Physics Letters B 761 (2016) 197–202
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Physics Letters B
www.elsevier.com/locate/physletb
Proof that Casimir force does not originate from vacuum energy
Hrvoje Nikoli
´
c
Theoretical Physics Division, Rudjer Boškovi´c Institute, P.O.B. 180, HR-10002 Zagreb, Croatia
a r t i c l e i n f o a b s t r a c t
Article history:
Received
27 May 2016
Received
in revised form 28 July 2016
Accepted
15 August 2016
Available
online 18 August 2016
Editor:
M. Cveti
ˇ
c
We present a simple general proof that Casimir force cannot originate from the vacuum energy of
electromagnetic (EM) field. The full QED Hamiltonian consists of 3 terms: the pure electromagnetic
term H
em
, the pure matter term H
matt
and the interaction term H
int
. The H
em
-term commutes with
all matter fields because it does not have any explicit dependence on matter fields. As a consequence,
H
em
cannot generate any forces on matter. Since it is precisely this term that generates the vacuum
energy of EM field, it follows that the vacuum energy does not generate the forces. The misleading
statements in the literature that vacuum energy generates Casimir force can be boiled down to the fact
that H
em
attains an implicit dependence on matter fields by the use of the equations of motion and to
the illegitimate treatment of the implicit dependence as if it was explicit. The true origin of the Casimir
force is van der Waals force generated by H
int
.
© 2016 The Author(s). 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
The Casimir force [1] is widely viewed as a force that origi-
nates
from the vacuum energy, which is a view especially popular
in the community of high-energy physicists [2–6]. Another view,
more popular in the condensed-matter community, is that Casimir
force has the same physical origin as van der Waals force [7–13],
which does not depend on energy of the vacuum. From a practical
perspective, the two points of view appear as two complementary
approaches, each with its advantages and disadvantages.
From
a fundamental perspective, however, one may be inter-
ested
to know which of the two approaches is more fundamental.
After all, the conceptual picture of the world in which the vacuum
energy has a direct physical role is very different from the picture
in which it does not. From such a fundamental perspective, Jaffe
argued [14] that the physically correct approach is the one based
on van der Waals force, while the approach based on vacuum en-
ergy
is merely a heuristic shortcut valid only as an approximation
in the limit of infinite fine structure constant. Similar doubts about
the vacuum-energy approach to Casimir force has been expressed
by Padmanabhan [15]. Nevertheless, it seems that a general con-
sensus
is absent [16,17]. The question of relevance of the vacuum
energy for Casimir force is still a source of controversy.
In
this paper we present a theoretical way to resolve the con-
troversy.
In short, similarly to Jaffe [14], we find that the approach
E-mail address: hnikolic@irb.hr.
based on vacuum energy is unjustified from a fundamental the-
oretical
perspective, leaving only the non-vacuum van der Waals-
like
approaches as physically viable. However, to arrive at that con-
clusion,
we use an approach very different from the approach used
by Jaffe. Our approach is rather mathematical in spirit, because our
central idea is to carefully distinguish explicit dependence from
implicit dependence in canonical equations of motion for classical
and quantum physics. In this way our approach is more abstract
and more general than the approach by Jaffe, but still sufficiently
simple to be accessible to a wide readership of theoretical physi-
cists.
2. Heuristic idea
Let us start with a brief overview of the standard calculation
[2] of Casimir force from vacuum energy. The energy of electro-
magnetic
(EM) field is
H
em
=
d
3
x
E
2
+ B
2
2
. (1)
In general, the fields E and B have Fourier transforms with con-
tributions
from all possible wave vectors k. However, in the ab-
sence
of electric currents in a conductor, the use of Maxwell equa-
tions
implies that the EM field must vanish at conducting plates.
Consequently, if there are two conducting plates separated by a
distance y, then the only wave vectors in the y-direction that
contribute to E and B are those which satisfy k
y
= nπ/ y (for
n = 1, 2, 3, ...). In this way E and B attain a dependence on y,
http://dx.doi.org/10.1016/j.physletb.2016.08.036
0370-2693/
© 2016 The Author(s). 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
.