Physics Letters B 756 (2016) 142–146
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
String theory as a Lilliputian world
J. Ambjørn
a,b,∗
, Y. Makeenko
a,c
a
The Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, DK-2100 Copenhagen, Denmark
b
IMAPP, Radboud University, Heyendaalseweg 135, 6525 AJ, Nijmegen, The Netherlands
c
Institute of Theoretical and Experimental Physics, B. Cheremushkinskaya 25, 117218 Moscow, Russia
a r t i c l e i n f o a b s t r a c t
Article history:
Received
22 January 2016
Received
in revised form 25 February 2016
Accepted
29 February 2016
Available
online 4 March 2016
Editor:
N. Lambert
Lattice regularizations of the bosonic string do not allow us to probe the tachyon. This has often been
viewed as the reason why these theories have never managed to make any contact to standard continuum
string theories when the dimension of spacetime is larger than two. We study the continuum string
theory in large spacetime dimensions where simple mean field theory is reliable. By keeping carefully
the cutoff we show that precisely the existence of a tachyon makes it possible to take a scaling limit
which reproduces the lattice-string results. We compare this scaling limit with another scaling limit
which reproduces standard continuum-string results. If the people working with lattice regularizations
of string theories are akin to Gulliver they will view the standard string-world as a Lilliputian world no
larger than a few lattice spacings.
© 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
A first quantization of the free particle using the path inte-
gral
requires a regularization. A simple such regularization is to
use a hypercubic D-dimensional lattice if the particle propagates
in D-dimensional spacetime. The allowed wordlines for a particle
propagating between two lattice points are link-paths connecting
the two points and the action used is the length of the path, i.e.
the number of links of the path multiplied with the link length a
.
This a
is the UV cutoff of the path integral. The lattice regulariza-
tion
works nicely and the limit a
→ 0can be taken such that one
obtains the standard continuum propagator.
Similarly,
the first quantization of the free bosonic string us-
ing
the path integral requires a regularization. It seemed natural
to repeat the successful story of the free particle and use a hyper-
cubic
D-dimensional lattice if the wordsheet of the string lived in
D-dimensional spacetime, the worldsheet being a connected pla-
quette
lattice surface [1]. The Nambu–Goto action, the area of the
worldsheet, would then be the number of worldsheet plaquettes
times a
2
, a
again denoting the lattice spacing. However, in this
case one could not reproduce the results obtained by standard
canonical quantization of the string. First, one did not obtain the
whole set of string masses, starting with the tachyon mass, but
*
Corresponding author.
E-mail
addresses: ambjorn@nbi.dk (J. Ambjørn), makeenko@nbi.dk
(Y. Makeenko).
only a single, positive mass state. Next, after having renormalized
the bare string coupling constant to obtain a finite lowest mass
state, this renormalization led to an infinite physical string tension
for strings with extended boundaries.
Although
it was not clear why the hypercubic lattice regular-
ization
did not work, the formalism known as dynamical triangu-
lation
(DT) was suggested as an alternative [2]. It discretized the
independent intrinsic worldsheet geometry used in the Polyakov
formulation of bosonic string theory [3] and the integration over
these geometries was approximated by a summation over triangu-
lations
constructed from equilateral triangles with link lengths a
t
,
where a
t
again was a UV cutoff. However, the results were identi-
cal
to the hypercubic lattice results. In contrast to the hypercubic
lattice model the DT model can be defined when the dimension
of spacetime is less than two, where one encounters the so-called
non-critical string theory. This string theory can be solved both
using standard continuum quantization and using the DT-lattice
regularization (and taking the limit a
t
→ 0). Agreement is found.
Thus a lattice regularization is not incompatible with string theory
as such. However, in the lattice regularized theories it is impossible
to have a tachyonic lowest mass state and such states appear pre-
cisely
when the dimension of spacetime exceeds two. It might ex-
plain
the failure of lattice strings to connect to continuum bosonic
string theory in dimensions D > 2.
The
purpose of this article is to highlight the role of the
tachyon in connecting the continuum bosonic string theory to lat-
tice
strings. In [4] we showed how one could technically make
http://dx.doi.org/10.1016/j.physletb.2016.02.075
0370-2693/
© 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.