Noncommutative fermionic field theories with smooth commutative limit
Dongsu BakPhysics Department, University of Seoul, Seoul 130-743, Korea Sung Ku Kim
Physics Department, Ewha Women’s University, Seoul 120-750, Korea Kwang-Sup Soh
Physics Department, Seoul National University, Seoul 151-742, Korea Jae Hyung Yee
Institute of Physics and Applied Physics, Yonsei University, Seoul 120-749, Korea 共Received 14 September 2000; published 26 January 2001兲
We consider two model field theories on a noncommutative plane that have a smooth commutative limit. One is the nonrelativistic single-component fermion theory with a quartic interaction that vanishes identically in the commutative limit. The other is a scalar-fermion theory, which extends the scalar field theory with a quartic interaction by adding a fermion. We show the smooth commutative limits by computing the bound state energies and the two particle scattering amplitudes exactly.
DOI: 10.1103/PhysRevD.63.047701 PACS number共s兲: 11.10.Kk, 11.10.St, 11.25.Db Quantum field theories on noncommutative space have
recently been studied in both the classical 关1–7兴 and quan-tum mechanical context关8–11兴. It is one of the most press-ing questions whether or not the noncommutative quantum field theories are well defined. In this regard, the nonrelativ-istic scalar field theory with a quartic interaction, recently presented in Ref.关12兴, may provide an interesting laboratory for understanding the nature of noncommutative field theory. One of the interesting features of the model is that the wave functions exhibit the dipole separation transverse to the total momentum in their relative coordinates and that the UV/IR mixing does not interfere with the renormalizability of the theory.
However, as in typical noncommutative field theories, the infrared divergences arise in the noncommutative scalar field theory due to the UV-IR mixing. Moreover, the commutative limit after quantization does not agree with the theory de-fined without noncommutativity, which turns out to be closely related to the IR divergence problem. The under-standing of any IR divergences in a given theory is of physi-cal importance. However, the issue of the infrared diver-gences arising in noncommutative field theories is not yet fully resolved 关8兴. Here we pursue an alternative direction. Namely, we look for noncommutative field theories that have a smooth commutative limit. The theories we consider then do not have any IR divergence problem. One obvious further virtue of the smooth commutative limit lies in the fact that one may better understand the commutative field theory it-self by studying the corresponding noncommutative theory or vice versa.
In this note, we shall present two model field theories that have smooth commutative limits. The first model consists of 共2⫹1兲-dimensional single component fermion field with quartic interaction. Although the theory is nontrivial on the noncommutative plane, it becomes free in the commutative limit due to the Pauli exclusion principle. The other model consists of the coupled nonrelativistic scalar and fermionic
fields with quartic interaction terms, which has analytic but nontrivial commutative limit. This theory is the unique ex-tension of the scalar field theory with a quartic interaction by the following restrictions. One is just adding only one fer-mion and the other is requiring that the resulting theory has its smooth commutative limit. As will be shown below, the dipole nature of the wave functions does not change. Namely, as in the case of the scalar theory, the wave func-tions of these models have again two center posifunc-tions in the relative coordinates, whose separations grow with the non-commutative scale and the total momentum.
Single component fermion model: We consider a nonrela-tivistic single component fermionic field theory on a non-commutative plane described by the Lagrangian
L⫽
冕
d2x冉
i†˙⫹1 2†ⵜ2⫺v 4
†*†**
冊
, 共1兲 where the Moyal product (* product兲 is defined bya共x兲*b共x兲⬅„ei⬘a共x兲b共x⬘兲…兩x⫽x⬘ 共2兲
with
⬘
⬅12⑀ i ji
⬘
j. In the ordinary commutativespace-time, the quantum interaction vanishes due to the Pauli ex-clusion principle and the theory becomes free. In the non-commutative plane, however, it is nontrivial due to the noncommutativity of the space. As in the case of the scalar field theory 关12兴, the scale invariance and the boost part of the Galilean symmetry are broken by the explicit scale de-pendence of the Moyal product. The global U共1兲 invariance under→ei␣ persists in the noncommutative case and the number operator N⫽兰d2x† is conserved, enabling one to study the system in each N-particle sector separately. One can quantize the system共1兲 by using the same procedure as that of the scalar field theory of Ref.关12兴. Namely, we shall adopt the canonical quantization scheme instead of the path PHYSICAL REVIEW D, VOLUME 63, 047701
integral quantization in order to incorporate the normal or-dering of the Hamiltonian operator,
H⫽
冕
d2x冉
⫺1 2†ⵜ2⫹v 4
†*†**
冊
, 共3兲 with the field satisfying the canonical anticommutation rela-tion,兵(x),†(x⬘
)其⫽␦(x⫺x⬘) . Since the Hamiltonian共3兲 is normal ordered, the two-point Green function does not re-ceive any perturbative corrections, as shown in Ref.关12兴, for the bosonic case. Writing the action in the momentum space, one finds the Feynman rule for the propagator,S0共k0,k兲⫽ i k0⫺k 2 2 ⫹i⑀
, 共4兲
and for the vertex,
⌫0共k1,k2;k3,k4兲⫽iv0sin共k1k2兲sin共k3k4兲, 共5兲 where kk
⬘⫽(
/2)⑀i jkikj⬘
. Note that the vertex vanishes asthe commutative parameter approaches zero, exhibiting the fermionic nature of the theory. The propagator and vertex are shown diagrammatically in Fig. 1.
We now compute the on-shell four-point function to ob-tain the two particle scattering amplitude. The one-loop bubble diagram and the full four-point function are depicted in Fig. 2.
Using the Feynman rules one obtains the contribution from the one-loop bubble diagram:
⌫b⫽v0 2 sin共kP兲sin共k
⬘P兲
8i冕
d2q 2 1⫺cos共2qP兲 q2⫺k2⫺i⑀ ⫽v0 2sin共kP兲sin共k⬘P兲 8i 关Z共⌳/k兲⫺K0共⫺ik P兲兴, 共6兲 where K0(x) is the modified Bessel function, Z(x)⬅ln x ⫹i/2, and the large momentum cutoff ⌳ is introduced to regulate the divergent integral. It is interesting to compare the result 共6兲 with that of the scalar theory 关12兴. The only difference between Eq.共6兲 and that of the scalar field theory is the replacement of the cosine function by the sine function and the overall negative sign, which come from thefermi-onic nature of our system. Summing all the bubble diagrams of Fig. 2, one obtains the full four-point function,
⌫⫽ 1 ⫺i sin共kP兲sin共k⬘P兲
v0
⫹ 1
8关Z共⌳/k兲⫺K0共⫺ik P兲兴
. 共7兲
This result also shows that, in the commutative limit→0, the four-point function vanishes and the theory becomes free. Thus, in this fermionic theory, the noncommutativity of the limits →0 and ⌳→⬁ does not occur.
One can remove the divergence appearing in Eq. 共7兲 by redefining the coupling constant as
1 v共兲 ⫽ 1 v0 ⫹ 1 8ln
冉
⌳ 冊
, 共8兲where is a renormalization scale. For the consistency of this relation, the bare coupling constant should be negative because ⌳ is supposedly much larger than the renormaliza-tion scale . The beta function is given by „v()… ⬅关v()兴/⫽关v2()兴/8, which is the same as in the scalar field theory. One then obtains the two particle scatter-ing amplitude, A⫽ 1 4
冑
k sin共kP兲sin共k⬘P兲 1 v共兲 ⫹ 1 8关Z共/k兲⫺K0共⫺ik P兲兴 , 共9兲where an appropriate kinematical factor is included. The imaginary pole of the scattering amplitude indicates the ex-istence of the bound states of the two particle system. Setting the bound state energy EB⫽⫺⑀B(⑀B⬎0), it is determined by
the solution of the equation 8
v共兲 ⫹ln
冉
␥P
2
冊
⫽F共冑
⑀BP兲, 共10兲 where ␥ is the Euler constant and F(x)⬅ln(x␥/2)⫹K0(x). Note that F(x) is monotonically increasing and covers the range 关0,⬁) for x⭓0. The bound state does not exist when the left-hand side of Eq.共10兲 is negative. In other words, the bound states cannot be formed ifP is below the threshold value of P. This reflects the fermionic nature of the theory that, in the→0 limit, the theory becomes free. There exists one bound state whenP⭓(2/␥) e⫺8/v(). IfP is slightly bigger than the threshold, i.e., ␥P⫽关2 ⫺ln(␥2/e2)兴e⫺8/v()with 0⬍Ⰶ1, the expression for the bound state energy is given by EB⫽⫺ ␥22e16/v(),where we have ignored the higher order terms in. For large P, on the other hand, the bound state energy is given by EB⫽⫺2e16/v(), whose expression is the same as the case
of the noncommutative scalar theory.
Since the number operator of the system共1兲 is conserved, one may treat the system quantum mechanically by con-structing the Schro¨dinger equation in each N-particle sector. The Schro¨dinger equation for the two particle wave function, ⌿(r,r⬘)⫽
具
0兩(r,t)(r⬘
,t)兩⌽典can be obtained from the op-erator Schro¨dinger equation; in the momentum space, it readsFIG. 1. The fermion propagator and the vertex.
FIG. 2. The bubble diagram and the full four-point function.
BRIEF REPORTS PHYSICAL REVIEW D 63 047701
i⌿˙共Q,q兲⫽
冉
1 4Q2⫹q2
冊
⌿共Q,q兲⫺sin共qQ兲B共Q兲, 共11兲 where Q and q are, respectively, the total and the relative momenta of the two particles and B(Q) ⫽⫺(v0/82)兰d2q⬘
sin(q⬘Q)⌿(Q,q⬘
). Here again, Eq. 共11兲 for the fermion system can be obtained by replacing the cosine function by sine function in the equation for the scalar theory. Writing⌿(Q,q)⫽␦(Q⫺P)˜ (P,q)e⫺i(14P2⫹Er)t, the momentum space equation is reduced to共Er⫺q2兲˜共P,q兲⫽⫺sin共qP兲C共P兲, 共12兲
with C(P)⫽⫺(v0/82)兰d2q
⬘
sin(q⬘P)
˜(P,q⬘
). For the bound state, we obtain
˜共P,q兲⫽sin共qP兲 q2⫹⑀B
C共P兲, 共13兲
with Er⫽⫺⑀B. Integrating both sides with weighting factor
sin(qP), one obtains the eigenvalue equation 共10兲 for the bound state after renormalization. Thus the result of the two particle Schro¨dinger equation agrees with that of the pertur-bative theory. The bound state wave function in the position space can be obtained from Eq.共13兲:
⌿⫽eiP•R关K0共
冑
⑀Bx⫹兲⫺K0共
冑
⑀Bx⫺兲兴, 共14兲where R⫽(r1⫹r2)/2, r⬅r1⫺r2and x⫾i ⬅ri⫾12⑀i jPj. As in
the scalar field theory, the wave function has two centers at ⫾1
2P˜ in the relative coordinates. The separation of these
centers grows linearly inP and is perpendicular to the total momentum P, exhibiting explicitly the stringy dipole nature of the theory关13兴. The scattering solution can also be found with Er⫽k2:
˜共P,q兲⫽共2兲2␦共q⫺k兲⫹C共P兲 sin共qP兲
q2⫺k2⫺i⑀, 共15兲 where C(P) is as given above. Proceeding similarly as in the case of the scalar field theory, one obtains the exact wave function for the two particle scattering:
˜共r兲⫽eik•r⫹1
8关H0 (1)共kx
⫹兲⫺H0(1)共kx⫺兲兴C共P兲, 共16兲 where H(1)(x) is the Hankel function of the first kind. Once C(P) is determined self-consistently, one may check that the resulting scattering amplitude agrees precisely with that of the perturbation theory. The scattering solution again has two dipole centers whose separation grows linearly with P in the perpendicular direction of P.
Coupled scalar-fermion system: As another nontrivial ex-ample of the noncommutative field theory that has the smooth commutative limit, we consider the coupled scalar-fermion theory described by the Lagrangian
L⫽
冕
d2x冉
i†˙⫹i†˙⫺1 2兩ⵜ兩 2⫺1 2兩ⵜ兩 2 ⫺v4关†*†⫹†*†兴*关*⫹*兴冊
, 共17兲 where (x,t) and (x,t) represent a scalar and a single-component fermion field operators, respectively. In the com-mutative limit, the fermionic interaction terms vanish and thus the theory reduces to the ordinary nonrelativistic scalar theory with quartic interaction 关14,15兴. The global U共1兲 in-variance under→ei␣ and→ei␣ insures the conserva-tion of total number operator N⫽兰d2x (†⫹†). How-ever, the scalar and fermionic number operators are, respectively, conserved only for ⫽0. The violation of the the fermion number conservation is of order , as will be seen below, and will disappear in the commutative limit. There is an additional Z2 symmetry that exchanges the fer-mion and boson. It is this symmetry that is partly responsible for the smooth commutative limit of the system.Feynman rules for the perturbative calculations can be obtained as in the fermionic theory. The scalar and fermion propagators are the same as the ones in Ref.关12兴 and Eq. 共4兲, respectively. There are now four kinds of vertices. One can evaluate the Green function and study the bound state and the scattering problem perturbatively as done in the earlier cases. Instead, we shall use the quantum mechanical ap-proach to compute the exact wave functions for the sector of two bosonic and two fermionic particles. The coupled Schro¨-dinger equation for the two boson state (r,r
⬘
) ⫽具
0兩(r,t)(r⬘
,t)兩⌽典 and two fermion state ⌿(r,r⬘) ⫽具
0兩(r,t)(r⬘
,t)兩⌽典, can be constructed from the operator equation of motion for the field operator and. The center of mass motion can be diagonalized easily with momentumP, and the resulting time independent Schro¨dinger equations
for the relative motion are
Er共P,q兲⫽q2共P,q兲⫺cos共qP兲G共P兲,
共18兲 Er˜共P,q兲⫽q2˜共P,q兲⫹isin共qP兲G共P兲,
with G(P)⫽⫺(v0/82)兰d2q
⬘
关 cos(q⬘P)(P,q⬘
)⫹isin(q⬘ P)˜(P,q⬘
)兴. The bound state energy is obtained by setting Er⫽⫺⑀B. Then, the Schro¨dinger equation implies that⫽cosk2共kP兲⫹⑀
B
G共P兲, ˜⫽⫺isin共kP兲 k2⫹⑀B
G共P兲. 共19兲
Substituting Eq.共19兲 into the above expression of G(P), one finds the eigenvalue equation for the bound state energy,
4⫽⫺v0ln共⌳/
冑
⑀B兲. 共20兲By redefining the bare coupling constant as 1 v共兲 ⫽ 1 v0⫹ 1 4ln
冉
⌳ 冊
, 共21兲BRIEF REPORTS PHYSICAL REVIEW D 63 047701
where is the renormalization scale, one obtains the bound state energy, Er⫽⫺2e8/v(). This bound state energy is
exactly the same as that of the ordinary scalar theory with the contact interaction 关14,15兴. The reason for this is that the nonplanar parts of fermionic loop contribution exactly cancel those of the bosonic loops, and that only the planar diagrams contribute to the four-point Green functions. Thus the theory has the analytic limit as →0. The beta function, (v)
⫽v2/4, also agrees with the commutative scalar theory 关14,15兴 for the same reason.
The scattering problem can also be dealt with as in the earlier cases. Consider the scattering state of two incident scalar particles with relative energy Er⫽k2; the solution of
the Schro¨dinger equation共18兲 reads 共r兲⫽eik•r⫹i 8关H0 (1)共kx ⫹兲⫹H0(1)共kx⫺兲兴G共P兲, 共22兲 ˜共r兲⫽1 8关H0 (1)共kx ⫹兲⫺H0(1)共kx⫺兲兴G共P兲,
where G(P)⫽⫺12 cos(qP)/关(1/v0)⫹(1/4)Z(⌳/k)兴. This
result again exhibits the dipole nature in the relative coordi-nate having two centers.
In the commutative limit, →0, the bosonic wave func-tion reduces to that of the ordinary scalar theory of Ref.关14兴, and the fermion wave function vanishes. Thus these results clearly indicate that the coupled scalar fermion theory in Eq. 共17兲 has analytic commutative limit. By taking the asymptotic limit of Eq. 共22兲, one finds the scattering ampli-tudes. For the outgoing two bosons,
Ab⫽⫺ 1 4
冑
k cos共kP兲cos共k⬘P兲 1 v共兲 ⫹ 1 4Z共/k兲 ; 共23兲and for the outgoing two fermions,
Af⫽⫺ 1 4
冑
k cos共kP兲sin共k⬘P兲 1 v共兲 ⫹ 1 4Z共/k兲 , 共24兲where k and k
⬘
are, respectively, the initial and the final relative momenta. The fermion production out of two boson scattering is zero when the final relative momentum is par-allel to the total momentum, and has a peak when the final relative momentum is perpendicular to the total momentum. Of course, the production vanishes when ⫽0, so the limit →0 is analytic.In this note, we have presented two noncommutative field theories that have smooth commutative limits. The single component fermion model is intriguing because the quartic interaction was identically vanishing on the ordinary plane, but produces nontrivial interactions due to the noncommuta-tivity of the geometry. In the noncommutative scalar theory with a quartic interaction of Ref. 关12兴, sending the momen-tum cutoff to infinity does not commute with the operation P→0, though this does not interfere with the renormaliz-ability of the theory. In other words, the P⫽0 theory dif-fers from the theory defined by taking P→0 limit of the regulated theory with a nonzeroP. In particular, the theory turns out to be unbounded below in the infrared limit be-cause the bound state energy can become indefinitely nega-tive 关12兴. Although not conclusive, this discontinuity might even spoil the renormalizability in the case of the relativistic quantum field theories.
Here we asked whether there exists a theory free of the discontinuity at P⫽0. One is the fermionic theory, whose spectrum is bounded from below. We also extend the scalar theory with a quartic interaction by adding an appropriate fermionic field. The supersymmetric extension of the scalar Schro¨dinger theory is not the desired one because it may be verified that there are still discontinuities although the theory is renormalizable. The answer is rather unique and it is noth-ing but the coupled scalar-fermion theory presented here. However, as remarked before, the approach in this note does not resolve the infrared divergence problem of generic non-commutative field theories, and further studies are required in this direction.
This work is supported in part by KOSEF 1998 Interdis-ciplinary Research Grant 98-07-02-07-01-5 and BK21 Project of Ministry of Education.
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