BRST Quantization of the Proca Model based on the BFT and the BFV Formalism

29 Pages·0.21 MB·English

by  Yong-Wan Kim

SOGANG-HEP 208/96
September 10, 1996
BRST Quantization of the Proca Model based on the BFT
and the BFV Formalism
7
9 Yong-Wan Kim†, Mu-In Park†, Young-Jai Park†, and Sean J. Yoon†,∗,
9
1
† Department of Physics and Basic Science Research Institute

n
a Sogang University, C.P.O. Box 1142, Seoul 100-611, Korea
J

9 and
2
∗ LG Electronics Research Center, Seoul 137-140, Korea

1
v
2
0
0
2 ABSTRACT
0
7
9 TheBRST quantization oftheAbelian Procamodel isperformedusing theBatalin-
/
h Fradkin-Tyutin and the Batalin-Fradkin-Vilkovisky formalism. First, the BFT Hamil-
t
-
p tonian method is applied in order to systematically convert a second class constraint
e
h system of the model into an effectively first class one by introducing new fields. In find-
:
v
ingtheinvolutive Hamiltonianweadoptanewapproachwhichismoresimpler thanthe
i
X
usual one. We also show that in our model the Dirac brackets of the phase space vari-
r
a
ables in the original second class constraint system are exactly the same as the Poisson
brackets of the corresponding modified fields in the extended phase space due to the
linear character of the constraints comparing the Dirac or Faddeev-Jackiw formalisms.
Then, according to the BFV formalism we obtain that the desired resulting Lagrangian
preserving BRST symmetry in the standard local gauge fixing procedure naturally in-
cludes the Stu¨ckelberg scalar related to the explicit gauge symmetry breaking effect
due to the presence of the mass term. We also analyze the nonstandard nonlocal gauge
fixing procedure.
PACS number : 11.10.Ef, 11.15.Tk
1
1 Introduction
The Dirac method has been widely used in the Hamiltonian formalism 1 to quantize
the first and the second class constraint systems generally, which do and do not form a
closed constraint algebra in Poisson brackets, respectively. However, since the resulting
Dirac brackets are generally field-dependent and nonlocal, and have a serious ordering
problem, the quantization is under unfavorable circumstances because of, essentially,
the difficulty in finding the canonically conjugate pairs. On the other hand, the quanti-
zation of first class constraint systems established by Batalin, Fradkin, and Vilkovisky
(BFV),2,3 whichdoesnothavethepreviously notedproblemsoftheDiracmethodfrom
the start, has been well appreciated in a gauge invariant manner with preserving Becci-
Rouet-Stora-Tyutin (BRST) symmetry. 4,5 After their works, this procedure has been
generalized to include the second class constraints by Batalin, Fradkin, and Tyutin
(BFT) 6,7 in the canonical formalism, and applied to various models 8−10 obtaining the
Wess-Zumino (WZ) actions. 11,12
Recently, Banerjee 13 has applied the BFT Hamiltonian method 7 to the second
class constraint system of the Abelian Chern-Simons (CS) field theory, 14−16 which
yields the strongly involutive first class constraint algebra in an extended phase space
by introducing new fields. As a result, he has obtained a new type of an Abelian WZ
action, which cannot be obtained in the usual path-integral framework. Very recently,
we have quantized several interesting models 17 as well as the non-Abelian CS case,
which yields the weakly involutive first class constraint system originating from the
non-Abelian nature of the second class constraints of the system, by considering the
generalized form of the BFT formalism. 18 As shown in all these works, the nature of
the second class constraint algebra originates from only the symplectic structure of the
CS term, not due to the local gauge symmetry breaking. Banerjee’s and Ghosh 19 have
also considered the Abelian and non-Abelian (incompletely) Proca model, 20 which
have the explicit gauge-symmetry breaking term by extending the BFT approach to
thecaseofrank-1non-AbelianconversioncomparedtotheAbelian(rank-0)conversion.
As a result, the extra field in this approach has identified with the Stu¨ckelberg scalar.
21 However, all these analysis do not carry out the covariant gauge fixing procedure
2
preserving the BRST symmetry based on the BFV formalism hence completing the
BRST quantization procedure. Furthermore, up to now all above authors do not
explicitly treat the Dirac brackets in this BFT formalism although the general but
classical relation between the Dirac bracket and the Poisson bracket in the extended
phase space are formally reported for the case of Abelian conversion. 7
In the present paper, the BRST quantization of the Abelian Proca model 20 is
performed completely by using the usual BFT 7 and the BFV 2,3 formalism. In section
2, we will apply the usual BFT formalism 7 to the Abelian Proca model in order to
convert the second class constraint system into a first class one by introducing new
auxiliary fields. Here, we newly obtain the relation that the well-known Dirac brackets
between the phase space variables in our starting second class constraint system of
the Abelian Proca model are the same as the Poisson brackets of the corresponding
modified ones in the extended phase space without Φ → 0 limiting procedure of the
general formula of BFT 7 due to essentially the linear character of the constraint.
It is also compared with the Dirac 1 or Faddeev-Jackiw (FJ) symplectic formalism,
22 which is to be regarded as the improved version of the Dirac one. Furthermore,
we adopt a new approach, which is more simpler than the usual one, in finding the
involutive Hamiltonian by using these modified variables. In section 3, we will consider
the completion of the BRST quantization based on the BFV formalism. As a result we
show that by identifying a new auxiliary field with the Stu¨ckelberg scalar we naturally
derive the Stu¨ckelberg scalar term related to the explicit gauge symmetry breaking
mass term through a BRST invariant and local gauge fixing procedure according to
the BFV formalism. We also analyze the nonlocal gauge fixing procedure which has
been recently studied by several authors. 23 Our conclusions are given in section 4.
2 BFT Formalism
2.1 Proca Model and Constraints
Now, we first apply theusual BFT formalismwhich assumes the Abelian conversion
of the second class constraint of the original system 7 to the Abelian Proca model of
3
the massive photon in four dimensions, 20 whose dynamics are given by
1 1
S = d4x [− F Fµν + m2A Aµ], (1)
µν µ
4 2
Z
where F = ∂ A −∂ A , and g = diag(+,−,−,−).
µν µ ν ν µ µν
The canonical momenta of gauge fields are given by
δS
π ≡ ≈ 0,
0 δA˙
0
δS
π ≡ = F (2)
i δA˙i i0
with the Poisson algebra {Aµ(x),π (y)} = δµδ(x−y). The weak equality ‘ ≈ ’ means
ν ν
the equality is not applied before all involved calculations are finished. 1 In contrast,
the strong equality ‘ = ’ means the equality can be applied at all the steps of the
calculations.
Then, Ω ≡ π ≈ 0 is a primary constraint. 1 The total Hamiltonian is
1 0
H = H + d3xuΩ (3)
T c 1
Z
with the multiplier u and the canonical Hamiltonian
1 1 1
H = d3x π2 + F Fij + m2{(A0)2 +(Ai)2}−A Ω , (4)
c 2 i 4 ij 2 0 2
Z (cid:20) (cid:21)
where Ω is the Gauss’ law constraint, which comes from the time evolution of Ω with
2 1
H , defined by
T
Ω = ∂iπ +m2A0 ≈ 0. (5)
2 i
Note that the time evolution of the Gauss’ law constraint with H generates no more
T
additional constraints but only determine the multipler u ≈ −∂ Ai. As a result, the full
i
constraints of this model are Ω (i,j = 1,2) which satisfy the second class constraint
i
algebra as follows
∆ (x,y) ≡ {Ω (x),Ω (y)} = −m2ǫ δ3(x−y), (6)
ij i j ij
det∆ (x,y) 6= 0,
ij
where we denote x = (t,x) and three-space vector x = (x1,x2,x3) and ǫ = −ǫ = 1.
12 21
4
We now introduce new auxiliary fields Φi to convert the second class constraints Ω
i
into first class ones in the extended phase space with the Poisson algebra
{Aµ(or π ),Φi} = 0,
µ
{Φi(x),Φj(y)} = ωij(x,y) = −ωji(y,x). (7)
Here, the constancy, i.e., the field independence of ωij(x,y), isconsidered forsimplicity.
AccordingtotheusualBFTmethod,7 themodifiedconstraintsΩ withtheproperty
i
e
{Ω ,Ω } = 0, (8)
i j
e e
which is called the Abelian conversion, which is rank-0, of the second class constraint
(6) are generally given by
∞
Ω (Aµ,π ;Φj) = Ω + Ω(n) = 0; Ω(n) ∼ (Φj)n (9)
i µ i i i
n=1
X
e e
satisfying the boundary conditions, Ω (Aµ,π ;0) = Ω . Note that the modified con-
i µ i
straints Ω become strongly zero by introducing the auxiliary fields Φi, i.e., enlarging
i e
the phase space, while the original constraints Ω are weakly zero. As will be shown
e i
later, essentially due to this property, the result of the Dirac formalism can be easily
read off from the BFT formalism. The first order correction terms in the infinite series
7 are simply given by
Ω(1)(x) = d3yX (x,y)Φj(y), (10)
i ij
Z
and the first class constraintealgebra (8) of Ω requires the following relation
i
e
△ (x,y)+ d3w d3z X (x,w)ωkl(w,z)X (y,z) = 0. (11)
ij ik jl
Z
However, as was emphasized in Refs. 13, 18, and 19, there is a natural arbitrariness
in choosing the matrices ωij and X from Eqs. (7) and (10), which corresponds to
ij
canonical transformation in the extended phase space. 6,7 Here we note that Eq. (11)
can not be considered as the matrix multiplication exactly unless X (y,z) is the sym-
jl
metric matrix, i.e., X (y,z) = X (z,y) because of the form of the last two product of
jl lj
the matrices d3zωkl(ω,z)X (y,z) in the right hand side of Eq. (11). Thus, using this
jl
R
5
arbitrariness we can take the simple solutions without any loss of generality, which are
compatible with Eqs. (7) and (11) as
ωij(x,y) = ǫijδ3(x−y),
X (x,y) = mδ δ3(x−y), (12)
ij ij
i.e., antisymmetric ωij(x,y) and symmetric X (x,y) such that Eq. (11) is the form of
ij
the matrix multiplication exactly 10,13,17−19
∆ (x,y)+ d3ωd3zX (x,ω)ωkl(ω,z)X (z,y) = 0.
ij ik lj
Z
Note that X (x,y) needs not be generally symmetric, while ωij(x,y) is always an-
ij
tisymmetric by definition of Eq. (7). However, the symmetricity of X (x,y) is, by
ij
experience, a powerful property for the solvability of (9) with finite iteration 13,17−19 or
with infinite regular iterations. 24
In our model with this proper choice, the modified constraints up to the first order
iteration term
(1)
Ω = Ω +Ω
i i i
e = Ω +mΦi (13)
i
strongly form a first class constraint algebra as follows
{Ω (x)+Ω(1)(x), Ω (y)+Ω(1)(y)} = 0. (14)
i i j j
e e
Then, the higher order iteration terms
1
Ω(n+1) = − Φlω XkjB(n) (n ≥ 1) (15)
i n+2 lk ji
e
with
n n−2
B(n) ≡ {Ω(n−m),Ω(m)} + {Ω(n−m),Ω(m+2)} (16)
ji j i (A,π) j i (Φ)
m=0 m=0
X X
e e e e
are found to be vanishing without explicit calculation. Here, ω and Xkj are the
lk
inverse of ωlk and X , and the Poisson brackets including the subscripts are defined
kj
6
by
∂A∂B ∂A∂B
{A,B} ≡ − ,
(q,p) ∂q ∂p ∂p ∂q
∂A ∂B ∂A ∂B
{A,B} ≡ − , (17)
(Φ) "∂Φi∂Φj ∂Φj ∂Φi#
i,j
X
where (q,p) and (Φi,Φj) are the conjugate pairs, respectively.
2.2 Physical Variables, First Class Hamiltonian, and Dirac
Brackets
Now, corresponding to the original variables Aµ and π , the physical variables
µ
within the Abelian conversion in the extended phase space, Aµ and π , which are
µ
strongly involutive, i.e.,
e e
{Ω ,Aµ} = 0, {Ω ,πµ} = 0, (18)
i i
e e e e
can be generally found as
∞
Aµ(Aν,π ;Φj) = Aµ + Aµ(n), Aµ(n) ∼ (Φj)n;
ν
n=1
X
e ∞ e e
π (Aν,π ;Φj) = π + π(n), π(n) ∼ (Φj)n (19)
µ ν µ µ µ
n=1
X
e e e
satisfying the boundary conditions, Aµ(Aν,π ;0) = Aµ and π (Aν,π ;0) = π . Here,
ν µ ν µ
the first order iteration terms are given by
e e
Aµ(1) = −Φjω Xkl{Ω ,Aµ}
jk l (A,π)
1
e = ( Φ2,m∂ Φ1),
i
m
π(1) = −Φjω Xkl{Ω ,π }
µ jk l µ (A,π)
= (mΦ1,0). (20)
e
(1)
Furthermore, since the modified variables up to the first iterations, Aµ + Aµ and
π + π(1) are found to involutive, i.e., to satisfy Eq. (18), the higher order iteration
ν ν f
e
7
terms
1
Aµ(n+1) = − Φjω Xkl(Gµ)(n),
n+1 jk l
1
eπ(n+1) = − Φjω Xkl(H )(n) (21)
µ n+1 jk µ l
e
with
n n−2
(Gµ)(n) = {Ω(n−m),Aµ(m)} + {Ω(n−m),Aµ(m+2)} +{Ω(n+1),Aµ(1)} ,
l i (A,π) i (Φ) i (Φ)
m=0 m=0
X X
n e n−2 e e
(H )(n) = {Ω(n−m),π(m)} + {Ω(n−m),π(m+2)} +{Ω(n+1),π(1)}
µ l i µ (A,π) i µ (Φ) i µ (Φ)
m=0 m=0
X X
e e e (22)
are also found to be automatically vanishing. Hence, the physical variables in the
extended phase space are finally found to be
Aµ = Aµ +Aµ(1)
1 1
e = (A0 +e Φ2,Ai + ∂iΦ1),
m m
π = π +π(1)
µ µ µ
= (π +mΦ1,π )
e 0 e i
= (Ω ,π ). (23)
1 i
e
Similar to the physical phase space variables Aµ and π , all other physical quanti-
µ
ties, which correspond to the functions of Aµ and π , can be also found in principle
e µ e
by considering the solutions like as Eq. (19). 7,13,17−19 However, it is expected that
this procedure of finding the physical quantities may not be simple depending on the
complexity of the functions.
In this paper, we consider a new approach using the property 7,25
K(Aµ,π ;Φi) = K(Aµ,π ) (24)
µ µ
f e
e
for the arbitrary function or functional K defined on the original phase space variables
unless K has the time derivatives: the following relation
{K(Aµ,π ),Ω } = 0 (25)
µ i
e e
e
8
is satisfied for any function K not having the time derivatives because Aµ and π and
µ
their spatial derivatives already commute with Ω at equal times by definition. On the
i e e
other hand, since the solution K of Eq. (24) is unique up to the power of the first class
e
constraints Ω , 17,19 K(Aµ,π ) can be identified with K(Aµ,π ;Φ) modulus the power
i µ µ
of the first class constraints Ω˜ . However, note that this property is not satisfied when
e e e i f
the time derivatives exist.
UsingthiselegantpropertywecandirectlyobtainthedesiredfirstclassHamiltonian
H corresponding to the total Hamiltonian H of Eq. (3) as follows
T T
g
H (Aµ,π ;Φi) = H (Aµ,π )
T ν T ν
1 1 1 1
g = HT(Aeµ,πeν)+ d3x (∂iΦ1)2 + (Φ2)2 + ∂i∂iΦ1Ω1 − Φ2Ω2 .
2 2 m m
Z (cid:20) (cid:21)
f e(26)
On the other hand, since the first class Hamiltonian H corresponding to the canonical
c
Hamiltonian H of Eq. (4) can be similarly obtained as follows
c f
H (Aµ,π ;Φi) = H (Aµ,π )
c ν c ν
1 1 1
f = Hc(Aeµ,πeν)+ d3x 2(∂iΦ1)2 + 2(Φ2)2 +mΦ1∂iAi − mΦ2Ω2 ,
Z (cid:20) (cid:21)
e (27)
Eq. (26) can be re-expressed as follows
H (Aµ,π ;Φi) = H (Aµ,π ;Φi)− d3x(∂ Ai)Ω (28)
T ν c ν i 1
Z
g f f e
as it should be according to Eq. (3).
This is the same result as the usual approach (See the Appendix A) and, by con-
struction, both H and H are automatically strongly involutive,
T c
g f
{Ω ,H } = 0, (29)
i T
{eΩ ,gH } = 0.
i c
e f
Notethatallourconstraintshavealreadythisproperty,i.e.,Ω (Aµ,π ;Φ) = Ω (Aµ,π ).
i µ i µ
In this way, the second class constraints system Ω (A ,πµ) ≈ 0 is converted into the
i µ e e e
first class constraints one Ω (A ,πµ;Φ) = 0 with the boundary conditions Ω | = Ω .
i µ i Φ=0 i
e e
9
On the other hand, in the Dirac formalism 1 one can make the second class con-
straint system Ω ≈ 0 into the first class constraint one Ω (A ,πµ) = 0 only by deform-
i i µ
ing the phase space (A ,πµ) without introducing any new fields. Hence, it seems that
µ
these two formalisms are drastically different ones. However, remarkably the Dirac
formalism can be easily read off from the usual BFT-formalism 7 by noting that the
Poisson bracket in the extended phase space with Φ → 0 limit becomes
{A,B}|Φ=0 = {A,B}−{A,Ωk}∆kk′{Ωk′,B}
e e = {A,B} (30)
D
where ∆kk′ = −Xlkωll′Xl′k′ is the inverse of ∆kk′ in Eq. (6). About this remarkable
relation, we note that this is essentially due to the Abelian conversion method of
the original second class constraint: In this case the Poisson brackets between the
constraints and the other things in the extended phase space are already strongly zero
{Ω ,A} = 0,
i
{Ωe,Ωe} = 0, (31)
i j
e e
which resembles the property of the Dirac bracket in the non-extended phase space
{Ω ,A} = 0,
i D
{Ω ,Ω } = 0, (32)
i j D
such that
{Ω ,A}| ≡ {Ω ,A}∗ = 0,
i Φ=0 i
{Ωe,Ωe}| ≡ {Ω ,Ω }∗ = 0 (33)
i j Φ=0 i j
e e
are satisfied for some bracket in the non-extended phase space { , }∗. However, due
to the uniqueness of the Dirac bracket 26 it is natural to expect the previous result (29)
is satisfied, i.e.,
{ , }∗ = { , } (34)
D
10