Dirac quantization of noncommutative Abelian Proca field

0.12 MB·English

by  F. Darabi

Dirac quantization of noncommutative Abelian Proca field
∗
F. Darabi and F. Naderi
Department of Physics, Azarbaijan University of Tarbiat Moallem,
Tabriz, Iran, P.O. Box: 53714-161, and
Research Institute for Astronomy and Astrophysics of
Maragha (RIAAM)- Maragha, Iran, P.O. Box: 55134-441.
1
1 (Dated: January 18, 2011)
0
2
Abstract: Dirac formalism of Hamiltonian constraint systems is studied for the
n
a
noncommutative Abelian Proca field. It is shown that the system of constraints
J

5
are of second class in agreement with the fact that the Proca field is not guage
1

] invariant. Then, the system of second class constraints is quantized by introducing
h
t Dirac brackets in the reduced phase space.
-
p
e Keywords: Noncommutative Proca field, Second class constraint.
h
[

2 PACS numbers: 11.10.Ef, 11.10.Nx
v
3
7
5
1
.
1
0
1
1
:
v
i
X
r
a
∗
Electronic address: [email protected]
2
I. INTRODUCTION
The noncommutative geometry, pioneered by alain Connes, aims at a generalization of
geometrical ideas to spaces whose coordinates fail to commute [1]. In this theory it is
postulated that space-time is noncommutative at very high energies and one tries to guess
the small scale structure of space-time from our present knowledge at the electroweak scale.
StringtheoryontheotherhandaimsatderivingthestandardmodeldirectlyfromthePlanck
scale physics. Thus, noncommutative geometry may describe the low energy dynamics in
stringtheoryasapictureforthestandardmodel, wheresymmetriesactdirectlybyagroupof
coordinate transformations on an underlying space-time manifold producing the electroweak
and strong forces as pseudo-forces. Moreover, it may place gravity and the other forces on
the same footing by obtaining all forces as pseudo-forces from some general coordinate
transformations acting on some general spacetime.
Quantum field theories on noncommutative spaces are usually formulated in terms of star
products of ordinary functions [3]. The different field models on noncommutative spaces has
been recently of particular interest due to the recent development of the superstring theory.
It was shown that noncommutative coordinates emerge naturally in the perturbative version
of the D-brane theory, namely low energy excitations of a D-brane, with the presence of the
external background magnetic field [4]. Therefore, noncommutative Yang-Mills theories
appear in the string theory in an effective way and that is why they are being so widely
investigated. It is known that there exists a Seiberg-Witten map between noncommutative
field theories and effective commutative theories in that both have the same degrees of
freedom. Moreover, noncommutative gauge theories can be represented as ordinary gauge
theories with the same degrees of freedom, and with the additional deformation parameter
θ [5]. The Seiberg-Witten map between field theory on noncommutative spaces and the
corresponding commutative field theory allows one to formulate an action principle in terms
of ordinary field. The effective Lagrangian of this action is expanded as series of ordinary
field and the noncommutative parameter θ which plays the role of coupling constant.
Attempts to quantize the Maxwell theory on noncommutative spaces have already been
done. First, the corresponding commutative action in terms of ordinary fields and linear in
the deformation parameter has been derived in [6, 7]. Afterwards, the Dirac’s quantization
of Hamiltonian constraint systems [9] has been applied on this commutative action in [10]. ,
3
In the present paper, motivated by the quantization of Maxwell theory on noncommutative
space, we attempt to quantize thenoncommutative massive Abelian Maxwell theory, namely
noncommutative Abelian Proca field, using the Dirac quantization procedure in a similar
wayasinRef.[10]. InsectionII,wefirstobtainthenoncommutativeactionfortheProcafield
using the Moyal product and then derive the corresponding commutative action in terms
of ordinary fields and linear in the deformation parameter θ. In section III, we study this
actioninthecontext oftheDirac’sHamiltonianconstraint systems tofindthecorresponding
constraints which turn out to be the second class type. In section IV, the system of second
classconstraintsarequantizedinthereducedphasespace. Thepaperendswithaconclusion.
II. NONCOMMUTATIVE ABELIAN PROCA FIELD
The action of Abelian noncommutative Proca field is written
1 1
S = (− Fˆ ∗Fˆ + m2Aˆ ∗Aˆ )d4x, (1)
µν µν µ ν
Z 4 2
where ∗ denotes the Star product, Aˆ and Fˆ are the vector potential and field strength
µ µν
tensor respectively, and m is the mass of the Aˆ field. The fields Fˆ and Aˆ may be
µ µν µ
expressed in terms of the corresponding commutative quantities as follows [7]
1
Aˆ = A − θ A (∂ A +F ) (2)
µ µ αβ α β µ βµ
2
Fˆ = ∂ Aˆ −∂ Aˆ −iAˆ ∗Aˆ +iAˆ ∗Aˆ , (3)
µν µ ν ν µ µ ν ν µ
where θ stands forthenoncommutativity parameter that characterizes non-commutativity
αβ
through the coordinate commutation relation [x ,x ] = iθ [1], [2]. It is known that the
α β αβ
integral over the star product of quantities is equal to the corresponding integral over the
ordinary product [11], then we may rewrite the action (1) as
1 1
S = (− Fˆ Fˆ + m2Aˆ Aˆ )d4x. (4)
µν µν µ ν
Z 4 2
Using (2) and (3) the Lagrangian theory of the above noncommutative action may be ex-
pandedasthecommutativetheorywiththesamedegreesoffreedom, andwiththeadditional
terms containing the noncommutative parameter θ of the first order
αβ
1 1 1
Lˆ = − F F + θ F F2 − θ F F F (5)
4 µν µν 8 αβ αβ µν 2 αβ µα νβ µν
1
2 2
+ m (A −θ A (∂ A +F )A ),
2 µ αβ α β µ βµ µ
4
where the commutative field strength tensor is
F = ∂ A −∂ A . (6)
µν µ ν ν µ
Now we define the followings
A = (A~,iA )
µ 0

Ei = iFi4 (7)

B = 1ǫ F
i 2 ijk jk
θ = 1ǫ θ .
 i 2 ijk jk



Then, the Lagrangian density (5) casts in the following form
1 m2
Lˆ = (E2 −B2)(1+~θ·B~)−(~θ·E~)(E~ ·B~)+ (−A2 +A2) (8)
2 2 0 i
m2 m2 m2
~ ~ ~ 2 ~ ~ ~ ~ ~ 2 ~ ~ ~ ~
+ (θ×A)·∇(A )− [(θ×A)·E]A +3 [(θ ·B)A −(θ·A)(A·B)].
4 0 2 0 4 j
The Euler-Lagrange equations for the noncummutative Proca field are obtained
1 1
2
∂ F − ∂ (θ F )− ∂ (θ F F )+∂ (θ F F ) (9)
ρ ρσ 4 ρ ρσ µν 2 ρ αβ αβ ρσ ρ σβ νβ ρν
1 1
− ∂ (θ F F )+ ∂ (θ F F )− ∂ (θ F F )
ρ ρβ νβ σν ρ αβ ρα σβ ρ αβ σα ρβ
2 2
m2 m2
− θ (∂ A +F )A − θ (∂ A +F )A
σβ β µ βµ µ αβ β σ βσ α
2 2
m2
2 2
+ m ∂ (θ A A )− ∂ (θ A )A +m A = 0,
ρ αρ α σ ρ ασ α ρ σ
2
where in the last line the Lorentz condition ∂ A = 0 has been used using the fact that
µ µ
this condition holds in the massive as well as massless Maxwell field [8]. Using (7), the field
equations are divided into the following two set of equations
~
∂D
−∇~ ×H~ = −J~, (10)
∂t
∇~ ·D~ = ρ, (11)
~ ~ ~ ~
where (∇×H) = ǫ ∂ H , ∇·D = ∂ D , ∂/∂t = i∂ and
i ijk j k i i 4
1 1
ρ = m2[A + ∇~ ·(~θ×A~)A + (~θ×A~)·E~], (12)
0 0
2 2
m2
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
D = E +(θ·B)E −(θ·E)B −(E ·B)θ− (θ ×A)A , (13)
0
2
1 3 3 3
J~ = m2[A~ − (E~ ×θ~)A + (~θ·B~)A~ − (~θ·A~)B~ − (A~ ·B~)~θ], (14)
0
2 2 4 4
5
1 1
~ ~ ~ ~ ~ ~ ~ ~ 2 2 ~ 2 2 2 ~ 2~ ~ ~
H = B +(θ·B)B +(θ·E)E − (E −B )θ −m ( A +A )θ+m A(θ·A). (15)
2 4 0 j
On the other hand, using the strength tensor F = ∂ A −∂ A the source-less equations
µν µ ν ν µ
are
˜
∂ F = 0, (16)
µ µν
where F˜ = 1ǫ F and ǫ is the Levi-Civita constant tensor (ǫ = −i). Therefore,
µν 2 µναβ αβ µναβ 1234
equations (16) may be written as
∂B~
+∇~ ×E~ = 0, (17)
∂t
~ ~
∇·B = 0. (18)
III. HAMILTONIAN CONSTRAINT SYSTEM APPROACH
In this section, following Dirac, we will study the dynamics of the noncommutative
Abelian Proca field in the context of Hamiltonian constraint systems [9]. In so doing,
we first obtain the conjugate momenta of A and A , respectively as
i 0
∂Lˆ m2
π = = −E (1+~θ·B~)+(~θ·E~)B +(E~ ·B~)θ + (~θ ×A~) A , (19)
i i i i i 0
∂(∂ A ) 2
0 i
∂Lˆ
π = = 0. (20)
0
∂(∂ A )
0 0
Equation (20) results in a primary constraint
φ ≈ 0, (21)
1
where φ ≡ π , and comparison of (13) with (19) leads to
1 0
π − = D . (22)
i i
Therefore, we obtain the following commutation relations
{A (x,t),D (y,t)} = −δ δ(x−y), (23)
i j ij
{B (x,t),D (y,t)} = ǫ ∂ δ(x−y). (24)
i j ijk k
6
Using the Legendre transformation as H = π ∂ A − L, we may obtain the Hamiltonian
µ 0 µ
density
1 m2 m2 3
Hˆ = (E2 +B2)(1+~θ ·B~)−(~θ·E~)(E~ ·B~)+ A2 − A2(1+ ~θ ·B~) (25)
0 2 2 0 2 i 2
m2 m2
− (~θ ×A~) (∂ A )A +3 (~θ·A~)(A~ ·B~)−π ∂ A .
i i 0 0 i i 0
2 4
Using (19), we may obtain E in terms of π to first order in θ as
i i
m2
E = −π (1−~θ ·B~)−(~π ·B~)B −(~π ·B~)θ + (~θ ×A~) A . (26)
i i i i i 0
2
Substituting for E in Eq.(25) in terms of π we obtain
i i
1 1 m2 m2
Hˆ = (π2 +B2)+ (B2 −π2)(~θ ·B~)+(~π ·θ~)(B~ ·~π)+ A2 − (~θ ×A~)·~πA (27)
0 0 0
2 2 2 2
m2 3 m2 m2
− A2(1+ ~θ·B~)− (~θ×A~) (∂ A )A +3 (~θ·A~)(A~ ·B~)−π ∂ A +O(θ2).
2 i 2 2 i i 0 0 4 i i 0
Now, we study the consistency condition for the primary constraint (21)
˙
φ = {φ (x),H } = 0, (28)
1 1 0
where
H = Hˆ (x)d3x. (29)
0 0
Z
The consistency condition (28) results in a secondary constraint
m2 m2
2 ~ ~ ~ ~ ~
φ = ∂ π +m A + ∇·(θ ×A)A − (θ ×A)·~π, (30)
2 i i 0 0
2 2
or
m2 m2
φ = ∂ π +m2A + ∇~ ·(~θ×A~)A + (~θ ×A~)·E~ +O(θ2). (31)
2 i i 0 0
2 2
If we put π = −D , this equation casts in the form of generalized Gauss law which has
i i
already been obtained in (11). Now, the extended Hamiltonian is constructed by adding the
primary constraint φ up to an arbitrary coefficient u(x)
1
3
H = H + u(x)φ (x)d x. (32)
T 0 1
Z
The consistency condition for the secondary constraint φ is now considered as
2
˙
φ = {φ (x),H } = 0, (33)
2 2 T
7
which introduces no new constraint and just fixes the unknown coefficient as
3 1
u(x) = −∇~ ·(~θ ×A~)A −∂ A 1+ ~θ·B~ − ∇~ ·(~θ×A~) (34)
0 i i
(cid:18) (cid:18) 2 2 (cid:19)(cid:19)
3
~ ~ ~ ~ ~ ~ ~ ~ 2
+ ∂ (θ (A·B)+B (θ·A))+(θ ×A)·(∇×B)+O(θ ),
i i i
4
where we have used the expansion 1 ≃ (1−ǫ) due to the smallness of ~θ. It is easy to show
1+ǫ
that the constraints φ and φ are second class, namely
1 2
1
{φ (x),φ (x′)} = −m2(1+ ∇~ ·(~θ×A~))δ(x−x′) (35)
1 2
2
or
0 −m2(1+ 1∇~ ·(~θ ×A~))
2
 
′ ′
{φ (x),φ (x)} = δ(x−x). (36)
i j
 
m2(1+ 1∇~ ·(~θ ×A~)) 0 
 2 
 
It is of great importance to distinction between first and second class constraints. The first
class constraints are defined as the constraints which commute (i.e. have vanishing Poisson
brackets) with all the other constraints. This situation brings to light the presence of some
gauge degrees of freedom in the Dirac formalism. On the other hand, the second class
constraints have at least one non vanishing bracket with some other constraints, like (35).
By manipulation and integration by parts in some appropriate terms in (27) the constraint
φ appears in the Hamiltonian as
2
1 1 m2
H = [ (π2 +B2)+ (B2 −π2)(~θ·B~)+(~π ·~θ)(B~ ·~π)− A2 (37)
0 0
Z 2 2 2
m2 3 m2 m2
2 ~ ~ ~ ~ ~ 2 ~ ~ ~ ~ 3
− A (1+ θ ·B)+ ∇·(θ ×A)A +3 (θ ·A)(A·B)+A φ ]d x.
2 i 2 4 0 4 0 2
Since the original Hamiltonian H includes the constraint φ with the known coefficient A ,
0 2 0
it is not necessary to add once again this constraint with an unknown coefficient to the
Hamiltonian H , and so we obtain the extended Hamiltonian
T
H = H¯ + [u(x)φ (x)+A (x)φ (x)]d3x, (38)
E 0 1 0 2
Z
where
1 1 m2
H¯ = [ (π2 +B2)+ (B2 −π2)(~θ·B~)+(~π ·~θ)(B~ ·~π)− A2 (39)
0 0
Z 2 2 2
m2 3 m2 m2
− A2(1+ ~θ ·B~)+ ∇~ ·(~θ ×A~)A2 +3 (~θ ·A~)(A~ ·B~)]d3x.
2 i 2 4 0 4
8
IV. QUANTIZATION OF SECOND CLASS CONSTRAINT SYSTEM
TheproblemofquantizationofHamiltonianconstraintsystemsistwofold: quantizationof
first class constraints and of second class constraints [9]. However, the problem of quantizing
theories with second class constraints is less ambiguous than quantizing theories with first
class constraints. In the following analysis we will not deal with first class constraints due
to the fact that the Proca field is not gauge invariant. In the case of second class constraints
we can switch to new canonical brackets in order to set all of the second class constraints
strongly equal to zero. This means that in any given quantity, such as the Hamiltonian, we
can set them to zero by hand. In such a case, we can safely change to the new canonical
brackets, the so called Dirac brackets, defined as follows
{A(x,t),B(y,t)} = {A(x,t),B(y,t)} (40)
DB
−1 3 3
− {A(x,t),φ (z,t)}C (z,ω){φ (ω,t),B(y,t)}d zd ω,
i ij j
Z
−1
where C (x,z) = {φ (x),φ (z)} is given by (36). We may obtain the inverse matrix C
ij i j ij
using the relation
−1 3
C (x,z)C (z,y)d z = δ δ(x−y), (41)
Z ij jk ik
which leads to
0 m−2(1+ 1∇~ ·(~θ×A~))−1
2
 
−1
C (z,y) = δ(z −y). (42)
 
−m−2(1+ 1∇~ ·(~θ ×A~))−1 0 
 2 
 
Therefore, the following Dirac brackets are obtained
{π (x,t),A (y,t)} = 0, (43)
0 0 DB
{π (x,t),A (y,t)} = 0, (44)
0 i DB
{π (x,t),π (y,t)} = 0, (45)
µ ν DB
{A (x,t),A (y,t)} = 0, (46)
0 0 DB
{π (x,t),A (y,t)} = −δ δ(x−y), (47)
i j DB ij
{π (x,t),B (y,t)} = −ǫ ∂ δ(x−y), (48)
i j DB ijk k
9
1
−2 ~ ~ ~
{A (x,t),A (y,t)} = m (1− ∇·(θ×A))∂ (x)δ(x−y) (49)
0 j DB j
2
1
− (~θ×A~)) (x)δ(x−y)+O(θ2),
j
2
1
{A (x,t),B (y,t)} = ǫ ∂ (y)[m−2(1− ∇~ ·(~θ×A~))∂ (x)δ(x−y) (50)
0 k DB klj l j
2
1
− (~θ×A~)) (x)δ(x−y)],
j
2
1
{π (x,t),A (y,t)} = [A (y)ǫ θ (y)∂ (y)δ(x−y) (51)
i 0 DB 0 jli l j
2
~ 2
− (~π ×θ) (y)δ(x−y)]+O(θ ).
i
It is known that the Diracbrackets of second class constraints with each arbitraryfunction f
is strongly zero, namely {φ (x,t),f(y,t)} = 0. Therefore, the second class constraints are
i DB
supposed to be strongly zero [9] and so we obtain the physical Hamiltonian in the reduced
phase space as
1 1 m2
H = H¯ = [ (π2 +B2)+ (B2 −π2)(~θ·B~)+(~π ·~θ)(B~ ·~π)− A2 (52)
Ph 0 0
Z 2 2 2
m2 3 m2 m2
− A2(1+ ~θ·B~)+ ∇~ ·(~θ×A~)A2 +3 (~θ·A~)(A~ ·B~)]d3x.
2 i 2 4 0 4
The equations of motion are obtained
A˙ = ∂ A = {A ,H } (53)
i 0 i i Ph DB
m2
~ ~ ~ ~ ~ ~ ~ ~ ~
= π (1−θ·B)+θ (B ·~π)+(~π ·θ)B −∂ (A −∇·(θ ×A))− A (θ ×A) ,
i i i i 0 0 i
2
A˙ = ∂ A = {A ,H } (54)
0 0 0 0 Ph DB
A 1 3
0~ ~ ~ ~ ~ ~ ~ ~
= ∇·(~π ×θ)− (θ×A)·(∇×B)−∂ (A (1+ θ ·B))
j j
2 2 2
1 3
~ ~ ~ ~ ~ ~ ~ 2
+ ∇·(θ×A)∂ A + ∂ (θ (A·B)+(θ ·A)B )+O(θ ).
j j j j j
2 4
π˙ = ∂ π = {π ,H } (55)
i 0 i i Ph DB
1 3 1 3
~ ~ ~ ~ ~ 2 2 ~ 2 2~ 2 ~ 2 ~ ~
= −∇×[B(1+θ ·B)+ θ(B −π )+~π(~π ·θ)− m A θ− m A θ− m (θ·A)A]
2 4 j 4 0 4 i
3 1 3
2 ~ ~ 2 2 ~ 2 ~ ~ ~ ~
+ m A (1+ (θ ·B))+ m A (~π ×θ) − m (θ (A·B)+(θ·A)B ).
i 0 i i i
2 2 4
10
Substituting for π in (55) in terms of E through(19), ignoring the terms of the order O(θ2),
i i
using (14), (15) and considering π = −D we obtain
i i
∂D~
−∇~ ×H~ = −J~, (56)
∂t
which is the Ampere’s Law. This means the equation of motion for π in noncommutative
i
theory obtained by Dirac formalism is in agreement with the Maxwell’s equation. On the
other hand, equation (54) may be rewritten as
A 3 3
∂ ( 0(~π ×~θ) −A − A (~θ ·B~) + (θ (A~ ·B~)+(~θ·A~)B )) (57)
j j j j j j
2 2 4
1 1
˙ ~ ~ ~ ~ ~ ~ ~
= A + (θ ×A)·(∇×B)− ∇·(θ×A)∂ A .
0 j j
2 2
The left hand side of above equation, using (14), is equal to −m−2∇~ ·J~. The time derivative
of ρ in (12) leads us to
ρ˙ = m2[A˙ + 1∂ (~θ×A~˙) A + 1∂ (~θ×A~) A~˙ + 1(~θ×A~)·E~˙ + 1(~θ ×A~˙)·E~], (58)
0 j j 0 j j 0
2 2 2 2
which becomes the following form using (26) and ignoring terms of the order O(θ2)
ρ˙ = m2[A˙ + 1∂ (~θ×A~˙) A + 1∂ (~θ ×A~) A~˙ − 1(~θ×A~)·~π˙ − 1(~θ×A~˙)·~π]. (59)
0 j j 0 j j 0
2 2 2 2
Finally, using the equations of motion (53) and (55) we obtain
1 1
ρ˙ = m2 A˙ + (~θ×A~)·(∇~ ×B~)− ∇~ ·(~θ×A~)∂ A . (60)
0 j j
(cid:18) 2 2 (cid:19)
Therefore equation (57) casts in the following form
∂ρ
+∇~ ·J~ = 0. (61)
∂t
This shows that the equation of motion for A in noncommutative theory obtained by Dirac
0
formalism is in complete agreement with the charge conservation. The quantization of
system is then achieved by the standard replacement in the classical equations (53), (54)
and (55), namely
1
{,} → [,] . (62)
DB i~ QM