Zero-point energy

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Liquid helium retains kinetic energy and does not freeze regardless of temperature at standard atmospheric pressure due to zero-point energy. When cooled below its Lambda point, it exhibits properties of superfluidity.

Zero-point energy (ZPE) is the lowest possible energy that a quantum mechanical system may have. Unlike in classical mechanics, quantum systems constantly fluctuate in their lowest energy state as described by the Heisenberg uncertainty principle. Therefore, even at absolute zero, atoms and molecules retain some vibrational motion. Apart from atoms and molecules, the empty space of a vacuum also has these properties. According to quantum field theory, the universe can be thought of not as isolated particles but continuous fluctuating fields: matter fields, whose quanta are fermions (in other words, leptons and quarks), and force fields, whose quanta are bosons (such as photons and gluons). 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 such that there is no contradiction with Albert Einstein's theory of special relativity.

The notion of a zero-point energy is also important for cosmology, 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. Yet according to Einstein's theory of general relativity, any such energy would gravitate, and the experimental evidence from the expansion of the universe, dark energy and the Casimir effect shows any such energy to be exceptionally weak. One 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 out each other. This idea would be true if supersymmetry were an exact symmetry of nature; however, the Large Hadron Collider at CERN 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, 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 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 (e.g., electromagnetic interactions between photons, electrons and the vacuum) or the QCD vacuum which deals with quantum chromodynamics (e.g., color charge 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 this combination of fields is called the vacuum state, and its associated zero-point energy is called the vacuum energy.

Overview

Kinetic energy vs temperature

In classical mechanics all particles can be thought of as having some energy made up of their potential energy and kinetic energy. Temperature, for example, arises from the intensity of random particle motion caused by kinetic energy (known as Brownian motion). As temperature is reduced to absolute zero, 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 of quantum mechanics.

Zero-point radiation continually imparts random impulses on an electron, so that it never comes to a complete stop. Zero-point radiation gives the oscillator an average energy equal to the frequency of oscillation multiplied by one-half of the Planck constant.

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, 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-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. This results in motion even at absolute zero. For example, liquid helium 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's E = mc², any point in space that contains energy can be thought of as having mass to create particles. Modern physics has developed quantum field theory (QFT) to understand the fundamental interactions between matter and forces; it treats every single point of space as a quantum harmonic oscillator. According to QFT the universe is made up of matter fields, whose quanta are fermions (i.e. leptons and quarks), and force fields, whose quanta are bosons (e.g. photons and gluons). All these fields have zero-point energy. Recent experiments support the idea that particles themselves can be thought of as excited states of the underlying quantum vacuum, 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 with 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 the absolute energy value of space is not an arbitrary constant and gives rise to the cosmological constant. For decades most physicists assumed that there was some undiscovered fundamental principle that will remove the infinite zero-point energy (discussed further below) 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 energy contained in any unit of empty space will decrease as the total energy spreads out to fill the volume of the universe; galaxies 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 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.

Many physical effects attributed to zero-point energy have been experimentally verified, such as spontaneous emission, Casimir force, Lamb shift, magnetic moment of the electron and Delbrück scattering. 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, chaos and emergence. Active areas of research include the effects of virtual particles, quantum entanglement, the difference (if any) between inertial and gravitational mass, variation in the speed of light, a reason for the observed value of the cosmological constant and the nature of dark energy.

History

Early aether theories

James Clerk Maxwell

Zero-point energy evolved from historical ideas about the vacuum. To Aristotle the vacuum was τὸ κενόν, "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 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, it was not until much later on with the beginning of the renaissance, that Otto von Guericke 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. 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 aether theory based upon Maxwell's electrodynamics, this all-encompassing aether was endowed with energy and hence very different from nothingness. The fact that electromagnetic and gravitational phenomena were transmitted in empty space was considered evidence that their associated aethers were part of the fabric of space itself. However Maxwell noted that for the most part these aethers were ad hoc:

To those who maintained the existence of a plenum as a philosophical principle, nature's abhorrence of a vacuum was a sufficient reason for imagining an all-surrounding aether ... Aethers were invented for the planets to swim in, to constitute electric atmospheres and magnetic effluvia, to convey sensations from one part of our bodies to another, and so on, till a space had been filled three or four times with aethers.

Moreover, the results of the Michelson–Morley experiment in 1887 were the first strong evidence that the then-prevalent aether theories were seriously flawed, explicitly confirming the invariance of the speed of light predicted by Maxwell's equations, as later axiomatized in special relativity that 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 temperature after evacuation. Absolute zero was technically impossible to achieve in the 19th century, so the debate remained unsolved.

Second quantum theory

Planck in 1918, the year he received the Nobel Prize in Physics for his work on quantum theory

In 1900, Max Planck derived the average energy ε of a single energy radiator, e.g., a vibrating atomic unit, as a function of absolute temperature: $${\displaystyle \epsilon =\frac{h\nu }{e^{h\nu /(kT)}-1},}$$ where h is the Planck constant, ν is the frequency, k is the Boltzmann constant, and T is the absolute temperature. 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 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 hν. 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 ⁠hν/2⁠, as an additional term dependent on the frequency ν, which was greater than zero (where h is the Planck 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: $${\displaystyle \epsilon =\frac{h\nu }{2}+\frac{h\nu }{e^{h\nu /(kT)}-1}.}$$

Einstein's official 1921 portrait after receiving the Nobel Prize in Physics

Soon, the idea of zero-point energy attracted the attention of Albert Einstein and his assistant Otto Stern. 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 of the same year Einstein declared zero-point energy "dead as a doornail". Zero-point energy was also invoked by Peter Debye, who noted that zero-point energy of the atoms of a crystal lattice would cause a reduction in the intensity of the diffracted radiation in X-ray diffraction even as the temperature approached absolute zero. In 1916 Walther Nernst proposed that empty space was filled with zero-point electromagnetic radiation. With the development of general relativity Einstein found the energy density of the vacuum to contribute towards a cosmological constant in order to obtain static solutions to his field equations; the idea that empty space, or the vacuum, could have some intrinsic energy associated with it had returned, with Einstein stating in 1920:

There is a weighty argument to be adduced in favour of the aether hypothesis. To deny the aether is ultimately to assume that empty space has no physical qualities whatever. The fundamental facts of mechanics do not harmonize with this view ... according to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an aether. According to the general theory of relativity space without aether is unthinkable; for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time (measuring-rods and clocks), nor therefore any space-time intervals in the physical sense. But this aether may not be thought of as endowed with the quality characteristic of ponderable media, as consisting of parts which may be tracked through time. The idea of motion may not be applied to it.

Heisenberg, 1924

Kurt Bennewitz [de] and Francis Simon (1923), who worked at Walther Nernst's laboratory in Berlin, studied the melting process of chemicals at low temperatures. Their calculations of the melting points of hydrogen, argon 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 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 in Werner Heisenberg's article "Quantum theoretical re-interpretation of kinematic and mechanical relations" the zero-point energy was derived from quantum mechanics.

In 1913 Niels Bohr had proposed what is now called the Bohr model 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 in 1915: "There would be a very real difficulty in supposing that the (force) law ⁠1/r²⁠ held down to the zero values of r. For the force 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 be adequate to separate them. [...] Thus the matter in the universe would tend to shrink into nothing or to diminish indefinitely in size." The resolution to this puzzle came in 1926 when Erwin Schrödinger introduced the Schrödinger 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 published the first attempt to quantize the electromagnetic field. In a joint paper with Max Born and Werner Heisenberg 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 renormalization, in quantum field theory".

Paul Dirac, 1933

Building on the work of Heisenberg and others, Paul Dirac'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. Dirac described the quantization of the electromagnetic field as an ensemble of harmonic oscillators with the introduction of the concept of creation and annihilation operators 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:

The light-quantum has the peculiarity that it apparently ceases to exist when it is in one of its stationary states, namely, the zero state, in which its momentum and therefore also its energy, are zero. When a light-quantum is absorbed it can be considered to jump into this zero state, and when one is emitted it can be considered to jump from the zero state to one in which it is physically in evidence, so that it appears to have been created. Since there is no limit to the number of light-quanta that may be created in this way, we must suppose that there are an infinite number of light quanta in the zero state ...

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 who in 1935 wrote:

From quantum theory there follows the existence of so called zero-point oscillations; for example each oscillator in its lowest state is not completely at rest but always is moving about its equilibrium position. Therefore electromagnetic oscillations also can never cease completely. Thus the quantum nature of the electromagnetic field has as its consequence zero point oscillations of the field strength in the lowest energy state, in which there are no light quanta in space ... The zero point oscillations act on an electron in the same way as ordinary electrical oscillations do. They can change the eigenstate of the electron, but only in a transition to a state with the lowest energy, since empty space can only take away energy, and not give it up. In this way spontaneous radiation arises as a consequence of the existence of these unique field strengths corresponding to zero point oscillations. Thus spontaneous radiation is induced radiation of light quanta produced by zero point oscillations of empty space

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 (QED), predicted a fluctuating zero-point or "vacuum" field existing even in the absence of sources.

Throughout the 1940s improvements in microwave technology made it possible to take more precise measurements of the shift of the levels of a hydrogen atom, now known as the Lamb shift, and measurement of the magnetic moment of the electron. Discrepancies between these experiments and Dirac's theory led to the idea of incorporating renormalization into QED to deal with zero-point infinities. Renormalization was originally developed by Hans Kramers and also Victor Weisskopf (1936), and first successfully applied to calculate a finite value for the Lamb shift by Hans Bethe (1947). As per spontaneous emission, these effects can in part be understood with interactions with the zero-point field. But in light of renormalization 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 eliminated. In Wolfgang Pauli'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".

Hendrik Casimir (1958)

In 1948 Hendrik Casimir 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. 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 – 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 in 1930,did not properly explain the experimental measurements on colloids. Overbeek therefore asked Casimir to investigate the problem. Working with Dirk Polder, 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 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 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 (FDT) which was originally formulated in classical form by Nyquist (1928) as an explanation for observed Johnson noise 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 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). 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 (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) and has been experimentally verified. This phenomenon can be understood, partly, in terms of the action of the vacuum field on the atom.

Uncertainty principle

Main article: Uncertainty principle

Zero-point 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 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, the Hamiltonian of a general system (the quantum-mechanical operator giving its energy) can be approximated as a quantum harmonic oscillator, $${\displaystyle \hat{H}=V_0+\frac{1}{2}k{\left(\hat{x}-x_0\right)}^2+\frac{1}{2m}{\hat{p}}^2,}$$ where V₀ is the minimum of the classical potential well.

The uncertainty principle tells us that $${\displaystyle \sqrt{⟨{\left(\hat{x}-x_0\right)}^2⟩}\sqrt{⟨{\hat{p}}^2⟩}\ge \frac{\hslash }{2},}$$ making the expectation values of the kinetic and potential terms above satisfy $${\displaystyle ⟨\frac{1}{2}k{\left(\hat{x}-x_0\right)}^2⟩⟨\frac{1}{2m}{\hat{p}}^2⟩\ge {\left(\frac{\hslash }{4}\right)}^2\frac{k}{m}.}$$

The expectation value of the energy must therefore be at least

$${\displaystyle ⟨\hat{H}⟩\ge V_0+\frac{\hslash }{2}\sqrt{\frac{k}{m}}=V_0+\frac{\hslash \omega }{2}}$$

where ω = √k/m 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 E₀ = V₀ + ⁠ħω/2⁠, requires solving for the ground state of the system.

Atomic physics

Main article: Ground state

The zero-point energy E = ⁠ħω/2⁠ causes the ground-state of a harmonic oscillator to advance its phase (color). This has measurable effects when several eigenstates are superimposed.

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 ν above, using angular frequency, denoted with ω and defined by ω = 2πν. This leads to a convention of writing the Planck constant h with a bar through its top (ħ) to denote the quantity ⁠h/2π⁠. In these terms, an example of zero-point energy is the above E = ⁠ħω/2⁠ 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. Many systems have degenerate ground states. Degeneracy occurs whenever there exists a unitary operator which acts non-trivially on a ground state and commutes with the Hamiltonian of the system.

According to the third law of thermodynamics, a system at absolute zero temperature 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, 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 temperature for systems that exhibit negative temperature.

The wave function of the ground state of a 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: $${\displaystyle \frac{h^2{n}^2}{8m{L}^2}}$$ where h is the Planck constant, m is the mass of the particle, n is the energy state (n = 1 corresponds to the ground-state energy), and L is the width of the well.

Quantum field theory

Quantum field theory
Feynman diagram

Main articles: Vacuum expectation value, Vacuum energy, and Vacuum state

In quantum field theory (QFT), the fabric of "empty" space is visualized as consisting of 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 (e.g. electrons and quarks), force fields whose quanta are bosons (i.e. photons and gluons) 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 is called the vacuum energy and the average expectation value of the Hamiltonian is called the vacuum expectation value (also called condensate or simply VEV). The QED vacuum is a part of the vacuum state which specifically deals with quantum electrodynamics (e.g. electromagnetic interactions between photons, electrons and the vacuum) and the QCD vacuum deals with quantum chromodynamics (e.g. color charge 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, 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 E = ⁠ħω/2⁠, 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 cosmology, the vacuum energy is one possible explanation for the cosmological constant and the source of dark energy.

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 claimed it is, like a 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 Heisenberg uncertainty principle 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 vacuum fluctuations, or the zero-point energy to the particle masses.

Quantum electrodynamic vacuum

Main article: QED vacuum

The oldest and best known quantized force field is the electromagnetic field. Maxwell's equations have been superseded by quantum electrodynamics (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 α and α* are replaced by operators a and a^† that satisfy: $${\displaystyle \left[a,a^{†}\right]=1}$$

The classical quantity |α|² appearing in the classical expression for the energy of a field mode is replaced in quantum theory by the photon number operator a^†a. The fact that: $${\displaystyle \left[a,a^{†}a\right]\ne 1}$$ 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 a^†a and a. 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" a^† and a associated with these classical modes.

The zero-point energy of the field arises formally from the non-commutativity of a and a^†. This is true for any harmonic oscillator: the zero-point energy ⁠ħω/2⁠ appears when we write the Hamiltonian: $${\displaystyle \begin{array}{rl}{H}_{cl}& =\frac{p^2}{2m}+\frac{1}{2}m{\omega }^2{q}^2\\ & =\frac{1}{2}\hslash \omega \left(a{a}^{†}+a^{†}a\right)\\ & =\hslash \omega \left(a^{†}a+\frac{1}{2}\right)\end{array}}$$

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:$${\displaystyle \begin{array}{rl}{H}_F-⟨0|H_F|0⟩& =\frac{1}{2}\hslash \omega \left(a{a}^{†}+a^{†}a\right)-\frac{1}{2}\hslash \omega \\ & =\hslash \omega \left(a^{†}a+\frac{1}{2}\right)-\frac{1}{2}\hslash \omega \\ & =\hslash \omega a^{†}a\end{array}}$$ without affecting any physical predictions of the theory. The new Hamiltonian is said to be normally ordered (or Wick ordered) and is denoted by a double-dot symbol. The normally ordered Hamiltonian is denoted :H_F, i.e.: $${\displaystyle :H_F:\equiv \hslash \omega \left(a{a}^{†}+a^{†}a\right):\equiv \hslash \omega a^{†}a}$$

In other words, within the normal ordering symbol we can commute a and a^†. Since zero-point energy is intimately connected to the non-commutativity of a and a^†, 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 a and a^† 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 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.

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: $${\displaystyle \begin{array}{rl}{H}_F& =\frac{1}{8\pi }\int d^{3}r\left({\mathbf{E}}^2+{\mathbf{B}}^2\right)\\ & =\frac{k^2}{2\pi }|\alpha (t){|}^2\end{array}}$$

We introduce the "mode function" A₀(r) that satisfies the Helmholtz equation: $${\displaystyle \left({\mathrm{\nabla }}^2+k^2\right){\mathbf{A}}_0(\mathbf{r})=0}$$ where k = ⁠ω/c⁠ and assume it is normalized such that: $${\displaystyle \int d^{3}r{\left|{\mathbf{A}}_0(\mathbf{r})\right|}^2=1}$$

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 |A₀(r)|² should be independent of r for each mode of the field. The mode function satisfying these conditions is: $${\displaystyle {\mathbf{A}}_0(\mathbf{r})=e_{\mathbf{k}}{e}^{i\mathbf{k}\cdot \mathbf{r}}}$$ where k · e_k = 0 in order to have the transversality condition ∇ · A(r,t) satisfied for the Coulomb gauge in which we are working.

To achieve the desired normalization we pretend space is divided into cubes of volume V = L³ and impose on the field the periodic boundary condition: $${\displaystyle \mathbf{A}(x+L,y+L,z+L,t)=\mathbf{A}(x,y,z,t)}$$ or equivalently $${\displaystyle \left(k_x,k_y,k_z\right)=\frac{2\pi }{L}\left(n_x,n_y,n_z\right)}$$ where n 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: $${\displaystyle {\mathbf{A}}_{\mathbf{k}}(\mathbf{r})=\frac{1}{\sqrt{V}}{e}_{\mathbf{k}}{e}^{i\mathbf{k}\cdot \mathbf{r}}}$$ which satisfies the Helmholtz equation, transversality, and the "box normalization": $${\displaystyle {\int }_V{d}^{3}r{\left|{\mathbf{A}}_{\mathbf{k}}(\mathbf{r})\right|}^2=1}$$ where e_k is chosen to be a unit vector which specifies the polarization of the field mode. The condition k · e_k = 0 means that there are two independent choices of e_k, which we call e_k1 and e_k2 where e_k1 · e_k2 = 0 and e²
_k1 = e²
_k2 = 1. Thus we define the mode functions: $${\displaystyle {\mathbf{A}}_{\mathbf{k}\lambda }(\mathbf{r})=\frac{1}{\sqrt{V}}{e}_{\mathbf{k}\lambda }{e}^{i\mathbf{k}\cdot \mathbf{r}},\lambda =\{\begin{array}{l}1\\ 2\end{array}}$$ in terms of which the vector potential becomes: $${\displaystyle {\mathbf{A}}_{\mathbf{k}\lambda }(\mathbf{r},t)=\sqrt{\frac{2\pi \hslash c^2}{{\omega }_{k}V}}\left[a_{\mathbf{k}\lambda }(0)e^{i\mathbf{k}\cdot \mathbf{r}}+a_{\mathbf{k}\lambda }^{†}(0)e^{-i\mathbf{k}\cdot \mathbf{r}}\right]e_{\mathbf{k}\lambda }}$$ or: $${\displaystyle {\mathbf{A}}_{\mathbf{k}\lambda }(\mathbf{r},t)=\sqrt{\frac{2\pi \hslash c^2}{{\omega }_{k}V}}\left[a_{\mathbf{k}\lambda }(0)e^{-i({\omega }_{k}t-\mathbf{k}\cdot \mathbf{r})}+a_{\mathbf{k}\lambda }^{†}(0)e^{i({\omega }_{k}t-\mathbf{k}\cdot \mathbf{r})}\right]}$$ where ω_k = kc and a_kλ, a^†
_kλ are photon annihilation and creation operators for the mode with wave vector k and polarization λ. This gives the vector potential for a plane wave mode of the field. The condition for (k_x, k_y, k_z) shows that there are infinitely many such modes. The linearity of Maxwell's equations allows us to write: $${\displaystyle \mathbf{A}(\mathbf{r}t)=\sum _{\mathbf{k}\lambda }\sqrt{\frac{2\pi \hslash c^2}{{\omega }_{k}V}}\left[a_{\mathbf{k}\lambda }(0)e^{i\mathbf{k}\cdot \mathbf{r}}+a_{\mathbf{k}\lambda }^{†}(0)e^{-i\mathbf{k}\cdot \mathbf{r}}\right]e_{\mathbf{k}\lambda }}$$ for the total vector potential in free space. Using the fact that: $${\displaystyle {\int }_V{d}^{3}r{\mathbf{A}}_{\mathbf{k}\lambda }(\mathbf{r})\cdot {\mathbf{A}}_{{\mathbf{k}}^{\prime }{\lambda }^{\prime }}^{\ast }(\mathbf{r})={\delta }_{\mathbf{k},{\mathbf{k}}^{\prime }}^3{\delta }_{\lambda ,{\lambda }^{\prime }}}$$ we find the field Hamiltonian is: $${\displaystyle H_F=\sum _{\mathbf{k}\lambda }\hslash {\omega }_k\left(a_{\mathbf{k}\lambda }^{†}{a}_{\mathbf{k}\lambda }+\frac{1}{2}\right)}$$

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: $${\displaystyle \begin{array}{rl}\left[a_{\mathbf{k}\lambda }(t),a_{{\mathbf{k}}^{\prime }{\lambda }^{\prime }}^{†}(t)\right]& ={\delta }_{\mathbf{k},{\mathbf{k}}^{\prime }}^3{\delta }_{\lambda ,{\lambda }^{\prime }}\\ \left[a_{\mathbf{k}\lambda }(t),a_{{\mathbf{k}}^{\prime }{\lambda }^{\prime }}(t)\right]& =\left[a_{\mathbf{k}\lambda }^{†}(t),a_{{\mathbf{k}}^{\prime }{\lambda }^{\prime }}^{†}(t)\right]=0\end{array}}$$

Clearly the least eigenvalue for H_F is: $${\displaystyle \sum _{\mathbf{k}\lambda }\frac{1}{2}\hslash {\omega }_k}$$

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 $${\displaystyle \frac{8\pi v^{2}dv}{c^3}V}$$ shows. The summation becomes approximately the integral: $${\displaystyle \frac{4\pi hV}{c^3}\int v^{3}dv}$$ for high values of v. It diverges proportional to v⁴ for large v.

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 V 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 n_kλ = 0 for all modes (k, λ). 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 Σ_kλ ⁠ħω_k/2⁠ is infinite. We can make the replacement: $${\displaystyle \sum _{\mathbf{k}\lambda }⟶\sum _{\lambda }{\left(\frac{1}{2\pi }\right)}^3\int d^{3}k=\frac{V}{8{\pi }^3}\sum _{\lambda }\int d^{3}k}$$ the zero-point energy density is: $${\displaystyle \begin{array}{rl}\frac{1}{V}\sum _{\mathbf{k}\lambda }\frac{1}{2}\hslash {\omega }_k& =\frac{2}{8{\pi }^3}\int d^{3}k\frac{1}{2}\hslash {\omega }_k\\ & =\frac{4\pi }{4{\pi }^3}\int dk{k}^2\left(\frac{1}{2}\hslash {\omega }_k\right)\\ & =\frac{\hslash }{2{\pi }^2{c}^3}\int d\omega {\omega }^3\end{array}}$$ or in other words the spectral energy density of the vacuum field: $${\displaystyle {\rho }_0(\omega )=\frac{\hslash {\omega }^3}{2{\pi }^2{c}^3}}$$

The zero-point energy density in the frequency range from ω₁ to ω₂ is therefore: $${\displaystyle {\int }_{{\omega }_1}^{{\omega }_2}d\omega {\rho }_0(\omega )=\frac{\hslash }{8{\pi }^2{c}^3}\left({\omega }_2^4-{\omega }_1^4\right)}$$

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: $${\displaystyle H=\frac{1}{2m}{\left(\mathbf{p}-\frac{e}{c}\mathbf{A}\right)}^2+\frac{1}{2}m{\omega }_0^2{\mathbf{x}}^2+H_F}$$

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 x and the canonical momentum p = mẋ +⁠eA/c⁠ of the oscillator are: $${\displaystyle \begin{array}{rl}\dot{\mathbf{x}}& =(i\hslash {)}^{-1}[\mathbf{x}.H]=\frac{1}{m}\left(\mathbf{p}-\frac{e}{c}\mathbf{A}\right)\\ \dot{\mathbf{p}}& =(i\hslash {)}^{-1}[\mathbf{p}.H]\begin{array}{rl}& =\frac{1}{2}\mathrm{\nabla }{\left(\mathbf{p}-\frac{e}{c}\mathbf{A}\right)}^2-m{\omega }_0^2\dot{\mathbf{x}}\\ & =-\frac{1}{m}\left[\left(\mathbf{p}-\frac{e}{c}\mathbf{A}\right)\cdot \mathrm{\nabla }\right]\left[-\frac{e}{c}\mathbf{A}\right]-\frac{1}{m}\left(\mathbf{p}-\frac{e}{c}\mathbf{A}\right)\times \mathrm{\nabla }\times \left[-\frac{e}{c}\mathbf{A}\right]-m{\omega }_0^2\dot{\mathbf{x}}\\ & =\frac{e}{c}(\dot{\mathbf{x}}\cdot \mathrm{\nabla })\mathbf{A}+\frac{e}{c}\dot{\mathbf{x}}\times \mathbf{B}-m{\omega }_0^2\dot{\mathbf{x}}\end{array}\end{array}}$$ or: $${\displaystyle \begin{array}{rl}m\ddot{\mathbf{x}}& =\dot{\mathbf{p}}-\frac{e}{c}\dot{\mathbf{A}}\\ & =-\frac{e}{c}\left[\dot{\mathbf{A}}-\left(\dot{\mathbf{x}}\cdot \mathrm{\nabla }\right)\mathbf{A}\right]+\frac{e}{c}\dot{\mathbf{x}}\times \mathbf{B}-m{\omega }_0^2\mathbf{x}\\ & =e\mathbf{E}+\frac{e}{c}\dot{\mathbf{x}}\times \mathbf{B}-m{\omega }_0^2\mathbf{x}\end{array}}$$ since the rate of change of the vector potential in the frame of the moving charge is given by the convective derivative $${\displaystyle \dot{\mathbf{A}}=\frac{\mathrm{\partial }\mathbf{A}}{\mathrm{\partial }t}+(\dot{\mathbf{x}}\cdot \mathrm{\nabla }){\mathbf{A}}^3.}$$

For nonrelativistic motion we may neglect the magnetic force and replace the expression for mẍ by: $${\displaystyle \begin{array}{rl}\ddot{\mathbf{x}}+{\omega }_0^2\mathbf{x}& \approx \frac{e}{m}\mathbf{E}\\ & \approx \sum _{\mathbf{k}\lambda }\sqrt{\frac{2\pi \hslash {\omega }_k}{V}}\left[a_{\mathbf{k}\lambda }(t)+a_{\mathbf{k}\lambda }^{†}(t)\right]e_{\mathbf{k}\lambda }\end{array}}$$

Above we have made the electric dipole approximation in which the spatial dependence of the field is neglected. The Heisenberg equation for a_kλ is found similarly from the Hamiltonian to be: $${\displaystyle {\dot{a}}_{\mathbf{k}\lambda }=i{\omega }_k{a}_{\mathbf{k}\lambda }+ie\sqrt{\frac{2\pi }{\hslash {\omega }_{k}V}}\dot{\mathbf{x}}\cdot e_{\mathbf{k}\lambda }}$$ in the electric dipole approximation.

In deriving these equations for x, p, and a_kλ 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, t = 0) when the matter-field interpretation is presumed to begin, together with the fact that a Heisenberg-picture operator A(t) evolves in time as A(t) = U^†(t)A(0)U(t), where U(t) is the time evolution operator satisfying $${\displaystyle i\hslash \dot{U}=HU,U^{†}(t)=U^{-1}(t),U(0)=1.}$$

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: $${\displaystyle a_{\mathbf{k}\lambda }(t)=a_{\mathbf{k}\lambda }(0)e^{-i{\omega }_{k}t}+ie\sqrt{\frac{2\pi }{\hslash {\omega }_{k}V}}{\int }_0^{t}d{t}^{\prime }{e}_{\mathbf{k}\lambda }\cdot \dot{\mathbf{x}}(t^{\prime })e^{i{\omega }_k\left(t^{\prime }-t\right)}}$$ and therefore the equation for ȧ_kλ may be written: $${\displaystyle \ddot{\mathbf{x}}+{\omega }_0^2\mathbf{x}=\frac{e}{m}{\mathbf{E}}_0(t)+\frac{e}{m}{\mathbf{E}}_{RR}(t)}$$ where $${\displaystyle {\mathbf{E}}_0(t)=i\sum _{\mathbf{k}\lambda }\sqrt{\frac{2\pi \hslash {\omega }_k}{V}}\left[a_{\mathbf{k}\lambda }(0)e^{-i{\omega }_{k}t}-a_{\mathbf{k}\lambda }^{†}(0)e^{i{\omega }_{k}t}\right]e_{\mathbf{k}\lambda }}$$ and $${\displaystyle {\mathbf{E}}_{RR}(t)=-\frac{4\pi e}{V}\sum _{\mathbf{k}\lambda }{\int }_0^{t}d{t}^{\prime }\left[e_{\mathbf{k}\lambda }\cdot \dot{\mathbf{x}}\left(t^{\prime }\right)\right]\cos {\omega }_k\left(t^{\prime }-t\right)}$$

It can be shown that in the radiation reaction field, if the mass m is regarded as the "observed" mass then we can take $${\displaystyle {\mathbf{E}}_{RR}(t)=\frac{2e}{3c^3}\ddot{\mathbf{x}}}$$

The total field acting on the dipole has two parts, E₀(t) and E_RR(t). E₀(t) 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 $${\displaystyle \left[{\mathrm{\nabla }}^2-\frac{1}{c^2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}\right]\mathbf{E}=0}$$ satisfied by the field in the (source free) vacuum. For this reason E₀(t) 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 t = 0. E_RR(t) is the source field, the field generated by the dipole and acting on the dipole.

Using the above equation for E_RR(t) we obtain an equation for the Heisenberg-picture operator ${\displaystyle \mathbf{x}(t)}$ that is formally the same as the classical equation for a linear dipole oscillator: $${\displaystyle \ddot{\mathbf{x}}+{\omega }_0^2\mathbf{x}-\tau \stackrel{\mathbf{.}\mathbf{.}\mathbf{.}}{\mathbf{x}}=\frac{e}{m}{\mathbf{E}}_0(t)}$$ where τ = ⁠2e²/3mc³⁠. 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 E₀(t).

According to our earlier equation for a_kλ(t) the free field is the only field in existence at t = 0 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 t = 0 is therefore of the form $${\displaystyle |\mathrm{\Psi }⟩=|\text{vac}⟩|{\psi }_D⟩,}$$ where |vac⟩ is the vacuum state of the field and |ψ_D⟩ is the initial state of the dipole oscillator. The expectation value of the free field is therefore at all times equal to zero: $${\displaystyle ⟨{\mathbf{E}}_0(t)⟩=⟨\mathrm{\Psi }|{\mathbf{E}}_0(t)|\mathrm{\Psi }⟩=0}$$ since a_kλ(0)|vac⟩ = 0. however, the energy density associated with the free field is infinite: $${\displaystyle \begin{array}{rl}\frac{1}{4\pi }⟨{\mathbf{E}}_0^2(t)⟩& =\frac{1}{4\pi }\sum _{\mathbf{k}\lambda }\sum _{{\mathbf{k}}^{\prime }{\lambda }^{\prime }}\sqrt{\frac{2\pi \hslash {\omega }_k}{V}}\sqrt{\frac{2\pi \hslash {\omega }_{k^{\prime }}}{V}}\times ⟨a_{\mathbf{k}\lambda }(0)a_{{\mathbf{k}}^{\prime }{\lambda }^{\prime }}^{†}(0)⟩\\ & =\frac{1}{4\pi }\sum _{\mathbf{k}\lambda }\left(\frac{2\pi \hslash {\omega }_k}{V}\right)\\ & ={\int }_0^{\mathrm{\infty }}dw{\rho }_0(\omega )\end{array}}$$

The important point of this is that the zero-point field energy H_F does not affect the Heisenberg equation for a_kλ since it is a c-number or constant (i.e. an ordinary number rather than an operator) and commutes with a_kλ. 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 Σ_kλ ⁠ħω_k/2⁠ 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: $${\displaystyle \begin{array}{rl}\left[z(t),p_z(t)\right]& =\left[U^{†}(t)z(0)U(t),U^{†}(t)p_z(0)U(t)\right]\\ & =U^{†}(t)\left[z(0),p_z(0)\right]U(t)\\ & =i\hslash U^{†}(t)U(t)\\ & =i\hslash \end{array}}$$

We can calculate [z(t),p_z(t)] from the formal solution of the operator equation of motion $${\displaystyle \ddot{\mathbf{x}}+{\omega }_0^2\mathbf{x}-\tau \stackrel{\mathbf{.}\mathbf{.}\mathbf{.}}{\mathbf{x}}=\frac{e}{m}{\mathbf{E}}_0(t)}$$

Using the fact that $${\displaystyle \left[a_{\mathbf{k}\lambda }(0),a_{{\mathbf{k}}^{\prime }{\lambda }^{\prime }}^{†}(0)\right]={\delta }_{\mathbf{k}{\mathbf{k}}^{\prime }}^3,{\delta }_{\lambda {\lambda }^{\prime }}}$$ and that equal-time particle and field operators commute, we obtain: $${\displaystyle \begin{array}{rl}& =\left[z(t),m\dot{z}(t)\right]+\left[z(t),\frac{e}{c}{A}_z(t)\right]\\ & =\left[z(t),m\dot{z}(t)\right]\\ & =\left(\frac{i\hslash e^2}{2{\pi }^{2}m{c}^3}\right)\left(\frac{8\pi }{3}\right){\int }_0^{\mathrm{\infty }}\frac{d\omega {\omega }^4}{{\left({\omega }^2-{\omega }_0^2\right)}^2+{\tau }^2{\omega }^6}\end{array}}$$

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., τω₀ ≪ 1. Then the integrand above is sharply peaked at ω = ω₀ and: $${\displaystyle \begin{array}{rl}\left[z(t),p_z(t)\right]& \approx \frac{2i\hslash e^2}{3\pi m{c}^3}{\omega }_0^3{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{dx}{x^2+{\tau }^2{\omega }_0^6}\\ & =\left(\frac{2i\hslash e^2{\omega }_0^3}{3\pi m{c}^3}\right)\left(\frac{\pi }{\tau {\omega }_0^3}\right)\\ & =i\hslash \end{array}}$$ the necessity of the vacuum field can also be appreciated by making the small damping approximation in $${\displaystyle \begin{array}{rl}& \ddot{\mathbf{x}}+{\omega }_0^2\mathbf{x}-\tau \stackrel{\mathbf{.}\mathbf{.}\mathbf{.}}{\mathbf{x}}=\frac{e}{m}{\mathbf{E}}_0(t)\\ & \ddot{\mathbf{x}}\approx -{\omega }_0^2\mathbf{x}(t)& & \stackrel{\mathbf{.}\mathbf{.}\mathbf{.}}{\mathbf{x}}\approx -{\omega }_0^2\dot{\mathbf{x}}\end{array}}$$ and $${\displaystyle \ddot{\mathbf{x}}+\tau {\omega }_0^2\dot{\mathbf{x}}+{\omega }_0^2\mathbf{x}\approx \frac{e}{m}{\mathbf{E}}_0(t)}$$

Without the free field E₀(t) in this equation the operator x(t) would be exponentially dampened, and commutators like [z(t),p_z(t)] would approach zero for t ≫ ⁠1/τω²
₀⁠. With the vacuum field included, however, the commutator is iħ 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 x, the spectral energy density of the vacuum field must be proportional to the third power of ω in order for [z(t),p_z(t)] to hold. In the case of a dissipative force proportional to ẋ, by contrast, the fluctuation force must be proportional to ${\displaystyle \omega }$ 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.

Quantum chromodynamic vacuum

Main article: QCD vacuum

The QCD vacuum is the vacuum state of quantum chromodynamics (QCD). It is an example of a non-perturbative vacuum state, characterized by a non-vanishing 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 gauge bosons that mediate strong interactions of quarks in quantum chromodynamics (QCD). Gluons themselves carry the color charge of the strong interaction. This is unlike the photon, which mediates the 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) as it deals with nonlinear equations to characterize such interactions.

Higgs field

Main article: Higgs mechanism

The potential for the Higgs field, plotted as function of ϕ⁰ and ϕ³. It has a Mexican-hat or champagne-bottle profile at the ground.

The Standard Model hypothesises a field called the Higgs field (symbol: ϕ), 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 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 ϕ⁰ in the ground state (the vacuum expectation value or VEV) is then ⟨ϕ⁰⟩ = ⁠v/√2⁠, where v = ⁠|μ|/√λ⁠. The measured value of this parameter is approximately 246 GeV/c². 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. 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

Main article: Casimir effect

Casimir forces on parallel plates

A phenomenon that is commonly presented as evidence for the existence of zero-point energy in vacuum is the Casimir effect, proposed in 1948 by Dutch physicist Hendrik Casimir, who considered the quantized electromagnetic field 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 c between two closely spaced conducting plates.

Lamb shift

Main article: Lamb shift

Fine structure of energy levels in hydrogen – relativistic corrections to the Bohr model

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 ²S_⁠1/2⁠ and ²P_⁠1/2⁠ (in term symbol notation) of the hydrogen atom 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 4.38×10⁻⁶ eV is roughly 10⁻⁷ 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

Main article: Fine-structure constant

Taking ħ (the Planck constant divided by 2π), c (the speed of light), and e² = ⁠q²
_e/4πε₀⁠ (the electromagnetic coupling constant i.e. a measure of the strength of the electromagnetic force (where q_e is the absolute value of the electronic charge and ${\displaystyle {\epsilon }_0}$ is the vacuum permittivity)) we can form a dimensionless quantity called the fine-structure constant: $${\displaystyle \alpha =\frac{e^2}{\hslash c}=\frac{q_e^2}{4\pi {\epsilon }_0\hslash c}\approx \frac{1}{137}}$$

The fine-structure constant is the coupling constant of quantum electrodynamics (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 ħ = ⁠h/2π⁠, and Δ_x, Δ_p are the standard deviations of position and momentum states that: $${\displaystyle {\mathrm{\Delta }}_x{\mathrm{\Delta }}_p\ge \frac{1}{2}\hslash }$$

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 Z⁰ boson rest energy, m_zc² ≈ 90 GeV, that: $${\displaystyle \alpha \approx \frac{1}{129}}$$ rather than the low-energy α ≈ ⁠1/137⁠.  The renormalization procedure of eliminating zero-point energy infinities allows the choice of an arbitrary energy (or distance) scale for defining α. All in all, α depends on the energy scale characteristic of the process under study, and also on details of the renormalization procedure. The energy dependence of α has been observed for several years now in precision experiment in high-energy physics.

Vacuum birefringence

Main articles: Lorentz-violating electrodynamics and Euler–Heisenberg Lagrangian

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. 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 and Hans Heinrich Euler in 1936 and independently the same year by Victor Weisskopf 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 birefringent medium, thus in principle a rotation of the polarization frame (the Faraday effect) can be observed in empty space.

Wide field view of the neutron star RX J1856.5-3754

The first concrete evidence for vacuum birefringence was published in 2017 when a team of astronomers looked at the light coming from the neutron 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 realized on a longer timescale." The team found that visible light from the star had a degree of polarization 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).

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 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 Lorentz violation by finding a rotation of the polarization plane of light coming from distant galaxies.

Speculated involvement in other phenomena

Dark energy

Main article: Dark energy

In the late 1990s it was discovered that very distant supernovae were dimmer than expected suggesting that the universe's expansion was accelerating rather than slowing down. This revived discussion that Einstein's cosmological constant, 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 pressure or energy.

There is no natural candidate for what might cause what has been called dark energy but the current best guess is that it is the zero-point energy of the vacuum, but this guess is known to be off by 120 orders of magnitude.

The European Space Agency's Euclid telescope, launched on 1 July 2023, 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, 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 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

Main article: Inflation (cosmology)

Unsolved problem in physics

Why does the observable universe have more matter than antimatter?

More unsolved problems in physics

Cosmic inflation is the phase of accelerated cosmic expansion just after the Big Bang. It explains the origin of the 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 asymmetry (imbalance) between baryons and antibaryons produced in the very early universe, but this is far from certain.

Cosmology

Paul S. Wesson examined the cosmological implications of assuming that zero-point energy is real. Among numerous difficulties, general relativity requires that such energy not gravitate, so it cannot be similar to electromagnetic radiation.

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? 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". 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."

Milonni has shown the necessity of the vacuum field for the formal consistency of QED. 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 unified theory. Nevertheless, as pointed out by Jaffe, "no known phenomenon, including the Casimir effect, demonstrates that zero point energies are "real""

Purported applications

Physicists overwhelmingly reject any possibility that the zero-point energy field can be exploited to obtain useful energy (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. Others have since claimed to exploit zero-point or vacuum energy with a large amount of 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 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 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 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 1999, Pinto, a former scientist at NASA's 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 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."

Garret Moddel at University of Colorado has highlighted that he believes such devices hinge on the assumption that the Casimir force is a 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. 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 Callen and Welton proved the quantum fluctuation-dissipation theorem (FDT) which was originally formulated in classical form by Nyquist (1928) as an explanation for observed Johnson noise in electric circuits. That 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 is that the vacuum could be treated as a heat bath coupled to a dissipative force, and therefore energy could 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 would 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 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. 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 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 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, 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 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 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 fluxoid quantization. In 1983, Dewitt's work was substantially expanded by Ross.

Between 1971 and 1974 Henry William Wallace, a scientist at GE Aerospace was issued three patents. Wallace used 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 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 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 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 inflationary energy, dark energy or zero-point radiation of the vacuum.

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 and Douglas Torr published a number of articles about gravitational effects in superconductors. One finding they derived is the source of gravitomagnetic flux in a type II superconductor material is due to 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 U.S. Department of Defense grant for $448,970 in 2001 to continue anti-gravity research. The grant period ended in 2002 but no results from this research were 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 Russian government, 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 2009, Giorgio Fontana and Bernd Binder presented a new method to potentially extract the Zero-point energy of the electromagnetic field and nuclear forces in the form of gravitational waves. In the spheron model of the nucleus, proposed by the two times Nobel laureate Linus Pauling, dineutrons are among the components of this structure. Similarly to a dumbbell put in a suitable rotational state, but with nuclear mass density, dineutrons are nearly ideal sources of gravitational waves at X-ray and gamma-ray frequencies. The dynamical interplay, mediated by nuclear forces, between the electrically neutral dineutrons and the electrically charged core nucleus is the fundamental mechanism by which nuclear vibrations can be converted to a rotational state of dineutrons with emission of gravitational waves. Gravity and gravitational waves are well described by General Relativity, that is not a quantum theory, this implies that there is no Zero-point energy for gravity in this theory, therefore dineutrons will emit gravitational waves like any other known source of gravitational waves. In Fontana and Binder paper, nuclear species with dynamical instabilities, related to the Zero-point energy of the electromagnetic field and nuclear forces, and possessing dineutrons, will emit gravitational waves. In experimental physics this approach is still unexplored.

In 2014 NASA's Eagleworks Laboratories announced that they had successfully validated the use of a Quantum Vacuum Plasma Thruster which makes use of the Casimir effect 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 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 out these potential errors.

United States constitutional criminal procedure

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/United_States_constitutional_criminal_procedure 

The Warren Court (1953–1969) issued several landmark constitutional decisions concerning criminal procedure, including Gideon v. Wainwright (1963), Brady v. Maryland (1963), and Duncan v. Louisiana (1968).

The United States Constitution contains several provisions regarding the law of criminal procedure.

Petit jury and venue provisions—both traceable to enumerated complaints in the Declaration of Independence—are included in Article Three of the United States Constitution. More criminal procedure provisions are contained in the United States Bill of Rights, specifically the Fifth, Sixth, Seventh and Eighth Amendments. With the exception of the Grand Jury Clause of the Fifth Amendment, the Vicinage Clause of the Sixth Amendment, and (maybe) the Excessive Bail Clause of the Eighth Amendment, all of the criminal procedure provisions of the Bill of Rights have been incorporated to apply to the state governments.

Several of these rights regulate pre-trial procedure: access to a non-excessive bail, the right to indictment by a grand jury, the right to an information (charging document), the right to a speedy trial, and the right to be tried in a specific venue. Several of these rights are trial rights: the right to compulsory process for obtaining witnesses at trial, the right to confront witnesses at trial, the right to a public trial, the right to a trial by an impartial petit jury selected from a specific geography, and the right not to be compelled to testify against oneself. Others, such as the assistance of counsel and due process rights, have application throughout the proceeding.

If a defendant is convicted, the usual remedy for a violation of one of these provisions is reversal of the conviction or modification of the defendant's sentence. With the exception of structural errors (such as the total denial of counsel), constitutional errors are subject to harmless error analysis, although they must be harmless beyond a reasonable doubt. With the exception of a Double Jeopardy or Speedy Trial violation, the government will usually be permitted to retry the defendant. Pursuant to the Antiterrorism and Effective Death Penalty Act of 1996 (AEDPA), these provisions are the source of nearly all reviewable errors in federal habeas review of state convictions.

Relevant text

The U.S. Bill of Rights

Article Three, Section Two, Clause Three of the United States Constitution provides that:

Trial of all Crimes, except in Cases of Impeachment, shall be by Jury; and such Trial shall be held in the State where the said Crimes shall have been committed; but when not committed within any State, the Trial shall be at such Place or Places as the Congress may by Law have directed.

The Fifth Amendment to the United States Constitution provides, in relevant part, that:

No person shall be held to answer for a capital, or otherwise infamous crime, unless on a presentment or indictment of a Grand Jury, except in cases arising in the land or naval forces, or in the Militia, when in actual service in time of War or public danger; nor shall any person be subject for the same offense to be twice put in jeopardy of life or limb; nor shall be compelled in any criminal case to be a witness against himself, nor be deprived of life, liberty, or property, without due process of law . . . .

The Sixth Amendment to the United States Constitution provides that:

In all criminal prosecutions, the accused shall enjoy the right to a speedy and public trial, by an impartial jury of the State and district wherein the crime shall have been committed, which district shall have been previously ascertained by law, and to be informed of the nature and cause of the accusation; to be confronted with the witnesses against him; to have compulsory process for obtaining witnesses in his favor, and to have the Assistance of Counsel for his defence.

The Eighth Amendment to the United States Constitution provides, in relevant part, that:

Excessive bail shall not be required . . . .

The Fourteenth Amendment to the United States Constitution provides, in relevant part, that:

[N]or shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws.

History

The Supreme Court of the United States issued almost no constitutional criminal procedure decisions for its first century of existence. Professor Akhil Amar highlights two reasons for this. First, the Court's decision in Barron v. Baltimore (1833) meant that the federal constitution did not apply in state proceedings until the incorporation of the Bill of Rights after the Fourteenth Amendment. Second, the Court lacked general appellate jurisdiction over federal criminal cases until 1891.

The Marshall Court possessed jurisdiction in criminal cases only via writs of error from state courts, original writs of habeas corpus, and certificates of division from the circuit courts. In three cases involving certificates of division, the Marshall Court decided issues of double jeopardy, but did not clearly rely on the Double Jeopardy Clause. Similarly, the Marshall Court discussed the level of detail required for a sufficient indictment without explicitly citing the Information Clause of the Sixth Amendment.

In two appeals from state courts, the Taney Court considered, and rejected, double jeopardy claims arising from the hypothetical prospect of prosecution by the federal and state governments for the same conduct.

The first Supreme Court decisions to reverse state criminal convictions for constitutional procedural reasons involved the exclusion of African-Americans for grand and petit juries—Strauder v. West Virginia (1880), Virginia v. Rives (1880), Neal v. Delaware (1881), Carter v. Texas (1900), Rogers v. Alabama (1904), and Norris v. Alabama (1935)—and the conviction African-American defendants for crimes involving white victims in the southern states: by a mob-dominated trial, as in Moore v. Dempsey (1923); and without counsel, as in Powell v. Alabama (1932).

Pre-trial procedure

Bail

Main article: Excessive Bail Clause

U.S. Const. amend. VIII provides:

Excessive bail shall not be required . . . .

Stack v. Boyle (1951) is the only case in which the Supreme Court has held the bail imposed to have been constitutionally excessive. There, the Court found $50,000 to be excessive in relation to the flight risk for impecunious defendants charged under the Smith Act. In United States v. Salerno (1987), the Court upheld the Bail Reform Act of 1984, which authorized the consideration of future dangerousness in the determination of the amount of, or the denial of, bail.

The incorporation status of the Excessive Bail Clause is unclear. In Schilb v. Kuebel (1971), the Court stated in dicta: "Bail, of course, is basic to our system of law, and the Eighth Amendment's proscription of excessive bail has been assumed to have application to the States through the Fourteenth Amendment." In Murphy v. Hunt (1982), the Court did not reach the issue because the case was dismissed as moot. Bail was included in the list of incorporated rights in McDonald v. Chicago (2010), citing Schilb.

Grand Jury

Further information: Grand juries in the United States

A grand jury in 1913

U.S. Const. amend. V provides:

No person shall be held to answer for a capital, or otherwise infamous crime, unless on a presentment or indictment of a Grand Jury, except in cases arising in the land or naval forces, or in the Militia, when in actual service in time of War or public danger . . . .

The Grand Jury Clause applies only to capital and "otherwise infamous" crimes. Any crime "punishable by imprisonment in the penitentiary" is infamous. Only those convicted of felonies, i.e. crimes punishable by greater than one year of imprisonment, are confined to a penitentiary. Any crime punishable by hard labor, regardless of the term or place of imprisonment, is also infamous. Contempt of court, even if punished by greater than one year imprisonment, is not infamous. In Hurtado v. California (1884), the Supreme Court held that the Grand Jury Clause was not incorporated to apply to the states by the Fourteenth Amendment.

If the grand jury right attaches, every element of the charged crime must be submitted to the grand jury. Thus, the prosecution cannot augment the indictment without returning to a grand jury. But, the government may narrow the indictment without so returning.

The Grand Jury Clause does very little, if anything, to regulate the procedures of the grand jury. For example, the Clause does not prohibit a grand jury indictment based solely on hearsay evidence. Non-fundamental flaws with the grand jury, such as a violation of the defendant's self-incrimination rights or a violation of grand jury secrecy do not trigger a right not to be tried. In United States v. Williams (1992), where the Court rejected a rule that would have required "substantial exculpatory evidence" to be presented to the grand jury, the defendant did not even argue a Fifth Amendment violation. The lack of a grand jury does not deprive the court of jurisdiction, and the defendant may waive the grand jury right.

Information

U.S. Const. amend. VI provides:

In all criminal prosecutions, the accused shall enjoy the right . . . to be informed of the nature and cause of the accusation . . . .

A charging instrument is constitutionally sufficient under this clause (and under the Grand Jury Clause) if it (1) "contains the elements of the offense intended to be charged, and sufficiently apprises the defendant of what he must be prepared to meet," and (2) "shows with accuracy to what extent he may plead" double jeopardy in a subsequent prosecution. This right has been incorporated.

In a case submitted to a grand jury, the indictment must satisfy this requirement. In cases not required to be submitted to a grand jury, the formal charging instrument is referred to as an "information" (in the federal system and in some states) or a "complaint."

Speedy trial

Main article: Speedy Trial Clause

U.S. Const. amend. VI provides:

In all criminal prosecutions, the accused shall enjoy the right to a speedy . . . trial . . . .

The Speedy Trial Clause regulates delay between the bringing of a formal criminal charge and/or the pre-trial deprivation of the accused's liberty and the start of trial. The Clause has been incorporated to apply in state prosecutions.

In Barker v. Wingo (1972), the Supreme Court announced four factors relevant to the determination of a Speedy Trial Clause violation: (1) the length of the delay, (2) the reason for the delay, (3) whether the defendant demanded a speedy trial, and (4) prejudice. Applying Barker, the Court found such a violation in Doggett v. United States (1992), which involved an over eight-year period between indictment and arrest. The only possible remedy for a Speedy Trial Clause violation is dismissal with prejudice.

Venue

The Declaration of Independence accused King George III of "transporting us beyond Seas to be tried"

Further information: Venue (law)

U.S. Const. Art. III, § 2, cl. 3 provides:

Trial of all Crimes . . . shall be held in the State where the said Crimes shall have been committed; but when not committed within any State, the Trial shall be at such Place or Places as the Congress may by Law have directed.

The perceived abuse of English criminal venue law was one of the enumerated grievances in the United States Declaration of Independence, which accused George III of the United Kingdom of "transporting us beyond Seas to be tried for pretended offenses."

The "where the said Crimes shall have been committed" language refers to the locus delicti. "[T]he locus delicti must be determined from the nature of the crime alleged and the location of the act or acts constituting it." Thus, a single crime may often give rise to several constitutionally permissible venues, and venue may be constitutionally permissible even if an individual defendant was never personally present in the relevant state. For example, conspiracy may be prosecuted wherever the agreement occurred or wherever any overt act was committed.

For the purposes of constitutional venue, the boundaries of the states are questions of law to be determined by the judge, but the location of the crime is a question of fact to be determined by the jury.

The venue provision of Article III (regulating the location of the trial) is distinct from the Vicinage Clause of the Sixth Amendment (regulating the geography from which the jury pool is selected). The unit of the former is the state; the unit of the later is the state and judicial district. Unlike judicial districts under the Vicinage Clause, consistent with Article III, Congress may "provide a place of trial where none was provided when the offense was committed, or change the place of trial after the commission of the offense."

Trial procedure

Compulsory process

Main article: Compulsory Process Clause

U.S. Const. amend. VI provides:

In all criminal prosecutions, the accused shall enjoy the right . . . to have compulsory process for obtaining witnesses in his favor . . . .

The Compulsory Process Clause guarantees the defendant the right to obtain favorable witnesses at trial. For example, the Clause prevents a jurisdiction from precluding defendants from calling their codefendants as witnesses. Similarly, the Clause prevents the government from deporting a witness whose testimony would have been both material and favorable to the defense. The right does not pre-empt reasonable procedural rules. Thus, the right does not prevent the preclusion of defense witnesses as a discovery sanction.

Confrontation

Main article: Confrontation Clause

Crawford v. Washington (2004) referred to Sir Walter Raleigh's (pictured) inability to cross-examine Henry Brooke, 11th Baron Cobham as one of the "most notorious instances of civil-law examination."

U.S. Const. amend. VI provides:

In all criminal prosecutions, the accused shall enjoy the right . . . to be confronted with the witnesses against him . . . .

In Crawford v. Washington (2004), the Supreme Court held that the Confrontation Clause bars the "admission of testimonial statements of a witness who did not appear at trial" unless pursuant to one of the "exceptions established at the time of the founding." "[W]hen the declarant appears for cross-examination at trial, the Confrontation Clause places no constraints at all on the use of his prior testimonial statements . . . so long as the declarant is present at trial to defend or explain it." In Davis v. Washington (2006), the Court held that the Clause places no restrictions on nontestimonial statements.

Crawford did not completely define the term "testimonial." But, Crawford held that, "[w]hatever else the term covers, it applies at a minimum to prior testimony at a preliminary hearing, before a grand jury, or at a former trial; and to police interrogations." Laboratory reports of forensic tests are also testimonial, conferring on the defendant a right to cross-examine the analyst who certifies them.

Statements made during police interrogation are nontestimonial if circumstances objectively indicate "that the primary purpose of the interrogation is to enable police assistance to meet an ongoing emergency" but are testimonial if circumstances objective indicate "that there is no such ongoing emergency, and that the primary purpose of the interrogation is to establish or prove past events potentially relevant to later criminal prosecution." "[T]he relevant inquiry is not the subjective or actual purpose of the individuals involved in a particular encounter, but rather the purpose that reasonable participants would have had, as ascertained from the individuals' statements and actions and the circumstances in which the encounter occurred."

One exception established at the founding is if the witness is "unavailable to testify, and the defendant had had a prior opportunity for cross-examination." Another such exception is "forfeiture by wrongdoing," i.e. where the defendant intends to obtain and obtains the absence of the witness by wrongdoing. Still another exception is "the use of testimonial statements for purposes other than establishing the truth of the matter asserted." Another possible exception is for dying declarations, i.e. statements made by a speaker on the brink of death while aware that he or she is dying.

Petit jury, impartiality, and vicinage

Further information: Juries in the United States § Petit jury

An empty jury box

U.S. Const. Art. III, § 2, cl. 3 provides:

Trial of all Crimes, except in Cases of Impeachment, shall be by Jury . . . .

U.S. Const. amend. VI provides:

In all criminal prosecutions, the accused shall enjoy the right to a . . . trial, by an impartial jury of the State and district wherein the crime shall have been committed, which district shall have been previously ascertained by law . . . .

One of the enumerated complaints in the Declaration of Independence accused King George III of "depriving us, in many Cases, of the Benefits of Trial by Jury."

Availability

Depending on the authorized and actual sentence, upon demand, a criminal defendant has a right to trial by jury. The defendant does not have a right, conversely, to a bench trial without the consent of the prosecution. If the defendant is charged with crimes for which the authorized sentence exceeds six months, whether in state or federal court, the defendant has a right to a jury. Further, the defendant has a right to a trial by jury if the actual sentence exceeds six months and the charged crime has no maximum authorized sentence (e.g. contempt of court).

But, the defendant does not have a right to a jury in stacked misdemeanor prosecutions, even if the cumulative authorized imprisonment exceeds six months, as long as the actual sentence does not. Factors other than actual and authorized sentences may be relevant to seriousness, but so far the Court has pushed back against expanding the jury right.

Impartiality

The trial judge has an obligation to ensure an impartial jury, especially vis-a-vis juror biases and media coverage by such means as jury selection (including voir dire and for-cause challenges), jury sequestration, and jury instructions. For example, this may require the court to permit voir dire on the subject of the juror's potential racial prejudice. In some circumstances, the Sixth Amendment even requires the trial judge to grant a defendant's change of venue motion if an impartial jury cannot be obtained otherwise.

The Sixth Amendment also regulates the availability and use of cause and peremptory challenges. For example, it precludes a jurisdiction from granting the prosecution for-cause removal of jurors who oppose the death penalty. "The most that can be demanded of a venireman in this regard is that he be willing to consider all of the penalties provided by state law, and that he not be irrevocably committed, before the trial has begun, to vote against the penalty of death regardless of the facts and circumstances that might emerge in the course of the proceedings." While a defendant is not obliged to use peremptory challenges to cure a trial court's erroneous denial of a defendant's for-cause challenge, if the defendant does so, the defendant may not rely on the error for automatic reversal.

Size and unanimity

The Supreme Court has held that six-member juries are sufficient and that five-member juries are not. In Ramos v. Louisiana, 590 U.S. ___ (2020), the Supreme Court overturned Apodaca v. Oregon, 406 U.S. 404 (1972), and held that all jury verdicts resulting in a conviction must be unanimous.

Vicinage

Main article: Vicinage Clause

The provision requiring that the jury be drawn "of the State and district wherein the crime shall have been committed, which district shall have been previously ascertained by law" is known as the Vicinage Clause. The Vicinage Clause places no limits on the prosecution of crimes not committed within a state. Nor does the Clause prevent a crime from being tried by a jury from a different division (a subset of a federal judicial district) within the same district in which the crime was committed. The Third, Fifth, and Sixth Circuits have held that the Vicinage Clause was not incorporated against the states by the Fourteenth Amendment.

Public trial

Further information: Public trial

A courtroom sketch, a common component of media coverage of trials

U.S. Const. amend. VI provides:

In all criminal prosecutions, the accused shall enjoy the right to a . . . public trial . . . .

The defendant has a right to have the courtroom open to the public, absent a showing of a substantial government interest that cannot be addressed by alternatives other than closure. The right to a public trial extends to pre-trial matters such as a suppression hearing and jury selection. The Public Trial Clause has its roots in the "traditional Anglo-American distrust for secret trials has been variously ascribed to the notorious use of this practice by the Spanish Inquisition, to the excesses of the English Court of Star Chamber, and to the French monarchy's abuse of the lettre de cachet."

The Sixth Amendment public trial right is held by the defendant, and the excluded public have no ability to assert it. Independently, however, the public has a substantially similar First Amendment right to attend.

Self-incrimination

Main article: Self-Incrimination Clause

U.S. Const. amend. V provides:

[N]or shall any person . . . be compelled in any criminal case to be a witness against himself . . . .

While the Self-Incrimination Clause primarily implicates the law of criminal investigations, the Clause also protects against self-incrimination that may occur at trial. Plainly, the Clause prevents the government from compelling the defendant to testify against himself or herself at trial. Further, if the defendant chooses to testify, the Clause prevents the state from requiring her to testify first. But, if the defendant testifies, he or she cannot claim the privilege against self-incrimination with respect to cross-examination within the scope of the direct examination.

Similarly, the Clause "forbids either comment by the prosecution on the accused's silence or instructions by the court that such silence is evidence of guilt." This principle applies at the sentencing phase, even after a plea of guilty. While the defendant is entitled to a jury instruction forbidding adverse inferences from his or her failure to testify, a defendant is not entitled to prevent such an instruction.

"Nothing in the Fifth Amendment privilege entitles a defendant as a matter of constitutional right to await the end of the State's case before announcing the nature of his defense, any more than it entitles him to await the jury's verdict on the State's case-in-chief before deciding whether or not to take the stand himself." For example, a jurisdiction may require the defendant to disclose intended alibi witnesses before trial.

Double jeopardy

Main article: Double Jeopardy Clause

U.S. Const. amend. V provides:

[N]or shall any person be subject for the same offense to be twice put in jeopardy of life or limb . . . .

The Double Jeopardy Clause encompasses four distinct prohibitions: subsequent prosecution after acquittal, subsequent prosecution after conviction, subsequent prosecution after certain mistrials, and multiple punishment in the same indictment. Jeopardy "attaches" when the jury is empaneled, the first witness is sworn, or a plea is accepted. The "dual sovereignty doctrine" permits the federal government and each state to proceed separately.

Prosecution after acquittal

The government is not permitted to appeal or try again after the entry of an acquittal, whether a directed verdict before the case is submitted to the jury, a directed verdict after a deadlocked jury, an appellate reversal for sufficiency (except by direct appeal to a higher appellate court), or an "implied acquittal" via conviction of a lesser included offense. In addition, the government is barred by collateral estoppel from re-litigating against the same defense a fact necessarily found by the jury in a prior acquittal, even if the jury hung on other counts.

This principle does not prevent the government from appealing a pre-trial motion to dismiss or other non-merits dismissal, or a directed verdict after a jury conviction, Nor does it prevent the trial judge from entertaining a motion for reconsideration of a directed verdict, if the jurisdiction has so provided by rule or statute. Nor does it prevent the government from retrying the defendant after a deadlocked jury, an appellate reversal other than for sufficiency, including habeas, or "thirteenth juror" appellate reversals notwithstanding sufficiency on the principle that jeopardy has not "terminated." There may also be an exception for judicial bribery, but not jury bribery.

Multiple punishment, including prosecution after conviction

In Blockburger v. United States (1932), the Supreme Court announced the following test: the government may separately try and punish the defendant for two crimes if each crime contains an element that the other does not. Blockburger is the default rule, unless the legislatively intends to depart; for example, Continuing Criminal Enterprise (CCE) may be punished separately from its predicates, as can conspiracy.

The Blockburger test, originally developed in the multiple punishments context, is also the test for prosecution after conviction. In Grady v. Corbin (1990), the Court held that a double jeopardy violation could lie even where the Blockburger test was not satisfied, but Grady was overruled in United States v. Dixon (1993).

Prosecution after mistrial

The rule for mistrials depends upon who sought the mistrial. If the defendant moves for a mistrial, there is no bar to retrial, unless the prosecutor acted in "bad faith," i.e. goaded the defendant into moving for a mistrial because the government specifically wanted a mistrial. If the prosecutor moves for a mistrial, there is no bar to retrial if the trial judge finds "manifest necessity" for granting the mistrial. The same standard governs mistrials granted sua sponte.

Assistance of Counsel

Main article: Assistance of Counsel Clause

U.S. Const. amend. VI provides:

In all criminal prosecutions, the accused shall enjoy the right . . . to have the Assistance of Counsel for his defence.

The Assistance of Counsel Clause includes, as relevant here, at least six distinct rights: the right to counsel of choice, the right to appointed counsel, the right not to be constructively denied counsel, the right to conflict-free counsel, the effective assistance of counsel, and the right to represent oneself pro se.

A defendant does not have a Sixth Amendment right to counsel in any civil proceeding, including a deportation hearing (even though deportability is often a collateral consequence of criminal conviction).

Choice of counsel

A defendant must be given an opportunity to retain counsel, even if not entitled to appointed counsel. Subject to considerations such as conflicts of interest, scheduling, counsel's authorization to practice law in the jurisdiction, and counsel's willingness to represent the defendant (whether pro bono or for a fee), criminal defendants have a right to be represented by counsel of their choice. The remedy for erroneous depravation of first choice counsel is automatic reversal.

In Caplin & Drysdale v. United States (1989), the Court held that there is no Sixth Amendment exception to criminal forfeiture; i.e., after conviction, the government can seek forfeiture of already paid legal fees under a forfeiture statute, notwithstanding the effect on the defendant's ability to retain counsel of choice.

Appointment of counsel

A defendant unable to retain counsel has the right to appointed counsel at the government's expense. While the Supreme Court recognized this right gradually, it currently applies in all federal and state criminal proceedings where the defendant faces authorized imprisonment greater than one year (a "felony") or where the defendant is actually imprisoned.

The right to appointed counsel does not extend when the defendant is not sentenced to actual imprisonment and could not have been sentenced for more than one year, even if that conviction is later used to enhance sentencing for another crime, or even if the revocation of probation may result in actual imprisonment. Nor does the defendant have the right to appointed counsel to raise frivolous arguments on direct appeal, or to raise any arguments on habeas or other collateral appeal, even if facing execution.

Constructive denial

Whether counsel are appointed or retained, the Clause protects the role of counsel and certain attributes of the attorney-client relationship. For example, the Clause requires that the defendant be given time to consult with counsel and that counsel be given time to investigate the case pre-trial. And, the Clause also prohibits a state from barring a defendant from being cross-examined by counsel, or restricting the order in which the defendant may be called as a witness. Further, the court may not prevent a defendant from consulting with her counsel during an overnight recess, even if the recess bisects direct- and cross-examination of the defendant. Similarly, the defendant has a right to have her counsel make a closing argument, even if a bench trial.

Conflict-free counsel

Whether counsel is retained or appointed, the defendant has a right to counsel without a conflict of interest. If an actual conflict of interest is present, and that conflict results in any adverse effect on the representation, the result is automatic reversal. The general rule is that conflicts can be knowingly and intelligently waived, but some conflicts are un-waiveable.

Ineffective assistance of counsel

Main article: Ineffective assistance of counsel

In Strickland v. Washington (1984), the Court held that, on collateral review, a defendant may obtain relief if the defendant demonstrates both (1) that defense counsel's performance fell below an objective standard of reasonableness (the "performance prong") and (2) that, but for the deficient performance, there is a reasonable probability that the result of the proceeding would have been different (the "prejudice prong").

To satisfy the prejudice prong of Strickland, a defendant who pleads guilty must show that there is a reasonable probability that, but for counsel's deficient performance, he or she would not have pleaded guilty. In Padilla v. Kentucky (2010), the Court held that counsel's failure to inform an alien pleading guilty of the risk of deportation fell below the objective standard of the performance prong of Strickland and permitted an alien who would not have pleaded guilty but for such failure to withdraw his guilty plea.

To satisfy the prejudice prong of Strickland, a defendant who rejects the prosecution's plea offer must show that there is a reasonable probability that, but for counsel's deficient performance, the offer would have been accepted by the defendant, not withdrawn by the prosecution, and accepted by the court, and that the sentence actually received exceeded that which would have been received under the plea.

Pro se representation

Further information: Pro se legal representation in the United States

In Faretta v. California (1975), the Court held that a criminal defendant has the right to knowingly and voluntarily opt for pro se representation at trial. This right is not per se violated by the appointment of standby counsel. There is no constitutional right to self-representation on appeal.

Clauses of general applicability

All of the foregoing constitutional provisions apply exclusively to criminal matters. In contrast, the due process and equal protection clauses have substantial application outside of the criminal law.

Due process

Further information: Due Process Clause

U.S. Const. amend. V provides:

[N]or shall any person . . . be deprived of life, liberty, or property, without due process of law . . . .

U.S. Const. amend. XIV, § 1 provides:

[N]or shall any State deprive any person of life, liberty, or property, without due process of law . . . .

The due process clauses of the Fifth and Fourteenth Amendments apply generally to all stages of criminal proceedings. The Due Process Clause of the Fourteenth Amendment was the vehicle for the incorporation of all of the foregoing rights (with the exception of the Grand Jury Clause, the Vicinage Clause, and maybe the Excessive Bail Clause) to apply in state criminal proceedings. Due process is also the catchall vehicle for the enforcement of fundamental fairness, even if the infirmities of a given prosecution do not neatly sound in another enumerated provision.

Proof beyond a reasonable doubt

Further information: Reasonable doubt

The due process clauses require that the burden of proof in criminal cases be placed on the government, and that the quantum of proof be beyond a reasonable doubt. In re Winship (1970) explicitly held that "the Due Process Clause protects the accused against conviction except upon proof beyond a reasonable doubt of every fact necessary to constitute the crime with which he is charged." But, the state may place the burden of proof for an affirmative defense on the defendant.

Erroneous denial of a reasonable doubt instruction is a structural error that entitles the defendant to automatic reversal. Erroneous definitions of reasonable doubt do not require reversal as long as "taken as a whole, the instructions correctly conveyed the concept of reasonable doubt to the jury." Instructions on certain evidentiary presumptions against the defendant, if interpreted as conclusive presumptions or as shifting the burden of proof to the defendant, are also unconstitutional; permissive presumptions are constitutional. In some circumstances, a trial court must separately instruct the jury on the presumption of innocence, in addition to giving a reasonable doubt instruction.

The reasonable doubt standard is primarily effectuated by jury instructions, but it retains its relevance when the trial judge considers a motion for a directed verdict of acquittal and when an appellate court reviews the sufficiency of the evidence. On federal habeas review of a state conviction for sufficiency of the evidence, to grant relief, the reviewing court must find that "upon the record evidence adduced at the trial no rational trier of fact could have found proof of guilt beyond a reasonable doubt." In a successive, abusive, or defaulted federal habeas review of a state conviction, a defendant claiming "actual innocence" must show that "it is more likely than not that no reasonable juror would have found petitioner guilty beyond a reasonable doubt."

Brady disclosure

Main article: Brady disclosure

Brady v. Maryland (1963) is another significant, specific criminal procedural right guaranteed by the due process clauses. Brady requires a criminal conviction to be reversed if the government withholds exculpatory (or impeachment) material, within the government's possession, from the defendant, and there is a reasonable probability that, if such material had been disclosed, the result of the proceeding would have been different ("materiality"). Brady is a holistic, rather than piece-by-piece, inquiry.

Whether the government acted in "good faith" or "bad faith" is irrelevant to Brady. But, if the defendant cannot prove that withheld evidence would have been exculpatory, because its import is unknown, to obtain relief, the defendant must instead show that the government acted in bad faith.

The government is not required to disclose impeachment material prior to plea bargaining. Whether the government must disclose exculpatory material during plea bargaining is an open question.

Mental competence

Further information: Adjudicative competence

"It has long been accepted that a person whose mental condition is such that he lacks the capacity to understand the nature and object of the proceedings against him, to consult with counsel, and to assist in preparing his defense may not be subjected to a trial" consistent with the Due Process Clause. The "test" is "whether he has sufficient present ability to consult with his lawyer with a reasonable degree of rational understanding—and whether he has a rational as well as factual understanding of the proceedings against him."

A state may place the burden on the defendant has to prove incompetence by the preponderance of the evidence, but the state cannot require the defendant to prove incompetence by a higher standard, such as clear and convincing evidence. The right to competence cannot be waived because waivers of constitutional rights are required to be knowing and voluntary. The state may involuntarily medicate the defendant in order to make her competent for trial, but only after factual showings that there is a state interest in punishment (as opposed to civil confinement), that the medication is likely to result in competence, and that the medication is necessary to restore competence.

A defendant who is competent to stand trial is therefore also competent to plead guilty, waiving the full panoply of trial rights, but not necessarily competent enough to represent herself at trial in the face of a state procedural rule requiring a higher standard of competence for pro se representation.

Prosecutorial misconduct

Due process prohibits the prosecution from knowingly using falsehood to convict the defendant, and requires reversal if there is a reasonable likelihood that the verdict was affected—whether the falsehood is inculpatory or goes the credibility of a witness.

Equal protection

Further information: Equal Protection Clause

U.S. Const. amend. XIV, § 1 provides:

[N]or shall any State . . . deny to any person within its jurisdiction the equal protection of the laws.

The equal protection clauses has at least three applications relevant to criminal proceedings: a prohibition on selective prosecution on invidious bases, a requirement that jury pools and venires represent a "fair cross section" of the community, and a prohibition on the discriminatory use of jury peremptory challenges.

Selective prosecution

Main article: Selective prosecution

The defendant may move to dismiss a criminal charge on the ground that he or she has been singled out for prosecution because of race, gender, religion, national origin, illegitimacy, or similar. In order to get discovery on a racial selective prosecution claim, the defendant must make the threshold showing that the government declined to prosecute similarly situated suspects of other races. The defendant is not entitled to a presumption of selective prosecution based on data regarding the overall population of convicts.

Discrimination in the jury pool and venire

Further information: Jury selection in the United States

A nineteenth-century painting of a jury composed exclusively of white men

The Equal Protection Clause prohibits the exclusion of persons from selection for a grand or petit jury on the basis of race, regardless of the race of the defendant.

Further, the defendant is entitled to a jury pool that represents a "fair cross section" of the community. In order to prove a "fair cross section" violation, the defendant must show that (1) a "distinctive" (i.e., cognizable) group (2) is not represented fairly and reasonably in the jury pool in proportion to the community (3) due to systematic exclusion.

Discriminatory peremptory challenges

Further information: Peremptory challenge

While a defendant is entitled to a fair cross section in the venire, the defendant is not guaranteed a fair cross section in the actual grand jury or petit jury. Yet, the equal protection clause does regulate the use of peremptory challenges in the selection of the petit jury from the venire. In the landmark case of Batson v. Kentucky (1986), the Supreme Court reversed a criminal conviction because of the prosecutor's racially motivated use of peremptory challenges.

There are three steps to a Batson inquiry. First, the party opposing the use of a peremptory challenge must make a prima facie case. This requires only an inference, not preponderance. Second, the party seeking the peremptory challenge must provide a permissible, neutral explanation for the challenge. Third, the trial court must decide whether the explanation is pretextual. A rationale is pretextual if it applies equally to a similarly situated juror who was seated.

If the trial judge erroneously permits the striking of a juror under Batson, and the error is preserved, the only remedy is automatic reversal. If the trial judge erroneously prevents the striking of a juror under Batson, and the juror is seated, the Constitution permits a jurisdiction to utilize harmless error analysis. The race of the defendant is irrelevant to a Batson claim. Batson also permits the prosecutor to challenge defense peremptory strikes ("reverse Batson"). And, Batson applies equally to race and gender.