Operatorial quantization of Born-Infeld Skyrmions

13 Pages·0.11 MB·English

by  Jorge Ananias Neto

Operatorial quantization of Born-Infeld
Skyrmions
9
9
9
1 J. Ananias Neto, W. Oliveira and Emanuel R. de Oliveira
∗ † ‡
n
Departamento de F´ısica, ICE
a
J
Universidade Federal de Juiz de Fora, 36036-330,

6
2 Juiz de Fora, MG, Brazil

1
v
3
3
Abstract
1
1
0 The SU(2) collective coordinates quantization of the Born-Infeld
9 Skyrmions Lagrangean is performed. The obtainment of the classi-
9
cal Hamiltonian from this special Lagrangean is made by usingan ap-
/
h
proximateway: itisderivedfromtheexpansionofthisnon-polynomial
t
- Lagrangean up to second-order variable in the collective coordinates,
p
e usingfor this some causality arguments. Because this system presents
h
constraints, we use the Dirac Hamiltonian method and the Faddeev-
:
v Jackiw Lagrangean approach to quantize this model.
i
X
r PACS number: 11.10.Ef; 12.39.Dc
a
Keywords: Skyrme model, constrained systems.
∗e-mail:jorge@fisica.ufjf.br
†e-mail:wilson@fisica.ufjf.br
‡e-mail:emanuel@fisica.ufjf.br
1
J.A.Neto, W.Oliveira and E.R.Oliveira, ‘Operatorial quantization of...’ 2
Until this moment, the Skyrme model[1] is the effective field theory for
baryons and their interactions. These hadronic particles are described from
solitonsolutionsinthenon-linearsigmamodel. InthisLagrangean,normally,
it is necessary to add the Skyrme term to stabilize the soliton solutions. The
physical spectrum is obtained by performing the collective coordinates quan-
tization. Then, using the nucleon and the delta masses as input parameters,
we get the principal phenomenological results[2].
In principle, the Skyrme term is arbitrary, having no concrete reason to
fix this particular choice[3]. Its importance resides in the fact that, maybe,
it is the most simple possible quartic derivative term that we need to put in
the static Hamiltonian to obtain the soliton solution1. However, it is possible
to get round this ambiguity by adopting a non-conventional Lagrangean also
based in a non-linear sigma model, given by
3
L = F d3rTr ∂ U∂µU+ 2 , (1)
− πZ h µ i
where F is the pion decay constant and U is an SU(2) matrix. This model
π
was proposed by Deser, Duff and Isham[5], based on the ideas of Born-Infeld
Electrodynamics[6]. ApplyingDerricktheoreminthestaticHamiltonian(de-
rived from the Lagrangean (1) ) we observe the existence of soliton solution.
The purpose of this paper is to apply both Dirac Hamiltonian method[7]
and the Faddeev-Jackiw Lagrangean procedure[11] to obtain the commuta-
tors between the canonical operators in which will define the quantum struc-
ture of the Born-Infeld Skyrmions system[8]. We will observe that when we
keep the non-causality sector of the soliton solution influencing the physical
values as minimum as possible, the commutators obtained are the same of
1 According to Derrick scale theorem[4].
J.A.Neto, W.Oliveira and E.R.Oliveira, ‘Operatorial quantization of...’ 3
the Skyrme model.
The dynamic system will be given by performing the SU(2) collective
semi-classical expansion[2]. Substituting U(r,t) by A(t)U(r)A+(t) in (1),
where A is an SU(2) matrix, we obtain
3
L = F d3r m ITr(∂ A∂ A−1) 2 . (2)
− πZ h − 0 0 i
In the last equation, m and I are some functional of the chiral angle F(r),
with the topological boundary conditions,F(0) = π and F( ) = 0. Here,
∞
we use the Hedgehog ansatz for U, i.e., U = exp(iτ rˆF(r)). The SU(2)
·
matrix A can be written as A = a +ia τ, with the constraint
0
·
i=3
a a = 1. (3)
i i
X
i=0
The Lagrangean (2) can be written as a function of the a as
i
3
L = F d3r[m Ia˙ a˙ ]2 . (4)
− πZ − i i
From the Eq. (4) we can obtain the conjugate momentum given by
∂L 1
π = = 3F a˙ d3rI[m Ia˙ a˙ ]2 . (5)
i ∂a˙ π iZ − k k
i
The algebraic expression for the Hamiltonian is obtained by applying the
Legendre transformation, H = π a˙ L. However, due to the momentum
i i
−
formula given in Eq. (5), maybe it is not possible to write the conjugate
Hamiltonian in terms of π and a for the Lagrangean of the Born-Infeld
i i
Skyrmions. An alternative procedure is to expand the original Lagrangean
(4) in collective coordinates variables. Thus, considering the binomial ex-
pansion variable I a˙ a˙ , the Lagrangean sum is given by
m i i
J.A.Neto, W.Oliveira and E.R.Oliveira, ‘Operatorial quantization of...’ 4
L = M +A(a˙ a˙ ) B(a˙ a˙ 2)+... , (6)
i i i i
− −
where
3
M = F d3r m2, (7)
πZ
A = 3F d3r I√m, (8)
πZ
3 I2
B = F d3r , (9)
2 πZ √m
.
.
. (10)
Inthisstepwewouldlike togiveaphysical argument thatpermitstheuse
ofthisprocedure. Thismodelisnotrelativisticinvariant. Thus, wehopethat
only soliton velocity much smaller than speed of light can reproduce, with
a good accordance, the experimental physical results. From the relation[9],
A+∂ A = i/2 k=3τ ω , where ω is the uniform soliton angular velocity,
0 k=1 k k k
P
it is possible to show that Tr[∂ A∂ A+] = 2a˙ a˙ = ω2/2. If we want that
0 0 i i
the soliton rotates with velocity smaller than c, then ωr 1, leading to
≪
a˙ a˙ = ω2 1, and consequently a˙ a˙ 1 for all space2. Thus, these results
i i 4 ≪ i i ≪
explain our procedure.
The Hamiltonian is obtained by using the Legendre transformation
H = π a˙ L
i i
−
= M +A(a˙ a˙ ) 3B(a˙ a˙ )2 +... . (11)
i i i i
−
2In the context of semi-classical expansion, it is expected that the product of a˙ia˙i by
I
the expression given by the Euler-Lagrange equation, does not modify sensitively this
m
result.
J.A.Neto, W.Oliveira and E.R.Oliveira, ‘Operatorial quantization of...’ 5
Obtaining the momentum,π = ∂L , from Eq. (6), writing the the La-
i ∂a˙i
grangean as, L = π a˙ H, and comparing with the expansion of the La-
i i
−
grangean (6), it is possible to derive the expression of the Hamiltonian (11)
as
H = M +απ π +β(π π )2 +... , (12)
i i i i
being α = 1 ,β = B ( A and B are defined in Eqs.(8) and (9) respec-
4A 16A4
tively). We will truncate the expression (12) in the second order variable3 ,
and we will use this approximate Hamiltonian to perform the quantization.
In order to apply the Dirac Hamiltonian method, we need to look for
secondaryconstraints, inwhichcanbecalculatebyconstructing thefollowing
Hamiltonian
H = H +λ φ
T 1 1
= M +απ π +β(π π )2 +λ (a a 1), (13)
i i i i 1 i i
−
where λ istheLagrangeanmultiplier. Imposingthattheprimaryconstraint
1
(3) must be conserved in time, we have
˙
φ = φ ,H = 0, (14)
1 1 T
{ }
where φ = a a 1 0 istheprimaryconstraint. Theconsistency condition
1 i i
− ≈
over the constraint (3) leads to
3 Due to the equation(5) together with the fact that a˙ia˙i 1 , we expect that terms
like (πiπi)3 or higher order degree do not alter our conclusion≪about the commutators of
the quantum Born-Infeld Skyrmions.
J.A.Neto, W.Oliveira and E.R.Oliveira, ‘Operatorial quantization of...’ 6
φ˙ = φ ,H = a a 1,M +απ π +β(π π )2
1 1 T i i j j j j
{ } { − }
= (4α+8βπ π )a π . (15)
j j i i
As the quantity inside the parenthesis, formula (15), can not be a negative
value, then we get the secondary constraint
a π = 0, φ = a π 0. (16)
i i 2 i i
⇒ ≈
If one goes on and imposing the consistency condition over (16), we observe
that no new constraints are obtained via this iterative procedure. The theory
has then the constraints
φ = a a 1 0, (17)
1 i i
− ≈
φ = a π 0, (18)
2 i i
≈
which are second class ones. To implement the Dirac brackets we need to
calculate the matrix elements of their Poisson brackets, read as
C = φ ,φ (19)
αβ α β
{ }
= 2ǫ a a , α,β = 1,2, (20)
αβ i i
−
with ǫ = ǫ12 = 1. Using the Dirac bracket
12
− −
A,B ∗ = A,B A,φ C−1 φ ,B , (21)
{ } { }−{ α} αβ{ β }
we obtain
J.A.Neto, W.Oliveira and E.R.Oliveira, ‘Operatorial quantization of...’ 7
a ,a = 0, (22)
i j
{ }
a ,π = δ a a , (23)
i j ij i j
{ } −
π ,π = a π a π . (24)
i j j i i j
{ } −
∗
By means of the well known canonical quantization rule , i[, ], we
{ } → −
get the commutators
[a ,a ] = 0, (25)
i j
[a ,π ] = i(δ a a ), (26)
i j ij i j
− −
[π ,π ] = i(a π a π ). (27)
i j j i i j
− −
These results show that the quantum mechanics commutators of the Born-
Infeld Skyrmions, when we expand its Lagrangean until to the second order
in collective coordinates, are equal to the Skyrme model[10].
To implement the Faddeev-Jackiw quantization procedure[12], let us con-
sider the first-order Lagrangean
L = π a˙ V, (28)
i i
−
where the potential V is
V = M +απ π +β(π π )2 +λ(a a 1), (29)
i i i i i i
−
with the sympletic enlarged variables given by ξ = (a ,π ,λ). To obtain the
j j j
Poisson brackets, we need to determine the sympletic tensor, defined by
∂A ∂A
j i
f = . (30)
ij
∂ξi − ∂ξj
J.A.Neto, W.Oliveira and E.R.Oliveira, ‘Operatorial quantization of...’ 8
A are functions of the sympletic variable ξ. From the Lagrangean(28), we
i
identify the coefficients
A = π ,
ai i
A = 0,
πi
A = 0, (31)
λ
Thesympletictensoriscalculateusingthedefinition(30)withthecoefficients
above leading to the matrix f
0 δ 0
ij
−
f =  δ 0 0  (32)
ij
 0 0 0 
 
where the elements of rows and columns follow the order: a , π , λ. The
i i
matrix above is obviously singular. Thus, the system has constraints in
the Faddeev-Jackiw formalism. The eigenvector corresponding to the zero
eigenvalue is
0
v =  0  (33)
 1 
 
The primary constraint is obtained from
∂V
Ω(1) = v
i
∂A
i
= a a 1 0, (34)
i i
− ≈
where the potential V is given by Eq. (29). Taking the time derivative of
this constraint and introducing the result into the previous Lagrangean by
means of a Lagrange multiplier ρ, we get a new Lagrangean L(1)
J.A.Neto, W.Oliveira and E.R.Oliveira, ‘Operatorial quantization of...’ 9
L(1) = (π +ρa )a˙ V(1), (35)
i i i
−
where
V(1) = M +απ π +β(π π )2. (36)
i i i i
The new coefficients are4
A(1) = π +ρa ,
ai i i
A(1) = 0,
πi
A(1) = 0. (37)
ρ
The matrix f(1) is then
0 δ a
ij i
− −
f(1) =  δ 0 0  (38)
ij
 a 0 0 
 i 
where rows and columns follow the order: a , π , ρ. The matrix f(1) is
i i
singular. An eigenvector corresponding to the zero eigenvalue is
0
v(1) =  a  (39)
i
 1 
 − 
The secondary constraint is
∂V(1)
Ω(2) = v(1)
α (1)
∂A
α
= a π 0. (40)
i i
≈
4We have imposed strongly aiai 1=0.
−
J.A.Neto, W.Oliveira and E.R.Oliveira, ‘Operatorial quantization of...’ 10
Here we must mention that the primary and secondary constraints, Eqs(34)
and(40), respectively, derived bytheFaddeev-Jackiw procedurearethesame
obtainedbytheDiracformalism. Takingthetimederivativeofthisconstraint
and introducing the result into the Lagrangean (35) by means of a Lagrange
multiplier η, we get a new Lagrangean L(2)
L(2) = (π +ρa +ηπ )a˙ +ηa π˙ V(2), (41)
i i i i i i
−
where V(2) = V(1). The new sympletic enlarged variables are ξ = (a ,π ,
j j j
ρ,η). The new coefficients are
A(2) = π +ρa +ηπ ,
ai i i i
A(2) = ηa ,
πi i
A(2) = 0,
ρ
A(2) = 0.
η
And the matrix f(2) read as
0 δ a π
ij i i
− − −
 δ 0 0 a 
f(2) = ij − i (42)
 ai 0 0 0 
 
 π a 0 0 
 i i 
where rows and columns follow the order: a , π ,ρ η. The matrix f(2) is not
i i
singular. Then we can identify it as the sympletic tensor of the constrained
theory. The inverse of f(2) will give us the Dirac brackets of the physical
fields and can be obtained in a straightforward calculation. The resulting
inverse matrix is