Consistent Interactions in terms of the Generalized Fields Method

12 Pages·0.11 MB·English

by  O. F. Dayi

MRC.PH.TH-6-96
hep-th/9604167
CONSISTENT INTERACTIONS IN TERMS OF THE
GENERALIZED FIELDS METHOD
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9 O¨mer F. DAYI1

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TUBITAK - Marmara Research Center, Research Institute for Basic Sciences,
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7 Department of Physics, P.O. Box 21, 41470 Gebze-TURKEY
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Abstract
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The interactions which preserve the structure of the gauge interactions of the free
h
: theory are introduced in terms of the generalized fields method of solving the
v
i Batalin–Vilkovisky master equation. It is shown that by virtue of this method
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the solution of the descent equations resulting from the cohomological analysis is
r
a provided straightforwardly. The general scheme is illustrated by applying it to
spin–1 gauge field in 3 and 4 dimensions, to free BF theory in 2–d and to the
antisymmetric tensor field in any dimension. It is shown that it reproduces the
results obtained by cohomological techniques.
1E-mail address: [email protected]
1
In a local gauge theory if the number of the physical degrees of freedom
is modified when the interactions are switched off, there may be some neg-
ative norm states which yield inconsistencies in the scattering theory where
one uses the asymptotic states[1]. To avoid it one deals with the local in-
teractions which preserve the structure of the gauge generators of the free
theory. These are the consistent interactions[2]. By using the cohomologi-
cal aspects of the Batalin–Vilkovisky (BV) method of quantization of gauge
theories[3] one obtains the descent equations whose solution yields the con-
sistent interactions[4]–[6].
The generalized fields method is proven to be powerful in the BV quan-
tization of a large class of gauge theories[7]–[8]. It offers the proper solution
of the (BV) master equation in a straightforward way.
We show that when one deals with a free gauge theory whose action can
be written first order in the exterior derivative d and bilinear in fields, the
generalized fields method can be applied. As far as the properties of gauge
generators are unaltered when interactions are introduced the generalized
fieldsmethodshouldstillbeapplicable. Hence,itcanbeutilizedtoobtainthe
consistent interactions. Moreover, solution of the related descent equations
is a by-product in this scheme, i.e. this method provides the solution of
the cohomology problem in a straightforward and compact manner. We do
not deal with the consistent interactions neither non-local nor possessing
derivatives and obtainable by field redefinitions.
The general scheme is illustrated by applying it to the spin–1 gauge field
ind = 4andd = 3,whoseconsistent interactionsyieldYang-MillsandChern-
Simons field theories. The consistent interactions of the free BF theory in
d = 2 are shown to lead to the gauge action of the quadratic Lie algebras[9].
In all of the examples the solutions of the descent equations are presented.
When we deal with the antisymmetric tensor field in D–dimension the ap-
plication of the method is shown to provide in a straightforward manner the
solution of the cohomology problem obtained in [6].
Let us deal with a free gauge theory whose action can be written as
1
A = (Ba ∧dAa + α Ba ∧Bb) (1)
0 Z 2 ab
where α can be a vanishing or a constant matrix which will be suppressed in
the following. It is invariant under the infinitesimal gauge transformations
δ Aa = dΛa, δ Ba = 0. (2)
Λ Λ
2
To quantize this theory in terms of the BV method, one introduces the ghost,
ghost of ghost fields and the antifields (minimal sector), by inspecting the
reducibility properties of the gauge transformations. One can attribute to
each of the fields a total form degree defined as the sum of the differential
form degree and the ghost number. To apply the generalized fields method
one collects the original fields and the ones introduced for applying the BV
method in the generalized fields A˜a, B˜a whose total form degrees are the
same with the differential form degrees of the original fields Aa, Ba. The
action written in terms of the generalized fields
1
S ≡ L˜ = (B˜adA˜a + B˜aB˜a) (3)
0 Z 0 Z 2 (D,0)
(the first of the numbers in the parentheses is the differential form degree,
the second is the ghost number and if there is only one number it denotes
the total form degree) is the proper solution of the master equation
δ S δ S δ S δ S
(S ,S ) ≡ r 0 l 0 − r 0 l 0 = 0. (4)
0 0 δA˜a ∂B˜a δB˜a ∂A˜a
To prove this observe that due to the gauge transformations (2), in B˜ there
is not any field possessing positive ghost number so that (B˜B˜) = B ∧B
(D,0)
and there is not B⋆ term in (B˜dA˜) .
(D,0)
BRST transformation of a functional is defined as
∂ F ∂ F
Ω F(A˜,B˜) = r Ω A˜a + r Ω B˜a (5)
0 ∂A˜a 0 ∂B˜a 0
where
∂ S ∂ S
Ω A˜a = − l 0, Ω B˜a = l 0. (6)
0 ∂B˜a 0 ∂A˜a
In terms of the BRST charge Ω (4) can equivalently be formulated as
0
Ω0L˜0 +dω(D−1,1) = 0, (7)
where ω(D−1,1) isa suitable(D−1)–form possessing ghost number one. More-
over, due to the fact that (Ω +d)2 = 0, there are the descent equations
0
Ω0ω(D−1,1) +dκ(D−2,2) = 0, Ω0κ(D−2,2) +dλ(D−3,3) = 0, ···, (8)
3
with suitable κ,λ,···, vanishing at a certain step.
Obviously Ω maps A˜ → d˜A and B˜ → d˜B except
0
Ω B⋆a = dAa +Ba.
0
However, there is not any B⋆ term in L˜ , so that, the BRST transformation
0
of the free (7) theory can be generalized as
(Ω +d)(B˜adA˜a) = 0. (9)
0 (D)
WedroppedtheBa∧Ba termwhich doesnot transformunder Ω .(9)written
0
incomponentsproduces(8)withtheidentificationω(D−1,1) ≡ (B˜adA˜a)(D−1,1),
κ(D−2,2) ≡ (B˜adA˜a)(D−2,2) and so on. A compact notation similar to (9) is
also presented in [10]
The consistent interactions will be introduced as cubic or higher order in
the generalized fields Φ˜ ≡ (A˜,B˜):
S ≡ gk S = (g Φ˜M +g2 Φ˜M+1 +···)
int M Mk Z M M (D,0)
X X
k=1,M=3 M=3
= (g Φ˜3 +g2Φ˜4 +···+g Φ˜4 +g2Φ˜5 +···) , (10)
3 3 4 4 (D,0)
such that
(S +S ,S +S ) = 0. (11)
0 int 0 int
g are independent coupling constants, hence, (11) leads to
M
(S ,S ) = 0, (12)
0 0
(S ,S ) = 0, (13)
0 M1
2(S ,S )+(S ,S ) = 0, (14)
0 M2 M1 M1
.
.
.
(13) can equivalently be written in terms of the BRST charge Ω as
0
Ω0Φ˜M(D,0) +dk(D−1,1) = 0, (15)
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where k is a suitable (D −1)–form possessing ghost number one. As it has
already mentioned Ω maps A˜ → d˜A and B˜ → d˜B except B⋆. Let the B⋆
0
dependent terms in S are
M1
(Φ˜M(D,0))B⋆ ∼ (B⋆Φ˜M−1 +B⋆2Φ˜M−2 +···+B⋆M−1Φ˜)(D,0).
Thus in (15) the terms which does not possess an exterior derivative will be
Ω0Φ˜M(D,0) +dkD−1,1 ∼ BΦ˜M−1 +2BB⋆Φ˜M−2 +···+(M −1)B⋆M−2Φ˜. (16)
There are two possibilities i) (16) vanishes due to symmetry properties, so
that (15) is satisfied with k(D−1,1) ≡ Φ˜3(D−1,1), ii) (16) does not vanish so that,
(15) is not satisfied. In the latter case the interactions are not consistent.
Therefore, we conclude that as far as we deal with the consistent interactions
(13) is equivalent to
Ω Φ˜M +dΦ˜M = 0, (17)
0 (D,0) (D−1,1)
and it can be generalized to D–total form:
(Ω +d)Φ˜M = 0, (18)
0 (D)
which leads to the descent equations
Ω Φ˜M +dΦ˜M = 0, (19)
0 (D,0) (D−1,1)
Ω Φ˜M +dΦ˜3 = 0, (20)
0 (D−1,1) (D−2,2)
.
.
Ω Φ˜M +dΦ˜M = 0, (21)
0 (1,D−1) (0,D)
Ω Φ˜M = 0. (22)
0 (0,D)
Obviously some of Φ˜M can be vanishing. We conclude that the generalized
(a,b)
fields method provide the consistent interactions as well as the solution of
the descent equations.
Gauge invariance of the interacting theory can be obtained by replacing
S with S +S in (6) on thesurface where the ghost fields andthe antifields
0 0 int
are vanishing.
Spin–1 Gauge Field in D=4: Free theory given by the first order action
−1 1
L = d4x (Baµν(∂ Aa −∂ Aa)− Ba Baµν), (23)
0 2 Z µ ν ν µ 2 µν
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is invariant under the infinitesimal gauge transformations
δ Aa = ∂ Λa , δ Ba = 0.
Λ µ µ Λ µν
Thetheoryisirreducible, sothatweneedtointroduce(intheminimalsector)
the ghost field η , and the antifields A⋆ , η⋆ , and B⋆ . Here the
(0,1) (3,−1) (4,−2) (2,−1)
star indicates the antifields as well as the Hodge-map.
The generalized fields are
A˜a = Aa +ηa +B⋆a ,
(1,0) (0,1) (2,−1)
B˜a = −A⋆a −η⋆a +Ba .
(3,−1) (4,−2) (2,0)
The proper solution of the master equation is
−1 1
S = d4x (B˜adA˜a − B˜aB˜a) , (24)
0 2 Z 2 (4.0)
1 1
= − d4x ( Baµν(∂ Aa −∂ Aa)−A⋆a∂µηa − Ba Baµν), (25)
Z 2 µ ν ν µ µ 4 µν
from which we read the BRST transformations
Ω Aa = ∂ ηa; Ω Baµν = 0; Ω ηa = 0;
0 µ µ 0 0
Ω A⋆aµ = ∂ Baµν; Ω B⋆a = ∂ Aa −∂ Aa −Ba ; Ω ηa⋆ = −∂ A⋆aµ.
0 ν 0 µν µ ν ν µ µν 0 µ
(26)
Theunique candidateforaconsistent cubicinteractionisthe4–totalform
1
κ ≡ f B˜aA˜bA˜c. (27)
(4) abc
2
Onecanshowthat,infact,(27)providesthesolutionofthedescent equations
Ω κ +dκ = 0,
0 (4,0) (3,1)
Ω κ +dκ = 0,
0 (3,1) (2,2)
Ω κ = 0,
0 (2,2)
κ = κ = 0,
(1,3) (0,4)
if f is totally antisymmetric. Written explicitly
abc
1 1
κ = f ( BaµνAbAc −A⋆aµAbηc +BaµνηbB⋆c − η⋆aηbηc),
(4,0) abc 2 µ ν µ µν 2
1
κ = f (BµνAbηc + A⋆aµηbηc),
(3,1) abc ν 2
1
κ = − f Baµνηbηc.
(2,2) abc
2
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If f satisfy theJacobiidentities, onecanshow thatS = g κ satisfies
abc 31 3 (4,0)
R
(S ,S ) = 0. (28)
31 31
Hence, there is no need of adding a quartic interaction with g2 coupling
3
constant. However, one can in principle add a 4–total form A˜4 with another
coupling constant g . For being a consistent interaction S should be BRST
4 41
invariant. But one can show that
Ω A˜4 6= dK ,
0 (4,0) (3,1)
for any K. Therefore, we cannot add this interaction term. Observe that
after a gauge fixing B⋆ = 0 and using the equations of motion related to
Bµν, Yang-Mills theory follows with the required quartic interaction.
Spin–1 Gauge Field in D=3: The free gauge theory is
A = Aa ∧dAa. (29)
0 Z
It is invariant under the infinitesimal gauge transformations δ Aa = dΛa. It
Λ
is an irreducible theory, so that the generalized field is
A˜a = Aa +ηa +A⋆a +η⋆a . (30)
µ(1,0) (0,1) µ(2,−1) (3,−2)
The solution of the master equation
S = A˜adA˜a, (31)
0 Z
leads to the BRST transformations
Ω Aa = ∂ ηa, Ω ηa = 0,
0 µ µ 0
Ω A⋆aµν = −ǫµνρ∂ Aa, Ω η⋆a = −∂ A⋆aµ.
0 ν ρ 0 µ
The unique candidate for a consistent interaction is the ghost zero com-
ponent of the 3–total form ξ ≡ A˜3. Indeed, one can show that if f are
abc
antisymmetric in all of the indices the descent equations
(Ω +d)(f A˜aA˜bA˜c) = 0, (32)
0 abc (3)
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are satisfied, where the components are
ξ = −f (Aa ∧Ab ∧Ac +6A⋆a ∧Ab ∧ηc +3η⋆aηbηc),
(3,0) abc
ξ = 3f (Aa ∧Ab ∧ηc +A⋆a ∧ηb ∧Ac),
(2,1) abc
ξ = 3f Aa ∧ηb ∧ηc,
(1,2) abc
ξ = −f ηa ∧ηb ∧ηc.
(0,3) abc
There is no need of any further interaction term because, the total action
S = S + g ξ satisfies the master equation if f satisfy the Jacobi
0 3 (3,0) abc
identities. R
Free BF Theory in D=2: Deal with the free gauge theory given by the La-
grange density
1
L = − ǫµνΦa(∂ ha −∂ ha), (33)
0 2 µ ν ν µ
which is invariant under the infinitesimal gauge transformations
δha = ∂ λa, δΦa = 0. (34)
µ µ
Although a careful treatment of the globalmodes showed thatat some points
of the target manifold the theory is reducible[11], we deal with the regions
of the target manifold which does not include these points. Hence, the gen-
eralized fields are
h˜a = ha +ηa +Φ⋆a , (35)
(1,0) (0,1) (2,−1)
Φ˜a = −h⋆a −η⋆a +Φa . (36)
(1,−1) (2,−2) (0,0)
The BV quantized free action
1
S = Φ˜adh˜a, (37)
0 2 Z
leads to the BRST transformations
Ω ha = ∂ ηa, Ω Φa = 0, Ω ηa = 0,
0 µ µ 0 0
Ω h⋆aµ = ǫµν∂ Φa, Ω Φ⋆a = ǫµν∂ ha, Ω η⋆a = −∂ h⋆aµ.
0 ν 0 µ ν 0 µ
There is a unique candidate for a consistent cubic interaction:
1
S = g σ ≡ g d2xf (Φ˜ah˜bh˜c) . (38)
31 3Z (2,0) 32 Z abc (2,0)
8
In fact, one can show that σ written in components
(D)
1 1
σ = f ( ǫµνhahbΦc −h⋆aµhbηc +Φ⋆aΦbηc − η⋆aηbηc),
(2,0) abc 2 µ ν µ 2
1
σ = f (haηbΦc − h⋆aηbηc),
(1,1) abc µ 2 µ
1
σ = − f ηaηbΦc,
(0,2) abc
2
satisfy the descent equations if f is antisymmetric in all of the indices.
abc
Moreover, if they satisfy the Jacobi identities one can show that
(S ,S ) = 0.
1 1
Although, there is no need of quartic interaction with a coupling constant
g2, one can in principle add quartic or higher interaction terms
3
g S ≡ g (Φ˜M−2h˜2) ,
M M1 M Z (2,0)
X X
M=4 M=4
if they satisfy the descent equations (19)–(22). Let us deal with the quartic
term
S = g d2xVad(Φ˜ Φ˜ h˜bh˜c) . (39)
41 4Z bc a d (2,0)
Indeed one can show that
(Ω +d)(VadΦ˜ Φ˜ h˜bh˜c) = 0, (40)
0 bc a d (2)
(S ,S ) = 0, (41)
1 2
(S ,S ) = 0, (42)
2 2
if the constants satisfy
Vcd = −Vcd, Vcd = Vdc, (43)
ab ba ab ab
f dVef +Vdff e +Vedf f = 0, (44)
[ab c]d [ab c]d [ab c]d
VdeVfg = 0, (45)
[ab c]d
where [ ] denotes that the indices within them are antisymmetrized.
The higher interactions can also be treated similarly.
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Antisymmetric Tensor Field in D–dimension: The first order free action of
the two form field B = B dxµ ∧dxν in any D dimensions is
µν
1
A = (dBa ∧Ha − Hˆa ∧Ha), (46)
0 Z 2
where H is a D−3 differential formandHˆ is its Hodgedual. (46)is invariant
under the gauge transformations
δ Ba = dΛa, (47)
Λ
where Λa is 1-form. The gauge transformations (47) are reducible because,
they vanish for Λa = dǫa, where ǫa is a scalar. Thus, in the minimal sector
there are ghost and ghost of ghost fields. The generalized fields are
B˜a = Ba +Ca +ηa −H⋆a ,
(2) (2,0) (1,1) (0,2) (3,−1)
H˜a = B⋆a +C⋆a +η⋆a +Ha ,
(D−3) (D−2,−1) (D−1,−2) (D,−3) (D−3,0)
where, now, star indicates also the Hodge map.
Solution of the master equation for the free theory is
1
S = (dB˜aH˜a − Hˆa ∧Ha) . (48)
0 Z 2 (D,0)
Hence, the BRST transformations are
Ω Ba = dCa, Ω Ha = 0, Ω Ca = dηa, Ω ηa = 0,
0 0 0 0
Ω B⋆a = dH, Ω H⋆a = dB −Hˆ, Ω C⋆a = −dB⋆a, Ω η⋆a = dC⋆a.
0 0 0 0
There are four combinations of the generalized fields which are cubic:
1) H˜3 is a 3(D−3)–totalform. For being a Lagrange density the form degree
should be D, which means that the dimension should be D = 9/2. Hence, it
is not permitted.
2) B˜3 is 6–total form so it is permitted in D = 6. In 6–dimension
B˜3 = f (Ba ∧Bb ∧Bc +6Ba ∧Cb ∧H⋆c +2η⋆a ∧H⋆b ∧H⋆c).
(6,0) abc
However, one can easily observe that
Ω B˜3 6= dκ
0 (6,0) (5,1)
10