Limits on Quantum Coherent Domains in Liquid Water

Abstract

A Quantum Field Theoretic analysis has led to the claim that liquid water supports coherent domains of almost millimeter size [E. Del Giudice, G. Preparata, G. Vitiello, Phys. Rev. Lett. 61, 1085 (1988)]. Such domains would be described by one quantum mechanical state function. Further analysis results in new characteristic frequencies and in the claim that a long-range (> 100 μm) structure emerges around a molecular size dipole. The quantum-physics-based claim that liquid water supports structures of over 100 micrometer in size at room temperature is irreconcilable with a well-known consensus of condensed matter physics: Brownian collisions make wave functions collapse and hot, wet environments do not allow for quantum entanglements to survive. Simulations, theory, and experiment agree on how a hydration shell of a few layers of directed water dipoles forms around an ion or a polar molecule. Such a shell extends to less than a nanometer. We reexamine the assumptions and theory behind the coherent domain dynamics in water. It appears likely that large, longlasting coherent domains do not emerge in liquid water.

Full text

Vol. 49 (2018) ACTA PHYSICA POLONICA B No 9
LIMITS ON QUANTUM COHERENT DOMAINS
IN LIQUID WATER
Martin Bier
Dept. of Physics, East Carolina University, Greenville, NC 27858, USA
and
Faculty of Mechanical Engineering and Institute of Mathematics and Physics
University of Technology and Life Sciences, 85-796 Bydgoszcz, Poland
David Pravica
Dept. of Mathematics, East Carolina University, Greenville, NC 27858, USA
(Received June 13, 2018; accepted July 18, 2018)
A Quantum Field Theoretic analysis has led to the claim that liquid
water supports coherent domains of almost millimeter size [E. Del Giudice,
G. Preparata, G. Vitiello, Phys. Rev. Lett. 61, 1085 (1988)]. Such domains
would be described by one quantum mechanical state function. Further
analysis results in new characteristic frequencies and in the claim that a
long-range (>100 µm) structure emerges around a molecular size dipole.
The quantum-physics-based claim that liquid water supports structures of
over 100 micrometer in size at room temperature is irreconcilable with a
well-known consensus of condensed matter physics: Brownian collisions
make wave functions collapse and hot, wet environments do not allow for
quantum entanglements to survive. Simulations, theory, and experiment
agree on how a hydration shell of a few layers of directed water dipoles
forms around an ion or a polar molecule. Such a shell extends to less
than a nanometer. We reexamine the assumptions and theory behind the
coherent domain dynamics in water. It appears likely that large, long-
lasting coherent domains do not emerge in liquid water.
DOI:10.5506/APhysPolB.49.1717
1. Introduction
In 1988, a Letter was published with the remarkable title “Water as a
Free Electric Dipole Laser” [1]. The guiding idea of the Letter is that a
rotating water molecule implies a rotating dipole. Such a rotating dipole
emits radiation. With that radiation, stimulated-emission dynamics can oc-
cur. The formalism developed for water was similar to one developed earlier
for a free electron laser [2]. Thus, the Free Electron Dipole Laser (FEDL).
(1717)

1718 M. Bier, D. Pravica
Working out the idea involves a body of nontrivial Quantum Field Theory.
The analysis resulted in a remarkable assertion: “In the study of the dynam-
ics of water, the neglect of the coherent interaction of water molecules with
the quantized radiation field is completely unjustified, for we have shown
that its coupling to collective quantum states of size 2π/ω0(a few hundreds
of microns) is indeed very large”. The found long-range interaction was the
basis for the claim that “a macroscopic permanent polarization can easily
arise in water in the presence of a small electric disturbance such as the local
field produced by a macromolecule”. New characteristic frequencies in the
absorption spectrum of water were also derived.
The statements are all the more remarkable as they contradict what
standard physical-chemistry textbooks describe [3,4]. The consensus picture
is that around a dissolved ion or dipole, the water molecules orient their
dipoles and form a nanometer-scale hydration shell. Around an ion, there
is a stiff inner shell of one layer of water molecules. Immediately beyond
this, an outer shell consists of one or two more layers that are less rigid.
Further away, the hydration bonds do not have sufficient energy to withstand
Brownian motion with its characteristic energy of kBT. Hydration shells
around proteins and other biomolecules are the subject of much research.
Experimental observation appears to be in a good agreement with the results
of theory and molecular dynamics simulation [5,6]. In Ref. [7], it is shown
that the hydration shell around a big protein also extends to no more than
about a nanometer away from the protein. The long-range interactions that
are claimed in Ref. [1] cover a distance that is more than 105times larger.
Google Scholar lists 434 citations for Ref. [1] (accessed May 7, 2018). The
claimed long-range order is regularly used to give plausibility to puzzling
phenomena in physics, chemistry, and biology (see e.g. Refs. [8–11]). In
addition, it has been put forward as the physics that underlies cold fusion
[12,13]. It has been presented as the basic science behind “water memory”
and homeopathy [14,15]. It has also been put forward as the scientific
validation behind nonconventional approaches to life and health [16].
At the end of Ref. [1], it is written: “Even though, it must be admitted,
our analysis is in many ways at a preliminary, rather rudimentary stage . . . ”
and “Of course much more work is needed in this direction”. However, in
spite of the spectacular claim about long-range structures in liquid water,
little further, independent research has been done into the quantum-field-
theoretic analysis that underlies the claim. There is an obvious reason for
this. Quantum Field Theory is mostly about interactions between subatomic
particles; liquid water is not generally in the realm of interest of quantum
field theorists. And while the behavior of water as a solvent is central to
much of Biochemistry and Biophysics, the standard toolbox in these fields
does not contain Quantum Field Theory. In 2007, Philip Ball wrote the

Limits on Quantum Coherent Domains in Liquid Water 1719
following about homeopathy, water memory, and Ref. [1] in his Nature News
column: “This ‘field’ has acquired its own deus ex machina, an unsubstanti-
ated theory of ‘quantum coherent domains’ in water proposed in 1988 that is
vague enough to fit anything demanded of it” [17]. The abstruseness of the
Letter has indeed been an impediment to a good discourse on the methods
and results of Ref. [1]. Below, we will summarize and critically reexamine
the main line of argument of Ref. [1].
There have been refinements, extensions, and variations of the original
quantum-coherent-domain theory of 1988. A longer article of 2006 centers
largely around the same equations, but presents a more polished interpre-
tation and discussion [18]. In 2007, the same formalism was applied to
the transition of water between the ground state and an excited state at
12.06 eV that is very close to the ionization threshold [19]. In 2009, a sim-
ilar theory with slightly differently equations was developed [20]. However,
as previously stated, Ref. [1] is still the central and most cited source on the
subject.
2. The dipole laser
Together with its dipole moment, the water molecule also has moments
of inertia. The quantum mechanical treatment of the rigid rotor [3] leads to
the discrete energy levels El=l(l+ 1)~2/(2I), where l= 0,1,2,3, . . . is the
quantum number, Iis the moment of inertia, and ~is Planck’s constant.
The analysis in Ref. [1] focusses on the l= 0 and l= 1 levels. The energy
difference between these levels equals ∆E=~2/I. With I≈2×10−47 kg m2
and ∆E=~ω0, it is found that transitions between the l= 0 and l= 1
levels are associated with photons of ω0≈5×1012 s−1,i.e. the far infrared
regime. The corresponding wavelength is about 400 µm.
With room temperature for T, we have ∆E≈0.1kBT, where kBis
Boltzmann’s constant. As ∆Eis apparently an order of magnitude smaller
than the basic “kBTquantum” of Brownian energy, there will be a 50–50
distribution over the l= 0 and l= 1 levels. Furthermore, to have signif-
icance, any dynamics that will be derived should be much faster than the
thermal equilibration to the aforementioned 50–50 distribution.
Two problems with the FEDL idea arise at this point. First of all, the
moment of inertia of a water molecule depends on the axis of rotation (see
Fig. 1). The largest of the three values is about three times the smallest
value. The relative width of the range is larger than that of the visible
window in the spectrum of light. Lasering generally requires very narrow
levels. Secondly, the rotational levels can be seen in the spectra of water
vapor [21]. For liquid water, however, they are not observed [22]. This
is because hydrogen bonds between neighboring water molecules restrict
rotation.

1720 M. Bier, D. Pravica
Fig. 1. The moments of inertia of a water molecule around three different axes.
The numbers are ×1047 and in units of kg m2.
The basic idea of Ref. [1] is that the 400 µm wavelength of the radiation
would for a very short time (Ref. [1] mentions 10−14 s, in that time the wave
moves only a few micrometers) act like a laser cavity. In that cavity, the
system then forms a “coherent domain”. That domain is described by one
quantum physical state function. Coherent domains are supposed to come
and go very rapidly, but in such a way that, at all times, there is always a
significant fraction of the water in the coherent-domain phase.
There is a precedent for this paradigm. In his “two fluid model” Landau
explains the superfluid behavior of 4He with a similar rapid coming and
going of coherent domains [23]. Superfluidity thus ultimately appears as a
macroscopic quantum phenomenon.
3. The dynamical system
In Ref. [1], an inventory is made of the different contributions to the
total energy of the system. Next, a Lagrangian is set up and equations that
describe the dynamics are derived. Below, we will just present the resulting
dynamics and make it intuitive.
The molecules have a ground state (γ0) and an excited state (γ1). There
is, furthermore, a population of photons (b) that is associated with the traffic
between the two states (see Fig. 2). The square norms of the state functions
γ0,γ1, and bgive the population in each state. Two conservation laws are
easily inferred. The conservation of the number of molecules gives
|γ0|2+|γ1|2= (λk)2.(1)
For every γ1→γ0transition, a photon appears and for every γ0→γ1, a
photon disappears. This leads to
|γ1|2+|b|2=λ2.(2)

Limits on Quantum Coherent Domains in Liquid Water 1721
Later in this section, it will become clear why λand kare used the way they
are in the above conservation laws.
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γ1=−bγ0(2)
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Fig. 2. The dynamics in a coherent domain as proposed in Ref. [1]. Water molecules
can be in two rotational states, l= 0 and l= 1. The wave functions corresponding
to these states are γ0and γ1. The photons that are associated with the transitions
between these states constitute state b.
In a classical context, Eq. (2) could be seen as a conservation of energy.
However, in Quantum Field Theory, the picture is more complex because of
an added energy term. The system also loses energy through the coupling
between the photons and the molecular dipoles. If there are Nmolecules in
the coherence domain, then we have Eγ1∝Nand Eb∝N, where Eγ1is the
energy in the excited state molecules and Ebis the energy in the photons.
It turns out that the energy in the aforementioned coupling amounts to
Eγ0,γ1,b ∝ −N√N. The coupling term essentially says that the system loses
energy whenever a photon interacts with a dipole, whether it is through
excitation or through stimulated emission. Because of the different pro-
portionalities, “∝N”vs. “∝ −N√N”, the coupling becomes relatively more
significant for a larger system. With a large enough N, and the (N√N)-term
thus being sufficiently dominant, the system will keep oscillating. In a “nor-
mal” chemical reaction with no energy input from the outside, one would
never see a sustained oscillation. The oscillations that we analyze below
are like Landau’s superfluidity in that they are a macroscopic quantum ef-
fect. However, paradoxically, our quantum effect actually only appears if
the system is large enough.
The dynamics in the phase space that is spanned by γ0,γ1, and bis
restricted to the curves that constitute the intersections of the cylinders
described by Eqs. (1) and (2) (see Fig. 3). The following autonomous system
describes the (γ0(t),γ1(t),b(t))-dynamics:
˙γ0=b∗γ1,
˙γ1=−bγ0,
˙
b=γ∗
0γ1,(3)

1722 M. Bier, D. Pravica
where ·≡d
dt. These equations are derived in Ref. [1], but it can be easily
intuited how the schematic in Fig. 2leads to these equations. Equations (3)
represent the bare bone dynamics of the system. Constants have been scaled
away. Time is dimensionless — it is the real time multiplied by ω0(the fre-
quency of the involved radiation) and by a dimensionless combination of
parameters G, that equals about 17. The asterisks denote complex conjuga-
tion. It is easily verified that Eqs. (3) imply the conservation laws (1) and (2).
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γ0=b∗γ1(1)
˙
γ1=−bγ0(2)
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b=γ∗
0γ1(3)
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γ0=b∗γ1(1)
˙
γ1=−bγ0(2)
˙
b=γ∗
0γ1(3)
ho
γ0(t)=λeiφγ0dn [λ(t−t0),k](4)
γ1(t)=λkeiφγ1sn [λ(t−t0),k
](5)
b(t)=λkeiφbcn [λ(t−t0),k](6)
1
)*$+&*"(="(>./3-(*1(:$$=?&'(@+"'&*"-(A#"B#/3*"='(*-'=$$#&*"-C(
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Fig. 3. The dynamics of the system described in Fig. 2leads to two conservation
laws given by Eqs. (1) and (2). These conservation laws describe cylinders in the
3D space that is spanned by the ground state γ0, the excited state γ1, and the
radiation b. The allowed trajectories are constituted by the intersections of the
cylinders.
Through substitution, the authors of Ref. [1] identify a periodic solution
of Eqs. (3). They claim this to be a limit cycle solution. However, this
cannot be the case. The existence of conservation laws makes clear that
Eqs. (3) cannot be a system where dissipation makes a phase space contract
to an eventual single limit cycle.
For γ0,γ1, and bbeing real, Eqs. (3) are actually the Euler equations for
rigid body rotation [24]. The Euler equations describe standard anharmonic
oscillations and they are solved in terms of elliptic functions. In the 1980s,
a complex generalization exactly like Eqs. (3) was derived in the context of
magnetic monopole dynamics. This generalization is known as the Nahm
equations [25].
The following solution is readily derived:
γ0(t) = λkeiφγ0cn [λ(t−t0), k],
γ1(t) = λkeiφγ1sn [λ(t−t0), k],
b(t) = λeiφbdn [λ(t−t0), k].(4)

Limits on Quantum Coherent Domains in Liquid Water 1723
Here, the dn, sn, and cn denote the standard elliptic functions [26,27].
As Eqs. (3) constitute an autonomous system (no explicit time dependence
on the right-hand sides), the “starting time” t0is a free parameter. Next,
it is easily verified that there is a scaling invariance in Eqs. (3): if any
(f1(t), f2(t), f3(t)) solves the system, then so does (λf1(λt), λf2(λt), λf3(λt)).
We thus have λas a next free parameter. The parameter kis a real number
between 0 and 1 that characterizes the elliptic function. For k= 0, the
elliptic functions are ordinary sines and cosines again. For k= 1, they are
hyperbolic secants (sech) and a hyperbolic tangent (tanh). If γ0,γ1, and b
are real, then the free parameters t0,λ, and kare all that is necessary to
match the initial conditions.
We took unusual symbols for the conserved quantities on the right-hand
sides in Eqs. (1) and (2). However, at this point, our choice conveys the
straightforward relation between the conserved quantities and the parame-
ters λand kof solution (4).
For the parameter k, we have 0< k < 1. This means that solution (4)
has to be rearranged in case the initial conditions are such that |γ0|2>|b|2.
In that case, we take for the conservation laws
|γ0|2+|γ1|2=λ2and |γ1|2+|b|2= (λk)2,(5)
and formulate the solution as
γ0(t) = λeiφγ0dn [λ(t−t0), k],
γ1(t) = λkeiφγ1sn [λ(t−t0), k],
b(t) = λkeiφbcn [λ(t−t0), k].(6)
The period Tof the elliptic functions sn(u, k), cn(u, k), and dn(u, k) is
given by the complete elliptic integral of the first kind [26]: T= 4K(k) =
4Rπ/2
0(p1−k2sin2θ)−1dθ. After some algebra, we can straightforwardly
express our period Tin terms of initial conditions γ0(0), γ1(0), b(0). With
k2= min |γ0(0)|2+|γ1(0)|2
|γ1(0)|2+|b(0)|2,|γ1(0)|2+|b(0)|2
|γ0(0)|2+|γ1(0)|2and
λ2= max |γ0(0)|2+|γ1(0)|2,|γ1(0)|2+|b(0)|2,(7)
we have for the period T(k, λ),
T(k, λ)=4K(k)/λ . (8)

1724 M. Bier, D. Pravica
Linear dynamical systems, like the harmonic oscillator ¨x=−ω2x, have
a characteristic frequency. Limit cycles, as commonly occurring in nonlin-
ear dynamical systems, also have a characteristic frequency. Equations (3),
however, have no characteristic frequency. Like in the case of the pendulum
(¨x=−sin x) or other anharmonic oscillators (like ¨x=−x3), the oscillations
described by Eqs. (3) have frequencies that are related to the amplitude. The
solutions of Eqs. (3) are closed curves (see Figs. 3and 4). Together, these
curves fill the entire phase space. Different curves correspond to different
frequencies.
If γ0,γ1, and bare complex valued, then the solution should contain
6 free parameters. Equations (4), therefore, contain the imaginary powers
of e. These terms represent constant phase factors. It is easily deduced
through substitution that, in order to satisfy Eqs. (3), the phases have to
satisfy the relation: φγ0−φγ1+φb=π. Thus, Eqs. (4) constitute a 5-free-
parameter solution. In Section 5, it will be illustrated that this solution does
exhibit the required gauge invariance, i.e., a mere phase shift does not affect
the relevant physics.
Though not trivial, it is ultimately feasible to write down a general 6-free-
parameter solution of Eqs. (3) [28]. For each of the variables γ0(t),γ1(t), and
b(t), a 2nd order nonlinear ordinary differential equation can be formulated.
For γ1, the equation is ¨γ1=−(A−2|γ1|2)γ1, where A=λ2+ (λk)2is a
constant. Next, we separate out γ1into its real and imaginary part, i.e.,
γ1(t) = x(t) + iy(t). The resulting differential equations for ¨xand ¨yare
¨x(t) = −A−2r2(t)x(t)and ¨y(t) = −A−2r2(t)y(t), where r2(t) =
x2(t) + y2(t). It is easily ascertained that these equations describe a motion
in the xy-plane that is ruled by a central force that derives from a potential
V(r) = Ar2−r4/2. With such a motion, the total energy and the angular
momentum are conserved, thus giving two free constants. It is interesting to
note that the solutions that we identified in Eqs. (4) and (6) have a constant
phase. That means that in the complex xy-plane, they stay on one and
the same line through the origin and thus have a zero angular momentum.
Two-dimensional motion in a central force field is a much studied problem
and our V(r) = Ar2−r4/2leads to general solutions for x(t)and y(t)in
terms of elliptic functions [29], though the forms are more complicated than
Eqs. (4) and (6).
In Ref. [1], harmonic oscillations are substituted into the dynamical
system. Indeed, upon substitution of γ0=Γ0exp [i(Ω0t+θ0)],γ1=Γ1
exp [i(Ω1t+θ1)], and b=B0exp [i(Ωbt+θb)] into Eqs. (3), five independent
algebraic equations relating the Γs, the B0, the Ωs, and the θs are derived.
However, the obtained 4-parameter family of solutions is ultimately not dy-
namic in nature. In this family of solutions, the phase factors change, but
the pertinent quantities |γ0|2,|γ1|2, and |b|2do not change. Any solution in

Limits on Quantum Coherent Domains in Liquid Water 1725
this family is characterized by a point in Fig. 3that remains stationary. In
the context of the motion in the central force field that was discussed in the
previous paragraph, this family of solutions corresponds to circular motion
and a nonzero angular momentum. There are no small oscillations around
the solutions in this family; if a solution is not represented by a stationary
point in Fig. 3, then the only way it can move is around on the curve that
is constituted by the intersection of the two cylinders.
4. The effect of Brownian collisions
For liquid water, the size of the molecule roughly equals the mean free
path — it is about 0.25 nm. At room temperature, the speed of a water
molecule is about 600 m/s. It is readily evaluated that this implies about a
trillion (1012) collisions per water molecule per second. A coherent domain
of 400 µm×400 µm×400 µm would contain about 1018 molecules. In
Ref. [1], the lifetime of a coherence domain is identified with approximately
one period of an oscillation described by Eqs. (3). If we take this to be the
10−14 s that is mentioned in Ref. [1], then we find that there are about 1016
collisions in the coherent domain during its lifetime. There is also a smaller
number of collisions between “domain molecules” and molecules from the
bath outside the domain.
Such Brownian collisions are not without consequence. Energy is ex-
changed in these collisions. For our system, the collisions cause added traffic
between the ground state, the excited state, and the radiation (cf. Fig. 2).
They also cause traffic between others energies (translational, vibrational,
etc.) and the rotational energies that are the focus of our analysis. Brownian
collisions are the underlying mechanism behind thermal equilibration. The
Brownian collisions add a noise-term and a dissipation-term to the coherent
domain dynamics of Eqs. (3). The dissipation coefficient is related to the
amplitude of the noise through the Fluctuation–Dissipation Theorem [30].
Brownian collisions and the ensuing fluctuation-dissipation are the molec-
ular mechanism that underlies the Second Law of Thermodynamics. In a
system without energy input from the outside (the system in Fig. 2indeed
has no such input), Brownian collisions will bring the system to a thermo-
dynamic equilibrium, i.e. a stationary state in which the energy is equally
distributed over the available degrees of freedom. It is in violation of the
Second Law if a closed system with 1018 densely packed water molecules in
the liquid state exhibits a sustained oscillation between the `= 0 and `= 1
state.
The anharmonic oscillator with just an added noise term has been stud-
ied recently [31,32] and it is at the limit of mathematical analyzability.
However, even without detailed mathematical analysis, it can be readily un-

1726 M. Bier, D. Pravica
derstood and intuited that even a small amount of noise and dissipation will
disrupt the nested-closed-curves topology that is depicted in Fig. 4. Brow-
nian motion will cause the curves in Fig. 4to not close upon themselves.
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01/02/
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Fig. 4. The solutions of Eqs. (3) that are depicted in Fig. 3are circles when pro-
jected on the (γ0, γ1) or (γ1, b) plane. Unlike for the harmonic oscillator, these
circular trajectories have a frequency that changes with the radius. Brownian col-
lisions between molecules add noise and dissipation to the system. The addition
of noise and dissipation will result in the trajectories not closing upon themselves
anymore.
Brownian noise would not destroy a limit-cycle topology. Limit cycles
are robust and structurally stable. With added noise, a limit-cycle system
would move in a narrow band around the limit cycle. It would still ex-
hibit the characteristic frequency. Limit cycles, however, only occur in open
nonequilibrium systems, i.e. systems that transport or convert incoming en-
ergy before they put it out again.
In the previous section, it was shown how the different closed curves
correspond to different frequencies. Without Brownian noise, the frequency
of the oscillation depends on the initial conditions. The effect of Brownian
noise is that that frequency is not maintained (see Fig. 4).
5. The permanent polarization
The most spectacular claim in Ref. [1] is that the coherent domain dy-
namics describes how a “sizable electric dipole” from a “macromolecule” can
lead to a polarization across the entire almost millimeter-magnitude coherent
domain. This claim has commonly been put forward as the theoretical un-
derpinning when anomalies in liquid water, with or without small concentra-
tions of solute, are observed and examined (see, for instance, Refs. [33,34]).
Below, we show how the pertinent derivation is done and assess the veracity
of the claim.

Limits on Quantum Coherent Domains in Liquid Water 1727
For the state functions γl(t), with l= 0 for the ground state and l= 1
for the excited state, we take
γ0(t) = λeiφ0cn [λ(t−t0), k], γ1(t) = λeiφ1sn [λ(t−t0), k].(9)
The orientation of the water dipole is given by the spherical harmonics Ym
l.
These spherical harmonics are generally known from their use in describing
the wave function of the electron in the hydrogen atom for the different
energy levels l[3]. For l= 0, there is one harmonic (m= 0) and for l= 1,
there are three (m=−1,m= 0, and m= 1)
Y0
0(θ, ϕ) = 1
2r1
π,
Y−1
1(θ, ϕ) = r3
8πsin θ e−iϕ ,
Y0
1(θ, ϕ) = r3
4πcos θ ,
Y1
1(θ, ϕ) = −r3
8πsin θ eiϕ .(10)
Here, θis the angle of the dipole with the z-axis and ϕis the angle with the
x-axis of the projection of the dipole on the xy-plane. As we saw before, the
energy difference between the ground state and the excited state is small
(about 0.1kBT). Thus, the four available states are taken to be equally
populated. For the not-normalized state function χ(t, θ, ϕ)of the entire
system, one then has
χ(t, θ, ϕ) = γ0(t)Y0
0(θ, ϕ) + γ1(t)Y−1
1(θ, ϕ) + Y0
1(θ, ϕ) + Y1
1(θ, ϕ)e−iωt .
(11)
Here, ωis the angular velocity associated with the l= 1 level as explained in
the first paragraph of Section 2 (though it should be realized that the time
“t” in Eq. (11) is the new dimensionless unit). To evaluate the polarization
Pzin the z-direction, we take the projection of the dipole on the z-axis and
average over the entire unit sphere, i.e. Pz∝ hχ|(u·ez)|χi. Here, udenotes
a vector in the direction of the dipole. The problem reduces to an integration
over the unit sphere, Pz∝Rχ(cos θ)χ∗dΩu, that can be evaluated to yield
Pz∝cos [φ0−φ1+ωt] sn [λ(t−t0), k] cn [λ(t−t0), k].(12)
The original physical setup is isotropic, i.e., the directions are not physically
distinguishable. So there should be no polarization. Equation (12) indeed
shows that Pzhas a zero average. The cosine term has a zero average and

1728 M. Bier, D. Pravica
so does the product of the sn and the cn. The latter can be easily seen after
realizing that Rsn ucn udu∝dn uand that dn uoscillates symmetrically
around its average.
It can also be ascertained from Eq. (12) how the φ0and φ1represent
gauge freedoms. These factors can produce a phase shift of the cosine, but
they will leave the physically relevant zero-average unaffected.
Next, we add the “sizable electric dipole” of Ref. [1]. Reference [1] models
the presence of the “sizable electric dipole” with the presence of a constant,
homogeneous electric field Ethat extends over the entire coherent domain.
The reality of such modeling is questionable. First of all, it is hard to imagine
how even a polymer of micrometer order length could give rise to an electric
field that is constant and homogeneous over several hundreds of micrometers.
Secondly, the movable water dipoles screen the fields of dissolved dipoles and
mitigate them. In biological solutions, there is the added effect of dissolved
small ions like sodium, potassium, and chloride. The so-called Debye–Hückel
screening (chapter 10 in Ref. [3] and chapter 12 in Ref. [4]) that results from
the presence of these ions is very effective and essentially eliminates any field
within nanometers. Thirdly, a macromolecule is also subject to Brownian
fluctuations. The individual charged groups twist and turn, and it is hard
to imagine them giving rise to a constant homogeneous electric field.
A constant homogeneous field that extends over the entire coherent do-
main breaks the isotropy of the system. With Curie’s Principle (a symmetry
or asymmetry of a cause is always preserved in its effects [35]) in mind, it is
no surprise that a constant homogeneous electric field in a medium of dipoles
will give rise to a net polarization. It would actually have been a surpris-
ing violation if there had been no ensuing polarization. By postulating a
macroscopic constant homogeneous electric field, the authors of Ref. [1] are
already assuming what they are trying to prove.
It does not require Quantum Field Theory, or even quantum physics or
any form of quantum entanglement, to have dipoles align themselves with
a constant homogeneous electric field. Already a century ago, Peter Debye
considered dipoles making up a classical gas or liquid. His dipoles are subject
only to Brownian motion. He derived analytical formulae for the polarization
in such a medium when a constant homogeneous electric field is imposed.
These formulae have been successful in accounting for experimental results
and are now a standard feature in many authoritative textbooks (chapter 15
of [3], chapter 5 of [4], and Appendix 13 of [36]). We will, nevertheless, follow
the idea of Ref. [1] and derive a formula for the polarization but we will enter
the solutions that we derived in Section 3.
The eventual polarization is calculated with a perturbation approach.
The presence of an electric field Eadds a term H=−d·Eto the Hamilto-
nian, where dis the dipole of a water molecule. Without loss of generality,

Limits on Quantum Coherent Domains in Liquid Water 1729
the field Eis assumed to be in the z-direction. The perturbation leads to a
“mixing” of the l= 0 state and the l= 1 state,
Y0
0
Y0
1new
=cos τsin τ
−sin τcos τ Y0
0
Y0
1,
where the parameter τdepends on the dipole strength d, the electric field E,
and the frequency ω. A good explanation of this “state mixing” is found in
Ref. [37]. It is a perturbation treatment and one is looking at the effect of the
perturbing Hamiltonian on the unperturbed state. Doing again the integral
Pz∝Rχ(cos θ)χ∗dΩu, but now with the “new” Y0
0and Y0
1substituted in
Eq. (11), it is found that
Pz∝cos(2τ) cos(φ0−φ1+ωt) sn [λ(t−t0), k] cn [λ(t−t0), k]
+ sin(2τ)1−2 sn2[λ(t−t0), k].(13)
The expression from Eq. (12) now comes with a prefactor cos(2τ). There
is a new term that goes with a sin(2τ)prefactor and that does not involve
the ω-oscillation. The long-time average of the term in curly brackets is
nonzero if k6= 0. By evaluating the integral of this term over one period
and dividing by this period, the long-time average can be obtained. This
can be done with standard methods [26,27] and yields
hPzi ∝ sin(2τ)1−2/k2{1−E(k)/K(k)}.(14)
Here, K(k)is again the elliptic integral of the first kind that we encoun-
tered before, and E(k)is the elliptic integral of the second kind (E(k) =
Rπ/2
0p1−k2sin2θdθ).
6. Discussion
The claim by way of quantum physics that liquid water supports struc-
tures of almost millimeter size at room temperature appears to be at odds
with a well-known fact of condensed matter physics: Brownian collisions
destroy quantum entanglement and make wave functions collapse. The is-
sue became particularly salient when, in the 1990s, Penrose and Hameroff
proposed that the units in a long biopolymer could be the binary units
in a quantum computer that produces “consciousness” [38]. In 2000, Max
Tegmark took issue with this idea and derived how collisions with water and
ions from the medium terminate any quantum entanglements between the
units of a biopolymer within 10−13 s [39]. There is currently a widespread
consensus among biophysicists that the aqueous environment in a living cell
is simply too hot and too wet to allow for quantum entanglement to play

1730 M. Bier, D. Pravica
a role in intermolecular interactions. This is because a collision localizes
the colliding particles and thus makes state functions collapse onto position
eigenfunctions (see e.g. Ref. [40]). In the first paragraph of Section 4, we
saw that, in liquid water, there are 1012 collisions per molecule per second
and that there are about 1018 molecules in a coherent domain. This implies
that, inside a coherent domain, there is a collision every 10−30 s. If the de-
coherence timescale is the collision timescale, then a coherent domain does
not survive beyond this completely inconsequential 10−30 s.
None of the conclusions of Ref. [1] warrants the idea of a “water memory”.
Reference [1] is silent about what happens when the macromolecules are
diluted away. There is nothing in Ref. [1] that hints at an “imprint” left by
a substance that is no longer there. Even if coherent domains were real, it
is impossible to see how they would create a “memory”.
All in all, there are good grounds to doubt the coherent domains that
Ref. [1] proposes. The theory is built on questionable premises. Next, there
are errors in the mathematical analysis. And finally, even if they are real, the
coherent domains do not give rise to the claimed characteristic frequencies
and to the long-range order.
We are grateful to Tomasz Brzeziński, Steven Yuvan, and Jan Willem
Nienhuys for valuable feedback.
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