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 many types 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.[3]

Each isolated 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]

Accompanying Postulate I is the composite system postulate:[5]

Composite system postulate

The Hilbert space of a composite system is the Hilbert space tensor product of the state spaces associated with the component systems. For a non-relativistic system consisting of a finite number of distinguishable particles, the component systems are the individual particles.

In the presence of quantum entanglement, the quantum state of the composite system cannot be factored as a tensor product of states of its local constituents; Instead, it is expressed as a sum, or superposition, of tensor products of states of component subsystems. A subsystem in an entangled composite system generally can't be described by a state vector (or a ray), but instead is described by a density operator; Such quantum state is known as a mixed state. The density operator of a mixed state is a trace class, nonnegative (positive semi-definite) self-adjoint operator normalized to be of trace 1. In turn, any density operator of a mixed state can be represented as a subsystem of a larger composite system in a pure state (see purification theorem).

In the absence of quantum entanglement, the quantum state of the composite system is called a separable state. The density matrix of a bipartite system in a separable state can be expressed as = k p k 1 k 2 k {displaystyle rho =sum _{k}p_{k}rho _{1}^{k}otimes rho _{2}^{k}} , where k p k = 1 {displaystyle ;sum _{k}p_{k}=1} . If there is only a single non-zero p k {displaystyle p_{k}} , then the state can be expressed just as = 1 2 , {textstyle rho =rho _{1}otimes rho _{2},} and is called simply separable or product state.

Physical observables are represented by Hermitian matrices on H. Since these operators are Hermitian, their eigenvalues are always real, and represent the possible outcomes/results from measuring the corresponding observable. 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 |A|psi rangle } . 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.

For a mixed state , the expected value of A in the state is tr ( A ) {displaystyle operatorname {tr} (Arho )} , and the probability of obtaining an eigenvalue a n {displaystyle a_{n}} in a discrete, nondegenerate spectrum of the corresponding observable A {displaystyle A} is given by P ( a n ) = tr ( | a n a n | ) = a n | | a n {displaystyle mathbb {P} (a_{n})=operatorname {tr} (|a_{n}rangle langle a_{n}|rho )=langle a_{n}|rho |a_{n}rangle } .

If the eigenvalue a n {displaystyle a_{n}} has degenerate, orthonormal eigenvectors { | a n 1 , | a n 2 , , | a n m } {displaystyle {|a_{n1}rangle ,|a_{n2}rangle ,dots ,|a_{nm}rangle }} , then the projection operator onto the eigensubspace can be defined as the identity operator in the eigensubspace:

Postulates II.a and II.b are collectively known as the Born rule of quantum mechanics.

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.

For a mixed state , after obtaining an eigenvalue a n {displaystyle a_{n}} in a discrete, nondegenerate spectrum of the corresponding observable A {displaystyle A} , the updated state is given by = P n P n tr ( P n P n ) {textstyle rho '={frac {P_{n}rho P_{n}^{dagger }}{operatorname {tr} (P_{n}rho P_{n}^{dagger })}}} . If the eigenvalue a n {displaystyle a_{n}} has degenerate, orthonormal eigenvectors { | a n 1 , | a n 2 , , | a n m } {displaystyle {|a_{n1}rangle ,|a_{n2}rangle ,dots ,|a_{nm}rangle }} , then the projection operator onto the eigensubspace is P n = | a n 1 a n 1 | + | a n 2 a n 2 | + + | a n m a n m | {displaystyle P_{n}=|a_{n1}rangle langle a_{n1}|+|a_{n2}rangle langle a_{n2}|+dots +|a_{nm}rangle langle a_{nm}|} .

Postulates II.c is sometimes called the "state update rule" or "collapse rule"; Together with the Born rule (Postulates II.a and II.b), they form a complete representation of measurements, and are sometimes collectively called the measurement postulate(s).

Note that the projection-valued measures (PVM) described in the measurement postulate(s) can be generalized to positive operator-valued measures (POVM), which is the most general kind of measurement in quantum mechanics. A POVM can be understood as the effect on a component subsystem when a PVM is performed on a larger, composite system (see Naimark's dilation theorem).

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 III

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 }

Equivalently, the time evolution postulate can be stated as:

Postulate III

The time evolution of a closed system is described by a unitary transformation on the initial state.

| ( t ) = U ( t ; t 0 ) | ( t 0 ) {displaystyle |psi (t)rangle =U(t;t_{0})|psi (t_{0})rangle }

For a closed system in a mixed state , the time evolution is ( t ) = U ( t ; t 0 ) ( t 0 ) U ( t ; t 0 ) {displaystyle rho (t)=U(t;t_{0})rho (t_{0})U^{dagger }(t;t_{0})} .

The evolution of an open quantum system can be described by quantum operations (in an operator sum formalism) and quantum instruments, and generally does not have to be unitary.

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.

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 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

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

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}},}

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 {frac {d}{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.

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" H E, 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 H E (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).

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.[6] 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

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

Whoever said, you cant get something from nothing must never have learned quantum physics. As long as you have empty space the ultimate in physical nothingness simply manipulating it in the right way will inevitably cause something to emerge. Collide two particles in the abyss of empty space, and sometimes additional particle-antiparticle pairs emerge. Take a meson and try to rip the quark away from the antiquark, and a new set of particle-antiparticle pairs will get pulled out of the empty space between them. And in theory, a strong enough electromagnetic field can rip particles and antiparticles out of the vacuum itself, even without any initial particles or antiparticles at all.

Previously, it was thought that the highest particle energies of all would be needed to produce these effects: the kind only obtainable at high-energy particle physics experiments or in extreme astrophysical environments. But in early 2022, strong enough electric fields were created in a simple laboratory setup leveraging the unique properties of graphene, enabling the spontaneous creation of particle-antiparticle pairs from nothing at all. The prediction that this should be possible is 70 year old: dating back to one of the founders of quantum field theory: Julian Schwinger. The Schwinger effect is now verified, and teaches us how the Universe truly makes something from nothing.

This chart of the particles and interactions details how the particles of the Standard Model interact according to the three fundamental forces that Quantum Field Theory describes. When gravity is added into the mix, we obtain the observable Universe that we see, with the laws, parameters, and constants that we know of governing it. Mysteries, such as dark matter and dark energy, still remain.

In the Universe we inhabit, its truly impossible to create nothing in any sort of satisfactory way. Everything that exists, down at a fundamental level, can be decomposed into individual entities quanta that cannot be broken down further. These elementary particles include quarks, electrons, the electrons heavier cousins (muons and taus), neutrinos, as well as all of their antimatter counterparts, plus photons, gluons, and the heavy bosons: the W+, W-, Z0, and the Higgs. If you take all of them away, however, the empty space that remains isnt quite empty in many physical senses.

For one, even in the absence of particles, quantum fields remain. Just as we cannot take the laws of physics away from the Universe, we cannot take the quantum fields that permeate the Universe away from it.

For another, no matter how far away we move any sources of matter, there are two long-range forces whose effects will still remain: electromagnetism and gravitation. While we can make clever setups that ensure that the electromagnetic field strength in a region is zero, we cannot do that for gravitation; space cannot be entirely emptied in any real sense in this regard.

Instead of an empty, blank, three-dimensional grid, putting a mass down causes what would have been straight lines to instead become curved by a specific amount. No matter how far away you get from a point mass, the curvature of space never reaches zero, but always remains, even at infinite range.

But even for the electromagnetic force even if you completely zero out the electric and magnetic fields within a region of space theres an experiment you can perform to demonstrate that empty space isnt truly empty. Even if you create a perfect vacuum, devoid of all particles and antiparticles of all types, where the electric and magnetic fields are zero, theres clearly something thats present in this region of what a physicist might call, from a physical perspective, maximum nothingness.

Travel the Universe with astrophysicist Ethan Siegel. Subscribers will get the newsletter every Saturday. All aboard!

All you need to do is place a set of parallel conducting plates in this region of space. Whereas you might expect that the only force theyd experience between them would be gravity, set by their mutual gravitational attraction, what actually winds up happening is that the plates attract by a much greater amount than gravity predicts.

This physical phenomenon is known as the Casimir effect, and was demonstrated to be true by Steve Lamoreaux in 1996: 48 years after it was calculated and proposed by Hendrik Casimir.

The Casimir effect, illustrated here for two parallel conducting plates, excludes certain electromagnetic modes from the interior of the conducting plates while permitting them outside of the plates. As a result, the plates attract, as predicted by Casimir in the 1940s and verified experimentally by Lamoreaux in the 1990s.

Similarly, in 1951, Julian Schwinger, already a co-founder of the quantum field theory that describes electrons and the electromagnetic force, gave a complete theoretical description of how matter could be created from nothing: simply by applying a strong electric field. Although others had proposed the idea back in the 1930s, including Fritz Sauter, Werner Heisenberg, and Hans Euler, Schwinger himself did the heavy lifting to quantify precisely under what conditions this effect should emerge, and henceforth its been primarily known as the Schwinger effect.

Normally, we expect there to be quantum fluctuations in empty space: excitations of any and all quantum fields that may be present. The Heisenberg uncertainty principle dictates that certain quantities cannot be known in tandem to arbitrary precision, and that includes things like:

While we normally express the uncertainty principle in terms of the first two entities, alone, the other applications can have consequences that are equally profound.

This diagram illustrates the inherent uncertainty relation between position and momentum. When one is known more accurately, the other is inherently less able to be known accurately. Every time you accurately measure one, you ensure a greater uncertainty in the corresponding complementary quantity.

Recall that, for any force that exists, we can describe that force in terms of a field: where the force experienced by a particle is its charge multiplied by some property of the field. If a particle passes through a region of space where the field is non-zero, it can experience a force, depending on its charge and (sometimes) its motion. The stronger the field, the greater the force, and the stronger the field, the greater the amount of field energy exists in that particular region of space.

Even in purely empty space, and even in the absence of external fields, there will still be some non-zero amount of field energy that exists in any such region of space. If there are quantum fields everywhere, then simply by Heisenbergs uncertainty principle, for any duration of time that we choose to measure this region over, there will be an inherently uncertain amount of energy present within that region during that time period.

The shorter the time period were looking at, the greater the uncertainty in the amount of energy in that region. Applying this to all allowable quantum states, we can begin to visualize the fluctuating fields, as well as fluctuating particle-antiparticle pairs, that pop in-and-out of existence due to all of the Universes quantum forces.

Even in the vacuum of empty space, devoid of masses, charges, curved space, and any external fields, the laws of nature and the quantum fields underlying them still exist. If you calculate the lowest-energy state, you may find that it is not exactly zero; the zero-point (or vacuum) energy of the Universe appears to be positive and finite, although small.

Now, lets imagine turning up the electric field. Turn it up, higher and higher, and what will happen?

Lets take an easier case first, and imagine theres a specific type of particle already present: a meson. A meson is made of one quark and one antiquark, connected to one another through the strong force and the exchange of gluons. Quarks come in six different flavors: up, down, strange, charm, bottom, and top, while the anti-quarks are simply anti-versions of each of them, with opposite electric charges.

The quark-antiquark pairs within a meson sometimes have opposite charges to one another: either + and - (for up, charm, and top) or + and - (for down, strange, and bottom). If you apply an electric field to such a meson, the positively charged end and the negatively charged end will be pulled in opposite directions. If the field strength is great enough, its possible to pull the quark and antiquark away from one another sufficiently so that new particle-antiparticle pairs are ripped out of the empty space between them. When this occurs, we wind up with two mesons instead of one, with the energy required to create the extra mass (via E = mc) coming from the electric field energy that ripped the meson apart in the first place.

When a meson, such as a charm-anticharm particle shown here, has its two constituent particles pulled apart by too great an amount, it becomes energetically favorable to rip a new (light) quark/antiquark pair out of the vacuum and create two mesons where there was one before. A strong enough electric field, for long-enough lived mesons, can cause this to occur, with the needed energy for creating more massive particles coming from the underlying electric field.

Now, with all of that as background in our minds, lets imagine weve got a very, very strong electric field: stronger than anything we could ever hope to make on Earth. Something so strong that it would be like taking a full Coulomb of charge around ~1019 electrons and protons and condensing each of them into a tiny ball, one purely of positive charge and one purely of negative charge, and separating them by only a meter. The quantum vacuum, in this region of space, is going to be extremely strongly polarized.

Strong polarization means a strong separation between positive and negative charges. If your electric field in a region of space is strong enough, then when you create a virtual particle-antiparticle pair of the lightest charged particle of all (electrons and positrons), you have a finite probability of those pairs being separated by large enough amounts due to the force from the field that they can no longer reannihilate one another. Instead, they become real particles, stealing energy from the underlying electric field in order to keep energy conserved.

As a result, new particle-antiparticle pairs come to exist, and the energy required to make them, from E = mc, reduces the exterior electric field strength by the appropriate amount.

As illustrated here, particle-antiparticle pairs normally pop out of the quantum vacuum as a consequences of Heisenberg uncertainty. In the presence of a strong enough electric field, however, these pairs can be ripped apart in opposite directions, causing them to be unable to reannihilate and forcing them to become real: at the expense of energy from the underlying electric field.

Thats what the Schwinger effect is, and unsurprisingly, its never been observed in a laboratory setting. In fact, the only places where it was theorized to occur was in the highest-energy astrophysical regions to exist in the Universe: in the environments surrounding (or even interior to) black holes and neutron stars. But at the great cosmic distances separating us from even the nearest black holes and neutron stars, even this remains conjecture. The strongest electric fields weve created on Earth are at laser facilities, and even with the strongest, most intense lasers at the shortest pulse times, we still arent even close.

Normally, whenever you have a conducting material, its only the valence electrons that are free to move, contributing to conduction. If you could achieve large enough electric fields, however, you could get all of the electrons to join the flow. In January of 2022, researchers at the University of Manchester were able to leverage an intricate and clever setup involving graphene an incredibly strong material that consists of carbon atoms bound together in geometrically optimal states to achieve this property with relatively small, experimentally accessible magnetic field. In doing so, they also witnesses the Schwinger effect in action: producing the analogue of electron-positron pairs in this quantum system.

Graphene has many fascinating properties, but one of them is a unique electronic band structure. There are conduction bands and valence bands, and they can overlap with zero band gap, enabling both holes and electrons to emerge and flow.

Graphene is an odd material in a lot of ways, and one of those ways is that sheets of it behave effectively as a two-dimensional structure. By reducing the number of (effective) dimensions, many degrees of freedom present in three-dimensional materials are taken away, leaving far fewer options for the quantum particles inside, as well as reducing the set of quantum states available for them to occupy.

Leveraging a graphene-based structure known as a superlattice where multiple layers of materials create periodic structures the authors of this study applied an electric field and induced the very behavior described above: where electrons from not just the highest partially-occupied energy state flow as part of the materials conduction, but where electrons from lower, completely filled bands join the flow as well.

Once this occurs, a lot of exotic behaviors arose in this material, but one was seen for the first time ever: the Schwinger effect. Instead of producing electrons and positrons, it produced electrons and the condensed-matter analogue of positrons: holes, where a missing electron in a lattice flows in the opposite directions to the electron flow. The only way to explain the observed currents were with this additional process of spontaneous production of electrons and holes, and the details of the process agreed with Schwingers predictions from all the way back in 1951.

Atomic and molecular configurations come in a near-infinite number of possible combinations, but the specific combinations found in any material determine its properties. Graphene, which is an individual, single-atom sheet of the material shown here, is the hardest material known to humanity, and in pairs-of-sheets it can create a type of material known as a superlattice, with many intricate and counterintuitive properties.

There are many ways of studying the Universe, and quantum analogue systems where the same mathematics that describes an otherwise inaccessible physical regime applies to a system that can be created and studied in a laboratory are some of the most powerful probes we have of exotic physics. Its very difficult to foresee how the Schwinger effect could be tested in its pure form, but thanks to the extreme properties of graphene, including its ability to withstand spectacularly large electric fields and currents, it arose for the very first time in any form: in this particular quantum system. As coauthor Dr. Roshan Krishna Kumar put it:

When we first saw the spectacular characteristics of our superlattice devices, we thought wow it could be some sort of new superconductivity. Although the response closely resembles those routinely observed in superconductors, we soon found that the puzzling behavior was not superconductivity but rather something in the domain of astrophysics and particle physics. It is curious to see such parallels between distant disciplines.

With electrons and positrons (or holes) being created out of literally nothing, just ripped out of the quantum vacuum by electric fields themselves, its yet another way that the Universe demonstrates the seemingly impossible: we really can make something from absolutely nothing!

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How the Universe really makes something from nothing - Big Think

The Standard Model of physics is the theory of particles, fields and the fundamental forces that govern them.

It tells us about how families of elementary particles group together to form larger composite particles, and how one particle can interact with another, and how particles respond to the fundamental forces of nature. It has made successful predictions such as the existence of the Higgs boson, and acts as the cornerstone for theoretical physics.

One way to think about the Standard Model is as a family tree for particles. For example, the Standard Model tells us how the atoms that make up our bodies are made of protons and neutrons, which in turn are made of elementary particles called quarks.

Related: What are bosons?

Keith Cooper is a freelance science journalist and editor in the United Kingdom, and has a degree in physics and astrophysics from the University of Manchester. He's the author of "The Contact Paradox: Challenging Our Assumptions in the Search for Extraterrestrial Intelligence" (Bloomsbury Sigma, 2020) and has written articles on astronomy, space, physics and astrobiology for a multitude of magazines and websites.

The Standard Model is considered by physicists, such as Glenn Starkman at Case Western Reserve University, as one of the most successful scientific theories (opens in new tab) of all time, but on the flip-side, scientists have also recognized that it is incomplete, in the same way that Isaac Newton's theory of universal gravitation derived from his laws of motion, while remarkably successful, was not the whole picture and required Albert Einstein's General Theory of Relativity to fill in the missing gaps.

The Standard Model was drawn together in the 1960s and early 1970s from the work of a cadre of pioneering scientists, but in truth its origins extend back almost 100 years earlier. By the 1880s, it was becoming apparent that there were positively and negatively charged particles produced when gasses are ionized, and that these particles must be smaller than atoms, which were the smallest known structures at the time. The first subatomic particle to be identified, in cathode rays (opens in new tab), was the negative electron in 1897 by the British physicist and subsequent Nobel Prize winner, J. J. Thomson (opens in new tab).

Then, in 1911, Hans Geiger and Ernest Madsen, under the supervision of the Nobel Laureate Ernest Rutherford (opens in new tab) at the University of Manchester, performed their famous 'gold foil' experiment, in which alpha particles (helium nuclei) were fired at a thin gold foil. Some of the alpha particles passed right through the atoms in the foil, while others were scattered left and right and a small fraction bounced right back.

Rutherford interpreted this as meaning that atoms contained a lot of empty space that the alpha particles were passing through, but that their positive charge was concentrated in a nucleus at their center, and on the occasions an alpha particle hit this nucleus dead on, it was scattered. Further experimentation by Rutherford in 191920 found that an alpha particle fired into air could knock a positively charged particle out of a nitrogen atom in the air, turning it into carbon in the process. That particle was the proton (opens in new tab), which gives the atomic nucleus its positive charge. The proton's neutrally charged partner, the neutron, was identified in 1932 by James Chadwick (opens in new tab) at Cambridge, who also won the Nobel Prize.

So, the picture of particle physics in the early 1930s seemed relatively straightforward atoms were made of two kinds of 'nucleons', in the guise of protons and neutrons, and electrons orbited them.

But things were already quickly starting to become more complicated. The existence of the photon was already known, so technically that was a fourth particle. In 1932 the American physicist Carl Anderson discovered the positron (opens in new tab), which is the antimatter equivalent of an electron. The muon was identified in 1936 by Anderson and Seth Neddermeyer (opens in new tab), and then the pion was discovered in 1947 (opens in new tab) by Cecil Powell. By the 1960s, with the advent of fledgling particle accelerators, hundreds of particles were being discovered, and the scientific picture was becoming very complicated indeed. Scientists needed a way of organizing and streamlining it all, and their answer to this was to create the Standard Model, which is the crowning glory of the cumulative work of the physics community of that era.

According to the Standard Model, there are three families of elementary particles. When we say 'elementary', scientists mean particles that cannot be broken down into even smaller particles. These are the smallest particles that together make up every other particle.

The three families are leptons, quarks and bosons. Leptons and quarks are known as Fermions because they have a half-integer spin. Bosons, on the other hand, have a whole-integer spin. What does this mean?

Spin, in the context of quantum physics, refers to spin angular momentum. This is different to orbital angular momentum, which describes Earth's spin around the sun, Earth's spin around its rotational axis, and even the spin of a spinning top. On the other hand, spin angular momentum is a quantum property intrinsic to each particle, even if that particle is stationary. Half-integer spin particles have spin values that are half-integers, so 1/2, 3/2, etc. The bosons have whole integer spin values, eg 1, 2, 3 etc.

Leptons include electrons, muons, tau particles and their associated neutrinos. Quarks are tiny particles that, when joined together, form composite particles such as protons and neutrons. Particles that are made of quarks are called hadrons (hence the Large Hadron Collider), with composite particles formed of odd numbers of quarks, usually three, being called baryons, and those made of two quarks called mesons. Bosons are force carriers they transfer the electromagnetic force (photons), the weak force (Z and W bosons), the strong nuclear force (gluons), and the Higgs force (Higgs boson).

Each 'family' consists of six known particles (except the bosons, which we'll explain later) that come in pairs called 'generations.' The most stable and least massive particles of the family form the first generation. Because of their stability, meaning that they don't decay quickly, all stable matter in the universe is made from first generation elementary particles. For example, protons are formed of two 'up' quarks and one 'down' quark, which are the two most stable quarks.

There are 17 known elementary particles 6 leptons, 6 quarks, but only 5 bosons. There's one force carrier missing the graviton. The Standard Model predicts that gravity should have a force-carrying boson, in the guise of the graviton. Gravitational waves are, in theory, formed from gravitons. However, detecting the graviton will be no mean feat. Gravity is the weakest of the four fundamental forces. You might not think so, after all it keeps your feet on the ground, but when you consider that it takes the entire mass of the planet to generate enough gravity to keep your feet on the ground, you might get a sense that gravity isn't as strong as, say, magnetism can be, which can pick up a paperclip against the gravitational pull of Earth. Consequently, individual gravitons do not interact with matter that easily they are said to have a low cross section of interaction (opens in new tab). Gravitons may have to remain hypothetical for the time being.

As wonderful as the Standard Model is, it describes only a small fraction of the universe. The European Space Agency's Planck spacecraft (opens in new tab) has confirmed that everything that we can see in the cosmos planets, stars and galaxies accounts for just 4.9% of all the mass and energy in the universe (opens in new tab). The rest is dark matter (26.8%) and dark energy (68.3%), the nature of which are completely unknown and which are definitely not predicted by the Standard Model.

That's not all that's unknown. One big question in physics is whether the elementary particles really are elementary, or whether there is hidden physics underlying them. For example, String Theory posits that elementary particles are made from tiny vibrating strings. Then there's the question of antimatter equal amounts of matter and antimatter (opens in new tab) should have been created in the Big Bang, but this would mean we should not be here at all, because all the matter and antimatter should have annihilated each other. Today we see that the universe contains mostly matter, with very little antimatter. Why is there this asymmetry?

Then there's the question of why particles have the masses that they do, and why the forces have the strengths that they have, and why particles are broken down into the three families of leptons, quarks and bosons. That they just are isn't a good enough answer for physicists they want to understand why, and the Standard Model does not tell them.

In an effort to bring the Standard Model up to speed to face these challenges, scientists have introduced the idea of supersymmetry. If true, then supersymmetry would mean that every particle in the Standard Model has a supersymmetric partner with a much greater mass, and a spin that is different by one-half to their Standard Model partners. This would unify fermions with bosons, since the integer-spin fermions would have half-integer-spin super-partners, and the half-integer-spin bosons would have integer-spin super-partners. The least massive and most stable supersymmetry particles would also have no electric charge and interact only very weakly with normal matter, which sounds very much like the properties of dark matter.

Meanwhile, at the very highest energies analogous to those that existed in the first moment after the Big Bang, supersymmetry predicts that the weak force, the strong force and the electromagnetic force would all have the same strength, and essentially be the same force. Scientists call such a concept a 'Grand Unified Theory'.

According to the CERN website, supersymmetry could also help explain the surprisingly small mass of the Higgs boson (opens in new tab), which is 125 GeV (125 billion electronvolts). While this is relatively high, it is not as high as expected. The existence of extremely massive supersymmetric partners would balance things out. And they must be extremely massive, because the Large Hadron Collider (LHC), nor any other particle accelerator before it, has found any evidence for the existence of supersymmetric partners so far, leading some scientists to doubt that supersymmetry is real. If supersymmetric particles exist, then they must be more massive than the LHC can detect; for example, the mass of the gluino (opens in new tab), which is the supersymmetric partner of the gluon that mediates the strong force binding quarks together inside protons and neutrons, has been ruled out up to 2 trillion eV.

So supersymmetry is in danger and physicists are now scrambling to find a replacement theory that can advance upon the Standard Model and explain the Higgs boson's mass, as well as dark matter, Grand Unified Theories and everything else. There are no strong candidates to replace supersymmetry yet, and supersymmetry may still win out, but for now physicists will have to make do with the imperfect world of the Standard Model.

CERN's website (opens in new tab) features more information about the Standard Model.

The U.S. Department of Energy explains the Standard Model (opens in new tab) on their own site.

The Institute of Physics also describes the Standard Model (opens in new tab) on their website.

Follow Keith Cooper on Twitter @21stCenturySETI (opens in new tab). Follow us on Twitter @Spacedotcom (opens in new tab) and on Facebook (opens in new tab).

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What is the standard model? - Space.com

Sabine Hossenfelder is a theoretical physicist and creator of the popular YouTube series Science Without the Gobbledygook. In her new book Existential Physics, she argues that some of her colleagues may have gotten a little too excited about wild ideas like multiverse theory or the simulation hypothesis.

If you want to discuss them on the level of philosophy, or maybe over a glass of wine with dinner because its fun to talk about, thats all fine with me, Hossenfelder says in Episode 525 of the Geeks Guide to the Galaxy podcast. I have a problem if they argue that its based on a scientific argument, which is not the case.

Multiverse theory states that an infinite number of alternate universes are constantly branching off from our own. Hossenfelder says its possible to create mathematical models that are consistent with multiverse theory, but that doesnt necessarily tell you anything about reality. I know quite a lot of cosmologists and astrophysicists who actually believe that other universes are real, and I think its a misunderstanding of how much mathematics can actually do for us, she says. There are certainly some people who have been pushing this line a little bit too farprobably deliberately, because it sellsbut I think for most of them theyre genuinely confused.

Hossenfelder is also skeptical of the simulation hypothesis, the idea that were living in a computer simulation. Its an idea thats been taken increasingly seriously by scientists and philosophers, but Hossenfelder says it really amounts to nothing more than a sort of techno-religion. If people go and spit out numbers like, I think theres a 50 percent chance were living in a simulation, Im not having it, she says. As a physicist who has to think about how you actually simulate the reality that we observe on a computer, Im telling you its not easy, and its not a problem that you can just sweep under the rug.

While theres currently no scientific evidence for multiverse theory or the simulation hypothesis, Hossenfelder says there are still plenty of cool ideas, including weather control, faster-than-light communication, and creating new universes, that dont contradict known science. This is exactly what I was hoping to achieve with the book, she says. I was trying to say, Physics isnt just something that tells you stuff that you cant do. It sometimes opens your mind to new things that we might possibly one day be able to do.'

Listen to the complete interview with Sabine Hossenfelder in Episode 525 of Geeks Guide to the Galaxy (above). And check out some highlights from the discussion below.

Sabine Hossenfelder on entropy:

Entropy is a very anthropomorphic quantity. The way its typically phrased is that entropy tells you something about the decrease of order or the increase of disorder, but this is really from our perspectivewhat we think is disorderly. I think that if you were not to use this human-centric notion of order and disorder, you would get a completely different notion of entropy, which brings up the question, Why is any one of them more tenable than any other? Theres just too much that we dont really understand about space and timeand entropy in particular, gravity, and so onto definitely make the statement. I dont think the second law of thermodynamics is as fundamental as a lot of physicists think it is.

Sabine Hossenfelder on creating a universe:

There is nothing in principle that would prevent us from creating a universe. When I talked about this the first time, people thought I was kidding, because Im kind of known to always say, No, this is bullshit. You cant do it. But in this case, its actually correct. I think the reason people get confused about it is, naively, it seems you would need a huge amount of mass or energy to create a universe, because where does all the stuff come from? And this just isnt necessary in Einsteins theory of general relativity. The reason is that if you have an expanding spacetime, it basically creates its own energy. How much mass youd need to create a new universe turns out to be something like 10 kilograms. So thats not all that much, except that you have to bring those 10 kilograms into a state that is very similar to the conditions in the early universe, which means you have to heat it up to dramatically high temperatures, which we just currently cant do.

Sabine Hossenfelder on faster-than-light communication:

I think that physicists are a little bit too fast to throw out faster-than-light communication, because theres a lot that we dont understand about locality. Im not a big fan of big wormholes, where you can go in one end and come out on the other end, but if spacetime has some kind of quantum structureand pretty much all physicists I know believe that it doesits quite conceivable that it would not respect the notion of locality that we enjoy in the macroscopic world. So on this microscopic quantum level, when youre taking into account the quantum properties of space and time, distance may just completely lose meaning. I find it quite conceivably possible that this will allow us to send information faster than light.

Sabine Hossenfelder on community:

When I was at the Perimeter Institute in Canada, they had a weekly public lecture. It was on the weekendso a time when people could actually come, not during work hoursand afterward there was a brunch that everyone would have together, and I know that the people who would attend those lectures would go there regularly, and they would appreciate the opportunity to just sit together and talk with other people who were interested in the same things. This is something that I think scientists take for granted. We have all our friends and colleagues that we talk to about the stuff that were interested in, but its not the case for everybody else. Some people are interested in, I dont know, quantum mechanics, and maybe they dont know anyone else whos interested in quantum mechanics. To some extent there are online communities that fulfill this task now, but of course its still better to actually meet with people in person.

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Have Some Scientists Gotten Too Excited About the Multiverse? - WIRED

Newswise In physics, Schroedingers cat is an allegory for two of the most awe-inspiring effects of quantum mechanics: entanglement and superposition. Researchers from Dresden and Munich have now observed these behaviors on a much larger scale than that of the smallest of particles. Until now, materials that display properties like, e.g., magnetism have been known to have so-called domains islands in which the materials properties are homogeneously either of one or a different kind (imagine them being either blackorwhite, for example). Looking at lithium holmium fluoride (LiHoF4), the physicists have now discovered a completely new phase transition, at which the domains surprisingly exhibit quantum mechanical features, resulting in their properties becoming entangled (being black and white at the same time). Our quantum cat now has a new fur because weve discovered a new quantum phase transition in LiHoF4which has not previously been known to exist, comments Matthias Vojta, Chair of Theoretical Solid State Physics at TUD.

Phase transitions and entanglement

We can easily observe the spontaneously changing properties of a substance if we look at water: at 100 degrees Celsius it evaporates into a gas, at zero degrees Celsius it freezes into ice. In both cases, these new states of matter form as a consequence of a phase transition where the water molecules rearrange themselves, thus changing the characteristics of the matter. Properties like magnetism or superconductivity emerge as a result of electrons undergoing phase transitions in crystals. For phase transitions at temperatures approaching the absolute zero at -273.15 degrees Celsius, quantum mechanical effects such as entanglement come into play, and one speaks of quantum phase transitions. Even though there are more than 30 years of extensive research dedicated to phase transitions in quantum materials, we had previously assumed that the phenomenon of entanglement played a role only on a microscopic scale, where it involves only a few atoms at a time,explains Christian Pfleiderer, Professor of Topology of Correlated Systems at the TUM.

Quantum entanglement is one of the most astonishing phenomena of physics, where the entangled quantum particles exist in a shared superposition state that allows for usually mutually exclusive properties (e.g., black and white) to occur simultaneously. As a rule, the laws of quantum mechanics only apply to microscopic particles. The research teams from Munich and Dresden have now succeeded in observing effects of quantum entanglement on a much larger scale, that of thousands of atoms. For this, they have chosen to work with the well-known compound LiHoF4.

Spherical samples enable precision measurements

At very low temperatures, LiHoF4acts as a ferromagnet where all magnetic moments spontaneously point in the same direction. If you then apply a magnetic field exactly vertically to the preferred magnetic direction, the magnetic moments will change direction, which is known as fluctuations. The higher the magnetic field strength, the stronger these fluctuations become, until, eventually, the ferromagnetism disappears completely at a quantum phase transition. This leads to the entanglement of neighboring magnetic moments. If you hold up a LiHoF4sample to a very strong magnet, it suddenly ceases to be spontaneously magnetic. This has been known for 25 years, summarizes Vojta.

What is new is what happens when you change the direction of the magnetic field. We discovered that the quantum phase transition continues to occur, whereas it had previously been believed that even the smallest tilt of the magnetic field would immediately suppress it, explains Pfleiderer. Under these conditions, however, it is not individual magnetic moments but rather extensive magnetic areas, so-called ferromagnetic domains, that undergo these quantum phase transitions. The domains constitute entire islands of magnetic moments pointing in the same direction. We have used spherical samples for our precision measurements. That is what enabled us to precisely study the behavior upon small changes in the direction of the magnetic field, adds Andreas Wendl, who conducted the experiments as part of his doctoral dissertation.

From fundamental physics to applications

We have discovered an entirely new type of quantum phase transitions where entanglement takes place on the scale of many thousands of atoms instead of just in the microcosm of only a few, explains Vojta. If you imagine the magnetic domains as a black-and-white pattern, the new phase transition leads to either the white or the black areas becoming infinitesimally small, i.e., creating a quantum pattern, bevor dissolving completely. A newly developed theoretical model successfully explains the data obtained from the experiments. For our analysis, we generalized existing microscopic models and also took into account the feedback of the large ferromagnetic domains to the microscopic properties, elaborates Heike Eisenlohr, who performed the calculations as part of her PhD thesis.

The discovery of the new quantum phase transitions is important as a foundation and general frame of reference for the research of quantum phenomena in materials, as well as for new applications. Quantum entanglement is applied and used in technologies like quantum sensors and quantum computers, amongst other things, says Vojta. Pfleiderer adds: Our work is in the area of fundamental research, which, however, can have a direct impact on the development of practical applications, if you use the materials properties in a controlled way.

Publikation:Emergence of mesoscale quantum phase transitions in a ferromagnetAndreas Wendl, Heike Eisenlohr, Felix Rucker, Christopher Duvinage, Markus Kleinhans, Matthias Vojta & Christian Pfleiderer,Nature609, 6570 (2022)https://www.nature.com/articles/s41586-022-04995-5

More information:The research has been financially supported by the Excellence Strategy of the German Federal and State Governments within the Wrzburg-Dresden Cluster of Excellence Complexity and Topology in Quantum Matter (ct.qmat) and the Cluster of Excellence Munich Center for Quantum Science and Technology (MCQST). In addition, the work has been supported by the European Research Council (ERC) via the Advanced Grant ExQuiSid and by the Deutsche Forschungsgemeinschaft (DFG) within the Collaborative Research Centers (SFB) 1143 und TRR80.

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Quantum materials: entanglement of many atoms discovered for the first time - Newswise