Large N Matrix Mechanics on the Light-Cone

23 Pages·0.23 MB·

by  M. B. Halpern

LBNL-49194
UCB-PTH-01/43
UFIFT-HEP-01-23
hep-th/0111280
Large N Matrix Mechanics on the Light-Cone
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J M. B. Halpern∗ and Charles B. Thorn†‡

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Department of Physics, University of California, Berkeley CA 94720

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Theoretical Physics Group, Lawrence Berkeley National Laboratory
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2 University of California, Berkeley CA 94720
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/ Abstract
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- We report a simplification in the large N matrix mechanics of light-cone matrix field theories.
p
The absence of pure creation or pure annihilation terms in the Hamiltonian formulation of
e
h these theories allows us to find their reduced large N Hamiltonians as explicit functions of the
: generators of the Cuntz algebra. This opens up a free-algebraic playground of new reduced
v
models – all of which exhibit new hidden conserved quantities at large N and all of whose
i
X eigenvalue problems are surprisingly simple. The basic tool we develop for the study of these
r models is the infinite dimensional algebra of all normal-ordered products of Cuntz operators,
a
and this algebra also leads us to a special number-conserving subset of these models, each of
which exhibits an infinite number of new hidden conserved quantities at large N.
∗E-mail address: [email protected]
†E-mail address: [email protected]
‡Visiting Miller Research Professor, on sabbatical leave from the Department of Physics, University of Florida,
Gainesville FL 32611.
1 Introduction
Light-conequantumfieldtheories[1–5]havebeenstudiedsince1949andlargeN methods[6,7]since
1974. See Refs.[8,9] for more complete referencing of the two developments. The idea of combining
light-cone quantization with large N was pursued in [6,10] in the context of summing planar
Feynman diagrams, and in [11,12] in the context of solving the large N energy eigenvalue problem
in the color singlet sector of the light-cone Fock space. Here we study a different combination of
the two developments–which leads to an algebraic reformulation of the light-cone theory at large
N.
In this paper we will focus on large N phase space methods [13–16] and in particular on the
developmentknownaslarge N matrix mechanics[16–18], [9], [19,20]. InlargeN matrixmechanics,
theHilbertspaceisspannedbythesingletgroundstate andthedominantadjointeigenstates which
saturate the traces of the theory at large N, and the output is the so-called reduced formulation for
both large N action [19] and Hamiltonian [9,19] theories in terms of free algebras. The simplest
free algebra found at large N is the Cuntz algebra, whose history in mathematics and physics can
be found in Ref. [9]. In fact, a broad variety of free algebras has been observed [16], [9,19] in large
N matrix mechanics, including the Cuntz algebra, the symmetric Cuntz algebra, the interacting
symmetric Cuntz algebras and infinite dimensional free algebras, as well as fermionic versions and
Cuntz superalgebras. Moreover, the algebraic formulation has led to the discovery of new hidden
conserved quantities [9] which appear only at large N, as well as various free-algebraic forms of the
master field (see also Ref.[21]), and a numerical approach [20] to computations in many- matrix
models at large N.
But there has always been an enigma, called the opacity phenomenon in Ref.[9], in the large
N matrix mechanics of equal-time quantized Hamiltonian formulations: Except for oscillators [9]
and one matrix models [17], the reduced free-algebraic Hamiltonian of a given large N system is
difficult to obtain in closed form.
We report here an important simplification in the large N matrix mechanics of discretized
light-cone matrix quantum field theories, including light cone models which are both local and
non-local in x−. The non-local models, including for example the number-conserving truncations
of local models, can be considered as quantum-mechanical models in their own right. Because of
the well-known absence of pure annihilation and purecreation terms in any light-cone Hamiltonian
H = P.− we are able to obtain the explicit form of the large N reduced Hamiltonian entirely in
·
terms of Cuntz operators.
This opens up a free-algebraic playground of new reduced models, all of which exhibit an unex-
pectedly simple algebraic structure. In particular, each of these models has new hidden conserved
quantities at large N and the free-algebraic eigenvalue problems of the reduced Hamiltonians are
surprisingly simple.
The basic tool which we develop for the study of these models is the infinite dimensional
algebra of all normal-ordered Cuntz operators, and this algebra also leads us to a special subset of
the number-conserving models which apparently becomes integrable at large N. In particular, each
of these models exhibits an infinite number of new hidden conserved quantities at large N, and
moreover the eigenvalue problems of these models exhibit oscillator-like spectra.
2 Light-cone Matrix Quantum Field Theories
Webeginwithsomegeneralremarksaboutlight-conequantizationofmatrixfieldtheories,deferring
consideration of large N matrix mechanics to Section 3.
1
2.1 The Light-Cone Field
Thematrixfieldtheoryofmostobviousphysicalinterestisnon-abeliangaugetheory. Buttosimply
illustratethemainideas,weshallfocushereonamuchsimplerexample,thehermitianscalarN N
×
matrixfieldφ intwospace-timedimensions. Wedefinelight-frontcoordinatesx± = (x0 x3)/√2,
rs
±
choosingx+ astime,andreferringthex− coordinatetoitsconjugatemomentumlabelledbyp+ > 0.
The field φ then has the expansion
rs
∞ dp+
φ (x−,x+) = a(p+,x+) e−ix−p+ +a†(p+,x+) e+ix−p+ (1)
rs rs rs
Z0+ 4πp+ (cid:16) (cid:17)
with r,s = 1,...,N. The quantumpnature of the field is fixed by imposing the equal time (equal
x+) commutation relations
a(p+) ,a†(q+) = δ δ δ(p+ q+)
rs tu st ru
−
h a(p+) ,a(q+) i = a†(p+) ,a†(q+) = 0. (2)
rs tu rs tu
h i
In the following we shall s(cid:2)uppress the x+ a(cid:3)rgument in expressions at equal time. A simple conse-
quence of these commutation relations is the well-known equal time light-cone field commutation
rule
i
φ (x−),φ (y−) = δ δ ǫ(y− x−) (3)
rs tu st ru
4 −
(cid:2) (cid:3)
whereǫ(x) = x/x,thesignofx. Inparticular,thefieldsatdifferentspatialpointsdonotcommute,
| |
even at equal time.
In our discussion we findit convenient to deal with discretely labelled operators so we discretize
p+ = mδ, m = 1,2,.... Putting a(mδ) a /√δ, the commutation relations then read
m
→
(a ) ,(a†) = δ δ δ
m rs n tu mn st ru
h i
[(a ) ,(a ) ] = (a† ) ,(a†) = 0 (4)
m rs n tu m rs n tu
(a ) 0. = h.0(a† ) =0i (5)
m rs| i h | m rs
where 0. is the Fock vacuum, and the expansion of the field reads
| i
∞
1
φ (x−) = (a ) e−ix−mδ +(a† ) e+ix−mδ (6)
rs √4πm m rs m rs
mX=1 (cid:16) (cid:17)
K
1
(a ) e−ix−mδ +(a† ) e+ix−mδ (7)
→ √4πm m rs m rs
mX=1 (cid:16) (cid:17)
where K is a large integer serving as a cutoff on the high p+ modes. With m an integer, we should
restrict π x−δ π. The expansion (7) can be inverted
− ≤ ≤
m π/δ
(a ) = δ e+ix−mδφ (x−). (8)
m rs rs
π
r Z−π/δ
We also need the contraction factor
dp+ ∞ 1 K 1
L (9)
4πp+ → 4πm → ≡ 4πm
Z m=1 m=1
X X
which arises in the process of normal ordering.
2
2.2 Partial Normal Ordering
For the purposes of this paper the most important feature of light-cone coordinates is that the
creation operators a†(p+) always create particles of positive p+. As seen in Eq. (1), we will assume
that p+ = 0 is excluded, either by a small low momentum cutoff ǫ, or in the context of discretized
p+ = mδ, by the exclusion of m = 0. Only annihilation operators carry negative p+. Therefore the
onlyp+ conservingtermsthatcanappearintheHamiltoniancontainbothcreationandannihilation
operators, i.e. there can be no terms Tr an or Tr a†n, n > 0. As we shall see, this feature is
responsible for a dramatic simplification in the free-algebraic description of the large N limit.
Since the creation and annihilation operators are N N matrices, as well as operators in
×
Hilbert space, the process of normal ordering each term in the Hamiltonian can lead to awkward
color descriptions. For example, the normal-ordered form of Tra†aa†a would be
Tr :a†aa†a := a† a† a a . (10)
rs tu st ur
Here there is a clash between the ordering for matrix multiplication and the ordering for Hilbert
space operator multiplication on the r.h.s. Thus we refrain from completely normal ordering such
terms.
But we can partially normal order the operators in a trace, so there is always at least one
annihilation operator on the extreme right or at least one creation operator on the extreme left
without disturbingthe coincidence of matrix ordering with quantum mechanical operator ordering.
† † † †
Forexample, apartially normal-orderedformofTra a a a wouldbeTra a a a . Suchreorderings
1 2 3 4 4 1 2 3
preserve the cyclic ordering of the operators within each trace. Note that in general a partially
† † † †
normal-ordered form is not unique, for example Tra a a a = Tra a a a although the operators
4 1 2 3 6 2 3 4 1
are each in the same cyclic order. We shall see in Secs. 4 and 5, however, that the non-uniqueness
disappears for N . Of course, in field theory the Hamiltonian is typically presented in a
→ ∞
(local) form which is not even partially normal-ordered. But by exploiting the known commutation
relations,wecanalwaysrearrangetheoperatorsinthepartiallynormal-orderedformjustdescribed.
Note that if there is at least one creation and one annihilation operator in a trace, partial normal
ordering will lead to a generic structure
Tr(a† A a ) (11)
m mn n
m,n
X
where A is any matrix function of a ,a†, not necessarily itself normal-ordered.
p q
As a concrete example we consider the U(N)-invariant light-cone Hamiltonian
µ2 λ
H. = P− = dx−Trφ2+ dx−Trφ4 (12)
· 2 4N
Z Z
of a local two dimensional scalar field theory. Then the commutation relations (4) are used to
obtain the partially normal-ordered form
K K
H. = E + Tra† A a + Tr(a† )B Tr(a ). (13)
0 m mn n m mn n
m,n=1 m,n=1
X X
The explicit forms of the quantities appearing here are
2π λL 1
E = [L2(N2+1)λ/4+N2L/2] , B =δ (14)
0 mn mn
δ N 2mδ
3
1
A = C +D a† a† +E a a +F a† a + a a† (15)
mn mn mnm2m3 m2 m3 mnm2m3 m2 m3 mnm2m3 m2 m3 2 m3 m2
(cid:18) (cid:19)
µ2 3λL λ 1
C = 1+ δ , D = δ (16)
mn 2mδ 2µ2 mn mnpq 8πδN √mpqn m+p+q,n
(cid:18) (cid:19)
λ 1 λ 1
E = δ , F = δ (17)
mnpq m,p+q+n mnpq m+p,q+n
8πδN √mpqn 8πδN √mpqn
where repeated indices are always summed. Since our Hamiltonian is now in partially normal-
ordered form, the Fock vacuum 0. is manifestly an energy eigenstate with
| i
H.0. = E 0. .
0
| i | i
In the above computation we have retained all the commutator terms arising from reordering the
a’s and a†’s of the original Hamiltonian. Because our starting Hamiltonian was at most quartic in
the fields the only such terms are c-numbers and quadratic. Furthermore, because of our treatment
of p+, there are never reordering terms with only annihilation or only creation operators. However
the reordering does alter the original color structure in the sense that we started with only a single
trace and ended up with reordering terms involving two traces. Partial normal ordering of the
general light-cone Hamiltonian is discussed in Sec. 5.
3 Reduced momentum and reduced number operator
Inthis section webegin our studyof thelarge N matrix mechanics [16–18], [9], [19,20] of light-cone
field theory, drawing heavily on the methods and results of Ref.[9]
We start with the simpler problems of the momentum operator P. P+ and the number
·
≡
operator Nˆ.
P.= P† mTra† a , Nˆ. = Nˆ.† Tra† a (18)
· ≡ m m ≡ m m
m m
X X
a a a† a†
P., m = m m , P., m = m m (19)
(cid:20) (cid:18)√N(cid:19)rs(cid:21) − (cid:18)√N(cid:19)rs " √N!rs# √N!rs
a a a† a†
Nˆ., m = m , Nˆ., m = m (20)
(cid:20) (cid:18)√N(cid:19)rs(cid:21) −(cid:18)√N(cid:19)rs " √N!rs# √N!rs
P.0. = Nˆ.0. = .0P. = .0Nˆ.= 0 (21)
| i | i h | h |
which are two examples of so-called trace class operators in the unreduced theory. In large N
matrix mechanics one then takes matrix elements of the commutation relations above, using only
the ground state 0. and the dominant time-independent adjoint eigenstates rs,A which saturate
| i | i
thetracesofthetheoryatlargeN.ThelargeN completenessrelationforthesestatesreads(t = x+)
d d
1.= 0. .0 + rs,A rs,A, 0. = rs,A = 0 (22)
N | ih | | ih | dt| i dt| i
rs,A
X
because large N factorization guarantees that the singlet ground state will be sufficient to saturate
the singlet channels of the traces.
4
Next one introduces reduced matrix elements§, for example
ρ (t ) ρ (t )
1 1 n n
pq,A st,B = P Aρ (t ) ρ (t )B (23)
qp,rt 1 1 n n
h | √N ··· √N | i h | ··· | i
(cid:18) (cid:19)rs
ρ (t ) ρ (t )
1 1 n n
.0Tr 0. = N 0ρ (t ) ρ (t )0 (24)
1 1 n n
h | √N ··· √N | i h | ··· | i
(cid:18) (cid:19)
1
P =δ δ δ δ , ρ = a or a† (25)
sr,pq rp sq sr pq
− N
where t = x+ and P is a Clebsch-Gordan coefficient. Matrix products (ρ ρ ) are called den-
i i 1··· n rs
sities in the unreduced theory. These “Wigner-Eckart” relations then define the reduced operators
a and a† as well as the reduced ground state 0 and the reduced adjoint eigenstates A . The
m m | i | i
reduced eigenstates satisfy the reduced completeness relation
d d
1 = 0 0 + A A, 0 = A = 0 (26)
| ih | | ih | dt| i dt| i
A
X
at large N.
The reduced forms of Eqs. (19) and (20) are
[P,a ]= ma , P,a† = ma† (27)
m − m m m
Nˆ,a = a , hNˆ,a†i = a† . (28)
m − m m m
h i h i
Here P and Nˆ are the reduced momentum operator and the reduced number operator respectively,
and the reduced operators a and a† satisfy the Cuntz algebra
m m
a a† = δ , a† a = 1 0 0 (29)
m n mn m m −| ih |
m
X
a 0 = 0a† = 0 (30)
m| i h | m
where 0 is the reduced ground state of the system. The Cuntz algebra is a simple example of a
| i
so-called free algebra, defined by products instead of commutators, and this free algebra is known
to describe classical or Boltzmann statistics, another classical aspect of the large N limit.
In fact the Cuntz algebra is a subalgebra of the so-called symmetric Cuntz algebra [9] which
includes the following further operators and relations
a˜ a˜† = δ , a˜† a˜ = 1 0 0 (31)
m n mn m m −| ih |
m
X
a˜ ,a† = a ,a˜† = δ 0 0, [a ,a˜ ] = [a† ,a˜†] =0 (32)
m n m n mn| ih | m n m n
h a˜ 0i = h0a˜† =i0, a˜† 0 = a† 0 , 0a˜ = 0a (33)
m| i h | m m| i m| i h | m h | m
Moreover, the tilde operators a˜ , a˜† are important in deriving the Cuntz algebra (29) from the
m m
unreduced theory: The reduction procedure (see Eq. (2.29) of Ref. [9]) gives only the relations in
Eq. (32) as a consequence of the unreduced commutators (4), and a completeness argument (see
Eq. (3.7) of Ref. [9]) on the general basis state is necessary to obtain the Cuntz algebra from (32)
and the ground state conditions. Beyond this application, the tilde operators are not used in this
paper.
§Reduced matrix elements were introduced by K.Bardakci in Ref.[14].
5
Thereduction proceduredoesnotgive usdirectly thecompositestructureof reducedtraceclass
operators such as P, Nˆ or the reduced Hamiltonian H. This is called the opacity phenomenon in
Ref.[9]. But one can in principle find the composite structure of such operators by solving reduced
commutation relations such as Eqs.(27) and (28).
In fact, the reduced commutators (27) and (28) are so simple that standard methods [9] suffice
to determine the form of the reduced operators P and Nˆ. To express these solutions succinctly, it
is convenient to introduce a standard tool called word notation:
a a a , w = m m (34)
w ≡ m1··· mn 1··· n
a † a† a† (35)
w ≡ mn··· m1
n
[w] n, w m . (36)
i
≡ { } ≡
i=1
X
Here w is an arbitrary word composed of some number of ordered letters, which label the lattice
sites in this case. In (36) we have also defined the length [w] of the word w and its weight w ,
{ }
which is the sum of its letters.
Then the reduced momentum and reduced number operators can be written as
P = P† = a † ma† a a
w m m w
w m !
X X
= ma† a + a† ma† a a + (37)
m m p m m! p ···
m p m
X X X
Nˆ = Nˆ† = a † a† a a
w m m w
!
w m
X X
= a† a + a† a† a a + (38)
m m m p p m m m! p ···
X X X
P 0 = Nˆ 0 = 0P = 0Nˆ = 0 (39)
| i | i h | h |
where means the sum over all words w, including the trivial word w = 0. To check that
w
these operators satisfy the commutation relations (27) and (28), write them out, e.g. [P,a ] =
P m
Pa a P, and use the Cuntz algebra (29) to cancel all but one of the terms in the difference.
m m
−
Later we shall discuss more sophisticated ways to handle operators of this type.
4 Reduced Hamiltonian
The reduced momentum and number operators discussed above are very simple examples, but the
opacity phenomenonhas beenparticularly vexing in thecase of models describedby theequal-time
quantized Hamiltonian
1 φ
H.= Tr πmπm+NV . (40)
2 √N
(cid:18) (cid:18) (cid:19)(cid:19)
For such systems one knows the reduced equations of motion [9,16]
φ˙ = i[H,φ ]= π , π˙ = i[H,π ]= V (φ) (41)
m m m m m m
−
ρ(t) = eiHtρ(0)e−iHt, ρ = φ or π (42)
m m
6
where H is the reduced Hamiltonian, and one knows the interacting Cuntz algebras [9,19]. But
beyond the case of matrix oscillators [9] and the general one-matrix model [17], the explicit form
of the reduced Hamiltonian H of such systems is not known (an implicit construction of H is given
in Subsec. 4.6 of Ref. [9]).
For quantized matrix theories of the light-cone type, i.e. with no pure creation or pure annihila-
tion terms in H., we shall see that we can in fact find the explicit form of the reduced Hamiltonian.
As an example we begin with a general unreduced quartic Hamiltonian of the light-cone type
1 a† a† a a 1 a† 1 a
H. E = N2 Tr m A , n + Tr m B Tr n (43)
0 mn mn
− (N m,n √N √N √N! √N! N √N! (cid:18)N √N(cid:19))
X
a† a C D C
A , = C + mnpa† + mnpa + mnpqa†a†
mn √N √N!rs (cid:18) mn √N p √N p N p q
D E F
+ mnpqa†a + mnpqa a† + mnpqa a (44)
N p q N p q N p q
(cid:19)rs
C = C∗ , D∗ = C , C∗ = F
mn nm mnp nmp mnpq nmqp
D∗ = D , E∗ = E , B∗ = B . (45)
mnpq nmqp mnpq nmqp mn nm
Here the quantities B,C,D,E,F are constants and the relations in (45) guarantee A† = A
mn nm
and hence H† = H.. In what follows, we often refer to the form A in (43) as the kernel of the
· mn
Hamiltonian. We emphasize that the general quartic form (43) includes the local quartic example
in (12) and (13) as well as many light-cone models which are non-local in x− (use Eq. (8)). The
non-local models, including for example the number-conserving truncations of local models, can be
considered as quantum-mechanical models in their own right.
Following the standard procedurein large N matrix mechanics, we compute the time derivative
of the annihilation operators
a˙ a
m m
= i H., = (G + + ) , (46)
√N √N m Em Fm rs
(cid:18) (cid:19)rs (cid:20) (cid:18) (cid:19)rs(cid:21)
A a
mn n
(G ) i (47)
m rs ≡ − √N
n (cid:18) (cid:19)rs
X
a† a a
p m q
( ) i ,(A ) (48)
Em rs ≡ − p,q √N!tu(cid:20)(cid:18)√N(cid:19)rs pq uv(cid:21)(cid:18)√N(cid:19)vt
X
1 a
q
( ) iδ B Tr . (49)
Fm rs ≡ − rs mqN √N
(cid:18) (cid:19)
The term called comes from “opening” a trace in the double trace term of the Hamiltonian.
m
F
To understand this term, we need the large N trace theorem [9,17]
1 a† a
Trf , = 0f a†,a 0 , f (50)
N √N √N! N h | | i ∀
(cid:16) (cid:17)
which gives the leading term of the trace at large N in terms of the reduced average on the right.
This implies in particular that
1 a† a 1 a† a
Tr a† f , = Tr g , a = 0 (51)
N m m √N √N!! N N m √N √N! m!
7
and so the term fails to contribute to the large N equation of motion.
m
F
The term called collects the contribution of all creation operators except the first in the
m
E
a†Aa term of the Hamiltonian. By direct computation, we find the explicit form of this term
1 a† a
p q
i( ) = C
Em rs N " pqm √N!ts(cid:18)√N(cid:19)rt
a† a† a a† a† a
+C p x q +C p x q
pqmx pqxm
√N!ts √N √N!rt √N √N!ts(cid:18)√N(cid:19)rt
a† a a a† a a
p x q p x q
+D +E . (52)
pqmx pqxm
√N!ts(cid:18)√N √N(cid:19)rt √N √N!ts(cid:18)√N(cid:19)rt#
Each of these terms has the form
V = E F ,
rs ts rt
t
X
called a “twisted density” in Ref.[9]. Twisted densities can be reduced directly in terms of the
tilde operators in Eq. (33), but we prefer to express the result first in terms of untwisted densities
(ABCD) . Thus we commute the factors into the form F E (FE) , keeping all correction
rs rt ts rs
→
terms.
The result of this algebra is
1 a a†
q p
i( ) = C C δ
Em rs N " pqm √N √N! − ppm rs!
rs
a† a a† a†
+ C x q p C x
pqmx ppmx
√N √N √N! − √N! !
rs rs
a a† a† a† 1 a†
+ C q p x C x C δ Tr p
pqxm ppxm pqqm rs
√N √N √N! − √N! − N √N!
rs rs
a a a† 1 a a
x q p q x
+ D D δ Tr D
pqmx pqmp rs ppmx
√N √N √N!rs− N √N − (cid:18)√N(cid:19)rs!
a a† a a
q p x q
+ E E (53)
pqxm ppqm
√N √N √N!ts− (cid:18)√N(cid:19)rs!#
where each term in parentheses corresponds to one of the twisted densities in Eq.(52). The two
trace terms in (53) can again be neglected by the large N trace theorem (51).
Now all surviving terms in the large N equation of motion are densities, so we may use reduced
matrix elements to obtain the reduced equation of motion
ia˙ = [H,a ] = A (a†,a)a +Extra (54)
m m mn n
Extra = C a a† C + C a†a a† C a† + C a a†a† C a†
pqm q p− ppm pqmx x q p− ppmx x pqxm q p x− ppxm x
+(cid:16) D a a a† (cid:17)D (cid:16) a + E a a†a (cid:17)E (cid:16) a (cid:17)(55)
pqmx x q p− ppmx x pqxm q p x− ppqm q
(cid:16) (cid:17) (cid:16) (cid:17)
where A (a†,a) is the reduced form of the kernel, given explicitly below. The parentheses here
mn
correspond to the parentheses of Eq.(53). But now we notice that each of the extra terms in
parentheses is separately zero, e.g.
E a a†a = E a (56)
pqxm q p x ppxm x
8
by the Cuntz algebra.
The results of our computation are the simple reduced equations of motion
a˙ = i[H,a ]= iA (a†,a)a
m m mn n
−
a˙† = i[H,a† ]= ia†A (a†,a) (57)
m m n nm
where H is the reduced Hamiltonian.
For such simple reduced equations of motion, standard methods again suffice to write down the
reduced Hamiltonian of the large N light-cone system
H E = a †( a† A (a†,a)a )a (58)
− 0 w m mn n w
w m,n
X X
= a† A (a†,a)a + a†( a† A (a†,a)a )a + (59)
m mn n p m mn n p ···
m,n p m,n
X X X
A (a†,a) = C +C a† +D a +C a†a†
mn mn mnp p mnp p mnpq p q
+D a†a +E a a† +F a a (60)
mnpq p q mnpq p q mnpq p q
(H E )0 = 0(H E )= 0 (61)
0 0
− | i h | −
where the form A (a†,a) is just the kernel we started with in the unreduced Hamiltonian (43).
mn
The reduced Hamiltonian is hermitian because A† = A . The Cuntz algebra also tells us that
mn nm
the E term of A can be further simplified
mn
E a a† = E (62)
mnpq p q mnpp
so that the kernel A can be considered as completely normal-ordered.
mn
Inthefollowingsectionwewillarguethattheresults(57)and(58)areinfacttheformsobtained
by the large N reduction of any light-cone system, where the reduced kernel A is always defined
mn
by the single trace term of the Hamiltonian. As a check on this conclusion, we note here that the
reduced equations of motion (57) are consistent for all A with the ground state conditions (30),
mn
for example
a˙ 0 = iA (a†,a)a 0 = 0. (63)
m mn n
| i − | i
Moreover the reduced equations of motion are consistent with the Cuntz algebra (29)
d d
a a† = a† a = 0 (64)
dt m n dt m m!
(cid:16) (cid:17) Xm
for all A . Note that the Cuntz algebra itself is necessary
mn
d
a a† = i( A (a†,a)a a† +a a†A (a†,a)) = 0 (65)
dt m n − mp p n m p pn
(cid:16) (cid:17)
to check the first of these relations from (57).
5 The General Reduced Light-Cone Hamiltonian
Now we extend this discussion to a general light-cone Hamiltonian. First consider a Hamiltonian,
not necessarily of the light-cone type, required to be a color singlet and behaving at large N no
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