We propose here a quantum hoop conjecture which states: the de Broglie
wavelength of a quantum system cannot be arbitrarily small, it must be larger
than the characterized Schwarzschild radius of the quantum system. Based on
this conjecture, we find an upper bound for the wave number (or the momentum)
of a particle, which offers a natural cutoff for the vacuum energy of a scalar
field.Comment: 3 pages, no figure

Yang, Rong-Jia

Results in Physics

English

arXiv

We propose here a quantum hoop conjecture which states: the de Broglie wavelength of a quantum system cannot be arbitrarily small, it must be larger than the characterized Schwarzschild radius of the quantum system. Based on this conjecture, we find an upper bound for the wave number (or the momentum) of a particle, which offers a natural cutoff for the vacuum energy of a scalar field. Keywords: Mass-energy relation, de Broglie relation, Vacuum energy, Black hol

Rongjia Yang

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Quantum hoop conjecture and a natural cutoff for vacuum energy of a scalar field

AbstractWe propose here a quantum hoop conjecture which states: the de Broglie wavelength of a quantum system cannot be arbitrarily small, it must be larger than the characterized Schwarzschild radius of the quantum system. Based on this conjecture, we find an upper bound for the wave number (or the momentum) of a particle, which offers a natural cutoff for the vacuum energy of a scalar field

Yang, Rongjia

Elsevier - Publisher Connector 

Results in Physics 6 (2016) 98–99Contents lists available at ScienceDirect
Results in Physics
journal homepage: www.journals .e lsevier .com/resul ts - in-physicsMicroarticleQuantum hoop conjecture and a natural cutoff for vacuum energy
of a scalar fieldhttp://dx.doi.org/10.1016/j.rinp.2016.02.005
2211-3797/ 2016 The Author. Published by Elsevier B.V.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
⇑ Address: College of Physical Science and Technology, Hebei University, Baoding
071002, China.
E-mail address: yangrongjia@tsinghua.org.cnRongjia Yang ⇑
College of Physical Science and Technology, Hebei University, Baoding 071002, China
Hebei Key Lab of Optic-Electronic Information and Materials, Hebei University, Baoding 071002, Chinaa r t i c l e i n f o
Article history:
Received 3 February 2016
Accepted 18 February 2016
Available online 26 February 2016
Keywords:
Mass-energy relation
de Broglie relation
Vacuum energy
Black holea b s t r a c t
We propose here a quantum hoop conjecture which states: the de Broglie wavelength of a quantum sys-
tem cannot be arbitrarily small, it must be larger than the characterized Schwarzschild radius of the
quantum system. Based on this conjecture, we find an upper bound for the wave number (or the momen-
tum) of a particle, which offers a natural cutoff for the vacuum energy of a scalar field.
 2016 The Author. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).Introduction 11 4 48In the past years, a lot of independent cosmological observations,
such as supernova (SN) Ia at high redshift [1,2], the cosmic micro-
wave background (CMB) anisotropy [3,4], and large-scale structure
[5], have confirmed that the Universe is undergoing an accelerated
expansion. In the framework of general relativity, an unknown
energy component, usually called dark energy, has to be introduced
to explain this phenomenon. The simplest and most theoretically
appealing scenario of dark energy is the vacuum energy which is
about qovac  ð103 eVÞ
4 ¼ 108 ergs=cm3 matched from observa-
tional data. However, this model is confronted with very difficult
problem–cosmological constant problem [6–10] (may suffer from
age problem as well [11]). To briefly illustrate this issue, we con-
sider, for example, the vacuum energy density of a scalar field. It
is well known that the total vacuum energy density of a scalar field
with mass m is quartically divergent in the ultraviolet (UV)
qtvac ¼ h0jq^tvacj0i ¼
Z
dk
k2h
4p2c2
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
k2c2 þm2c4=h2
q
: ð1Þ
A usually used regularization for this divergence is to artificially
take a UV cutoff. But if we take different UV cutoffs, such as elec-
troweak scale, grand unification scale, or Planck scale, we can get
different values of vacuum energy density. Furthermore the differ-
ences between these values are huge, see for example, takingelectroweak scale, we get qtvac  ð10 eVÞ ¼ 10 ergs=cm3; tak-
ing Planck scale, we have qtvac  ð1027 eVÞ
4 ¼ 10112 ergs=cm3. The
ratio of theoretical to observational value of the vacuum energy
ranges from 1056 to 10120. This is the well known cosmological con-
stant problem [6–10]. Which scale we should take is still an open
problem. Can we find a UV cutoff from fundamental laws of phy-
sics? This is the major issue we will consider in this letter.
Here, combining with quantum and black hole physics, we find
an upper bound for the wave number of a quantum particle, which
gives a natural cutoff for the vacuum energy of a scalar field.Upper bound for wave number and a natural cutoff for vacuum
energy
For a quantum particle with mass m, the de Broglie relation
reads E ¼ hx; ~p ¼ h~k. According to the mass-energy relation in
special relativity, the total energy of a particle is E2 ¼ p2c2 þm2c4.
Combining the de Broglie relation and the mass-energy relation,
then we have
E2 ¼ h2k2c2 þm2c4: ð2Þ
This equation indicates that E ! 1 for k ! 1. A natural
question rises: is this result reasonable? In other words, because
x ¼
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
k2c2 þm2c4=h2
q
, the question can also be stated as: can a
particle oscillate arbitrarily fast (or, can the de Broglie wavelength
of a particle be arbitrarily small)? If we take into account the effect
of gravitation, the answer may be not.
R. Yang / Results in Physics 6 (2016) 98–99 99Think of black hole physics, a system with total energy E has an
effective mass E=c2, so it will be characterized with a Sch-
warzschild radius which is given by
rc ¼ 2Gc3
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
h2k2 þm2c2
q
: ð3Þ
The hoop conjecture in black hole physics states: if matter is
enclosed in sufficiently small region, then the system should
collapse to a black hole [12,13]. Similar assumptions were also sug-
gested in [14–16]: for example, it argued that the energy of a sys-
tem of size Lmust have an upper bound not to collapse into a black
hole [14]. Here we generalize the hoop conjecture to the quantum
case: the de Broglie wavelength of a quantum system cannot be
arbitrarily small, it should be larger than the characterized Sch-
warzschild radius of the quantum system. This can be called quan-
tum hoop conjecture.
This quantum hoop conjecture can get supports from earlier
works in literature. Possible connection between gravitation and
the fundamental length was discussed in [17]. From quantum
mechanics and classical general relativity, it was shown in
[18,19] that any primitive probe or target used in an experiment
must be larger than the Planck length, which implies a device inde-
pendent limit on possible position measurements. Researches from
string theory, black hole physics, and quantum gravity also predict
that there exists a minimum measurable length scale which is
approximately equivalent to the Planck length lp [20–24]. Based
on these researches, we can conclude that the de Broglie wave-
length of any quantum system must not be less than the minimum
length scale. This conclusion is consistent with the quantum hoop
conjecture proposed here: the de Broglie wavelength of a quantum
system should be larger than its characterized Schwarzschild
radius. In [25], a quantum hoop conjecture was also suggested by
constructing the horizon wave-function for quantum mechanical
states representing two highly boosted non-interacting particles,
which is different from the conjecture we proposed here.
The quantum hoop conjecture suggested here provides: k > rc,
which gives an upper bound for the wave number
k ¼ 2p
k
<
2p
rc
¼ pc
3
G
h2k2 þm2c2
h i12
<
pc3
Gkh
: ð4Þ
It is easy to get
k <
ﬃﬃﬃ
p
p
l1p ; ð5Þ
where lp ¼
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Gh=c3
p
is the Planck length. This bound only holds in
the observer’s reference frame. Bound (5) also gives an upper limit
for the momentum of the particle: p <
ﬃﬃﬃ
p
p
hl1p . Obviously, the wave
number of a massive particle is less than that of a massless particle.
As an application, we apply the bound for the wave number (5)
to the vacuum energy of a scalar field. For a quantum particle of a
scalar field, there are three freedoms for oscillation:
k ¼
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
k2x þ k2y þ k2z
q
. So we have k < 2
ﬃﬃ
3
p
p
rc
<
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
3
p
p
p
l1p which offers a
natural cutoff for the vacuum energy of a scalar field (1)
qtvac ¼ h0jq^tvacj0i ¼
Z kmax
0
dk
k2h
4p2c2
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
k2c2 þm2c4=h2
q
: ð6Þ
For k m, integration (6) is approximatively equivalent to
3h=ð16cl4pÞwhich is close to the value obtained by taking the Planck
scale cutoff. Also based on black hole physics, a cutoff for vacuum
energy of a scalar field was found in [26].
Conclusions and discussions
In this letter, we suggested a quantum hoop conjecture: the de
Broglie wavelength of a quantum system cannot be arbitrarilysmall, it must be larger than the characterized Schwarzschild
radius of the quantum system. This conjecture gives an upper
bound for the wave number or the momentum of the quantum sys-
tem. For application, we found a natural cutoff for the vacuum
energy of a scalar field.
Appendix A. Supplementary data
Supplementary data associated with this article can be found, in
the online version, at http://dx.doi.org/10.1016/j.rinp.2016.02.005.
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