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Duality between constraints and gauge conditions.

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Ann. Phys. (Leipzig) 16, No. 7–8, 529 – 542 (2007) / DOI 10.1002/andp.200710247
Duality between constraints and gauge conditions
M. N. Stoilov∗
Institute for Nuclear Research and Nuclear Energy, 72 Tzarigradsko Chausseé, Sofia 1784, Bulgaria
Received 11 January 2007, revised 16 April 2007, accepted 12 May 2007 by A. Wipf
Published online 3 July 2007
Key words gauge theory, constrained systems, BRST charge.
PACS 11.15.-q; 11.30.-j
There are two important sets of seemingly absolutely different objects in any gauge theory: the set of
constraints, which generate the local symmetry and the set of gauge conditions, which fix this symmetry;
the first one is determined by the Lagrangean of the model, the second is a matter of choice. However, in the
transition amplitude constraints and gauge conditions participate in exactly the same way. This suggests the
possibility for existence of a model with the same transition amplitude and in which gauge conditions and
constraints are interchanged. We investigate the conditions that gauge fixing terms should satisfy so that this
dual picture is allowed. En route, we propose to add new terms in the constraints which would generate the
gauge transformation of the Lagrange multipliers and construct two BRST charges – one, as usual, for the
constraints, and one for the gauge conditions.
c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Introduction
Gauge theories are in the heart of contemporary particle physics. There are several methods to treat such
theories, all originating from the Hamiltonian approach to dynamical systems with constraints proposed
in [1] and all of which are equivalent for the class of models we shall discuss. In the Hamiltonian approach
a constrained system is characterized by its Hamiltonian and by constraints (which are an inherent part of
the model and generate the gauge symmetry in it). Additional gauge fixing conditions have to be specified
in order to obtain well defined quantities.
Using the Hamiltonian, constraints, and gauge conditions one can write down the transition amplitude
(or the S-matrix) of the model as a functional integral over phase space [2]. In this expression there is a
symmetry, which is of primary interest for us – constraints and gauge conditions participate in exactly the
same way and we cannot distinguish them. The reason is that, loosely speaking, in a particular coordinate
system the gauge conditions are part of the dynamical coordinates and the constraints are the corresponding
momenta, both forming the unphysical sector of the theory. But one can perform canonical transformation
in the phase space and any such transformation preserves the transition amplitude. The simplest example of
canonical transformation is the mutual interchange of the coordinates and their momenta. Performing exactly
this transformation in the unphysical sector, we in fact interchange the constraints and gauge conditions.
The constraints-gauge conditions symmetry allows us to view the transition amplitude as originating
from a different gauge theory in which the local symmetry is generated by the gauge conditions of the
initial model and in which the former constraints play the role of gauge conditions. Moreover, in this dual
picture we can change the gauge conditions (former constraints) because they are now in our hands and
end up with a theory totally different from the initial one. This gives us strong evidence to believe that the
constraints-gauge conditions symmetry is connected to the important electric–magnetic duality.
∗
E-mail: mstoilov@inrne.bas.bg
c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
530
M. N. Stoilov: Duality between constraints and gauge conditions
The symmetry between constraints and gauge conditions gives us a new way to abelianize the constraints.
The possibility for local abelianization of the gauge symmetry in any model is a consequence of the BatalinVilkovisky theorem [3]. Usually it is not possible to achieve global abelianization. It turns out that the reason
for this is with very fundamental origin – it is the Gribov ambiguity which does not allow us to construct
Abelian constraints in the entire phase space [4]. The Gribov ambiguity determines the boundaries (the
Gribov horizon) of a region in the phase space in which the transition amplitude can be written as a
functional integral. But the constraints-gauge conditions duality is a feature of the very same functional
integral. Therefore, the Gribov ambiguity sets also the limit of validity of the proposed duality. Inside
any given Gribov region we can find using the constraints-gauge conditions duality (and under conditions
specified in the paper) another gauge model with the same transition amplitude, and moreover, the gauge
symmetry of the new model can be Abelian. However, we cannot prove the equivalence between the two
dual models outside the Gribov region we have started with which is in agreement with the results of [4].
Unfortunately, the constraints-gauge conditions symmetry is explicit only in the Hamiltonian functional
integral representation in which the gauge conditions are functions of the phase space variables only. But
the gauge fixing conditions can be any set of independent gauge non-invariant functions and in general
can depend on Lagrange multipliers too. These multipliers are independent variables, have no Poisson
brackets with constraints, and gauge conditions involving them cannot be handled within the canonical
Hamiltonian approach. As a result, when more general gauge conditions are used the expression for the
transition amplitude is a little bit different and the symmetry we are talking about is missing (or at least is not
obvious). That is why we want to find a variant of the Hamiltonian procedure allowing an uniform treatment
of different gauges, so that all nice properties of the Hamiltonian transition amplitude hold for more general
situations. For this we consider a larger phase space adding to the initial one the space of the Lagrange
multipliers and their momenta. In this enlarged space we construct constraints, which generate the gauge
transformation of any function in it. This construction solves another aspect of the symmetry problem – we
achieve that both constraints and gauge conditions depend on same variables. There is yet another reason
to search for constraints in the enlarged phase space. As we shall see below, gauge conditions with and
without Lagrange multipliers are not equivalent and do not always fix the gauge freedom completely. Only
gauge conditions in the enlarged phase space possess the universal property to fix the gauge completely for
any gauge model.
The line we follow is very similar to the Batalin-Fradkin-Vilkovisky approach to the BRST symmetry [5].
So, finding the constraints in the enlarged phase space, our next task is to construct the BRST charge
corresponding to them. It turns out that our BRST charge differs from the BFV–BRST one. There is also a
difference in the BRST invariant Hamiltonian. Finally we construct second BRST charge connected to the
gauge conditions and construct the corresponding double BRST invariant action, making the constraints–
gauge conditions symmetry transparent in the BRST approach too. Thus, a purely quantum observation for
an existing symmetry in the transition amplitude becomes more or less a classical problem.
The paper is organized as follows: In Sect. 2 we specify the class of theories we shall work with and
recall the essence of the Hamiltonian approach to constrained systems. In Sect. 3 we prove that there is
a symmetry in the transition amplitude between constraints and gauge conditions. The symmetry holds
under some conditions on the gauge fixing terms. The basic limitation is dictated by the very nature of the
Hamiltonian approach in which one can handle gauges depending only on dynamical phase space variables.
We also show that such gauges are in general nonequivalent to the gauges involving Lagrange multipliers.
In Sect. 4 we propose how to apply the canonical Hamiltonian procedure to any gauge. We enlarge the
dynamical phase space adding to it the space of Lagrange multipliers and their momenta. In the space
thus constructed we find constraints and Hamiltonian corresponding to the initial ones. Using them the key
result of Sect. 3 is immediately extended over arbitrary gauge. In Sect. 5 we obtain a new BRST charge
corresponding to the constraints found in the previous section. We discuss the differences between this
BRST charge and the BFV one. In Sect. 6 we construct second BRST charge connected to the gauge fixing
terms and write down an action which is invariant with respect to both BRST charges. We show that in
certain limit this action coincides with the Faddeev-Popov one.
c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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2 Hamiltonian approach to gauge theories
The Hamiltonian approach to dynamical systems is equivalent to first order Lagrangean formalism in which
dynamical coordinates qi and their momenta pi , i = 1, . . . , n are treated as independent variables (both
forming the dynamical phase space of the model). The procedure how to obtain the first order Lagrangean
for models with gauge symmetry from the more familiar second order Lagrangean is described in details
in [6]. Here we suppose that all necessary steps are done, the constraints ϕa , a = 1, . . . , m, and the canonical
Hamiltonian1 H are identified and we have ended up with the following Lagrangean:
L = pq̇ − H − λa ϕa .
(1)
An implicit summation over all degrees of freedom (which could be discrete as well as continuance)
is understood. Here λa are the Lagrange multipliers, which are new independent ( arbitrary) variables.
As usual, the Hamiltonian and constraints are functions on the phase space only. Therefore, the Poisson
brackets amongst H, ϕ, and any other function g on the phase space are well defined while these with the
Lagrange multipliers vanish.
Let us now specify exactly the class of models we are dealing with. Hereafter we assume that all constraints
are Bose and first class and that the model in consideration is a rank one theory. All these requirements
mean that the constraints and Hamiltonian satisfy the following Poisson bracket relations:
[ϕa , ϕb ] = Cabc ϕc ,
(2)
[H, ϕa ] = Uab ϕb ,
(3)
where Uab and Cabc do not depend on dynamical variables (this is the order one requirement). The later
requirement is not crucial. Most of our results can be easily generalized to higher rank theories. In the latter
case the procedure resembles the construction of BRST charge for such theories.
Having a theory with first class constraints it is very easy to find the gauge transformations and time
evolution of any dynamical quantity g(q, p): gauge transformations are generated by constraints ϕ and
dynamics (up to a gauge transformation) is determined by the Hamiltonian H through the following Poisson
bracket relations:
δ g = [g, a ϕa ] ,
ġ = [g, H] .
(4)
Here a are arbitrary gauge parameters and ġ = ∂t g. The dynamical quantity g in (4) could be any
but a Lagrange multiplier. Lagrangean multipliers are the only exception of the rule – neither their time
evolution nor their gauge transformations are determined by eqs. (4). The Lagrange multipliers are absolutely
arbitrary and the same are their time derivatives. However, their gauge variations are well defined and can
be determined from the requirement that the action, corresponding to the Lagrangean (1) is gauge invariant.
The result is:
δ λa = ∂t a − b Uba + c Ccba λb .
(5)
In order to obtain eqs. (5) we assume that gauge variation and time derivative of the canonical coordinates
commute, i.e.:
δ ∂t q = ∂t (δ q).
(6)
The above formula can be derived if we use the equations of motion, or, said in other words, if we temporarily
switch to second order Lagrangean description of the theory, find the desired variation, and then go back to
1 The Hamiltonian of any constrained system is determined up to weakly zero terms (terms proportional to constraints). The
Hamiltonian for which H = H|ϕ=0 is the canonical one.
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M. N. Stoilov: Duality between constraints and gauge conditions
the first order Lagrangean (1). In the case of field theory eq. (6) means that the gauge transformations are
‘internal’, i.e., they affect the dynamical fields but do not affect the space-time variables.
A short note concerning eq. (3) should be added at this point. It can be proved that if the algebra (2) is
an Abelian or a semisimple Lie algebra then the matrix U in eq. (3) is zero. The proof of this statement
is given in Appendix A. Despite this result we shall continue to write our formulae for arbitrary U . There
are two reasons for that. First, this allows easier generalization of our results for higher order theories.
Second, and this is more important, keeping U is essential to compare our results with the well known
formulae of the BFV approach to the BRST symmetry. Note that keeping U means that we work not with
the canonical Hamiltonian but with Hamiltonian with weakly zero terms. This is not a problem at all because
any weakly zero term in the Hamiltonian can be absorbed in the λa ϕa term in (1) after redefinition (with
unity Jacobian) of the Lagrange multipliers. Moreover, working with non-canonical Hamiltonian appears
to be more convenient for large number of models.
3 Gauge fixing
The proper treatment of any model with gauge symmetry requires supplementary gauge fixing conditions. If
we stick to the Hamiltonian approach we have to pick some independent and gauge non-invariant functions
χa of the dynamical phase space variables and to impose the following conditions:
(7)
χa (q, p) = 0.
The exact meaning of the words ‘independent and gauge non-invariant’ is that
det [χb , ϕc ] = 0.
(8)
In addition, one demands that χa form an Abelian algebra under Poisson bracket relations
[χa , χb ] = 0 ∀a, b.
(9)
Both requirements (8) and (9) are important.
Having the gauge conditions (7) the transition amplitude for a model with a canonical Hamiltonian H
and constraints ϕa is given by the following expression [2]:
δ(ϕa )δ(χa )| det ∆|
Z = DpDq exp i {pq̇ − H) dt
=
t
a
DpDqDλDπ exp i {pq̇ − H − λϕ − πχ) dt | det ∆|.
(10)
t
The operator ∆ which determinant takes place in Z is
∆ab ≡ [χa , ϕb ]
(11)
and by eq. (8) is invertible. Eq. (10) plays very important role in the gauge theory because for it the unitarity of
the S-matrix can be proved. To do this one chooses χa as dynamical coordinates in the unphysical subsector
of the phase space (which is allowed by eqs. (9)), and then uses eqs.(8) and (11) to resolve constraints with
respect to the momenta conjugated to χa . As a result the integration over unphysical subspace is trivial and
one obtains a description of the constrained system entirely in terms of the physical phase space. We stress
again that in this proof the form (11) of the operator ∆ is essential.
Having ∆ defined by eq. (11) we see that constraints and gauge conditions participates in a same way
in eq. (10). There are however two questions we have to answer in order to prove that there is a symmetry
between constraints and gauge fixing terms in the transition amplitude. The first one concerns the algebra
c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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Ann. Phys. (Leipzig) 16, No. 7–8 (2007)
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formed by gauge conditions through Poisson brackets. The second is about the Poisson brackets between
the Hamiltonian and gauge conditions.
Eqs. (9) say that χa form an Abelian algebra. We want to relax this requirement because the constraints
may form non-Abelian algebra and if we insist on the symmetry constraints – gauge conditions the same
possibility should be opened for the gauge conditions too. It is enough for our purposes to prove that the
transition amplitude is still given by eq. (10) provided χa obey the following relations
[χa , χb ] = Dabc χc .
(12)
Here Dabc , as Cabc , are independent of the dynamical variables. The proof of this assertion is given in
Appendix B.
Another condition which gauge conditions have to satisfy provided we want to consider them as constraints in some dual model is that they should be closed under time evolution. Therefore, they have to
satisfy the following relations, analogous to those given in eqs. (3) for the initial constraints
[H, χa ] = Vab χb
(13)
with Vab independent of the phase space variables. There are indications that adding to the Hamiltonian
proper weakly zero terms we can ensure the fulfilment of eq. (13) for any χ but we have not rigorous proof
of this statement. So, eq. (13) becomes the crucial condition which distinguishes the gauges for which the
constraints-gauge conditions duality is fulfiled.
Having transition amplitude given by eq. (10), operator ∆ defined by eq. (11), constraints satisfying
eqs. (2), (3), and gauge conditions – eqs. (12), (13), the assertion that there is a symmetry between constraints
and gauge conditions is obvious. It is clear that if we consider a new gauge model in which the constraints
are the former gauge conditions χa and if we use as gauge conditions the former constraints ϕa this model
will possess the same transition amplitude as (10) and will describe the same physics.
Together eqs. (3) and (13) impose severe restrictions on the Hamiltonian. For example, as a result of
these equations we get the following necessary condition on the Hamiltonian form:
H = ϕa Fab χb + Hind
(14)
where Hind has zero Poisson brackets both with ϕa and χa for each a. As it should be expected from the
proposition in Appendix A, the term ϕa Fab χb in eq. (14) is weakly zero no matter which of the functions ϕa
and χa we considered as constraints. From eqs. (3), (14) we get that U T = ∆F on the hypersurface χ = 0;
from eqs. (13), (14) we get ∆F = −V on the hypersurface ϕ = 0. Therefore, on the physical subspace we
have the following relation between the matrices U and V :
U T = ∆F = −V.
(15)
Note that when ϕa and/or χa form non-Abelian algebra, ∆ depends on the dynamical variables and therefor
eqs. (15) are highly nontrivial.
Examples:
1. The simplest U (1) example for which the Hamiltonian H has not vanishing Poisson bracket with the
constraint ϕ can be constructed in 2D phase space. In this case H = qp and ϕ = p. The gauge condition
χ = q fixes the gauge completely and H = χϕ as it has to be expected from eq. (14).
A slightly more complicated U (1) example is based on a model in which the gauge generator is the
positive step operator of sl(2) algebra and the Hamiltonian is the Cartan element of the same algebra.
The common 2 × 2 matrix representations of these operators are
0 1
1 0
ϕ=
, H=
.
(16)
0 0
0 −1
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M. N. Stoilov: Duality between constraints and gauge conditions
Having in mind this realization we can decide in a hurry that our assertion that the Hamiltonian is
proportional to the constraint is wrong. In order to see that everything is correct let us construct a
representation of ϕ and H in some phase space. Using eqs. (16) we can realize the representation we
are looking for in the 4D phase space spanned by q1 , q2 , p1 , and p2 . In this space the constraint and
the Hamiltonian are given as follows:
ϕ = −q1 p2
H = q 2 p2 − q 1 p1 .
It is easy to show that χ = p1 /p2 −q2 /q1 fixes the gauge ([χ, ϕ] = 2) and that H = χϕ which formally
proves our assertion. However, we have to be a little bit more precise in the analysis of this model. Let
us consider χ = q2 as a gauge condition. For this gauge the Faddeev–Popov determinant is
ϕ, χ = |q1 |
and so, we have a Gribov horizon q1 = 0 which has to be excluded from our considerations. But when
q1 = 0 the constraint ϕ is equivalent to ϕ = p2 . Thus in both Gribov regions q1 > 0 and q1 < 0
the Hamiltonian is χ ϕ + gauge invariant term which is exactly the structure suggested by eq. (14).
The consideration of the alternative gauge χ = p1 follows the same line. In this case the Gribov
horizon is p2 = 0, the equivalent constraint is φ = q1 and the Hamiltonian is again in the form
(14). In the exceptional point q1 = 0; p2 = 0 we have enlarged gauge symmetry, we need two gauge
conditions (p1 = 0; q2 = 0) and the Hamiltonian is a sum of gauge fixing terms multiplied by the
corresponding constraints.
2. Our non-Abelian example is pureYang–Mills for simple Lie group in four dimensions. The components
of the Yang–Mills field are denoted by Aaµ and the components of the field strength tensor are denoted
a
by Fµν
. The first order Lagrangean density for the model is [2]
L = Eia Ȧai − H + Aa0 ϕa .
(18)
a
Here Eia = Fi0
are the momenta conjugated to the coordinates Aai , H is the Hamiltonian, Aa0 are the
Lagrange multipliers and ϕa are the constraints. The Hamiltonian and the constraints are given by the
following expressions:
2
2
(19)
H = 12 (Eia ) + (Gai )
ϕa = ∂i Eia − Cabc Abi Eic
(20)
a
where Gai = 12 ijkFkj
. The constraints form closed algebra with structure constants Cabc and have
vanishing Poisson brackets with the Hamiltonian. As gauge conditions we use Coulomb gauge χa =
∂i Aai . Unfortunately, these gauge terms have non-vanishing Poisson brackets with the Hamiltonian
(19) even on the surface χa = 0. This can be corrected subtracting from the Hamiltonian weakly zero
terms. Let us introduce longitudinal component of the momentum (which is weakly zero)
a
b
E L i = ∂i ∆−1
ab ϕ
and use instead of the Hamiltonian (19) the following one:
2
L
2
1
E−E
+G .
H = 2
The Hamiltonian H has vanishing Poisson brackets with χa and its Poisson brackets with ϕa are
proportional to the constraints. Therefore H , ϕa and χb fit in our scheme (except the unessential
c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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Ann. Phys. (Leipzig) 16, No. 7–8 (2007)
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technical requirement for order one) and the dual picture we are speaking about is given by an U (1)m
gauge model with the following Lagrangean density
2 D
a a
a 2
a
La
1
+ Aa0 ∂i Eia .
(23)
L = Ei Ȧi − 2 (Di ) + Ai − A i
Here
∂j Eka − ∂k Eja + Cabc Ekb Ejc ,
¯ −1 ∂i Aai + Cbcd Aci Eid ) .
= ∂i ∆
ab
Dia =
a
AL i
1
2 ijk
(24)
If we use as gauge conditions ∂i Aai −Cabc Abi Eic = 0 then the transition amplitudes for the U (1)d gauge
model with Lagrangean density (23) will coincides with the transition amplitudes for pure Yang–Mills
with Lagrangean density (18).
Let us come back to the general considerations. A problem we have to consider concerns the functional
form of the gauge conditions. Eqs. (7) are not the most general gauge conditions, even if they form nonAbelian algebra. It is possible to fix the gauge with functions not only of the dynamical phase space variables
but of the Lagrange multipliers too. The ultimate case, which is in fact very common, is when the gauge
involves Lagrange multipliers only and sets them to some constants λ0
λ = λ0 .
(25)
In any case when one uses gauges different from (7) the operator ∆ is not given by eq. (11) but is the
matrix of independent gauge variations of the gauge conditions. Therefore, gauge (25), and any one which
involve Lagrange multipliers, goes beyond the Hamiltonian approach, because the gauge variations (5) of
λa are not expressible as Poisson brackets. Moreover, and this is important for us, the usage of such gauge
conditions spoils the constraints-gauge conditions symmetry. Again the reason is that in this case ∆ is not
given by eq. (11). We believe that, because the expression (10) is so fundamental in the theory, the symmetry,
observed in it, is not incidental. Our goal here is to make it explicit in any gauge. This task is solved trivially
if all gauges are equivalent: for any given gauge fixing conditions find the corresponding gauge (7) and use
it instead of the initial one. However, it turns out that gauges (7) and (25) are in general nonequivalent. That
is why the way we see to achieve our aim is to extend the applicability of the Hamiltonian approach.
The gauge conditions (7) and (25) are equivalent if eq. (7) implies eq. (25) and vise versa. For a large class
of models imposing eqs. (7) we can determine the Lagrange multipliers , i.e. we can find corresponding to
(7) equations of the type (25). In order to do this we use the equations of motion for χ
χ̇a = [χa , H + λb ϕb ] .
(26)
χ̇a have to be zero, and because ∆ is invertable we get
λ = ∆−1 [χ, H] .
(27)
Thus, in general eq. (27) gives a solution of the (7) to (25) direction of the equivalence. However, if the
canonical Hamiltonian is zero (as in the string models for instance) we encounter a problem. In this case
the unique solution of eqs. (27) is λa = 0 and we end up with an empty theory. Therefore, not for any gauge
theory conditions (7) are the good ones.
The situation is even more serious with the gauge (25). Using eqs. (5) it is easy to see that for this gauge
the operator ∆ has zero modes, i.e. the gauge conditions do not fix the gauge freedom entirely – we can
freely make transformation with parameter
= e−(U +Cλ
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0
)t
ζ
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M. N. Stoilov: Duality between constraints and gauge conditions
and eqs. (25) still hold. Note that in the case of field theory ζa are in fact arbitrary functions of the spatial
coordinates. This is a huge residual freedom in striking contrast to the situation when we use gauge conditions
(7). As a consequence, eqs. (25) are in general not enough to determine the physical degrees of freedom,
the transition amplitude is ill defined, and there are no conditions of the type (7) corresponding to the
gauge (25).
A possibility to get rid of all these contradictions is to use gauge conditions with explicit time dependence,
i.e.
χa (q, p, t) = 0,
(28)
or to make χa functions of all variables including λ and/or their momenta π
χa (q, p, λ, π) = 0.
(29)
An example of such gauge is the Lorentz gauge ∂µ Aµ in Electrodynamics (A0 is Lagrange multiplier).
4 Constraints in the Lagrange multiplier phase space
In order to use the transition amplitude as given by eq. (10) for gauges like (29) we have to be able to obtain
the gauge transformation of the Lagrange multipliers (5) in the same way as we get the gauge transformation
of any other dynamical quantity, namely, generated by constraints via Poisson brackets [9]. We are working
in an enlarged phase space formed as a direct sum of the dynamical phase space and the auxiliary phase
space spanned by the Lagrange multipliers λa and the momenta conjugated to them πa . In this space we
are looking for a realization of the constraints ϕ̂a , such that
δ λ = [λ, a ϕ̂a ] .
(30)
ϕ̂a = ϕa + . . .
(31)
Here
and dots stands for terms involving λb and their momenta πb . In turns out that once we step on this way we
have to modify not only the constraints but also the other two terms in the Lagrangean (1) – the kinetic term
and the Hamiltonian. However, we do not want to modify neither the dynamics nor the gauge freedom in
the theory. This means that we want the modified Hamiltonian Ĥ = H + h(λ, π) and modified constraints
ϕ̂a to satisfy an algebra like (2), (3), and eqs. (4) still to hold. Therefor, we impose:
[ϕ̂a , ϕ̂b ] = Cabc ϕ̂c ,
Ĥ, ϕ̂a = Uab ϕ̂b ,
[f (q, p), ϕ̂a ] = [f (q, p), ϕa ]
(32)
for any function f (q, p). Using the Jacobi identities among the structure constants Cabc and between them
and Uab we find the following expression for ϕ̂a :
←
ϕ̂a = ϕa + ∂tπa − Uab πb + λb Cabc πc .
(33)
It is important to stress, that the time derivative in the second term in the r.h.s. of the above expression
acts on the gauge parameters and not on the phase space variables. Thus we can freely calculate Poisson
brackets involving ϕ̂.
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Ann. Phys. (Leipzig) 16, No. 7–8 (2007)
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Our example to illustrate the above construction is again from Electrodynamics. There the total constraint
reads
←
(34)
ϕ̂ =∂tE0 + ∂i Ei
and using the Lorenz gauge ∂µ Aµ , we get ∆ = ∂µ ∂ µ , det ∆ = 0, and full gauge fixing. The constraint
(34) is also considered (in a different context) in [7].
Having ϕ̂a we find the Hamiltonian Ĥ:
(35)
Ĥ = H + λa Uab πb .
The procedure of derivation of eqs. (33), (35) is very similar to the construction of the ghost terms in the
constraints and Hamiltonian. Note, however, that we need the matrix U in ϕ̂a , which is not the case when
one "prolongs" the constraints with ghost terms.
The first order Lagrangean L̂ which involves ϕ̂a and Ĥ instead of ϕ and H is invariant under gauge
transformations provided we add also a kinetic term πa λ̇a for the Lagrange multipliers. Putting all things
together we surprisingly get that all extra terms cancel out
L̂ = pq̇ + π λ̇ − Ĥ − λa ϕ̂a = L.
(36)
As a result the dynamics of the Lagrange multipliers is not determined and they are completely arbitrary
as they should be.
5 BRST charge
Everything in the construction of the BRST charge Q is quite standard, except that we shall need extra
ghosts at a particular point. Let ca and P̄a are the ghost variables, {ca , P̄b } = −δab , ca are real, and P̄a are
imaginary and with ghost numbers 1 and −1 respectively. For a theory with constraints ϕa which form an
algebra with structure constants Cabc the so called ‘minimal BRST charge’ [5] is
Qmin = ca ϕa + 12 ca cb Cabc P̄c .
(37)
In our case we have to use constraints ϕ̂a , so the BRST charge reads
Q = ca ϕ̂a + 12 ca cb Cabc P̄c =
= ca (ϕa − Uab πb + Cabc λb πc ) + 12 ca cb Cabc P̄c + ċa πa .
(38)
There is little use of this expression because of the term ċa πa whose Poisson bracket with other quantities
we cannot calculate. So, we introduce another set of ghost–antighost pairs {c̄a , P} (with opposite to ca and
P̄a ghost numbers), substitute ċa in (38) with iPa , and choose such gauge fixing conditions as to ensure the
following equation of motion
Pa = −iċa .
(39)
The BRST charge now reads
Q = ca (ϕa − Uab πb + Cabc λb πc ) + 12 ca cb Cabc P̄c + iPa πa .
(40)
However, the BRST charge thus constructed is not nilpotent. We need an additional ghost term to ensure
[Q, Q] = 0. The BRST charge we finally found is:
Q = ca (ϕa − Uab πb + Cabc λb πc + Cabc Pb c̄c ) + 12 ca cb Cabc P̄c + iPa πa
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c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
538
M. N. Stoilov: Duality between constraints and gauge conditions
= ca ϕa + 12 ca cb Cabc P̄c + iPa πa − ca Uab πb + ca Cabc λb πc + ca Cabc Pb c̄c .
(41)
In order to compare this expression with the one in the BFV formalism we write down the BFV-BRST
charge as given in [5].
QBF V = ca ϕa + 12 ca cb Cabc P̄c + iPa πa .
(42)
Both charges act in one and the same phase space. The difference is in the last three terms in eq. (41). Note
that, first, we need the matrix U to construct Q, while we do not need it for QBF V . This difference however
could be significant only for higher rank theories. Second, QBF V is a sum of two independent minimal
BRST charges which is not the case with Q:
QBF V = Qmin + Qext
(43)
where Qmin is given by eq. (37). It corresponds to the initial constraints ϕa in the theory and acts in the
dynamical phase space and the phase space of the first set of ghost–antighost pairs. The BRST charge Qext
can be viewed as other minimal BRST charge but based on the Abelian constraints πa = 0 and acts in the
phase space of the Lagrange multipliers and the second set of ghost–antighost pairs. The BRST charge Q,
which contains QBF V , cannot be presented as a sum of two independent charges. As a result Q and QBF V
are not in one class of equivalence, i.e., their difference cannot be expressed as a Poisson bracket of one
of the charges with some appropriate Bose function with ghost number zero. This assertion can be proved
by
showing
that the direct iterative construction of such function fails, but is much simpler just to calculate
Q, QBF V which should be zero if they are equivalent. The result is
QBF V , Q = 12 (ca cb Cabc Ucd πd − ca cb Cabc Ccde cd πe − ca cb Cabc Ccde Pd c̄e )
(44)
which is zero only if Cabc = 0, i.e., in the case of Abelian gauge algebra.
Using eq. (41) we find the BRST invariant Hamiltonian and it is
H = H + ca Uab P̄b + λa Uab πb + Pa Uab c̄b .
(45)
The difference between (45) and the BRST invariant Hamiltonian in the BFV formalism is in the last two
terms. They depend on the matrix U and for rank one theories are not important according to the proposition
in Appendix 1.
For a gauge condition we choose the basic BFV one and it is:
ψ = ic̄a χa + P̄a λa ,
(46)
Note that here χa are functions of q and p only, which satisfy eqs. (9), i.e. they form an Abelian algebra.
The BRST invariant action for a model with BRST charge Q, BRST invariant Hamiltonian H and gauge
fixing function ψ is:
˙ − H + [Q, ψ] ,
S = q̇p + λ̇π + ċP̄ + c̄P
(47)
t
In our case it reads
˙ − H − PU c̄ + ic̄∆c + πχ − λϕ − λCP c̄ + iP̄P + icCχc̄ .
q̇p + λ̇π + ċP̄ + c̄P
S =
t
(48)
Note that the variation of S with respect to P̄ gives eqs. (39), so our gauge is correct. The corresponding
action in the BFV approach is:
˙ − H − cU P̄ + ic̄∆c + πχ − λϕ − λCcP̄ + iP̄P . (49)
q̇p + λ̇π + ċP̄ + c̄P
S BF V =
t
The only significant difference between S and S BF V is in the ghost term ica Cabc χb c̄c in eq. (48), all other
differences are just redefinitions of the ghosts.
c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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Ann. Phys. (Leipzig) 16, No. 7–8 (2007)
539
6 Second BRST charge and double BRST invariant action
We want to stressing again that deriving BRST action (47) we use gauge (46) in which functions χa form an
Abelian algebra under Poisson bracket relations. In the spirit of our previous considerations, it is natural to
recognize in eq. (46) after a canonical change of variables λ → π, π → −λ the (multiplied by i) BFV-BRST
charge for the Abelian ‘constraints’ χa . This suggests, when gauge conditions χa form non-Abelian algebra
(12), to use instead (46) the corresponding to them BRST charge. This charge according to eq. (41) is:
(50)
Q̄ = c̄a χa − Vab λb − Dabc πb λc + Dabc P̄b cc + 12 c̄a c̄b Dabc Pc − iP̄a λa ,
and using it we construct the following action:
˙ − H + i Q, Q̄ .
S = q̇p + λ̇π + ċP̄ + c̄P
(51)
The Fradkin-Vilkovisky theorem [3, 8] guarantees
that the action (51) is as correct as (47). Note the
beautiful cohomological character of the Q, Q̄ term in (51): The functions over the full phase space
formed by the dynamical variables, Lagrange multipliers and their momenta, and the two systems of ghosts
is an associative supercommutative algebra F [10]. This algebra has a natural grading with respect to the
ghost number operator. The action of the BRST charge Q on F gives to this superalgebra the structure of a
graded differential algebra. Here Q, which has ghost number 1, plays the same role as the operator of the
exterior derivative d in the case of differential forms.
The operator Q̄ which has ghost number −1 plays the
role of d∗ – the Hodge dual to d. The term i Q, Q̄ is, in fact, the Hodge operator for F.
A note should be added at this place before proceeding further. In order eq. (51) to be invariant under
both BRST charges Q andQ̄ the Hamiltonian H has to be Q̄ invariant. Initially it was constructed to be Q
invariant. We expect that additional ghost terms may be needed if we want also Q̄ invariance. However, it
turns out that, as a consequence of the eqs. (15), H is also Q̄ invariant, and so eq. (51) describes a double
BRST invariant action.
Substituting all our formulae in eq. (51) we obtain a simple expression for the double–BRST invariant action
˙ − H − λϕ + πχ + ic̄∆c + λU π − iχCcc̄
S=
q̇p + λ̇π + ċP̄ + c̄P
− iϕDc̄c + iP̄P + Lext ,
(52)
where Lext is a ghost term which we separate for a reason which shall become clear later. Note that λ̇π term
can be attached either to λϕ or to πχ thus producing the ‘Lorenz’ gauge for dual theories.
Using the action (52) we can obtain the corresponding transition amplitude as a functional integral of eiS
over all phase space variables (matter coordinates, Lagrange multipliers, ghosts, and their momenta). We
want to compare thus obtained expression with the one given by eq. (10). In order to do this we exploit an
idea of [5] to perform a rescaling of the gauge conditions χa plus a change of variables whose Berezinian is
equal to 1. However in our case we need to rescale also the constraints ϕa . All together our manipulations
look as follow:
χa →
1
χa ; πa = βπa , c̄a = βc̄a ,
β
ϕa →
1
ϕa ; λa = αλa , ca = αca .
α
(53)
Note that as a consequence of eqs. (53) the structure constants as well as ∆ are modified
Dabc →
www.ann-phys.org
1
1
1
Dabc ; Cabc → Cabc ; Vab → Vab ; ∆ab →
∆ab .
β
α
αβ
(54)
c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
540
M. N. Stoilov: Duality between constraints and gauge conditions
˙ go to zero and the transition
We consider the limit α → ∞, β → ∞ in which Lext , λ̇π, ċP̄, and c̄P
amplitude for the model with action (52) takes the form (for clearness we omit the sign in some of the
variable notations)
Z̄ = D pqπλP̄Pcc̄ exp i q̇p − H + ic̄∆c + πχ − iχCcc̄ − λϕ − iϕDc̄c + iP̄P . (55)
The integral over momenta P̄ and P is trivial giving an overall normalization constant. The term iχa Cabc cb c̄c
can be absorbed in πa χa by a redefinition of πa and the same is possible for iϕa Dabc c̄b cc which can be
absorbed in λa ϕa . After that integrations over λ, π, c, and c̄ are easily performed giving
(56)
Z̄ = Z.
7 Conclusions
We have shown that for a model with first order gauge symmetry it is possible to find an equivalent model
with different gauge symmetry provided some general assumptions (13) are fulfiled. For this dual model the
former gauge conditions are constraints, i.e., generators of the local symmetry. Changing gauge conditions
in the dual model (which are constraints in the theory we have started with) we obtain a system which looks
totally different but is equivalent to the initial one. We associate with the gauge conditions second BRST
charge and its Poisson brackets with the original BRST charge is the Hodge operator of the corresponding
cohomology complex and plays the role of the Hamiltonian in the phase space of the Lagrange multipliers
and ghosts.
Acknowledgements It is a pleasure to thank V. Dobrev, A. Ganchev, and O. Stoytchev for useful discussions. The
author is much obliged to V. Dobrev also for reading the manuscript. This work is supported by the Bulgarian National
Science Foundation, Grant Ph-1010/00.
Appendix A
Proposition: If the gauge algebra is an Abelian or a semisimple Lie algebra and we work with the
canonical Hamiltonian the coefficients Uab in eq. (3) are zeros [11].
P r o o f. Let us start our consideration with the case of U (1) gauge algebra. In this case we have one
gauge generator ϕ, a Hamiltonian H and eq. (3) has the following form:
(57)
[H, ϕ] = U ϕ
where U is some constant. It is always possible (after suitable canonical transformation) to identify the
Abelian constraint ϕ with one of the momenta of the system, say p1 . In the corresponding coordinate
system eq. (57) takes the form
∂H
= U p1 .
∂q1
(58)
Now consider eq. (58) as a partial differential equation which determines H. Its general solution is
H = U q 1 p1 + h
(59)
where h is an undefined function which does not depend on q1 . The term U q1 p1 in eq. (59) is weakly zero
(it is proportional to the constraint p1 ) and therefor can be neglected. Thus we get the following expression
for the canonical Hamiltonian
H=h
c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
(60)
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Ann. Phys. (Leipzig) 16, No. 7–8 (2007)
541
But h does not depend on q1 and so [H, p1 ] = 0. This proves that in the case of an U (1) gauge algebra
and for the canonical Hamiltonian the coefficient U in eq. (57) is zero . The generalization of the result to
U (1)m is straightforward.
Now suppose that the gauge algebra is semisimple. Note that eq. (3) represents not only the requirement
that the time evolution preserves the gauge algebra but it also asserts that the commutator with H is a
derivation of the gauge algebra. It is known (see, e.g. [12]) that any nontrivial derivation of a semisimple
algebra is an inner derivation. As a consequence the Hamiltonian H must be a combination of the generators
ϕa (plus, eventually, terms which commute with the gauge algebra). But we work with the canonical
Hamiltonian with all weakly zero terms removed from it. Therefore, H contains only terms which commute
with the gauge algebra and so, the coefficients Uab in eq. (3) are zeros.
Appendix B
Proposition: For a gauge theory with Hamiltonian H, constraints ϕ and gauge conditions χ such that
eqs. (2) and (12) hold the transition amplitude is given again with eq. (10).
P r o o f. We introduce notations allowing symmetric treatment of both ϕa and χa . Let φa denote the
set of constraints and gauge conditions
φa = {ϕa , χa }.
So, in our gauge fixed model we have the following relations
φa = 0 ∀a
(61)
det | [φa , φb ] | = 0.
(62)
such that
(Note that as a consequence of the eq. (62) we can view the relations (61) as second class constraints and
to use the results of [13] for a direct solution of the problem in consideration.)
Under some general nondegeneracy condition (see below) the 2m relations (61) determine 2n − 2m
dimensional physical submanifold in the entire 2n dimensional phase space. The normals to this submanifold
correspond to the unphysical degrees of freedom ζa . The nondegeneracy condition mentioned above is
det|
∂φa
| = 0
∂ζb
(63)
on the hypersurface φa = 0. The physical coordinates (which we denote by ζ ∗ ) are complementary to ζa
and they describe the directions tangential to the surface (61), i.e.
∂φa
= 0.
∂ζ ∗
(64)
Eq. (63) allows us to write down the transition amplitude for the considered model and it reads
∗
∗ ∗
∗
Z = Dζ exp i (p q̇ − H(ζ )) dt
=
www.ann-phys.org
t
∂φa ,
DpDq exp i (pq̇ − H(p, q)) dt
δ(φa ) det ∂ζb t
a
(65)
c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
542
M. N. Stoilov: Duality between constraints and gauge conditions
where H(ζ ∗ ) is the Hamiltonian H(p, q) in which ζa are substituted with their solutions determined from
eqs. (61). It is easy to show that
∂φa 2
[ϕ,
ϕ]
[ϕ,
χ]
u
u
= det
det ,
(66)
∂ζb [χ, χ]u [χ, ϕ]u
where [·, ·]u is the Poisson bracket in the unphysical phase space. Using eqs. (64) we get
∂φa = det |[χa , ϕb ]| ,
det ∂ζb (67)
and so, the transition amplitude given by eq. (65) coincides with that in eq. (10). This proves that it is possible
to use non-Abelian gauge conditions provided det | [χb , ϕc ] | = 0.
References
[1] P.A. M. Dirac, Lectures on Quantum Mechanics (Yeshiva Univ. Academic Press, New York, 1967).
[2] L. D. Faddeev and A.A. Slavnov, Gauge Fields: An Introduction to Quantum Field Theory (Addison-Wesley
Publ., London, 1991).
[3] I.A. Batalin and G.A. Vilkovisky, Phys. Lett. B 69, 309 (1977).
[4] S. Hwang, Nucl. Phys. B 351, 425 (1991).
[5] M. Henneaux, Phys. Rep. 126, 1 (1985).
[6] D. M. Gitman and I.V. Tyutin, Quantization of Fields with Constraints (Springer-Verlag, Berlin, 1990).
[7] J. M. Pons and J.A. Garcia, Int. J. Mod. Phys. A 15, 4681 (2000).
[8] E. S. Fradkin and G.A. Vilkovisky, Phys. Lett. B 55, 224 (1975).
[9] M. Stoilov, CanonicalApproach to Lagrange Multipliers, in: Proceedings of QTS-4,Vol. 2, edited byV. K. Dobrev
(Heron Press, Sofia, 2006), p. 540.
[10] E. S. Fradkin and V.Y. Linetsky, Nucl. Phys. B 431, 569 (1994).
[11] M. Stoilov, Note on the structure of constraint algebras, hep-th/0611100.
[12] V. S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations (Springer-Verlag, 1984).
[13] E. Egorian E and R. Manvelyan, Theor. Math. Phys. 94, 173 (1993).
c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
www.ann-phys.org
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