Fock space methods and large N

0.15 MB·English

by  M. Bonini

Fock space methods and large N
M. Bonini, G.M. Cicuta and E. Onofri
Dipartimento di Fisica, Univ. di Parma, v. G.P. Usberti 7A, 43100 Parma, Italy
and INFN, Sezione di Milano Bicocca, Gruppo di Parma
email: name(at)fis.unipr.it
7
0
0 Abstract
2 Ideas and techniques (asymptotic decoupling of single-trace subspace, asymp-
n totic operator algebras, duality and role of supersymmetry) relevant in current
a Fockspaceinvestigationsofquantumfieldtheorieshaveverysimplerolesinaclass
J
of toy models.

9

1
v
Hamiltonian methods have a long history in the attempts to understand the
6
bound states spectrum of a strongly interacting relativistic quantum field theory.
7
0 This is perhaps the hardest and most important problem in a strongly interacting
1 quantum field theory and analytic and numerical efforts were devoted to invent
0 reliablemethods. For severalyearslight-frontquantization[1] wasa promisingap-
7
proachbecauseoftheverydifferentnatureofthegroundstateandsomeimportant
0
simplifications of the operators occurring in the hamiltonian of non-abelian mod-
/
h els.
t
- The analysis of the large-N limit, at t’Hooft coupling fixed, of the models with
p
SU(N) or U(N) invariance, provided additional insights, indicating features of
e
string theory in non-abelian gauge models and the existence of symmetries, con-
h
: served quantum numbers and operator algebras occurring only in the asymptotic
v
theory,atN = ,[2],[3],[6],[13]. Muchworkwasdevotedtotheevaluationofthe
i ∞
X spectrum of the Hamiltonian for states in the lowest representation of the group :
r the singlet sector and the adjoint sector.
a A color-singletstate of n free bosons, with total momentum P~ is represented in
the Fock space by a linear superposition of states of the form
(1) tr a†(k ) a†(k ) tr a†(p ) a†(p ) tr a†(q ) a†(q ) 0
1 ··· n1 1 ··· n2 ··· 1 ··· ns
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:12) (cid:11)
where tr() denotes the trace on U(N) colour indices, n=n +n + +n(cid:12), P =
1 2 s
···
k + p + q and a†(k) are creation operators. These states are called
j j ··· j
mPulti-traPce states. PMatrix-valued operators may be written in terms of the group
generators a(k)= λ a (k), a†(k)= λ a†(k), and all matrix-related coeffi-
a a a a a a
cients are efficientlyPevaluatedby graphicPmethods [5,9]. Severalremarkableprop-
ertieswere foundin the large-N limit. Operatorsnormally–orderedinside a single-
trace,thatisoftheformγ =N−(r+s−2)/2 tr a†(k ) a†(k )a(p ) a(p ) ,act-
1 ··· r 1 ··· s
ing onsingle-tracestates generatesingle-trac(cid:0)estates [6], providedr 1 and(cid:1)s 1.
≥ ≥
Then if the Hamiltonian is a linear combination of such operators,the subspace of
the Fock space spanned by single-trace states is invariant under the action of the
Hamiltonian. These operators and analogous ones involving fermion operators act
on single-trace states in a way reminiscent of the coupling of strings. Single-trace
1
2
states like n = N−n/2n−1/2tr a†(k ) a†(k ) 0 form an orthonormal basis
1 n
···
in the color(cid:12)-si(cid:11)ngletand single-tra(cid:0)ceFock space [4](cid:1). M(cid:12) (cid:11)ulti-trace states like in eq.(1)
(cid:12) (cid:12)
provide an orthonormal basis in the color-singletFock space at N = [4].
∞
Recently G. Veneziano and J. Wosiek [7] suggested a supersymmetric model in
D = 1 space-time dimension, that is a matrix quantum mechanical model. It is
notsurprisingthatthemodelisanalyticallysolvableinthe large-N limitinseveral
sectors of the Fock space and reliable numerical evaluations may be performed in
other sectors. Several features of the bound states are very interesting.
The goal of this letter is to use results derived in recent analysis of the bosonic
sector of the model [8] and in a simple generalization presented here, to comment
on the properties of V-W hamiltonian with a view to the role they may have in
models in more realistic space-time dimension.
In the bosonic sector the Hamiltonian of the V-W model is
(2) H =tr a†a+g a†2a+a†a2 +g2a†2a2 , λ=g2N
(cid:0) (cid:0) (cid:1) (cid:1)
In the single trace sector of the singlet states, for large N, the Hamiltonian is a
tridiagonalrealsymmetricinfinite matrix. The bound states spectrumwasanalyt-
ically and numerically solved in the large-N limit for every λ 0 and it presents
≥
remarkable features:
the infinitely many eigenvalues of the discrete spectrum of the model, for
•
0 λ < 1 decrease in a monotonous way as λ increases and all vanish at
≤
λ = 1, where a phase transition occurs. For λ > 1 one eigenvalue remains
atzeroenergy,itisanewgroundstate,andtheinfinitelymanyeigenvalues
increase in a monotonous way as λ increases.
The non-zero eigenvalues at the pairs of values λ and 1/λ are related by a
•
duality property
1 1 1
(3) (E (λ) λ))=√λ E
n n
√λ − (cid:18) (cid:18)√λ(cid:19)− λ(cid:19)
Let us consider the trivial generalization of eq.(2) by allowing two coupling con-
stants
H =tr a†a+g (a†2a+a†a2)+g2a†2a2 , λ =g √N, λ =g2N
3 4 3 3 4 4
(cid:0) (cid:1) p
Here too the single-tracesector in the Fock space decouples in the large-N limit
andthehamiltonianisrepresentedbythetridiagonalrealsymmetricinfinitematrix
H(λ ,λ )
3 4
H =H = λ j(j+1),
j,j+1 j+1,j 3
(4) H =(1+λ (1 pδ ))pj, j =1,2,...
j,j 4 1j
−
The eigenvalue equation Hx = Ex is a system of recurrent relations which trans-
latesintoaneasydifferentialequationforthe generatingfunctionG(z)= ∞y zj
1 j
where xj =yj√j. P
λ ω(z)G′(z) EG(z) zλ + λ G′(0)=0,
3 4 3
p − −(cid:16) p (cid:17)
where ω(z)=z2+1+z(1+λ )/ λ
4 3
p
If(1+λ )2 4λ >0,ω(z)has2distinctrealroots. Theclosesttotheoriginz =0
4 3
−
translates into the asymptotic behaviour of the coefficients x . The integration
j
3
constant of the differential equation may be chosen to kill the closest singularity
thenobtaininganexponentiallydecreasingsequencex , hence normalizablebound
j
states.
The region of the quadrant λ 0 and λ 1 where the spectrum is discrete
3 4
≥ ≥−
lies abovethe parabolaof equation 4λ =(1+λ )2, shownby the black line in the
3 4
plot.
Λ
4
3
2
1
Λ
0.5 1 1.5 2 2.5 3 3
Fig.1. The line λ =λ corresponds to the V-W model.
3 4
The solution has a compact form in terms of the variables σ and η
1+λ (1+λ )2 4λ λ (1+λ )+λ (1+λ )2 4λ 2λ
4 4 3 4 4 4 4 3 3
σ = − − , η = − −
1+λ +p(1+λ )2 4λ 2λ (p1+λ )2 4λ
4 4 3 4 4 3
− −
p p
The eigenvalues E of normalizable states are the infinitely many solution of the
n
equation
F (α,1;1+α;σ)=η , 0<σ <1, where
∞ σk E
F (α,1;1+α;σ)=1+α , α=
α+k − (1+λ )2 4λ
Xk=1 4 − 3
p
It is easy to check that the hypergeometric function F (α,1;1+α;σ) satisfies the
translation identity
ασ
(5) F(α,1;1+α;σ)=1+ F(α+1,1;2+α;σ)
1+α
The plot shows the Hypergeometricfunction F (α,1;1+α;σ) versusα for fixed
σ and 6<α<2.
−
4
FHΑ,1;1+Α;ΣL
6
4
2
Α
-6 -4 -2 2
-2
-4
Fig.2. The graph shows F (α,1;1+α;σ) for 3 values σ = 0.3, 0.5, 0.7 depicted
with increasingly dark color.
In the space of parameters corresponding to bound states, let us consider the
2
parabolasofequationλ =σ¯ 1+λ4 ,seeFig.1. Ateachpointofthespaceofcou-
3 1+σ¯
(cid:16) (cid:17)
pling constants corresponding to the discrete spectrum there is a unique σ¯. For a
givenparabola,itiseasytodescribethe spectrumaswemovealongitfromλ =0
4
to λ = . Indeed along this path, the value of η increases in a monotonous way
4
∞
from η = to η = 0 at the point where it first crosses the line λ = λ , next it
4 3
−∞
reaches η = 1 at the second crossing with the same line, and continues increasing
up to its asymptotic limit η =1/(1 σ¯2).
−
At η = the eigenvalue equation has infinitely many solution of the form
−∞
α = n+ ǫ where n = 1,2, and ǫ are small positive numbers. That is
n n n
− ···
E =n(1 σ¯)/(1+σ¯) ǫ . As η increasesallrootsα moveina monotonousway
n n n
− −
to the right, that is each E decrease. At η = 1, the value of α = 0 and E = 0.
n 1 1
Thisboundstatebecamesdegeneratewiththevacuumstate 0>oftheFockspace,
|
whichhasE =0. As η increasesbeyondη =1,α increasesto positivevaluesand
0 1
E has increasingly negative values [11].
1
At the first crossing λ = λ = λ < 1, η = 0, σ¯ = λ and we may consider the
3 4
roots α +1 of the eigenvalue equation F(α +1,1; α +2; λ) = 0. Because of
n n n
the translationidentity eq.(5) they are simply relatedto the roots α of the eigen-
n
value equation at the second crossing λ = λ = λ > 1, η = 1, σ¯ = 1/λ, which is
3 4
F(α ,1; α +1; 1/λ)=1. This is the V-W duality of eq.(3).
n n
ThispicturelooksverydifferentfromtheV-Wspectrumbut,ofcourse,itisfully
compatible : in the V-W model the unique coupling moves along the line λ = λ
4 3
and it touches at λ=1 the boundary of normalizable eigenstates.
We now briefly describe the solution of the 2-couplings model by use of a non–
compact Lie algebra which arises in the large N limit.
5
Let us consider the Hamiltonian H˜(α,β)
H˜(α,β)=αD+ 1β(J +J ),
2 + −
D =jδ , (J ) = j(j+1)δ , J =(J )†
ij ij + ij i,j+1 − +
p
J =J iJ , α=1+λ , β =2 λ
± x y 4 3
±
p
The Hamiltonian H˜(α,β) differs from the asymptotic Hamiltonian H(λ ,λ )
3 4
given in eq.(4) only for one matrix element H˜ =H +λ . One easily computes
11 11 4
the commutators [D,J ] = J and [J ,J ] = 2D, showing that D,J ,J
± ± + − + −
± − { }
form a basis for the Lie algebra SO(2,1) in the degenerate Bargmann’s discrete
series representation n, (n = 0),characterized by a vanishing Casimir operator.
D+
Hence the generator J has a continuous spectrum filling the whole real axis [12].
x
Previous results about the spectrum can be re–derived as follows: if α>β, one
writes
1 α
H˜ =coshyD+sinhyJ , coshy =
x
α2 β2 α2 β2
− −
and a unitarypoperator U (a boost) exists such that U 1 pH˜ U−1 =D and the
√α2−β2
spectrum of H˜ is simply E =n α2 β2.
n
−
If α<β, upon writing p
1 β
H˜ =sinhyD+coshyJ , coshy =
x
β2 α2 β2 α2
− −
aunitaryopepratorboostingH˜ toJ exists,henceH˜ hasaconptinuousspectrum. On
x
the border α=β, H˜ coincides with a light–cone generator which has a continuous
(positive) spectrum [12].
FinallythespectrumoftheasymptoticHamiltonianH(λ ,λ )maybecomputed
3 4
from the spectrum of H˜ by the method outlined in the appendix of [8], based on
an exact perturbation formula (rank–one perturbation).
Notice that the operators closing the SO(2,1) algebra represent the restriction
tothesingle–tracestatesofmoregeneraloperatorsactingongeneralsingletstates.
For instance J tr a†2a /√N. The operators (H,J ,J ) close a Lie algebra
+ + −
→
up to terms of order (cid:0)1/N,(cid:1)hence the spectra can be discussed as arising from a
dynamical symmetry breaking.
Let us summarize our conclusions :
Hamiltonian models where each operator has the form Tr[(a†)nam] with
•
n 1andm 1leaveeachsectorofk-tracestatesinvariantinthelarge-N
≥ ≥
limit. By representing the hamiltonian in the basis of single-trace states,
one obtains a real symmetric fixed-width band matrix. The eigenvalue
equation is a system of recurrent equations which usually allows analytic
solution. Stillmulti-tracesingletsectorsarenotirrelevantbecausethemass
of bound states in these sectors is similar to that of the single-trace states.
As indicated in [8] the V-W bosonic hamiltonian may be evaluated in each
sector obtaining the same eigenvalues, therefore changing (in infinite way)
the degeneracy of eigenvalues found in the single-trace analysis.
In the simple models in D =1 supersymmetry is not necessary for the as-
•
ymptotic decoupling ofthe single-tracesectornor for the exactasymptotic
solution of the model. Howeverthe duality property of the spectrum has a
striking simple form only on the susy line λ =λ .
3 4
6
Light-frontquantizationseemsrelevantforrealisticmodels,thatisinspace-
•
time dimension 1 < D 4, because it makes possible to represent a local
≤
Hamiltonian in a partial normally ordered form [13] such that the single-
trace sector of Fock space may be asymptotically invariant. This is a prac-
tical necessity for the Tamm-Dancoff approach. Exact or approximate al-
gebrasofthe operatorswhichoccurinthe asymptoticHamiltonianprovide
a precious tool for the understanding of the spectrum.
When considering a local quantum mechanical hamiltonian H = Tr[p2 +
•
x2+V(x)] in the large-N limit, the Fock space methods seems inappropri-
ate: one cannot avoidoperator terms that couple the single-tracestates to
multiple-tracestatesinleadingorder. Thismakesanyevaluationrestricted
to the single-trace Fock states totally unreliable [8][10].
AtlargeN anewdynamicalsymmetryshowsup,simplifyingthecalculation
•
of the spectrum. The interplay of this symmetry with supersymmetry as
in V–W model deserves further study.
We add a last comment to indicate that the bosonic two-couplings model dis-
cussedinthisletteristheasymptoticgenericformofinfinitelymanyHamiltonians.
Let us consider the operators A , A†, D , j =1,2,
j j j ···
1 1
A† = tr a†(a†a)j , A = tr (a†a)ja ,
j Nj−1/2 j Nj−1/2
(cid:0) (cid:1) (cid:0) (cid:1)
1
D = tr (a†a)j
j Nj−1
(cid:0) (cid:1)
Itiseasytoverifythatasymptoticallytheyleavethesectorofthesingle-tracestates
invariant
A n n(n 1) n 1 +O(1/N),
j
∼ − −
A†(cid:12)(cid:12)n(cid:11) pn(n+1)(cid:12)(cid:12)n+1(cid:11)+O(1/N),
j ∼
(cid:12) (cid:11) p (cid:12) (cid:11)
Dj(cid:12)n n n +O(1(cid:12)/N)
∼
(cid:12) (cid:11) (cid:12) (cid:11)
Then all the Hamiltonians(cid:12) (cid:12)
H =tr a†a + g tr A†+A + g tr(D )
j,3 j j j,4 j
(cid:0) (cid:1) Xj≥1 (cid:16) (cid:17) Xj≥2
asymptoticallyleavethesectorofthesingle-tracestatesinvariantandinthissector
are all represented by the tridiagonal real symmetric infinite matrix H(λ ,λ ) in
3 4
eq.(4) with λ = √N g and λ = N g . The sums may be finite or
3 j j,3 4 j j,4
infinite. P P
Acknowledgments
E.O. would like to thank G. Veneziano, J. Wosiek and M. Trzetrzelewskifor inter-
esting discussions.
References
[1] AnintroductiontothevastliteratureonLightconequantizationmaybefoundontheILCAC
webpageshttp://www.slac.stanford.edu/xorg/ilcac.
[2] S. Dalley and I.R. Klebanov, String spectrum of a (1+1)-dimensional large-N QCD with
adjoint matter,Phys.Rev.D47(1993)2517.
[3] M.B.HalpernandC.Schwartz, The algebras of large N matrix mechanics Int.J.ModPhys.
A14 (1999) 3059 hep-th/9809197. M.B. Halpern On the large N limit of conformal field
theoryhep-th/0208150.
7
[4] C.B. Thorn, Fock space description of the 1/Nc expansion of quantum chromodynamics,
Phys.Rev.D 20(1979) 1435-1441.
[5] P.Cvitanovic,GrouptheoryforFeynmandiagramsinnon-abeliangaugetheories,Phys.Rev.
D 14 (1976) 1536, Group Theory Nordita notes 1984, the webbook : Group Theory, Bird-
tracks, Lie’sand Exceptional Groups. availableathttp://chaosbook.org/GroupTheory.
[6] C.-W.H. Lee and S.G. Rajeev Integrability of Supersymmetric Quantum Matrix Models in
the Large-N Limit, Phys.Lett. B 436 (1998) 91, hep-th/9806019 , C.-W.H. Lee Symmetry
Algebras of Quantum Matrix Models inthe Large-N Limit,hep-th/9907130.
[7] G. Veneziano and J. Wosiek, Planar Quantum Mechanics: an Intriguing Supersymmetric
Example , JHEP 0601 (2006) 156 hep-th/0512301. G. Veneziano and J. Wosiek, Large N,
Supersymmetry...and QCD hep-th/0603045. E. Onofri, G. Veneziano and J. Wosiek, Su-
persymmetryandCombinatoricshep-th/0603082.G.VenezianoandJ.Wosiek,Asupersym-
metric matrix model : II. Exploring higher fermion-number sectors JHEP 10 (2006) 033
hep-th/0607198.G.VenezianoandJ.Wosiek,Asupersymmetric matrixmodel : III. Hidden
SUSY in statistical systems JHEP 11 (2006) 030, hep-th/0609210. J. Wosiek Solving some
gauge systems at infinite N hep-th/0610172.
[8] R. DePietri, S. Moriand E .Onofri,The planar spectrum in U(N)-invariant quantum me-
chanics by Fock space methods: I. The bosonic case,JHEP01(2007) 018(hep-th/0610045).
[9] M.Trzetrzelewski,LargeN behavioroftwodimensionalsupersymmetricYang-Millsquantum
mechanicshep-th/0608147.
[10] However several methods are well known for local potentials, see for instance E. Brezin,
C. Iztykson, G. Parisi, J.B. Zuber, Comm. Math. Phys. 59 (1978) 35. G. Marchesini and
E. Onofri, Planar limit for SU(N) symmetric quantum dynamical systems, J. Math. Phys.
21 (1980) 1103. M.B. Halpern and C. Schwartz, Phys.Rev. D24 (1981) 2146. C.M. Canali,
G.M. Cicuta, L. Molinari, E. Montaldi, The quantum mechanical planar propagator, from
vectormodels to planar graphs, Nucl.Phys.B 265(1986)485.
[11] Somereaders mayappreciate understanding themodel without need ofextensive numerical
work.
[12] V.Bargmann,Ann.ofMath.,48(3)568–640(1947).
[13] M.B.Halpern and C.B.Thorn, Large N Matrix Mechanics on the Light-ConeInt. J. Mod.
Phys.A17(2002) 1517,hep-th/0111280.