We present a nonperturbative framework for collective field theory. Two methods are given. In the first exact Fock spac
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English Pages 24 Year 1992
NUCLEAR
P H Y S I CS B
Nuclear Physics B 376 (1992) 75—98 NorthHolland
Nonperturbative collective field theory
*
Antal Jevicki Department of Physics, Brown University, Providence, RI 02912, USA Received 1 July 1991 (Revised 23 December 1991) Accepted for publication 30 December 1991
We present a nonperturbative framework for collective field theory. Two methods are given. In the first exact Fock space states diagonalizing the hamiltonian are described. Based on these, effective lagrangians are introduced. An extra singleparticle branch is pointed out. It is shown that the states of this new branch correspond to solitons, the semiclassical soliton solutions are seen to take the form of a single eigenvalue effect. In general we argue that single (complex) eigenvalues will lead to solitons while real eigenvalues give instanton configurations.
1. Introduction There has been significant progress in understanding of lowerdimensional string theories [1—5].At present the theory with a most interesting physical structure is that for D 1 which represents c 1 matter coupled to twodimensional gravity [3,5,9]. It has a nontrivial physical spectrum consisting of a massless tachyon and possibly other states. The collective theory of refs. [3,4] offers a field theoretic framework for onedimensional strings. It is based on a scalar field but contains structure which could possibly give a complete theory of closed strings. Its properties have been investigated in some detail recently [6—11].Perturbative calculations were performed [7,81 giving expressions for scattering amplitudes. These are seen [71to contain poles exhibiting an infinite sequence of discrete states [13,14]. It is clear that there are very interesting physical phenomena which have nonperturbative origin [15,16]. One has, furthermore, the possibility of under=
*
=
Work supported in part by the Department of Energy under contract DEACO276ER03 130Task A.
05503213/92/$05.00 © 1992
—
Elsevier Science Publishers B.V. All rights reserved
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Nonperturbative collective field theory
standing black holes and their quantum meaning in the onedimensional theory [17—19]. Of major interest are nontrivial classical solutions of collective theory [8—101 and the structure of exact physical states. There is the possibility that this theory contains the complete dynamical effects of all string degrees of freedom and this issue has to be investigated. In this paper we begin the study of nonperturbative phenomena in collective field theory. In general the D 1 string theory corresponds to a —x2 (inverted oscillator) potential which induces a spatially noninvariant background field ~ 0(x). One of the interesting properties which was shown in ref. [9] is that a backgroundindependent formulation is possible. It is given by the purely cubic hamiltonian, the oscillator background can be induced through a reparametrization [9]. It is of interest, therefore, to study the general features of the purely cubic hamiltonian. The theory with constant background will serve as a laboratory where many of the interesting phenomena can be easily studied in exact terms. Among them is the question of the completness of the collective representation [19—211. We first describe the exact Fock space spectrum of states for the cubic theory in constant background. These states are most easily obtained by rescaling the relationship of the collective theory to the original matrix model. In this case the corresponding matrix hamiltonian is just a laplacian on infinitedimensional matrices. The Fock space states come in exact correspondence with the original matrix model states and are given by the character polynomials of U(N) [23,241.One of the first basic questions that one can answer then concerns the exact collective hamiltonian and the question of extra higher h terms. It will follow from the above analysis that the cubic hamiltonian gives a complete theory. We use the exact Fock space states to point out that they describe two branches of single particle states. Their energies are split exhibiting a 1/N (rather than 2) correction. 1/N We proceed to describe the same phenomena in semiclassical terms. This leads =
to the notion of the effective lagrangians. Here explicit higherorder terms are introduced and the degeneracy is split by them. The second branch of states is now shown to correspond to classical solutions with finite energy: solitons. We analytically construct the timedependent soliton configurations. We therefore exhibit the following basic phenomena. The cubic collective theory shows nonperturbative features when exactly solved while in the effective theory these are exhibited through a semiclassical analysis. The nontrivial soliton solutions are seen to take a form of a singleeigenvalue effect. This can be generalized to nontrivial backgrounds and therefore D 1 string theory. We argue that complex eigenvalues give solitons while real eigenvalue solutions lead to tunnelling solutions and instantons. The content of the paper is as follows. In sect. 2 we describe the exact Fock space states of the purely cubic theory. In sect. 3 the effective lagrangians are =
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77
discussed. In sect. 4 soliton solutions are their physical interpretation is given. Sect. 5 describes general singleeigenvalue dynamics and the corresponding instanton solutions.
2. Fock states As we have explained in the introduction we are motivated to study the cubic collective field theory H [~(x),
fdx (~11~411~ + 1~.2~3
—
11(y)] =i~(xy).
(1)
The Lagrange multiplier ~ induces a constant background =
=
kF,
representing the Fermi momentum. After the shift
4
=
çb0 +
ij
the hamiltonian
separates into a free and interaction part. H
=
H0
+
H3, 2s12),
Hofdx ~0([I~+1T H 3=fdx(~H~~H~+ ~
(2)
It is useful to introduce the linear combinations a±(x) =H~±i~n(x), [a~(x),
~~(y)]
=
[a±(x),
a(y)]
=0,
(3)
in terms of which Ho=~cbofdx(a~.+ct~),
H3(l/l2Tr)fdx (a~a~).
(4)
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In this representation it becomes obvious that the free and interaction term commute with each other [H0, H3] =0.
(5)
This implies that they are simultaneously diagonalizable. The basic feature of the problem is that the eigenstates of H0 are degenerate; linear combinations are then formed which diagonalize H3. It is easy to describe these in explicit terms, they follow from the connection to the matrix model. In the present case the matrix hamiltonian is just a laplacian on large (oo X Go) matrices (6) M denotes the matrix variable M= V Diag(A~)V’.
(7)
The collective variables are the invariants (traces) given by N
=
Tr(eIkM) dk
=
i= ~ 1 e~
(8)
In a box with k~ 2~rn/Lwe have =
(9) with
2nU=exp i_jj_M
.
(10)
The matrixmodel laplacian is then a laplacian on U(oo)
2ir HM=
2
~
E~=Tr(taU~j).
(11)
It is well known that the invariant eigenstates of the laplacian are the characters.
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79
Let ~ ~ /3,.. .} denote a Young tableau with 11 boxes in the first row, ~2 in the second, /3 in the third etc. One can easily express the lowest characters in terms of traces which are our collective variables. We list the characters for the lowest Young tableaux
x1~,
=
Tr U, 2+Tr U) ~[(Tr U)2 Tr
x12121(Tr X{1,1}
—
u21, U2],
3+2Tr U3+3Tr U Tr
~
U2],
131=~{(TrU) X{2.1)
Tr U~1,
=
~[(TrU)3
=
~[(TrU)3 + 2 Tr U3 —3 Tr U Tr
—
X{4} = ~[(Tr U)4
+
6 Tr U4
3(Tr
+
U2j,
U2)2
+6Tr U2(Tr U)2+8Tr U3 Tr
X{3,1}
=
~F(TrU)4 —2 Tr U4
X{2,2)
=
~ [Tr U)4 —4 Tr U3 Tr U + 3(Tr
U2)2j,
X(2,1,1}
=
~[(TrU)4 + 2 Tr U4
2 Tr U2 (Tr
=
~{(Tr U)
4
X 1i,i,i,i1
—
—
—
6 Tr U4
(Tr
(Tr
+
U2)2 +
U2)2
3(Tr
—
2 Tr U2 (Tr
U)21,
U)2],
U2)2
—6Tr U2 (Tr U)2+8Tr U3 Tr
The simplest way to generate the characters in invariant form is actually the collective method. They are formed from linear combinations of polynomials in çb~ ~flI~~fl2
~~/:1fl.
(12)
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One considers the subspace of polynomials of given order n1 + n2 The laplacian acts as a differential operator and
+ ... +flk
EaEafl(fl,fl’)~_~f+W(fl)~f,
=
(13)
with
(Ea43n)(Ea4~n~) =nn’4~~_~~
Q(n, n’)
(14)
n—(n)
~
~
(15)
~
n’=e(n)
These formulas follow by simple matrix differentiation, they are also the basic ingredients in deriving the collective hamiltonian (see ref. [3]). Acting on a general polynomial of order h we have
k
+ ~ntL
+
fl
~
~ n~n1~ ~
. . .
. . .
. . . ~.
..
..
.
~flk)~
(16)
0 n ,m
=
(a~ (a~)2)jo>,
{2, 1)>
=
(a~ (a~)3)10>,


{‘~1, i~>= (2a~+(a~)33a~(a~))Io>,
~
(20)
One can easily convince oneself that these are indeed exact eigenstates. The eigenvalues are
~ E 2~—2N+
2,
E3~
3N + 6,
30
3N,
~
=
3N— 6,
2. In fact for a Fock state corresponding to a where denote = ~(2~r/L) Young we tableau Y= E{1~, 12, 13,. . } one will have the eigenvalues
{/~,‘2,
/3,...)
=
1 1(N
—
1)
+
l~+ 12(N —3)
+
/~+ 13(N —5)
+
/~+
.
One can give a general proof that the characters give Fock eigenstates. It follows from a direct correspondence between the action of the collective hamilto
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Nonperturbative collectivefield theory
nian on Fock space and the action of the U(N) laplacian on traces of U(N) matrices. We have already explained the procedure for the laplacian. Consider now the Fock space. The eigenstates of H0 are degenerate. The states
(21) with fixed total momentum (22)
fll+fl2+...+flkfl,
all have the same eigenvalue (23) H3 acts on the above states as
H3(a~a~...a~k)lo>= ~n1~(ã~
...
+ ~
(24)
i t,~I~g~a~n.
(31)
A general ‘rfunction is r(401, 402,...;g) =(0I eL~~I0),
(32)
with H(40)=
(33)
~—40j~(II~tff~+j.
I~0
n
It was shown in ref. [251 that characters are given by rfunction. Consequently Tfunctions are a natural arena for states of collective field theory. We give now a conformal field theory representation of the states. Since the free collective theory describes a massless scalar one can represent it in terms of a conformal field ~(z), ~(z) related to ~ and a.... A transition from a strip to the complex plane is given by 2n~ z=exp i with r = tensor
—
it being the euclidean time. The components of the energy—momentum 2
2
T=(d~40) ~(2’~40) T=
(040)2
define Virasoro generators L71, L71. Denote the primary fields by &o
~—i’P’
~oi
~—i’P~
We note that a cubic perturbation of the form + (~2~o1)}
was studied recently in ref. [26] in connection with the tifunction Bose gas. The lowlying eigenstates [22] which read
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Nonperturbative collective field theory
(L2
+
85
0),
~
.1. —
(L3
(
2’~~—i~Poi
0),
+ ~(~2~1o))
1~)~
ID),
+ ~(~2~1o)) 2(~ —
3!7~.Z
29’OI
are in onetoone correspondence with the Fock space states given above (eq. (20)). In conclusion the main message which we want to draw from the above analysis is twofold. First of all we observe an agreement of the cubic theory with the states of the matrix model itself and also with the Nfermion problem. More importantly the analysis of exact states provides us with the following physical picture: one sees two branches of singleparticle states that follow from a single boson by degenerate perturbation theory. In sect. 3 we will discuss the implications of this phenomenon.
3. Effective lagrangians In sect. 2 we have background. We now We have seen that branches. The first is
studied in exact terms the collective field theory is constant zero in on some interesting features of this solution. the exact analysis of the theory produces two singleparticle given by
(34)
~
with the energy 2 Ni
~
2 2
—
2 N—i
(35)
‘
and the second
I~m)
Qm(~,
a~,...)Io)=
~~dzz~1
ex~{E ~71atnzn}~
(36)
with N+12 2
N—i 2 —m
2 .
(37)
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This is unusual. In standard field theory we expect a singleparticle branch associated with the field itself. Here it is the dispersion of the first kind that would naturally correspond to the collective field. The meaning of the new set of states will be given latter. The appearance of two singleparticle branches is natural in terms of the fermions. One has a Fermi ground state with momenta between —kF ~ k ~ k~all filled. There are positive and negative momentum excitations coming from the vicinity of the upper (+kF) and the lower (—kF) Fermi surface respectively. Consider the upper surface for example (that is what we have concentrated on in the above collective analysis). A single particle excitation is obtained by moving a state from the top of the Fermi surface (+kF) to a higher level kF + k. The in agreement with (35). A second excitation energy is w(k) = ~(kF + k)2 ~ singleparticle excitation [27] is obtained by moving a state from below the Fermi surface (at momentum kF k) to the Fermi surface. The energy of this (hole) state is ~(k)= ~(kF k)2 in agreement with (37). Note than that in general this second branch tests the structure from inside the Fermi surface. Returning to collective field theory in the continuum notation we have —
—
—
—
I~(k))
=
(a~+ ljkdkf
a~a~k+ .,.)
ID),
w(k)=kFIkl+~k2, —Go