Gauge Systems With Noncommutative Phase Space

Gauge systems with noncommutative phase space
Vladimir Cuesta∗ and Merced Montesinos†
Departamento de F´ısica, Cinvestav, Av. Instituto Politćnico Nacional 2508,
e
San Pedro Zacatenco, 07360, Gustavo A. Madero, Ciudad de M´xico, M´xico
e e

Jos´ David Vergara‡
e
Instituto de Ciencias Nucleares, Universidad Nacional Autńoma de M´xico, 70-543, Ciudad de M´xico, M´xico
o e e e
(Dated: February 7, 2008)
Some very simple models of gauge systems with noncanonical symplectic structures having sl(2, r)
as the gauge algebra are given. The models can be interpreted as noncommutative versions of the
usual SL(2, R) model of Montesinos-Rovelli-Thiemann. The symplectic structures of the noncom-
mutative models, the first-class constraints, and the equations of motion are those of the usual
SL(2, R) plus additional terms that involve the parameters θµν which encode the noncommutativity
arXiv:hep-th/0611333v1 29 Nov 2006

among the coordinates plus terms that involve the parameters Θµν associated with the noncommu-
tativity among the momenta. Particularly interesting is the fact that the new first-class constraints
get corrections linear and quadratic in the parameters θµν and Θµν . The current constructions show
that noncommutativity of coordinates and momenta can coexist with a gauge theory by explicitly
building models that encode these properties. This is the first time models of this kind are reported
which might be significant and interesting to the noncommutative community.

PACS numbers: 02.40.Gh, 11.10.Nx, 11.15.-q

Hamiltonian constrained systems with a finite num- {γa , χα }ω = Caα b γb + Caα β χβ , (4b)
ber of degrees of freedom can be written in a Hamilto-
nian form by means of the dynamical equations of mo-
tion (summation convention over repeated indices is used {H, γa }ω = Va b γb + Va αβ χα χβ , (4c)
throughout)
∂H ∂γa ∂χα {H, χα }ω = Vα b γb + Vα β χβ . (4d)
xµ = ω µν (x)
˙ + λa ν + λα ν
∂xν ∂x ∂x
∂HE Also,
= ω µν (x) , µ, ν = 1, 2, . . . , 2N, (1)
∂xν {χα , χβ }ω = Cαβ , det (Cαβ ) = 0, (5)
where HE = H + λa γa + λα χα is the extended Hamilto-
nian, where the Poisson brackets { , }ω involved in Eqs. (4)
and (5) are computed using the symplectic structure ω
γa (x) ≈ 0, χα (x) ≈ 0, (2) on Γ:
are the constraints which define the constraint surface ∂f µν ∂g
{f, g}ω = ω (x) ν . (6)
Σ embedded in the phase space Γ. Γ is a symplectic ∂xµ ∂x
manifold endowed with the symplectic structure
The dynamical equations of motion (1) and the con-
1 straints (2) can be obtained from the action principle
ω = ωµν (x)dxµ ∧ dxν , (3)
2 [1, 2]
where (xµ ) are coordinates which locally label the points τ2
p of Γ. It is important to emphasize that the phase space S[xµ , λa , λα ] = (θµ (x)xµ − HE ) dτ,
˙ (7)
τ1
Γ is considered as a single entity, i.e., Γ need not be nec-
essarily interpreted as the cotangent bundle of a configu- where HE = H +λa γa +λα χα as before, ω = dθ with θ =
ration space C. H is taken to be a first-class Hamiltonian, θµ (x)dxµ the symplectic potential 1-form. It is important
the γ’s are first-class constraints while the χ’s are second to emphasize that this way of formulating Hamiltonian
class, i.e., constrained systems is beyond Dirac’s method which, by
{γa , γb }ω = Cab c γc + Tab αβ χα χβ , (4a) construction, employs canonical symplectic structures [3,
4, 5, 6].
In order to set down the problem addressed in the pa-
per which links the use of noncanonical symplectic struc-
∗ Electronic address: vcuesta@fis.cinvestav.mx tures in gauge theories to noncommutative geometry, it is
† Electronic address: merced@fis.cinvestav.mx convenient to say some words about the various notions
‡ Electronic address: vergara@nucleares.unam.mx of noncommutative geometry used by physicists.

2

1. First of all, there exists the so-called field theory group (more precisely, the gauge algebra of the first-class
defined on spacetimes endowed with noncommuting co- constraints) be the same than that of the original gauge
ordinates [7, 8], where the noncommutativity of the co- system. Even though the analysis will be focused on the
ordinates requires a noncommutativity among the field item 2, it will result clear at the end of the paper, that
variables. In this sense, most of the work in this field of the current approach can also be used in the context of
research consists in to mix consistently the noncommuta- item 1.
tivity of the coordinates of spacetime with the noncom- The current ideas will be implemented in a nontriv-
mutativity among the field variables [9, 10]. ial gauge system, the SL(2, R) model. The phase space
2. On the other hand, there exists a version where the Γ = R8 , whose points are locally labeled by (xµ ) =
noncommutativity of the coordinates of spacetime mod- (u1 , u2 , v 1 , v 2 , p1 , p2 , π1 , π2 ), is endowed with the sym-
ifies the dynamics of test particles moving in it, in the plectic structure
sense that the equations of motion for the test particles
get corrections that involve the parameters attached to Ω = dpi ∧ dui + dπi ∧ dv i , i = 1, 2, (11)
the noncommuting coordinates [11]. There, the situation
usually studied involves a Hamiltonian system without which has the canonical form, in total agreement with
constraints, endowed with a canonical symplectic struc- Dirac’s method. The model is defined by the action prin-
ture Ω = dpi ∧ dq i , or, equivalently, ciple [12]
{q i , q j }Ω = 0, {q i , pj }Ω = δj , {pi , pj }Ω = 0,
i
(8) τ2
S[ui , v i , pi , πi , N, M, λ] = ui pi + v i πi
˙ ˙
and with an action principle of the form τ1
−N H1 − N H2 − λD) dτ, i = 1, 2, (12)
t2
S[q i , pi ] = pi q i − H(q, p, t) dt,
˙ (9) where the constraints
t1

as a starting point. Next, the dynamical equations ob- 1
H1 := (p1 )2 + (p2 )2 − (v 1 )2 − (v 2 )2 ≈ 0,
tained from the action of Eq. (9) are ad hoc modified 2
by replacing the original symplectic structure Ω with a 1
H2 := (π1 )2 + (π2 )2 − (u1 )2 − (u2 )2 ≈ 0,
new one ωnc which involves noncommutativity among 2
the coordinates q’s and among the p’s D := ui pi − v i πi ≈ 0, i = 1, 2, (13)
i
{q i , q j }ωnc = θij , {q i , pj }ωnc = δj , {pi , pj }ωnc = Θij . satisfy the relations
(10)
θij = −θji and Θij = −Θji are, usually, assumed to be {H1 , H2 }Ω = D,
constant parameters. The conceptual (or physical) expla-
{H1 , D}Ω = −2H1 ,
nation about the origin of noncommutativity depends on
the particular theory assumed to be the fundamental one. {H2 , D}Ω = 2H2 . (14)
Independently of the explanation, some consequences of
the introduction of ωnc are: (1) the original symmetries which turn out to be isomorphic to those defining the
of the various geometrical objects involved might change, sl(2, r) Lie algebra.
in particular, the algebra of observables Aωnc (computed Before going on, it is convenient to say that in or-
using ωnc ) might be completely different to the original der to make clear the concepts involved in the paper,
one AΩ (computed using Ω), (2) the elements that form the original SL(2, R) model will be gradually modified,
a complete set of commuting observables (csco) which by first introducing one parameter, next two parameters,
are in involution with respect to Ω might be completely and finally, the generic case which allows noncommuta-
different to the elements that form a complete set of com- tivity among the coordinates and among the momenta
muting observables computed with respect to ωnc , in fact involved. It will be shown for the case of the SL(2, R)
the elements of the former set are not in involution with model that it is always possible, at least locally, to main-
respect to ωnc in the generic case. tain the same algebra and obtain as a consequence a new
Therefore, it is rather natural to ask if there exists set of constraints that corresponds to the twisted gener-
analogs for gauge systems of the situation previously ators of the original symmetry [9].
mentioned in the item 2. In this paper, it is shown that Model with one noncommutative parameter θ. As it
the answer is in the affirmative and the term “gauge sys- was mentioned, the idea behind the current theoretical
tems with noncommutative phase space” is used to refer framework is to modify the symplectic structure (11)
to this fact, that is to say, that it is technically possible of the original SL(2, R) model by introducing a pa-
to start from a gauge system endowed with a canonical rameter θ that encodes the noncommutativity among
symplectic structure Ω and then to replace the original the coordinates, the phase space of the system is still
symplectic structure Ω in Γ with a new one ωnc to build a Γ = R8 , its points are still locally labeled by (xµ ) =
new gauge system in such a way that the resulting gauge (u1 , u2 , v 1 , v 2 , p1 , p2 , π1 , π2 ), but now the inverse of the

3

new symplectic structure on it is given by leads to
u1 = N p1 − M θu2 + λ u1 + 2θp2 ,
˙
0 θ 0 0 1 0 0 0
 
 −θ 0 0 0 0 1 0 0  u2
˙ = N p2 + λu2 ,
0 0 0 0 0 0 1 0 v1
˙ = M π1 − λv 1 ,
 
 
0 0 0 0 0 0 0 1
 
µν
(ωnc ) = 

.

(15) v2
˙ = M π2 − λv 2 ,
−1 0 0 0 0 0 0 0
˙
p1 = M u1 + θp2 − λp1 ,
 

 0 −1 0 0 0 0 0 0 

 0 0 −1 0 0 0 0 0  p2 = M u2 − λp2 ,
˙
0 0 0 −1 0 0 0 0 π1 = N v 1 + λπ1 ,
˙
π2 = N v 2 + λπ2 ,
˙ (19)
Nevertheless, the original constraints (13) do not close
under the bracket {A, B}ωnc computed with (15). In or- which are different from those corresponding to the origi-
der to preserve the SL(2, R) symmetry of the original nal SL(2, R) system because there are terms that involve
system in the new model, it is mandatory to modify the θ. Note that when θ = 0, the symplectic structure (15),
original constraints. The more efficient way to build the the constraints (17), and the dynamical equations of mo-
new first class constraints is by means of Darboux’s the- tion (19) acquire the same form of those corresponding
orem [13]. According to it, it is always possible, at least to the original SL(2, R) model.
locally, to find a map from the original coordinate system Using (19), the evolution of the constraints (17) yields
and symplectic structure (15) to a new coordinate system ˙
H1 = M D − 2λH1 ,
(Darboux’s variables) in terms of which the symplectic
structure acquires the ordinary canonical form. In the ˙ 2 = −N D + 2λH2 ,
H
current case this map is given by ˙
D = −2M H2 + 2N H1 , (20)
in agreement with the sl(2, r) Lie algebra (18).
u1 = u1 + θp2
˜ Observables. It can be shown that the following six
u2 = u2 , . . .
˜ (16) observables
O12 = u1 p2 − p1 u2 + θ(p2 )2 ,
Now, with respect to Darboux’s variables
(˜1 , u2 , v 1 , v 2 , p1 , p2 , π1 , π2 ), the new constraints ex-
u ˜ ˜ ˜ ˜ ˜ ˜ ˜ O13 = u1 v 1 − p1 π 1 + θv 1 p2 ,
pressed in terms of the new variables have the same O14 = u1 v 2 − p1 π2 + θv 2 p2 ,
analytical form than the old constraints have in terms of O23 = u2 v 1 − p2 π1 ,
the old variables. In terms of the original variables, the
O24 = u2 v 2 − p2 π2 ,
new constraints acquire the form
O34 = π1 v 2 − v 1 π2 , (21)
1 are all gauge invariant. The resulting Lie algebra of their
H1 := (p1 )2 + (p2 )2 − (v 1 )2 − (v 2 )2 ≈ 0,
2 Poisson brackets computed with respect to the symplec-
1 tic structure (15) is isomorphic to the Lie algebra of the
H2 := (π1 )2 + (π2 )2 − (u1 )2 − (u2 )2
2 Lie group SO(2, 2), i.e., the algebra of observables is ex-
1 2 actly the same than that of the original SL(2, R) model.
− θu1 p2 − θ2 (p2 ) ≈ 0, The four relations among the observables are the same
2
D := ui pi − v i πi + θp1 p2 ≈ 0, i = 1, 2. (17) than those of the original SL(2,R) model but now the
expression of the observables are
2
It turns out that the constraints (17) are also first-class, u1 p2 − p1 u2 + θ (p2 )
but now with respect to the Poisson brackets computed ǫ = 2 , (22)
| u1 p2 − p1 u2 + θ (p2 ) |
with the new symplectic structure of Eq. (15). The re-
sulting algebra of constraints is π1 v 2 − v 1 π2
ǫ′ = , (23)
| π1 v 2 − v 1 π2 |
2
{H1 , H2 }ωnc = D, J = | u1 p2 − p1 u2 + θ (p2 ) |, (24)
{H1 , D}ωnc = −2H1 , u1 v 2 − p1 π 2 + θv 2 p2
α = arctan . (25)
{H2 , D}ωnc = 2H2 , (18) u1 v 1 − p1 π1 + θp2 v 1
Model with two noncommutative parameters θ and
which is isomorphic to the sl(2, r) Lie algebra, as ex- φ. Now, the symplectic structure (11) of the origi-
pected. By inserting (15) and (17) with (λ1 , λ2 , λ3 ) = nal SL(2, R) model is modified by introducing two pa-
(N, M, λ) and (γ1 , γ2 , γ3 ) = (H1 , H2 , D) into Eqs. (1) rameters θ and φ which encode the noncommutativity

4

among the coordinates, the phase space of the system in agreement with the sl(2, r) Lie algebra (28).
is still Γ = R8 , its points are still locally labeled by Observables. The following six observables
(xµ ) = (u1 , u2 , v 1 , v 2 , p1 , p2 , π1 , π2 ), but now the inverse 2
of the new symplectic structure on it is given by O12 = u1 p2 − p1 u2 + θ (p2 ) ,
O13 = u1 v 1 − p1 π 1 + θv 1 p2 + θφp2 π2 + φu1 π2 ,
0 θ 0 0 1 0 0 0
 
 −θ 0 0 0 0 1 0 0  O14 = u1 v 2 − p1 π2 + θv 2 p2 ,
 0 0 0 φ 0 0 1 0 O23 = u2 v 1 − p2 π1 + φu2 π2 ,
 
 0 0 −φ 0 0 0 0 1 
 
µν
(ωnc ) =   . (26) O24 = u2 v 2 − p2 π2 ,
 −1 0 0 0 0 0 0 0  2
 0 −1 0 0 0 0 0 0  O34 = π1 v 2 − v 1 π2 − φ (π2 ) , (31)
 
 0 0 −1 0 0 0 0 0 
are all gauge invariant. Again, the resulting Lie algebra
0 0 0 −1 0 0 0 0 of their Poisson brackets computed with respect to the
As for the case of the model with one parameter θ, the symplectic structure (26) is isomorphic to the Lie algebra
original constraints of Eq. (13) get also modiﬁed, by of the Lie group SO(2, 2), i.e., the algebra of observables
terms linear and quadratic in the noncommutative pa- is exactly the same than that of the original SL(2, R)
rameters θ and φ. The new constraints are model. The four relations among the observables is the
same than those of the original SL(2,R) model but now
1
C1 := (p1 )2 + (p2 )2 − (v 1 )2 − (v 2 )2 the expression of the observables are
2
1 2
− φv 1 π2 − φ2 (π2 ) ≈ 0,
2 u1 p2 − p1 u2 + θ (p2 )
2 ǫ = 2 , (32)
| u1 p2 − p1 u2 + θ (p2 ) |
1
C2 := (π1 )2 + (π2 )2 − (u1 )2 − (u2 )2 π1 v 2 − v 1 π2 − φ (π2 )
2
2 ǫ′ = , (33)
2
1 2 | π1 v 2 − v 1 π2 − φ (π2 ) |
− θu1 p2 − θ2 (p2 ) ≈ 0,
2 J = | u1 p2 − p1 u2 + θ (p2 )2 |, (34)
V := ui pi − v i πi + θp1 p2 − φπ1 π2 ≈ 0, (27) u1 v 2 − p1 π 2 + θv 2 p2
α = arctan . (35)
with i = 1, 2 [cf. with Eqs. (17)]. It turns out that the u1 v 1 − p1 π1 + θp2 v 1
constraints of Eq. (27) are also ﬁrst-class but now with
respect to the Poisson brackets computed with the new Model with noncommuting momenta and noncommut-
symplectic structure of Eq. (26). The resulting algebra ing coordinates. The more general case corresponds
of constraints is to consider noncommutativity among all the momenta
pµ = (p1 , p2 , π1 , π2 ) and among all the coordinates xµ =
{C1 , H2 }ωnc = V, (u1 , u2 , v 1 , v 2 ). In this case the noncommutative brackets
{C1 , V}ωnc = −2C1 , are given by:
{C2 , V}ωnc = 2C2 , (28) {xµ , xν }ωnc = θµν ,
which is isomorphic to the sl(2, r) Lie algebra. {pµ , pν }ωnc = Θµν , (36)
By plugging (26), (27), (λ1 , λ2 , λ3 ) = (N, M, λ), and {xµ , pν }ωnc = δ µν ,
(γ1 , γ2 , γ3 ) = (C1 , C2 , V) into Eqs. (1), the dynamical
equations (1) acquire the form where, in order that could be possible to get a repre-
˙
u 1 2 1
= N p1 − M θu + λ u + 2θp2 , sentation in terms of commutative variables [11], the
noncommutative parameters are restricted by θµν Θνρ =
u = N p2 + λu2 ,
˙ 2
4αβ(αβ − 1)δ µρ , with (α, β) arbitrary real parameters.
v 1 = M π1 − λ v 1 + 2φπ2 − φN v 2 ,
˙ Taking this relation into account the commutative vari-
ables are
v 2 = M π2 − λv 2 ,
˙
p1 = M u1 + θp2 − λp1 ,
˙ β µ θµν
xµ =
˜ x + pν ,
ρ 2αρ
p2 = M u2 − λp2 ,
˙
α Θµν ν
π1 = N v 1 + φπ2 + λπ1 ,
˙ ˜
pµ = pµ − x , (37)
ρ 2βρ
2
π2 = N v + λπ2 .
˙ (29)
with ρ = 2αβ − 1 and the new constraints are
[cf. with Eqs. (19)]. Using (29), the evolution of the
constraints (27) yields 1
γ1 := (˜1 )2 + (˜2 )2 − (˜1 )2 − (˜2 )2 ≈ 0,
p p v v
2
˙
C1 = M V − 2λC1 , 1
˙ γ2 := (˜1 )2 + (˜2 )2 − (˜1 )2 − (˜2 )2 ≈ 0,
π π u u
C2 = −N V + 2λC2 , 2
˙
V = −2M C2 + 2N C1 , (30) d := ui pi − v i πi ≈ 0.
˜ ˜ ˜˜ (38)

5

The algebra of these constraints will be isomorphic to the tative quantum mechanics, to twist the gauge symmetry
sl(2, r) Lie algebra by construction. However, when these in agreement with a recent proposal in field theory[15].
constraints are written explicitly in terms of the noncom- The generalization of these results to noncommutative
muting variables (36), they become the twist generators field theory (item 1 of this paper) would mean that it is
of the sl(2, r) algebra. Also, for this noncommutative possible to modify the usual symplectic structure among
model, it is possible to construct Dirac’s observables the field variables by symplectic structures involving non-
commuting field variables. Due to the fact that the
O12 = u 1 p2 − p1 u 2 ,
˜ ˜ ˜ ˜ noncommutativity among the gauge fields is related to
O13 = u 1 v 1 − p1 π 1 ,
˜ ˜ ˜ ˜ the noncommutativity of the coordinates of spacetime,
O14 = 1 2
u v − p1 π2 ,
˜ ˜ ˜ ˜ a mechanism to combine consistently the two types of
2 1 noncommutativity must be required, for instance incor-
O23 = u v − p2 π1 ,
˜ ˜ ˜ ˜ porating the coordinates as dynamical variables (e.g., pa-
O24 = u2 v 2 − p2 π2 , rameterizing the field theory) to set noncommuting co-
O34 = π1 v 2 − v 1 π2 ,
˜ ˜ ˜ ˜ (39) ordinates of spacetime and noncommuting fields on the
same footing [16]. By doing things in this way, both
which are all gauge invariant under the transformation types of noncommutativity would emerge in an extended
generated by the first class constraints. phase space endowed with noncommutative symplectic
Furthermore, the resulting Lie algebra of their Pois- structures, situation that can be handled with the same
son brackets computed with respect to the symplectic ideas developed in this work.
structure (36) is isomorphic to the Lie algebra of the Lie
group SO(2, 2), i.e., the algebra of observables is exactly
the same than that of the original SL(2, R) model.
The relevance of the noncommutative models of this
Acknowledgements
paper lies in the fact that it has been explicitly shown
that it possible to introduce noncommutativity among
coordinates and among momenta which are both com- This work was supported in part by CONACyT Grant
patible with the gauge principle in the sense that alge- Nos. SEP-2003-C02-43939 and 47211. J.D. Vergara also
bra of generators closes [cf. with Ref. [14]]. The result acknowledges support from DGAPA-UNAM Grant No.
shows that it is possible, in the context of noncommu- IN104503.

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