1. Introduction
Non-Abelian gauge theory offer the greatest promise to describe the elementary forces in nature [1]. We investigate here the solutions to the classical Yang-Mills
equations, they are non-Abelian field equations. However, solutions to non-linear
field equations are notoriously difficult to fine since there exists no general method
for discovering them. The usual approach is to make some guess as to the form of
the solution and insert it in to the field equations to see if it solves them. According to this approach, one have found some exact solutions to the Yang-Mills field
equations [2-6]. Besides the above approach one sees that there exists a connection
between the SU(2) Yang-Mills theory and scalar φ4 theory. From this connection
one reduces the complicated equations of motion of the Yang-Mills theory to the single equation of motion for scalar φ4 theory. Therefore one can find some properties
of the Yang-Mills theory [7-9].

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JOURNAL OF SCIENCE OF HNUE
Natural Sci., 2011, Vol. 56, No. 7, pp. 72-77
SOME PROPERTIES OF SU(2) YANG-MILLS THEORY
CONNECTING WITH φ4 THEORY
Nguyen Van Thuan
Hanoi National University of Education
E-mail: thuanvatli@yahoo.com
Abstract. In this paper we consider some properties of the SU(2) Yang-
Mills theory connecting with scalar φ4 theory. In the case when scalar field
φ is time independent, we find explicit solutions to equations of motion for
corresponding φ4 theory. The spatial component of the SU(2) Yang-Mills
potential corresponding with this solution is like potential of point magnetic
monopole.
Keywords: Gauge field theory, scalar field theory.
1. Introduction
Non-Abelian gauge theory offer the greatest promise to describe the elemen-
tary forces in nature [1]. We investigate here the solutions to the classical Yang-Mills
equations, they are non-Abelian field equations. However, solutions to non-linear
field equations are notoriously difficult to fine since there exists no general method
for discovering them. The usual approach is to make some guess as to the form of
the solution and insert it in to the field equations to see if it solves them. Accord-
ing to this approach, one have found some exact solutions to the Yang-Mills field
equations [2-6]. Besides the above approach one sees that there exists a connection
between the SU(2) Yang-Mills theory and scalar φ4 theory. From this connection
one reduces the complicated equations of motion of the Yang-Mills theory to the sin-
gle equation of motion for scalar φ4 theory. Therefore one can find some properties
of the Yang-Mills theory [7-9].
2. Content
2.1. Equations of motion of the SU(2) Yang-Mills field
Yang-Mills fields can be introduced in the following fashion. Consider a multi-
plet ψ(x) which transforms locally under the action of some gauge groupG according
to the rule
ψ(x)→ ψ′(x) = ω(x)ψ(x), (2.1)
72
Some properties of SU(2) Yang-Mills theory connecting with φ4 theory
where ω(x) belongs to the relevant representation of G. Let us define a derivative
Dµψ of ψ which has this same simple transformation property. We make the ansatz
Dµψ =
(
∂µ − igWµ(x)
)
ψ(x), (2.2)
where Wµ(x) is a matrix function. By assumption, Dµψ transforms like
Dµψ → D′µψ′ = ωDµψ, (2.3)
where D′µ = ∂µ − igW ′µ(x). This lead to(
∂µ − igW ′µ
)
ω = ω
(
∂µ − igWµ
)
, (2.4)
which can be rewritten
W ′µ = ωWω
−1 − (i/g)
(
∂µω
)
ω−1. (2.5)
Wµ(x) is the Yang-Mills potential in matrix form, and eq. (2.5) is the local
transformation rule for this potential under the gauge group G. If the potential
Wµ(x) is not introduced, then one can not define the covariant derivative Dµψ with
its simple transformation property (2.3).
Wµ(x) is analogous to the four-potential in electromagnetism. The Yang-Mills
forces, or field strengths, are defined as follows
Gµν = ∂µWν − ∂νWµ + (g/i)
[
Wµ,Wν
]
. (2.6)
Gµν has the simple transformation property
Gµν → G′µν = ωGµνω−1, (2.7)
under the gauge group G, as follows from eq. (2.5). Gµν is therefore thenatura
generalization of the field strength tensor Fµν in eletromagnetism. Note that Gµν is
not invariant under gauge transformations, whereas Fµν is gauge invariant. This is
an important difference between Abelian and non-Abelian gauge theories.
Let us now particularize the discussion to the gauge group G = SU(2). In the
2× 2 representation
Wµ =
1
2
σaW
a
µ ; Gµν =
1
2
σaG
a
µν , (2.8)
where
Gaµν = ∂µW
a
ν − ∂νW aµ + gεabcW bµW cν , (2.9)
73
Nguyen Van Thuan
g is coupling constant; µ, ν = 0, 1, 2, 3 are space-time indices; a, b, c = 1, 2, 3 are
SU(2) group indices.
A Lagrangian density which is invariant with any local real or complex SU(2)
transformation is easily constructed
L = −1
4
GaµνG
µν
a . (2.10)
The equations of motion of the SU(2) Yang-Mills obtained from Lagrangian
density (2.10) are
∂µGaµν = gεabcG
b
µνW
ν
c . (2.11)
Here we see two important new features of non-Abelian gauge theories which
are absent in electromagnetism: (i) The equations of motion are nonlinear in the
gauge potential. (ii) The gauge potential appears explicitly in the equations of
motion. It seems that the Yang-Mills potential plays a more basic role than the
potential in an Abelian gauge theory. At least, this is true in the conventional
formulation of the theory which directly involves the potential W aµ . In electromag-
netism one can work exclusively with the field strengths ~E and ~B. Things are not
so simple when the gauge group is non-Abelian.
2.2. Connection between Yang-Mills theory and scalar φ4
theory
In Minkowski space the ansatz for the SU(2) gauge potential is
gW a0 = ±i
∂iφ
φ
,
gW ai = εian
∂iφ
φ
± iδai∂0φ
φ
, (2.12)
where φ is scalar funtion.
Inserting ansatz (2.12) into eq. (2.11) yields the following equation for φ
1
φ
∂µ2φ =
3
φ2
∂µφ2φ. (2.13)
Equation (2.13) can be integrated once to give
2φ + λφ3 = 0, (2.14)
where λ is an arbitrary integration constant. Suppose that solutions of eq. (2.14)
are known, and in the ansatz (2.12) these lead automatically to explicit solutions
74
Some properties of SU(2) Yang-Mills theory connecting with φ4 theory
of the SU(2) gauge theory. We think that φ can be interpreted as a physical field,
and not merely as an ansatz function. When one considers the extension to massive
fields, it becomes apparent that φ has a natural interpretation as a Higgs-like field.
Let the SU(2) field have real mass m. Then the Yang-Mills equations of
motion are:
∂νGaµν = gεabcG
b
µνW
ν
c +m
2W aµ . (2.15)
These are reduced by the ansatz (2.12) to
1
φ
(
2+m2
)
∂µφ =
3
φ2
∂µφ2φ, (2.16)
which in turn is satisfied if φ satisfies
2φ− 1
2
m2φ+ λφ3 = 0. (2.17)
Equation (2.17) is the equation of motion for the scalar φ4 theory with spon-
taneous symmetry breakdown.
We now give formulas for several quantities of interest, following from the
ansatz (2.12). The Yang-Mills strengths are
gEan = gG
a
0n = εnam
[1
φ
∂0∂mφ− 2
φ2
∂0φ∂mφ
]
± iδna
[ 1
φ
∂20φ−
1
φ2
(
∂0φ∂0φ+ ∂mφ∂mφ
)]
∓ i
[ 1
φ
∂n∂aφ− 2
φ2
∂nφ∂aφ
]
, (2.18)
gBan = −
1
2
εnijG
a
ij = ±igEan + δan
(1
φ
)
2φ. (2.19)
From eq. (2.19) we see that the self-duality condition Ban = ±iEan evidently
implies 2φ = 0, or m2 = 0, λ = 0 in eq. (2.17). The field strengths Ean and B
a
n
are in general complex. However their squares are both real, and this means that
the energy and Lagrangian densities obtained from the ansatz (2.12) are real, even
though the potential W aµ is complex.
The Lagrangian density is given explicitly by
Ls = 1
2
(
EanE
a
n − BanBan
)
=
1
g2φ2
[
− 1
2
2φ2φ− ∂α∂βφ∂α∂βφ
]
+
1
g2φ3
[
− 2φ∂αφ∂αφ+ 4∂α∂βφ∂αφ∂βφ
]
+
1
g2φ4
[
− 3∂αφ∂αφ∂βφ∂βφ
]
. (2.20)
75
Nguyen Van Thuan
For m2 = 0 we have
Ls = 1
2g2
[
2∂α
(
∂αφ
φ
)
− 3λ2φ4
]
. (2.21)
Another quantity of primary interest is the energy-momentum tensor Tµν(x).
This is easily shown to be
Tµν =
2φ
g2φ
{ 4
φ2
∂µφ∂νφ− 2
φ
∂µ∂νφ+ gµν
[ 1
2φ
2φ − 1
φ2
∂αφ∂αφ
]}
. (2.22)
With self-duality field then Tµν = 0 because 2φ = 0. The total energy for the
case m2 = 0 is
E =
∫
T00d
3x =
6λ
g2
∫ [1
2
(
∂0φ
)2
+
1
2
(
▽ φ
)2
+
1
4
λφ4
]
d3x, (2.23)
where we have neglected surface terms at infinite.
2.3. Particular case
In the case when scalar function φ is time independent and λ = 0, eq. (2.14)
has the spherically symmetric particular solution.
φ =
(
a
r
)
, (2.24)
where a is constant.
For simplicity, we choose + in ansatz (2.12). Then the SU(2) Yang-Mills
potential obtained from φ in eq. (2.24) is
gW a0 = −i
ra
r2
,
gW ai = −εian
rn
r2
. (2.25)
The spatial component of the SU(2) gauge potential has the potential form
of point magnetic monopole with magnetic charge
m =
1
g
. (2.26)
For the field strengths we find
gEan = −iδna
1
r2
, (2.27)
and Ban given by eq. (2.19), here B
a
n = ±iEan because 2φ = 0. We easily verify that
the square of the electric field intensity is real.
76
Some properties of SU(2) Yang-Mills theory connecting with φ4 theory
3. Conclusion
Using specific ansatz for the SU(2) Yang-Mills potential W aµ in terms of the
scalar field φ, we reduce the complicated equations of motion of the Yang-Mills
theory to the single equation of motion for the φ4 theory. Therefore one can find
explicit solutions of the SU(2) Yang-Mills equations by solving the much simpler
scalar field equations. Some properties of the SU(2) Yang-Mills theory connecting
with φ4 theory are considered.
In the case when scalar function φ is time independent, we have found ex-
plicit solution to equation of motion for corresponding φ4 theory. For this solution,
the spatial component of the SU(2) Yang-Mills potential is like potential of point
magnetic monopole.
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