We show that the spectrum of a bosonic open 2-brane does not contain any massless states to take the role of gravitons. Moreover, the spectrum of this open 2-brane only contains half integer mass squared values

Ding-Fang Zeng

M Abdul Wasay

CiteSeerX

Research Article
An Inconsistency in the Spectrum of Bosonic Open 2-Brane
M. Abdul Wasay1,2 and Ding-fang Zeng1
1 Institute of Theoretical Physics, Beijing University of Technology, Beijing 100124, China
2Department of Physics, University of Agriculture, Faisalabad 38040, Pakistan
Correspondence should be addressed to M. Abdul Wasay; wasay 31@hotmail.com
Received 14 July 2015; Revised 3 October 2015; Accepted 7 October 2015
Academic Editor: Anastasios Petkou
Copyright © 2015 M. A. Wasay and D.-f. Zeng. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited. The publication of this article was funded by SCOAP3.
We show that the spectrum of a bosonic open 2-brane does not contain any massless states to take the role of gravitons. Moreover,
the spectrum of this open 2-brane only contains half integer mass squared values.
1. Introduction
Besides the impressive progress of string theories in the
past few decades which compelled physicists to believe that
a consistent theory of quantum gravity based on string
philosophy would be built soon has slowed down. The
early success of string theory also sparked the motivation
to study higher dimensional extended objects. Indeed if
one can give up 0-dimensional particles in favor of strings
which are one-dimensional objects, then why not strings in
favor of two-dimensional membranes? The basic idea that
elementary particles could be interpreted as vibrating modes
of a membrane originally came in 1962 by Dirac [1].
Since the task of achieving a consistent theory of quan-
tum gravity based on string philosophy became more and
more subtle, physicists tried to find alternative candidates to
construct a theory of fundamental interactions; one of these
candidates was supermembrane theory [2–4]. However, it
was soon realized that the supermembrane has an unstable
ground state [5].
Compared to the works on string quantization, works
on membrane quantization are very few, precisely because
of the difficulty the Nambu action meets when one tries to
generalize it to 2+ 1 dimensions or general 𝑝+1 dimensions.
However, it is possible to quantize a bosonic membrane
theory in a flat background [6, 7].
We will analyze the mass squared values in the spectrum
of the bosonic open 2-brane. The bosonic membranes could
be related to the bosonic string theory via dimensional
reduction [8]. The critical spacetime dimension in analogy
with bosonic string theory is taken to be 𝐷 = 26. We
will focus on the spectrum of states after reviewing their
quantization.
The organization of this paper is as follows: Section 2 is a
brief review of the quantization scheme in a flat background
[6]. Section 3 deals with the spectrum of this open 2-brane
and Section 4 is summary and conclusion.
2. A Brief Review
The quantization of bosonic open branes was studied in [6, 7,
9]. The open 2-brane dynamics is captured by the Nambu-
Goto action, which physically describes the world-volume
swept out by the brane. However, for quantization purposes it
is not suitable because of the square root. However, one may
follow [10] and write a Polyakov-type action for the general
bosonic open 2-brane given by
𝑆 = −
1
4𝜋𝛼
󸀠
∫𝑑
3
𝜎√−ℎ [ℎ
𝛼𝛽
𝜕
𝛼
𝑋
𝜇
𝜕
𝛽
𝑋
𝜇
+ 𝑅 − 2Λ] , (1)
where 𝑅 and Λ give the contribution from the cosmological
constant term. A 2-brane sweeps out a 3-dimensional world-
volume which is parameterized by 𝜏, 𝜎1, 𝜎2.
Hindawi Publishing Corporation
Advances in High Energy Physics
Volume 2015, Article ID 360356, 4 pages
http://dx.doi.org/10.1155/2015/360356
2 Advances in High Energy Physics
The energy-momentum tensor is given by the variation of
the action (1), with respect to the world-volume metric ℎ𝛼𝛽,
as
𝑇
𝛼𝛽
=
−2𝜋𝛼
󸀠
√−ℎ
𝛿𝑆
𝛿ℎ
𝛼𝛽
= 𝜕
𝛼
𝑋
𝜇
𝜕
𝛽
𝑋
𝜇
−
1
2
ℎ
𝛼𝛽
ℎ
𝛾𝛿
𝜕
𝛾
𝑋
𝜇
𝜕
𝛿
𝑋
𝜇
+
1
2
ℎ
𝛼𝛽
= 0.
(2)
Under the flat metric condition ℎ
𝛼𝛽
= 𝜂
𝛼𝛽
, the compo-
nents of this energy-momentum tensor can be written as
𝑇
𝛼𝛽
= 𝜕
𝛼
𝑋
𝜇
𝜕
𝛽
𝑋
𝜇
−
1
2
⋅ ℎ
𝛼𝛽
[−𝜕
𝜏
𝑋
𝜇
𝜕
𝜏
𝑋
𝜇
+ 𝜕
𝜎
1𝑋
𝜇
𝜕
𝜎
1𝑋
𝜇
+ 𝜕
𝜎
2𝑋
𝜇
𝜕
𝜎
2𝑋
𝜇
]
+
1
2
ℎ
𝛼𝛽
(3)
with
𝑇
00
=
1
2
[𝜕
𝜏
𝑋
𝜇
𝜕
𝜏
𝑋
𝜇
+ 𝜕
𝜎
1𝑋
𝜇
𝜕
𝜎
1𝑋
𝜇
+ 𝜕
𝜎
2𝑋
𝜇
𝜕
𝜎
2𝑋
𝜇
− 1]
= 0,
𝑇
11
=
1
2
[𝜕
𝜏
𝑋
𝜇
𝜕
𝜏
𝑋
𝜇
+ 𝜕
𝜎
1𝑋
𝜇
𝜕
𝜎
1𝑋
𝜇
− 𝜕
𝜎
2𝑋
𝜇
𝜕
𝜎
2𝑋
𝜇
+ 1]
= 0,
𝑇
22
=
1
2
[𝜕
𝜏
𝑋
𝜇
𝜕
𝜏
𝑋
𝜇
− 𝜕
𝜎
1𝑋
𝜇
𝜕
𝜎
1𝑋
𝜇
+ 𝜕
𝜎
2𝑋
𝜇
𝜕
𝜎
2𝑋
𝜇
+ 1]
= 0,
(4)
and the Euler-Lagrange equation for the𝑋𝜇 fields is given by
(𝜕
2
𝜏
− 𝜕
2
𝜎
1
− 𝜕
2
𝜎
2
)𝑋
𝜇
(𝜏, 𝜎
1
, 𝜎
2
) = 0. (5)
By imposing Neumann boundary condition
𝜕
𝜎
1𝑋
𝜇
(𝜏, 0, 𝜎
2
) = 𝜕
𝜎
1𝑋
𝜇
(𝜏, 𝜋, 𝜎
2
) = 0,
𝜕
𝜎
2𝑋
𝜇
(𝜏, 𝜎
1
, 0) = 𝜕
𝜎
2𝑋
𝜇
(𝜏, 𝜎
1
, 𝜋) = 0,
(6)
one can get the following modes expansion for the𝑋𝜇 fields:
𝑋
𝜇
(𝜏, 𝜎
1
, 𝜎
2
) =
𝑥
𝜇
√𝜋
+
2?́?𝑝
𝜇
√𝜋
𝜏
+ 𝑖√2?́?
+∞
∑
𝑚,𝑛=0
(𝑛
2
+ 𝑚
2
)
−1/4
⋅ (𝑋
𝜇
𝑛𝑚
𝑒
𝑖𝜏√𝑛
2
+𝑚
2
− 𝑋
†𝜇
𝑛𝑚
𝑒
−𝑖𝜏√𝑛
2
+𝑚
2
) cos 𝑛𝜎1 cos𝑚𝜎2
(7)
and the canonical momentum, as
𝑃
𝜇
(𝜎) =
𝑝
𝜇
𝜋√𝜋
+
1
𝜋
√
2
?́?
+∞
∑
𝑚,𝑛=0
(𝑛
2
+ 𝑚
2
)
1/4
⋅ (𝑃
𝜇†
𝑛𝑚
𝑒
𝑖𝜏√𝑛
2
+𝑚
2
+ 𝑃
𝜇
𝑛𝑚
𝑒
−𝑖𝜏√𝑛
2
+𝑚
2
) cos 𝑛𝜎1
⋅ cos𝑚𝜎2.
(8)
According to the standard commutation relation
[𝑋
𝜇
, 𝑃
]
] = 𝜂
𝜇]
𝛿 (𝜎
1
− 𝜎
󸀠1
) 𝛿 (𝜎
2
− 𝜎
󸀠2
) (9)
the following commutation relations between the creation/
annihilation operators can be obtained:
[𝑋
𝜇
𝑛𝑚
, 𝑃
]
𝑛
󸀠
𝑚
󸀠]
=
1
𝜋
𝜂
𝜇]
[𝛿
𝑛𝑛
󸀠𝛿
𝑚𝑚
󸀠 −
1
2
𝛿
−𝑛,𝑛
󸀠𝛿
𝑚𝑚
󸀠 −
1
2
𝛿
𝑛𝑛
󸀠𝛿
−𝑚,𝑚
󸀠] .
(10)
Using these creation/annihilation operators, theHamiltonian
of the system is expressed as
4𝜋𝛼
󸀠
𝐻 = ∫
𝜋
0
𝑑𝜎
1
∫
𝜋
0
𝑑𝜎
2
(𝑃
𝜇
?̇?
𝜇
−L)
= 2𝛼
󸀠
𝜋
2
𝜂
𝜇]
∞
∑
𝑛=1
𝑛 [𝑋
𝜇†
𝑛0
𝑋
]
𝑛0
+ 𝑋
𝜇
𝑛0
𝑋
]†
𝑛0
]
+ 2𝛼
󸀠
𝜋
2
𝜂
𝜇]
∞
∑
𝑚=1
𝑚[𝑋
𝜇†
0𝑚
𝑋
]
0𝑚
+ 𝑋
𝜇
0𝑚
𝑋
]†
0𝑚
]
+ 𝛼
󸀠
𝜋
2
𝜂
𝜇]
∞
∑
𝑛,𝑚=1
(𝑛
2
+ 𝑚
2
)
1/2
[𝑋
𝜇†
𝑛𝑚
𝑋
]
𝑛𝑚
+ 𝑋
𝜇
𝑛𝑚
𝑋
]†
𝑛𝑚
]
+ 4𝜋𝛼
󸀠2
𝑝
2
.
(11)
Under the tensorial assumption,
𝑋
𝜇
𝑛𝑚
= √
2
𝜋
𝜙
𝜇†
𝑛
⊗ 𝜙
𝜇†
𝑚
,
𝑋
𝜇†
𝑛𝑚
= √
2
𝜋
𝜙
𝜇
𝑛
⊗ 𝜙
𝜇
𝑚
,
(12)
the Hamiltonian (11) is translated into
𝐻 = 𝑁
1
+ 𝑁
2
+ 𝑁
12
+ 𝑎 + 𝑏 − 𝛼
󸀠
𝑀
2
, (13)
Advances in High Energy Physics 3
where
𝑁
1
= 𝜂
𝜇]
∞
∑
𝑛=1
𝑛 {𝜙
𝜇†
𝑛
⊗ 𝜙
𝜇†
0
} {𝜙
𝜇
𝑛
⊗ 𝜙
𝜇
0
} , (14)
𝑁
2
= 𝜂
𝜇]
∞
∑
𝑚=1
𝑚{𝜙
𝜇†
0
⊗ 𝜙
𝜇†
𝑚
} {𝜙
𝜇
0
⊗ 𝜙
𝜇
𝑚
} , (15)
𝑁
12
= 𝜂
𝜇]
∞
∑
𝑛,𝑚=1
(𝑛
2
+ 𝑚
2
)
1/2
{𝜙
𝜇†
𝑛
⊗ 𝜙
𝜇†
𝑚
} {𝜙
𝜇
𝑛
⊗ 𝜙
𝜇
𝑚
} , (16)
𝑎 = 𝜂
𝜇
𝜇
∞
∑
𝑛=1
𝑛, (17)
𝑏 =
1
2
𝜂
𝜇
𝜇
∞
∑
𝑛,𝑚=1
√𝑛
2
+ 𝑚
2
. (18)
3. Spectrum
We will employ the Zeta function regularization scheme to
regularize the infinite divergent summation in (17) and (18).
The contributions arising from the infinite sum (18) have not
been considered carefully in earlier works and have usually
been taken as some background field effects.
In fact, there is another scheme [11], the so-called Epstein
Zeta functions to regularize the infinite sums of type (18):
∞
∑
𝑛,𝑚=1
√(
𝑛
𝑐
1
)
2
+ (
𝑚
𝑐
2
)
2
=
1
24
(
1
𝑐
1
+
1
𝑐
2
) −
𝜁 (3)
8𝜋
2
(
𝑐
1
𝑐
2
2
+
𝑐
2
𝑐
2
1
)
−
𝜋
3/2
2√𝑐1𝑐2
[exp(−2𝜋𝑐1
𝑐
2
) (1 + O (10
−3
))]
(19)
with 𝜁(3) ≈ 1.202 and 𝑐
1
≤ 𝑐
2
. The infinite sums (17) and
(18) appearing in the quantization of the open 2-brane can be
written as
𝑎 = 𝜂
𝜇
𝜇
∞
∑
𝑛=1
𝑛 = (𝐷 − 2)
∞
∑
𝑛=1
𝑛 = (𝐷 − 2) 𝜁 (−1) ,
𝑏 =
1
2
𝜂
𝜇
𝜇
∞
∑
𝑛,𝑚=1
√𝑛
2
+ 𝑚
2
=
𝐷 − 2
2
∞
∑
𝑛,𝑚=1
√𝑛
2
+ 𝑚
2
.
(20)
Numerically, 𝑎 = −2 while 𝑏 ≈ 1/2, given that the spacetime
dimension𝐷 = 26.
According to the above calculation and the fact that 𝑇
00
=
0 = 𝐻, we can rewrite (13), the mass formula of the open 2-
brane spectrum, as
𝛼
󸀠
𝑀
2
= 𝑁
1
+ 𝑁
2
+ 𝑁
12
−
3
2
. (21)
By definition, the ground state of the system is annihilated by
the operators 𝜙𝜇
𝑛⩾0
and 𝜙𝜇
𝑚⩾0
, with the condition that 𝑛⋅𝑚 ̸= 0:
𝜙
𝑛
⊗ 𝜙
𝑚
|0, 0⟩ = 0. (22)
A general state |𝜑, 𝜒⟩, in the Fock space, can be obtained
by application of the creation operators 𝜙𝜇†
𝑛
and 𝜙𝜇†
𝑚
on the
ground state:
󵄨󵄨󵄨󵄨𝜑, 𝜒⟩ = Π𝑛,𝑚 {𝜙
𝜇†
𝑛
𝑖
⊗ 𝜙
𝜇†
𝑚
𝑖
} |0, 0⟩ (23)
with∑
𝑖
𝑛
𝑖
= 𝜑 and∑
𝑖
𝑚
𝑖
= 𝜒.The states |𝜑, 𝜒⟩ and |𝜒, 𝜑⟩ have
the same mass squared.
At the lowest mass level, the number operators 𝑁
1
, 𝑁
2
,
and𝑁
12
are zero, so
𝛼
󸀠
𝑀
2
= −
3
2
. (24)
Thefirst excited level contains two kinds of states correspond-
ing to 𝑛 = 1,𝑚 = 0 and 𝑛 = 0,𝑚 = 1; that is,
|1, 0⟩ = 𝑘
𝜇
(𝜙
𝜇†
1
⊗ 𝜙
𝜇†
0
) |0, 0⟩ ,
|0, 1⟩ = 𝑙
𝜇
(𝜙
𝜇†
0
⊗ 𝜙
𝜇†
1
) |0, 0⟩
(25)
whose mass-square operator reads
𝛼
󸀠
𝑀
2
𝑘
= 𝑁
1
+ 𝑁
2
+ 𝑁
12
−
3
2
= 1 + 0 + 0 −
3
2
= −
1
2
,
𝛼
󸀠
𝑀
2
𝑙
= 𝑁
1
+ 𝑁
2
+ 𝑁
12
−
3
2
= 0 + 1 + 0 −
3
2
= −
1
2
.
(26)
The ground and first excited states are both tachyonic.
However the value of mass squared operator in [6] is an
integer, while we show that it is half integer valued. The
ground states are scalar states. We have two different kinds
of vector states at the first excited state level corresponding to
|1, 0⟩ and |0, 1⟩.
At the second excited level, there are four kinds of tensor
states, featured by
𝑛 = 0
𝑚 = 2,
𝑛 = 2
𝑚 = 0,
𝑛
1
= 𝑛
2
= 1
𝑚
1
= 𝑚
2
= 0,
𝑛
1
= 𝑛
2
= 0
𝑚
1
= 𝑚
2
= 1
(27)
with the creation operators arranging at this level as follows:
|2, 0⟩
1
= 𝑘
𝜇] (𝜙
𝜇†
1
⊗ 𝜙
𝜇†
0
) (𝜙
]†
1
⊗ 𝜙
]†
0
) |0, 0⟩ ,
|2, 0⟩
2
= 𝑘
𝜇
(𝜙
]†
2
⊗ 𝜙
]†
0
) |0, 0⟩ ,
|0, 2⟩
1
= 𝑙
𝜇] (𝜙
𝜇†
0
⊗ 𝜙
𝜇†
1
) (𝜙
]†
0
⊗ 𝜙
]†
1
) |0, 0⟩ ,
|0, 2⟩
2
= 𝑙
𝜇
(𝜙
]†
0
⊗ 𝜙
]†
2
) |0, 0⟩ .
(28)
4 Advances in High Energy Physics
For these states, the number operators are 𝑁
1
= 2, 𝑁
2
=
𝑁
12
= 0 or𝑁
2
= 2,𝑁
1
= 𝑁
12
= 0. Themass squared operator
will become
𝛼
󸀠
𝑀
2
= 𝑁
1
+ 𝑁
2
+ 𝑁
12
−
3
2
= 0 + 2 + 0 −
3
2
or 2 + 0 + 0 − 3
2
=
1
2
.
(29)
We can see that the spectrum of the bosonic 2-brane does
not contain any massless states to play the role of gravitons.
Moreover, the mass squared operator for the open 2-brane
only gives half integer mass squared values.
4. Summary and Conclusion
In this paper, firstly we review the quantization of open 2-
brane, starting from the Polyakov action. We then investigate
the spectrum of the open 2-brane by taking into account the
contributions of both the infinite sums (17) and (18). On the
basis of this we can draw the conclusion that there are no
gravitons present in the bosonic open 2-brane at the massless
level, or there are no massless states in the bosonic open 2-
brane spectrum. This further implies that the bosonic open
2-brane theory is not a theory of quantum gravity. This is
reminiscent of the fact that, in the string case [12], simple
open strings do not form self-complete system of quantum
gravity.
Conflict of Interests
The authors declare that there is no conflict of interests
regarding the publication of this paper.
Acknowledgments
Theauthors thank Jian-FengWu for suggesting to them litera-
tures on the special zeta functions [11].The work is supported
by the Open Project Program of State Key Laboratory of
Theoretical Physics, Institute of Theoretical Physics, Chinese
Academy of Sciences, China. M. Abdul Wasay would also
like to thank Yasir Jamil for providing the facilities at Physics
Department, University of Agriculture, Faisalabad, where
part of this work was carried out.
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An Inconsistency in the Spectrum of Bosonic Open 2-Brane

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An Inconsistency in the Spectrum of Bosonic Open 2-Brane

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