∣∣∣∣∣∣∣ .
Any generator Λ′ of those transformations must obey the relation
Uk = eiaΛ′
k , (Λ′
k)
+η = ηΛ′
k. k ∈ {1, . . . , 15}.
The matrix η is a linear combination
η = 1
2(2iσ12 − 2σ03 − γ5 + I) = 1
2(iγ1γ2 − γ0γ3 + iγ0γ1γ2γ3 + I).
All generators of the group SU(3, 1) can readily be constructed on the basis of the known
generators λk of SU(4) (see (6.2))
λ1, λ2, λ3, λ4, λ5, λ6, λ7, λ8, iλ9, iλ10,
iλ11, iλ12, iλ13, iλ14, λ15; (10.1)
generator λ9, . . . , λ14 are multiplied by imaginary unit i. Instead of λ8, λ15 one can introduce
other generator λ′8, λ′15 see (6.4) (diagonal generators are the same for group SU(4), SU(2, 2),
and SU(3, 1).
42 V.M. Red’kov, A.A. Bogush and N.G. Tokarevskaya
11 Discussion
Let us summarize the main point of the present treatment.
Parametrization of 4 × 4 matrices G of the complex linear group GL(4, C) in terms of four
complex 4-vector parameters (k, m, n, l) is investigated. Additional restrictions separating some
subgroups of GL(4, C) are given explicitly. In the given parametrization, the problem of in-
verting any 4 × 4 matrix G is solved. Expression for determinant of any matrix G is found:
det G = F (k, m, n, l). Unitarity conditions G+ = G−1 have been formulated in the form of
non-linear cubic algebraic equations including complex conjugation. Several simplest solutions
of these unitarity equations have been found: three 2-parametric subgroups G1, G2, G3 – each
of subgroups consists of two commuting Abelian unitary groups; 4-parametric unitary subgroup
consisting of a product of a 3-parametric group isomorphic SU(2) and 1-parametric Abelian
group.
The Dirac basis of generators Λk, being of Gell-Mann type, substantially differs from the
basis λi used in the literature on SU(4) group, formulas relating them are found – they permit
to separate SU(3) subgroup in SU(4). Special way to list 15 Dirac generators of GL(4, C)
can be used {Λk} = {αi ⊕ βj ⊕ (αiβj = K ⊕ L ⊕ M)}, which permit to factorize SU(4)
transformations according to S = ei~a~αei~b~βeikKeilLeimM , where two first factors commute with
each other and are isomorphic to SU(2) group, the three last are 3-parametric groups, each of
them consists of three Abelian commuting unitary subgroups. Besides, the structure of fifteen
Dirac matrices Λk permits to separate twenty 3-parametric subgroups in SU(4) isomorphic
to SU(2); those subgroups might be used as bigger elementary blocks in constructing a general
transformation SU(4). It is shown how one can specify the present approach for the unitary
group SU(2, 2) and SU(3, 1).
In principle, all different approaches used in the literature are closely related so that any
result obtained within one technique may be easily translated to any other. There is no sense
to persist in exploiting only one representation, thinking that it is much better than all others.
Success should lie in combining different techniques. For instance, Euler angles-based approach
provides us with the group elements in the separated variables form, which may be of a supreme
importance at calculating matrix elements of the group. In turn, a factorized subgroup- based
structure is of special interest in the particle physics and gauge theory of fundamental interaction.
Geometrical properties of the groups, their global structure, differences between orthogonal
groups and their double covering, and so on, seem to be most easily understood in terms of
bilinear functions in space of linear parameters: G = xjΛj , x′′j = ejklx
′
kxl.
We have no ground to think that only exponential functions eiΛ are suitable for exploration
into group structures. We may expect that in addition to Euler angles many other curvilinear
coordinates might be of value for studying of the group structure. For instance, in the case of
the group SO(4, C) we have known 34 such coordinate systems owing to Olevskiy investigation
[58] on 3-orthogonal coordinates in real Lobachevski space.
In conclusion, several words about possible application areas of the obtained results. The
main argument in favor of constructing the theory of unitary groups SU(4) (and related to it)
in terms of Dirac matrices is the role of spinor methods being widely adopted in physics. Let us
mention several problems most attractive for authors:
SU(2, 2) and conformal symmetry, massless particles;
classical Yang-Mills equations and gauge fields;
geometric phases for multi-level quantum systems;
composite structure of quarks and leptons;
SU(4) gauge models.
On Parametrization of GL(4, C) and SU(4) 43
In particular, description of the group SU(2, 2) in terms of matrices αj , βj should be of
great benefit in investigation of conformal symmetry in massless particles theory. For instance,
classical Maxwell equations in a medium can be presented in 4-dimensional complex matrix form
with the use of two sets of matrices, exploited above:(
−i
∂
∂x0
+ αj ∂
∂xj
)
M +
(
−i
∂
∂x0
+ βj ∂
∂xj
)
N =
1
ε0
∣∣∣∣ ρ
j/c
∣∣∣∣ ,
where
M =
∣∣∣∣ 0
M
∣∣∣∣ , M =
1
ε0
(D + iH/c) + (E + icB),
N =
∣∣∣∣ 0
N
∣∣∣∣ , N =
1
ε0
(D − iH/c)− (E − icB).
Acknowledgements
Authors are grateful to participants of the seminar of Laboratory of Physics of Fundamental
Interaction, National Academy of Sciences of Belarus for discussion. Authors are grateful to the
anonymous reviewer for many comments and advice improving the paper.
This work was supported by Fund for Basic Research of Belarus F07-314. We wish to thank
the Organizers of the Seventh International Conference “Symmetry in Nonlinear Mathematical
Physics” (June 24–30, 2007, Kyiv) and ICTP Office of External Activities for having given us
the opportunity to talk on this subject as well as for local and travel support.
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1 Introduction
2 On parameters of inverse transformations G-1
3 Unitarity condition
4 2-parametric subgroups in SU(4)
5 4-parametric unitary subgroup
6 On subgroups GL(3,C) and SU(3), expressions for Gell-Mann matrices through the Dirac basis
7 On the multiplication law for GL(4,C) in Dirac basis
8 On factorization SU(4) and the group fine-structure
9 On pseudo-unitary group SU(2,2)
10 On pseudo-unitary group SU(3,1)
11 Discussion
References
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