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On this log scale, we did not consider the ultimate – shortest – time of the universe, what is now known as Planck's time. In his attempt to give a universality to constants of nature, Planck in 1899 proposed that natural units of mass, length, time, and temperature can be constructed from the most fundamental constants: the gravitation constant G, the speed of light c, and the constant of action h (which now bears his name). By dimensional analysis, the shortest possible time becomes: t(Planck) = (h G/c 5 ) 1/2 which is 10 -43 seconds, and the corresponding length is 10 - 33 cm , obtained simply by multiplying by c. Even before 1900, the year quantum mechanics began to emerge, this unity in defining Planck's time is telling of “relationships” between quantum mechanics (h), gravity (G), and relativity (c). Implicit in this unification is the meaning of physical laws at scales below these values, and the nature spacetime with a universal speed of light – Einstein enters here! Top of Page
Time, Light and Relativity Before Einstein, the great contribution by James Clerk Maxwell gave us a universal description of the nature of light. By a unification of electricity and magnetism, light, as a wave, propagates in space and time with electric and magnetic (electromagnetic) disturbances. This was a brilliant contribution expressed quantitatively in Maxwell's equations. Einstein in 1905 was concerned about two issues that relate to the nature of light – Is it really a wave? and, What happens to these waves if you can imagine running with them near the speed c? The first issue is not our concern here, but the second one is. Something is special. Whichever direction a beam of light is coming from, independent of our own velocity for observation on Earth, we will always measure c for light. Einstein, in his special theory of relativity, gave the correct picture for adding velocities: For a motion of an object (say a moving ball) with velocity v in a reference frame (say a moving train) with a velocity u, an observer will see a motion not by the expected v + u velocity, but by v + u divided by the factor (1+vu/c 2 ); when the speeds v and u are the “normal” ones, i.e., much less than c, then the total velocity is the expected (Newtonian) v + u. However, if instead of the ball we have light with speed c, then the total velocity becomes c + u divided by (1 + u/c) which is exactly c. The speed of light is the same in all reference frames, in all directions, for all observers, and every observer will experience the same natural laws. The consequences of these findings for time, length and mass require some philosophical interpretation. As the speed of light is approached, the length of a spaceship will shrink and approach zero in the direction of the motion. Similarly, moving objects become more massive and approach infinity when the object velocity becomes near the speed of light. For time, the mystery continues. Moving clocks must slow down and stop when the object velocity reaches the speed of light. In this “Dilation of Time”, time becomes relative: t(moving) = t(stationary) / (1-v 2 /c 2 ) 1/2 where the velocity of the moving clock is v. From the expression, we note that the time of the moving clock gets longer (slowing down) as v increases, but we also note that if v is made to exceed c, we enter an imaginary world of time! Thus, within the framework of this theory, the speed of light is the ultimate speed in our world and universe. Top of Page In approaching these large scales of speed and mass, what happens to light? In his 1911-1916 papers on the General Theory of Relativity, Einstein addressed the effect of gravity on light. Gravity is described as distortions in the four dimensions of space and time (3 dimensions for space and one for time) and such distortions define Newton's “force” of gravity – spacetime is actually curved. Because of this curvature, a beam of light passing near the sun would bend in the gravity of a massive object. Experimentally, it was found by Arthur Eddington during the 1919 eclipse that indeed light was bent as it passed by the sun, as predicted by the theory. In 1922, Einstein received the 1921 Nobel Prize, not for his theory of relativity but for the photoelectric effect, a contribution which elucidated one of the two characteristics (duality) of light - a bundle of particles of quantized energy.
Symmetry of Time Even if we consider the “normal world” when velocities, masses, lengths, and time are with no corrections – Newtonian Limits – and spacetime with no curvature, we still have problems with time, its direction and uncertainty. First let us consider the symmetry of time. Can time go forward and backward, or does it have a direction, an arrow? In Newton's world, the motion of objects, like you and me, should follow “symmetry of time”, i.e., the equations describing motion on say the human scale, or that of the Earth around the sun, are time symmetric. There is no difference in the way they work if we make the direction of time go “forward” or “backward”. Newtonian mechanics are deterministic and time symmetric. Because the force is related to the mass and the acceleration (F=ma=m(d 2 x/dt 2 )), the equation works equally well for positive and negative time. So, calculating the future of a physical system from its present situation is the same as calculating its past physical situation from its present one – weird and contrary to our common sense. What about microscopic systems; e.g., the world invisible to the eye – the atom. For quantum systems, the equation of motion also has invariance under time reversal insofar as the positions of microscopic particles are concerned. This is true despite the deceptive appearance of a first derivative in the Schrödinger wave equation which would imply time reversal. If you can magnify a box containing a gas and see the atoms hitting each other individually you will conclude that there is no arrow of time for every pair of collisions. So, in Newton's mechanics and quantum mechanics, time flows in both directions, making the apparent confusion for the meaning of past, present and future! In our life, we feel the passage of time and we also know that matter is made of atoms, so we have a dilemma. Top of Page
Arrow of Time Phenomena in our life follow an arrow of time. A cup of hot water with a piece of ice displays melting of the ice – the ice does not spontaneously reform again; heat always flows from a hotter object to a cooler one, and not the reverse. An egg breaks when it hits the floor, but it cannot be reformed from the floor. These and similar phenomena are described by the most powerful law, or what Arthur Eddington called the “supreme law of Nature” – the second law of thermodynamics. In one way it describes the arrow of time. In another way, it tells us about the content of information – there is a natural tendency for systems in change to become less ordered or more disordered. A measure of this change is called entropy which is defined as a negative measure of information. Entropy always increases (or at best does not change), order decreases, information decreases, and complexity decreases. But this loss of information and increase in entropy is for the socalled closed systems (the ice and hot water form a closed system). In some cases, order is created of disorder, and it appears at first that this is in violation of the law of entropy. The tree is a good example – light from the sun, soil and water, and by photosynthesis we have an ordered tree. The Earth is not a closed system and is a part of the solar system – the local decrease in entropy for the tree is compensated for by the way solar (and other) energy change its entropy, and for the solar system on a whole, entropy is increasing according to the second law. If entropy is always increasing in our universe and the arrow of time is well defined from past to future, why do individual particles, constituents of matter, follow trajectories which are symmetric in time? Put in another way, why for a collection of particles each obeying time-reversal symmetry the ensemble as a whole defines an arrow of time? Imagine a box divided into two halves with a partition, one half contains a gas and the other empty. If we remove the partition the gas will move and fill the whole box. Entropy increased and it appears that we can never reverse the process – we cannot make the gas go into one half and then reinstall the partition to acquire the originally ordered state. In the gas box each particle has a trajectory that follows Newton's mechanics, with time being symmetric, why then does the collection of these particles make the time unsymmetrical? This is a debatable subject and there are different views on the subject, one I find particularly interesting. Top of Page
Time Scales and Recurrences in Time In the nineteenth century, Henri Poincaré considered this problem of a gas in a box, with all possible arrangements of the particles. He concluded that the system, if we wait long enough, will return back to the initial state. The time for this Poincaré recurrence is vastly different depending on the system under consideration. For the gas in the box, the recurrence time for reordering all particles is longer than the age of the universe, but for the vibrational motions of atoms and molecules it could be a millionth of a millionth of a second. This concept of time scale could explain the apparent behavior of systems, reversible or irreversible, depending on complexity and the number of possible arrangements or configurations. This view is perhaps most clearly demonstrated on quantum systems with time scales short enough that we can experiment with them. If we take the same gas in the box and replace the hypothetical particles with shaped molecules we can perform an interesting experiment. To start with, we already know that there is no order in orientation of molecules and entropy is maximum. We now preferentially excite some of these molecules with their head and tails oriented roughly north and south of the box (we can do so in the laboratory with lasers). If the laser is ultrashort in duration (this too we can achieve in the laboratory) the induced ordered orientation of the molecules will ultimately be maximum at time zero and will decay with time. We call this process of degrading order rephrasing , as the whole ensemble of millions of molecules prepared becomes out of step (phase) with each other. However, if we wait for some time, the molecules will acquire back the initial orientation giving rise to Poincaré's recurrences. Such recurrences have been observed in our laboratory and on an ensemble of millions of molecules. Furthermore, these molecules are complex in their structure and internal motions and experts will tell you that these recurrences should not occur in such systems. But this is not true, as the energy levels are commensurate or nearly so even in complex systems. The recurrences are spaced long enough in time that depending on the time scale of observation the behavior of the system will appear differently – if the time scale of observation is too short, the system would appear irreversible in its decay behavior, but if we wait until recurrences occur we can then see the reversibility behavior. Top of Page Irreversibility becomes apparent if the system is not isolated. When the system interacts with a foreign perturber (such as collisions with other molecules – a heat bath) then such recurrences become weak and the system appears irreversibly disordered. Thus without designed methods for introducing order (coherence) to the system and/or without probes for observing its time evolution of disorder (dephasing) we may be misled about the nature of the dynamics. This is critical for defining the meaning and control of complexity and the time scale for reversible/irreversible behavior. We shall come back to this point when we consider measurement of time and matter's time scale. Irreversibility becomes apparent if the system is not isolated. When the system interacts with a foreign perturber (such as collisions with other molecules – a heat bath) then such recurrences become weak and the system appears irreversibly disordered. Thus without designed methods for introducing order (coherence) to the system and/or without probes for observing its time evolution of disorder (dephasing) we may be misled about the nature of the dynamics. This is critical for defining the meaning and control of complexity and the time scale for reversible/irreversible behavior. We shall come back to this point when we consider measurement of time and matter's time scale. The above consideration of microscopic/macroscopic behavior considers the origin of irreversible behavior in large ensembles as due to statistical “averaging”. As such the law of entropy increase becomes a statistical law. To Ilya Prigogine, however, the second law of thermodynamics is a fundamental law describing irreversibility of nature – the gas in the box will never rearrange again and the ice in the hot cup will never reform, no matter how long we wait. We are now entering a risky area of interpretations and I prefer to stop here until we see further experimental proofs! What about the behavior of individual atoms in molecules and their time scale? And, can we observe them moving with order in the ensemble?
At the Limit of Time – Democritus’ Atom The motion of atoms in molecules is fundamental to all dynamical changes of matter, whether the change is physical, chemical, or biological. But these atoms move with awesome rapidity and on ultrashort scales of time and length. On these scales, it is not clear that we can treat them as real, classical objects. Clearly, we must measure the passage of time for atoms on the time scale of the motion and we must develop the concepts for understanding localization of atoms in space and time. Can this be achieved at the limit of time for quantum atomic motions? If we do, we will then observe Democritus' atom in motion and as a real object, making the transformation from the microscopic (wave function description) to the macroscopic (particle description) a reality in real time. At Caltech, we have been interested in this endeavor of developing ultrafast laser light to freeze the motion of atoms, to make a motion-picture film of molecules with a frame resolution of a femtosecond. A femtosecond is a millionth of a billionth of a second, i.e., 0.000 000 000 000 001 second. In one second, light travels 300,000 kilometers ( 186,000 miles ) almost the distance from the Earth to the moon; in one femtosecond, light travels 300 nanometers (0.000 000 3 meter ), the dimension of a bacterium, or a small fraction of the thickness of a human hair. In principle, with femtosecond timing, the atom's motion becomes visible, but how can we advance stop-motion photography to reach the scale of the atom? In the nineteenth century, the motion of animals was recorded for the first time using light shutters and flashes. In France , Étienne-Jules Marey, a professor at the Collège de France, was working (1894) on a solution to the problem of action photography using chronophotography, a regularly timed sequence of images. Marey's idea was to use a single camera and a rotating slotted-disk shutter, with exposures on a single film plate or strip that was similar to modern motion picture photography. Marey applied his chronophotographic apparatus in particular to humans and animals in motion, and to a subject that had puzzled people for many years: the righting of a cat as it falls so that it lands on its feet. How does the cat do it? Does its motion violate Newton's laws of mechanics? Does the cat have some special, magical physiology or a command of some weird new physics or what? Top of Page By “slicing time” and freezing the motion during the fall, in the transition state of the righting, Marey was able to answer the questions. First, the cat rotates the front of its body clockwise and the rear part counterclockwise, a motion that conserves energy and maintains the lack of spin, in accordance with Newton's laws. It then pulls in its legs, reverses the twist, and with a little extension of the legs, it is prepared for final landing. The cat instinctively knows how to move, and high divers, dancers, and some other athletes learn how to move in the absence of torque (the pushing force that gives you momentum in one direction or another), but scientists needed photographic evidence of the individual stopped-action steps to understand the mystery. The answer to the puzzle was that the moving body was not rigid, and Newton's laws prevailed. At the time, these observations were thought-provoking and renowed scientists discussed in public their meaning and significance. J. Willard Gibbs gave a talk on December 4, 1894 before the Mathetical Club at Yale with the title “On motions by which falling animals may be able to fall on their feet”. Marey's work and that of Eadweard Muybridge on the horse have changed the way we think of the behavior of animals (and humans) in motion. For the world of atoms in molecules, if the above ideas of stop-motion photography can be carried over in a straightforward manner, then the requirements can be identified for experiments in femtochemistry—the field of studying molecular motions on the femtosecond time scale. The contrast in length and time scales for the motion of the cat and the atom is awesome (Figure 2). For a definition of 1 cm , a cat speeding at 2 m/s requires a time resolution of 0.005 second. But for a molecular structure in which atomic motions of a few angstroms (an angstrom, Å, is 10 - 8 cm ) typically characterize chemical change, a detailed mapping of the motion will require a spatial resolution of less than 1 Å (about 0.1 Å). Therefore, the shutter time, or time resolution, required to observe with high definition atoms in motion at a speed of one kilometer per second (1000 m/s) is 0.1 Å divided by 1000 m/s, which equals 10 -14 second or 10 femtoseconds—a million million times shorter than what was needed for Marey's (or Muybridge's) stop-motion photography. Although this was a central idea in the development of femtochemistry, we had to overcome a major dogma regarding the uncertainty principle! Top of Page
Solving the Riddle of Uncertainty - Physics For the atom such minute time and distance scales mean that molecular-scale phenomena should be governed by the principles, or language, of quantum mechanics, which are quite different from the familiar laws of Newton's mechanics that were used in the description of the motion of the cat and horse. Werner Heisenberg in the 1920s discovered that for quantum systems we are not allowed to make a precise measurement of both the position (x) and the momentum (p) of a particle at the same time. This tells us that we are losing knowledge – we do not know exactly where it is and where it is going (future), simultaneously, i.e., the more accurately we determine one of these conjugates the more information we lose on the other. There is intrinsic uncertainty! Similarly, if we can measure the energy (E) of a system very precisely we cannot obtain the same precision for time (t) simultaneously. There is uncertainty in the measurement of time depending on how accurate the energy is, and the consequences are important for all sciences on the ultrashort time scale. Top of Page These considerations of uncertainties led initially to the belief that the femtosecond time resolution would not be useful. Moreover, predictions suggested that localization of atoms in space—wave packets—would not be possible to sustain for a long time, even on the femtosecond scale. Finally, there is a fundamental difference in the analogy between femtosecond stop-motion action of atoms and the millisecond photography of a cat or horse - in femtochemistry experiments one probes typically millions to trillions of molecules, and/or repeats the experiment many times to provide a signal strong enough for adequate images. Unlike experiments on one cat or one horse, the picture for an ensemble of molecules would be blurred. We accommodate this by recognizing two of the most powerful and yet indigestible concepts: the uncertainty principle and the particle-wave duality of matter (de Broglie, 1924). The complementary aspect of these two descriptions is interwoven with the concept of coherence. Two or more waves can produce interference patterns when their amplitudes add up coherently. For matter, superpositions analogous to those of light waves can be formed from matter wavefunctions. The Schrödinger equation yields wavefunctions together with their probability distributions, which are diffuse over position space. But if these waves are added up coherently with well-defined phases, the probability distribution becomes localized in space. The resultant wave packet and its associated de Broglie wavelength has the essential character of a classical particle: a trajectory in space and time with a well-defined (group) velocity and position – a moving classical marble but at atomic scale! Top of Page To see motion in real systems, localized wave packets must form in every molecule, and there must also be a limited spread in position among the wave packets formed in the millions of molecules studied. This is achieved by the well-defined initial equilibrium configuration of the molecules before excitation and by the “instantaneous” femtosecond launching of the packet. The spatial confinement of the initial ground state, typically 0.05 Å, ensures that all molecules, each with its own coherence, begin their motion in a bond-distance range much smaller than that of the actual motion, typically 5-10 Å. The femtosecond launching ensures that this narrow range of bond distance is maintained throughout preparation. With coherent and synchronous preparation, the motion of the ensemble becomes that of a single-molecule trajectory. In 1987, we reached our goal of observing, for the first time, Democritus' atom—theorized by the Greek philosopher some 2500 years ago—in motion, and we could describe it on the femtosecond time scale as a classical object like the cat and horse (Figure 3). In reaching the femtosecond domain of the atom, with a scale of a millionth of a billionth of a second, the time resolution of today compared to that of a century ago, with a scale of a thousandth of a second, is like one day compared to the age of the universe.
Eugene Wigner and Edward Teller debated the uncertainty paradox for picosecond time-resolution in a lively exchange at the Welch Conference in 1972. But, because of coherence, the uncertainty paradox is not a paradox even for femtoscience, and certainly not for the dynamics of physical, chemical, and biological changes. Charles Townes encountered objections in the realization of the maser because of concern about the uncertainty principle, but coherence was again the key to success. As we cross the femtosecond barrier into the attosecond regime for studies of electron dynamics, we must recall this vital role of coherence. Otherwise the spectre of quantum uncertainty might veil the path to new discoveries. Top of Page In retrospect, this vital role of coherence in the uncertainty paradox and the fog that surrounded its utility should have been clear (Figure 3, and bibliography). We and others have considered in detail the theoretical quantum calculations of molecular systems and indeed confirmed the localized motions of atoms. But, the physical origin of the behavior is simple to understand. Considering the uncertainty in the position to be D x, and similarly for the other variables, the two uncertainty relations, Δ x Δ ≥ ħ /2 Δ t Δ E ≥ ħ /2 show that the only way to localize atoms (small Δx) is by shortening time ( Δt). Moreover, when Δt is on the femtosecond time scale, even a discrete quantum system, if excited coherently, becomes effectively a continuum or quasi-continuum of energy states, which represents a transition to the classical world. Given that we can localize a system to an initial distance of Δx o at time zero, why does the system remain coherent and behave as a classical object? And, does the time for the loss of coherence depend on the size of the object? Because the value of ħ is very small, this time depends crucially on the size. To see this clearly, we must recall that the uncertainty relation relates the uncertainty in position ( Δx) to the uncertainty in momentum ( Δp), but it is the velocity, and not momentum per se, which tells us the future position. Since Δp=m Δv, it follows, from the uncertainty relation, that Δ v= ħ /(2m Δx o ) – the larger the size (the larger the mass m and also the larger the scale of precision in position Δx o ) the smaller the uncertainty in velocity ( Δv) and the better we are in predicting the future. Now it is straightforward to calculate the “time of uncertainty” which tells us how long it will be before the uncertainty in velocity will contribute as much to our lack of knowledge of where the object is as that which came from the original position uncertainty ( Δx o ): t (uncertainty) = Δx o / Δ v = 2m Δx 2 o / ħ Beyond this time scale, the uncertainty, due to our lack of knowledge of velocity, makes us less certain of the future and the description of the object becomes quantum, not a classical one. This simplified equation can be obtained from a more rigorous treatment of wave packet motion, and elsewhere we did so. The size of ħ , 1 x 10 -27 erg-sec, means that the fuzziness required by the uncertainty principle is imperceptible on the normal scales of size and momentum, but becomes important at atomic scales. For example, if the position of a stationary 200-g apple is initially determined to a small fraction of a wavelength of light, say ħ x o = 10 nm, the apple's position uncertainty will spread by about 40% only after 4 x 10 17 s, or 12 billion years, that is, the age of the universe! On the other hand, an electron with a mass 29 orders of magnitude smaller would spread by 40% from an initial 1-Å localization after only 0.2 femtosecond. Top of Page From atom to man, the time and length of uncertainty determine the classical-quantum description of motion (Figure 4). The time scale for future uncertainty runs from femtoseconds for the hydrogen atom, to 300 years for biological cells, and to more than the age of the universe for humans – we have 300 years (or more) to behave in a deterministic classical world, so biotechnologists can be sure to improve the human life expectancy by at least three times from the current one without the need of new mechanics!
The Molecular World – Chemistry Conceptually, our work in the late 1970s on coherence phenomena and in the mid-1980s closing in to resolve reaction dynamics in real time provided the foundation for thinking about the issues raised above. It became clear that molecules can be made to vibrate coherently and ensembles of molecules can be made to behave in unison. Experimentally, we needed a whole new apparatus, a whole new “camera” with unprecedented time resolution. We needed to interface femtosecond lasers and molecular-beam technology, which required not only a new initiative but also a major effort at Caltech. In a relatively short time, femtochemistry research became active in many laboratories around the world. The breadth of applications emerging spans the very small to very complex molecular assemblies, and all phases of matter. An example that demonstrates the unity of concepts from small to large molecular systems came from a paradigmatic study made at Caltech on a sibling of table salt (two atoms) and another at Berkeley on the protein molecule of vision (hundreds of atoms). In both, the primary step involves femtosecond motion of the atoms, and we now understand better the remarkably coherent and highly efficient first step of vision at the atomic level. Top of Page
Complexity – Biology An especially exciting frontier for femtoscience is in biology. At Caltech we now have the National Science Foundation's Laboratory for Molecular Sciences (LMS) for interdisciplinary research on very complex systems. Among the recent new studies published in femtobiology are those concerned with the conduction of electrons in the genetic material, the binding of oxygen to hemoglobin (myglobin) and its mimics, molecular recognition of protein by drugs, and the molecular basis for the cytotoxicity of anticancer drugs, and of digestion. A current major problem of interest is the role of water in biological systems – biological water. The pertinent question is: How does the interaction of water molecules with proteins and DNA influence the biological function? In a series of papers we have reported our studies of this interfacial water dynamics and the unique role they play in the function. We are also developing new techniques to observe the behavior and architecture of these complex molecules—in space and time—using diffraction images, which give the 3-D location of all the atoms, all at once! But now a fourth dimension – time – is introduced to see how complex systems behave during the function. The new methodology, which we termed ultrafast electron crystallography (and microscopy), is now established with many applications (see bibliography). The impact on biology and medicine is clear. Life is a manifestation of complexity in which atoms of the microscopic world combine in different ways to form functional systems with enormous diversity and unique information. And that is what makes the human “intermediate scale” (Figure 1) special – on one hand simple in function and on the other hand rich in complexity. Deciphering this complexity and reducing its meaning to the atomic motions involved is one of the most fundamental problems of this century.
Technology of Femtoscience As for technology developments—femtotechnology—there are exciting new developments in microelectronics (femtomachining), femtodentistry, and femtoimaging (microscopy) of cells and tumors, not to mention possible new developments with intensities reaching that of the sun (in femtoseconds!) and duration going beyond the femtosecond (attosecond), and the interface with nanoscience and technology—marrying scales of time and length. The ability to count optical oscillations of more than 10 15 cycles per second will lead to the construction of all-optical atomic clocks, which are expected to outperform today's state-of-the-art cesium clocks, with a new precision limit in metrology. There is also the potential for using powers reaching 10 20 watts/cm 2 to induce nuclear fusion in clusters of atoms through Coulomb explosion. And, the possibility for controlling matter on the femtosecond time scale—one day we may direct chemical reactions into specific or new products. Top of Page
Epilogue Let me conclude by conjecturing on some future mysteries and miracles of time. In the physicial sciences, one advance that surely will allow us to reach the electron domain involves measurements on the sub-femtosecond time scale. Now the average energy is nearing the x-ray region, much above chemical and biological energies, and the pulse width is larger than chemical binding energies. Nonetheless, such advances will make it possible to study electron dynamics in many domains of physics and related areas. In the life sciences, the advent of diffraction and microscopy techniques with atomic-scale spatial and temporal resolution will undoubtedly lead to a revolution in structural dynamics of biomolecules, building real bridges between structures, dynamics and functions (bibliography). In cosmology, Planck's scale of time, the nature of spacetime, and the arrow of time are subjects that will remain in need of further discovery and search for meaning. From the very small (atom), to the very complex (life), to the very big (universe), despite some mysteries, new frontiers will be reached with time defining a fundamental dimension. Perhaps the biggest of all challenges is reversal of time. Ever since H. G. Well's novel “The Time Machine”, man's imagination has considered the possibility of reversing the arrow of time, going back in time. In theory we could, but the paradoxes are many. A time traveler may go back in time and alter circumstances leading to his own existence or lack thereof. Two-way time travel is indeed weird, and may force an entry to the world of weird physics! So despite its miracles and the impact on our life, we still struggle with the meaning of time. Bibliography
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