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European Journal of Applied Sciences – Vol. 12, No. 1
Publication Date: February 25, 2024
DOI:10.14738/aivp.121.16514
Cerveró, J. M. (2024). A Brief Review on the Conformal Scalar Equation. European Journal of Applied Sciences, Vol - 12(1). 470-
472.
Services for Science and Education – United Kingdom
A Brief Review on the Conformal Scalar Equation
José M. Cerveró
8-12 San Patricio Street, 37002. Salamanca, Spain
ABSTRACT: In this short note we develop a method to fully integrate the conformal scalar
equation in dimensions (N −1,1) where N is the total number of dimensions.
The Conformal Group in this flat space is SO(N,2) and the Conformal Scalar equation is readily
given by [1]:
{
∂
2
∂x1
2 +
∂
2
∂x2
2 +
∂
2
∂x3
2 + ⋯ +
∂
2
∂xN−1
2 −
∂
2
∂t2
} Φ+ Φ
N+2
N−2 = 0
where the exceptional case N = 2 has been fully discussed in reference [2]. Let us write down
the new set of conformal coordinates as in [3]. Always we set c=1:
r = √x1
2 + x2
2 + x3
2 ... + xN−1
2 and t; t± = t ± r
sin ω =
t+ + t−
(1 + t+
2
)
1
2 (1 + t−
2)
1
2
cos ω =
1 − t+t−
(1 + t+
2
)
1
2 (1 + t−
2)
1
2
sin ω̂ =
t+ − t−
(1 + t+
2
)
1
2 (1 + t−
2)
1
2
cos ω̂ =
1 + t+t−
(1 + t+
2
)
1
2 (1 + t−
2)
1
2
ω = arctan {
t+ + t−
1 − t+t−
}; ω̃ = arctan {
t+ − t−
1 + t+t−
};
In these coordinates, the D’Alambertian takes the following form:
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471
Cerveró, J. M. (2024). A Brief Review on the Conformal Scalar Equation. European Journal of Applied Sciences, Vol - 12(1). 470-472.
URL: http://dx.doi.org/10.14738/aivp.121.16514
{
∂
2
∂x1
2 +
∂
2
∂x2
2 +
∂
2
∂x3
2 + ⋯ +
∂
2
∂xN−1
2 −
∂
2
∂t2
} = −(cosω + cosω̃)
2
{
∂
2
∂ω2 +
∂
2
∂ω̃ 2
}+
+(N − 2)
(cosω + cos ω̃)
sin ω̃
{−(sin ω sin ω̃)
∂
∂ ω
+ (1 + cos ω cos ω̃)
∂
∂ ω̃
} +
+ {θ1...θN−2} → {derivatives of the harmonic spherical angles}
For example, in the important physical case N = 4 one easily obtains:
−(cosω + cosω̃)
2 {
∂
2
∂ω2 +
∂
2
∂ω̃
2
}
+ 2
(cosω + cos ω̃)
sin ω̃
{−(sin ω sin ω̃)
∂
∂ ω
+ (1 + cos ω cos ω̃)
∂
∂ ω̃
}
+
(cosω + cos ω̃)
2
sin2 ω̃
{
1
sin θ
∂
∂θ (sinθ
∂
∂θ) +
1
sin2θ
∂
∂ φ2
}
The theory is now being analyzed in a compact manifold independent of the harmonic angles
which ensures in turn compact solutions with a well-defined topology
SN−1 ⊗ S1 ≃ {θ1, θ2, θN−2, ω ̃} ⊗{ω}
Notice that ω and ω̃are the two angles responsible of the geometrical union of the two spheres.
Let us now try a solution of the form:
Φ = (cos ω + cos ω̃)
N−2
2 F(ω, ω̃)
After some algebra, one obtains for F (ω, ω̃) the following two-dimensional non-linear partial
differential equation:
{
∂
2
∂ω2 +
∂
2
∂ω̃
2
} F(ω, ω̃) − (N − 2) cot ω̃
∂
∂ω̃
F(ω, ω̃) +
(N − 2)
2
4
F(ω, ω̃) − F
N+2
N−2(ω, ω̃) = 0
to the physical N = 4 case:
{
∂
2
∂ω2 +
∂
2
∂ω̃
2
} F(ω, ω̃) − 2 cot ω̃
∂
∂ω̃
F(ω, ω̃) + F(ω, ω̃) − F
3
(ω, ω̃) = 0
For F being just a function of ω, the only compact solution is:
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Services for Science and Education – United Kingdom 472
European Journal of Applied Sciences (EJAS) Vol. 12, Issue 1, February-2024
F (ω) = √
2k
2
1 + k
2
cn
dn [
ω − ω0
√1 + k
2
‖k
2
]
No general solution of the two-dimensional non-linear partial differential equation has been
found. Such an equation does not pass the Painlevé test [4].
One can now define the Energy-Momentum Tensor and the Lagrangian for the dynamical
variables. Upon integration we obtain finite values for energy and action ([1], [3] and [5]):
ε =
6k
2
(1 + k
2)
2
π
2 A =
6k
2
(1 + k
2)
2
π
3
The last expressions are the main result of the conformal scalar equation as the vacuum is full
of finite energy balls in space and time. This property is in sharp difference with
electromagnetism. Here the vacuum is not trivial and has to be treated in a suitable manner
different as the main physical theory that we fully understand: Classical and Quantum
Electrodynamics.
References
[1] José M. Cerveró, Lawrence Jacobs and Craig Nohl “Elliptic solutions of Classical Yang-Mills Theory”, Physics
Letters, 69B, 351-353 (1977) and José M. Cerveró, “Explicit solutions of the conformal scalar equations in
arbitrary dimensions”, Journal of Mathematical Physics 23, 1466-1470, (1982)
[2] Edward Witten. “Some Exact Multipseudoparticle Solutions of Classical Yang-Mills Theory”, Physical
Review Letters 38, 121-125 (1977)
[3] José M. Cerveró. “On electric and magnetic configurations in the Yang-Mills Field Equations”, Letters in
Mathematical Physics 8, 233-237, (1984)
[4] P. Painlevé, “Sur les equations differentielles du second ordre et d’ordre superieur dont l’integrale
generale est uniforme”, Acta Mathematica 25, 1–85, (1902)
[5] Bruce M. Schechter, “Yang-Mills theory in the hypertorus” Physical Review 16D, 3015-3020, (1977) and M.
Lüscher, “SO (4) Symmetric solutions of the Yang-Mills Field Equations”, Physics Letters, 70B, 321-324,
(1977)