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Zero-point energy (ZPE) is the lowest possible
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...
that a
quantum mechanical Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
system may have. Unlike in
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
, quantum systems constantly fluctuate in their lowest energy state as described by the
Heisenberg uncertainty principle In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
. Therefore, even at
absolute zero Absolute zero is the lowest limit of the thermodynamic temperature scale, a state at which the enthalpy and entropy of a cooled ideal gas reach their minimum value, taken as zero kelvin. The fundamental particles of nature have minimum vibra ...
, atoms and molecules retain some vibrational motion. Apart from
atom Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas, a ...
s and
molecule A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and b ...
s, the empty space of
the vacuum ''The'' () is a grammatical article in English, denoting persons or things already mentioned, under discussion, implied or otherwise presumed familiar to listeners, readers, or speakers. It is the definite article in English. ''The'' is the ...
also has these properties. According to
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, the universe can be thought of not as isolated particles but continuous fluctuating fields:
matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic part ...
fields, whose quanta are
fermions In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
(i.e.,
lepton In particle physics, a lepton is an elementary particle of half-integer spin (spin ) that does not undergo strong interactions. Two main classes of leptons exist: charged leptons (also known as the electron-like leptons or muons), and neutr ...
s and
quark A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly ...
s), and force fields, whose quanta are
boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer spi ...
s (e.g.,
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
s and
gluon A gluon ( ) is an elementary particle that acts as the exchange particle (or gauge boson) for the strong force between quarks. It is analogous to the exchange of photons in the electromagnetic force between two charged particles. Gluons bind ...
s). All these fields have zero-point energy. These fluctuating zero-point fields lead to a kind of reintroduction of an aether in physics since some systems can detect the existence of this energy. However, this aether cannot be thought of as a physical medium if it is to be
Lorentz invariant In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of ...
such that there is no contradiction with Einstein's theory of
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
. The notion of a zero-point energy is also important for
cosmology Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', and in 1731 taken up in Latin by German philosopher ...
, and physics currently lacks a full theoretical model for understanding zero-point energy in this context; in particular, the discrepancy between theorized and observed vacuum energy in the universe is a source of major contention. Physicists
Richard Feynman Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superfl ...
and John Wheeler calculated the zero-point radiation of the vacuum to be an order of magnitude greater than nuclear energy, with a single light bulb containing enough energy to boil all the world's oceans. Yet according to Einstein's theory of
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, any such energy would gravitate, and the experimental evidence from the
expansion of the universe The expansion of the universe is the increase in distance between any two given gravitationally unbound parts of the observable universe with time. It is an intrinsic expansion whereby the scale of space itself changes. The universe does not ex ...
,
dark energy In physical cosmology and astronomy, dark energy is an unknown form of energy that affects the universe on the largest scales. The first observational evidence for its existence came from measurements of supernovas, which showed that the univ ...
and the
Casimir effect In quantum field theory, the Casimir effect is a physical force acting on the macroscopic boundaries of a confined space which arises from the quantum fluctuations of the field. It is named after the Dutch physicist Hendrik Casimir, who pre ...
shows any such energy to be exceptionally weak. A popular proposal that attempts to address this issue is to say that the fermion field has a negative zero-point energy, while the boson field has positive zero-point energy and thus these energies somehow cancel each other out. This idea would be true if
supersymmetry In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories ...
were an exact symmetry of nature; however, the LHC at
CERN The European Organization for Nuclear Research, known as CERN (; ; ), is an intergovernmental organization that operates the largest particle physics laboratory in the world. Established in 1954, it is based in a northwestern suburb of Gen ...
has so far found no evidence to support it. Moreover, it is known that if supersymmetry is valid at all, it is at most a
broken symmetry In physics, symmetry breaking is a phenomenon in which (infinitesimally) small fluctuations acting on a system crossing a critical point decide the system's fate, by determining which branch of a bifurcation is taken. To an outside observ ...
, only true at very high energies, and no one has been able to show a theory where zero-point cancellations occur in the low-energy universe we observe today. This discrepancy is known as the
cosmological constant problem In cosmology, the cosmological constant problem or vacuum catastrophe is the disagreement between the observed values of vacuum energy density (the small value of the cosmological constant) and theoretical large value of zero-point energy sugge ...
and it is one of the greatest unsolved mysteries in physics. Many physicists believe that "the vacuum holds the key to a full understanding of nature".


Etymology and terminology

The term zero-point energy (ZPE) is a translation from the German Nullpunktsenergie. Sometimes used interchangeably with it are the terms zero-point radiation and ground state energy. The term zero-point field (ZPF) can be used when referring to a specific vacuum field, for instance the QED vacuum which specifically deals with
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
(e.g., electromagnetic interactions between photons, electrons and the vacuum) or the
QCD vacuum In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type o ...
which deals with
quantum chromodynamics In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type ...
(e.g.,
color charge Color charge is a property of quarks and gluons that is related to the particles' strong interactions in the theory of quantum chromodynamics (QCD). The "color charge" of quarks and gluons is completely unrelated to the everyday meanings of colo ...
interactions between quarks, gluons and the vacuum). A vacuum can be viewed not as empty space but as the combination of all zero-point fields. In
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
this combination of fields is called the
vacuum state In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. The word zero-point field is sometimes used as ...
, its associated zero-point energy is called the
vacuum energy Vacuum energy is an underlying background energy that exists in space throughout the entire Universe. The vacuum energy is a special case of zero-point energy that relates to the quantum vacuum. The effects of vacuum energy can be experimental ...
and the average energy value is called the
vacuum expectation value In quantum field theory the vacuum expectation value (also called condensate or simply VEV) of an operator is its average or expectation value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle ...
(VEV) also called its condensate.


Overview

In
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
all
particle In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from ...
s can be thought of as having some
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...
made up of their
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potenti ...
and
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acce ...
.
Temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
, for example, arises from the intensity of random particle motion caused by kinetic energy (known as
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
). As temperature is reduced to
absolute zero Absolute zero is the lowest limit of the thermodynamic temperature scale, a state at which the enthalpy and entropy of a cooled ideal gas reach their minimum value, taken as zero kelvin. The fundamental particles of nature have minimum vibra ...
, it might be thought that all motion ceases and particles come completely to rest. In fact, however, kinetic energy is retained by particles even at the lowest possible temperature. The random motion corresponding to this zero-point energy never vanishes; it is a consequence of the
uncertainty principle In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
of
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. The uncertainty principle states that no object can ever have precise values of position and velocity simultaneously. The total energy of a quantum mechanical object (potential and kinetic) is described by its Hamiltonian which also describes the system as a harmonic oscillator, or
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
, that fluctuates between various energy states (see wave-particle duality). All quantum mechanical systems undergo fluctuations even in their ground state, a consequence of their
wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
-like nature. The uncertainty principle requires every quantum mechanical system to have a fluctuating zero-point energy greater than the minimum of its classical
potential well A potential well is the region surrounding a local minimum of potential energy. Energy captured in a potential well is unable to convert to another type of energy ( kinetic energy in the case of a gravitational potential well) because it is ca ...
. This results in motion even at
absolute zero Absolute zero is the lowest limit of the thermodynamic temperature scale, a state at which the enthalpy and entropy of a cooled ideal gas reach their minimum value, taken as zero kelvin. The fundamental particles of nature have minimum vibra ...
. For example,
liquid helium Liquid helium is a physical state of helium at very low temperatures at standard atmospheric pressures. Liquid helium may show superfluidity. At standard pressure, the chemical element helium exists in a liquid form only at the extremely low temp ...
does not freeze under atmospheric pressure regardless of temperature due to its zero-point energy. Given the equivalence of mass and energy expressed by
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
's , any point in
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consi ...
that contains energy can be thought of as having mass to create particles. Virtual particles spontaneously flash into existence at every point in space due to the energy of quantum fluctuations caused by the uncertainty principle. Modern physics has developed
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
(QFT) to understand the fundamental interactions between matter and forces, it treats every single point of space as a
quantum harmonic oscillator 量子調和振動子 は、 古典調和振動子 の 量子力学 類似物です。任意の滑らかな ポテンシャル は通常、安定した 平衡点 の近くで 調和ポテンシャル として近似できるため、最� ...
. According to QFT the universe is made up of matter fields, whose quanta are
fermions In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
(i.e.
lepton In particle physics, a lepton is an elementary particle of half-integer spin (spin ) that does not undergo strong interactions. Two main classes of leptons exist: charged leptons (also known as the electron-like leptons or muons), and neutr ...
s and
quark A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly ...
s), and force fields, whose quanta are
boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer spi ...
s (e.g.
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
s and
gluon A gluon ( ) is an elementary particle that acts as the exchange particle (or gauge boson) for the strong force between quarks. It is analogous to the exchange of photons in the electromagnetic force between two charged particles. Gluons bind ...
s). All these fields have zero-point energy. Recent experiments advocate the idea that particles themselves can be thought of as excited states of the underlying
quantum vacuum In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. The word zero-point field is sometimes used as ...
, and that all properties of matter are merely vacuum fluctuations arising from interactions of the zero-point field. The idea that "empty" space can have an intrinsic energy associated to it, and that there is no such thing as a "true vacuum" is seemingly unintuitive. It is often argued that the entire universe is completely bathed in the zero-point radiation, and as such it can add only some constant amount to calculations. Physical measurements will therefore reveal only deviations from this value. For many practical calculations zero-point energy is dismissed by fiat in the mathematical model as a term that has no physical effect. Such treatment causes problems however, as in Einstein's theory of
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
the absolute energy value of space is not an arbitrary constant and gives rise to the
cosmological constant In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant, is the constant coefficient of a term that Albert Einstein temporarily added to his field eq ...
. For decades most physicists assumed that there was some undiscovered fundamental principle that will remove the infinite zero-point energy and make it completely vanish. If the vacuum has no intrinsic, absolute value of energy it will not gravitate. It was believed that as the universe expands from the aftermath of the
Big Bang The Big Bang event is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models of the Big Bang explain the evolution of the observable universe from the ...
, the energy contained in any unit of empty space will decrease as the total energy spreads out to fill the volume of the universe;
galaxies A galaxy is a system of stars, stellar remnants, interstellar gas, dust, dark matter, bound together by gravity. The word is derived from the Greek ' (), literally 'milky', a reference to the Milky Way galaxy that contains the Solar System ...
and all matter in the universe should begin to decelerate. This possibility was ruled out in 1998 by the discovery that the expansion of the universe is not slowing down but is in fact accelerating, meaning empty space does indeed have some intrinsic energy. The discovery of
dark energy In physical cosmology and astronomy, dark energy is an unknown form of energy that affects the universe on the largest scales. The first observational evidence for its existence came from measurements of supernovas, which showed that the univ ...
is best explained by zero-point energy, though it still remains a mystery as to why the value appears to be so small compared to the huge value obtained through theory - the
cosmological constant problem In cosmology, the cosmological constant problem or vacuum catastrophe is the disagreement between the observed values of vacuum energy density (the small value of the cosmological constant) and theoretical large value of zero-point energy sugge ...
. Many physical effects attributed to zero-point energy have been experimentally verified, such as
spontaneous emission Spontaneous emission is the process in which a quantum mechanical system (such as a molecule, an atom or a subatomic particle) transits from an excited energy state to a lower energy state (e.g., its ground state) and emits a quantized amount ...
,
Casimir force In quantum field theory, the Casimir effect is a physical force acting on the macroscopic boundaries of a confined space which arises from the quantum fluctuations of the field. It is named after the Dutch physicist Hendrik Casimir, who pre ...
,
Lamb shift In physics, the Lamb shift, named after Willis Lamb, is a difference in energy between two energy levels 2''S''1/2 and 2''P''1/2 (in term symbol notation) of the hydrogen atom which was not predicted by the Dirac equation, according to which th ...
, magnetic moment of the electron and
Delbrück scattering Delbrück scattering, the deflection of high-energy photons in the Coulomb field of nuclei as a consequence of vacuum polarization, was observed in 1975. The related process of the scattering of light by light, also a consequence of vacuum polariz ...
. These effects are usually called "radiative corrections". In more complex nonlinear theories (e.g. QCD) zero-point energy can give rise to a variety of complex phenomena such as multiple stable states,
symmetry breaking In physics, symmetry breaking is a phenomenon in which (infinitesimally) small fluctuations acting on a system crossing a critical point decide the system's fate, by determining which branch of a bifurcation is taken. To an outside observe ...
,
chaos Chaos or CHAOS may refer to: Arts, entertainment and media Fictional elements * Chaos (''Kinnikuman'') * Chaos (''Sailor Moon'') * Chaos (''Sesame Park'') * Chaos (''Warhammer'') * Chaos, in ''Fabula Nova Crystallis Final Fantasy'' * Cha ...
and
emergence In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergenc ...
. Many physicists believe that "the vacuum holds the key to a full understanding of nature" and that studying it is critical in the search for the
theory of everything A theory of everything (TOE or TOE/ToE), final theory, ultimate theory, unified field theory or master theory is a hypothetical, singular, all-encompassing, coherent theoretical framework of physics that fully explains and links together all asp ...
. Active areas of research include the effects of virtual particles,
quantum entanglement Quantum entanglement is the phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of ...
, the difference (if any) between inertial and gravitational mass,See for proposal and , and for comment. variation in the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
, a reason for the observed value of the
cosmological constant In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant, is the constant coefficient of a term that Albert Einstein temporarily added to his field eq ...
and the nature of
dark energy In physical cosmology and astronomy, dark energy is an unknown form of energy that affects the universe on the largest scales. The first observational evidence for its existence came from measurements of supernovas, which showed that the univ ...
.


History


Early aether theories

Zero-point energy evolved from historical ideas about the
vacuum A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often ...
. To
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ...
the vacuum was , "the empty"; i.e., space independent of body. He believed this concept violated basic physical principles and asserted that the elements of fire, air, earth, and water were not made of atoms, but were continuous. To the
atomists Atomism (from Greek , ''atomon'', i.e. "uncuttable, indivisible") is a natural philosophy proposing that the physical universe is composed of fundamental indivisible components known as atoms. References to the concept of atomism and its atoms a ...
the concept of emptiness had absolute character: it was the distinction between existence and nonexistence. Debate about the characteristics of the vacuum were largely confined to the realm of
philosophy Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. ...
, it was not until much later on with the beginning of
the renaissance The Renaissance ( , ) , from , with the same meanings. is a period in European history marking the transition from the Middle Ages to modernity and covering the 15th and 16th centuries, characterized by an effort to revive and surpass idea ...
, that
Otto von Guericke Otto von Guericke ( , , ; spelled Gericke until 1666; November 20, 1602 – May 11, 1686 ; November 30, 1602 – May 21, 1686 ) was a German scientist, inventor, and politician. His pioneering scientific work, the development of experimental me ...
invented the first vacuum pump and the first testable scientific ideas began to emerge. It was thought that a totally empty volume of space could be created by simply removing all gases. This was the first generally accepted concept of the vacuum. Late in the 19th century, however, it became apparent that the evacuated region still contained
thermal radiation Thermal radiation is electromagnetic radiation generated by the thermal motion of particles in matter. Thermal radiation is generated when heat from the movement of charges in the material (electrons and protons in common forms of matter) i ...
. The existence of the aether as a substitute for a true void was the most prevalent theory of the time. According to the successful
electromagnetic In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
aether theory based upon
Maxwell's Maxwell's, last known as Maxwell's Tavern, was a bar/restaurant and music club in Hoboken, New Jersey. Over several decades the venue attracted a wide variety of acts looking for a change from the New York City concert spaces across the river. Ma ...
electrodynamics In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
, this all-encompassing aether was endowed with energy and hence very different from nothingness. The fact that electromagnetic and gravitational phenomena were easily transmitted in empty space indicated that their associated aethers were part of the fabric of space itself. Maxwell himself noted that: However, the results of the
Michelson–Morley experiment The Michelson–Morley experiment was an attempt to detect the existence of the luminiferous aether, a supposed medium permeating space that was thought to be the carrier of light waves. The experiment was performed between April and July 188 ...
in 1887 were the first strong evidence that the then-prevalent aether theories were seriously flawed, and initiated a line of research that eventually led to
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
, which ruled out the idea of a stationary aether altogether. To scientists of the period, it seemed that a true vacuum in space might be created by cooling and thus eliminating all radiation or energy. From this idea evolved the second concept of achieving a real vacuum: cool a region of space down to
absolute zero Absolute zero is the lowest limit of the thermodynamic temperature scale, a state at which the enthalpy and entropy of a cooled ideal gas reach their minimum value, taken as zero kelvin. The fundamental particles of nature have minimum vibra ...
temperature after evacuation. Absolute zero was technically impossible to achieve in the 19th century, so the debate remained unsolved.


Second quantum theory

In 1900,
Max Planck Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918. Planck made many substantial contributions to theoretical p ...
derived the average energy of a single ''energy radiator'', e.g., a vibrating atomic unit, as a function of absolute temperature: : \varepsilon = \frac \,, where is Planck's constant, is the
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
, is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
, and is the absolute
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
. The zero-point energy makes no contribution to Planck's original law, as its existence was unknown to Planck in 1900. The concept of zero-point energy was developed by
Max Planck Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918. Planck made many substantial contributions to theoretical p ...
in Germany in 1911 as a corrective term added to a zero-grounded formula developed in his original quantum theory in 1900. In 1912, Max Planck published the first journal article to describe the discontinuous emission of radiation, based on the discrete quanta of energy. In Planck's "second quantum theory" resonators absorbed energy continuously, but emitted energy in discrete energy quanta only when they reached the boundaries of finite cells in phase space, where their energies became integer multiples of . This theory led Planck to his new radiation law, but in this version energy resonators possessed a zero-point energy, the smallest average energy a resonator could take on. Planck's radiation equation contained a residual energy factor, one , as an additional term dependent on the frequency , which was greater than zero (where is Planck's constant). It is therefore widely agreed that "Planck's equation marked the birth of the concept of zero-point energy." In a series of papers from 1911 to 1913, Planck found the average energy of an oscillator to be: : \varepsilon =\frac 2 + \frac ~. Soon, the idea of zero-point energy attracted the attention of
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
and his assistant
Otto Stern :''Otto Stern was also the pen name of German women's rights activist Louise Otto-Peters (1819–1895)''. Otto Stern (; 17 February 1888 – 17 August 1969) was a German-American physicist and Nobel laureate in physics. He was the second most ...
. In 1913 they published a paper that attempted to prove the existence of zero-point energy by calculating the specific heat of hydrogen gas and compared it with the experimental data. However, after assuming they had succeeded, they retracted support for the idea shortly after publication because they found Planck's second theory may not apply to their example. In a letter to
Paul Ehrenfest Paul Ehrenfest (18 January 1880 – 25 September 1933) was an Austrian theoretical physicist, who made major contributions to the field of statistical mechanics and its relations with quantum mechanics, including the theory of phase transition ...
of the same year Einstein declared zero-point energy "dead as a doornail" Zero-point energy was also invoked by
Peter Debye Peter Joseph William Debye (; ; March 24, 1884 – November 2, 1966) was a Dutch-American physicist and physical chemist, and Nobel laureate in Chemistry. Biography Early life Born Petrus Josephus Wilhelmus Debije in Maastricht, Netherland ...
, who noted that zero-point energy of the atoms of a
crystal lattice In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by : \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n ...
would cause a reduction in the intensity of the diffracted radiation in
X-ray diffraction X-ray crystallography is the experimental science determining the atomic and molecular structure of a crystal, in which the crystalline structure causes a beam of incident X-rays to diffract into many specific directions. By measuring the angles ...
even as the temperature approached absolute zero. In 1916
Walther Nernst Walther Hermann Nernst (; 25 June 1864 – 18 November 1941) was a German chemist known for his work in thermodynamics, physical chemistry, electrochemistry, and solid state physics. His formulation of the Nernst heat theorem helped pave the w ...
proposed that empty space was filled with zero-point
electromagnetic radiation In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...
. With the development of
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
Einstein found the energy density of the vacuum to contribute towards a
cosmological constant In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant, is the constant coefficient of a term that Albert Einstein temporarily added to his field eq ...
in order to obtain static solutions to his field equations; the idea that empty space, or the vacuum, could have some intrinsic energy associated to it had returned, with Einstein stating in 1920: Kurt Bennewitz and
Francis Simon Sir Francis Simon (2 July 1893 – 31 October 1956), was a German and later British physical chemist and physicist who devised the gaseous diffusion method, and confirmed its feasibility, of separating the isotope Uranium-235 and thus made a m ...
(1923) who worked at
Walther Nernst Walther Hermann Nernst (; 25 June 1864 – 18 November 1941) was a German chemist known for his work in thermodynamics, physical chemistry, electrochemistry, and solid state physics. His formulation of the Nernst heat theorem helped pave the w ...
's laboratory in Berlin, studied the melting process of chemicals at low temperatures. Their calculations of the melting points of
hydrogen Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-to ...
,
argon Argon is a chemical element with the symbol Ar and atomic number 18. It is in group 18 of the periodic table and is a noble gas. Argon is the third-most abundant gas in Earth's atmosphere, at 0.934% (9340 ppmv). It is more than twice a ...
and mercury led them to conclude that the results provided evidence for a zero-point energy. Moreover, they suggested correctly, as was later verified by Simon (1934), that this quantity was responsible for the difficulty in solidifying helium even at absolute zero. In 1924
Robert Mulliken Robert Sanderson Mulliken Note Longuet-Higgins' amusing title for reference B238 1965 on page 354 of this Biographical Memoir. The title should be "Selected papers of Robert S Mulliken." (June 7, 1896 – October 31, 1986) was an American ph ...
provided direct evidence for the zero-point energy of molecular vibrations by comparing the band spectrum of 10BO and 11BO: the isotopic difference in the transition frequencies between the ground vibrational states of two different electronic levels would vanish if there were no zero-point energy, in contrast to the observed spectra. Then just a year later in 1925, with the development of
matrix mechanics Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It was the first conceptually autonomous and logically consistent formulation of quantum mechanics. Its account of quantum j ...
in
Werner Heisenberg Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a Über quantentheoretische Umdeutung kinematis ...
's famous article " Quantum theoretical re-interpretation of kinematic and mechanical relations" the zero-point energy was derived from quantum mechanics. In 1913
Niels Bohr Niels Henrik David Bohr (; 7 October 1885 – 18 November 1962) was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922 ...
had proposed what is now called the
Bohr model In atomic physics, the Bohr model or Rutherford–Bohr model, presented by Niels Bohr and Ernest Rutherford in 1913, is a system consisting of a small, dense nucleus surrounded by orbiting electrons—similar to the structure of the Solar Syst ...
of the atom, but despite this it remained a mystery as to why electrons do not fall into their nuclei. According to classical ideas, the fact that an accelerating charge loses energy by radiating implied that an electron should spiral into the nucleus and that atoms should not be stable. This problem of classical mechanics was nicely summarized by
James Hopwood Jeans Sir James Hopwood Jeans (11 September 187716 September 1946) was an English physicist, astronomer and mathematician. Early life Born in Ormskirk, Lancashire, the son of William Tulloch Jeans, a parliamentary correspondent and author. Jeans ...
in 1915: "There would be a very real difficulty in supposing that the (force) law held down to the zero values of . For the forces between two charges at zero distance would be infinite; we should have charges of opposite sign continually rushing together and, when once together, no force would tend to shrink into nothing or to diminish indefinitely in size." The resolution to this puzzle came in 1926 with Schrödinger's famous equation. This equation explained the new, non-classical fact that an electron confined to be close to a nucleus would necessarily have a large kinetic energy so that the minimum total energy (kinetic plus potential) actually occurs at some positive separation rather than at zero separation; in other words, zero-point energy is essential for atomic stability.


Quantum field theory and beyond

In 1926
Pascual Jordan Ernst Pascual Jordan (; 18 October 1902 – 31 July 1980) was a German theoretical and mathematical physicist who made significant contributions to quantum mechanics and quantum field theory. He contributed much to the mathematical form of matri ...
published the first attempt to quantize the electromagnetic field. In a joint paper with
Max Born Max Born (; 11 December 1882 – 5 January 1970) was a German physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a ...
and
Werner Heisenberg Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a Über quantentheoretische Umdeutung kinematis ...
he considered the field inside a cavity as a superposition of quantum harmonic oscillators. In his calculation he found that in addition to the "thermal energy" of the oscillators there also had to exist an infinite zero-point energy term. He was able to obtain the same fluctuation formula that Einstein had obtained in 1909. However, Jordan did not think that his infinite zero-point energy term was "real", writing to Einstein that "it is just a quantity of the calculation having no direct physical meaning". Jordan found a way to get rid of the infinite term, publishing a joint work with Pauli in 1928, performing what has been called "the first infinite subtraction, or renormalisation, in quantum field theory". Building on the work of Heisenberg and others
Paul Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
's theory of emission and absorption (1927) was the first application of the quantum theory of radiation. Dirac's work was seen as crucially important to the emerging field of quantum mechanics; it dealt directly with the process in which "particles" are actually created:
spontaneous emission Spontaneous emission is the process in which a quantum mechanical system (such as a molecule, an atom or a subatomic particle) transits from an excited energy state to a lower energy state (e.g., its ground state) and emits a quantized amount ...
. Dirac described the quantization of the
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical ...
as an ensemble of
harmonic oscillator In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force ''F'' proportional to the displacement ''x'': \vec F = -k \vec x, where ''k'' is a positive const ...
s with the introduction of the concept of
creation and annihilation operators Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually d ...
of particles. The theory showed that spontaneous emission depends upon the zero-point energy fluctuations of the electromagnetic field in order to get started. In a process in which a photon is annihilated (absorbed), the photon can be thought of as making a transition into the vacuum state. Similarly, when a photon is created (emitted), it is occasionally useful to imagine that the photon has made a transition out of the vacuum state. In the words of Dirac: Contemporary physicists, when asked to give a physical explanation for spontaneous emission, generally invoke the zero-point energy of the electromagnetic field. This view was popularized by
Victor Weisskopf Victor Frederick "Viki" Weisskopf (also spelled Viktor; September 19, 1908 – April 22, 2002) was an Austrian-born American theoretical physicist. He did postdoctoral work with Werner Heisenberg, Erwin Schrödinger, Wolfgang Pauli, and Niels Boh ...
who in 1935 wrote: This view was also later supported by Theodore Welton (1948), who argued that spontaneous emission "can be thought of as forced emission taking place under the action of the fluctuating field." This new theory, which Dirac coined
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
(QED) predicted a fluctuating zero-point or "vacuum" field existing even in the absence of sources. Throughout the 1940s improvements in
microwave Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency ra ...
technology made it possible to take more precise measurements of the shift of the levels of a
hydrogen atom A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. Atomic hydrogen cons ...
, now known as the
Lamb shift In physics, the Lamb shift, named after Willis Lamb, is a difference in energy between two energy levels 2''S''1/2 and 2''P''1/2 (in term symbol notation) of the hydrogen atom which was not predicted by the Dirac equation, according to which th ...
, and measurement of the
magnetic moment In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electroma ...
of the electron. Discrepancies between these experiments and Dirac's theory led to the idea of incorporating
renormalisation Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering va ...
into QED to deal with zero-point infinities. Renormalization was originally developed by
Hans Kramers Hendrik Anthony "Hans" Kramers (17 December 1894 – 24 April 1952) was a Dutch physicist who worked with Niels Bohr to understand how electromagnetic waves interact with matter and made important contributions to quantum mechanics and statistical ...
and also
Victor Weisskopf Victor Frederick "Viki" Weisskopf (also spelled Viktor; September 19, 1908 – April 22, 2002) was an Austrian-born American theoretical physicist. He did postdoctoral work with Werner Heisenberg, Erwin Schrödinger, Wolfgang Pauli, and Niels Boh ...
(1936), and first successfully applied to calculate a finite value for the Lamb shift by
Hans Bethe Hans Albrecht Bethe (; July 2, 1906 – March 6, 2005) was a German-American theoretical physicist who made major contributions to nuclear physics, astrophysics, quantum electrodynamics, and solid-state physics, and who won the 1967 Nobel ...
(1947). As per spontaneous emission, these effects can in part be understood with interactions with the zero-point field. But in light of renormalisation being able to remove some zero-point infinities from calculations, not all physicists were comfortable attributing zero-point energy any physical meaning, viewing it instead as a mathematical artifact that might one day be fully eliminated. In
Wolfgang Pauli Wolfgang Ernst Pauli (; ; 25 April 1900 – 15 December 1958) was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after having been nominated by Albert Einstein, Pauli received the Nobel Prize in Physics ...
's 1945 Nobel lecture he made clear his opposition to the idea of zero-point energy stating "It is clear that this zero-point energy has no physical reality". In 1948
Hendrik Casimir Hendrik Brugt Gerhard Casimir (15 July 1909 – 4 May 2000) was a Dutch physicist best known for his research on the two-fluid model of superconductors (together with C. J. Gorter) in 1934 and the Casimir effect (together with D. Polder) in 1 ...
showed that one consequence of the zero-point field is an attractive force between two uncharged, perfectly conducting parallel plates, the so-called
Casimir effect In quantum field theory, the Casimir effect is a physical force acting on the macroscopic boundaries of a confined space which arises from the quantum fluctuations of the field. It is named after the Dutch physicist Hendrik Casimir, who pre ...
. At the time, Casimir was studying the properties of "colloidal solutions". These are viscous materials, such as paint and mayonnaise, that contain micron-sized particles in a liquid matrix. The properties of such solutions are determined by
Van der Waals forces In molecular physics, the van der Waals force is a distance-dependent interaction between atoms or molecules. Unlike ionic or covalent bonds, these attractions do not result from a chemical electronic bond; they are comparatively weak and th ...
– short-range, attractive forces that exist between neutral atoms and molecules. One of Casimir's colleagues, Theo Overbeek, realized that the theory that was used at the time to explain Van der Waals forces, which had been developed by
Fritz London Fritz Wolfgang London (March 7, 1900 – March 30, 1954) was a German physicist and professor at Duke University. His fundamental contributions to the theories of chemical bonding and of intermolecular forces ( London dispersion forces) are today ...
in 1930, did not properly explain the experimental measurements on colloids. Overbeek therefore asked Casimir to investigate the problem. Working with
Dirk Polder Dirk Polder (23 August 1919 – 18 March 2001) was a Dutch physicist who, together with Hendrik Casimir, first predicted the existence of what today is known as the Casimir-Polder force, sometimes also referred to as the ''Casimir effect ...
, Casimir discovered that the interaction between two neutral molecules could be correctly described only if the fact that light travels at a finite speed was taken into account. Soon afterwards after a conversation with
Bohr Niels Henrik David Bohr (; 7 October 1885 – 18 November 1962) was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922. B ...
about zero-point energy, Casimir noticed that this result could be interpreted in terms of vacuum fluctuations. He then asked himself what would happen if there were two mirrors – rather than two molecules – facing each other in a vacuum. It was this work that led to his famous prediction of an attractive force between reflecting plates. The work by Casimir and Polder opened up the way to a unified theory of Van der Waals and Casimir forces and a smooth continuum between the two phenomena. This was done by Lifshitz (1956) in the case of plane parallel dielectric plates. The generic name for both Van der Waals and Casimir forces is dispersion forces, because both of them are caused by dispersions of the operator of the dipole moment. The role of relativistic forces becomes dominant at orders of a hundred nanometers. In 1951 Herbert Callen and Theodore Welton proved the quantum
fluctuation-dissipation theorem The fluctuation–dissipation theorem (FDT) or fluctuation–dissipation relation (FDR) is a powerful tool in statistical physics for predicting the behavior of systems that obey detailed balance. Given that a system obeys detailed balance, the th ...
(FDT) which was originally formulated in classical form by Nyquist (1928) as an explanation for observed
Johnson noise Johnson is a surname of Anglo-Norman origin meaning "Son of John". It is the second most common in the United States and 154th most common in the world. As a common family name in Scotland, Johnson is occasionally a variation of ''Johnston'', a ...
in electric circuits. The fluctuation-dissipation theorem showed that when something dissipates energy, in an effectively irreversible way, a connected heat bath must also fluctuate. The fluctuations and the dissipation go hand in hand; it is impossible to have one without the other. The implication of FDT being that the vacuum could be treated as a heat bath coupled to a dissipative force and as such energy could, in part, be extracted from the vacuum for potentially useful work. FDT has been shown to be true experimentally under certain quantum, non-classical, conditions. In 1963 the
Jaynes–Cummings model The Jaynes–Cummings model (sometimes abbreviated JCM) is a theoretical model in quantum optics. It describes the system of a two-level atom interacting with a quantized mode of an optical cavity (or a bosonic field), with or without the prese ...
was developed describing the system of a two-level atom interacting with a quantized field mode (i.e. the vacuum) within an optical cavity. It gave nonintuitive predictions such as that an atom's spontaneous emission could be driven by field of effectively constant frequency (
Rabi frequency The Rabi frequency is the frequency at which the probability amplitudes of two atomic energy levels fluctuate in an oscillating electromagnetic field. It is proportional to the Transition Dipole Moment of the two levels and to the amplitude (''not ...
). In the 1970s experiments were being performed to test aspects of quantum optics and showed that the rate of spontaneous emission of an atom could be controlled using reflecting surfaces. These results were at first regarded with suspicion in some quarters: it was argued that no modification of a spontaneous emission rate would be possible, after all, how can the emission of a photon be affected by an atom's environment when the atom can only "see" its environment by emitting a photon in the first place? These experiments gave rise to
cavity quantum electrodynamics Cavity quantum electrodynamics (cavity QED) is the study of the interaction between light confined in a reflective cavity and atoms or other particles, under conditions where the quantum nature of photons is significant. It could in principle be u ...
(CQED), the study of effects of mirrors and cavities on radiative corrections. Spontaneous emission can be suppressed (or "inhibited") or amplified. Amplification was first predicted by Purcell in 1946 (the
Purcell effect Henry Purcell (, rare: September 1659 – 21 November 1695) was an English composer. Purcell's style of Baroque music was uniquely English, although it incorporated Italian and French elements. Generally considered among the greatest Eng ...
) and has been experimentally verified. This phenomenon can be understood, partly, in terms of the action of the vacuum field on the atom.


The uncertainty principle

Zero-point energy is fundamentally related to the Heisenberg
uncertainty principle In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
. Roughly speaking, the uncertainty principle states that complementary variables (such as a particle's position and momentum, or a field's value and derivative at a point in space) cannot simultaneously be specified precisely by any given quantum state. In particular, there cannot exist a state in which the system simply sits motionless at the bottom of its potential well, for then its position and momentum would both be completely determined to arbitrarily great precision. Therefore, the lowest-energy state (the ground state) of the system must have a distribution in position and momentum that satisfies the uncertainty principle, which implies its energy must be greater than the minimum of the potential well. Near the bottom of a
potential well A potential well is the region surrounding a local minimum of potential energy. Energy captured in a potential well is unable to convert to another type of energy ( kinetic energy in the case of a gravitational potential well) because it is ca ...
, the Hamiltonian of a general system (the quantum-mechanical operator (physics), operator giving its energy) can be approximated as a quantum harmonic oscillator, :\hat = V_0 + \tfrac k \left(\hat - x_0\right)^2 + \frac \hat^2 \,, where is the minimum of the classical potential well. The uncertainty principle tells us that :\sqrt \sqrt \geq \frac \,, making the expectation value (quantum mechanics), expectation values of the kinetic energy, kinetic and potential energy, potential terms above satisfy :\left\langle \tfrac k \left(\hat - x_0\right)^2 \right\rangle \left\langle \frac \hat^2 \right\rangle \geq \left(\frac\right)^2 \frac \,. The expectation value of the energy must therefore be at least :\left\langle \hat \right\rangle \geq V_0 + \frac \sqrt = V_0 + \frac where is the angular frequency at which the system oscillates. A more thorough treatment, showing that the energy of the ground state actually saturates this bound and is exactly , requires solving for the ground state of the system.


Atomic physics

The idea of a quantum harmonic oscillator and its associated energy can apply to either an atom or a subatomic particle. In ordinary atomic physics, the zero-point energy is the energy associated with the ground state of the system. The professional physics literature tends to measure frequency, as denoted by above, using angular frequency, denoted with and defined by . This leads to a convention of writing Planck's constant with a bar through its top () to denote the quantity . In these terms, the most famous such example of zero-point energy is the above associated with the ground state of the quantum harmonic oscillator. In quantum mechanical terms, the zero-point energy is the expectation value of the Hamiltonian of the system in the ground state. If more than one ground state exists, they are said to be degenerate energy level, degenerate. Many systems have degenerate ground states. Degeneracy occurs whenever there exists a unitary operator which acts non-trivially on a ground state and commutator, commutes with the Hamiltonian of the system. According to the third law of thermodynamics, a system at
absolute zero Absolute zero is the lowest limit of the thermodynamic temperature scale, a state at which the enthalpy and entropy of a cooled ideal gas reach their minimum value, taken as zero kelvin. The fundamental particles of nature have minimum vibra ...
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
exists in its ground state; thus, its entropy is determined by the degeneracy of the ground state. Many systems, such as a perfect
crystal lattice In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by : \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n ...
, have a unique ground state and therefore have zero entropy at absolute zero. It is also possible for the highest excited state to have
absolute zero Absolute zero is the lowest limit of the thermodynamic temperature scale, a state at which the enthalpy and entropy of a cooled ideal gas reach their minimum value, taken as zero kelvin. The fundamental particles of nature have minimum vibra ...
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
for systems that exhibit negative temperature. The
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
of the ground state of a particle in a box, particle in a one-dimensional well is a half-period sine wave which goes to zero at the two edges of the well. The energy of the particle is given by: :\frac where is the Planck constant, is the mass of the particle, is the energy state ( corresponds to the ground-state energy), and is the width of the well.


Quantum field theory

In
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
(QFT), the fabric of "empty" space is visualized as consisting of field (physics), fields, with the field at every point in space and time being a quantum harmonic oscillator, with neighboring oscillators interacting with each other. According to QFT the universe is made up of matter fields whose quanta are
fermions In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
(e.g. electrons and
quark A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly ...
s), force fields whose quanta are
boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer spi ...
s (i.e.
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
s and
gluon A gluon ( ) is an elementary particle that acts as the exchange particle (or gauge boson) for the strong force between quarks. It is analogous to the exchange of photons in the electromagnetic force between two charged particles. Gluons bind ...
s) and a Higgs field whose quantum is the Higgs boson. The matter and force fields have zero-point energy. A related term is ''zero-point field'' (ZPF), which is the lowest energy state of a particular field. The vacuum can be viewed not as empty space, but as the combination of all zero-point fields. In QFT the zero-point energy of the
vacuum state In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. The word zero-point field is sometimes used as ...
is called the
vacuum energy Vacuum energy is an underlying background energy that exists in space throughout the entire Universe. The vacuum energy is a special case of zero-point energy that relates to the quantum vacuum. The effects of vacuum energy can be experimental ...
and the average expectation value of the Hamiltonian is called the
vacuum expectation value In quantum field theory the vacuum expectation value (also called condensate or simply VEV) of an operator is its average or expectation value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle ...
(also called condensate or simply VEV). The QED vacuum is a part of the vacuum state which specifically deals with
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
(e.g. electromagnetic interactions between photons, electrons and the vacuum) and the
QCD vacuum In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type o ...
deals with
quantum chromodynamics In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type ...
(e.g.
color charge Color charge is a property of quarks and gluons that is related to the particles' strong interactions in the theory of quantum chromodynamics (QCD). The "color charge" of quarks and gluons is completely unrelated to the everyday meanings of colo ...
interactions between quarks, gluons and the vacuum). Recent experiments advocate the idea that particles themselves can be thought of as excited states of the underlying
quantum vacuum In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. The word zero-point field is sometimes used as ...
, and that all properties of matter are merely vacuum fluctuations arising from interactions with the zero-point field. Each point in space makes a contribution of , resulting in a calculation of infinite zero-point energy in any finite volume; this is one reason renormalization is needed to make sense of quantum field theories. In physical cosmology, cosmology, the vacuum energy is one possible explanation for the
cosmological constant In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant, is the constant coefficient of a term that Albert Einstein temporarily added to his field eq ...
and the source of
dark energy In physical cosmology and astronomy, dark energy is an unknown form of energy that affects the universe on the largest scales. The first observational evidence for its existence came from measurements of supernovas, which showed that the univ ...
. Scientists are not in agreement about how much energy is contained in the vacuum. Quantum mechanics requires the energy to be large as
Paul Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
claimed it is, like a Dirac sea, sea of energy. Other scientists specializing in General Relativity require the energy to be small enough for curvature of space to agree with observed astronomy. The Werner Heisenberg, Heisenberg
uncertainty principle In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
allows the energy to be as large as needed to promote quantum actions for a brief moment of time, even if the average energy is small enough to satisfy relativity and flat space. To cope with disagreements, the vacuum energy is described as a virtual energy potential of positive and negative energy. In quantum perturbation theory, it is sometimes said that the contribution of one-loop and multi-loop Feynman diagrams to elementary particle propagators are the contribution of quantum fluctuation, vacuum fluctuations, or the zero-point energy to the particle masses.


The quantum electrodynamic vacuum

The oldest and best known quantized force field is the
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical ...
. Maxwell's equations have been superseded by
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
(QED). By considering the zero-point energy that arises from QED it is possible to gain a characteristic understanding of zero-point energy that arises not just through electromagnetic interactions but in all quantum field theories.


Redefining the zero of energy

In the quantum theory of the electromagnetic field, classical wave amplitudes and are replaced by operators and that satisfy: :\left[a,a^\dagger\right] = 1 The classical quantity appearing in the classical expression for the energy of a field mode is replaced in quantum theory by the photon number operator . The fact that: :\left[a,a^\dagger a\right] \ne 1 implies that quantum theory does not allow states of the radiation field for which the photon number and a field amplitude can be precisely defined, i.e., we cannot have simultaneous eigenstates for and . The reconciliation of wave and particle attributes of the field is accomplished via the association of a probability amplitude with a classical mode pattern. The calculation of field modes is entirely classical problem, while the quantum properties of the field are carried by the mode "amplitudes" and associated with these classical modes. The zero-point energy of the field arises formally from the non-commutativity of and . This is true for any harmonic oscillator: the zero-point energy appears when we write the Hamiltonian: :\begin H_ &= \frac + \tfrac m \omega^2 ^2 \\ &= \tfrac \hbar \omega \left(a a^\dagger + a^\dagger a\right) \\ &=\hbar \omega \left(a^\dagger a +\tfrac\right) \end It is often argued that the entire universe is completely bathed in the zero-point electromagnetic field, and as such it can add only some constant amount to expectation values. Physical measurements will therefore reveal only deviations from the vacuum state. Thus the zero-point energy can be dropped from the Hamiltonian by redefining the zero of energy, or by arguing that it is a constant and therefore has no effect on Heisenberg equations of motion. Thus we can choose to declare by fiat that the ground state has zero energy and a field Hamiltonian, for example, can be replaced by: :\begin H_F - \left\langle 0, H_F, 0\right\rangle &=\tfrac \hbar \omega \left(a a^\dagger + a^\dagger a\right)-\tfrac\hbar \omega \\ &= \hbar \omega \left(a^\dagger a + \tfrac \right)-\tfrac\hbar \omega \\ &= \hbar \omega a^\dagger a \end without affecting any physical predictions of the theory. The new Hamiltonian is said to be Normal order, normally ordered (or Wick ordered) and is denoted by a double-dot symbol. The normally ordered Hamiltonian is denoted , i.e.: ::H_F : \equiv \hbar \omega \left(a a^\dagger + a^\dagger a\right) : \equiv \hbar \omega a^\dagger a In other words, within the normal ordering symbol we can commute and . Since zero-point energy is intimately connected to the non-commutativity of and , the normal ordering procedure eliminates any contribution from the zero-point field. This is especially reasonable in the case of the field Hamiltonian, since the zero-point term merely adds a constant energy which can be eliminated by a simple redefinition for the zero of energy. Moreover, this constant energy in the Hamiltonian obviously commutes with and and so cannot have any effect on the quantum dynamics described by the Heisenberg equations of motion. However, things are not quite that simple. The zero-point energy cannot be eliminated by dropping its energy from the Hamiltonian: When we do this and solve the Heisenberg equation for a field operator, we must include the vacuum field, which is the homogeneous part of the solution for the field operator. In fact we can show that the vacuum field is essential for the preservation of the commutators and the formal consistency of Quantum electrodynamics, QED. When we calculate the field energy we obtain not only a contribution from particles and forces that may be present but also a contribution from the vacuum field itself i.e. the zero-point field energy. In other words, the zero-point energy reappears even though we may have deleted it from the Hamiltonian.


The electromagnetic field in free space

From Maxwell's equations, the electromagnetic energy of a "free" field i.e. one with no sources, is described by: :\begin H_F &= \frac\int d^3r \left(\mathbf^2 +\mathbf^2\right) \\ &=\frac, \alpha (t), ^2 \end We introduce the "mode function" that satisfies the Helmholtz equation: : \left( \nabla^2 + k^2 \right) \mathbf_0(\mathbf) = 0 where and assume it is normalized such that: :\int d^3r \left, \mathbf_0(\mathbf)\^2 = 1 We wish to "quantize" the electromagnetic energy of free space for a multimode field. The field intensity of free space should be independent of position such that should be independent of for each mode of the field. The mode function satisfying these conditions is: : \mathbf_0(\mathbf) = e_e^ where in order to have the transversality condition satisfied for the Coulomb gauge in which we are working. To achieve the desired normalization we pretend space is divided into cubes of volume and impose on the field the periodic boundary condition: :\mathbf(x+L,y+L,z+L,t)=\mathbf(x,y,z,t) or equivalently : \left(k_x,k_y,k_z\right)=\frac\left(n_x,n_y,n_z\right) where can assume any integer value. This allows us to consider the field in any one of the imaginary cubes and to define the mode function: :\mathbf_\mathbf(\mathbf)= \frac\sqrt e_e^ which satisfies the Helmholtz equation, transversality, and the "box normalization": :\int_V d^3r \left, \mathbf_\mathbf(\mathbf)\^2 = 1 where is chosen to be a unit vector which specifies the polarization of the field mode. The condition means that there are two independent choices of , which we call and where and . Thus we define the mode functions: :\mathbf_(\mathbf)=\frac\sqrte_e^ \, , \quad \lambda = \begin 1\\2 \end in terms of which the vector potential becomes: :\mathbf_(\mathbf,t)=\sqrt\left[a_(0)e^+a_^\dagger(0)e^\right]e_ or: :\mathbf_(\mathbf,t)=\sqrt\left[a_(0)e^+a_^\dagger(0)e^\right] where and , are photon annihilation and creation operators for the mode with wave vector and polarization . This gives the vector potential for a plane wave mode of the field. The condition for shows that there are infinitely many such modes. The linearity of Maxwell's equations allows us to write: :\mathbf(\mathbft)=\sum_\sqrt\left[a_(0)e^+a_^\dagger(0)e^\right]e_ for the total vector potential in free space. Using the fact that: :\int_V d^3r \mathbf_(\mathbf)\cdot \mathbf_^\ast(\mathbf)=\delta_^3\delta_ we find the field Hamiltonian is: :H_F=\sum_\left(\hbar\omega_k\left(a_^\dagger a_\right) + \tfrac\right ) This is the Hamiltonian for an infinite number of uncoupled harmonic oscillators. Thus different modes of the field are independent and satisfy the commutation relations: :\begin \left[a_(t),a_^\dagger(t)\right]&=\delta_^3\delta_ \\[10px] \left[a_(t),a_(t)\right]&=\left[a_^\dagger(t),a_^\dagger(t)\right]=0 \end Clearly the least eigenvalue for is: :\sum_\tfrac\hbar\omega_k This state describes the zero-point energy of the vacuum. It appears that this sum is divergent – in fact highly divergent, as putting in the density factor :\fracV shows. The summation becomes approximately the integral: :\frac\int v^3 \, dv for high values of . It diverges proportional to for large . There are two separate questions to consider. First, is the divergence a real one such that the zero-point energy really is infinite? If we consider the volume is contained by perfectly conducting walls, very high frequencies can only be contained by taking more and more perfect conduction. No actual method of containing the high frequencies is possible. Such modes will not be stationary in our box and thus not countable in the stationary energy content. So from this physical point of view the above sum should only extend to those frequencies which are countable; a cut-off energy is thus eminently reasonable. However, on the scale of a "universe" questions of general relativity must be included. Suppose even the boxes could be reproduced, fit together and closed nicely by curving spacetime. Then exact conditions for running waves may be possible. However the very high frequency quanta will still not be contained. As per John Wheeler's "geons" these will leak out of the system. So again a cut-off is permissible, almost necessary. The question here becomes one of consistency since the very high energy quanta will act as a mass source and start curving the geometry. This leads to the second question. Divergent or not, finite or infinite, is the zero-point energy of any physical significance? The ignoring of the whole zero-point energy is often encouraged for all practical calculations. The reason for this is that energies are not typically defined by an arbitrary data point, but rather changes in data points, so adding or subtracting a constant (even if infinite) should be allowed. However this is not the whole story, in reality energy is not so arbitrarily defined: in general relativity the seat of the curvature of spacetime is the energy content and there the absolute amount of energy has real physical meaning. There is no such thing as an arbitrary additive constant with density of field energy. Energy density curves space, and an increase in energy density produces an increase of curvature. Furthermore, the zero-point energy density has other physical consequences e.g. the Casimir effect, contribution to the Lamb shift, or anomalous magnetic moment of the electron, it is clear it is not just a mathematical constant or artifact that can be cancelled out.


Necessity of the vacuum field in QED

The vacuum state of the "free" electromagnetic field (that with no sources) is defined as the ground state in which for all modes . The vacuum state, like all stationary states of the field, is an eigenstate of the Hamiltonian but not the electric and magnetic field operators. In the vacuum state, therefore, the electric and magnetic fields do not have definite values. We can imagine them to be fluctuating about their mean value of zero. In a process in which a photon is annihilated (absorbed), we can think of the photon as making a transition into the vacuum state. Similarly, when a photon is created (emitted), it is occasionally useful to imagine that the photon has made a transition out of the vacuum state. An atom, for instance, can be considered to be "dressed" by emission and reabsorption of "virtual photons" from the vacuum. The vacuum state energy described by is infinite. We can make the replacement: :\sum_\longrightarrow\sum_\left (\frac \right )^3 \int d^3 k = \frac \sum_\lambda \int d^3 k the zero-point energy density is: :\begin \frac\sum_\tfrac\hbar\omega_k &=\frac\int d^3 k \tfrac\hbar\omega_k \\ &= \frac \int dk\,k^2 \left(\tfrac\hbar\omega_k\right) \\ &=\frac \int d\omega\,\omega^3 \end or in other words the spectral energy density of the vacuum field: :\rho_0(\omega)=\frac The zero-point energy density in the frequency range from to is therefore: :\int_^ d\omega\rho_0(\omega) = \frac\left(\omega_2^4-\omega_1^4\right) This can be large even in relatively narrow "low frequency" regions of the spectrum. In the optical region from 400 to 700 nm, for instance, the above equation yields around 220 erg/cm3. We showed in the above section that the zero-point energy can be eliminated from the Hamiltonian by the normal ordering prescription. However, this elimination does not mean that the vacuum field has been rendered unimportant or without physical consequences. To illustrate this point we consider a linear dipole oscillator in the vacuum. The Hamiltonian for the oscillator plus the field with which it interacts is: :H=\frac\left(\mathbf-\frac\mathbf\right)^2 + \tfracm\omega_0^2\mathbf^2 + H_F This has the same form as the corresponding classical Hamiltonian and the Heisenberg equations of motion for the oscillator and the field are formally the same as their classical counterparts. For instance the Heisenberg equations for the coordinate and the canonical momentum of the oscillator are: :\begin \mathbf&=(i\hbar)^[\mathbf.H] = \frac\left(\mathbf-\frac\mathbf\right) \\ \mathbf&=(i\hbar)^[\mathbf.H] \begin&=\tfrac\nabla\left(\mathbf-\frac\mathbf\right)^2-m\omega_0^2\mathbf \\ &=-\frac \left[\left(\mathbf-\frac\mathbf\right) \cdot \nabla\right] \left[-\frac\mathbf\right] - \frac \left(\mathbf-\frac\mathbf\right) \times \nabla \times \left[-\frac\mathbf\right] -m\omega_0^2 \mathbf \\ &= \frac(\mathbf\cdot\nabla)\mathbf + \frac\mathbf \times \mathbf -m\omega_0^2 \mathbf \end\end or: :\begin m \mathbf &= \mathbf - \frac \mathbf \\ &= -\frac \left[\mathbf - \left(\mathbf \cdot \nabla\right) \mathbf\right] + \frac \mathbf \times \mathbf - m\omega_0^2\mathbf \\ &= e\mathbf + \frac \mathbf \times \mathbf - m\omega_0^2\mathbf \end since the rate of change of the vector potential in the frame of the moving charge is given by the convective derivative :\mathbf=\frac + (\mathbf \cdot \nabla) \mathbf^3 \,. For nonrelativistic motion we may neglect the magnetic force and replace the expression for by: :\begin \mathbf+\omega_0^2\mathbf &\approx \frac\mathbf \\ &\approx \sum_ \sqrt \left[a_(t) + a_^\dagger(t)\right] e_ \end Above we have made the electric dipole approximation in which the spatial dependence of the field is neglected. The Heisenberg equation for is found similarly from the Hamiltonian to be: :\dot_ = i \omega_k a_ + ie \sqrt\frac \mathbf \cdot e_ In the electric dipole approximation. In deriving these equations for , , and we have used the fact that equal-time particle and field operators commute. This follows from the assumption that particle and field operators commute at some time (say, ) when the matter-field interpretation is presumed to begin, together with the fact that a Heisenberg-picture operator evolves in time as , where is the time evolution operator satisfying :i\hbar\dot = HU \,,\quad U^\dagger(t) = U^(t) \,,\quad U(0) = 1 \,. Alternatively, we can argue that these operators must commute if we are to obtain the correct equations of motion from the Hamiltonian, just as the corresponding Poisson brackets in classical theory must vanish in order to generate the correct Hamilton equations. The formal solution of the field equation is: :a_(t)=a_(0)e^+ie \sqrt \int^t_0dt'\,e_\cdot\mathbf(t')e^ and therefore the equation for may be written: :\mathbf+\omega^2_0\mathbf=\frac\mathbf_0(t)+\frac\mathbf_(t) where: :\mathbf_0(t)=i\sum_ \sqrt\left[a_(0)e^-a^\dagger_(0)e^\right]e_ and: :\mathbf_(t)=-\frac \sum_ \int^t_0dt'\left[e_\cdot\mathbf\left(t'\right)\right]\cos\omega_k\left(t'-t\right) It can be shown that in the Abraham–Lorentz force, radiation reaction field, if the mass is regarded as the "observed" mass then we can take: :\mathbf_(t)=\frac\mathbf The total field acting on the dipole has two parts, and . is the free or zero-point field acting on the dipole. It is the homogeneous solution of the Maxwell equation for the field acting on the dipole, i.e., the solution, at the position of the dipole, of the wave equation :\left[\nabla^2-\frac\frac\right]\mathbf=0 satisfied by the field in the (source free) vacuum. For this reason is often referred to as the "vacuum field", although it is of course a Heisenberg-picture operator acting on whatever state of the field happens to be appropriate at . is the source field, the field generated by the dipole and acting on the dipole. Using the above equation for we obtain an equation for the Heisenberg-picture operator \mathbf(t) that is formally the same as the classical equation for a linear dipole oscillator: : \mathbf + \omega^2_0\mathbf-\tau \mathbf=\frac\mathbf_0(t) where . in this instance we have considered a dipole in the vacuum, without any "external" field acting on it. the role of the external field in the above equation is played by the vacuum electric field acting on the dipole. Classically, a dipole in the vacuum is not acted upon by any "external" field: if there are no sources other than the dipole itself, then the only field acting on the dipole is its own radiation reaction field. In quantum theory however there is always an "external" field, namely the source-free or vacuum field . According to our earlier equation for the free field is the only field in existence at as the time at which the interaction between the dipole and the field is "switched on". The state vector of the dipole-field system at is therefore of the form :, \Psi\rangle=, \text\rangle, \psi_D\rangle \,, where is the vacuum state of the field and is the initial state of the dipole oscillator. The expectation value of the free field is therefore at all times equal to zero: :\langle\mathbf_0(t)\rangle=\langle\Psi, \mathbf_0(t), \Psi\rangle=0 since . however, the energy density associated with the free field is infinite: :\begin \frac \left\langle \mathbf^2_0(t) \right\rangle &= \frac \sum_ \sum_ \sqrt \sqrt \times \left\langle a_(0)a^\dagger_(0)\right\rangle \\ &= \frac\sum_\left (\frac \right )\\ &= \int^\infin_0dw\,\rho_0(\omega) \end The important point of this is that the zero-point field energy does not affect the Heisenberg equation for since it is a c-number or constant (i.e. an ordinary number rather than an operator) and commutes with . We can therefore drop the zero-point field energy from the Hamiltonian, as is usually done. But the zero-point field re-emerges as the homogeneous solution for the field equation. A charged particle in the vacuum will therefore always see a zero-point field of infinite density. This is the origin of one of the infinities of quantum electrodynamics, and it cannot be eliminated by the trivial expedient dropping of the term in the field Hamiltonian. The free field is in fact necessary for the formal consistency of the theory. In particular, it is necessary for the preservation of the commutation relations, which is required by the unitary of time evolution in quantum theory: :\begin \left[z(t),p_z(t)\right]&=\left[U^\dagger(t)z(0)U(t),U^\dagger(t)p_z(0)U(t)\right]\\ &=U^\dagger(t)\left[z(0),p_z(0)\right]U(t)\\ &=i\hbar U^\dagger(t)U(t)\\ &=i\hbar \end We can calculate from the formal solution of the operator equation of motion :\mathbf + \omega^2_0\mathbf-\tau \mathbf=\frac\mathbf_0(t) Using the fact that :\left[a_(0),a^\dagger_(0)\right]=\delta^3_\mathbf,\delta_ and that equal-time particle and field operators commute, we obtain: :\begin [z(t),p_z(t)]&=\left[z(t),m\dot(t)\right]+\left[z(t),\fracA_z(t)\right] \\ &=\left[z(t),m\dot(t)\right] \\ &= \left (\frac \right ) \left (\frac \right ) \int^\infin_0\frac \end For the dipole oscillator under consideration it can be assumed that the radiative damping rate is small compared with the natural oscillation frequency, i.e., . Then the integrand above is sharply peaked at and: :\begin \left[z(t),p_z(t)\right]&\approx \frac\omega^3_0 \int^\infin_ \frac \\ &= \left (\frac \right )\left (\frac \right ) \\ &=i\hbar \end the necessity of the vacuum field can also be appreciated by making the small damping approximation in :\begin &\mathbf + \omega^2_0\mathbf-\tau \mathbf=\frac\mathbf_0(t) \\ &\mathbf\approx-\omega^2_0\mathbf(t) && \mathbf\approx-\omega^2_0\mathbf \end and :\mathbf+\tau\omega^2_0\mathbf+\omega^2_0\mathbf\approx\frac\mathbf_0(t) Without the free field in this equation the operator would be exponentially dampened, and commutators like would approach zero for . With the vacuum field included, however, the commutator is at all times, as required by unitarity, and as we have just shown. A similar result is easily worked out for the case of a free particle instead of a dipole oscillator. What we have here is an example of a "fluctuation-dissipation elation". Generally speaking if a system is coupled to a bath that can take energy from the system in an effectively irreversible way, then the bath must also cause fluctuations. The fluctuations and the dissipation go hand in hand we cannot have one without the other. In the current example the coupling of a dipole oscillator to the electromagnetic field has a dissipative component, in the form of the zero-point (vacuum) field; given the existence of radiation reaction, the vacuum field must also exist in order to preserve the canonical commutation rule and all it entails. The spectral density of the vacuum field is fixed by the form of the radiation reaction field, or vice versa: because the radiation reaction field varies with the third derivative of , the spectral energy density of the vacuum field must be proportional to the third power of in order for to hold. In the case of a dissipative force proportional to , by contrast, the fluctuation force must be proportional to \omega in order to maintain the canonical commutation relation. This relation between the form of the dissipation and the spectral density of the fluctuation is the essence of the fluctuation-dissipation theorem. The fact that the canonical commutation relation for a harmonic oscillator coupled to the vacuum field is preserved implies that the zero-point energy of the oscillator is preserved. it is easy to show that after a few damping times the zero-point motion of the oscillator is in fact sustained by the driving zero-point field.


The quantum chromodynamic vacuum

The QCD vacuum is the
vacuum state In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. The word zero-point field is sometimes used as ...
of
quantum chromodynamics In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type ...
(QCD). It is an example of a ''non-perturbative'' vacuum state, characterized by a non-vanishing condensate (quantum field theory), condensates such as the gluon condensate and the quark condensate in the complete theory which includes quarks. The presence of these condensates characterizes the confined phase of quark matter. In technical terms, gluons are Vector boson, vector gauge bosons that mediate strong interactions of
quark A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly ...
s in
quantum chromodynamics In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type ...
(QCD). Gluons themselves carry the
color charge Color charge is a property of quarks and gluons that is related to the particles' strong interactions in the theory of quantum chromodynamics (QCD). The "color charge" of quarks and gluons is completely unrelated to the everyday meanings of colo ...
of the strong interaction. This is unlike the
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
, which mediates the Electromagnetic force, electromagnetic interaction but lacks an electric charge. Gluons therefore participate in the strong interaction in addition to mediating it, making QCD significantly harder to analyze than QED (
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
) as it deals with nonlinear equations to characterize such interactions.


The Higgs field

The Standard Model hypothesises a field called the Higgs field (symbol: ), which has the unusual property of a non-zero amplitude in its ground state (zero-point) energy after renormalization; i.e., a non-zero vacuum expectation value. It can have this effect because of its unusual "Mexican hat" shaped potential whose lowest "point" is not at its "centre". Below a certain extremely high energy level the existence of this non-zero vacuum expectation Spontaneous symmetry breaking, spontaneously breaks electroweak gauge symmetry which in turn gives rise to the Higgs mechanism and triggers the acquisition of mass by those particles interacting with the field. The Higgs mechanism occurs whenever a charged field has a vacuum expectation value. This effect occurs because scalar field components of the Higgs field are "absorbed" by the massive bosons as degrees of freedom, and couple to the fermions via Yukawa coupling, thereby producing the expected mass terms. The expectation value of in the ground state (the
vacuum expectation value In quantum field theory the vacuum expectation value (also called condensate or simply VEV) of an operator is its average or expectation value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle ...
or VEV) is then , where . The measured value of this parameter is approximately . It has units of mass, and is the only free parameter of the Standard Model that is not a dimensionless number. The Higgs mechanism is a type of superconductivity which occurs in the vacuum. It occurs when all of space is filled with a sea of particles which are charged and thus the field has a nonzero vacuum expectation value. Interaction with the vacuum energy filling the space prevents certain forces from propagating over long distances (as it does in a superconducting medium; e.g., in the Ginzburg–Landau theory).


Experimental observations

Zero-point energy has many observed physical consequences. It is important to note that zero-point energy is not merely an artifact of mathematical formalism that can, for instance, be dropped from a Hamiltonian by redefining the zero of energy, or by arguing that it is a constant and therefore has no effect on Heisenberg equations of motion without latter consequence. Indeed, such treatment could create a problem at a deeper, as of yet undiscovered, theory. For instance, in general relativity the zero of energy (i.e. the energy density of the vacuum) contributes to a cosmological constant of the type introduced by Einstein in order to obtain static solutions to his field equations. The zero-point energy density of the vacuum, due to all quantum fields, is extremely large, even when we cut off the largest allowable frequencies based on plausible physical arguments. It implies a cosmological constant larger than the limits imposed by observation by about 120 orders of magnitude. This "cosmological constant problem" remains one of the greatest unsolved mysteries of physics.


Casimir effect

A phenomenon that is commonly presented as evidence for the existence of zero-point energy in vacuum is the
Casimir effect In quantum field theory, the Casimir effect is a physical force acting on the macroscopic boundaries of a confined space which arises from the quantum fluctuations of the field. It is named after the Dutch physicist Hendrik Casimir, who pre ...
, proposed in 1948 by Netherlands, Dutch physicist
Hendrik Casimir Hendrik Brugt Gerhard Casimir (15 July 1909 – 4 May 2000) was a Dutch physicist best known for his research on the two-fluid model of superconductors (together with C. J. Gorter) in 1934 and the Casimir effect (together with D. Polder) in 1 ...
, who considered the quantized
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical ...
between a pair of grounded, neutral metal plates. The vacuum energy contains contributions from all wavelengths, except those excluded by the spacing between plates. As the plates draw together, more wavelengths are excluded and the vacuum energy decreases. The decrease in energy means there must be a force doing work on the plates as they move. Early experimental tests from the 1950s onwards gave positive results showing the force was real, but other external factors could not be ruled out as the primary cause, with the range of experimental error sometimes being nearly 100%. That changed in 1997 with Lamoreaux conclusively showing that the Casimir force was real. Results have been repeatedly replicated since then. In 2009, Munday et al. published experimental proof that (as predicted in 1961) the Casimir force could also be repulsive as well as being attractive. Repulsive Casimir forces could allow quantum levitation of objects in a fluid and lead to a new class of switchable nanoscale devices with ultra-low static friction. An interesting hypothetical side effect of the Casimir effect is the Scharnhorst effect, a hypothetical phenomenon in which light signals travel slightly Faster-than-light, faster than between two closely spaced conducting plates.See and


Lamb shift

The quantum fluctuations of the electromagnetic field have important physical consequences. In addition to the Casimir effect, they also lead to a splitting between the two energy levels and (in term symbol notation) of the
hydrogen atom A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. Atomic hydrogen cons ...
which was not predicted by the Dirac equation, according to which these states should have the same energy. Charged particles can interact with the fluctuations of the quantized vacuum field, leading to slight shifts in energy, this effect is called the Lamb shift. The shift of about is roughly of the difference between the energies of the 1s and 2s levels, and amounts to 1,058 MHz in frequency units. A small part of this shift (27 MHz ≈ 3%) arises not from fluctuations of the electromagnetic field, but from fluctuations of the electron–positron field. The creation of (virtual) electron–positron pairs has the effect of screening the Coulomb field and acts as a vacuum dielectric constant. This effect is much more important in muonic atoms.


Fine-structure constant

Taking ( Planck's constant divided by ), (the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
), and (the electromagnetic coupling constant i.e. a measure of the strength of the electromagnetic force (where is the absolute value of the Electron charge, electronic charge and \varepsilon_0 is the vacuum permittivity)) we can form a dimensionless quantity called the fine-structure constant: :\alpha = \frac = \frac \approx \frac The fine-structure constant is the coupling constant of
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
(QED) determining the strength of the interaction between electrons and photons. It turns out that the fine-structure constant is not really a constant at all owing to the zero-point energy fluctuations of the electron-positron field. The quantum fluctuations caused by zero-point energy have the effect of screening electric charges: owing to (virtual) electron-positron pair production, the charge of the particle measured far from the particle is far smaller than the charge measured when close to it. The Heisenberg inequality where , and , are the standard deviations of position and momentum states that: :\Delta_x\Delta_p\ge\frac\hbar It means that a short distance implies large momentum and therefore high energy i.e. particles of high energy must be used to explore short distances. QED concludes that the fine-structure constant is an increasing function of energy. It has been shown that at energies of the order of the Z boson, Z0 boson rest energy, 90 GeV, that: :\alpha\approx\frac rather than the low-energy . The renormalization procedure of eliminating zero-point energy infinities allows the choice of an arbitrary energy (or distance) scale for defining . All in all, depends on the energy scale characteristic of the process under study, and also on details of the renormalization procedure. The energy dependence of has been observed for several years now in precision experiment in high-energy physics.


Vacuum birefringence

In the presence of strong electrostatic fields it is predicted that virtual particles become separated from the vacuum state and form real matter. The fact that electromagnetic radiation can be transformed into matter and vice versa leads to fundamentally new features in
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
. One of the most important consequences is that, even in the vacuum, the Maxwell equations have to be exchanged by more complicated formulas. In general, it will be not possible to separate processes in the vacuum from the processes involving matter since electromagnetic fields can create matter if the field fluctuations are strong enough. This leads to highly complex nonlinear interaction - gravity will have an effect on the light at the same time the light has an effect on gravity. These effects were first predicted by
Werner Heisenberg Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a Über quantentheoretische Umdeutung kinematis ...
and Hans Heinrich Euler in 1936 and independently the same year by
Victor Weisskopf Victor Frederick "Viki" Weisskopf (also spelled Viktor; September 19, 1908 – April 22, 2002) was an Austrian-born American theoretical physicist. He did postdoctoral work with Werner Heisenberg, Erwin Schrödinger, Wolfgang Pauli, and Niels Boh ...
who stated: "The physical properties of the vacuum originate in the "zero-point energy" of matter, which also depends on absent particles through the external field strengths and therefore contributes an additional term to the purely Maxwellian field energy". Thus strong magnetic fields vary the energy contained in the vacuum. The scale above which the electromagnetic field is expected to become nonlinear is known as the Schwinger limit. At this point the vacuum has all the properties of a Birefringence, birefringent medium, thus in principle a rotation of the polarization frame (the Faraday effect) can be observed in empty space. Both Einstein's theory of special and general relativity state that light should pass freely through a vacuum without being altered, a principle known as Lorentz invariance. Yet, in theory, large nonlinear self-interaction of light due to quantum fluctuations should lead to this principle being measurably violated if the interactions are strong enough. Nearly all theories of quantum gravity predict that that Lorentz invariance is not an exact symmetry of nature. It is predicted the speed at which light travels through the vacuum depends on its direction, polarization and the local strength of the magnetic field. There have been a number of inconclusive results which claim to show evidence of a Modern searches for Lorentz violation, Lorentz violation by finding a rotation of the polarization plane of light coming from distant galaxies. The first concrete evidence for vacuum birefringence was published in 2017 when a team of astronomers looked at the light coming from the star RX J1856.5-3754, the closest discovered neutron star to Earth. Roberto Mignani at the National Institute for Astrophysics in Milan who led the team of astronomers has commented that "When Einstein came up with the theory of general relativity 100 years ago, he had no idea that it would be used for navigational systems. The consequences of this discovery probably will also have to be realised on a longer timescale." The team found that visible light from the star had undergone linear polarisation of around 16%. If the birefringence had been caused by light passing through interstellar gas or plasma, the effect should have been no more than 1%. Definitive proof would require repeating the observation at other wavelengths and on other neutron stars. At X-ray wavelengths the polarization from the quantum fluctuations should be near 100%. Although no telescope currently exists that can make such measurements, there are several proposed X-ray telescopes that may soon be able to verify the result conclusively such as China's Hard X-ray Modulation Telescope (HXMT) and NASA's Imaging X-ray Polarimetry Explorer (IXPE).


Speculated involvement in other phenomena


Dark energy

In the late 1990s it was discovered that very distant supernova were dimmer than expected suggesting that the universe's expansion was accelerating rather than slowing down. This revived discussion that Einstein's
cosmological constant In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant, is the constant coefficient of a term that Albert Einstein temporarily added to his field eq ...
, long disregarded by physicists as being equal to zero, was in fact some small positive value. This would indicate empty space exerted some form of Negative energy, negative pressure or energy. There is no natural candidate for what might cause what has been called
dark energy In physical cosmology and astronomy, dark energy is an unknown form of energy that affects the universe on the largest scales. The first observational evidence for its existence came from measurements of supernovas, which showed that the univ ...
but the current best guess is that it is the zero-point energy of the vacuum. One difficulty with this assumption is that the zero-point energy of the vacuum is absurdly large compared to the observed cosmological constant. In
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, mass and energy are equivalent; both produce a gravitational field and therefore the theorized vacuum energy of quantum field theory should have led to the universe ripping itself to pieces. This obviously has not happened and this issue, called the
cosmological constant problem In cosmology, the cosmological constant problem or vacuum catastrophe is the disagreement between the observed values of vacuum energy density (the small value of the cosmological constant) and theoretical large value of zero-point energy sugge ...
, is one of the greatest unsolved mysteries in physics. The European Space Agency is building the Euclid (spacecraft), Euclid telescope. Due to launch in 2023, it will map galaxies up to 10 billion light years away. By seeing how dark energy influences their arrangement and shape, the mission will allow scientists to see if the strength of dark energy has changed. If dark energy is found to vary throughout time it would indicate it is due to Quintessence (physics), quintessence, where observed acceleration is due to the energy of a scalar field, rather than the cosmological constant. No evidence of quintessence is yet available, but it has not been ruled out either. It generally predicts a slightly slower acceleration of the expansion of the universe than the cosmological constant. Some scientists think that the best evidence for quintessence would come from violations of Einstein's equivalence principle and equivalence principle#Tests of the Einstein equivalence principle, variation of the fundamental constants in space or time. Scalar fields are predicted by the ''Standard Model of particle physics'' and string theory, but an analogous problem to the cosmological constant problem (or the problem of constructing models of cosmological inflation) occurs: renormalization theory predicts that scalar fields should acquire large masses again due to zero-point energy.


Cosmic inflation

Cosmic inflation is a faster-than-light expansion of space just after the
Big Bang The Big Bang event is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models of the Big Bang explain the evolution of the observable universe from the ...
. It explains the origin of the Observable universe#Large-scale structure, large-scale structure of the cosmos. It is believed quantum vacuum fluctuations caused by zero-point energy arising in the microscopic inflationary period, later became magnified to a cosmic size, becoming the gravitational seeds for galaxies and structure in the Universe (see galaxy formation and evolution and structure formation). Many physicists also believe that inflation explains why the Universe appears to be the same in all directions (isotropic), why the cosmic microwave background radiation is distributed evenly, why the Universe is flat, and why no magnetic monopoles have been observed. The mechanism for inflation is unclear, it is similar in effect to dark energy but is a far more energetic and short lived process. As with dark energy the best explanation is some form of vacuum energy arising from quantum fluctuations. It may be that inflation caused baryogenesis, the hypothetical physical processes that produced an symmetry, asymmetry (imbalance) between baryons and antibaryons produced in the Big Bang, very early universe, but this is far from certain.


Alternative theories

There has been a long debate over the question of whether zero-point fluctuations of quantized vacuum fields are "real" i.e. do they have physical effects that cannot be interpreted by an equally valid alternative theory? Julian Schwinger, Schwinger, in particular, attempted to formulate QED without reference to zero-point fluctuations via his "source theory". From such an approach it is possible to derive the Casimir Effect without reference to a fluctuating field. Such a derivation was first given by Schwinger (1975) for a scalar field, and then generalized to the electromagnetic case by Schwinger, DeRaad, and Milton (1978). in which they state "the vacuum is regarded as truly a state with all physical properties equal to zero". More recently Robert Jaffe, Jaffe (2005) has highlighted a similar approach in deriving the Casimir effect stating "the concept of zero-point fluctuations is a heuristic and calculational aid in the description of the Casimir effect, but not a necessity in QED." Nevertheless, as Jaffe himself notes in his paper, "no one has shown that source theory or another S-matrix based approach can provide a complete description of QED to all orders." Furthermore, Peter W. Milonni, Milonni has shown the necessity of the vacuum field for the formal consistency of QED. In QCD, color confinement has led physicists to abandon the source theory or S-matrix based approach for the strong interactions. The Higgs mechanism, Hawking Radiation and the Unruh effect are also theorized to be dependent on zero-point vacuum fluctuations, the field contribution being an inseparable parts of these theories. Jaffe continues "Even if one could argue away zero-point contributions to the quantum vacuum energy, the problem of spontaneous symmetry breaking remains: condensates [ground state vacua] that carry energy appear at many energy scales in the Standard Model. So there is good reason to be skeptical of attempts to avoid the standard formulation of quantum field theory and the zero-point energies it brings with it." It is difficult to judge the physical reality of infinite zero-point energies that are inherent in field theories, but modern physics does not know any better way to construct gauge-invariant, renormalizable theories than with zero-point energy and they would seem to be a necessity for any attempt at a Grand Unified Theory, unified theory.


Chaotic and emergent phenomena

The mathematical models used in classical electromagnetism,
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
(QED) and the Standard Model all view the QED vacuum, electromagnetic vacuum as a linear system with no overall observable consequence. For example, in the case of the Casimir effect, Lamb shift, and so on these phenomena can be explained by alternative mechanisms other than action of the vacuum by arbitrary changes to the normal ordering of field operators. See the #Alternative Theories, alternative theories section. This is a consequence of viewing electromagnetism as a U(1) gauge theory, which topologically does not allow the complex interaction of a field with and on itself. In higher symmetry groups and in reality, the vacuum is not a calm, randomly fluctuating, largely immaterial and passive substance, but at times can be viewed as a turbulent virtual Plasma (physics), plasma that can have complex vortices (i.e. solitons vis-à-vis particles), entangled states and a rich nonlinear structure. There are many observed nonlinear physical electromagnetic phenomena such as Aharonov–Bohm effect, Aharonov–Bohm (AB) and Altshuler–Aronov–Spivak (AAS) effects, Geometric phase, Berry, Aharonov–Anandan, Pancharatnam and Chiao–Wu phase rotation effects, Josephson effect, Quantum Hall effect, the De Haas–Van Alphen effect, the Sagnac effect and many other physically observable phenomena which would indicate that the electromagnetic potential field has real physical meaning rather than being a mathematical artifact and therefore an all encompassing theory would not confine electromagnetism as a local force as is currently done, but as a SU(2) gauge theory or higher geometry. Higher symmetries allow for nonlinear, aperiodic behaviour which manifest as a variety of complex non-equilibrium phenomena that do not arise in the linearised U(1) theory, such as multiple stable states,
symmetry breaking In physics, symmetry breaking is a phenomenon in which (infinitesimally) small fluctuations acting on a system crossing a critical point decide the system's fate, by determining which branch of a bifurcation is taken. To an outside observe ...
,
chaos Chaos or CHAOS may refer to: Arts, entertainment and media Fictional elements * Chaos (''Kinnikuman'') * Chaos (''Sailor Moon'') * Chaos (''Sesame Park'') * Chaos (''Warhammer'') * Chaos, in ''Fabula Nova Crystallis Final Fantasy'' * Cha ...
and
emergence In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergenc ...
. What are called Maxwell's equations today, are in fact a simplified version of the original equations reformulated by Oliver Heaviside, Heaviside, George Francis FitzGerald, FitzGerald, Oliver Lodge, Lodge and Heinrich Hertz, Hertz. The original equations used William Rowan Hamilton, Hamilton's more expressive quaternion notation, a kind of Clifford algebra, which fully subsumes the standard Maxwell vectorial equations largely used today. In the late 1880s there was a debate over the relative merits of vector analysis and quaternions. According to Heaviside the electromagnetic potential field was purely metaphysical, an arbitrary mathematical fiction, that needed to be "murdered". It was concluded that there was no need for the greater physical insights provided by the quaternions if the theory was purely local in nature. Local vector analysis has become the dominant way of using Maxwell's equations ever since. However, this strictly vectorial approach has led to a restrictive topological understanding in some areas of electromagnetism, for example, a full understanding of the energy transfer dynamics in Nikola Tesla, Tesla's oscillator-shuttle-circuit can only be achieved in quaternionic algebra or higher SU(2) symmetries. It has often been argued that quaternions are not compatible with special relativity, but multiple papers have shown ways of incorporating relativity. A good example of nonlinear electromagnetics is in high energy dense plasmas, where Vortex, vortical phenomena occur which seemingly violate the second law of thermodynamics by increasing the energy gradient within the electromagnetic field and violate Maxwell's laws by creating ion currents which capture and concentrate their own and surrounding magnetic fields. In particular Lorentz force, Lorentz force law, which elaborates Maxwell's equations is violated by these force free vortices. These apparent violations are due to the fact that the traditional conservation laws in classical and quantum electrodynamics (QED) only display linear U(1) symmetry (in particular, by the extended Noether theorem, conservation laws such as the laws of thermodynamics need not always apply to dissipative systems, which are expressed in gauges of higher symmetry). The second law of thermodynamics states that in a closed linear system entropy flow can only be positive (or exactly zero at the end of a cycle). However, negative entropy (i.e. increased order, structure or self-organisation) can spontaneously appear in an open nonlinear thermodynamic system that is far from equilibrium, so long as this emergent order accelerates the overall flow of entropy in the total system. The 1977 Nobel Prize in Chemistry was awarded to thermodynamicist Ilya Prigogine for his theory of dissipative systems that described this notion. Prigogine described the principle as "order through fluctuations" or "order out of chaos". It has been argued by some that all emergent order in the universe from galaxies, solar systems, planets, weather, complex chemistry, evolutionary biology to even consciousness, technology and civilizations are themselves examples of thermodynamic dissipative systems; nature having naturally selected these structures to accelerate entropy flow within the universe to an ever-increasing degree. For example, it has been estimated that human body is 10,000 times more effective at dissipating energy per unit of mass than the sun. One may query what this has to do with zero-point energy. Given the complex and adaptive behaviour that arises from nonlinear systems considerable attention in recent years has gone into studying a new class of phase transitions which occur at absolute zero temperature. These are quantum phase transitions which are driven by EM field fluctuations as a consequence of zero-point energy. A good example of a spontaneous phase transition that are attributed to zero-point fluctuations can be found in superconductors. Superconductivity is one of the best known empirically quantified macroscopic electromagnetic phenomena whose basis is recognised to be quantum mechanical in origin. The behaviour of the electric and magnetic fields under superconductivity is governed by the London equations. However, it has been questioned in a series of journal articles whether the quantum mechanically canonised London equations can be given a purely classical derivation. Bostick, for instance, has claimed to show that the London equations do indeed have a classical origin that applies to superconductors and to some collisionless plasmas as well. In particular it has been asserted that the Beltrami vector field, Beltrami vortices in the plasma focus display the same paired Flux tube, flux-tube morphology as Type II superconductors. Others have also pointed out this connection, Fröhlich has shown that the hydrodynamic equations of compressible fluids, together with the London equations, lead to a macroscopic parameter (\mu = electric charge density / mass density), without involving either phase factor, quantum phase factors or Planck's constant. In essence, it has been asserted that Beltrami plasma vortex structures are able to at least simulate the morphology of Type-I superconductor, Type I and Type-II superconductor, Type II superconductors. This occurs because the "organised" dissipative energy of the vortex configuration comprising the ions and electrons far exceeds the "disorganised" dissipative random thermal energy. The transition from disorganised fluctuations to organised helical structures is a phase transition involving a change in the condensate's energy (i.e. the ground state or zero-point energy) but ''without any associated rise in temperature''. This is an example of zero-point energy having multiple stable states (see Quantum phase transition, Quantum critical point, Topological degeneracy, Topological order) and where the overall system structure is independent of a reductionist or deterministic view, that "classical" macroscopic order can also causally affect quantum phenomena. Furthermore, the pair production of Beltrami vortices has been compared to the morphology of pair production of virtual particles in the vacuum. The idea that the vacuum energy can have multiple stable energy states is a leading hypothesis for the cause of cosmic inflation. In fact, it has been argued that these early vacuum fluctuations led to the expansion of the universe and in turn have guaranteed the non-equilibrium conditions necessary to drive order from chaos, as without such expansion the universe would have reached thermal equilibrium and no complexity could have existed. With the continued accelerated expansion of the universe, the cosmos generates an energy gradient that increases the "free energy" (i.e. the available, usable or potential energy for useful work) which the universe is able to utilize to create ever more complex forms of order. The only reason Earth's environment does not decay into an equilibrium state is that it receives a daily dose of sunshine and that, in turn, is due to the sun "polluting" interstellar space with decreasing entropy. The sun's fusion power is only possible due to the gravitational disequilibrium of matter that arose from cosmic expansion. In this essence, the vacuum energy can be viewed as the key cause of the negative entropy (i.e. structure) throughout the universe. That humanity might alter the morphology of the vacuum energy to create an energy gradient for useful work is the subject of much controversy.


Purported applications

Physicists overwhelmingly reject any possibility that the zero-point energy field can be exploited to obtain useful energy (work (physics), work) or uncompensated momentum; such efforts are seen as tantamount to perpetual motion machines. Nevertheless, the allure of free energy has motivated such research, usually falling in the category of fringe science. As long ago as 1889 (before quantum theory or discovery of the zero point energy) Nikola Tesla proposed that useful energy could be obtained from free space, or what was assumed at that time to be an all-pervasive Aether (classical element), aether. Others have since claimed to exploit zero-point or vacuum energy with a large amount of Pseudoscience, pseudoscientific literature causing ridicule around the subject. Despite rejection by the scientific community, harnessing zero-point energy remains an interest of research, particularly in the US where it has attracted the attention of major aerospace/defence contractors and the DoD, U.S. Department of Defense as well as in China, Germany, Russia and Brazil.


Casimir batteries and engines

A common assumption is that the Casimir effect, Casimir force is of little practical use; the argument is made that the only way to actually gain energy from the two plates is to allow them to come together (getting them apart again would then require more energy), and therefore it is a one-use-only tiny force in nature. In 1984 Robert L. Forward, Robert Forward published work showing how a "vacuum-fluctuation battery" could be constructed. The battery can be recharged by making the electrical forces slightly stronger than the Casimir force to reexpand the plates. In 1995 and 1998 Maclay et al. published the first models of a microelectromechanical system (MEMS) with Casimir forces. While not exploiting the Casimir force for useful work, the papers drew attention from the MEMS community due to the revelation that Casimir effect needs to be considered as a vital factor in the future design of MEMS. In particular, Casimir effect might be the critical factor in the stiction failure of MEMS. In 1999, Pinto, a former scientist at NASA's Jet Propulsion Laboratory, Jet Propulsion Laboratory at Caltech in Pasadena, published in ''Physical Review'' his thought experiment (Gedankenexperiment) for a "Casimir engine". The paper showed that continuous positive net exchange of energy from the
Casimir effect In quantum field theory, the Casimir effect is a physical force acting on the macroscopic boundaries of a confined space which arises from the quantum fluctuations of the field. It is named after the Dutch physicist Hendrik Casimir, who pre ...
was possible, even stating in the abstract "In the event of no other alternative explanations, one should conclude that major technological advances in the area of endless, by-product free-energy production could be achieved." In 2001, Capasso et al. showed how the force can be used to control the mechanical motion of a MEMS device, The researchers suspended a polysilicon plate from a torsional rod – a twisting horizontal bar just a few microns in diameter. When they brought a metallized sphere close up to the plate, the attractive Casimir force between the two objects made the plate rotate. They also studied the dynamical behaviour of the MEMS device by making the plate oscillate. The Casimir force reduced the rate of oscillation and led to nonlinear phenomena, such as hysteresis and bistability in the frequency response of the oscillator. According to the team, the system's behaviour agreed well with theoretical calculations. Despite this and several similar peer reviewed papers, there is not a consensus as to whether such devices can produce a continuous output of work. Garret Moddel at University of Colorado Boulder, University of Colorado has highlighted that he believes such devices hinge on the assumption that the Casimir force is a Conservative force, nonconservative force, he argues that there is sufficient evidence (e.g. analysis by Scandurra (2001)) to say that the Casimir effect is a conservative force and therefore even though such an engine can exploit the Casimir force for useful work it cannot produce more output energy than has been input into the system. In 2008, DARPA solicited research proposals in the area of Casimir Effect Enhancement (CEE). The goal of the program is to develop new methods to control and manipulate attractive and repulsive forces at surfaces based on engineering of the Casimir force. A 2008 patent by Haisch and Moddel details a device that is able to extract power from zero-point fluctuations using a gas that circulates through a Casimir cavity. As gas atoms circulate around the system they enter the cavity. Upon entering the electrons spin down to release energy via electromagnetic radiation. This radiation is then extracted by an absorber. On exiting the cavity the ambient vacuum fluctuations (i.e. the zero-point field) impart energy on the electrons to return the orbitals to previous energy levels, as predicted by Senitzky (1960). The gas then goes through a pump and flows through the system again. A published test of this concept by Moddel was performed in 2012 and seemed to give excess energy that could not be attributed to another source. However it has not been conclusively shown to be from zero-point energy and the theory requires further investigation.


Single heat baths

In 1951 Herbert Callen, Callen and Welton proved the quantum
fluctuation-dissipation theorem The fluctuation–dissipation theorem (FDT) or fluctuation–dissipation relation (FDR) is a powerful tool in statistical physics for predicting the behavior of systems that obey detailed balance. Given that a system obeys detailed balance, the th ...
(FDT) which was originally formulated in classical form by Nyquist (1928) as an explanation for observed
Johnson noise Johnson is a surname of Anglo-Norman origin meaning "Son of John". It is the second most common in the United States and 154th most common in the world. As a common family name in Scotland, Johnson is occasionally a variation of ''Johnston'', a ...
in electric circuits. Fluctuation-dissipation theorem showed that when something dissipates energy, in an effectively irreversible way, a connected heat bath must also fluctuate. The fluctuations and the dissipation go hand in hand; it is impossible to have one without the other. The implication of FDT being that the vacuum could be treated as a heat bath coupled to a dissipative force and as such energy could, in part, be extracted from the vacuum for potentially useful work. Such a theory has met with resistance: Macdonald (1962) and Harris (1971) claimed that extracting power from the zero-point energy to be impossible, so FDT could not be true. Grau and Kleen (1982) and Kleen (1986), argued that the Johnson noise of a resistor connected to an antenna must satisfy Planck's thermal radiation formula, thus the noise must be zero at zero temperature and FDT must be invalid. Kiss (1988) pointed out that the existence of the zero-point term may indicate that there is a renormalization problem—i.e., a mathematical artifact—producing an unphysical term that is not actually present in measurements (in analogy with renormalization problems of ground states in quantum electrodynamics). Later, Abbott et al. (1996) arrived at a different but unclear conclusion that "zero-point energy is infinite thus it should be renormalized but not the 'zero-point fluctuations'". Despite such criticism, FDT has been shown to be true experimentally under certain quantum, non-classical conditions. Zero-point fluctuations can, and do, contribute towards systems which dissipate energy. A paper by Armen Allahverdyan and Theo Nieuwenhuizen in 2000 showed the feasibility of extracting zero-point energy for useful work from a single bath, without contradicting the laws of thermodynamics, by exploiting certain quantum mechanical properties. There have been a growing number of papers showing that in some instances the classical laws of thermodynamics, such as limits on the Carnot efficiency, can be violated by exploiting negative entropy of quantum fluctuations. Despite efforts to reconcile quantum mechanics and thermodynamics over the years, their compatibility is still an open fundamental problem. The full extent that quantum properties can alter classical thermodynamic bounds is unknown


Space travel and gravitational shielding

The use of zero-point energy for space travel is speculative and does not form part of the mainstream scientific consensus. A complete Theory of everything, quantum theory of gravitation (that would deal with the role of quantum phenomena like zero-point energy) does not yet exist. Speculative papers explaining a relationship between zero-point energy and gravitational shielding effects have been proposed, but the interaction (if any) is not yet fully understood. Most serious scientific research in this area depends on the theorized anti-gravitational properties of antimatter (currently being tested at the Antiproton Decelerator#ALPHA, alpha experiment at
CERN The European Organization for Nuclear Research, known as CERN (; ; ), is an intergovernmental organization that operates the largest particle physics laboratory in the world. Established in 1954, it is based in a northwestern suburb of Gen ...
) and/or the effects of non-Newtonian forces such as the gravitomagnetic field under specific quantum conditions. According to the general theory of relativity, rotating matter can generate a new force of nature, known as the gravitomagnetic interaction, whose intensity is proportional to the rate of spin. In certain conditions the gravitomagnetic field can be repulsive. In Neutron star, neutrons stars for example it can produce a gravitational analogue of the Meissner effect, but the force produced in such an example is theorized to be exceedingly weak. In 1963 Robert L. Forward, Robert Forward, a physicist and aerospace engineer at Hughes Research Laboratories, published a paper showing how within the framework of general relativity "anti-gravitational" effects might be achieved. Since all atoms have Spin (physics), spin, gravitational permeability may be able to differ from material to material. A strong toroidal gravitational field that acts against the force of gravity could be generated by materials that have Chaos theory, nonlinear properties that enhance time-varying gravitational fields. Such an effect would be analogous to the nonlinear electromagnetic permeability of iron making it an effective core (i.e. the doughnut of iron) in a transformer, whose properties are dependent on magnetic permeability. In 1966 Bryce DeWitt, Dewitt was first to identify the significance of gravitational effects in superconductors. Dewitt demonstrated that a magnetic-type gravitational field must result in the presence of Magnetic flux quantum, fluxoid quantization. In 1983, Dewitt's work was substantially expanded by Ross. From 1971 to 1974 Henry William Wallace, a scientist at GE Aerospace (1960s), GE Aerospace was issued with three patents. Wallace used Bryce DeWitt, Dewitt's theory to develop an experimental apparatus for generating and detecting a secondary gravitational field, which he named the kinemassic field (now better known as the gravitomagnetic field). In his three patents, Wallace describes three different methods used for detection of the gravitomagnetic field – change in the motion of a body on a pivot, detection of a transverse voltage in a semiconductor crystal, and a change in the specific heat of a crystal material having spin-aligned nuclei. There are no publicly available independent tests verifying Wallace's devices. Such an effect if any would be small. Referring to Wallace's patents, a New Scientist article in 1980 stated "Although the Wallace patents were initially ignored as cranky, observers believe that his invention is now under serious but secret investigation by the military authorities in the USA. The military may now regret that the patents have already been granted and so are available for anyone to read." A further reference to Wallace's patents occur in an electric propulsion study prepared for the Air Force Research Laboratory, Astronautics Laboratory at Edwards Air Force Base which states: "The patents are written in a very believable style which include part numbers, sources for some components, and diagrams of data. Attempts were made to contact Wallace using patent addresses and other sources but he was not located nor is there a trace of what became of his work. The concept can be somewhat justified on general relativistic grounds since rotating frames of time varying fields are expected to emit gravitational waves." In 1986 the U.S. Air Force's then Rocket Propulsion Laboratory (RPL) at Edwards Air Force Base solicited "Non Conventional Propulsion Concepts" under a small business research and innovation program. One of the six areas of interest was "Esoteric energy sources for propulsion, including the quantum dynamic energy of vacuum space..." In the same year BAE Systems launched "Project Greenglow" to provide a "focus for research into novel propulsion systems and the means to power them". In 1988 Kip Thorne et al. published work showing how traversable Lorentzian wormhole, wormholes can exist in spacetime only if they are threaded by quantum fields generated by some form of exotic matter that has negative energy. In 1993 Scharnhorst and Barton showed that Scharnhorst effect, the speed of a photon will be increased if it travels between two Casimir plates, an example of negative energy. In the most general sense, the exotic matter needed to create wormholes would share the repulsive properties of the Inflation (cosmology), inflationary energy,
dark energy In physical cosmology and astronomy, dark energy is an unknown form of energy that affects the universe on the largest scales. The first observational evidence for its existence came from measurements of supernovas, which showed that the univ ...
or zero-point radiation of the vacuum. Building on the work of Thorne, in 1994 Miguel Alcubierre proposed a method for changing the geometry of space by creating a wave that would cause the fabric of space ahead of a spacecraft to contract and the space behind it to expand (see Alcubierre drive). The ship would then ride this wave inside a region of flat space, known as a ''warp bubble'' and would not move within this bubble but instead be carried along as the region itself moves due to the actions of the drive. In 1992 Evgeny Podkletnov published a heavily debated journal article claiming a specific type of rotating superconductor could shield gravitational force. Independently of this, from 1991 to 1993 Ning Li (physicist), Ning Li and Douglas Torr published a number of articles about gravitational effects in superconductors. One finding they derived is the source of Gravitomagnetic field, gravitomagnetic flux in a type II superconductor material is due to Spin (physics), spin alignment of the lattice ions. Quoting from their third paper: "It is shown that the coherent alignment of lattice ion spins will generate a detectable gravitomagnetic field, and in the presence of a time-dependent applied magnetic vector potential field, a detectable gravitoelectric field." The claimed size of the generated force has been disputed by some but defended by others. In 1997 Li published a paper attempting to replicate Podkletnov's results and showed the effect was very small, if it existed at all. Li is reported to have left the University of Alabama in 1999 to found the company ''AC Gravity LLC''. AC Gravity was awarded a United States Department of Defense, U.S. DOD grant for $448,970 in 2001 to continue anti-gravity research. The grant period ended in 2002 but no results from this research were ever made public. In 2002 Phantom Works, Boeing's advanced research and development facility in Seattle, approached Evgeny Podkletnov directly. Phantom Works was blocked by Russian technology transfer controls. At this time Lieutenant General George Muellner, the outgoing head of the Boeing Phantom Works, confirmed that attempts by Boeing to work with Podkletnov had been blocked by Moscow, also commenting that "The physical principles – and Podkletnov's device is not the only one – appear to be valid... There is basic science there. They're not breaking the laws of physics. The issue is whether the science can be engineered into something workable" Froning and Roach (2002) put forward a paper that builds on the work of Puthoff, Haisch and Alcubierre. They used fluid dynamic simulations to model the interaction of a vehicle (like that proposed by Alcubierre) with the zero-point field. Vacuum field perturbations are simulated by fluid field perturbations and the aerodynamic resistance of viscous drag exerted on the interior of the vehicle is compared to the Lorentz force exerted by the zero-point field (a Casimir-like force is exerted on the exterior by unbalanced zero-point radiation pressures). They find that the optimized negative energy required for an Alcubierre drive is where it is a saucer-shaped vehicle with toroidal electromagnetic fields. The EM fields distort the vacuum field perturbations surrounding the craft sufficiently to affect the permeability and permittivity of space. In 2014 Lyndon B. Johnson Space Center, NASA's Advanced Propulsion Physics Laboratory, Eagleworks Laboratories announced that they had successfully validated the use of a Quantum Vacuum Plasma Thruster which makes use of the
Casimir effect In quantum field theory, the Casimir effect is a physical force acting on the macroscopic boundaries of a confined space which arises from the quantum fluctuations of the field. It is named after the Dutch physicist Hendrik Casimir, who pre ...
for propulsion. In 2016 a scientific paper by the team of NASA scientists passed peer review for the first time. The paper suggests that the zero-point field acts as De Broglie–Bohm theory, pilot-wave and that the thrust may be due to particles pushing off the quantum vacuum. While peer review doesn't guarantee that a finding or observation is valid, it does indicate that independent scientists looked over the experimental setup, results, and interpretation and that they could not find any obvious errors in the methodology and that they found the results reasonable. In the paper, the authors identify and discuss nine potential sources of experimental errors, including rogue air currents, leaky electromagnetic radiation, and magnetic interactions. Not all of them could be completely ruled out, and further peer reviewed experimentation is needed in order to rule these potential errors out.


See also

*
Casimir effect In quantum field theory, the Casimir effect is a physical force acting on the macroscopic boundaries of a confined space which arises from the quantum fluctuations of the field. It is named after the Dutch physicist Hendrik Casimir, who pre ...
* Ground state *
Lamb shift In physics, the Lamb shift, named after Willis Lamb, is a difference in energy between two energy levels 2''S''1/2 and 2''P''1/2 (in term symbol notation) of the hydrogen atom which was not predicted by the Dirac equation, according to which th ...
* QED vacuum *
QCD vacuum In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type o ...
* Quantum fluctuation * Quantum foam * Scalar field * Time crystal * Topological order * Unruh effect * Vacuum energy * Vacuum expectation value * Vacuum state * Virtual particle


References


Notes


Articles in the press

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Via Calphysics Institute
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Bibliography

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Further reading


Press articles

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Journal articles

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Books

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External links

* Nima Arkani-Hamed o
the issue of vacuum energy and dark energy
* Steven Weinberg o
the cosmological constant problem
{{DEFAULTSORT:Zero-Point Energy Energy (physics) Quantum field theory Quantum electrodynamics Concepts in physics Mathematical physics Condensed matter physics Materials science Quantum phases Non-equilibrium thermodynamics Perpetual motion Physical paradoxes Thermodynamics