Zeropoint energy is the lowest possible
energyIn physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
that a quantum mechanical
physical systemIn physics, the word system has a technical meaning, namely, it is the portion of the physical universe chosen for analysis. Everything outside the system is known as the environment, which in analysis is ignored except for its effects on the system. The cut between system and the world is a free...
may have; it is the energy of its
ground stateThe ground state of a quantum mechanical system is its lowestenergy state; the energy of the ground state is known as the zeropoint energy of the system. An excited state is any state with energy greater than the ground state...
. All quantum mechanical systems undergo fluctuations even in their ground state and have an associated zeropoint energy, a consequence of their
waveIn physics, a wave is a disturbance that travels through space and time, accompanied by the transfer of energy.Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass...
like nature. The
uncertainty principleIn quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...
requires every physical system to have a zeropoint energy greater than the minimum of its classical
potential wellA 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 because it is captured in the local minimum of a potential well...
, even at
absolute zeroAbsolute zero is the theoretical temperature at which entropy reaches its minimum value. The laws of thermodynamics state that absolute zero cannot be reached using only thermodynamic means....
. For example,
liquid heliumHelium exists in liquid form only at extremely low temperatures. The boiling point and critical point depend on the isotope of the helium; see the table below for values. The density of liquid helium4 at its boiling point and 1 atmosphere is approximately 0.125 g/mL Helium4 was first liquefied...
does not freeze under atmospheric pressure at any temperature because of its zeropoint energy.
The concept of zeropoint energy was developed in Germany by
Albert EinsteinAlbert Einstein was a Germanborn theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
and
Otto SternOtto Stern was a German physicist and Nobel laureate in physics.Biography:Stern was born in Sohrau, now Żory in the German Empire's Kingdom of Prussia and studied at Breslau, now Wrocław in Lower Silesia....
in 1913, using a formula developed by
Max PlanckMax Karl Ernst Ludwig Planck, ForMemRS, was a German physicist who actualized the quantum physics, initiating a revolution in natural science and philosophy. He is regarded as the founder of the quantum theory, for which he received the Nobel Prize in Physics in 1918.Life and career:Planck came...
in 1900. The term
zeropoint energy originatesIn linguistics, a calque or loan translation is a word or phrase borrowed from another language by literal, wordforword or rootforroot translation.Calque:...
from the German
Nullpunktsenergie. The German name is also spelled
Nullpunktenergie (without the "s").
Vacuum energyVacuum energy is an underlying background energy that exists in space even when the space is devoid of matter . The concept of vacuum energy has been deduced from the concept of virtual particles, which is itself derived from the energytime uncertainty principle...
is the zeropoint energy of all the
fieldIn physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...
s in space, which in the
Standard ModelThe Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...
includes the
electromagnetic fieldAn electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...
, other gauge fields,
fermionic fieldIn quantum field theory, a fermionic field is a quantum field whose quanta are fermions; that is, they obey FermiDirac statistics. Fermionic fields obey canonical anticommutation relations rather than the canonical commutation relations of bosonic fields....
s, and the Higgs field. It is the energy of the
vacuumIn quantum field theory, the vacuum state is the quantum state with the lowest possible energy. Generally, it contains no physical particles...
, which in
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and manybody systems. It is the natural and quantitative language of particle physics and...
is defined not as empty space but as the ground state of the fields. In
cosmologyPhysical cosmology, as a branch of astronomy, is the study of the largestscale structures and dynamics of the universe and is concerned with fundamental questions about its formation and evolution. For most of human history, it was a branch of metaphysics and religion...
, the vacuum energy is one possible explanation for the
cosmological constantIn physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe...
. A related term is
zeropoint field, which is the lowest energy state of a particular field.
History
In 1900,
Max PlanckMax Karl Ernst Ludwig Planck, ForMemRS, was a German physicist who actualized the quantum physics, initiating a revolution in natural science and philosophy. He is regarded as the founder of the quantum theory, for which he received the Nobel Prize in Physics in 1918.Life and career:Planck came...
derived the formula for the energy of a single
energy radiator, e.g. a vibrating atomic unit:
where
is Planck's constant,
is the
frequencyFrequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
,
k is Boltzmann's constant, and
T is the absolute
temperatureTemperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
.
Then in 1913, using this formula as a basis,
Albert EinsteinAlbert Einstein was a Germanborn theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
and
Otto SternOtto Stern was a German physicist and Nobel laureate in physics.Biography:Stern was born in Sohrau, now Żory in the German Empire's Kingdom of Prussia and studied at Breslau, now Wrocław in Lower Silesia....
published a paper of great significance in which they suggested for the first time the existence of a residual energy that all oscillators have at absolute zero. They called this residual energy
Nullpunktsenergie (German), later
translatedIn linguistics, a calque or loan translation is a word or phrase borrowed from another language by literal, wordforword or rootforroot translation.Calque:...
as
zeropoint energy. They carried out an analysis of the specific heat of hydrogen gas at low temperature, and concluded that the data are best represented if the vibrational energy is
According to this expression, an atomic system at
absolute zeroAbsolute zero is the theoretical temperature at which entropy reaches its minimum value. The laws of thermodynamics state that absolute zero cannot be reached using only thermodynamic means....
retains an energy of ½
hν.
In 1916
Walther NernstWalther Hermann Nernst FRS was a German physical chemist and physicist who is known for his theories behind the calculation of chemical affinity as embodied in the third law of thermodynamics, for which he won the 1920 Nobel Prize in chemistry...
postulated that the vacuum of space is filled with zeropoint electromagnetic radiation.
Relation to the uncertainty principle
Zeropoint energy is fundamentally related to the Heisenberg uncertainty principle. 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 defined precisely by any given quantum state. In particular, there cannot be a state in which the system 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 lowestenergy 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 wellA 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 because it is captured in the local minimum of a potential well...
, the
HamiltonianIn quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
of a system (the quantummechanical
operatorIn physics, an operator is a function acting on the space of physical states. As a resultof its application on a physical state, another physical state is obtained, very often along withsome extra relevant information....
giving its energy) can be approximated as

where
is the minimum of the classical potential well. The uncertainty principle tells us that

making the
expectation valueIn quantum mechanics, the expectation value is the predicted mean value of the result of an experiment. Despite the name, it is not the most probable value of a measurement...
s of the
kineticThe kinetic energy of an object is the energy which 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 acceleration, the body maintains this kinetic energy unless its speed changes...
and
potentialIn physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...
terms above satisfy

The expectation value of the energy must therefore be at least

where
is the
angular frequencyIn physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
at which the system oscillates.
A more thorough treatment, showing that the energy of the ground state actually is
requires solving for the ground state of the system. See
quantum harmonic oscillatorThe quantum harmonic oscillator is the quantummechanical analog of the classical harmonic oscillator. Because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics...
for details.
Varieties
The concept of zeropoint energy occurs in a number of situations.
In ordinary quantum mechanics, the zeropoint energy is the energy associated with the
ground stateThe ground state of a quantum mechanical system is its lowestenergy state; the energy of the ground state is known as the zeropoint energy of the system. An excited state is any state with energy greater than the ground state...
of the system. The professional physics literature tends to measure frequency, as denoted by
above, using
angular frequencyIn physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
, 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 those terms, the most famous such example of zeropoint energy is
associated with the ground state of the
quantum harmonic oscillatorThe quantum harmonic oscillator is the quantummechanical analog of the classical harmonic oscillator. Because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics...
. In quantum mechanical terms, the zeropoint energy is the expectation value of the
HamiltonianIn quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
of the system in the ground state.
In
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and manybody systems. It is the natural and quantitative language of particle physics and...
, the fabric of space is visualized as consisting of
fieldsIn physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...
, with the field at every point in space and time being a
quantum harmonic oscillatorThe quantum harmonic oscillator is the quantummechanical analog of the classical harmonic oscillator. Because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics...
, with neighboring oscillators interacting. In this case, one has a contribution of
from every point in space, resulting in a calculation of infinite zeropoint energy in any finite volume; this is one reason
renormalizationIn quantum field theory, the statistical mechanics of fields, and the theory of selfsimilar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
is needed to make sense of quantum field theories. The zeropoint energy is again the expectation value of the Hamiltonian; here, however, the phrase
vacuum expectation valueIn quantum field theory the vacuum expectation value of an operator is its average, expected value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle...
is more commonly used, and the energy is called the
vacuum energyVacuum energy is an underlying background energy that exists in space even when the space is devoid of matter . The concept of vacuum energy has been deduced from the concept of virtual particles, which is itself derived from the energytime uncertainty principle...
.
In quantum
perturbation theoryPerturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...
, it is sometimes said that the contribution of oneloop and multiloop
Feynman diagramFeynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prizewinning American physicist Richard Feynman, and first introduced in 1948...
s to
elementary particleIn particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...
propagatorIn quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. Propagators are used to represent the contribution of virtual particles on the internal...
s are the contribution of vacuum fluctuations or the zeropoint energy to the particle
massMass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
es.
Experimental observations
A phenomenon that is commonly presented as evidence for the existence of zeropoint energy in vacuum is the
Casimir effectIn quantum field theory, the Casimir effect and the Casimir–Polder force are physical forces arising from a quantized field. The typical example is of two uncharged metallic plates in a vacuum, like capacitors placed a few micrometers apart, without any external electromagnetic field...
, proposed in 1948 by
DutchThe Netherlands is a constituent country of the Kingdom of the Netherlands, located mainly in NorthWest Europe and with several islands in the Caribbean. Mainland Netherlands borders the North Sea to the north and west, Belgium to the south, and Germany to the east, and shares maritime borders...
physicistA physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from subatomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...
Hendrik B. G. CasimirHendrik Brugt Gerhard Casimir FRS was a Dutch physicist best known for his research on the twofluid model of superconductors in 1934 and the Casimir effect Hendrik Brugt Gerhard Casimir FRS (July 15, 1909 in The Hague, Netherlands – May 4, 2000 in Heeze) was a Dutch physicist best known...
(
Philips ResearchThe Philips Natuurkundig Laboratorium or NatLab was the Dutch section of the Philips research department, which did research for the product divisions of that company. Originally located in the Strijp district of Eindhoven, the facility moved to Waalre in the early 1960s...
), who considered the quantized
electromagnetic fieldAn electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...
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. This force has been measured and found to be in good agreement with the theory. However, there is still some debate on whether vacuum energy is necessary to explain the Casimir effect.
Robert JaffeRobert L. Jaffe is an American physicist and the Jane and Otto Morningstar Professor of Physics at the Massachusetts Institute of Technology . He was formerly director of the MIT Center for Theoretical Physics.Biography:...
of MIT argues that the Casimir force should not be considered evidence for vacuum energy, since it can be derived in
QEDQuantum electrodynamics 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 special relativity is achieved...
without reference to vacuum energy by considering chargecurrent interactions (the radiationreaction picture).
The experimentally measured
Lamb shift has been argued to be, in part, a zeropoint energy effect.
Gravitation and cosmology
In
cosmologyPhysical cosmology, as a branch of astronomy, is the study of the largestscale structures and dynamics of the universe and is concerned with fundamental questions about its formation and evolution. For most of human history, it was a branch of metaphysics and religion...
, the zeropoint energy offers an intriguing possibility for explaining the speculative positive values of the proposed
cosmological constantIn physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe...
. In brief, if the energy is "really there", then it should exert a gravitational force. In
general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
,
massMass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
and energy are equivalent; both produce a gravitational field. One obvious difficulty with this association is that the zeropoint energy of the vacuum is absurdly large. Naively, it is infinite, because it includes the energy of waves with arbitrarily short wavelengths. But since only
differences in energy are physically measurable, the infinity can be removed by
renormalizationIn quantum field theory, the statistical mechanics of fields, and the theory of selfsimilar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
. In all practical calculations, this is how the infinity is handled. It is also arguable that undiscovered physics relevant at the
Planck scaleIn particle physics and physical cosmology, the Planck scale is an energy scale around 1.22 × 1019 GeV at which quantum effects of gravity become strong...
reduces or eliminates the energy of waves shorter than the
Planck length, making the total zeropoint energy finite.
Freeenergy devices
As a scientific concept, the existence of zeropoint energy is not controversial although the ability to harness it is. In particular, perpetual motion machines and other power generating devices supposedly based on zeropoint energy are highly controversial and, in many cases, in violation of some of the fundamental laws of physics. No device claimed to operate using zeropoint energy has been demonstrated to operate as claimed. No plausible description of a device drawing useful power from a source of zeropoint energy has been given. Thus, current claims to zeropointenergybased power generation systems have the status of
pseudosciencePseudoscience is a claim, belief, or practice which is presented as scientific, but which does not adhere to a valid scientific method, lacks supporting evidence or plausibility, cannot be reliably tested, or otherwise lacks scientific status...
.
The discovery of zeropoint energy did not alter the implausibility of perpetual motion machines. Much attention has been given to reputable science suggesting that zeropointenergy density is infinite, but in quantum theory, zeropoint energy is a minimum energy below which a
thermodynamic systemA thermodynamic system is a precisely defined macroscopic region of the universe, often called a physical system, that is studied using the principles of thermodynamics....
can never go. Thus according to the standard quantumtheoretic viewpoint, none of this energy can be withdrawn without altering the system to a different form in which the system has a lower zeropoint energy.
It is possible that the discovery of new physics will alter this conclusion. For example, in
stochastic electrodynamicsIn theoretical physics, Stochastic Electrodynamics is a variant of Classical Electrodynamics which posits the existence of a classical Lorentz Invariant radiation field having statistical properties similar to that of the electromagnetic zeropoint field of Quantum Electrodynamics...
, the zeropoint field is viewed as simply a classical background isotropic noise wave field which excites all systems present in the vacuum and thus is
responsible for their minimumenergy or "ground" states. The requirement of Lorentz invariance at a statistical level then implies that the energy density spectrum must increase with the third power of frequency, implying infinite energy density when integrated over all frequencies. If this theory is correct, there is no reason that energy, or for that matter, momentum, could not be extracted, and would of course still leave infinite energy density and infinite momentum density, isotropic in all directions simultaneously, remaining in the wave field.
According to NASA, "the concept of accessing a significant amount of useful energy from the ZPE gained much credibility when a major article on this topic was recently [March 1st, 2004] published in Aviation Week & Space Technology, a leading aerospace industry magazine".
The calculation that underlies the Casimir experiment, a calculation based on the formula predicting infinite vacuum energy, shows the zeropoint energy of a system consisting of a vacuum between two plates will decrease at a finite rate as the two plates are drawn together. The vacuum energies are predicted to be infinite, but the changes are predicted to be finite. Casimir combined the projected rate of change in zeropoint energy with the principle of conservation of energy to predict a force on the plates. The predicted force, which is very small and was experimentally measured to be within 5% of its predicted value, is finite. Even though the zeropoint energy is theoretically infinite, there is as yet no practical evidence to suggest that infinite amounts of zeropoint energy are available for use, that zeropoint energy can be withdrawn for free, or that zeropoint energy can be used in violation of conservation of energy.
External links