Daily Archives: March 18, 2022

Mathematical structures that allow quantum mechanics to be explained

The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. This mathematical formalism uses mainly a part of functional analysis, especially Hilbert spaces, which are a kind of linear space. Such are distinguished from mathematical formalisms for physics theories developed prior to the early 1900s by the use of abstract mathematical structures, such as infinite-dimensional Hilbert spaces (L2 space mainly), and operators on these spaces. In brief, values of physical observables such as energy and momentum were no longer considered as values of functions on phase space, but as eigenvalues; more precisely as spectral values of linear operators in Hilbert space.[1]

These formulations of quantum mechanics continue to be used today. At the heart of the description are ideas of quantum state and quantum observables, which are radically different from those used in previous models of physical reality. While the mathematics permits calculation of many quantities that can be measured experimentally, there is a definite theoretical limit to values that can be simultaneously measured. This limitation was first elucidated by Heisenberg through a thought experiment, and is represented mathematically in the new formalism by the non-commutativity of operators representing quantum observables.

Prior to the development of quantum mechanics as a separate theory, the mathematics used in physics consisted mainly of formal mathematical analysis, beginning with calculus, and increasing in complexity up to differential geometry and partial differential equations. Probability theory was used in statistical mechanics. Geometric intuition played a strong role in the first two and, accordingly, theories of relativity were formulated entirely in terms of differential geometric concepts. The phenomenology of quantum physics arose roughly between 1895 and 1915, and for the 10 to 15 years before the development of quantum mechanics (around 1925) physicists continued to think of quantum theory within the confines of what is now called classical physics, and in particular within the same mathematical structures. The most sophisticated example of this is the SommerfeldWilsonIshiwara quantization rule, which was formulated entirely on the classical phase space.

In the 1890s, Planck was able to derive the blackbody spectrum, which was later used to avoid the classical ultraviolet catastrophe by making the unorthodox assumption that, in the interaction of electromagnetic radiation with matter, energy could only be exchanged in discrete units which he called quanta. Planck postulated a direct proportionality between the frequency of radiation and the quantum of energy at that frequency. The proportionality constant, h, is now called Planck's constant in his honor.

In 1905, Einstein explained certain features of the photoelectric effect by assuming that Planck's energy quanta were actual particles, which were later dubbed photons.

All of these developments were phenomenological and challenged the theoretical physics of the time. Bohr and Sommerfeld went on to modify classical mechanics in an attempt to deduce the Bohr model from first principles. They proposed that, of all closed classical orbits traced by a mechanical system in its phase space, only the ones that enclosed an area which was a multiple of Planck's constant were actually allowed. The most sophisticated version of this formalism was the so-called SommerfeldWilsonIshiwara quantization. Although the Bohr model of the hydrogen atom could be explained in this way, the spectrum of the helium atom (classically an unsolvable 3-body problem) could not be predicted. The mathematical status of quantum theory remained uncertain for some time.

In 1923, de Broglie proposed that waveparticle duality applied not only to photons but to electrons and every other physical system.

The situation changed rapidly in the years 19251930, when working mathematical foundations were found through the groundbreaking work of Erwin Schrdinger, Werner Heisenberg, Max Born, Pascual Jordan, and the foundational work of John von Neumann, Hermann Weyl and Paul Dirac, and it became possible to unify several different approaches in terms of a fresh set of ideas. The physical interpretation of the theory was also clarified in these years after Werner Heisenberg discovered the uncertainty relations and Niels Bohr introduced the idea of complementarity.

Werner Heisenberg's matrix mechanics was the first successful attempt at replicating the observed quantization of atomic spectra. Later in the same year, Schrdinger created his wave mechanics. Schrdinger's formalism was considered easier to understand, visualize and calculate as it led to differential equations, which physicists were already familiar with solving. Within a year, it was shown that the two theories were equivalent.

Schrdinger himself initially did not understand the fundamental probabilistic nature of quantum mechanics, as he thought that the absolute square of the wave function of an electron should be interpreted as the charge density of an object smeared out over an extended, possibly infinite, volume of space. It was Max Born who introduced the interpretation of the absolute square of the wave function as the probability distribution of the position of a pointlike object. Born's idea was soon taken over by Niels Bohr in Copenhagen who then became the "father" of the Copenhagen interpretation of quantum mechanics. Schrdinger's wave function can be seen to be closely related to the classical HamiltonJacobi equation. The correspondence to classical mechanics was even more explicit, although somewhat more formal, in Heisenberg's matrix mechanics. In his PhD thesis project, Paul Dirac[2] discovered that the equation for the operators in the Heisenberg representation, as it is now called, closely translates to classical equations for the dynamics of certain quantities in the Hamiltonian formalism of classical mechanics, when one expresses them through Poisson brackets, a procedure now known as canonical quantization.

To be more precise, already before Schrdinger, the young postdoctoral fellow Werner Heisenberg invented his matrix mechanics, which was the first correct quantum mechanics the essential breakthrough. Heisenberg's matrix mechanics formulation was based on algebras of infinite matrices, a very radical formulation in light of the mathematics of classical physics, although he started from the index-terminology of the experimentalists of that time, not even aware that his "index-schemes" were matrices, as Born soon pointed out to him. In fact, in these early years, linear algebra was not generally popular with physicists in its present form.

Although Schrdinger himself after a year proved the equivalence of his wave-mechanics and Heisenberg's matrix mechanics, the reconciliation of the two approaches and their modern abstraction as motions in Hilbert space is generally attributed to Paul Dirac, who wrote a lucid account in his 1930 classic The Principles of Quantum Mechanics. He is the third, and possibly most important, pillar of that field (he soon was the only one to have discovered a relativistic generalization of the theory). In his above-mentioned account, he introduced the braket notation, together with an abstract formulation in terms of the Hilbert space used in functional analysis; he showed that Schrdinger's and Heisenberg's approaches were two different representations of the same theory, and found a third, most general one, which represented the dynamics of the system. His work was particularly fruitful in all kinds of generalizations of the field.

The first complete mathematical formulation of this approach, known as the Diracvon Neumann axioms, is generally credited to John von Neumann's 1932 book Mathematical Foundations of Quantum Mechanics, although Hermann Weyl had already referred to Hilbert spaces (which he called unitary spaces) in his 1927 classic paper and book. It was developed in parallel with a new approach to the mathematical spectral theory based on linear operators rather than the quadratic forms that were David Hilbert's approach a generation earlier. Though theories of quantum mechanics continue to evolve to this day, there is a basic framework for the mathematical formulation of quantum mechanics which underlies most approaches and can be traced back to the mathematical work of John von Neumann. In other words, discussions about interpretation of the theory, and extensions to it, are now mostly conducted on the basis of shared assumptions about the mathematical foundations.

The application of the new quantum theory to electromagnetism resulted in quantum field theory, which was developed starting around 1930. Quantum field theory has driven the development of more sophisticated formulations of quantum mechanics, of which the ones presented here are simple special cases.

A related topic is the relationship to classical mechanics. Any new physical theory is supposed to reduce to successful old theories in some approximation. For quantum mechanics, this translates into the need to study the so-called classical limit of quantum mechanics. Also, as Bohr emphasized, human cognitive abilities and language are inextricably linked to the classical realm, and so classical descriptions are intuitively more accessible than quantum ones. In particular, quantization, namely the construction of a quantum theory whose classical limit is a given and known classical theory, becomes an important area of quantum physics in itself.

Finally, some of the originators of quantum theory (notably Einstein and Schrdinger) were unhappy with what they thought were the philosophical implications of quantum mechanics. In particular, Einstein took the position that quantum mechanics must be incomplete, which motivated research into so-called hidden-variable theories. The issue of hidden variables has become in part an experimental issue with the help of quantum optics.

A physical system is generally described by three basic ingredients: states; observables; and dynamics (or law of time evolution) or, more generally, a group of physical symmetries. A classical description can be given in a fairly direct way by a phase space model of mechanics: states are points in a symplectic phase space, observables are real-valued functions on it, time evolution is given by a one-parameter group of symplectic transformations of the phase space, and physical symmetries are realized by symplectic transformations. A quantum description normally consists of a Hilbert space of states, observables are self-adjoint operators on the space of states, time evolution is given by a one-parameter group of unitary transformations on the Hilbert space of states, and physical symmetries are realized by unitary transformations. (It is possible, to map this Hilbert-space picture to a phase space formulation, invertibly. See below.)

The following summary of the mathematical framework of quantum mechanics can be partly traced back to the Diracvon Neumann axioms. The postulates are canonically presented in six statements, though there are many important points to each.[3]

Each physical system is associated with a (topologically) separable complex Hilbert space H with inner product |. Rays (that is, subspaces of complex dimension 1) in H are associated with quantum states of the system.

In other words, quantum states can be identified with equivalence classes of vectors of length 1 in H, where two vectors represent the same state if they differ only by a phase factor. Separability is a mathematically convenient hypothesis, with the physical interpretation that countably many observations are enough to uniquely determine the state. "A quantum mechanical state is a ray in projective Hilbert space, not a vector. Many textbooks fail to make this distinction, which could be partly a result of the fact that the Schrdinger equation itself involves Hilbert-space "vectors", with the result that the imprecise use of "state vector" rather than ray is very difficult to avoid."[4]

The Hilbert space of a composite system is the Hilbert space tensor product of the state spaces associated with the component systems (for instance, J. M. Jauch, Foundations of quantum mechanics, section 11.7). For a non-relativistic system consisting of a finite number of distinguishable particles, the component systems are the individual particles.

Physical observables are represented by Hermitian matrices on H. Since these operators are Hermitian, the measurement is always a real value. If the spectrum of the observable is discrete, then the possible results are quantized.

By spectral theory, we can associate a probability measure to the values of A in any state . We can also show that the possible values of the observable A in any state must belong to the spectrum of A. The expectation value (in the sense of probability theory) of the observable A for the system in state represented by the unit vector H is A {displaystyle langle psi mid Amid psi rangle } .

Postulate III

The result of measuring a physical quantity A {displaystyle {mathcal {A}}} must be one of the eigenvalues of the corresponding observable A {displaystyle A} .

In the special case A has only discrete spectrum, the possible outcomes of measuring A are its eigenvalues. More precisely, if we represent the state in the basis formed by the eigenvectors of A, then the square of the modulus of the component attached to a given eigenvector is the probability of observing its corresponding eigenvalue.

More generally, a state can be represented by a so-called density operator, which is a trace class, nonnegative self-adjoint operator normalized to be of trace 1. The expected value of A in the state is tr ( A ) {displaystyle operatorname {tr} (Arho )} .

When a measurement is performed, only one result is obtained (according to some interpretations of quantum mechanics). This is modeled mathematically as the processing of additional information from the measurement, confining the probabilities of an immediate second measurement of the same observable. In the case of a discrete, non-degenerate spectrum, two sequential measurements of the same observable will always give the same value assuming the second immediately follows the first. Therefore the state vector must change as a result of measurement, and collapse onto the eigensubspace associated with the eigenvalue measured.

If is the orthogonal projector onto the one-dimensional subspace of H spanned by |, then tr ( A ) = A {displaystyle operatorname {tr} (Arho _{psi })=leftlangle psi mid Amid psi rightrangle } .

Though it is possible to derive the Schrdinger equation, which describes how a state vector evolves in time, most texts assert the equation as a postulate. Common derivations include using the DeBroglie hypothesis or path integrals.

Postulate VI

The time evolution of the state vector | ( t ) {displaystyle |psi (t)rangle } is governed by the Schrdinger equation, where H ( t ) {displaystyle H(t)} is the observable associated with the total energy of the system (called the Hamiltonian)

i d d t | ( t ) = H ( t ) | ( t ) {displaystyle ihbar {frac {d}{dt}}|psi (t)rangle =H(t)|psi (t)rangle }

One can in this formalism state Heisenberg's uncertainty principle and prove it as a theorem, although the exact historical sequence of events, concerning who derived what and under which framework, is the subject of historical investigations outside the scope of this article.

Furthermore, to the postulates of quantum mechanics one should also add basic statements on the properties of spin and Pauli's exclusion principle, see below.

The time evolution of the state is given by a differentiable function from the real numbers R, representing instants of time, to the Hilbert space of system states. This map is characterized by a differential equation as follows:If |(t) denotes the state of the system at any one time t, the following Schrdinger equation holds:

i d d t | ( t ) = H | ( t ) {displaystyle ihbar {frac {d}{dt}}left|psi (t)rightrangle =Hleft|psi (t)rightrangle }

where H is a densely defined self-adjoint operator, called the system Hamiltonian, i is the imaginary unit and is the reduced Planck constant. As an observable, H corresponds to the total energy of the system.

Alternatively, by Stone's theorem one can state that there is a strongly continuous one-parameter unitary map U(t): H H such that

for all times s, t. The existence of a self-adjoint Hamiltonian H such that

is a consequence of Stone's theorem on one-parameter unitary groups. It is assumed that H does not depend on time and that the perturbation starts at t0 = 0; otherwise one must use the Dyson series, formally written as

where T {displaystyle {mathcal {T}}} is Dyson's time-ordering symbol.

(This symbol permutes a product of noncommuting operators of the form

into the uniquely determined re-ordered expression

The result is a causal chain, the primary cause in the past on the utmost r.h.s., and finally the present effect on the utmost l.h.s..)

It is then easily checked that the expected values of all observables are the same in both pictures

and that the time-dependent Heisenberg operators satisfy

d d t A ( t ) = i [ H , A ( t ) ] + A ( t ) t , {displaystyle {frac {d}{dt}}A(t)={frac {i}{hbar }}[H,A(t)]+{frac {partial A(t)}{partial t}},}

which is true for time-dependent A = A(t). Notice the commutator expression is purely formal when one of the operators is unbounded. One would specify a representation for the expression to make sense of it.

i d d t | ( t ) = H i n t ( t ) | ( t ) {displaystyle ihbar {frac {d}{dt}}left|psi (t)rightrangle ={H}_{rm {int}}(t)left|psi (t)rightrangle }

i d d t A ( t ) = [ A ( t ) , H 0 ] . {displaystyle ihbar {d over dt}A(t)=[A(t),H_{0}].}

The interaction picture does not always exist, though. In interacting quantum field theories, Haag's theorem states that the interaction picture does not exist. This is because the Hamiltonian cannot be split into a free and an interacting part within a superselection sector. Moreover, even if in the Schrdinger picture the Hamiltonian does not depend on time, e.g. H = H0 + V, in the interaction picture it does, at least, if V does not commute with H0, since

So the above-mentioned Dyson-series has to be used anyhow.

The Heisenberg picture is the closest to classical Hamiltonian mechanics (for example, the commutators appearing in the above equations directly translate into the classical Poisson brackets); but this is already rather "high-browed", and the Schrdinger picture is considered easiest to visualize and understand by most people, to judge from pedagogical accounts of quantum mechanics. The Dirac picture is the one used in perturbation theory, and is specially associated to quantum field theory and many-body physics.

Similar equations can be written for any one-parameter unitary group of symmetries of the physical system. Time would be replaced by a suitable coordinate parameterizing the unitary group (for instance, a rotation angle, or a translation distance) and the Hamiltonian would be replaced by the conserved quantity associated with the symmetry (for instance, angular or linear momentum).

Summary:

The original form of the Schrdinger equation depends on choosing a particular representation of Heisenberg's canonical commutation relations. The Stonevon Neumann theorem dictates that all irreducible representations of the finite-dimensional Heisenberg commutation relations are unitarily equivalent. A systematic understanding of its consequences has led to the phase space formulation of quantum mechanics, which works in full phase space instead of Hilbert space, so then with a more intuitive link to the classical limit thereof. This picture also simplifies considerationsof quantization, the deformation extension from classical to quantum mechanics.

The quantum harmonic oscillator is an exactly solvable system where the different representations are easily compared. There, apart from the Heisenberg, or Schrdinger (position or momentum), or phase-space representations, one also encounters the Fock (number) representation and the SegalBargmann (Fock-space or coherent state) representation (named after Irving Segal and Valentine Bargmann). All four are unitarily equivalent.

The framework presented so far singles out time as the parameter that everything depends on. It is possible to formulate mechanics in such a way that time becomes itself an observable associated with a self-adjoint operator. At the classical level, it is possible to arbitrarily parameterize the trajectories of particles in terms of an unphysical parameter s, and in that case the time t becomes an additional generalized coordinate of the physical system. At the quantum level, translations in s would be generated by a "Hamiltonian" HE, where E is the energy operator and H is the "ordinary" Hamiltonian. However, since s is an unphysical parameter, physical states must be left invariant by "s-evolution", and so the physical state space is the kernel of HE (this requires the use of a rigged Hilbert space and a renormalization of the norm).

This is related to the quantization of constrained systems and quantization of gauge theories. Itis also possible to formulate a quantum theory of "events" where time becomes an observable (see D. Edwards).

In addition to their other properties, all particles possess a quantity called spin, an intrinsic angular momentum. Despite the name, particles do not literally spin around an axis, and quantum mechanical spin has no correspondence in classical physics. In the position representation, a spinless wavefunction has position r and time t as continuous variables, = (r, t). For spin wavefunctions the spin is an additional discrete variable: = (r, t, ), where takes the values;

That is, the state of a single particle with spin S is represented by a (2S + 1)-component spinor of complex-valued wave functions.

Two classes of particles with very different behaviour are bosons which have integer spin (S=0,1,2...), and fermions possessing half-integer spin (S=12,32,52,...).

The property of spin relates to another basic property concerning systems of N identical particles: Pauli's exclusion principle, which is a consequence of the following permutation behaviour of an N-particle wave function; again in the position representation one must postulate that for the transposition of any two of the N particles one always should have

( , r i , i , , r j , j , ) = ( 1 ) 2 S ( , r j , j , , r i , i , ) {displaystyle psi (dots ,,mathbf {r} _{i},sigma _{i},,dots ,,mathbf {r} _{j},sigma _{j},,dots )=(-1)^{2S}cdot psi (dots ,,mathbf {r} _{j},sigma _{j},,dots ,mathbf {r} _{i},sigma _{i},,dots )}

i.e., on transposition of the arguments of any two particles the wavefunction should reproduce, apart from a prefactor (1)2S which is +1 for bosons, but (1) for fermions.Electrons are fermions with S=1/2; quanta of light are bosons with S=1. In nonrelativistic quantum mechanics all particles are either bosons or fermions; in relativistic quantum theories also "supersymmetric" theories exist, where a particle is a linear combination of a bosonic and a fermionic part. Only in dimension d = 2 can one construct entities where (1)2S is replaced by an arbitrary complex number with magnitude 1, called anyons.

Although spin and the Pauli principle can only be derived from relativistic generalizations of quantum mechanics the properties mentioned in the last two paragraphs belong to the basic postulates already in the non-relativistic limit. Especially, many important properties in natural science, e.g. the periodic system of chemistry, are consequences of the two properties.

The picture given in the preceding paragraphs is sufficient for description of a completely isolated system. However, it fails to account for one of the main differences between quantum mechanics and classical mechanics, that is, the effects of measurement.[5] The von Neumann description of quantum measurement of an observable A, when the system is prepared in a pure state is the following (note, however, that von Neumann's description dates back to the 1930s and is based on experiments as performed during that time more specifically the ComptonSimon experiment; it is not applicable to most present-day measurements within the quantum domain):

For example, suppose the state space is the n-dimensional complex Hilbert space Cn and A is a Hermitian matrix with eigenvalues i, with corresponding eigenvectors i. The projection-valued measure associated with A, EA, is then

where B is a Borel set containing only the single eigenvalue i. If the system is prepared in state

Then the probability of a measurement returning the value i can be calculated by integrating the spectral measure

over Bi. This gives trivially

The characteristic property of the von Neumann measurement scheme is that repeating the same measurement will give the same results. This is also called the projection postulate.

A more general formulation replaces the projection-valued measure with a positive-operator valued measure (POVM). To illustrate, take again the finite-dimensional case. Here we would replace the rank-1 projections

by a finite set of positive operators

whose sum is still the identity operator as before (the resolution of identity). Just as a set of possible outcomes {1...n} is associated to a projection-valued measure, the same can be said for a POVM. Suppose the measurement outcome is i. Instead of collapsing to the (unnormalized) state

after the measurement, the system now will be in the state

Since the Fi Fi* operators need not be mutually orthogonal projections, the projection postulate of von Neumann no longer holds.

The same formulation applies to general mixed states.

In von Neumann's approach, the state transformation due to measurement is distinct from that due to time evolution in several ways. For example, time evolution is deterministic and unitary whereas measurement is non-deterministic and non-unitary. However, since both types of state transformation take one quantum state to another, this difference was viewed by many as unsatisfactory. The POVM formalism views measurement as one among many other quantum operations, which are described by completely positive maps which do not increase the trace.

In any case it seems that the above-mentioned problems can only be resolved if the time evolution included not only the quantum system, but also, and essentially, the classical measurement apparatus (see above).

An alternative interpretation of measurement is Everett's relative state interpretation, which was later dubbed the "many-worlds interpretation" of quantum physics.

Part of the folklore of the subject concerns the mathematical physics textbook Methods of Mathematical Physics put together by Richard Courant from David Hilbert's Gttingen University courses. The story is told (by mathematicians) that physicists had dismissed the material as not interesting in the current research areas, until the advent of Schrdinger's equation. At that point it was realised that the mathematics of the new quantum mechanics was already laid out in it. It is also said that Heisenberg had consulted Hilbert about his matrix mechanics, and Hilbert observed that his own experience with infinite-dimensional matrices had derived from differential equations, advice which Heisenberg ignored, missing the opportunity to unify the theory as Weyl and Dirac did a few years later. Whatever the basis of the anecdotes, the mathematics of the theory was conventional at the time, whereas the physics was radically new.

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Mathematical formulation of quantum mechanics - Wikipedia

March 18, 2022

Physicists have worked and wrestled with quantum theory for more than a century now, applying it to explore and help solve the profound mysteries of Albert Einsteins theory of relativity and cosmological conundrums such as black holes, gravity and the origins of the universe.

But for Arizona State University theoretical chemist Vladimiro Mujica, there is still a vast, secret and fascinating world to explore but rather than out there in the vastness of space time, at the nexus between everyday life on Earth and the quantum world. Cellular mutations in the molecule of life, DNA, happen randomly and are governed by quantum probability rules. Download Full Image

Recently, quantum mechanics has been found to play an essential role in our understanding of chemistry and biology, and the molecular theory of evolution.

Now, Mujica will get a chance to further explore this quantum world by leading a three-year, $1 million award from the prestigious Keck Foundation. Their goal is build a foundational understanding of how the sometimes weird, exotic features of quantum physics influence the very stuff that makes life work.

To do so, Mujica will lead a multi-institutional quantum biology team that includes ASU colleague William Petuskey and leading experimentalists, including Northwestern University co-investigators Michael Wasielewski and University of California Los Angeles professors Paul Weiss and Louis Bouchard.

To be successful, we really needed to think outside of the box, with a good foundation, said Mujica, a professor in the School of Molecular Sciences. So, we put this team together of leading experimentalists, but also with a firm grasp of theory top-ranking people to take a quantum leap in this field of science.

The awards initiative, titled Chirality, Spin Coherence, and Entanglement in Quantum Biology,will explore fundamental quantumeffects in biological systems.

For example, two key processes necessary for life: photosynthesis in plants and respiration in animals, are driven by reactions that involve the transfer of electrons in molecules and across boundaries within the cell.

Electrons themselves, in addition to carrying a negative charge, have key quantum properties, including spin, that plays a fundamental role in the molecular electron transfer processes that make life possible.

Vladimiro Mujica. Photo courtesy Mary Zhu

Chiral is the Greek word for hand. No matter how hard one tries, a left hand and right hand are non-superimposable mirror images of each other. Ever try to shake a persons hand with the opposite hand? That awkward encounter simply because the thumbs are in different positions is an everyday demonstration of chirality.

It turns out molecules, and life, have the same chiral properties. But how does that help their biological function?

We're trying to decipher in a way, a mystery of nature and evolution, Mujica said. Because it turns out that biological systems use these chiral molecules in proteins, DNA and RNA. These are some of the most important molecules in biology. For example, DNA is a double-helix ladder that is intrinsically chiral. And so are the proteins encoded by these fundamental biological molecules, which are the bricks and mortars of the cell, doing all the work that makes us alive.

Quantum mechanics is all-across biology: Photosynthesis. Cellular respirationc. Oxygen transport.Cellular mutations.

Are all governed by quantum effects.

These happen randomly and are governed by quantum probability rules.

One can zoom in further on life, under the skin all the way to the molecules at the atomic level and clouds of electrons in quantum states. In everyday life, we are used to electrons being transported through copper wires to deliver electricity to our homes.

But what are the wires that deliver electrons in living system, a process that involves substantial amounts of energy and heat? And how do they avoid frying life, or by proxy, us?

In living systems, how electrons are transferred or transported depends on organic molecules, Mujica said. Now, organic molecules are far less efficient than copper wires or anything like that to transport or transfer electrons. But nevertheless, evolution chose this in a way.

Mujica refers to this as a real mystery as to why Mother Nature chose these lousy molecules for transferring electrons.

Yet, as Jeff Goldblums quirky scientist character in "Jurassic Park" famously once said: "Life finds a way.

It turns out electrons are transported in organic molecules primarily by tunneling, not diffusion as in copper wires.

The mechanism electrons going through organic molecules is to a large extent a quantum phenomenon, Mujica said Its a mechanism called tunneling, and what it implies is that electrons can go from one region of the molecule to the other, even if they do not have enough energy to overcome intrinsic barriers.

The research team wants to investigate why and how electrons use this tunneling mechanism for biological function essential to life. First, they have designed a series of experiments using synthetic pairs of right or left-handed DNA structures. Next, they will custom tailor electron donors andacceptors as part of their structures to probe this chirality-dependentelectron transfer. All this experimental effort is guided by a predictive theoretical and computational effort.

Some of themodelsystems tweaks they will examine are the effect of the electron donor-acceptordistance, the temperature, redox properties and the coupling to their surrounding environment.

An electron transfer process with the electron-vibration (phonon) interaction. The process is essential to understanding and controlling charge and energy flow in various electronic, photonic and energy conversion devices or, in this case, a biomolecule. The "IN" and "OUT" have either the same or distorted phase, depending on whether the transport is coherent or incoherent.

A fundamental quantum electron property is spin. Electrons can be like spinning tops, rotating on their own axis.

Mujica explains that because electrons are charged particles, "this rotation creates a magnetic moment, which only has two components; one component aligns in the direction of transport and the other component is aligned in the opposite direction to transport.

"As they tunnel through chiral organic molecules, they have a preferential orientation due to the spin orbit interaction and the loss of time-inversion symmetry.

This is known as spin polarization.

It turns out, when electron spin is polarized, electrons can tunnel much easier and farther because one of the two spin components has a larger transmission probability.

Mujica likens it to a bullet going through the barrel of a gun. The first guns that were ever made all had smooth, hollowed-out barrels. But when grooves were etched, it gave the bullet a spin that allowed it to travel straighter and farther. Also, it is easy to understand with this simple analogy that bullets rotating clockwise will not go through counter-clockwise designed barrels, and vice versa. A classical analogy to what happens with electron spins.

And so, for their second set of experiments, they willuse magnetic substrates, nanoscale chemical patterning, andmultimodalspin-polarized scanningtunneling microscopyand spectroscopieswith orientedenantiomeric pairs of DNAandintercalated metalstoelucidate and to quantifythe molecular and interface contributionstochirality-induced spin selectivity.

Since most biological molecules, including amino acids inproteins and nucleotides in RNA and DNA, are chiral, thecriticalroles of spin polarization inelectron transport within and between biological molecules will be determined.

Finally, electrons have a dual particle-wave quantum nature; they have particle-like properties such as mass and charge, but their dynamics and propagation follows the rules of wave quantum mechanics.

In biology, as the electrons encounter other molecules or molecular barriers like cell membranes, they are scattered, and their wave properties are modified. Two wave sources arecoherentif their frequency and waveform are identical. If not, the waves can be canceled or enhanced due to interference. This interference can be destructive and leads to noise, which can also be due to thermal interactions.

Spin coherence can coexist with spin polarization Mujica said. What it means is that you have in-phase transport, so you're not reducing the intensity of the wave, and we're not changing the phase of a wave associated to that transfer.

Spin coherence is intimately associated to another quantum process, entanglement, that is of fundamental importance in quantum information and quantum computing.

Mujica says this is a high-risk, high-reward project that may upset the current conventional wisdom in quantum biology.

I mean, the common knowledge was that you couldn't have coherence in a quantum biological system, because the environmental effects would destroy coherence in a very short time.

They will try to put it all together by determining how chirality influences theelectronic, vibrational and spin-polarized electron transferfrom electrondonors to acceptor sites as spin-coherent electron pairs are generated in photo-induced electron transfer reactions.

Essentially, the grant focuses on the role of spin-polarized electrons and how it influences the behavior of biological systems, especially the length and temperature dependence, and how spin polarization and spin coherence can coexist, Mujica said. These are key unsolved issues in biological electron-transfer reactions.

In addition tostudying the unexplored roles of spin coherence in quantum biology, Mujicas team will study how it can coexist with spinpolarization and how, or if, it can create what is referred to as the spooky "action at a distance," or quantum entangled states.

The overarching Keck grant goal is to answer these questions, and the contributions of three key ingredients: tunneling, spin and coherence. These are central to discovering the underpinnings of the emerging field of quantumbiology.

By exploring these questions, Mujicas team ultimately hopes to use the Keck grant as a catalyst to create an ASU center for quantum biology, and further down the road, practical applications, such as quantum information and computing. All this could help position ASU in quantum technologies and information efforts, which are of strategic importance for the U.S.

If we can provide enough evidence, we hope to unveil some very important questions that will be crucial for an ASU effort in quantum information sciences, and this is something that we are starting with efforts in engineering and physics, Mujica said.

We want to weigh in on the roadmap to be able to use molecules for quantum information. From our perspective, we really think of this as a step in the direction of defining our capabilities of using quantum biology in molecular quantum information sciences, a field that is experiencing a true renaissance.

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Keck award will help scientists take quantum leap to explore the mysteries of life - ASU News Now

Fragments of the interior of a proton have been shown by scientists from Mexico andPoland to exhibit maximum quantum entanglement.

The discovery, already confronted with experimental data, allows us to suppose that in some respects thephysics of the inside of a proton may have much in common not only with well-known thermodynamic phenomena, but even with the physics of... black holes.

Various fragments of the inside of a proton must be maximally entangled with each other, otherwise theoretical predictions would not agree with the data collected in experiments, it was shown in European Physical Journal C. The theoretical model (which extends the original proposal by physicists Dimitri Kharzeev and Eugene Levin) makes it possible to suppose that, contrary to current belief, the physics operating inside protons may be related to such concepts as entropy or temperature, which in turn may relate it to such exotic objects as black holes. The authors of the discovery are Dr. Martin Hentschinski from the Universidad de las Americas Puebla in Mexico and Dr. Krzysztof Kutak from the Institute of Nuclear Physics of the Polish Academy of Sciences (IFJ PAN) in Cracow, Poland.

The Mexican-Polish theorists analysed the situation in which electrons are fired at protons. When an incoming electron carrying a negative electric charge approaches a positively charged proton, it interacts with it electromagnetically and deflects its path. Electromagnetic interaction means that a photon has been exchanged between the electron and the proton. The stronger the interaction, the greater the change in momentum of the photon and therefore the shorter the associated electromagnetic wave.

"If a photon is 'short' enough to 'fits' inside a proton, it begins to 'resolve' details of its internal structure. The result of interacting with this sort of photon can be the decay of the proton into particles. We have shown that there is entanglement between the two situations. If the observation by the photon of the interior part of the proton leads to its decay into a number of particles, let's say three, then the number of particles originating from the unobserved part of the proton is determined by the number of particles seen in the observed part of the proton," explains Dr. Kutak.

We can speak of quantum entanglement of various quantum objects, if certain characteristic of the objects are related to each other in a particular way. The classical analogy of the phenomenon can be represented by the toss of a coin. Let's assume that one object is one side of the coin, and the other object is its other side. When we flip a coin, there is the same probability that the coin will land heads or tails facing up. If it lands heads up, we know for sure that the other side is tails. We can then speak of maximum entanglement since the probability which determines the value of an object's characteristic does not favour any possible value: we have a 50% chance of heads and the same for tails. A smaller than maximum entanglement occurs when the probability starts to favour one of the possible outcomes to a greater or lesser extent.

"Our study shows that the interior of a proton seen by a passing photon must be entangled with the unseen part in just this maximal manner, as suggested by Kharzeev and Levin. In practice, this means that we have no chance of predicting whether, due to interaction with the photon, the proton will decay into three, four or any other number of particles," explains Dr. Hentschinski.

The new theoretical predictions have already been verified. If entanglement inside the proton were not maximal, there would be discrepancies between theoretical calculations and the results of the H1 experiment at the HERA accelerator at the DESY centre in Hamburg, where positrons (i.e. antiparticles of the electrons) were collided with protons until 2007. Such discrepancies were not observed.

The success of the Polish-Mexican tandem is due to the fact that the researchers managed to correctly identify the factors responsible for the maximum entanglement of the proton interior.

In the naive schoolbook view, the proton is a system of three elementary particles: two up quarks and one down quark. However, the strong interactions between these quarks, carried by gluons, can be so strong that they lead to the creation of virtual particle-antiparticle pairs. These can be not only pairs of virtual gluons (which are their own antiparticles), but also pairs made up of any quark and its corresponding antiparticle (even one as massive as charm). All this means that inside the proton, apart from three valence quarks, there are constantly 'boiling' seas of virtual gluons and virtual quarks and antiquarks.

"In earlier publications, physicists dealing with the subject assumed that the source of entanglement should be a sea of gluons. Later, attempts were made to show that quarks and antiquarks are the dominant source of entanglement, but even here the proposed methods of description did not stand the test of time. Meanwhile, according to our model, verified by confrontation with experimental data, the sea of virtual gluons is responsible for about 80% of the entanglement, while the sea of virtual quarks and antiquarks is responsible for the remaining 20%," emphasizes Dr. Kutak.

Most recently, quantum physicists have been associating entropy with the state inside a proton. This is a quantity well known from classical thermodynamics, where it is used to measure the degree of disordered motion of particles in an analysed system. It is assumed that when a system is disordered, it has high entropy, whereas an ordered system has low entropy. It has recently been shown that in the case of the proton, we can successfully talk about entanglement entropy. However, many physicists have considered the proton to be a pure quantum state in which one should not speak of entropy at all. The consistency of the Mexican-Polish model with experiment is a strong argument for the fact that the concept of entanglement inside the proton as proposed by Kharzeev and Levin has a point. Last but not least, since entanglement entropy is also related to concepts such as the surface area of black holes, the latest result opens an interesting field for further research.

Source:https://www.ifj.edu.pl/en/

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Fragments of the Inside of a Proton Exhibit Maximum Quantum Entanglement - AZoQuantum

What does a six-dimensional figure look like? Theoretical physicist Richard Nally cant show you exactly, but he does have a sculpture a pink shape the size of a grapefruit that can help you imagine a piece of one.

Its called a K3 surface, said Nally, a Klarman Fellow in physics in the College of Arts and Sciences (A&S). Of course, we cant make sculptures of things that live in six dimensions, but you can take little slices of them to see what they look like. This is a slice of a four-dimensional shape that is really important to the history and practice of string theory.

Researchers have known about the shapes in string theory for decades, Nally said. But in the past few years, he and others have started to take the shapes seriously as number theoretic objects and to study them in that framework. Nally will spend his three-year Klarman Postdoctoral Fellowship seeking to understand the mathematical structures at the root of gravity and quantum mechanics.

We want to find a nice shape that lets us keep the solution to quantum gravity, while getting the features such as an expanding universe and only having four dimensions that we see in the world around us, he said.

Read the full story on the College of Arts and Sciences website.

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Klarman fellow blends physics and math to explore string theory | Cornell Chronicle - Cornell Chronicle

Dr. Parker, he said, was happy when people pointed out a mistake in his calculations but not pleased when people accepted prevalent scientific assumptions without question.

He had little patience for Its well known that Dr. Turner said.

Even though Dr. Chandrasekhar, a future Nobel laureate, disagreed with Dr. Parkers conclusions, he overruled the reviewers, and the paper was published.

Four years later, Dr. Parker was vindicated when Mariner 2, a NASA spacecraft en route to Venus, observed energetic particles streaming through interplanetary space exactly what he had predicted.

When Dr. Zurbuchen joined NASA in 2016, the agency had been working for years on a mission called Solar Probe Plus, which was to swoop close to the sun repeatedly. Dr. Zurbuchen said he disliked the name Solar Probe Plus and wrote to the National Academies of Sciences, Engineering and Medicine asking it to suggest a person to name the mission after.

The unequivocal response: Eugene Parker.

NASA had never before named a spacecraft after a living person. But Dr. Zurbuchen, who had met Dr. Parker years earlier, said he did not have much trouble getting Robert Lightfoot, the acting administrator of NASA at the time, to approve the change in 2017. Dr. Zurbuchen then called Dr. Parker to ask if that would be all right with him. He said, Absolutely. It will be my honor, Dr. Zurbuchen recalled.

Dr. Parker later said he was surprised that NASA had asked for his permission.

A few months afterward, Dr. Parker went to visit the Johns Hopkins Applied Physics Laboratory in Maryland, where the spacecraft was built and tested. Dr. Fox, then project scientist for the mission, recalled saying, Parker, meet Parker.

The next year, Dr. Parker and his family traveled to Florida to watch the launch of his namesake spacecraft.

Originally posted here:

Eugene N. Parker, 94, Dies; Predicted the Existence of Solar Wind - The New York Times

The higher education sector today has seen loads of changes. However, the watershed moment came a few years back with the introduction of online learning. Such is the paramount impact of the e-learning option that most people today are embracing at a rapid pace. The huge shift in the scenario has also led to access to quality education which otherwise was not possible. Although the online classroom seems like a new phase, below are a few reasons why the concept will see more growth in the future.

It Provides For A Customized Learning ApproachIrrespective of what people may say, online learning is the future of higher education. One of the prime benefits of this new approach towards learning is that it is flexible as per every students requirement. Online classes are smaller than the conventional teaching approaches which give more space to teacher and student interaction. Other than that study materials can be accessed at any time which not only improves learning but also offers a dynamic approach.

FlexibilityEver since the online learning approach has been introduced, flexibility is one benefit that has been discussed time and again. Not only this method is giving students to set their own learning space but is also helpful for those who live far away. Apart from this studying online also cultivates vital time management skills in a person which are a common requisite nowadays.

Easily AccessibleLet us be honest here! The introduction of an internet-based approach has pushed new boundaries for the students and as well as higher education. It is mostly because online learning enables you to access study matter from anywhere. This means there is no need to follow a rigid schedule or move from one place to another. The onset of online learning has been a great opportunity for students who want to pursue higher education. For example, if you are interested to study abroad and take up a new job then many online universities are providing courses for you.

Learn From A Wide Range of ProgramsInternet today is blessed with courses that not only help in your career path but also improve your overall skill set. This is possible due to a growing number of top-notch universities which are offering programs for all aspirants out there. From data science to quantum physics, every candidate can choose the niche accordingly and get ready to hone their skill. And the main takeaway is the e-learning option helps you to do this from the comfort of your home.

Cost-EffectiveOnline education is more cost-effective than conventional methods. Why? Well, there are ample reasons for it. E-learning platforms today are so well-versed that every student can avail of a wide range of payment options. Also, some platforms have discounts and scholarships which makes it pocket-friendly also. In other words, one can easily say that online platforms help in cost-cutting as there are blessed with better results with less investment.

Suits Every StudentThe learning graph of every individual is different. Some students grasp lessons quickly while others might take some more time to learn. Similarly, many candidates are visual learners while there are people who love to listen through audio. In such type of scenario, the online education system is a boon. The concept of e-learning is equipped with various options and resources which can be personalized according to the choice of the student. Candidates can easily learn the lessons while being in segue with their daily routine. Also, since online classes are taken from home, students will never want to miss any lessons intentionally.

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How Higher Education Is Turning A New Leaf Through An Online Approach - CEOWORLD magazine

The CUNY Dance Initiative and the Gerald W. Lynch Theater at John Jay College announce a performance by Jon Lehrer Dance Company featuring the world premiere of Through The Storm, a full company work set to an original score by composer Zeno Pittarelli.

The program will include additional works, including four New York City premieres. The performance will take place on Friday, May 6, 2022 at 7pm at The Gerald W. Lynch Theater, John Jay College, 524 W 59th St, New York, NY. Tickets are $35 and are available at http://www.jonlehrerdance.com.

Jon Lehrer Dance Company (JLDC)'s trademark artistic, athletic, and accessible style is known for striking the elusive balance between art and entertainment. Taking the energy and chaos of a storm as inspiration, Through The Storm represents the past two years of struggle, hardship, but most of all the creativity and perseverance artists held on to throughout the pandemic. Visually stunning, evocative, and physically impressive, Through The Storm will be performed both internationally and nationally as part of JLDC's 22/23 season following the premiere. The program also highlights the range of JLDC's repertory with four New York City premieres (full details below).

JLDC is freshly off a successful four-week European tour where the company "captivated the audience with its grandiose choreography and the ensemble's top dance performance without fail. Dynamism, elegance, acrobatics, power and an absolute passion for dance - these are the attributes that could be used to describe the performance." (Zollern-Alb Kurier - Balingen, Germany)

Program

"Through The Storm" (2022) - world premiere

A full company work set to an original score by Zeno Pittarelli.

"Sum of Us" (2021) - New York premiere

An upbeat showcase for the dancers as individuals and as a strong, cohesive group.

"Murmur" (2012) - New York premiere

A lyrically infused quartet based on the phenomenon of murmuration and quantum entanglement.

"Pulp" (2016) - New York premiere

Inspired by the silent films of Buster Keaton and Scooby-Doo cartoons, this zany romp highlights the company's theatrical comedic side.

"Hearth" (2010) - New York premiere

Performed by JLDC Apprentices, this work was inspired by the many strong, caring, and remarkable women in Jon Lehrer's life.

"Solstice"(2019)

A joyous, uplifting tribute to the first day of Summer, commissioned by Artpark in Lewiston, NY for their "Summer Solstice Celebration."

Jon Lehrer Dance Company (JLDC) began in 2007 in Buffalo, NY and has been based in NYC since 2019. Under Jon Lehrer's artistic direction, the company showcases his unique choreography and definitive style. Jon's extensive background in both the modern and jazz dance idioms fosters choreography that is organic, artistic, accessible and often humorous, reflecting life experience and the human condition. Dance Magazine praised "the company took the house not so much by storm as by quantum physics," and Galerie Ortenau in Offenburg, Germany declares, "Dynamic, Powerful, Elegant - brings an absolute passion for the dance." The company was brought to Russia from 2012-2016 sponsored by the US State Department in order to, as they said, "bring the best in American modern dance to the people of Russia." JLDC tours worldwide, to Europe every other year, and made their 4th trip in January 2022 for another world premiere performance.

JLDC's expressive technique and style is a combination of modern, jazz, and physics. It is based on three main elements of movement - Circularity, 3-Dimensionality, and Momentum, which combine to create a form that is best described as "Organically Athletic." Jon Lehrer Dance Company is dedicated to maintaining the impact of dance through performance, education, outreach, community involvement, and collaboration. The professional dancers of JLDC work collaboratively towards a common creative goal, while maintaining and developing their own artistic voice to bring edge-of-your-seat excitement to audiences. JLDC operates with integrity at all levels and respects and honors ideas from constituent groups including dancers, directors, and the public. Jon Lehrer Dance Company actively promotes and values equity, diversity, inclusion, and anti-racism on an institutional and individual level. http://www.jonlehrerdance.com

Jon Lehrer (Founder & Artistic Director of JLDC), who was raised in Queens, New York, took his very first dance class on a dare at age 19 at the University at Buffalo. While dating a dancer, Jon teased her about how easy it must be to get an A in a dance class. The girl dared Jon to try a beginning level modern dance class and his life was changed. Jon ultimately received his BFA in Dance from the University at Buffalo.

As a professional, Jon danced with the Erick Hawkins Dance Company, Paul Sanasardo, John Passafiume Dancers, in Merv Griffin's "Funderful" in Atlantic City, NJ, and the Radio City Rockettes Christmas Spectacular. In 1997, Jon was hired by Giordano Dance Chicago, the world's preeminent jazz dance company. After only three years he was promoted to Rehearsal Director and became the Associate Director two years later. During his ten years with Giordano, Jon also became the resident choreographer, creating seven original works on the company that received rave reviews around the world.

Jon has choreographed for several professional dance companies and universities all over the country. He teaches master classes throughout the U.S. and around the world, having been on faculty at Dance Masters of America, Jazz Dance World Congress, Dance Teacher Summer Conference, Chicago National Association of Dance Masters (CNADM), Chautauqua Institution, Florida Dance Educators Organization, and Florida Dance Masters to name a few. Jon has received many awards and honors, including the University at Buffalo Zodiaque Dance Company Distinguished Alumni Award, CNADM's Artistic Achievement Award, University at Buffalo College of Arts and Sciences Distinguished Alumni Award, and the "Rising Star" SPARK Award for being an integral part of Buffalo's thriving arts and cultural community.

About Zeno Pittarelli

Composer/Musician Zeno Pittarelli is a musician, engineer, and artist living in New York City. Founder of Newlywed Records and produces, records, mixes + masters in his home studio and remotely. Some of his projects have been featured in major publications including Rolling Stone, NPR, and Stereogum. Pittarelli strives to make moving, exciting, and unique records.

JLDC's residency and performance at the Gerald W. Lynch Theater are part of the CUNY Dance Initiative (CDI), a transformative incubator that secures two vital yet scarce resources-rehearsal time and performance space-for New York City choreographers and dance companies across the five boroughs. Housed within the City University of New York (CUNY)-the nation's largest public urban university system-CDI is a residency program that supports local artists, enhances the cultural life and education of college students, and builds new dance audiences at CUNY performing arts centers.

CDI receives major support from The Mertz Gilmore Foundation and Howard Gilman Foundation. Additional support is provided by the Jerome Robbins Foundation, SHS Foundation, Harkness Foundation for Dance, the National Endowment for the Arts, and public funds from the New York City Department of Cultural Affairs in partnership with the City Council. CDI is part of Dance/NYC's New York City Dance Rehearsal Space Subsidy Program, made possible by The Andrew W. Mellon Foundation. CDI is spearheaded by The Kupferberg Center for the Arts at Queens College. http://www.cuny.edu/danceinitiative

Additional funding for this residency and performance is provided by the John Jay College Student Activities Association, Inc.

John Jay College of Criminal Justice of The City University of New York, an international leader in educating for justice, offers a rich liberal arts and professional studies curriculum to upwards of 15,000 undergraduate and graduate students from more than 135 nations. In teaching, scholarship and research, the College approaches justice as an applied art and science in service to society and as an ongoing conversation about fundamental human desires for fairness, equality and the rule of law. For more information, visit http://www.jjay.cuny.edu.

Since opening its doors in 1988, the Gerald W. Lynch Theater has been an invaluable cultural resource. The Theater is a member of CUNY Stages, a consortium of 16 performing arts centers located on CUNY campuses across New York City and the CUNY Dance Initiative. The Theater is home to the Lincoln Center's Mostly Mozart Festival & White Light Festival, as well as the New Yorker Festival, Mannes Opera, the World Science Festival, and the revival of Mummunschanz. The Theater has hosted live and recorded events including David Letterman's My Next Guest Needs No Introduction, Inside the Actor's Studio, Carnegie Hall Neighborhood Concerts, Comedy Central Presents one-hour specials, the American Justice Summit, the NYC Mayoral Democratic Debates, and the launch of Jay-Z's REFORM initiative. The Theater welcomes premiere galas, conferences, international competitions, and graduations. For more information, and a schedule of events, please visit http://www.GeraldWLynchTheater.com.

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Jon Lehrer Dance Company To Present The World Premiere of THROUGH THE STORM, May 6 - Broadway World

Ethan Lous latest book is Once a Bitcoin Miner: Scandal and Turmoil in the Cryptocurrency Wild West.

In the cryptocurrency world, theres an event similar to a central bank rate adjustment in the mainstream economy: bitcoins halving.

Its far from the same as changing the benchmark interest rate, but it is also a high-level, wide-impact event that foretells which way the market winds will blow. And much like how central bank rate hikes and the talk of them have dampened the past years surging stocks, bitcoins halving trends are now also pointing to a bear market for cryptocurrency.

The halving is a reduction of the rate at which new bitcoins are introduced into the system. That process happens through so-called miners, who process transactions by solving mathematical puzzles through special computers. The miners are rewarded with bitcoins for their work. Every four years, that reward gets cut in half. The last halving was in 2020, when the reward for processing a block of transactions became 6.25 bitcoins.

Those reductions are significant for price because bitcoin does not have much of a circulating supply. The creator Satoshi Nakamotos more-than 1 million bitcoins, for example, have never moved. Many long-term investors do not sell. And examples abound of people losing their private keys, the equivalent of a password, rendering their coins forever lost.

Research published in the industrys Bitcoin Magazine shows that 76 per cent of the close to 19 million bitcoins currently in the system can be considered illiquid. The new bitcoins that miners introduce into the system are important in satisfying demand.

So, when one day, miners pump out only half as many new bitcoins as before, its a big blow to supply. It does take a while for the impact to trickle into the market, but past halvings have always been followed by sharp price rallies. By the end of 2020, bitcoin was up four times its price at the beginning of that year. Bitcoin prices usually lift the wider cryptocurrency market, too.

At the same time, though, rapid growth brings hype and absurdity the sort that eventually becomes sobering. A year or so after the halving, bitcoin usually reaches an unprecedented price. And there comes a point when people tire of it all and wonder, What am I doing with this $300,000 picture of an ape?

Brutal bust cycles have also always followed the cryptocurrency booms, with drops as much as 80 per cent until the next halving. Such has been the market cycles for the past 13 years bitcoin has been in existence.

Now, we are some two years after the last halving, which could be a cause for concern because this is usually around the time the bust starts.

Of course, like all market phenomena, this is the sort of quantum-physics situation in which having an observer affects the outcome. If everyone knows about this mechanism and thinks that bitcoins prices are affected by the halving, it will be baked in, and then prices will no longer be so affected by the halving. This time around, will we see history repeat?

It might be helpful to look further into history. There is this saying that past performance is no guarantee of future results, but for every such platitude theres an opposite: History doesnt repeat, but it often rhymes, those that fail to learn from history are doomed to repeat it and so on.

The latest bitcoin price peak of nearly US$69,000 was a little more than triple of the last peak of US$20,000, in 2017. Yet that earlier peak was about 20 times of the peak before to it. And before the first halving that sent bitcoin to US$1,000, bitcoins peak was only US$20 that rally was 50-fold.

The numbers might seem bigger now, but in percentage terms, bitcoins price movement has become less volatile. That does validate the view that the impact of the halvings has been increasingly baked in.

Its much like the central bank rate hikes these days. Bankers have not so much been telegraphing them, but beating them in with a stick. That blunts the otherwise sharp shocks, but the downward pressure on stocks is still there.

The bottom line is that, for a decentralized cryptocurrency ecosystem designed with no single point of failure, the bitcoin halving is perhaps the only factor that can have such a big and across-the-board impact. Like a central bank rate adjustment, its a factor that every investor must take into account.

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Opinion: The best predictor of bitcoin's price is foretelling a bear market - The Globe and Mail