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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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γ0=b∗γ1(1)

˙

γ1=−bγ0(2)

˙

b=γ∗

0γ1(3)

1

C'

!',#-.-/'''''/'-1'&-5+%1-#%6+11'.5+H'/#,#0$"1''902IJ<'

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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)

˙

b=γ∗

0γ1(3)

1

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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)|2and

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/

03/

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

1new

=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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