Table Of ContentA NEW CURRENT REGURALIZATION OF
THIRRING MODEL
Hidenori TAKAHASHI1
Laboratory of Physics, College of Science and Technology
Nihon University, Funabashi Chiba 274-8501,Japan
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0 and
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n Akihiro OGURA2
a Laboratory of Physics, School of Dentistry at Matsudo
J
Nihon University, Matsudo, Chiba 271-8587,Japan
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2
2
v
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ABSTRACT
1
2
5 We study an ambiguity of the current regularizationin the Thirring model.
0 We find a new current definition which enables to make a comprehensive treat-
0
ment of the current. Our formulation is simpler than Klaiber’s formulation.
0
We compareourresultwith otherformulationsandfinda verygoodagreement
/
h with their result. We also obtain the Schwinger term and the general formula
t
- for any current regularization.
p
e
h 1 Introduction
:
v
Xi The Thirring model has been investigated by many people. It is well-known
that the Thirring model is an exactly solvable quantum field theory model in
r
a (1+1) dimensions [1]. An extensive investigation of the model was given by
Hagen [2] and Klaiber [3]. Hagen introduced an external field and gave the
general solution of the Thirring model. Klaiber analyzed the Thirring model
andfoundtheoperatorsolutionswhichareexpressedintermsofafreemassless
Dirac field. He constructed the solution to fulfill the positive definiteness. On
the other hand, Nakanishi expressed the solution in terms of the free massless
bosonic field [4]. He asserted that all Heisenberg operators should be expressed
in terms of asymptotic fields from the standpoint of the general principle of
quantum field theory. In the present paper, we use the bosonic expression
(bosonization) [5, 6].
One ofthe methods forsolvingthe quantumfield theoryisto determine the
OperatorProductExpansion(OPE)[7,8]. Inthisformalism,theshortdistance
1e-mail: [email protected]
2e-mail: [email protected]
behavior for products of the two local fields is important. For some case, we
candeterminetheOPEexactly,e.g. (1+1)-dimensionalConformalFieldTheory
[9]. Concerningtheoperatorproductsofthequantumfield,therearedifficulties
with respect to the currentregularization. In most cases, the currentis defined
by the limiting procedures as ψ¯(x+ǫ)γ ψ(x). In the Thirring model, there are
µ
several definitions of the current. For example, the Schwinger current [10] is
defined by limiting from spacelike direction only. The Johnson current [11] is
defined by limiting from not only spacelike directionbut also timelike direction
symmetrically as
1
jµ(x)= jµ(x; ǫ)+jµ(x; ǫ) , (1)
2
(cid:20) (cid:21)ǫ,ǫ→0
where ǫ and ǫ are a timelike andespacelike vectors, respectively. Both current
definitions are consistent with the soelution of the Thirring model. However,
the couplingeconstant is affected by the current definition. Therefore, in the
Thirring model, the coupling constant is determined only when we define the
currentregularization[3]. It is also noted that these coupling constants are not
independent. The coupling constant of the Schwinger definition g is given by
S
g
J
g = , (2)
S
1 g /2π
J
−
whereg isthecouplingconstantoftheJohnsondefinition. Thesecurrentambi-
J
guities also appear in the massive Thirring model, which we do not understand
yet [12].
Ontheotherhand,Dell’Antonio,FrishmanandZwanziger[14]analyzedthe
Thirring model without looking into the structure of the current. They extend
the Johnson result [11]. They start with defining the commutation relations of
the current, current algebra formulation [7, 8]. There are three parameters and
we have two relations among them. Therefore, we can construct the current
algebrafromthe lagrangianandthe suitabledefinitionofthe currentwhichhas
one parameter.
Inthispaper,wepresentanewcurrentregularizationoftheThirringmodel.
We introduce one parameter in the definition. Our formulation is simpler than
Klaiber’s one and the new current definition is consistent with other formula-
tions. TheThirringcurrentandfieldcanbewrittenintermsofthefreemassless
bosonic field. Therefore, we can analyze the model exactly.
In this paper, we employ the following notation:
x± =x0 x1, x =x∓/2, ∂ =(∂ ∂ )/2, ∂± =2∂ (3)
± ± 0 1 ∓
± ±
and gamma matrices are
0 1 0 1 1 0
γ0 = , γ1 = − , γ5 =γ0γ1 = . (4)
1 0 1 0 0 1
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) − (cid:19)
The anti-symmetric tensor ǫ is taken to be ǫ =ǫ01 = 1.
µν 10
−
2
2 Thirring model
The Thirring model is (1+1) dimensional field theory with the current-current
interaction. The lagrangianof the Thirring model is given by
g
=ψ¯i∂ γµψ j jµ, j =ψ¯γ ψ, (5)
µ µ µ µ
L − 2
where g is a coupling constant. Then, the equations of motion become
i∂ ψ =gj ψ , i∂ ψ =gj ψ , (6)
+ R L R − L R L
where
1 1
ψT =(ψ , ψ ), j j0+j1 , j j0 j1 . (7)
R L R L
≡ 2 ≡ 2 −
(cid:0) (cid:1) (cid:0) (cid:1)
Fromeq.(6),thecurrentjµ anditsdual(axial)currentjµ =ǫµνj isconserved,
ν
∂ jµ =0, ∂ jµ =0. (8)
µ µ e
TheThirringmodelisexactlysolvable. Thirring[1]constructedthe eigenstates
e
while Klaiber [3]found the operatorsolution. Onthe other hand, Nakanishi[4]
described the quantum operator solution of eq.(6) in terms of the free massless
bosonic field ϕ=ϕ (x−)+ϕ (x+) as
R L
: :
Z eisϕR−is¯ϕL
ψ(x)= : : , (9)
√2π eis¯ϕR−isϕL
(cid:18) (cid:19)
where s,s¯ are constant parameters and Z is a normalization factor. The free
bosonic field satisfies
∂ ∂µϕ=0 (10)
µ
and we can regularize [13] as
1
ϕ↓(x−),ϕ↑(y−) = lni(x− y− i0), (11)
R R −4π − −
h i
1
ϕ↓(x+),ϕ↑(y+) = lni(x+ y+ i0), (12)
L L −4π − −
h i
where ϕ↓ and ϕ↑ are the positive and the negative frequency part respec-
R,L R,L
tively. Therefore, we have the Operator Product Expansion (OPE) of the
Thirring operator,
ψR(x)ψR(y) = |Z|2is24+πs¯2(x− y− i0)s2/4π(x+ y+ i0)s¯2/4π
2π − − − −
: :
eisϕR(x)−is¯ϕL(x)+isϕR(y)−is¯ϕL(y) , (13)
×
ψR†(y)ψR(x) = |Z2π|2i−s24+πs¯2(y−−x−−i0)−s2/4π(y+−x+−i0)−s¯2/4π
: :
e−isϕR(y)+is¯ϕL(y)+isϕR(x)−is¯ϕL(x) (14)
×
3
and so on. We have the similar relationfor ψ if s¯ s. For the massless Dirac
L
↔
field case (g =0), we find s=2√π and s¯=0.
To solvethe model, we mustdetermine the parameterss,s¯. The firstcondi-
tion of s and s¯is given by the aniti-commutativity of ψ and we have
s2 s¯2
− =1. (15)
4π
3 Current regularization
Next, we insert the operator solution into the field equation eq.(6). To do this,
we propose the following current definition,
1
jµ(x) = ψ¯(x+ε)γµψ(x)+ψ¯(x+ε)γµψ(x)
2
(cid:20) (cid:21)
σ
αµ ψ¯(x+ε)γνψ(x) eψ¯(x+ε)γνψ(x) , (16)
−2 ν −
(cid:20) (cid:21)
where ε(ε) is an infinitesimal timelike (spacelike) vecetor and σ is a parameter
and α0 = α1 = 1, α0 = α1 = 0. Here, we get timelike vector εµ close to
0 − 1 1 0
zero with ε1 0 firstly, whereas the spacelike vector ε is done in an opposite
e
→
way. In our formulation, the current is written by
s σs¯ s σs¯ e
j = − ∂ ϕ , j = − ∂ ϕ . (17)
R − R L + L
− 2π 2π
Therefore, the operator solution (9) is valid if
g
s¯= (s σs¯). (18)
2π −
Finally,we havethe equationswhichmustbe satisfiedby the parametersofthe
solution
s2 s¯2 g
− =1, s¯= (s σs¯). (19)
4π 2π −
First, we consider σ =0 and σ =1 case. The second equation becomes
g g
σ=0 σ=1
s¯= s, s¯= (s s¯). (20)
2π 2π −
They can identify with the map,
g
σ=0
g = . (21)
σ=1
1 g /2π
σ=0
−
ThisisnothingbuttherelationbetweenthecouplingconstantoftheSchwinger
definition and that of the Johnson definition. This is consistent with Klaiber’s
result [3]. Therefore, σ = 1 corresponds to Schwinger’s current definition and
σ =0 is Johnson’s one in our formulation.
4
We can also calculate the commutation rules between the current and the
spinor field ψ,
(s σs¯)(s+s¯)
ψ(x1,t), j0(y1,t) = − δ(x1 y1)ψ(x) (22)
4π −
(cid:20) (cid:21)
and
(s σs¯)(s s¯)
ψ(x1,t), j1(y1,t) = − − δ(x1 y1)γ5ψ(x). (23)
4π −
(cid:20) (cid:21)
4 Comparison with other formulations
Dell’Antonio, Frishman and Zwanziger [14] analyzed the Thirring model in a
different way. They consider the commutation relations of the current, the
Schwinger term. We can identify their result with
2
(s σs¯)(s+s¯) (s σs¯)(s s¯) s σs¯
a= − , a¯= − − , c= − , (24)
4π 4π 2π
(cid:18) (cid:19)
where a, a¯ and c are parameters in their formulation (Note that in [14], ǫ is
µν
defined by ǫ = 1). It is easy to check the consistency condition, eq.(6.1) in
10
[14],
a a¯=gc. (25)
−
c is written in terms of the coupling constant g as
g g
1/c=π 1+ (σ 1) 1+ (σ+1) . (26)
2π − 2π
h ih i
If σ = 0, it becomes eq.(6.3) in [14]. Therefore, our result perfectly agrees
withDell’Antonioetal. andtheparameterofthecurrentcommutationrelation
is determined by the coupling constant and the parameter σ appeared in the
current definition.
Taguchi, Tanaka and Yamamoto [15] consider the Thirring model with the
Tomonaga-Schwinger equation. They consider the deformed hamiltonian and
calculate the commutation relations between the current and the spinor field.
In this case, we have eq.(22) and (23) in a similar way.
It is well-known that the Thirring model is c = 1 (c is the central charge)
Conformal Field Theory (CFT) [16]. Klassen and Melzer [17] argued that the
Thirring model is equivalent to the fermionic Gaussian CFT. They show the
relation between the compactification radius of the fermionic Gaussian CFT
and the Thirring coupling constant. We give their result with σ = 0 (Johnson
current). More generally, the compactification radius R is written by
1 g 4π
R= 1 1 σ σ2 . (27)
± 2π − g −
g g (cid:20) r (cid:21)
1+ (σ 1) 1+ (σ+1)
s 2π − 2π
(cid:20) (cid:21)(cid:20) (cid:21)
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5 Conclusion
WehavepresentedthegeneralizationofthecurrentregularizationintheThirring
model. Thedefinitionofthecurrentiscomplicated,butitbecomessimplewhen
itisexpressedintermsofthefreemasslessbosonicfield. Thepresentdescription
is consistent with known results of the Thirring model. The present formula-
tion is simpler than Klaiber’s formulation. Klaiber defines the commutator of
the current jµ and the field ψ, and makes the anzatz about the current while
we employ the operator solution which is given by Nakanishi [4]. The solution
is written in terms of the free massless bosonic field, and thus we can easily
evaluatevariousquantities. Thisisthe maindifferencebetweenKlaiber’streat-
ment and ours. Further, we obtain the general formula for arbitrary current
regularization.
TheshortdistancebehavioroftheThirringmodelismorecomplicatedthan
the free massless Dirac field. For the Dirac field, the limiting procedures of
ε 0 and ε 0 are the same. On the other hand, they are different for the
→ →
Thirringfield. This is the consequenceofthe fact thatthe Dirac field(s¯=0)is
written in terms of the bosonic field ϕ and ϕ separately in contrastwith the
e R L
Thirring field.
In the present description, we introduce a new parameter σ in our current
definition. ThiscurrentbecomesLorentzcovariantlimitingoperator,jµ ∂µϕ.
∼
Here, ϕ is the dual massless field of ϕ[13]. We obtain the Johnson current for
σ = 0 and the Schwinger current for σ = 1. Note that, for σ = 1, the current
e
definition is given by
e
j0(x)=ψ¯(x+ε)γ0ψ(x), j1(x)=ψ¯(x+ε)γ1ψ(x). (28)
Therefore,in our formulation, j0 is defined by limiting from spacelike direction
e
while j1 is defined by limiting from timelike direction. This is in contrast with
the originalSchwinger’sdefinition [10] whichis defined by the spacelikesepara-
tion only. However, we can show that j0 and j1 commute with each other for
the Thirring case if we adopt the original prescription. On the other hand, in
our formulation, the Schwinger term [18, 10] appear as
j (x1,t), j (y1,t) =icδ′(x1 y1), (29)
0 1
−
where(cid:2)cisgivenbyeq.(26(cid:3)). Accordingly,thecurrentmustbedefinedbylimiting
from both spacelike and timelike direction in the Thirring model. Note that
the parameter σ determines the current algebra. Unfortunately, we do not
understand the physical meaning of the parameter σ yet.
ACKNOWLEDGMENTS
We would like to thank T. Fujita and M. Hiramoto for helpful discussions
and comments.
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