QED vacuum

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

The QED vacuum or quantum electrodynamic vacuum is the field-theoretic vacuum of quantum electrodynamics. It is the lowest energy state (the ground state) of the electromagnetic field when the fields are quantized. When the Planck constant is hypothetically allowed to approach zero, QED vacuum is converted to classical vacuum, which is to say, the vacuum of classical electromagnetism.

Another field-theoretic vacuum is the QCD vacuum of the Standard Model.

A Feynman diagram (box diagram) for photon-photon scattering, one photon scatters from the transient vacuum charge fluctuations of the other

Fluctuations

Main article: Quantum fluctuation

The QED vacuum is subject to fluctuations about a dormant zero average-field condition; Here is a description of the quantum vacuum:

The quantum theory asserts that a vacuum, even the most perfect vacuum devoid of any matter, is not really empty. Rather the quantum vacuum can be depicted as a sea of continuously appearing and disappearing [pairs of] particles that manifest themselves in the apparent jostling of particles that is quite distinct from their thermal motions. These particles are ‘virtual’, as opposed to real, particles. ...At any given instant, the vacuum is full of such virtual pairs, which leave their signature behind, by affecting the energy levels of atoms.

— Joseph Silk On the Shores of the Unknown, p. 62

Virtual particles

Main articles: Virtual particles and Quantum fluctuation

It is sometimes attempted to provide an intuitive picture of virtual particles based upon the Heisenberg energy-time uncertainty principle: $${\displaystyle \mathrm{\Delta }E\mathrm{\Delta }t\ge \frac{\hslash }{2},}$$ (where ΔE and Δt are energy and time variations, and ħ the Planck constant divided by 2π) arguing along the lines that the short lifetime of virtual particles allows the "borrowing" of large energies from the vacuum and thus permits particle generation for short times.

This interpretation of the energy-time uncertainty relation is not universally accepted, however. One issue is the use of an uncertainty relation limiting measurement accuracy as though a time uncertainty Δt determines a "budget" for borrowing energy ΔE. Another issue is the meaning of "time" in this relation, because energy and time (unlike position q and momentum p, for example) do not satisfy a canonical commutation relation (such as [q, p] = iħ). Various schemes have been advanced to construct an observable that has some kind of time interpretation, and yet does satisfy a canonical commutation relation with energy. The many approaches to the energy-time uncertainty principle are a continuing subject of study.

Quantization of the fields

Main article: Quantization of the electromagnetic field

The Heisenberg uncertainty principle does not allow a particle to exist in a state in which the particle is simultaneously at a fixed location, say the origin of coordinates, and has also zero momentum. Instead the particle has a range of momentum and spread in location attributable to quantum fluctuations; if confined, it has a zero-point energy.

An uncertainty principle applies to all quantum mechanical operators that do not commute. In particular, it applies also to the electromagnetic field. A digression follows to flesh out the role of commutators for the electromagnetic field.

The standard approach to the quantization of the electromagnetic field begins by introducing a vector potential A and a scalar potential V to represent the basic electromagnetic electric field E and magnetic field B using the relations:$${\displaystyle \begin{array}{rl}\mathbf{B}& =\mathrm{\nabla }\times \mathbf{A},\\ \mathbf{E}& =-\frac{\mathrm{\partial }}{\mathrm{\partial }t}\mathbf{A}-\mathrm{\nabla }V.\end{array}}$$ The vector potential is not completely determined by these relations, leaving open a so-called gauge freedom. Resolving this ambiguity using the Coulomb gauge leads to a description of the electromagnetic fields in the absence of charges in terms of the vector potential and the momentum field Π, given by: $${\displaystyle \mathbf{\Pi }={\epsilon }_0\frac{\mathrm{\partial }}{\mathrm{\partial }t}\mathbf{A},}$$ where ε₀ is the electric constant of the SI units. Quantization is achieved by insisting that the momentum field and the vector potential do not commute. That is, the equal-time commutator is: $${\displaystyle [{\mathrm{\Pi }}_i(\mathbf{r},t),A_j({\mathbf{r}}^{\prime },t)]=-i\hslash {\delta }_{ij}\delta (\mathbf{r}-{\mathbf{r}}^{\prime }),}$$ where r, r′ are spatial locations, ħ is the reduced Planck constant, δ_ij is the Kronecker delta and δ(r − r′) is the Dirac delta function. The notation [ , ] denotes the commutator.

Quantization can be achieved without introducing the vector potential, in terms of the underlying fields themselves: $${\displaystyle \left[{\hat{E}}_k(\mathit{r}),{\hat{B}}_{k^{\prime }}({\mathit{r}}^{\prime })\right]=-{ϵ}_{k{k}^{\prime }m}\frac{i\hslash }{{\epsilon }_0}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_m}\delta (\mathit{r}\mathbf{-}{\mathit{r}}^{\mathbf{\prime }}),}$$ where the circumflex denotes a Schrödinger time-independent field operator, and ε_ijk is the antisymmetric Levi-Civita tensor.

Because of the non-commutation of field variables, the variances of the fields cannot be zero, although their averages are zero. The electromagnetic field has therefore a zero-point energy, and a lowest quantum state. The interaction of an excited atom with this lowest quantum state of the electromagnetic field is what leads to spontaneous emission, the transition of an excited atom to a state of lower energy by emission of a photon even when no external perturbation of the atom is present.

Electromagnetic properties

See also: Lamb shift, Casimir effect, and Spontaneous emission

The polarization of the observed light in the extremely strong magnetic field suggests that the empty space around the neutron star RX J1856.5−3754 is subject to the vacuum birefringence.

As a result of quantization, the quantum electrodynamic vacuum can be considered as a material medium. It is capable of vacuum polarization. In particular, the force law between charged particles is affected. The electrical permittivity of quantum electrodynamic vacuum can be calculated, and it differs slightly from the simple ε₀ of the classical vacuum. Likewise, its permeability can be calculated and differs slightly from μ₀. This medium is a dielectric with relative dielectric constant > 1, and is diamagnetic, with relative magnetic permeability < 1. Under some extreme circumstances in which the field exceeds the Schwinger limit (for example, in the very high fields found in the exterior regions of pulsars), the quantum electrodynamic vacuum is thought to exhibit nonlinearity in the fields. Calculations also indicate birefringence and dichroism at high fields. Many of electromagnetic effects of the vacuum are small, and only recently have experiments been designed to enable the observation of nonlinear effects. PVLAS and other teams are working towards the needed sensitivity to detect QED effects.

Attainability

A perfect vacuum is itself only attainable in principle. It is an idealization, like absolute zero for temperature, that can be approached, but never actually realized:

One reason [a vacuum is not empty] is that the walls of a vacuum chamber emit light in the form of black-body radiation...If this soup of photons is in thermodynamic equilibrium with the walls, it can be said to have a particular temperature, as well as a pressure. Another reason that perfect vacuum is impossible is the Heisenberg uncertainty principle which states that no particles can ever have an exact position ...Each atom exists as a probability function of space, which has a certain nonzero value everywhere in a given volume. ...More fundamentally, quantum mechanics predicts ...a correction to the energy called the zero-point energy [that] consists of energies of virtual particles that have a brief existence. This is called vacuum fluctuation.

— Luciano Boi, "Creating the physical world ex nihilo?" p. 55

Virtual particles make a perfect vacuum unrealizable, but leave open the question of attainability of a quantum electrodynamic vacuum or QED vacuum. Predictions of QED vacuum such as spontaneous emission, the Casimir effect and the Lamb shift have been experimentally verified, suggesting QED vacuum is a good model for a high quality realizable vacuum. There are competing theoretical models for vacuum, however. For example, quantum chromodynamic vacuum includes many virtual particles not treated in quantum electrodynamics. The vacuum of quantum gravity treats gravitational effects not included in the Standard Model. It remains an open question whether further refinements in experimental technique ultimately will support another model for realizable vacuum.

Position and momentum spaces

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Position_and_momentum_spaces

In physics and geometry, there are two closely related vector spaces, usually three-dimensional but in general of any finite dimension. Position space (also real space or coordinate space) is the set of all position vectors r in Euclidean space, and has dimensions of length; a position vector defines a point in space. (If the position vector of a point particle varies with time, it will trace out a path, the trajectory of a particle.) Momentum space is the set of all momentum vectors p a physical system can have; the momentum vector of a particle corresponds to its motion, with dimension of mass ⋅ length ⋅ time⁻¹.

Mathematically, the duality between position and momentum is an example of Pontryagin duality. In particular, if a function is given in position space, f(r), then its Fourier transform obtains the function in momentum space, φ(p). Conversely, the inverse Fourier transform of a momentum space function is a position space function.

These quantities and ideas transcend all of classical and quantum physics, and a physical system can be described using either the positions of the constituent particles, or their momenta, both formulations equivalently provide the same information about the system in consideration. Another quantity is useful to define in the context of waves. The wave vector k (or simply "k-vector") has dimensions of reciprocal length, making it an analogue of angular frequency ω which has dimensions of reciprocal time. The set of all wave vectors is k-space. Usually, the position vector r is more intuitive and simpler than the wave vector k, though the converse can also be true, such as in solid-state physics.

Quantum mechanics provides two fundamental examples of the duality between position and momentum, the Heisenberg uncertainty principle ΔxΔp ≥ ħ/2 stating that position and momentum cannot be simultaneously known to arbitrary precision, and the de Broglie relation p = ħk which states the momentum and wavevector of a free particle are proportional to each other. In this context, when it is unambiguous, the terms "momentum" and "wavevector" are used interchangeably. However, the de Broglie relation is not true in a crystal.

Classical mechanics

Lagrangian mechanics

Most often in Lagrangian mechanics, the Lagrangian L(q, dq/dt, t) is in configuration space, where q = (q₁, q₂,..., q_n) is an n-tuple of the generalized coordinates. The Euler–Lagrange equations of motion are $${\displaystyle \frac{d}{dt}\frac{\mathrm{\partial }L}{\mathrm{\partial }{\dot{q}}_i}=\frac{\mathrm{\partial }L}{\mathrm{\partial }{q}_i},{\dot{q}}_i\equiv \frac{d{q}_i}{dt}.}$$

(One overdot indicates one time derivative). Introducing the definition of canonical momentum for each generalized coordinate $${\displaystyle p_i=\frac{\mathrm{\partial }L}{\mathrm{\partial }{\dot{q}}_i},}$$ the Euler–Lagrange equations take the form $${\displaystyle {\dot{p}}_i=\frac{\mathrm{\partial }L}{\mathrm{\partial }{q}_i}.}$$

The Lagrangian can be expressed in momentum space also, L′(p, dp/dt, t), where p = (p₁, p₂, ..., p_n) is an n-tuple of the generalized momenta. A Legendre transformation is performed to change the variables in the total differential of the generalized coordinate space Lagrangian; $${\displaystyle dL=\sum _{i=1}^n\left(\frac{\mathrm{\partial }L}{\mathrm{\partial }{q}_i}d{q}_i+\frac{\mathrm{\partial }L}{\mathrm{\partial }{\dot{q}}_i}d{\dot{q}}_i\right)+\frac{\mathrm{\partial }L}{\mathrm{\partial }t}dt=\sum _{i=1}^n({\dot{p}}_{i}d{q}_i+p_{i}d{\dot{q}}_i)+\frac{\mathrm{\partial }L}{\mathrm{\partial }t}dt,}$$ where the definition of generalized momentum and Euler–Lagrange equations have replaced the partial derivatives of L. The product rule for differentials allows the exchange of differentials in the generalized coordinates and velocities for the differentials in generalized momenta and their time derivatives, $${\displaystyle {\dot{p}}_{i}d{q}_i=d(q_i{\dot{p}}_i)-q_{i}d{\dot{p}}_i}$$ $${\displaystyle p_{i}d{\dot{q}}_i=d({\dot{q}}_i{p}_i)-{\dot{q}}_{i}d{p}_i}$$ which after substitution simplifies and rearranges to $${\displaystyle d\left[L-\sum _{i=1}^n(q_i{\dot{p}}_i+{\dot{q}}_i{p}_i)\right]=-\sum _{i=1}^n({\dot{q}}_{i}d{p}_i+q_{i}d{\dot{p}}_i)+\frac{\mathrm{\partial }L}{\mathrm{\partial }t}dt.}$$

Now, the total differential of the momentum space Lagrangian L′ is $${\displaystyle d{L}^{\prime }=\sum _{i=1}^n\left(\frac{\mathrm{\partial }{L}^{\prime }}{\mathrm{\partial }{p}_i}d{p}_i+\frac{\mathrm{\partial }{L}^{\prime }}{\mathrm{\partial }{\dot{p}}_i}d{\dot{p}}_i\right)+\frac{\mathrm{\partial }{L}^{\prime }}{\mathrm{\partial }t}dt}$$ so by comparison of differentials of the Lagrangians, the momenta, and their time derivatives, the momentum space Lagrangian L′ and the generalized coordinates derived from L′ are respectively $${\displaystyle L^{\prime }=L-\sum _{i=1}^n(q_i{\dot{p}}_i+{\dot{q}}_i{p}_i),-{\dot{q}}_i=\frac{\mathrm{\partial }{L}^{\prime }}{\mathrm{\partial }{p}_i},-q_i=\frac{\mathrm{\partial }{L}^{\prime }}{\mathrm{\partial }{\dot{p}}_i}.}$$

Combining the last two equations gives the momentum space Euler–Lagrange equations $${\displaystyle \frac{d}{dt}\frac{\mathrm{\partial }{L}^{\prime }}{\mathrm{\partial }{\dot{p}}_i}=\frac{\mathrm{\partial }{L}^{\prime }}{\mathrm{\partial }{p}_i}.}$$

The advantage of the Legendre transformation is that the relation between the new and old functions and their variables are obtained in the process. Both the coordinate and momentum forms of the equation are equivalent and contain the same information about the dynamics of the system. This form may be more useful when momentum or angular momentum enters the Lagrangian.

Hamiltonian mechanics

In Hamiltonian mechanics, unlike Lagrangian mechanics which uses either all the coordinates or the momenta, the Hamiltonian equations of motion place coordinates and momenta on equal footing. For a system with Hamiltonian H(q, p, t), the equations are $${\displaystyle {\dot{q}}_i=\frac{\mathrm{\partial }H}{\mathrm{\partial }{p}_i},{\dot{p}}_i=-\frac{\mathrm{\partial }H}{\mathrm{\partial }{q}_i}.}$$

Quantum mechanics

In quantum mechanics, a particle is described by a quantum state. This quantum state can be represented as a superposition of basis states. In principle one is free to choose the set of basis states, as long as they span the state space. If one chooses the (generalized) eigenfunctions of the position operator as a set of basis functions, one speaks of a state as a wave function ψ(r) in position space. The familiar Schrödinger equation in terms of the position r is an example of quantum mechanics in the position representation.

By choosing the eigenfunctions of a different operator as a set of basis functions, one can arrive at a number of different representations of the same state. If one picks the eigenfunctions of the momentum operator as a set of basis functions, the resulting wave function ${\displaystyle \varphi (\mathbf{k})}$ is said to be the wave function in momentum space.

A feature of quantum mechanics is that phase spaces can come in different types: discrete-variable, rotor, and continuous-variable. The table below summarizes some relations involved in the three types of phase spaces.

Comparison and summary of relations between conjugate variables in discrete-variable (DV), rotor (ROT), and continuous-variable (CV) phase spaces (taken from arXiv:1709.04460). Most physically relevant phase spaces consist of combinations of these three. Each phase space consists of position and momentum, whose possible values are taken from a locally compact Abelian group and its dual. A quantum mechanical state can be fully represented in terms of either variables, and the transformation used to go between position and momentum spaces is, in each of the three cases, a variant of the Fourier transform. The table uses bra-ket notation as well as mathematical terminology describing Canonical commutation relations (CCR).

Reciprocal relation

See also: Wave packet

The momentum representation of a wave function and the de Broglie relation are closely related to the Fourier inversion theorem and the concept of frequency domain. Since a free particle has a spatial frequency ${\displaystyle k=|\mathbf{k}|=2\pi /\lambda }$ proportional to the momentum ${\displaystyle p=|\mathbf{p}|=\hslash k}$, describing the particle as a sum of frequency components is equivalent to describing it as the Fourier transform of a "sufficiently nice" wave function in momentum space.

Position space

Further information: Position operator

Suppose we have a three-dimensional wave function in position space ψ(r), then we can write this functions as a weighted sum of orthogonal basis functions ψ_j(r): $${\displaystyle \psi (\mathbf{r})=\sum _j{\varphi }_j{\psi }_j(\mathbf{r})}$$ or, in the continuous case, as an integral $${\displaystyle \psi (\mathbf{r})={\int }_{\mathbf{k}\text{-space}}\varphi (\mathbf{k}){\psi }_{\mathbf{k}}(\mathbf{r}){\mathrm{d}}^3\mathbf{k}}$$ It is clear that if we specify the set of functions ${\displaystyle {\psi }_{\mathbf{k}}(\mathbf{r})}$, say as the set of eigenfunctions of the momentum operator, the function ${\displaystyle \varphi (\mathbf{k})}$ holds all the information necessary to reconstruct ψ(r) and is therefore an alternative description for the state ${\displaystyle \psi }$.

In coordinate representation the momentum operator is given by $${\displaystyle \hat{\mathbf{p}}=-i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }\mathbf{r}}}$$ (see matrix calculus for the denominator notation) with appropriate domain. The eigenfunctions are $${\displaystyle {\psi }_{\mathbf{k}}(\mathbf{r})=\frac{1}{(\sqrt{2\pi }{)}^3}{e}^{i\mathbf{k}\cdot \mathbf{r}}}$$ and eigenvalues ħk. So $${\displaystyle \psi (\mathbf{r})=\frac{1}{(\sqrt{2\pi }{)}^3}{\int }_{\mathbf{k}\text{-space}}\varphi (\mathbf{k})e^{i\mathbf{k}\cdot \mathbf{r}}{\mathrm{d}}^3\mathbf{k}}$$ and we see that the momentum representation is related to the position representation by a Fourier transform.

Momentum space

Further information: Momentum operator

Conversely, a three-dimensional wave function in momentum space ${\displaystyle \varphi (\mathbf{k})}$ can be expressed as a weighted sum of orthogonal basis functions ${\displaystyle {\varphi }_j(\mathbf{k})}$, $${\displaystyle \varphi (\mathbf{k})=\sum _j{\psi }_j{\varphi }_j(\mathbf{k}),}$$ or as an integral, $${\displaystyle \varphi (\mathbf{k})={\int }_{\mathbf{r}\text{-space}}\psi (\mathbf{r}){\varphi }_{\mathbf{r}}(\mathbf{k}){\mathrm{d}}^3\mathbf{r}.}$$

In momentum representation the position operator is given by $${\displaystyle \hat{\mathbf{r}}=i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }\mathbf{p}}=i\frac{\mathrm{\partial }}{\mathrm{\partial }\mathbf{k}}}$$ with eigenfunctions $${\displaystyle {\varphi }_{\mathbf{r}}(\mathbf{k})=\frac{1}{{\left(\sqrt{2\pi }\right)}^3}{e}^{-i\mathbf{k}\cdot \mathbf{r}}}$$ and eigenvalues r. So a similar decomposition of ${\displaystyle \varphi (\mathbf{k})}$ can be made in terms of the eigenfunctions of this operator, which turns out to be the inverse Fourier transform, $${\displaystyle \varphi (\mathbf{k})=\frac{1}{(\sqrt{2\pi }{)}^3}{\int }_{\mathbf{r}\text{-space}}\psi (\mathbf{r})e^{-i\mathbf{k}\cdot \mathbf{r}}{\mathrm{d}}^3\mathbf{r}.}$$

Unitary equivalence

The position and momentum operators are unitarily equivalent, with the unitary operator being given explicitly by the Fourier transform, namely a quarter-cycle rotation in phase space, generated by the oscillator Hamiltonian. Thus, they have the same spectrum. In physical language, p acting on momentum space wave functions is the same as r acting on position space wave functions (under the image of the Fourier transform).

Reciprocal space and crystals

Main article: Reciprocal lattice

For an electron (or other particle) in a crystal, its value of k relates almost always to its crystal momentum, not its normal momentum. Therefore, k and p are not simply proportional but play different roles. See k·p perturbation theory for an example. Crystal momentum is like a wave envelope that describes how the wave varies from one unit cell to the next, but does not give any information about how the wave varies within each unit cell.

When k relates to crystal momentum instead of true momentum, the concept of k-space is still meaningful and extremely useful, but it differs in several ways from the non-crystal k-space discussed above. For example, in a crystal's k-space, there is an infinite set of points called the reciprocal lattice which are "equivalent" to k = 0 (this is analogous to aliasing). Likewise, the "first Brillouin zone" is a finite volume of k-space, such that every possible k is "equivalent" to exactly one point in this region.