On Abelian Bosonization of Free Fermi Fields in Three Space Dimensions

5 Pages·0.08 MB·English

by  Ramesh Abhiraman

YCTP-P1-95
On Abelian Bosonization of Free Fermi Fields
in Three Space Dimensions
Ramesh Abhiraman and Charles M. Sommerfield*
5
9 Center for Theoretical Physics, Yale University,New Haven,Connecticut, USA
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1

n
a
J
Abstract. Oneofthe methods usedto extendtwo-dimensionalbosonizationto
3
fourspace-timedimensionsinvolvesatransformationto newspatialvariablesso

1 that only one of them appears kinematically. The problem is then reduced to
v an Abelian version of two-dimensional bosonization with extra “internal” coor-
8
dinates. On a formal level, putting these internal coordinates on a finite lattice
0
seemstoprovideawell-definedprescriptionforcalculatingcorrelationfunctions.
0
1 However, in the infinite-lattice or continuum limits, certain difficulties appear
0 that require very delicate specification of all of the many limiting procedures
5 involved in the construction.
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/
h Introduction
t
- After the initial success of bosonization in 1+1 dimensions [1,2] Alan Luther
p
e presented a heuristic formula [3] for bosonization of free, massless relativistic
h fermions in 3+1 dimensions. A configuration-spacetransform[4] was later used
v: by H. Aratyn[5] to put this into a prettier but no less heuristic form. For other
i approaches to this problem the reader is referred to References [6] and [7].
X
The basic ideaofthe methoddescribedhere is to viewfield theoriesin3+1
r
a dimensions as 1+1 dimensional theories with an internal, non-kinetic degree of
freedom. Themainsubjectofthispaperistoexaminecarefullythelimitingpro-
cedures involved in bosonizing a 1+1 dimensional free fermion with an internal
degree of freedom that takes values that are in a continuum.
Tomographic transform
We beginbyreviewingthe transformationto a1+1dimensionaltheorybothfor
bosons and for fermions. We then present the standard Abelian technique for
bosonizing the 1+1 dimensional fermion when the internal degrees of freedom
lie on a finite lattice. There follows a careful analysis of the continuum limit of
this lattice in the context of a simplified model.
* Based on a talk given by C. Sommerfield at Gu¨rsey Memorial Conference I,
Boˇgazi¸ci University,Istanbul,Turkey,June 1994.
Thetomographictransformofafreescalarfieldφ(x,t)withmassmin3+1
dimensions is defined as
1
φ˜(y,n,t)= d3x∂ δ(y n x)φ(x,t).
2π Z y − ·
Here n is a unit vector and d2n an element of solid angle in the direction of n.
The field equation satisfied by φ˜(y,n,t) and its equal time commutation
relations are essentially those of a 1+1 dimensional scalar field, also free and
with mass m.
The corresponding transform for a fermion field is
1
ψ˜a(y,n,t)= d3x∂ δ(y n x) u a(n)ψ (x,t).
2π Z y − · †α α
X
α
It satisfies the 1+1dimensional Dirac equationfor a right-movingfermion field,
(that is, depending only on y t) and has the expected equal-time anticommu-
−
tation relations. Here ua(n), a = 1,2, are orthonormal four-component spinors
that satisfy (α n)ua(n)=ua(n) where α are the Dirac alpha matrices.
·
The strategy is to use standard Abelian 1+1 dimensional bosonization to
construct the right-moving ψ˜a(y,n,t), in terms of the right-moving part of the
transformed boson φ˜(y,n,t), assuming, to begin with, that n is restricted to a
lattice. This is given formally, with the t dependence suppressed, as
φ˜a(y,n)= 1[φ˜a(y,n) 1 ∞ dy ǫ(y y )∂ φ˜a(y ,n)],
r 2 − 2Z ′ − ′ t ′
−∞
where ǫ(y y )=(y y )/y y . It follows that
′ ′ ′
− − | − |
1
0φ˜a(y,n)φ˜b(y ,n)0 = δabδ(n,n)ln(µ[α i(y y )])
h | r r ′ ′ | i −4π ′ − − ′
where µ and α are infrared and ultraviolet cutoffs, respectively.
Wefirsttakentobediscretelydistributedonsomelatticeofdirectionsand
reinterpret the Dirac delta function as a Kronecker delta function. Replacing
the pairaandnby asingleindexA, wefindthe formalbosonizationexpression
ψ˜A(y)= 1 eiφ˜Ar(y)KA
√2πα
where
i
KA =exp √π ǫAC[φC( ) φC( )]
2 XC r ∞ − r −∞
and where ǫAB = ǫBA, with ǫAB 2 = 1. The Klein factors KA are needed
− | |
to make sure that fermion fields with different indices anticommute rather than
commute. In what follows we will ignore these factors. Their only effect is to
correct an occasional sign to its proper value.
The correlation functions for the fermion field are determined from the
matrix elements of their bosonic representation in the bosonic vacuum. The
two-point function is computed as
1 µ
0ψ˜A(y)ψ˜ B(y )0 = .
h | † ′ | i 2π µ[α+i(y y )] δAB
{ − ′ }
If A = B we get the known fermion function. If A = B then this vanishes
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in the limit µ 0, so that the result is indeed proportional to δAB. When
→
reinterpretedintermsofacontinuousnwedogetthecorrecttwo-pointfunction
for the transformed fermion field:
1 δabδ(n,n)
0ψ˜a(y,n)ψ˜†b(y′,n′)0 = ′ .
h | | i 2π α i(y y )
− − ′
All of the other fermion correlationfunctions come out properly as well.
To test the consistency of the bosonization formula we must consider the
important fermion bilinear operators in the 3+1 dimensional theory such as
the Poincar´e group generators and the chiral charge operators. If Ω is one of
these, does the sequence Ω[ψ ] Ω[ψ˜ ] Ω[φ˜ ] Ω[φ ] make sense?
3+1 1+1 1+1 3+1
↔ ↔ ↔
Trouble arises when we consider the rotation and boost generators. The proper
treatment of the continuum limit of the lattice plays a fundamental role here.
Simplified model and smoothing functions
In order to get at the heart of the problem of the continuum limit we will treat
a simple case in which the the internal variable is one-dimensional. We thus
considerasinglecomponent1+1dimensionalmasslesschiralfermionfieldψ(x,u)
that depends on a continuous internal variable u whose domain is the real line.
The bosonizationof the fermion field in the Dirac equation(∂ +∂ )ψ(x,u)=0
t x
willbeintermsofaright-movingchiralmasslessbosonfieldφ (x,u). Assuming
r
for the moment that u is a discrete variable, we write, up to Klein factors,
ψ(x,u) = (2πα) 1/2exp[i√4πφ (x,u)] and obtain for the fermion two-point
− r
function
1 δu,u′
0ψ(x,u)ψ (x,u)0 =
† ′ ′
h | | i 2πα i(x x)
− − ′
For the bosonization procedure to work, it is clear that the exponent that
appears in the evaluation of the fermion two-point function must be associated
with a Kronecker delta function, rather than a Dirac delta function so that
the logarithmic singularity in the bosonic two-point function exponentiates to
a simple pole or a simple zero. We need a more analytic method of converting
one type delta function to the other. Returning to a continuous u,we introduce
a new boson field Φ(x,u) = duf(u u)φ (x,u). We take f(u) to be a real,
′ ′ r ′
−
evenfunction. TheDiracdeltRafunctionisreplacedinthecommutationrelations
among the Φ operators, as well as in the bosonic two-point function, by the
convolution g(u u) = du f(u u )f(u u). It is thereby softened. We
′ ′′ ′′ ′′ ′
− − −
R
will choose f(u) so that g(u) 0, g(0) = 1, and ǫ(u)g (u) < 0 if g(u) = 0.
′
≥ 6
To make sure that g(u) goes to 0 very quickly as u increases we take it to
| |
be a function of u/λ and eventually take the scale parameter λ to zero. This
simulates the properties of a Kronecker delta function.
We define the field ψˆ(x,u) = C(2πα) 1/2exp[i√4πΦ(x,u)] which will be
−
the candidate for the canonicalfermion field ψ in the appropriatelimit. Here C
is some constant, to be specified later, that depends on µ and on the choice of
g(u). We then find for the fermionic two-point function
0ψˆ(x,u)ψˆ (x,u)0 = C 2µ[1 g(u,u′)] 1 1 .
h | † ′ ′ | i | | − 2π[α i(x x)]g(u u′)
− − ′ −
Astheinfra-redcutoffµgoestozero,thisvanishesunlessu=u. ChoosingC to
′
dependonµsothatC divergessuitablyinthislimit,weobtainthedesiredDirac
delta function and the same answer as in the discrete case, with the Kronecker
delta replaced by the Dirac delta.
There is a surprise when we consider the four-point function. One must
analyze the behavior of the quantity C 4µ(2+g12+g34 g13 g23 g14 g24) as µ 0.
− − − −
| | →
Here g =g u u ) with u and u assigned to the fermion fields and u and
ij ( i j 1 2 3
−
u to the adjoint fields. In order to obtain the correct behavior when all four
4
internalindicesareclosewefindthatwecannotchoosef(u)suchthatg (0)=0.
′
In fact g(u) must have a cusp at u = 0. Thus Gaussian functions for f(u) and
g(u)areruledout. Atriangularpulseisacceptableforg(u)andthiscorresponds
to a rectangular pulse for f(u).
Fermion bilinears
We go on to discuss the fermion bilinears. This is where we had trouble in the
discretecase. Agenericformofsuchabilinearoperatoris[ψ (x ,u ),ψ(x ,u )]
† 2 2 1 1
wheretheargumentsareallowedtocometogether,possiblyaftervariousderiva-
tiveshavebeentaken. Thisis the nature ofthe spatiallypoint-splitstructureof
the chargeoperator,andofthe generatorsoftranslationsinx, t andu. We pro-
ceed to evaluate it using operator-product expansion methods (which are exact
in this model). We have
C 2
[ψˆ2†,ψˆ1]=−|2π| µ(1−g12)(R12−g12 −R21−g21):ei√4π(Φ1−Φ2):.
Here R =α i(x x ). The normal ordering is with respect to the bosonic
12 1 2
− −
creationand annihilation operators. We introduce u= 1(u +u ), v =u u ,
2 1 2 1− 2
x= 1(x +x ) and ξ =x x , and obtain
2 1 2 1− 2
C 2
[ψˆ2†,ψˆ1]=−|2π| µ[1−g12][R(ξ)−g(v)−R(−ξ)−g(v)]Ω(ξ,v)
where Ω(ξ,v) = :expi√4π[Φ(x+ 1ξ,u+ 1v) Φ(x 1ξ,u 1v)]:. Taking
2 2 − − 2 − 2
v and ξ small, (while keeping α ξ), and sending µ to 0 we discover that
≪
[ψˆ2†,ψˆ1]→−iδ(v)Ω(ξ,0)/(πξ).
The chiralchargeoperatorQ is givenin terms of the canonicalfermions by
Qˆ = 1 dxdu du δ(u u )[ψ (x,u ),ψ(x,u )].
2Z 1 2 1− 2 † 2 1
We choose to smooth out the u delta function and replace it by N g(u u )
λ 1 2
−
whereN ischosensothatN du g(u u )=1. ThenN g(u u )becomes
λ λ 1 1 2 λ 1 2
− −
a Dirac delta function as λ R0. We construct the candidate chiral chargeQˆ in
terms of ψˆ as Qˆ =limξ→0N→λ dxdudv g(v)21[ψˆ2†,ψˆ1] so that
R
iN g(0) N ∂
λ λ
Q= lim − dxdu[Ω(ξ,0) Ω( ξ,0)]= dxdu Φ(x,u).
ξ 0 4πξ Z − − √π Z ∂x
→
The charge density in u space is thus Qˆ(u) = N π 1/2[Φ( ,0) Φ( ,0)].
λ −
∞ − −∞
We then have [ψˆ(x,u),Qˆ(u)] = N g(u u)ψˆ(x,u) which becomes the correct
′ λ ′
−
canonical relation as λ 0.
→
The canonicaloperator J = i1 dxdu[ψ (x,u), ∂ ψ(x,u)]+(h. c.) gen-
− 4 † ∂u
eratestranslationsinu. ItistheanaloRgueoftherotationandboostoperatorsin
thetomographicrepresentationofthe3+1dimensionalcase. Ifweweretofollow
the procedure used for the chiral charge we would find that when expressed in
terms of bosonic fields, the candidate generator would vanish. The remedy for
thisistobackupastepanddelaytakingµtozero. Thenthespreadingfunction
usedtodefine thebilinear,(whichneednotbe the sameaswhatweusedforthe
chiral charge, namely N g(u)), can be taken to depend on µ in just such a way
λ
as to yield the correct answer in terms of canonical boson fields in the limit.
Acknowledgment. This work was partially supported by the United States
Department of Energy under grant DE-FG02-92ER40704.
References
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[2] S. Mandelstam, Phys. Rev. D 11 (1975) 3026.
[3] A.Luther, Phys. Rev. B 19 (1979) 320.
[4] C. M. Sommerfield, in “Symmetries in Particle Physics,” A. Chodos, I. Bars and
C.-H. Tze, eds., Plenum Press (1984) pp.127-140.
[5] H.Aratyn,Nuc. Phys. B 227 (1983) 172.
[6] E. C. Marino, Phys. Lett. B263 (1991) 63.
[7] A.Kovnerand P. S.Kurzepa, Phys. Lett. B328 (1994) 506.