Degrees of freedom in two dimensional string theory

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    P:I.SI%'II~:R Nuclear Physics B (Proc. Suppl.) 45B,C (1996) 224233

    Degrees of freedom in two dimensional string theory Sumit R. Das ~

    a Tara Institute of Fundamental Research, IIomi Bhabha Road, Bombay 400005, INDIA

    We discuss two issues regarding the question of degrees of freedom in two dimensional string theory. The fiIst issue relates to the classical limit of quantum string theory. In the classical limit one requires an infinite nmnber of fields wj(z, t) in addition to the standard collective field whenever a "fold" forms on the fermi surface of the underlying fermionic field theory. We show that in the quantum theory these are not additional degrees of freedom, but represent quantum dispersions of the collective field. These dispersions become O(1) rather than O(h) precisely after fold formation, thus giving additional classical quantities and leading to a nontrivial classical lirait. The second issue relates to the ultraviolet properties of the geometric entropy. We argue that the geometric entropy is finite in the ultraviolet due to nonperturbative effects. This indicates that the true degrees of freedom of the two dimensional string at high energies is much smaller than what one naively expects.

    1. In t roduct ion

    Recent developments seem to indicate that the true degrees of freedom in string theory are quite different from what is expected from perturba- tion theory. In fact we already know this from our experience with noncritical strings : the fun- demental formulation of two dimensional string theory is in terms of nonrelativistic fermions via matrix models and a stringy description in terms of the massless tachyon emerges as a low energy perturbative picture [1]. For higher dimensional strings we do not have a nonperturbative formu- lal;ion, but the remarkable duality properties in- di,:ate that string theory has to be formulated in terms of some yet unknown entities which are not the strings of perturbation theory.

    In this talk I will summarize some results which clarify the nature of degrees of freedom of the two dimensional string. This is a good labora- tory since some quantities can be calculated non- perturbatively. , even though the space-time in- terpretation of the theory is rather involved and not yet fully understood. The first set of results, obtained in collaboration with S. Mathur [2] indi- cate that the classical limit of the theory is rather nontrivial and requires more degrees of freedom than the quantum theory itself. The second set of results [3] indicate that a suitably defined geomet-

    ric entropy (or entropy of entanglement), which provides a measure of the degrees of freedom, is ultraviolet finite due to essentially nonperturba- tive effects This is relevant to the question of black hole entropy in string theory.

    2. Folds and the Classical l imit

    Two dimensional string theory is described by N mutually nointeracting nonrelativistic fermions in 1 + 1 dimensions in an external in- verted harmonic oscillator potential with the sec- ond quantized action in terms of fermi fields (x,t)

    Af=A/dt/dx~t[-1--~-02+ttc-~z2]~L 2A2 , (1)

    and A = v and g is the quartic coupling in the underlying matrix model. If # denotes the fermi level of the single particle hamiltonian in (1), the continuum string theory is described by the double scaling limit N ~ oo and # -~ #c with n = A(#c - #) = fixed.

    2.1. Classical Dynamics Let us first consider the classical dynamics of

    the fermion system in a manner which is indepen- dent of the form of the hamiltonian. The classi- cal state is specified by a distribution function u(x, p, t ) in phase space- in fact u(x, p, t) - - 1 -2-g

    0920-5632/961515.00 1996 Elsevier Science B.V. All rights reserved. SSDI: 0920-5632(95)00640-0

  • S.R. Das/Nuclear Physics B (Proc. Suppl.) 45B, C (1996) 224 233 225

    in some region bounded by the fermi surface and zero elsewhere. The fermi sea may be conve- niently paramterized in terms of two functions fl+ and an infinite set of additional fields w,~,+ [4]

    f dp u(x,p,t) - 1 2~h [fl+ - / L ]

    f dppu(x,p,t ) - 1 1 2

    -~(W.I_ 1 -- W_ l ) ]

    f dpp2 u(x ,p , t ) - 1 1 3

    +(/~+ ~+1 - fl- ~-1) - (2)

    and so on. Let a line of constant x in phase space cut the upper edge of the fermi sea at points pi(x,t) and the lower edges of the fermi sea at qi(x,t) where (i = 1,2, - , . im(x, t ) ) . Then

    @ = 1) (3) i=1

    i , , (x , t ) - 1 denotes the number of "folds" at the point x at t ime t. In the absence of folds, one can set fl+(x,t) = pl(x,t) and f l-(x,t) = ql(x,t) which then implies that all the w+,~ = 0. This is the standard bosonization in terms of the collec- tive field theory. As emphasized in [5] only in the absence of folds is the state described by a scalar field r/(x, t) and its momentum conjugate I I , (x, t), which are related to fl+ by fl+ = II~ =~ 0~/.

    In the deep asymptotic region, the string the- ory massless tachyon is a nonlocal transform of 7j(x, t) [6]. In the presence of folds we do not know how to extract the string theory space-time from the matrix model, since the w+.,~ 0 while in the far asymptotic region the collective field configu- rations alone exhaust the possible configurations of the string theory tachyon field 1

    As shown i n [4] the Poisson bracket algebra for fl+ is that of a chiral (antichiral) boson, while the w,~. commute with the fl+ and themselves satisfy a woo algebra, for the =t= components separately. In the following we will omit the + subscript.

    1 Away from the asymptot ic region higher moments of u(x,p,t) are required for determin ing the str ing theory tachyon. See Dhar et. al. in [6].

    We will also concentrate on free fermions. A background potential can be easily incorporated and is not essential to the main point.

    For free relativistic right moving fermions one has a hamiltonian H = f dx f dp p u(x, p, t). Tile Poisson brackets lead to the equations of motion

    0tfl = -a~f l 0tw.~ = -O~w.~ (4)

    For free nonrelativistic fermions one has H = f dx f dp 5Pl 2 u(x,p,t) and using the Poisson brackets one gets the evolution equations

    a, fl = - f lOxf l - O.wl

    Otwm = --2win &f l - - f i G~' .~-Gw.~+l (5)

    The fl, wn are expressible in terms of the (pi, qi) introduced in (3). We choose a parametrization associated to the upper edge of the fermi surface, i.e. we will choose all the w-n = 0 and

    im i,n--1

    = 9+(x , t ) = (6) i=1 i=1

    For relativistic fermions the fermi sea has no lower edge and one has to set all fl_ = w-n = 0. Using (6), (2) and (3) we can now easily calculate all the w+ m 's. Clearly, the wn 's are independent of h.

    Each of the Pi,qi satisfy an evolution equa- tion determined by the single particle hamilto- nian: Otpi .= -Oxpi for relatvisitic fermions and Otpi = -piO~pi for nonrelativistic fermions (and similarly for qi's). It may be checked that these equations then imply the corresponding evolu- tions for the fl and wm 's.

    Let the profile of the fermi surface at t = 0 be given by p(x, O) = a(x). Then at a later time t it is easy to show that p(x,t) = a(x-t ) for relativis- tic chiral fermions and p(x, t) = a(x-p(x, t) t) for nonrelativistic fermions. If there is a fold, there must be some point where dx/dp - O. Since the profile in a relativistic system is unchanged in time a fold cannot develop from a profile with no folds. On the other hand for a nonrelativistic system, even if we start with a single valued a(x) one will have dx/dp = 0 at some point on the fermi surface at some time. We give the result for the time of fold formation tf for two initial

  • 226 S.R. Das/Nuclear Physics B (Proc. Suppl.) 45B, C (1996) 224 233


    1 _ (~_ . )2 Ce 7

    p(x,O) : b e ~ --(a----*--a), Q ~, bv~

    1 p(x,0) : k ae(c t j - (7) (IGW) where a >> c. Thus fold formation can occur fl~r pulses of arbitrarily small energy density (and total energy), provided the width is sufficiently small. In the following we will only consider low energy pulses so that we can ignore the presence cf the bottom edge of the fermi sea.

    It may be noted that the notion of folds de- pends on a particular choice of coordinates and momenta in phase space. In fact for nonrelativis- tic fermions in an inverted harmonic oscillator potential, one may perform a transformation to l!ight cone like coordinates x = p 4- x in phase space in which the fermions appear relativistic and the resulting bosonic fields fl+ would be free (see O'Loughlin in [6]). In such a parametriza- tion a non-folded profile cannot evolve into a fold. tlowever, in the physical scattering process in- coming states are defined in terms of quanta us- ing x_ as the "space" while for the outgoing state x+ has to be used as "space". The classical scat- tering matrix is in fact given in terms of a Bo- goluibov transformation Since a nonfolded profile in the x_ parametrization generically becomes a folded profile in the x+ configuration one cannot evade the question of fold formation.

    It is now clear what happens if we set Wn = 0 in the time evolution equations (5). The evolution of/3 proceeds smoothly till t = tf at which point 0a;/3 diverges. The equations cannot be evolved beyond this point and the collective field the- ory fails. The time evolution of u(p,q,t) is of course unambiguous. The point is that one needs nonzero values of w,~ beyond this time, whose presence render the classical time evolution of the entire/3, w,~ system completely well defined. The same phenomenon happens in the presence of a background potential.

    2.2. The quantum theory Let us now turn to the quantum theory. Since

    the bottom edge of the fermi sea is irreleva