Physics Letters B 798 (2019) 135010
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Physics Letters B
www.elsevier.com/locate/physletb
Solutions for mixed states in open bosonic string theory
Dimitri Polyakov
a,b,∗
a
Center for Theoretical Physics, College of Physical Science and Technology, Sichuan University, Chengdu 6100064, China
b
Institute of Information Transmission Problems (IITP), Bolshoi Karetny per. 19/1, Moscow 127994, Russia
a r t i c l e i n f o a b s t r a c t
Article history:
Received
22 August 2019
Received
in revised form 2 October 2019
Accepted
4 October 2019
Available
online 8 October 2019
Editor:
M. Cveti
ˇ
c
We describe the family of normalizable solutions in linearized open string field theory, defined by Q
0
=
0(Q is BRST charge) understood in the sense << Q
0
, >>= 0for an arbitrary string field . The
solutions depend on shifted partition numbers and are parametrized in terms of values of ζ -function at
pairs of positive numbers greater than 2. We argue that the operators, defined by these solutions, create
mixed quantum-mechanical states by acting on the vacuum (as opposed to standard vertex operators,
creating the pure states with definite masses and spins).
© 2019 The Author. 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
In D-dimensional open bosonic string theory [1]the action in conformal gauge is given by
S ∼
d
2
z{∂ X
m
¯
∂
X
m
+b
¯
∂
c +
¯
b∂
¯
c}+S
Liou ville
m = 0, ..., D − 1 (1.1)
and the nilpotent BRST operator Q
2
= 0[7]is given by
Q =
dz
2iπ
{cT −bc ∂c}≡
dz
2iπ
{−
1
2
c
∂ X
m
∂ X
m
+bc ∂c + ...} (1.2)
where T is the full stress-energy tensor and we skipped the Liouville terms in the second integral (as they will play no role in the rest of
the paper). The physical spectrum of string theory, modulo BRST-exact states, is defined by vertex operators {V } satisfying
QV = 0 (1.3)
(this equation involves anticommutators or commutators, for unintegrated and integrated pictures of the vertices respectively). The equa-
tion
(1.3), defining the physical spectrum of bosonic string: {| >}={V |0 >} can also be viewed as a linearized limit of open string field
theory equation of motion [5,6](e.g. Q + = 0for the cubic OSFT). The equation (1.3) has two well-known classes of solutions: the
local operators -dimension 0 primary fields of ghost number +1:
V = cP(∂ X,∂
2
X, ...)e
ip X
ϕ(p) (1.4)
and also the worldsheet integrals of dimension 1 primaries with ghost number zero:
V = ϕ(p)
dz P(∂ X,∂
2
X, ...)e
ip X
(1.5)
*
Correspondence to: Center for Theoretical Physics, College of Physical Science and Technology, Sichuan University, Chengdu 6100064, China.
E-mail
addresses: polyakov@scu.edu.cn, polyakov@sogang.ac.kr.
https://doi.org/10.1016/j.physletb.2019.135010
0370-2693/
© 2019 The Author. 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
.