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501
Progress of Theoretical Physics, Vol. 89, No. 2, February 1993
How to Solve the Covariant Operator Formalism
of Gauge Theories and Quantum Gravity
in the Heisenberg Picture. III
— Two-Dimensional Nonabelian BF Theory —
Mitsuo Abe and Noboru Nakanishi
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-01
(Received September 10, 1992)
The two-dimensional nonabelian BF theory is studied, which turns out to be essentially the
zeroth-order approximation to QCD in the framework of the newly proposed method of solving
quantum field theory in the Heisenberg picture. The exact covariant operator solution and some
lower-point exact Wightman functions are constructed explicitly. It is shown that there is no
ultraviolet divergence in the Wightman functions. In the context of the BF theory, the system
coupled with the Dirac field is also studied.
§1.
Introduction
In our previous paper,1) we have proposed a new method of solving gauge theories
in the covariant operator formalism in the Heisenberg picture, so as to respect both
Lorentz covariance and gauge (BRS) invariance, and have demonstrated how we can
solve quantum electrodynamics (QED) by our method. It has been shown there that
our method reproduces the results of the conventional perturbation theory of QED.
This is due to the speciality of the abelian gauge theory; in general, our method
gives approximations different from the conventional perturbative ones.
We consider the following form of the Lagrangian density of gauge theories:1)
L=−
1 µν a
F
Fµν a + L GF+FP + L matter (ϕ, Dµ ϕ),
4g 2
(1.1)
with
Fµν a ≡ ∂µ Aν a − ∂ν Aµ a + (Aµ × Aν )a ,
Dµ ϕ ≡ (∂µ − iAµ a T a )ϕ
(1.2)
(1.3)
in the obvious notation. In contrast with the conventional expression for the Lagrangian density, (1·1)∼(1·3) are more natural in the sense that gauge invariance (more
precisely, BRS invariance) holds independently of the coupling constant g. This
means that the BRS transformation should be defined in the form without including g,
whence the gauge-fixing plus Faddeev-Popov ghost Lagrangian density L GF+FP in (1·1)
is also independent of g. Therefore, the coupling constant in (1·1) is present only in
the coefficient of the first term.
After calculating all field equations and equal-time (anti)commutation relations
based on the canonical quantization, we expand all fields in powers of g 2 (but not of
502
M. Abe and N. Nakanishi
g). In the Landau gauge, we have
(Dµ Fµν )a = O(g 2 ),
[ Aµ a (x), Aν b (y) ]|0 = 0,
[ Aµ a (x), ϕ(y) ]|0 = 0,
∂ µ Aµ a = 0,
[ A˙ µ a (x), Aν b (y) ]|0 = O(g 2 ),
[ A˙ µ a (x), ϕ(y) ]|0 = 0,
(1.4)
(1.5)
(1.6)
where |0 denotes the equal-time. Hence, expanding Aµ and ϕ as
Φ(x) ≡
∞
P
g 2N Φ(N ) (x)
for Φ = Aµ a and ϕ,
(1.7)
N =0
we can show that
[ (∂0 )n Aµ a (0) (x), Φ(0) (y) ]|0 = 0
for n ≥ 0,
Φ = Aν b and ϕ.
(1.8)
In the sense of a formal expansion in powers of x0 − y 0 , (1·8) implies the fourdimensional commutation relation
[ Aµ a (0) (x), Φ(0) (y) ] = 0.
(1.9)
Although there is a zeroth-order operator having a nontrivial commutator with Aµa(0),
it is given in a closed form; thus Aµ a (0) is quite a manageable operator.
Some time ago,2) we showed that the zeroth-order approximation, in the sense of
the above approach, to quantum Einstein gravity is nothing but the four-dimensional
extra-polation of the exact covariant solution to the two-dimensional quantum gravity, which was formulated and solved completely in Refs.3)∼7). In the present
paper, we consider the model which is exactly solvable and corresponds to the
zeroth-order approximation to the nonabelian gauge theory. This model is the
so-called BF theory.8) Since the above correspondence is essentially independent of
the spacetime dimensionality, we discuss the simplest case, i.e., two-dimensional BF
theory. Although this model is physically trivial, the purpose of the present paper is
to solve it as the first step of solving quantum chromodynamics (QCD). We find the
operator solution, i.e., the closed operator algebra for the primary fields, and then
construct the Wightman functions from it.
The present paper is organized as follows. In §2, after discussing the symmetry
properties of the two-dimensional BF theory, we carry out canonical quantization to
obtain the equal-time (anti)commutation relations between the primary fields. In §3,
we construct the operator solution on the basis of the field equations and the equaltime (anti)commutation relations. In §4, we construct some lower-point Wightman
functions and see that there is no ultraviolet divergence. In §5, we consider the Dirac
field minimally coupled with the gauge field of the BF theory and solve the coupled
system. Then, we find that there can appear ultraviolet divergence in the Wightman
functions consisting of the fields peculiar to the BF theory owing to the breakdown of
a certain symmetry caused by the presence of the Dirac field. The final section is
devoted to discussion.
How to Solve the Covariant Operator Formalism
§2.
503
Formulation and BRS-type symmetries
The Lagrangian denstiy of the two-dimensional nonabelian BF theory in the
Landau gauge is defined by8)
1
L BF = − Bea µν Fµν a + B a ∂ µ Aµ a − i∂ µ C¯ a · Dµ ab C b ,
2
(2.1)
where
Fµν a ≡ ∂µ Aν a − ∂ν Aµ a + f abc Aµ b Aν c ,
Dµ ab ≡ δ ab ∂µ + f acb Aµ c ,
µν = −νµ ,
10 = 01 = 1,
(2.2)
(2.3)
(2.4)
with f abc being the structure constant of a (semisimple compact) Lie algebra of the
internal gauge symmetry considered. Here, we use the symbol Be for the multiplier
field peculiar to the BF theory, while the symbol B is reserved for the B-field of gauge
fixing. Since Bea is essentially the conjugate field of B a in the context of the twodimensional massless scalar field as seen below, this usage of symbols is legitimate.
We first note the invariance of (2·1) (up to total divergence) under the BRS and
anti-BRS transformations, δ b and δ¯b , defined by
δ b Aµ a = Dµ ab C b ,
δb Ca = −
δ b C¯ a = iB a ,
δ b Bea = −f abc C bBec ;
δ b B a = 0,
δ¯b Aµ a = Dµ ab C¯ b ,
δ¯b C a = −iB a − f abc C¯ b C c ,
δ¯b Bea = −f abc C¯ bBec .
1 abc b c
f C C ,
2
(2.5)
1
δ¯b C¯ a = − f abc C¯ b C¯ c ,
2
δ¯b B a = −f abc C¯ b B c ,
(2.6)
e 2 (from C and C)=0
¯
Since the ghost counting of (2·1) is 2 (from Aµ or B and B)−
for
each component of the adjoint representation of the Lie algebra, the Kugo-Ojima
subsidiary condition guarantees the physical triviality of this theory under the postulate of asymptotic completeness.
One should note that (2·1) is invariant (up to total divergence) also under the
conjugate BRS and anti-BRS transformations, δ b˜ and δ¯b˜ , defined by
δ b˜ Aµ a = −µν ∂ ν C a ,
δ b˜ C a = 0,
δ b˜ C¯ a = iBea ,
δ b˜ Bea = 0,
δ b˜ B a = f abc C bBec ;
∗)
(2·7)∗)
This transformation was considered previously in the conventional two-dimensional pure Yang-Mills
theory.9)
504
M. Abe and N. Nakanishi
δ¯b˜ Aµ a = −µν ∂ ν C¯ a ,
δ¯b˜ C a = −iBea ,
δ¯b˜ B a = 0.
δ¯b˜ C¯ a = 0,
δ¯b˜ Bea = 0,
(2.8)
Here δ b˜ and δ¯b˜ are nilpotent (δ b˜ 2 = δ¯b˜ 2 = 0), and any two of δ b , δ¯b , δ b˜ and δ¯b˜ anticom¯ a , defined by
mute with each other. If we introduce the anti B-field, B
¯ a − if abc C¯ b C c = 0,
Ba + B
(2.9)
(2·7) becomes parallel to (2·8) because of
¯ a = 0.
δ b˜ B
(2.10)
It is possible to rewrite (2·1) into the manifestly conjugate (anti-)BRS invariant form
as follows:
i
a
µν
a
a
µ
a
¯ ∂ Aµ + δ b˜
C¯ Fµν − i∂ µ (C¯ a Dµ ab C b )
L BF = −B
2
i a µν
a µ
a
a
¯
= B ∂ Aµ − δ b˜
C Fµν + i∂ µ (C a ∂µ C¯ a ).
(2.11)
2
Here, if we add a gauge term (α/2)B a B a to (2·1) with α being a real constant, the
conjugate BRS invariance is explicitly broken; that is, the conjugate BRS invariance
is a symmetry peculiar to the Landau gauge. As for the conjugate anti-BRS invariance, it is kept unspoiled under the above modification of the gauge fixing term.
Owing to the rather simple structure of (2·8), the Kugo-Ojima quartet mechanism
would be more transparent if we set up the subsidiary condition by using the conjugate anti-BRS symmetry.
Now, we consider the field equations. The Euler derivatives of (2·1) with respect
to Bea , B a , Aµ a , C¯ a and C a yield
Fµν a = 0,
∂ µ Aµ a = 0,
µν Dν abBeb + ∂µ B a − if abc ∂µ C¯ b · C c = 0,
∂ µ Dµ ab C b = 0
(2.12)
(2.13)
(2.14)
(2.15)
Dµ ab ∂µ C¯ b = 0,
(2.16)
and
respectively. From (2·13), we have an identity
∂ µ Dµ ab = Dµ ab ∂ µ ,
(2.17)
whence the field equations for C a and C¯ a have the same form. As seen from (2·14),
Bea is essentially the conjugate field of B a as mentioned above, although the relation
between them is modified in the BRS invariant way. By taking the covariant
divergence of (2·14) and by using (2·12) and (2·16), we obtain
How to Solve the Covariant Operator Formalism
Dµ ab ∂µ B b − if abc ∂µ C¯ b · Dµ cd C d = 0,
505
(2.18)
while the operation of ρµ ∂ρ on (2·14) yields
∂ρ Dρ abBeb − if abc ρµ ∂µ C¯ b · ∂ρ C c = 0.
(2.19)
[ Aµ a (x), B b (y) ]|0 = iδµ 0 δ ab δ(x1 − y 1 ),
[ A˙ µ a (x), B b (y) ]|0 = −[ Aµ a (x), B˙ b (y) ]|0
= iδµ 1 (D1 ab )x δ(x1 − y 1 ),
{ C˙ a (x), C¯ b (y) }|0 = −{ C a (x), C¯˙ b (y) }|0
(2.20)
= δ ab δ(x1 − y 1 ),
[ C˙a (x), B b (y) ]|0 = −[ C a (x), B˙ b (y) ]|0
= −if abc C c (x)δ(x1 − y 1 ),
[ Aµ a (x), Beb (y) ]|0 = iδµ 1 δ ab δ(x1 − y 1 ),
˙
[ A˙ µ a (x), Beb (y) ]|0 = −[ Aµ a (x), Beb (y) ]|0
= i(δµ 0 δ ab ∂1 − δµ 1 f acb A0 c )x δ(x1 − y 1 ),
˙
[ Bea (x), B b (y) ]|0 = −[ Bea (x), B˙ b (y) ]|0
= −if abcBec (x)δ(x1 − y 1 ),
(2.22)
By setting up the canonical (anti)commutation relations and by using the field
equations (2·12)∼(2·14), it is straightforward to obtain the following equal-time
(anti)commutation relations:
(2.21)
(2.23)
(2.24)
(2.25)
(2.26)
where the symbol |0 indicates to take equal time. All other equal-time (anti)commutation relations vanish. Especially, we have
[Aµ a (x), Φ(y)]|0 = 0,
[A˙ µ a (x), Φ(y)]|0 = 0
for Φ = Aν b , C b and C¯ b . (2.27)
From the field equations (2·15)∼(2·19), we can construct the following conserved
currents:
i abc ¯ a
f ∂µ C · C b C c ,
2
i
J¯bµ ≡ B a Dµ ab C¯ b − ∂µ B a · C¯ a + f abc C¯ a C¯ b Dµ cd C d ,
2
i
a
ab
b
a
a
J˜bµ ≡ Be Dµ C − ∂µBe · C + f abc µν ∂ ν C¯ a · C b C c ,
2
i
J¯˜bµ ≡ Bea Dµ ab C¯ b − ∂µBea · C¯ a − f abc C¯ a C¯ b µν ∂ ν C c .
2
Jbµ ≡ B a Dµ ab C b − ∂µ B a · C a +
(2.28)
(2.29)
(2.30)
(2.31)
¯b,
The conserved charges corresponding to (2·28)∼(2·31), which are denoted by Qb , Q
¯
¯
¯
Q˜b and Q˜b , are the generators of δ b , δ b , δ b˜ and δ b˜ , respectively, as easily confirmed
506
M. Abe and N. Nakanishi
from (2·20)∼(2·26).
§3.
Exact operator solution
In this section, we construct the exact operator solution on the basis of the field
equations and the equal-time (anti)commutation relations presented in §2.
From (2·12), (2·13) and (2·27), we have
[ (∂0 )n Aµ a (x), Φ(y) ]|0 = 0
for n ≥ 0,
Φ = Aν b , C b and C¯ b ,
(3.1)
which implies
[ Aµ a (x), Φ(y) ] = 0
for Φ = Aν b , C b and C¯ b .
(3.2)
This full commutativity between Aµ a (x) and any one of Aν b (y), C b (y) and C¯ b (y) is a
remarkable property of the BF theory, just as in the two-dimensional quantum
gravity.4)
In order to write down the other two-dimensional (anti)commutators, we need the
nonabelian extension, Dab (x, y), of the q-number Pauli-Jordan D-function introduced
previously in QCD,10) which is defined by the followig Cauchy problem:
(∂ µ Dµ ac )x Dcb (x, y) = 0,
Dab (x, y)|0 = 0,
∂0 x Dab (x, y)|0 = −δ ab δ(x1 − y 1 ).
(3.3)
(3.4)
(3.5)
Based on the unique solvability of Cauchy problem and (3·2), we can derive the
following properties of Dab (x, y):
[ Dab (x, y), Φ(z) ] = 0
for Φ = Aν c , C c and C¯ c ,
[ Dab (x, y), Dcd (z, w) ] = 0,
Dab (x, y) = −Dba (y, x),
[Dab (x, y)]† = Dab (x, y).
(3.6)
(3.7)
(3.8)
(3.9)
If we know (∂ µ Dµ ab )x F b (x, y) together with initial conditions, we can generally
express F a (x, y) explicitly with the help of Dab (x, y), that is, we have an integral
representation
Z
a
F (x, y) = − d2 u (x, y; u)Dab (x, u)(∂ µ Dµ bc )u F c (u, y)
Z
− du1 [Dab (x, u)(D0 bc )u F c (u, y) − ∂0 u Dab (x, u)·F b (u, y)]|u0 =y0 ,
(3.10)
where
How to Solve the Covariant Operator Formalism
(x, y; u) ≡ θ(x0 − u0 ) − θ(y 0 − u0 )

0
0
0
 1 for x > u > y
=
−1 for y 0 > u0 > x0

0 otherwise.
507
(3·11)
For the two-dimensional anticommutator between C a (x) and C¯ b (y), the definition
of Dab (x, y) together with (3·2) and the unique solvability of Cauchy problem implies
{ C a (x), C¯ b (y) } = −Dab (x, y).
(3.12)
Likewise, we have
{ C a (x), C b (y) } = 0,
{ C¯ a (x), C¯ b (y) } = 0.
(3.13)
(3.14)
As for the two-dimensional commutators involving B a and/or Bea , it is necessary
to use (3·10).
From (2·18) and (3·2), we have
(∂ ν Dν bc )y [ Aµ a (x), B c (y) ] = 0.
(3.15)
Then, applying the formula (3·10) with respect to y in (3·15), and using (2·20) and
(2·21), we obtain
[ Aµ a (x), B b (y) ] = −i(Dµ ac )x Dcb (x, y).
(3.16)
One should note that (3·16) is consistent with (2·12) and (2·13). Indeed, (3·16) yields
[ Fµν a (x), B b (y) ] = −if acd Fµν c (x)Ddb (x, y),
(3.17)
and Dcb (x, y) satisfies (3·3). We can also derive (3·16) from the BRS transform of
(3·2) for Φ = C¯ b and (3·12).
From (2·15) and (3·16), we have
(∂ µ Dµ ac )x [ C c (x), B b (y) ] = −if acd ∂ µ C c (x) · (Dµ de )x Deb (x, y).
(3.18)
Applying (3·10) to (3·18) with (2·23), we obtain
[ C a (x), B b (y) ] = −if bcd Dac (x, y)C d (y)
Z
+i d2 u (x, y; u)Dag (x, u)f gcd ∂ µ C c (u) · (Dµ de )u Deb (u, y).
(3.19)
Integrating (3·19) by parts four times with the help of (2·15), (2·17) and (3·3)∼(3·5),
we can rewrite it into
[ C a (x), B b (y) ] = −if acd C c (x)Ddb (x, y) − iDb ab (x, y),
(3.20)
where Db ab (x, y) is defined by
Z
Db ab (x, y) ≡ d2 u (x, y; u)Dag (x, u)f gcd Dµ ce C e (u) · (∂ µ )u Ddb (u, y). (3.21)
508
M. Abe and N. Nakanishi
It is nothing but the BRS transform of Dab (x, y), that is, Db ab (x, y) = δ b Dab (x, y).
Indeed, it satisfies10)
(∂ µ Dµ ac )x Db cb (x, y) = −f acd Dµ ce C e (x) · (∂ µ )x Ddb (x, y),
Db ab (x, y)|0 = 0,
∂0 x Db ab (x, y)|0 = 0,
(3.22)
(3.23)
(3.24)
that is, the BRS transformation of the Cauchy problem for Dab (x, y). Note that
(3·20) is obtained also as the BRS transformation of (3·12).
Likewise for [ C¯ a (x), B b (y) ], from (2·16), (2·17), (3·16) and (3·10), we obtain
Z
[ C¯ a (x), B b (y) ] = i d2 u (x, y; u)Dag (x, u)f gcd ∂ µ C¯ c (u) · (Dµ de )u Deb (u, y)
= −if acd C¯ c (x)Ddb (x, y) − if bcd C¯ c (y)Dad (x, y) − iD¯b ab (x, y),
(3.25)
where
D¯b ab (x, y) ≡
Z
d2 u (x, y; u)Dag (x, u)f gcd Dµ ce C¯ e (u) · (∂ µ )u Ddb (u, y). (3.26)
It is the anti-BRS transform of Dab (x, y), that is, D¯b ab (x, y) = δ¯b Dab (x, y). Indeed, it
satisfies the Cauchy problem defined by replacing C by C¯ in (3·22)∼(3·24). Note
that (3·25) is nothing but the anti-BRS transformation of (3·12).
Next, with the aid of (3·3)∼(3·5), (3·16) and (3·10), we obtain
[ Dab (x, y), B c (z) ]
Z
= i d2 u (x, y; u)Dag (x, u)f gde (∂ µ )u Ddb (u, y) · (Dµ eh )u Dhc (u, z).
(3.27)
From (3·6), (3·12), (3·21), (3·26) and (3·27), identities
[ Dab (x, y), B c (z) ] = i{ Db ab (x, y), C¯ c (z) }
= −i{ D¯b ab (x, y), C c (z) }
(3.28)
follow, as it should be owing to the (anti-)BRS invariance. Integrating (3·27) by
parts twice with the help of (3·3)∼(3·5) and (2·17), we can rewrite it into
Z
i
d2 u (x, y; u)[Dag (x, u)(∂ µ )u Ddb (u, y)
[Dab (x, y), B c (z)] =
2
−(∂ µ )u Dag (x, u) · Ddb (u, y)] · f gde (Dµ eh )u Dhc (u, z)
i
i
= − f ade Ddb (x, y)Dec (x, z) − f bde Dad (x, y)Dec (y, z)
2 Z
2
i
2
dg u ag
−
d u (x, y; u)[(Dµ ) D (x, u) · (∂ µ )u Deb (u, y)
2
−(∂ µ )u Dad (x, u)·(Dµ eg )u Dgb (u, y)]·f deh Dhc (u, z). (3.29)
How to Solve the Covariant Operator Formalism
509
To calculate [ B a (x), B b (y) ], from (2·18), (3·16), (3·20) and (3·25), we have
(∂ µ Dµ ac )x [ B c (x), B b (y) ]
= −if acd ∂ µ B c (x) · (Dµ de )x Deb (x, y)
+f acd (∂ µ )x [f cgh C¯ g (x)Dhb (x, y) + f bgh C¯ g (y)Dch (x, y) + D¯b cb (x, y)]
· Dµ de C e (x)
+f acd ∂ µ C¯ c (x) · [(Dµ de )x Db eb (x, y) + f deg Dµ eh C h (x) · Dgb (x, y)]. (3.30)
After using (3·10) and carrying out integrations by parts many times with the aid of
(3·21) and the equations for B a , C a , C¯ a , Dab and Db ab , we obtain
[ B a (x), B b (y) ] = −if acd B c (x)Ddb (x, y) − if bcd B c (y)Dad (x, y)
+f acd C¯ c (x)Db db (x, y) + f bcd C¯ c (y)Db ad (x, y) − iDb¯b ab (x, y),
(3.31)
where
Db¯b ab (x, y) ≡
Z
d2 u (x, y; u)Dag (x, u)
×f gcd [(Dµ ce B e (u) − if ceh C¯ e (u)Dµ hk C k (u)) · (∂ µ )u Ddb (u, y)
+iDµ ce C¯ e (u)·(∂ µ )u Db db (u, y) − iDµ ce C e (u)·(∂ µ )u D¯b db (u, y)].
(3.32)
The formulae (3·31) and (3·32) are obtained also by BRS-transforming (3·25) and
(3·26), respectively, by setting δ b D¯b ab (x, y) = −δ¯b Db ab (x, y) ≡ iDb¯b ab (x, y).
The other nonvanishing commutators are the ones concerning Bea . Since the way
of obtaining them is similar to the above, we write the results only as follows:
[ Aµ a (x), Beb (y) ] = iµν (∂ ν )x Dab (x, y),
(3.33)
Z
[ Φa (x), Beb (y) ] = −i d2 u (x, y; u)Dag (x, u)f gcd ∂ µ Φc (u) · µν (∂ ν )u Ddb (u, y)
¯
for Φ = C and C,
(3.34)
[ Dab (x, y), Bec (z) ]
Z
= −i d2 u (x, y; u)Dag (x, u)f gde (∂ µ )u Ddb (u, y) · µν (∂ ν )u Dec (u, z), (3.35)
[ B a (x), Beb (y) ] = −iδ b [ C¯ a (x), Beb (y) ]
+if bcd {Dac (x, y)Bed (y) − C c (y)[ C¯ a (x), Bed (y) ]},
(3.36)
Z
[ Bea (x), Beb (y) ] = i d2 u (x, y; u)Dag (x, u)f gcd {[ ∂ µ C¯ c (u) · µν ∂ ν C d (u), Beb (y) ]
−∂ µBec (u) · µν (∂ ν )u Ddb (u, y)}.
(3.37)
eab (x, y),
If we introduce the conjugate nonabelian Pauli-Jordan D-function, D
510
M. Abe and N. Nakanishi
satisfying
ecb (x, y) = −µν (∂ ν )x Dab (x, y),
(Dµ ac )x D
(3.38)
we can rewrite (3·35) into the following form similar to (3·29):
i
eec (x, z) − i f bde Dad (x, y)D
eec (y, z)
[Dab (x, y), Bec (z)] = − f ade Ddb (x, y)D
2 Z
2
i
=−
d2 u (x, y; u)[(Dµ dg )u Dag (x, u) · (∂ µ )u Deb (u, y)
2
ehc (u, z). (3.39)
= −(∂ µ )u Dad (x, u)·(Dµ eg )u Dgb (u, y)]·f deh D
eab (x, y),
Although, unfortunately, we cannot solve (3·38) explicitly with respect to D
it is natural to assume the following properties:
eab (x, y), Φ(z) ] = 0 for Φ = Aν c , C c and C¯ c ,
[D
eab (x, y), Dcd (z, w) ] = 0,
[D
eab (x, y), D
ecd (z, w) ] = 0.
[D
(3.40)
(3.41)
(3.42)
eab (x, y) is, of course, not necessary for the discussion
However, the introduction of D
in the following sections.
To make the conjugate (anti-)BRS symmetry manifest, it is convenient to introduce the conjugate BRS and anti-BRS transforms of Dab (x, y), denoted by D˜b ab (x, y)
and D¯˜b ab (x, y), respectively, as follows:
D˜b ab (x, y) ≡ δ b˜ Dab (x, y)
Z
= d2 u (x, y; u)Dag (x, u)f gcd ∂ µ C c (u) · µν (∂ ν )u Ddb (u, y), (3.43)
D¯˜b ab (x, y) ≡ δ¯b˜ Dab (x, y)
Z
= d2 u (x, y; u)Dag (x, u)f gcd ∂ µ C¯ c (u) · µν (∂ ν )u Ddb (u, y), (3.44)
which are nothing but the quantities in the right-hand side of (3·34) apart from a
factor −i. Indeed, it is straightforward to confirm that (3·43) and (3·44) satisfy the
Cauchy problem obtained by the conjugate (anti-)BRS transformation of (3·3)∼(3·5).
Thus, we have
[ C a (x), Beb (y) ] = −iD˜b ab (x, y),
[ C¯ a (x), Beb (y) ] = −iD¯˜b ab (x, y),
(3.45)
(3.46)
which are obtained also by the conjugate BRS and anti-BRS transformations of
(3·12), respectively. Likewise, we can also obtain (3·33), (3·36) and (3·37) by the
conjugate BRS transformation of (3·2) for Φ = C¯ b , (3·25) and (3·46), respectively.
From (3·12), (3·35), (3·43) and (3·44), we obtain identities
[ Dab (x, y), Bec (z) ] = i{ D˜b ab (x, y), C¯ c (z) }
How to Solve the Covariant Operator Formalism
= −i{ D¯˜b ab (x, y), C c (z) },
511
(3.47)
as it should be owing to the conjugate (anti-)BRS invariance.
In this way, we have obtained the exact covariant operator solution to the
two-dimensional BF theory in the manifestly invariant way under the four BRS-type
symmetries. We note that this operator algebra expressed by the full (anti)commutation relations satisfies the Jacobi identities, as it should be.
§4.
Wightman functions
In this section, we construct Wightman functions in such a way that they are
consistent with the operator solution presented in §3 and that they satisfy the
energy-positivity condition. Since it is very complicated to construct higher-point
functions explicitly owing to the nonabelian character, we restrict ourselves to
considering one-, two- and three-point functions only in the present paper.
In the first place, we briefly summarize our method of constructing Wightman
functions based on the operator solution in the general context.6) We denote an
arbitrary (n−1)-ple (anti)commutator of Φ1 (x1 ), · · · , Φn (xn ) by [[Φ1 (x1 ), · · · , Φn (xn )]].
Rule 1:
h [[ Φ1 (x1 ), · · · , Φn (xn ) ]] i = 0 for all [[ Φ1 (x1 ), · · · , Φn (xn ) ]]
⇒ h Φ1 (x1 ) · · · Φn (xn ) iT = 0,
(4.1)
where a subscript T stands for the truncated function. Especially, we have
[[ Φ1 (x1 ), · · · , Φn (xn ) ]] = 0 =⇒ h Φ1 (x1 ) · · · Φn (xn ) iT = 0.
(4.2)
Rule 2: If h[[Φ1 (x1 ),· · ·,Φn (xn )]]i 6= 0, we construct hΦ1 (x1 )· · ·Φn (xn )iT and all its
permuted ones so as to reproduce all h [[Φ1 (x1 ),· · ·,Φn (xn )]] i and to satisfy the
energy-positivity condition [see Refs.6) and 1)].
Rule 3: For the Wightman function including composite fields, i.e., products of
the primary fields at the same spacetime points, we define it from the nontruncated higher-point function consisting of the same array of the primary fields
by identifying spacetime points belonging to each composite field and deleting all
resulting singular terms.
The following generalized Wick contraction follows from Rules 1 and 3. If all
double or higher multiple (anti)commutators including at least two of Φi (xi)(i = 1,· · ·,
m) vanish, we have
m
Q
h Φi (x1 ) · Φm+1 (xm+1 ) · · · Φn (xn ) i
i=1
P
= σF (P )h Φ1 (x1 ) · · · iT h Φ2 (x1 ) · · · iT · · · h Φm (x1 ) · · · iT · · · ,
(4.3)
P
where the summation runs over all partitions of {m+1, · · · , n} and σF (P ) denotes the
sign factor due to fermion exchange. One should note that Rule 3 dissolves only the
512
M. Abe and N. Nakanishi
divergence trouble due to the product of fields at one and the same spacetime point but
not the ordinary ultraviolet divergence, which arises from two (or more) different
spacetime points and is crucially dependent on the dimensionality of spacetime.
Now, we construct the Wightman functions of the two-dimensional nonabelian
BF theory.
Since the one-point functions, which characterize the representation of the operator solution, are arbitrary in principle, we define them in such a way that Poincar´e
invariance, ghost number conservation and BRS invariance are unbroken. Then, all
one-point functions of the primary fields are set equal to zero:
h Φ(x) i = 0
for Φ = Aµ a , B a , Bea , C a and C¯ a .
(4.4)
Next, we consider the vacuum expectation value of Dab (x, y). Taking vacuum
expectation value of (3·3)∼(3·5) and using Rules 1 and 3 together with (3·6) and (4
·4) for Φ = Aµ a , we have
x
h Dab (x, y) i = 0,
h Dab (x, y) i|0 = 0,
∂0 x h Dab (x, y) i|0 = −δ ab δ(x1 − y 1 ),
(4.5)
(4.6)
(4.7)
that is,
h Dab (x, y) i = δ ab D(x − y),
(4.8)
where
D(ξ) ≡ −
1
1
−ξ 2 + i0ξ 0
(ξ 0 )θ(ξ 2 ) = −
log
2
4πi
−ξ 2 − i0ξ 0
(4.9)
is the usual two-dimensional Pauli-Jordan D-function. The positive-energy part,
D(+) , of D satisfies
D(+) (ξ) = 0,
D(+) (ξ) − D(+) (−ξ) = iD(ξ),
[D(+) (ξ)]∗ = D(+) (−ξ);
(4.10)
(4.11)
(4.12)
we know its explicit expression
D(+) (ξ) = −
1
log (−µ2 ξ 2 + i0ξ 0 ),
4π
(4.13)
µ being a positive constant (infrared cutoff). Likewise, from (3·38) and (3·40) for Φ
= Aµ c , we obtain
eab (x, y) i = δ ab D(x
e − y),
hD
(4.14)
where
1
1
ξ 2 + i0ξ 1
e
D(ξ)
≡ − (ξ 1 )θ(−ξ 2 ) = −
log 2
.
2
4πi
ξ − i0ξ 1
(4.15)
How to Solve the Covariant Operator Formalism
513
We also know
0
1
e(+) (ξ) = 1 log ξ − ξ − i0 ,
D
4π
ξ 0 + ξ 1 − i0
ν
e + µν ∂ D(ξ) = 0.
∂µ D(ξ)
(4.16)
(4.17)
With the aid of Rules 1 and 3 together with (3·6), (3·9), (4·4), (4·8) and (4·17), the
(anti)commutation relations (3·12), (3·16), (3·29), (3·28) with (3·20), (3·33), (3·35) and
(3·47) with (3·45) yield
h { C a (x), C¯ b (y) } i = −δ ab D(x − y);
h [ Aµ a (x), B b (y) ] i = −iδ ab ∂µ x D(x − y);
i
h [ { C a (x), C¯ b (y) }, B c (z) ] i = f abc D(x − y)[D(x − z) − D(y − z)],
2
i
h { [ C a (x), B c (z) ], C¯ b (y) } i = f abc D(x − z)[D(x − y) − D(y − z)];
2
e − y);
h [ Aµ a (x), Beb (y) ] i = −iδ ab ∂µ x D(x
h [ { C a (x), C¯ b (y) }, Bec (z) ] i
Z
e − z)
= −if abc d2 u (x, y; u)D(x − u)(∂ µ )u D(u − y) · ∂µ u D(u
i abc
e − z) − D(y
e − z)],
f D(x − y)[D(x
2
h { [ C a (x), Bec (z) ], C¯ b (y) } i
Z
abc
e − z)
= −if
d2 u (x, z; u)D(x − u)(∂ µ )u D(u − y) · ∂µ u D(u
=
=−
i abc
e − y) + D(y
e − z)],
f D(x − z)[D(x
2
(4.18)
(4.19)
(4.20)
(4.21)
(4.22)
(4.23)
(4.24)
respectively, where use has been made of (4·17) in (4·24). We note that (4·23) and
(4·24) are derived also directly from (3·39) and (3·47) with the help of (3·40) for Φ
= Aµ c , (3·41) and (4·14).
Applying Rule 2 to (4·18)∼(4·24), we obtain the following truncated Wightman
functions:1,11)
h C a (x)C¯ b (y) iT = iδ ab D(+) (x−y);
(4.25)
h Aµ a (x)B b (y) iT = −δ ab ∂µ x D(+) (x−y);
(4.26)
i
h C a (x)C¯ b (y)B c (z) iT = − f abc [D(+) (x−y)D(+) (x−z)
2
−D(+) (x−y)D(+) (y−z)+D(+) (x−z)D(+) (y−z)], (4.27)
i
h C¯ b (y)C a (x)B c (z) iT = f abc [D(+) (y−x)D(+) (x−z)
2
514
M. Abe and N. Nakanishi
−D(+) (y−x)D(+) (y−z)+D(+) (x−z)D(+) (y−z)],
(4.28)
i abc (+)
f [D (x−y)D(+) (x−z)
2
−D(+) (x−y)D(+) (z−y)+D(+) (x−z)D(+) (z−y)];
(4.29)
h C a (x)B c (z)C¯ b (y) iT = −
e(+) (x − y);
h Aµ a (x)Beb (y) iT = −δ ab ∂µ x D
(4.30)
h C a (x)C¯ b (y)Bec (z) iT
Z
abc
e(+) (u − z)
=f
d2 u [θ(u0 − x0 )D(+) (u − x)(∂ µ )u D(+) (u − y) · ∂µ u D
e(+) (u − z)
+(x, y; u)D(+) (x − u)(∂ µ )u D(+) (u − y) · ∂µ u D
e(+) (u − z)
+(y, z; u)D(+) (x − u)(∂ µ )u D(+) (y − u) · ∂µ u D
e(+) (z − u)],
+θ(z 0 − u0 )D(+) (x − u)(∂ µ )u D(+) (y − u) · ∂µ u D
(4.31)
h C¯ b (y)C a (x)Bec (z) iT
Z
abc
e(+) (u − z)
= −f
d2 u [θ(u0 − y 0 )D(+) (u − x)(∂ µ )u D(+) (u − y) · ∂µ u D
e(+) (u − z)
+(y, x; u)D(+) (u − x)(∂ µ )u D(+) (y − u) · ∂µ u D
e(+) (u − z)
+(x, z; u)D(+) (x − u)(∂ µ )u D(+) (y − u) · ∂µ u D
e(+) (z − u)],
+θ(z 0 − u0 )D(+) (x − u)(∂ µ )u D(+) (y − u) · ∂µ u D
(4.32)
h C a (x)Bec (z)C¯ b (y) iT
Z
abc
e(+) (u − z)
=f
d2 u [θ(u0 − x0 )D(+) (u − x)(∂ µ )u D(+) (u − y) · ∂µ u D
e(+) (u − z)
+(x, z; u)D(+) (x − u)(∂ µ )u D(+) (u − y) · ∂µ u D
e(+) (z − u)
+(z, y; u)D(+) (x − u)(∂ µ )u D(+) (u − y) · ∂µ u D
e(+) (z − u)]
+θ(y 0 − u0 )D(+) (x − u)(∂ µ )u D(+) (y − u) · ∂µ u D
(4.33)
and the remaining reordered functions are given by taking complex conjugates of
the above ones. Here, we cannot rewrite (4·31)∼(4·33) into forms similar to (4·27)
e
e
∼(4·29) because of the behavior of D(ξ)
at ξ 0 = 0 different from that of D(ξ): D(ξ)|
0
1
1
e
= − 2 (ξ ), ∂0 D(ξ)|0 = 0. Likewise, from (3·16), (3·29), (3·33) and (3·39), we have
h [ [ Aµ a (x), B b (y) ], B c (z) ] i
1
= − f abc [∂µ x D(x−y)·(D(x−z)−D(y−z))−D(x−y)∂µ x D(x−z)];
2
h [ [ Aµ a (x), B b (y) ], Bec (z) ] i
1
x e
e
e
= − f abc [∂µ x D(x−y)·(D(x−z)−
D(y−z))−D(x−y)∂
µ D(x−z)]
2
(4.34)
How to Solve the Covariant Operator Formalism
1
= − f abc µν (∂ ν )x [D(x−y)(D(x−z)−D(y−z))],
2
h [ [ Aµ a (x), Bec (z) ], B b (y) ] i
1
= − f abc µν (∂ ν )x [D(x−z)(D(x−y)+D(y−z))];
2
h [ [ Aµ a (x), Beb (y) ], Bec (z) ] i
1 abc
e
e
f µν (∂ ν )x [D(x−y)(D(x−z)−
D(y−z))]
2
= −h [ [ Aµ a (x), B b (y) ], B c (z) ] i,
515
(4.35)
(4.36)
=
(4.37)
where use has been made of (4·17) and an identity
x e
e
e
∂µ x D(x−y)·(D(x−z)−
D(y−z))+∂
µ D(x−y)·(D(x−z)−D(y−z)) = 0 (4.38)
in (4·35) and (4·37). Therefore, we obtain
h Aµ a (x)B b (y)B c (z) iT =
1 abc x (+)
f [∂µ D (x−y) · (D(+) (x−z) − D(+) (y−z))
2
−(D(+) (x−y) − D(+) (y−z))∂µ x D(+) (x−z)],
(4.39)
h B b (y)Aµ a (x)B c (z) iT =
1 abc x (+)
f [∂µ D (y−x) · (D(+) (x−z) − D(+) (y−z))
2
−(D(+) (y−x) − D(+) (y−z))∂µ x D(+) (x−z)];
(4.40)
h Aµ a (x)B b (y)Bec (z) iT
1
= f abc µν (∂ ν )x [D(+) (x − y)D(+) (x − z) − D(+) (x − y)D(+) (y − z)
2
−D(+) (x − z)D(+) (y − z)],
(4.41)
h B b (y)Aµ a (x)Bec (z) iT
1
= f abc µν (∂ ν )x [D(+) (y − x)D(+) (x − z) − D(+) (y − x)D(+) (y − z)
2
−D(+) (x − z)D(+) (y − z)],
(4.42)
h Aµ a (x)Bec (z)B b (y) iT
1
= f abc µν (∂ ν )x [D(+) (x − y)D(+) (x − z) − D(+) (x − y)D(+) (z − y)
2
−D(+) (x − z)D(+) (z − y)]
(4.43)
and h Aµ a (x)Beb (y)Bec (z) iT and h Beb (y)Aµ a (x)Bec (z) iT are given by (4·39) and (4·40),
respectively, apart from the sign factor.
Now, it is straightforward to confirm the consistency of (4·25)∼(4·37) and (4·39)
∼(4·43) with the four BRS-type invariances. For example, we have
516
M. Abe and N. Nakanishi
h δ b (Aµ a (x)C¯ b (y)) i = h Dµ ac C c (x) · C¯ b (y) i + ih Aµ a (x)B b (y) i
= ∂µ x h C a (x)C¯ b (y) iT + ih Aµ a (x)B b (y) iT
= iδ ab ∂µ x D(+) (x − y) − iδ ab ∂µ x D(+) (x − y)
= 0,
(4.44)
h δ b (C a (x)C¯ b (y)C¯ c (z)) i
1
= − f ade h C d (x)C e (x)C¯ b (y)C¯ c (z) i
2
−ih C a (x)B b (y)C¯ c (z) iT + ih C a (x)C¯ b (y)B c (z) iT
= −f abc D(+) (x − y)D(+) (x − z)
1
+ f abc [D(+) (x − z)D(+) (x − y)
2
−D(+) (x − z)D(+) (y − z) + D(+) (x − y)D(+) (y − z)]
1
+ f abc [D(+) (x − y)D(+) (x − z)
2
−D(+) (x − y)D(+) (y − z) + D(+) (x − z)D(+) (y − z)]
= 0,
(4.45)
h δ b˜ (C a (x)C¯ b (y)C¯ c (z)) i
= −ih C a (x)Beb (y)C¯ c (z) iT + ih C a (x)C¯ b (y)Bec (z) iT
Z
e(+) (u − y)
= if abc d2 u [θ(u0 − x0 )D(+) (u − x)(∂ µ )u D(+) (u − z) · ∂µ u D
e(+) (u − y)
+(x, y; u)D(+) (x − u)(∂ µ )u D(+) (u − z) · ∂µ u D
e(+) (y − u)
+(y, z; u)D(+) (x − u)(∂ µ )u D(+) (u − z) · ∂µ u D
e(+) (y − u)]
+θ(z 0 − u0 )D(+) (x − u)(∂ µ )u D(+) (z − u) · ∂µ u D
Z
e(+) (u − z)
+if abc d2 u [θ(u0 − x0 )D(+) (u − x)(∂ µ )u D(+) (u − y) · ∂µ u D
e(+) (u − z)
+(x, y; u)D(+) (x − u)(∂ µ )u D(+) (u − y) · ∂µ u D
e(+) (u − z)
+(y, z; u)D(+) (x − u)(∂ µ )u D(+) (y − u) · ∂µ u D
e(+) (z − u)]
+θ(z 0 − u0 )D(+) (x − u)(∂ µ )u D(+) (y − u) · ∂µ u D
= 0,
(4.46)
e(+) (ζ)
where use has been made of Rule 3, (4·4), and an identity ∂ µ D(+) (ξ) · ∂µ D
e(+) (ξ)·∂µ D(+) (ζ). From Rule 3, therefore, the composite-field Wightman func= −∂ µ D
tions constructed from (4·25)∼(4·33) and (4·39)∼(4·43) are also consistent with the
four BRS-type invariances. Indeed, we have
How to Solve the Covariant Operator Formalism
h δ b (C¯ c (x)Ddb (x, y)) i
= ih B c (x)Ddb (x, y) i − h C¯ c (x)Db db (x, y) i
= −ih B c (x){ C d (x), C¯ b (y) } i − ih C¯ c (x)[ C d (x), B b (y) ] i
+f deg h C¯ c (x)C e (x)Dgb (x, y) i
i
i
= − f cdb D(+) (x − y)D(x − y) + f cdb D(+) (x − y)D(x − y)
2
2
= 0,
517
(4.47)
where use has been made of (3·12), (3·20) and (4·27)∼(4·29). Likewise, we have
h δ b (C¯ c (y)Ddb (x, y)) i = ih B c (y)Ddb (x, y) i − h C¯ c (y)Db db (x, y) i = 0,
h δ b D¯b (x, y) i = ih Db¯b (x, y) i = 0.
(4.48)
(4.49)
Thus, from (3·31), we obtain
h [ B a (x), B b (y) ] i = 0,
(4.50)
which leads us to
h δ b (B a (x)C¯ b (y)) iT = ih B a (x)B b (y) iT = 0
(4.51)
according to (4·1). In the same way, we obtain
h δ¯b˜ (B a (x)C¯ b (y)) iT = −ih B a (x)Beb (y) iT = 0,
h δ¯b˜ (Bea (x)C¯ b (y)) iT = −ih Bea (x)Beb (y) iT = 0.
(4.52)
(4.53)
e they should also vanish
As for the three-point functions consisting only of B and/or B,
owing to the BRS-type invariance. To confirm it explicitly, however, it is necessary
to construct nontrivial truncated four-point functions such as h C a C¯ b B c B d iT .
In concluding this section, we note that there is no ultraviolet divergence in the
Wightman functions in this theory. As seen in §3, the (multiple) (anti)commutators
e are expressible
of the primary fields, in which at least one of them is neither B nor B,
by only the fully commutative operators. Therefore, the corresponding Wightman
functions are represented by tree graphs in the sense of Feynman graphs, whence they
are free of ultraviolet divergence. On the contrary, for the Wightman functions
e divergences can, in principle, appear.∗) As shown by
consisting only of B and/or B,
(4·51)∼(4·53), however, such divergences cancel one another completely, in conformity with the fact that the BRS-type invariances are unspoiled.
§5.
Introduction of the Dirac field
It is straightforward to extend the previous results to the model defined by the
Lagrangian density
∗)
In the sense of the conventional perturbation theory, this divergence comes from individual one-loop
graphs.11)
518
M. Abe and N. Nakanishi
¯ D
/ − m)ψ,
L = L BF + ψ(i
(5.1)
where ψ is the two-dimensional massive Dirac field and
/ ≡ γ µ (∂µ − iAµ a T a )
D
(5.2)
with T a being a certain representation matrix of the Lie algebra. In contrast with
(2·1), however, (5·1) is no longer invariant under the conjugate (anti-)BRS transformation, though it is, of course, (anti-)BRS!invariant,where the (anti-)BRS transformation of ψ is defined by
δ b ψ = iC a T a ψ,
δ¯b ψ = iC¯ a T a ψ.
(5.3)
The field equations newly obtained from (5·1) are as follows:
µν Dν abBeb + ∂µ B a − if abc ∂µ C¯ b · C c + jµ a = 0,
¯ µT aψ
jµ a ≡ −ψγ
(5.4)
instead of (2·14) and
/ − m)ψ = 0.
(i D
(5.5)
Since jµ a satisfies
Dµ ab jµ b = 0,
(5.6)
(2·18) remains unchanged, while (2·19) is modified into
∂ρ Dρ abBeb − if abc ρµ ∂µ C¯ b · ∂ρ C c + ρµ ∂ρ jµ a = 0.
(5.7)
¯ }|0 = γ 0 δ(x1 − y 1 ),
{ ψ(x), ψ(y)
(5.8)
˙
[ ψ(x), B˙ b (y) ]|0 = −[ ψ(x),
B b (y) ]|0
= T b ψ(x)δ(x1 − y 1 ).
(5.9)
According to the canonical quantization and the field equation (5·4), the nonvanishing equal-time (anti)commutation relations concerning ψ are given by
Since (3·1) is also satisfied for Φ = ψ, we obtain
[ Aµ a (x), ψ(y) ] = 0.
(5.10)
Thus, it is possible to construct the two-dimensional (anti)commutators including ψ
on the basis of the unique solvability of Cauchy problem as done in §3.
Firstly, we define the q-number nonabelian S-function, S(x, y), by setting up the
following Cauchy problem:
/ − m)x S(x, y) = 0,
(i D
S(x, y)|0 = −iγ 0 δ(x1 − y 1 ).
(5.11)
(5.12)
In analogy to (3·6)∼(3·9), we can show that
[ S(x, y), Φ(z) ] = 0 for Φ = Aν c , C c , C¯ c and ψ,
[ Dab (x, y), ψ(z) ] = 0,
(5.13)
(5.14)
How to Solve the Covariant Operator Formalism
[ S(x, y), S(z, w) ] = 0,
[ Dab (x, y), S(z, w) ] = 0,
[S(x, y)]† = −γ 0 S(y, x)γ 0 .
519
(5.15)
(5.16)
/ − m)x V (x, y) together with an initial condition, we can generally
If we know (i D
express V (x, y) as
Z
/ − m)u V (u, y)
V (x, y) =
d2 u (x, y; u)S(x, u)(i D
Z
+i du1 S(x, u)γ 0 V (u, y)|u0 =y0 ,
(5.17)
where (x, y; u) is defined by (3·11).
Now, it is straightforward to construct the following two-dimensional (anti)
commutators as in a way similar to those in §3:
¯ } = iS(x, y),
{ ψ(x), ψ(y)
(5.18)
[ ψ(x), Φ(y) ]∓ = 0 for Φ = C b , C¯ b and ψ,
(5.19)
[ ψ(x), B b (y) ] = T a ψ(x)Dab (x, y),
(5.20)
[ S(x, y), B c (z) ] = T a S(x, y)Dac (x, z) − S(x, y)T a Dac (y, z),
(5.21)
Z
[ ψ(x), Beb (y) ] = −i d2 u (x, y; u)S(x, u)γ µ T a ψ(u)µν (∂ ν )u Dab (u, y), (5.22)
Z
c
e
[ S(x, y), B (z) ] = −i d2 u (x, y; u)S(x, u)γ µ T a S(u, y)µν (∂ ν )u Dac (u, z).
(5.23)
Corresponding to the modification of the field equation for Bea , (3·37) is also modified
into
[ Bea (x), Beb (y) ]
Z
= d2 u (x, y; u)Dag (x, u){µν [ if gcd ∂ µ C¯ c (u) · ∂ ν C d (u) + ∂ µ j ν g (u), Beb (y) ]
−if gcd ∂ µ B c (u) · µν (∂ ν )u Ddb (u, y)},
(5.24)
while the other two-dimensional (anti)commutators presented in §3 remain unchanged.
In a way similar to §4, we can construct the following nonvanishing Wightman
functions from (5·18)∼(5·23):1)
¯ iT = S(+) (x − y);
h ψ(x)ψ(y)
c
¯
h ψ(x)ψ(y)B
(z) iT = −iT c S(+) (x − y)(D(+) (x − z) − D(+) (y − z)),
t ¯
h ψ(y)ψ(x)B c (z) iT = −iT c S(−) (x − y)(D(+) (x − z) − D(+) (y − z)),
¯ iT = −iT c S(+) (x − y)(D(+) (x − z) − D(+) (z − y));
h ψ(x)B c (z)ψ(y)
(5.25)
(5.26)
(5.27)
(5.28)
520
M. Abe and N. Nakanishi
¯ Bec (z) iT
h ψ(x)ψ(y)
Z
c
= iT
d2 u [θ(u0 − x0 )S(−) (x − u)γ µ S(+) (u − y)µν (∂ ν )u D(+) (u − z)
−(x, y; u)S(+) (x − u)γ µ S(+) (u − y)µν (∂ ν )u D(+) (u − z)
+(y, z; u)S(+) (x − u)γ µ S(−) (u − y)µν (∂ ν )u D(+) (u − z)
+θ(z 0 − u0 )S(+) (x − u)γ µ S(−) (u − y)µν (∂ ν )u D(+) (z − u)],
(5.29)
¯
¯ iT , where a superscript t
and likewise for h ψ(y)ψ(x)
Bec (z) iT and h ψ(x)Bec (z)ψ(y)
stands for the transposition. Here, from (5·13) for Φ = Aν c , we have
h S(x, y) i = S(x − y),
(5.30)
which is the usual S-function for the free massive Dirac field defined by
(iγ µ ∂µ − m)S(ξ) = 0,
S(ξ)|0 = −iγ 0 δ(ξ 1 )
(5.31)
(5.32)
and the positive(negative)-energy part, S(±) (ξ), of S(ξ) satisfies the following:
(iγ µ ∂µ − m)S(±) (ξ) = 0,
S(+) (ξ) + S(−) (ξ) = iS(ξ),
t (±)
[S (ξ)]∗ = γ 0 S(±) (−ξ)γ 0 .
(5.33)
(5.34)
(5.35)
Almost all the Wightman functions presented in §4 remain unchanged. There is,
however, one exception, which is the two-point function of Bea . This is due to the
breakdown of the conjugate (anti-)BRS invariance caused by the presence of the Dirac
field. Since the vacuum expectation value of the right-hand side of (3·37) vanishes,
from (5·24), we have
Z
a
b
e
e
h [ B (x), B (y) ] i = d2 u (x, y; u)h Dac (x, u)µν [ ∂ µ j ν c (u), Beb (y) ] i. (5.36)
Substituting
¯ 5T cψ
µν ∂ µ j ν c = µν f cde Aµ d j ν e − 2imψγ
(5.37)
with γ 5 ≡ γ 0 γ 1 into (5·36), and applying Rule 3 of §4 with the help of (3·33) and
(5·22), we obtain
h [ Bea (x), Beb (y) ] i
Z
= 2imδ ab d2 u d2 v (x, y; u)(u, y; v)D(x − u)
×tr [γ 5 S(+)(u−v)γ µ S(−)(v−u)−γ 5 S(−)(u−v)γ µ S(+)(v−u)]µν (∂ ν )v D(v−y),
(5.38)
where use has been made of tr(T a T b ) = δ ab . Carrying out the integration by parts
and using an identity
How to Solve the Covariant Operator Formalism
µν (∂ ν )v [S(±)(u−v)γ µ S(∓)(v−u)] = −2imS(±)(u−v)γ 5 S(∓)(v−u),
521
(5.39)
we rewrite (5·38) into
h [ Bea (x), Beb (y) ] i
Z
2 ab
= −4m δ
d2 u d2 v (x, y; u)(u, y; v)D(x − u)
×tr [γ 5 S(+)(u−v)γ 5 S(−)(v−u)−γ 5 S(−)(u−v)γ 5 S(+)(v−u)]D(v−y),
(5.40)
Therefore, from the energy-positivity condition, we obtain1,11)
h Bea (x)Beb (y) iT
Z
= 4m2 δ ab d2 u [θ(u0 −x0 )D(+) (u−x)F (+) (u, y)
+(x, y; u)D(+) (x−u)F (+) (u, y)−θ(y 0 −u0 )D(+) (x−u)F (−) (u, y)]
Z
= 4m2 δ ab d2 u [θ(u0 −x0 )D(+) (u−y)F (+) (u, x)
−(x, y; u)D(+) (u−y)F (−) (u, x)−θ(y 0 −u0 )D(+) (y−u)F (−) (u, x)],
(5.41)
where
F (+) (u, y) ≡
Z
d2 v [θ(v 0 −u0 )tr(γ 5 S(−) (u−v)γ 5 S(+) (v−u))D(+) (v−y)
+(u, y; v)tr(γ 5 S(+) (u−v)γ 5 S(−) (v−u))D(+) (v−y)
+θ(y 0 −v 0 )tr(γ 5 S(+) (u−v)γ 5 S(−) (v−u))D(+) (y−v)],
(5.42)
Z
(−)
F (u, y) ≡ − d2 v [θ(v 0 −y 0 )tr(γ 5 S(−) (u−v)γ 5 S(+) (v−u))D(+) (v−y)
+(y, u; v)tr(γ 5 S(−) (u−v)γ 5 S(+) (v−u))D(+) (y−v)
+θ(u0 −v 0 )tr(γ 5 S(+) (u−v)γ 5 S(−) (v−u))D(+) (y−v)]
= [F (+) (u, y)]∗ .
(5.43)
The second equality in (5·41) is proved by using identities such as (x, y; u)(u, y; v)
= (x, y; v)(x, v; u) and by exchanging the integration variables u and v.
In order to deal with the ultraviolet divergence appearing in (5·41)∼(5·43), we
have to consider the renormalized theory. We do not, however, go into such a
renormalization problem in the present paper, since (5·41) is a quantity peculiar to the
model considered and has nothing to do with the zeroth-order approximation to the
four-dimensional gauge theories in which we are most interested.
§6.
Discussion
In the present paper, we have successfuly constructed the exact covariant
operator solution to the two-dimensional nonabelian BF theory and then its two-point
522
M. Abe and N. Nakanishi
and three-point Wightman functions explicitly in closed form. Since, in our
approach, the BF theory, apart from the part involving the Be field, is essentially
equivalent to the zeroth-order approximation to the nonabelian gauge theory (QCD),
the results obtained in the present paper can be interpreted as those of the manifestly
covariant and BRS invariant zeroth-oder QCD.
As for the ultraviolet divergence, we have encountered none in the pure BF
theory. When the Dirac field is introduced, we encounter ultraviolet divergence in
h BeaBeb i, but it has no counterpart in the zeroth-order QCD. The situation is quite
analogous to the quantum-gravity case: The two-dimensional quantum gravity, which
can be regarded as the zeroth-order quantum Einstein gravity, exhibits anomaly in the
part involving the Weyl B-field b, which has no counterpart in quantum Einstein
gravity.6)
In 1985, Kanno and Nakanishi10) proposed “local-gauge commutation relation” in
the nonabelian gauge theory. Let Φ(x) be any local quantity involving none of B a ,
C a and C¯ a . Suppose that under an infinitesimal local gauge transformation characterized by εa (x), Φ(x) transforms like
- a (Φ)εa (x),
Φ0 (x) = Φ(x) + L
(6.1)
- a (Φ) is generally a differential operator. Then the local-gauge commutation
where L
relation is written as
- a (Φ)Dab (x, y) + N.G.
[ Φ(x), B b (y) ] = −iL
(6.2)
where N.G. is a certain quantity involving either B a or C a and C¯ a . This proposal is
reconfirmed in the BF theory, in which the local-gauge commutation relation holds
exactly without N.G., as is seen from (3·16), (3·17) and (5·20), unless Φ involves Bea .
For Φ = Bea , however, N.G. is nonvanishing, as is seen from (3·36) with (3·46).
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