## zero point energy (ZPE)

### INTRODUCTION

ZPE links the basic concepts of inertia, gravity, energy, electromagnetic fields, and scalar technology. Some of the effects of electromagnetic instruments and devices that act at a distance can only be explained starting with an understanding of zero point energy.

As Zpower Corporation points out: Physicists recognize that we are immersed in an energetic field!

The existence of the zero point electromagnetic energy was discovered in 1958 by the Dutch physicist M. J. Sparnaay. Mr Sparnaay continued the experiments carried out by Hendrick B. G. Casimir in 1948 which showed the existence of a force between two uncharged plates which arose from electromagnetic energy surrounding the plates in a vacuum.

Mr Sparnaay discovered that the forces acting on the plates arose from not only thermal energy (heat) but also from another type of radiation now known as classical electromagnetic zero point energy.

Mr Sparnaay determined that not only did the zero point electromagnetic energy exist in a vacuum but also that it persisted even at a temperature of **absolute zero**. This term Zero Point Energy (ZPE) has been based on the concept that even if matter were cooled down to absolute zero (minus 273oC), in terms of its temperature, this energy field still remains.

Because it exists in a vacuum, ZPE is homogeneous (uniform) and isotropic (identical in all directions) as well as ubiquitous (exists everywhere). In addition, the intensity of the energy at any frequency is proportional to the cube of that frequency. Consequently, the intensity of the energy field increases without limit as the frequency increases resulting in an infinite energy density for the radiation spectrum.

With the introduction of the ZPE into the classical electron theory, a vacuum at a temperature of absolute zero is no longer considered empty of all electromagnetic fields. Instead, the vacuum is now considered as filled with randomly fluctuating fields having the ZPE spectrum.

The special characteristics of ZPE are that it has a virtually infinite energy density and that it is ubiquitous (even present in outer space), which make it very desirable as an energy source. However, because high energy densities exist at very high frequencies, and because conventional methods are only able to convert or extract energy effectively or efficiently at lower frequencies, effectively tapping this energy source has been unavailable using conventional techniques. Consequently, ZPE which may be used to provide for society’s demanding energy needs has remained unharnessed.

Even in its ground state, a quantum system possesses fluctuations with associated zero-point energy; otherwise, the uncertainty principle would be violated. For example, electric and magnetic fields in the electromagnetic vacuum of a quantum state are fluctuating quantities.

Einstein, Hopf and Stern studied the properties of zero-point energy. Shortly thereafter Nernst proposed that empty space was filled with zero-point electromagnetic radiation – a combination of oscillating electric and magnetic fields in perpendicular orientation to each other, moving through space, and effectively transporting energy from one place to another. Then in 1925 the existence of zero-point energy was shown to be “required by quantum mechanics, as a direct consequence of **Heisenberg’s uncertainty principle**” (Sciama 1991).

In the case of zero-point energy, there are reasons for believing that a cutoff does exist in the zero-point spectrum corresponding to the Planck scale.

This results in an enormous amount of zero-point energy whose existence is assumed to be negated (in spite of the unmistakable mandate of the Heisenberg uncertainty principle) by the claim that the mass equivalent of the energy should gravitate, resulting in an absurdly large cosmological constant, contrary to observations.

In response to the question “Do Zero-Point Fluctuations Produce a Gravitational Field?” Sciama (1991) writes:

*“We now wish to comment on the unsolved problem of the relation between zero-point fluctuations and gravitation. If we ascribe an energy to each mode of the vacuum radiation field, then the total energy of the vacuum is infinite. It would clearly be inconsistent with the original assumption of a background Minkowski space-time to suppose that this energy produces gravitation in a manner controlled by Einstein’s field equations of general relativity. It is also clear that the space-time of the real world approximates closely to the Minkowski state, at least on macroscopic scales. It thus appears that we must regularize the zero-point energy of the vacuum by subtracting it out according to some systematic prescription. At the same time, we would expect zero-point energy differences to gravitate. For example, the (negative) Casimir energy between two plane-parallel perfect conductors would be expected to gravitate; otherwise, the relativistic relation between a measured energy and gravitation would be lost.”*

**Localizable differences in the zero-point energy may prove to be of some practical use** and the basis of dark energy phenomena. Asymmetries in the zero-point field that appear upon acceleration may be associated with certain properties of inertia, gravitation and the principle of equivalence.

### GENERAL BACKGROUND

Quantum mechanics predicts the existence of what are usually called ”zero-point” energies for the strong, the weak and the electromagnetic interactions, where ”zero-point” refers to the energy of the system at temperature T=0, or the lowest quantized energy level of a quantum mechanical system. Although the term ”zero-point energy” applies to all three of these interactions in nature, customarily (and hereafter in this article) it is used in reference only to the electromagnetic case.

In conventional quantum physics, the origin of zero-point energy is the Heisenberg uncertainty principle, which states that, for a moving particle such as an electron, the more precisely one measures the position, the less exact the best possible measurement of its momentum (mass times velocity), and vice versa. The least possible uncertainty of position times momentum is specified by Planck’s constant, h. A parallel uncertainty exists between measurements involving time and energy (and other so-called conjugate variables in quantum mechanics). This minimum uncertainty is not due to any correctable flaws in measurement, but rather reflects an intrinsic quantum fuzziness in the very nature of energy and matter springing from the wave nature of the various quantum fields. This leads to the concept of zero-point energy.

Zero-point energy is the energy that remains when all other energy is removed from a system. This behaviour is demonstrated by, for example, liquid helium. As the temperature is lowered to absolute zero, helium remains a liquid, rather than freezing to a solid, owing to the irremovable zero-point energy of its atomic motions. (Increasing the pressure to 25 atmospheres will cause helium to freeze.)

A harmonic oscillator is a useful conceptual tool in physics. Classically a harmonic oscillator, such as a mass on a spring, can always be brought to rest. However a quantum harmonic oscillator does not permit this. A residual motion will always remain due to the requirements of the Heisenberg uncertainty principle, resulting in a zero-point energy, equal to 1/2 hf, where f is the oscillation frequency.

Electromagnetic radiation can be pictured as waves flowing through space at the speed of light. The waves are not waves of anything substantive, but are ripples in a state of a theoretically defined field. However these waves do carry energy (and momentum), and each wave has a specific direction, frequency and polarization state. Each wave represents a ”propagating mode of the electromagnetic field.”

Each mode is equivalent to a harmonic oscillator and is thus subject to the Heisenberg uncertainty principle. From this analogy, every mode of the field must have 1/2 hf as its average minimum energy. That is a tiny amount of energy in each mode, but the number of modes is enormous, and indeed increases per unit frequency interval as the square of the frequency. The spectral energy density is determined by the density of modes times the energy per mode and thus increases as the cube of the frequency per unit frequency per unit volume. The product of the tiny energy per mode times the huge spatial density of modes yields a very high theoretical zero-point energy density per cubic centimeter.

From this line of reasoning, quantum physics predicts that all of space must be filled with electromagnetic zero-point fluctuations (also called the zero-point field) creating a universal sea of zero-point energy. The density of this energy depends critically on where in frequency the zero-point fluctuations cease. Since space itself is thought to break up into a kind of quantum foam at a tiny distance scale called the Planck scale (10-33 cm), it is argued that the zero point fluctuations must cease at a corresponding Planck frequency (1043 Hz). If that is the case, the zero-point energy density would be 110 orders of magnitude greater than the radiant energy at the center of the Sun.

How could such an enormous energy not be wildly evident? There is one major difference between zero-point electromagnetic radiation and ordinary electromagnetic radiation. Turning again to the Heisenberg uncertainty principle one finds that the lifetime of a given zero-point photon, viewed as a wave, corresponds to an average distance traveled of only a fraction of its wavelength. Such a wave ”fragment” is somewhat different than an ordinary plane wave and it is difficult to know how to interpret this.

On the other hand, zero-point energy appears to have been directly measured as current noise in a resistively shunted Josephson junction by Koch, van Harlingen and Clarke up to a frequency of about 0.6 Tz (see Abstract).

### LORENTZ INVARIANCE OF THE SPECTRUM

That the spectrum of zero-point radiation has a frequency-cubed dependence is of great significance. That is the only kind of spectrum that has the property of being Lorentz invariant. The effect of motion is to Doppler shift detected electromagnetic radiation, but a frequency-cubed spectrum has the property that up- and down-shifting of the radiation is exactly compensated, i.e. there is as much radiation Doppler shifted into a given frequency interval as there is shifted out by uniform motion.

A remarkably different phenomenon occurs when accelerating through zero-point radiation. The zero-point radiation acts upon an accelerating detector as if the detector were immersed in a thermal spectrum, even though heat and temperature are not involved. The perceived ”temperature” is directly proportional to the acceleration.

### CASIMIR EFFECT

In 1947 Hendrik Casimir, once an assistant of Pauli, was working in applied industrial research at the Philips Laboratory in the Netherlands along with physicist J. T. G. Overbeek. They were analyzing the theory of van der Waals forces when Casimir had the opportunity to discuss ideas with Niels Bohr on a walk. According to Casimir, Bohr ”mumbled something about zero-point energy” being relevant. This led Casimir to an analysis of zero-point energy effects in the related problem of forces between perfectly conducting parallel plates.

The cavity between such plates cannot sustain all modes of the electromagnetic field. In particular wavelengths comparable to the plate separation and longer are excluded from the region between the plates. This fact leads to the situation that there is a zero-point radiation overpressure outside the plates which acts to push the plates together.

This can be considered analogous to radiation pressure (radiation pressure from the Sun pushes comet tails away from the comet nucleus), and the resulting effect is now called the Casimir force. It has the property of increasing in strength with the inverse fourth power of the plate separation. The force ceases when elements of the plates come into contact, the surface smoothness of the plates being a limiting factor, or when the plates are so close that the corresponding zero-point radiation wavelengths no longer ”see” a perfectly conducting surface. The actual noncontinuous nature of the plates, as opposed to the true surface and molecular nature of the materials, becomes an important factor for very short distances.

The Casimir force was not measured to high precision until the mid 1990s, when measurements by S. Lamoreux at the University of Washington verified Casimir’s predictions to within five percent in the size range of a few microns. It has since been verified even more precisely, by U. Mohideen at the University of California at Riverside, again in agreement with Casimir’s formula. Moreover the Casimir force (also called Casimir effect) has become relevant to micro-electro-mechanical structures in which it is both a problem (termed ”stiction”) and a possible mechanism for control.

The Casimir force is widely cited as evidence that underlying the universe there must be a sea of real zero-point energy. This argument follows from Casimir’s analysis and prediction. It is not necessarily true, however. It is perfectly possible to explain the Casimir effect by taking into account the quantum-induced motions of atoms in each plate and examining the retarded potential interactions of atoms in one plate with those in the other.

### FORWARD THOUGHT EXPERIMENT

There is growing interest concerning the possibility of tapping zero-point energy and many claims exist of ”over unity devices” (gadgets yielding a greater output than the required input for operation) driven by zero-point energy. In spite of the dubious nature of these claims (to date no such device has passed a rigorous, objective test), the concept of converting some amount of zero-point energy to usable energy cannot be ruled out in principle. Zero-point energy is not a thermal reservoir, and therefore does not suffer from the thermodynamic injunction against extracting energy from a lower temperature reservoir.

In 1993 Cole and Puthoff published a thermodynamic analysis, ”Extracting energy and heat from the vacuum” (see below), in which they concluded that ”extracting energy and heat from electromagnetic zero-point radiation via the use of the Casimir force” is in principle possible without violating the laws of thermodynamics.

A thought experiment for a device that readily demonstrates how the Casimir force could be put to use in principle was proposed by physicist Robert Forward in 1984 (see below). A ”vacuum fluctuation battery” could be constructed consisting of stacked conducting plates. Applying the same polarity charge to all the plates would yield a repulsive force between plates, thereby opposing the Casimir force which is acting to push the plates together. Adjusting the electrostatic force so as to permit the Casimir force to dominate will result in adding energy to the electric field between the plates, thereby converting zero-point energy to electric energy.

One can imagine an even simpler microdevice in which the Casimir force pushes two plates together thereby engaging some kind of lever which does work.

There is no practical application in these examples since ideally it would take just as much energy, and in practice somewhat more energy owing to frictional and other losses, to separate the plates for a second cycle. Nevertheless, this would demonstrate the concept of conversion of zero-point energy in principle if the Casimir effect attribution to zero-point energy is correct (which is debatable).

### DARK ENERGY

A major discovery in astrophysics in the late 1990s was the finding from type Ia supernovae redshift-luminosity observations that the expansion of the universe is accelerating. This led to the concept of dark energy, which is in effect a resurrection of Einstein’s cosmological constant. (The universe now appears to consist of about 70 percent dark energy, 25 percent dark matter and five percent ordinary matter.) Zero-point energy has the desired property of driving an accelerated expansion, and thus having the requisite properties of dark energy, but to an absurdly greater degree than required, i.e. 120 orders of magnitude.

According to relativity theory, energy is equivalent to mass as a source of gravity, thus zero-point energy should gravitate, which according to general relativity means producing a positive curvature in space-time. At first glance one might assume that if there is an enormous amount of zero-point energy underlying the universe, its effect would be to dramatically curve the universe to a minute size. Indeed, if the spectrum of zero-point energy extends to the Planck scale, its energy density would be the mass equivalent of about 1093 grams per cubic centimeter which would reduce the universe to a size smaller than an atomic nucleus.

Zero-point energy behaves differently. For ordinary radiation, the ratio of pressure to energy density is w=1/3c2, which is customarily expressed in units whereby c=1, and thus the ratio is expressed as w=+1/3. But for zero-point energy the ratio is w=-1. This is owing to the circumstance that the zero-point energy density is assumed to be constant: no matter how much the universe expands it does not become diluted, but instead more zero-point energy is assumed to be created out of nothing.

A further peculiarity is that a ratio of w=-1 implies that the zero-point energy exerts a negative pressure which, counter-intuitively, leads to an expansion of space-time.

Thus zero-point energy would appear to be identical with the mysterious dark energy, but unfortunately if the energy spectrum does continue up to the Planck frequency, there may be 120 orders of magnitude more energy per cubic centimeter than the observations of cosmic acceleration permit. Indeed, this amount of zero-point energy, interpreted this way, would have accelerated the universe into oblivion in microseconds.

Recent work by Christian Beck at the University of London and Michael Mackey at McGill University may have resolved the 120 order of magnitude problem. In that case dark energy is nothing other than zero-point energy. In Measureability of vacuum fluctuations and dark energy and Electromagnetic dark energy they propose that a phase transition occurs so that zero-point photons below a frequency of about 1.7 THz are gravitationally active whereas above that they are not. If this is the case, then the dark energy problem is solved: dark energy is the low frequency gravitationally active component of zero-point energy.

Zero-point photons continue to exist above the 1.7 THz phase transition, consistent with measurable QED effects such as the Casimir effect, the Lamb shift, etc. The proposed phase transition should be testable in the near future when the Koch et al. experiment is extended from 0.6 Tz to the proposed cutoff.

### STOCHASTIC ELECTRODYNAMICS THEORY

Although zero-point energy is usually regarded as a quantum phenomenon and a consequence of the Heisenberg uncertainty relationship, the existence of zero-point energy was inferred by Einstein, Planck, Nernst and others in the context of blackbody radiation prior to the discovery of quantum mechanics. Einstein and Otto Stern came close to deriving the blackbody function without assuming quantization but with the presence of zero-point energy.

Nernst in particular claimed in 1916 that the universe was filled with zero-point energy. This line of investigation was abandoned with the advent of quantum mechanics, but the concept of zero-point energy soon reemerged with a quantum interpretation.

In the 1960s British physicist Trevor Marshall and, separately, American Timothy Boyer were two of the principal investigators who essentially took up the abandoned approach and pushed it much farther by asking the question: which quantum phenomena might be explained using solely classical physics plus an assumed classical representation of a zero-point field with zero-point energy? For the contribution of other researchers, see the book “The Quantum Dice” by de la Pena and Cetto (below). This became the discipline known as stochastic electrodynamics (SED, earlier sometimes referred to as random electrodynamics). In the SED representation the zero-point field is taken to be a given, and is treated as an ensemble of ordinary electromagnetic plane waves having an energy 1/2 hf in each and every mode. There is no quantum physics involved.

This theory has had some success, although it is far from explaining most quantum effects. Apart from its ontological aspirations of possibly doing away with quantum physics in favor of solely classical physics, SED is useful as a computational tool since it involves well-known classical electrodynamics in place of more esoteric quantum laws and processes.

Two noteworthy successes of SED are its derivation of the Planck blackbody function without assuming quantization and its suggestion that the Bohr orbit of hydrogen could arise without a quantum law. In the latter case, the ground state electron is assumed to emit Larmor radiation which causes it to spiral inward, but this does not lead to collapse of the orbit because the electron also absorbs zero-point energy. The calculation of the absorption was done by Boyer and later by Puthoff by treating the electron as undergoing harmonic oscillation rather than true motion in a Coulomb potential. This is a weakness in the analysis but nonetheless it is striking that the Larmor emission and harmonic-oscillator-type absorption prove to be in balance exactly at the Bohr radius. The fact that the orbital angular momentum is zero in the quantum ground state is mirrored in the SED orbiting-electron interpretation by random changes in the orbital plane (due to the zero-point fluctuations) yielding a time averaged zero net angular momentum.

Recent simulations by Cole have successfully modeled the electron motion in the Coulomb potential of a hydrogen atom and have thereby replicated the electron probability density predicted by the Schroedinger wave function. In the SED case, the electron in a Coulomb field is jostled by its emission and absorption to a range of radial distances which reproduce the Schroedinger probability. This is an intriguing extension of the earlier result, but problems still remain such as the need to cut off the particle-field interactions to avoid autoionization, i.e. a single very high frequency, hence very energetic, zero-point fluctuation could free the electron.

The representation of the zero-point field as an ensemble of plane waves each with an energy of precisely 1/2 hf in all possible directions and random phases was modified in 1995 by Ibison and Haisch. They added a parameter having a random distribution of energies with 1/2 hf as the mean, thereby yielding a closer formal correspondence with the quantum behaviour.

### ZITTERBEWEGUNG

Schroedinger was apparently the first to note that solving the Dirac equation for the motion of the electron resulted in a necessary component that could be interpreted as random, speed-of-light fluctuations of a point-like particle. He dubbed this motion ”zitterbewegung” (German for ”jitter motion”). In SED theory, the phenomenon of zitterbewegung is caused by the electromagnetic zero-point fluctuations.

Several things are interesting about zitterbewegung. First, since the fluctuations occur at the speed of light, then at this level the electron would have to be massless, mass arising at some higher level of motion. Secondly, the fluctuations smear out the average position over a volume the Compton radius in size, which suggests a physical interpretation of the wave function and the associated probability density. (Scattering experiments indicate that the electron is far smaller than its Compton size, indeed point-like for all we know.) Thirdly, simulations that have recently been done show that if such a massless, fluctuating point particle is accelerated in an electric field, the zitterbewegung acquires a helical motion suggestive of spin.

The possible association of zitterbewegung with spin has been made by a number of authors over the years such as Barut and Zanghi, Hestenes, Huang, Weisskopf, etc.

Zitterbewegung thus suggests possibly deep connections between zero-point energy and the mass-energy relationship of matter and with the quantum properties of particles.

### SPECULATIVE CONNECTION TO INERTIAL AND GRAVITATIONAL MASS

Click here for a new popular-level overview on this topic by Marcus Chown.

The connection in SED theory between zitterbewegung and the zero-point fluctuations have led to speculative investigations of a possible mass-generating role as an alternative to the Higgs field. The Higgs field was first proposed in 1964 and is still a key element of the Standard Model of particle physics; it is needed to confer the property of mass on the fundamental particles. In the theory, all particles are intrinsically massless until acted upon by the Higgs field. The quantum of the Higgs field is the Higgs boson.

Attempts to detect the Higgs boson, and therefore to verify the Higgs field as the mass-generating mechanism of the Standard Model, have been unsuccessful. The current best hope is on the forthcoming Large Hadron Collider at CERN scheduled to go on line in May 2008.

Even if the Higgs field is experimentally discovered, however, that will still not explain the origin of inertial mass of ordinary matter. The Higgs field applies only to the electro-weak sector of the Standard Model. The mass of ordinary matter is overwhelmingly due to the protons and neutrons in the nuclei of atoms. Protons and neutrons are comprised of the two lightest quarks: the up and down quarks. The masses of their constituent quarks (approx. 0.005 and 0.010 GeV/c2 for the up and down quarks respectively) comprise only about one percent of the masses of the protons and neutrons (0.938 and 0.940 GeV/c2 respectively). The remainder of the mass would have to be due to the gluon fields and strong interaction energies. The quark masses, the gluon fields and other strong interaction energies would not be affected by a Higgs field. The origin of inertial mass of ordinary matter is thus a wide open question.

SED studies published in the 1990s showed that a massless point-charge oscillator accelerating through the zero-point field will experience a Lorentz force (from the magnetic components of the zero-point fluctuations) that turns out to be directly proportional to acceleration, allowing the derivation of the fundamental F=ma relationship of mechanics from electrodynamics. This points to the electromagnetic quantum vacuum as the origin of forces which appear as inertial mass. The same result can be derived by considering the transformation properties of the electromagnetic field when experienced in an accelerating coordinate system, and in that case the proper four-vector relativistic equation of motion can be derived.

A recent study showed that such a zero-point field based mass-generating approach would explain the origin of Einstein’s principle of equivalence. These as yet still speculative concepts suggest that zero-point energy may be involved in some of the most fundamental properties of matter. It should be noted that this unorthodox approach to mass based upon electrodynamics is not taken very seriously by the mainstream physics community, whose efforts remain focussed on superstring- and M-theory.

### IS IT POSSIBLE TO TAP ZERO POINT ENERGY?

As to whether zero-point energy may become a source of usable energy, this is considered extremely unlikely by most physicists, and none of the claimed devices are taken seriously by the mainstream science community. Nevertheless, SED interpretation of the Bohr orbit (above) does suggest a way whereby energy might be extracted. Based upon this a patent has been issued and experiments have been underway at the University of Colorado (U.S. Patent 7,379,286).

http://www.frequencyfoundation.com/2008/05/biodisks-and-zero-point-energy.html

### LITERATURE

(See Scientific Articles for additional articles.

Click here for new popular-level overview by Marcus Chown.)

Gravity and the Quantum Vacuum Inertia Hypothesis

Alfonso Rueda & Bernard Haisch, Annalen der Physik, Vol. 14, No. 8, 479-498 (2005).

Review of Experimental Concepts for Studying the Quantum Vacuum Fields

E. W. Davis, V. L. Teofilo, B. Haisch, H. E. Puthoff, L. J. Nickisch, A. Rueda and D. C. Cole, Space Technology and Applications International Forum (STAIF 2006), p. 1390 (2006).

Analysis of Orbital Decay Time for the Classical Hydrogen Atom Interacting with Circularly Polarized Electromagnetic Radiation

Daniel C. Cole & Yi Zou, Physical Review E, 69, 016601, (2004).

Inertial mass and the quantum vacuum fields

Bernard Haisch, Alfonso Rueda & York Dobyns, Annalen der Physik, Vol. 10, No. 5, 393-414 (2001).

Stochastic nonrelativistic approach to gravity as originating from vacuum zero-point field van der Waals forces

Daniel C. Cole, Alfonso Rueda, Konn Danley, Physical Review A, 63, 054101, (2001).

The Case for Inertia as a Vacuum Effect: a Reply to Woodward & Mahood

Y. Dobyns, A. Rueda & B.Haisch, Foundations of Physics, Vol. 30, No. 1, 59 (2000).

On the relation between a zero-point-field-induced inertial effect and the Einstein-de Broglie formula

B. Haisch & A. Rueda, Physics Letters A, 268, 224, (2000).

Contribution to inertial mass by reaction of the vacuum to accelerated motion

A. Rueda & B. Haisch, Foundations of Physics, Vol. 28, No. 7, pp. 1057-1108 (1998).

Inertial mass as reaction of the vacuum to acccelerated motion

A. Rueda & B. Haisch, Physics Letters A, vol. 240, No. 3, pp. 115-126, (1998).

Reply to Michel’s “Comment on Zero-Point Fluctuations and the Cosmological Constant”

B. Haisch & A. Rueda, Astrophysical Journal, 488, 563, (1997).

Quantum and classical statistics of the electromagnetic zero-point-field

M. Ibison & B. Haisch, Physical Review A, 54, pp. 2737-2744, (1996).

Vacuum Zero-Point Field Pressure Instability in Astrophysical Plasmas and the Formation of Cosmic Voids

A. Rueda, B. Haisch & D.C. Cole, Astrophysical Journal, Vol. 445, pp. 7-16 (1995).

Inertia as a zero-point-field Lorentz force

B. Haisch, A. Rueda & H.E. Puthoff, Physical Review A, Vol. 49, No. 2, pp. 678-694 (1994).