Found Phys (2011) 41: 843–862DOI 10.1007/s107010109527y
Bipartite Entanglement Induced by a CommonBackground (ZeroPoint) Radiation Field
A. ValdésHernández
·
L. de la Peña
·
A.M. Cetto
Received: 27 May 2010 / Accepted: 13 December 2010 / Published online: 22 December 2010© Springer Science+Business Media, LLC 2010
Abstract
This paper deals with an (otherwise classical) two(noninteracting) particle system immersedin a commonstochastic zeropointradiationﬁeld. The treatmentis an extension of the oneparticle case for which it has been shown that the quantumproperties of the particle emerge from its interaction with the background ﬁeld under stationary and ergodic conditions. In the present case we show that nonclassicalcorrelations—describable only in terms of entanglement—arise between the (nearby)particles whenever both of them resonate to a common frequency of the ﬁeld. Foridentical particles the entanglement becomes maximum and must be described bytotally (anti)symmetric states.
Keywords
Entanglement
·
Zeropoint radiation ﬁeld
·
Foundations of quantummechanics
·
Symmetrization
·
Stochastic processes
1 Introduction
In previous work [1] it was shown that the quantum behaviour of oneparticle systems can be understood as an emergent property arising from the interaction of an
A. ValdésHernández
·
L. de la Peña (
)
·
A.M. CettoInstituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20364,Mexico City, Mexicoemail: luis@ﬁsica.unam.mxA. ValdésHernándezemail: andreavh@ﬁsica.unam.mxA.M. Cetto (
)email: ana@ﬁsica.unam.mx
Present address:
A.M. CettoIAEA, P.O. Box 200, Vienna Austria
844 Found Phys (2011) 41: 843–862
otherwise classical particle with the background zeropoint radiation ﬁeld, the latter being a continuous (not quantized) ﬁeld, whose energy per mode of frequency
ω
is proportional to
ω
. The statistical description of the mechanical system in thestationary, ergodic and radiationless regime corresponds to nonrelativistic quantummechanics, a result that leads to important new insight into the srcin of quantization.However, for a more complete understanding of the mechanisms underlying quantummechanics it is necessary to inquire into a distinctive and unique trait of the quantumcomposite systems, namely entanglement.This phenomenon,which holds currently acentral place in physical research, should be regarded not only as an important sourceof possible revolutionary technological applications, but also as a key resource intothe search for a better understanding of quantum mechanics. Hence, in this paper weextend the oneparticle theory developed in [1] to systems composed of two nearbynoninteracting particles embedded in a common background (zeropoint) stochasticﬁeld, to delve into the mechanism that generates and supports their entanglement.
1
We ﬁnd that the mechanism responsible for entanglement is the coupling of theparticles through the common modes of the ﬁeld to which they resonate, once thestationary, ergodic and radiationless regime has been attained. Speciﬁcally, wheneverthe particles share a resonance frequency, correlations arise between their motions,induced via the background ﬁeld.When the description is reduced to one in termsof state vectors in an appropriate Hilbert space, the entangled states emerge naturally as the ones that reproduce such correlations. Moreover, for systems of identicalparticles the invariance properties of the ﬁeld imply that entanglement is maximumand must be described by totally (anti)symmetric states. With these results the theoryreveals the mechanism and the srcin—both of them foreign to the usual quantumdescription—of the entanglement between noninteracting particles and the quantumsymmetrization postulate.For the sake of clarity the ﬁrst part of the paper contains a schematic expositionof the oneparticle theory; for a comprehensive discussion of it we refer the readerto [1]. In the second part the twoparticle problem is discussed in detail.
2 The OneParticle System
2.1 The Background Field in Interaction with MatterWe consider a particle of (renormalized) mass
m
and electric charge
e
subject to anexternal force
f
(
x
)
and immersed in an (otherwise classical, that is, continuous) stochastic radiation ﬁeld, arbitrary except that it always includes the zeropoint contribution of average energy per normal mode
E
(ω)
=
ω/
2.
2
The equation of motion for
1
The very ﬁrst version of the oneparticle theory can be seen in [2], a short communication on the twoparticle system in [3]; an extended version of the present theory is contained in [4].
2
The mere introduction of this ﬁeld already constitutes a clear departure from usual classical physics, itszeropoint energy being contrary to classical equipartition of energy. It should be stressed that the zeropoint spectrum requires no quantum assumptions (see, for example, [5]).
Found Phys (2011) 41: 843–862 845
the particle is given, in the nonrelativistic approximation, by the AbrahamLorentzequation
m
··
x
=
f
(
x
)
+
mτ
···
x
+
e
E
(
x
,t), τ
=
2
e
2
3
mc
3
.
(1)The electric ﬁeld is expanded as usual in terms of plane waves
E
(
x
,t)
=
k
˜
E(ω
k
)
a
k
(x)e
iω
k
t
+
c.c., (2)where
a
k
(x)
is a stochastic vector (with random components) that results from an angular and polarization averaging over all ﬁeld modes having frequency
ω
k
. Assumingthat the longwavelength approximation is valid and restricting the study to the onedimensional case (along an arbitrary direction
ˆ
R
) (1) and (2) reduce to (
a
k
=
a
k
· ˆ
R
)
m
¨
x
=
f(x)
+
mτ
...
x
+
eE(t),
(3a)
E(t)
=
k
˜
E(ω
k
)a
k
e
iω
k
t
+
c.c. (3b)We will further assume that
f(x)
can be expanded as a power series of
x
.2.2 The Family of Stationary Resonant SolutionsThe solutions of (3a) constitute a stochastic process
x
(i)
(t)
where the index
i
—whichwill be omitted except when it is necessary—denotes the speciﬁc realization
i
of theﬁeld. The ensemble
{
i
}
of all the realizations is a statistical set that can be reproducedby considering an ensemble of individual particles, each one subject to a differentrealization of the ﬁeld. The average over the realizations (
(
·
)
(i)
) can therefore bealternatively obtained by averaging over the ensemble of particles (
·
).A stationary state is reached when the mean power radiated by the particle balances the mean power absorbed by it from the ﬁeld. Thus, according to (3a), thestationary solutions satisfy the global condition
e
˙
xE
=
mτ
··
x
2
. We assume that
{
i
}
can be decomposed as
{
i
} =
α
{
i
}
α
in such a way that the balance condition holdsseparately for each of the subensembles
{
i
}
α
, where
α
denotes the corresponding stationary state, and hence each
{
i
}
α
corresponds to those particles that have reached thestate
α.
Since we are dealing with a onedimensional singleparticle problem, suchstate is completely characterized by the energy attained by the particle, and so theindex
α
is directly associated with the mechanical energy.In what follows we focus on a given subensemble
{
i
}
α
. The quantities
x(t), E(t),f(t)
, and in general all the timedependent dynamical variables
A(t)
that can beexpressed as a power series of
x
and
˙
x
in the form
h(x)
+
g(
˙
x)
, must be thereforecharacterized with the index
α
, so we write the generic expression (with
˜
A
αβ
(t)
=
846 Found Phys (2011) 41: 843–862
˜
A
αβ
e
iω
αβ
t
)
:
3
A
α
(t)
=
β
˜
A
αβ
a
αβ
e
iω
αβ
t
=
β
˜
A
αβ
(t)a
αβ
.
(4)The index
β
enumeratesthosefrequencies
ω
αβ
thatare relevantfor aspeciﬁcproblemgiven the (ﬁxed) state
α
. We refer to these frequencies, which are determined by thetheory (see (7)), as
relevant frequencies
. The indices
αβ
appearing in the amplitudes
˜
A
αβ
(which in principle depend on the ﬁeld variables
{
a
(i)αβ
}
) indicate that any suchquantity is associated with the frequency
ω
αβ
.
Byintroducing
x
α
(t),E
α
(t)
and
f
α
(t)
into(3a)(assumedtobesatisﬁedseparatelyfor each relevant frequency) it can be shown that the response
x
α
behaves resonantlyat certain frequencies, called
resonance frequencies
, which are determined by theequation
αβ
≡
ω
2
αβ
−
iτω
3
αβ
+
˜
f
αβ
m
˜
x
αβ
0
.
The resonances are extremely sharp due tothe small value of
τ
(10
−
23
s for electrons). Nonresonant contributions, corresponding to the remaining frequencies, constitute a small noise that will be disregarded.2.3 Ergodic Hypothesis and Matrix MechanicsBy demanding the stationary solutions to comply with the ergodic conditions (here
(
·
)
t
denotes a time average over long enough times)

x
(i)α

2
= 
x
(i)α

2
t
and
˙
x
(i)α

2
=˙
x
(i)α

2
t
,
both
˜
f
αβ
and
{
ω
αβ
}
become independent of the ﬁeld variables
{
a
(i)αβ
}
,
andconsequently
f
α
(t)
becomes a
linear
expansion in such variables (see (4)). The ergodic condition ultimately leads to the
chain rule
,
ω
αβ
1
+
ω
β
1
β
2
+
ω
β
2
β
3
+ ··· +
ω
β
n
β
=
ω
αβ
,
(5a)
a
(i)αβ
1
a
(i)β
1
β
2
a
(i)β
2
β
3
···
a
(i)β
n
β
=
a
(i)αβ
=
exp
(iϕ
(i)αβ
),
(5b)where the
ϕ
(i)αβ
also satisfy the rule (5a). The solutions for
ω
αβ
and
ϕ
(i)αβ
are therefore(here
φ
(i)α
are random variables and
α
nonstochastic numbers)
ω
αβ
=
α
−
β
, ϕ
(i)αβ
=
φ
(i)α
−
φ
(i)β
.
(6)A major implication of (5) is that they lead to the matrix multiplication rule for theamplitudes
˜
x
αβ
, whence the coefﬁcients in the expansion for
x
α
(t)
deﬁne a hermiteanmatrix
ˆ
x
. This allows us to associate a hermitean matrix
ˆ
A(t)
to the dynamical variable
A,
whose elements
αβ
are precisely the amplitudes
˜
A
αβ
(t)
appearing in (4). Asa result the srcinal stochastic equation (3a) transforms into a nonstochastic matrixequation.
3
Note that the c.c. term has been omitted. Thus, the quantities
A
α
are not exactly the physical (real)variables, but complex variables associated to them.
Found Phys (2011) 41: 843–862 847
2.4 Radiationless Approximation: Establishing Contact with the Formalismof Quantum MechanicsOnce the background ﬁeld and the radiation reaction have played their fundamental role in leading the system to the stationary regime, the corresponding terms inthe matrix version of (3a) represent just radiative corrections that can be neglectedin a ﬁrst approximation. Though this eliminates from the description any explicitreference to the zeropoint ﬁeld, its footprint reappears—via the constant
thatdetermines the zeropoint energy—in the canonical commutator which is found tobe [1]
[ˆ
x,
ˆ
p
] =
i
I
.
For the matrix
ˆ
A(t)
one obtains the Heisenberg evolution law
i
d
ˆ
Adt
= [ ˆ
A,
ˆ
H
]
,
with
ˆ
H
the matrix representing the mechanical Hamiltonian.
4
Further, the energy
E
α
attained by the particle in state
α
, is found to satisfy the relation
ω
αβ
=
E
α
−
E
β
,
(7)which allows us to identify the resonance frequencies with transition frequencies.
5
On the other hand, the matrix
ˆ
A(t)
can be expanded in terms of a canonical basis
{
e
α
e
β
}
as usual,
˜
A
αβ
(t)
being the corresponding coefﬁcients of the expansion. Byperforming the transformation

e
α
→ 
α(t)
=
e
−
i(
E
α
/
)t

e
α
,
(8)the evolution of the system is transferred to the state vectors
{
α(t)
}
, directly related to
E
α
. From (8) and the orthonormal property of the basis it can be shown
that
˜
A
αβ
(t)
=
e
α
 ˆ
A(t)

e
β
=
α(t)
 ˆ
A(
0
)

β(t)
,
or in a more condensed notation,
˜
A
αβ
=
α
 ˆ
A

β
.
These results allow us to recover the whole of the basic Hilbertspace formalism of quantum mechanics.
3 The Bipartite System
3.1 The Field in the Vicinity of the ParticlesWe consider a pair of particles located at
x
1
and
x
2
and subject to the externalforces
f
1
(
x
1
)
and
f
2
(
x
2
)
respectively, with no external interaction potential betweenthem. The equations of motion, again in the nonrelativistic approximation, are (here
i,j
=
1
,
2 with
i
=
j
)
m
i
··
x
i
=
f
i
(
x
i
)
+
m
i
τ
i
···
x
i
+
e
i
e
j
m
j
τ
j
···
x
j
+
e
i
E
(
x
i
,t).
(9)
4
It can be shown [4] that
ˆ
A
follows the more general evolution law
i
d
ˆ
Adt
= [ ˆ
A,
ˆ
D
ˆ
H
]
,
where the elements of
ˆ
D
,
D
αβ
=
d
α
δ
αβ
,
relate the mechanical energy
E
α
to the parameters
α
deﬁning the relevant frequencies(see (6)) according to
α
=
d
α
E
α
. It is in the radiationless approximation that
ˆ
D
reduces to
ˆ
D
=
−
1
I
.
5
Those relevant frequencies that are not resonance frequencies, though important for the description of dynamical variables, do not induce transitions. This observation serves to understand why the selectionrules (or Einstein’s
A
coefﬁcients) involve the elements of
ˆ
x
and not other matrix elements.