ERCEIRA QUANTIZAÇÃO PELO SDCTIE GRACELI

TRANS-QUÂNTICA SDCTIE GRACELI, TRANSCENDENTE, RELATIVISTA SDCTIE GRACELI, E TRANS-INDETERMINADA.

FUNDAMENTA-SE EM QUE TODA FORMA DE REALIDADE SE ENCONTRA EM TRANSFORMAÇÕES, INTERAÇÕES, TRANSIÇÕES DE ESTADOS [ESTADOS DE GRACELI], ENERGIAS E FENÔMENOS DENTRO DE UM SISTEMA DE DEZ OU MAIS DIMENSÕES DE GRACELI, E CATEGORIAS DE GRACELI.

FUNÇÃO GERAL GRACELI DA TRANS- INDETERMINALIDADE PELO SDCTIE GRACELI

FUNÇÃO FUNDAMENTAL E GERAL DO SISTEMA [SDCTIE GRACELI] DE  INTERAÇÕES, TRANSFORMAÇÕES EM CADEIAS, DECADIMENSIONAL E CATEGORIAL GRACELI.  E DE ESTADOS TRANSICIONAIS =

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número atômico, estrutura eletrônica, níveis de energia 

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PARA TODA E QUALQUER FORMA DE EQUAÇÃO E FUNÇÃO EM:

Classical fields[edit]

See also: Classical field theory

A classical field is a function of spatial and time coordinates.^[18] Examples include the gravitational field in Newtonian gravity g(x, t) and the electric field E(x, t) and magnetic field B(x, t) in classical electromagnetism. A classical field can be thought of as a numerical quantity assigned to every point in space that changes in time. Hence, it has infinitely many degrees of freedom.^[18][19]

Many phenomena exhibiting quantum mechanical properties cannot be explained by classical fields alone. Phenomena such as the photoelectric effect are best explained by discrete particles (photons), rather than a spatially continuous field. The goal of quantum field theory is to describe various quantum mechanical phenomena using a modified concept of fields.

Canonical quantisation and path integrals are two common formulations of QFT.^[20]:61 To motivate the fundamentals of QFT, an overview of classical field theory is in order.

The simplest classical field is a real scalar field — a real number at every point in space that changes in time. It is denoted as ϕ(x, t), where x is the position vector, and t is the time. Suppose the Lagrangian of the field is

${\displaystyle L=\int d^{3}x\,{\mathcal {L}}=\int d^{3}x\,\left[{\frac {1}{2}}{\dot {\phi }}^{2}-{\frac {1}{2}}(\nabla \phi )^{2}-{\frac {1}{2}}m^{2}\phi ^{2}\right],}$

where ${\displaystyle {\dot {\phi }}}$ is the time-derivative of the field, ∇ is the gradient operator, and m is a real parameter (the "mass" of the field). Applying the Euler–Lagrange equation on the Lagrangian:^[1]:16

${\displaystyle {\frac {\partial }{\partial t}}\left[{\frac {\partial {\mathcal {L}}}{\partial (\partial \phi /\partial t)}}\right]+\sum _{i=1}^{3}{\frac {\partial }{\partial x^{i}}}\left[{\frac {\partial {\mathcal {L}}}{\partial (\partial \phi /\partial x^{i})}}\right]-{\frac {\partial {\mathcal {L}}}{\partial \phi }}=0,}$

we obtain the equations of motion for the field, which describe the way it varies in time and space:

${\displaystyle \left({\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}+m^{2}\right)\phi =0.}$

This is known as the Klein–Gordon equation.^[1]:17

The Klein–Gordon equation is a wave equation, so its solutions can be expressed as a sum of normal modes (obtained via Fourier transform) as follows:

${\displaystyle \phi (\mathbf {x} ,t)=\int {\frac {d^{3}p}{(2\pi )^{3}}}{\frac {1}{\sqrt {2\omega _{\mathbf {p} }}}}\left(a_{\mathbf {p} }e^{-i\omega _{\mathbf {p} }t+i\mathbf {p} \cdot \mathbf {x} }+a_{\mathbf {p} }^{*}e^{i\omega _{\mathbf {p} }t-i\mathbf {p} \cdot \mathbf {x} }\right),}$

where a is a complex number (normalised by convention), * denotes complex conjugation, and ω_p is the frequency of the normal mode:

${\displaystyle \omega _{\mathbf {p} }={\sqrt {|\mathbf {p} |^{2}+m^{2}}}.}$

Thus each normal mode corresponding to a single p can be seen as a classical harmonic oscillator with frequency ω_p.^[1]:21,26

Canonical quantisation[edit]

Main article: Canonical quantisation

The quantisation procedure for the above classical field is analogous to the promotion of a classical harmonic oscillator to a quantum harmonic oscillator.

The displacement of a classical harmonic oscillator is described by

${\displaystyle x(t)={\frac {1}{\sqrt {2\omega }}}ae^{-i\omega t}+{\frac {1}{\sqrt {2\omega }}}a^{*}e^{i\omega t},}$

where a is a complex number (normalised by convention), and ω is the oscillator's frequency. Note that x is the displacement of a particle in simple harmonic motion from the equilibrium position, which should not be confused with the spatial label x of a field.

For a quantum harmonic oscillator, x(t) is promoted to a linear operator ${\displaystyle {\hat {x}}(t)}$:

${\displaystyle {\hat {x}}(t)={\frac {1}{\sqrt {2\omega }}}{\hat {a}}e^{-i\omega t}+{\frac {1}{\sqrt {2\omega }}}{\hat {a}}^{\dagger }e^{i\omega t}.}$

Complex numbers a and a^* are replaced by the annihilation operator ${\displaystyle {\hat {a}}}$ and the creation operator ${\displaystyle {\hat {a}}^{\dagger }}$, respectively, where † denotes Hermitian conjugation. The commutation relation between the two is

${\displaystyle [{\hat {a}},{\hat {a}}^{\dagger }]=1.}$

The vacuum state ${\displaystyle |0\rangle }$, which is the lowest energy state, is defined by

${\displaystyle {\hat {a}}|0\rangle =0.}$

Any quantum state of a single harmonic oscillator can be obtained from ${\displaystyle |0\rangle }$ by successively applying the creation operator ${\displaystyle {\hat {a}}^{\dagger }}$:^[1]:20

${\displaystyle |n\rangle =({\hat {a}}^{\dagger })^{n}|0\rangle .}$

By the same token, the aforementioned real scalar field ϕ, which corresponds to x in the single harmonic oscillator, is also promoted to an operator ${\displaystyle {\hat {\phi }}}$, while a_p and a_p^* are replaced by the annihilation operator ${\displaystyle {\hat {a}}_{\mathbf {p} }}$ and the creation operator ${\displaystyle {\hat {a}}_{\mathbf {p} }^{\dagger }}$ for a particular p, respectively:

${\displaystyle {\hat {\phi }}(\mathbf {x} ,t)=\int {\frac {d^{3}p}{(2\pi )^{3}}}{\frac {1}{\sqrt {2\omega _{\mathbf {p} }}}}\left({\hat {a}}_{\mathbf {p} }e^{-i\omega _{\mathbf {p} }t+i\mathbf {p} \cdot \mathbf {x} }+{\hat {a}}_{\mathbf {p} }^{\dagger }e^{i\omega _{\mathbf {p} }t-i\mathbf {p} \cdot \mathbf {x} }\right).}$

Their commutation relations are:^[1]:21

${\displaystyle [{\hat {a}}_{\mathbf {p} },{\hat {a}}_{\mathbf {q} }^{\dagger }]=(2\pi )^{3}\delta (\mathbf {p} -\mathbf {q} ),\quad [{\hat {a}}_{\mathbf {p} },{\hat {a}}_{\mathbf {q} }]=[{\hat {a}}_{\mathbf {p} }^{\dagger },{\hat {a}}_{\mathbf {q} }^{\dagger }]=0,}$

where δ is the Dirac delta function. The vacuum state ${\displaystyle |0\rangle }$ is defined by

${\displaystyle {\hat {a}}_{\mathbf {p} }|0\rangle =0,\quad {\text{for all }}\mathbf {p} .}$

Any quantum state of the field can be obtained from ${\displaystyle |0\rangle }$ by successively applying creation operators ${\displaystyle {\hat {a}}_{\mathbf {p} }^{\dagger }}$, e.g.^[1]:22

${\displaystyle ({\hat {a}}_{\mathbf {p} _{3}}^{\dagger })^{3}{\hat {a}}_{\mathbf {p} _{2}}^{\dagger }({\hat {a}}_{\mathbf {p} _{1}}^{\dagger })^{2}|0\rangle .}$

Although the field appearing in the Lagrangian is spatially continuous, the quantum states of the field are discrete. While the state space of a single quantum harmonic oscillator contains all the discrete energy states of one oscillating particle, the state space of a quantum field contains the discrete energy levels of an arbitrary number of particles. The latter space is known as a Fock space, which can account for the fact that particle numbers are not fixed in relativistic quantum systems.^[21] The process of quantising an arbitrary number of particles instead of a single particle is often also called second quantisation.^[1]:19

The preceding procedure is a direct application of non-relativistic quantum mechanics and can be used to quantise (complex) scalar fields, Dirac fields,^[1]:52 vector fields (e.g. the electromagnetic field), and even strings.^[22] However, creation and annihilation operators are only well defined in the simplest theories that contain no interactions (so-called free theory). In the case of the real scalar field, the existence of these operators was a consequence of the decomposition of solutions of the classical equations of motion into a sum of normal modes. To perform calculations on any realistic interacting theory, perturbation theory would be necessary.

The Lagrangian of any quantum field in nature would contain interaction terms in addition to the free theory terms. For example, a quartic interaction term could be introduced to the Lagrangian of the real scalar field:^[1]:77

${\displaystyle {\mathcal {L}}={\frac {1}{2}}(\partial _{\mu }\phi )(\partial ^{\mu }\phi )-{\frac {1}{2}}m^{2}\phi ^{2}-{\frac {\lambda }{4!}}\phi ^{4},}$

where μ is a spacetime index, ${\displaystyle \partial _{0}=\partial /\partial t,\ \partial _{1}=\partial /\partial x^{1}}$, etc. The summation over the index μ has been omitted following the Einstein notation. If the parameter λ is sufficiently small, then the interacting theory described by the above Lagrangian can be considered as a small perturbation from the free theory.

Path integrals[edit]

Main article: Path integral formulation

The path integral formulation of QFT is concerned with the direct computation of the scattering amplitude of a certain interaction process, rather than the establishment of operators and state spaces. To calculate the probability amplitude for a system to evolve from some initial state ${\displaystyle |\phi _{I}\rangle }$ at time t = 0 to some final state ${\displaystyle |\phi _{F}\rangle }$ at t = T, the total time T is divided into N small intervals. The overall amplitude is the product of the amplitude of evolution within each interval, integrated over all intermediate states. Let H be the Hamiltonian (i.e. generator of time evolution), then^[20]:10

${\displaystyle \langle \phi _{F}|e^{-iHT}|\phi _{I}\rangle =\int d\phi _{1}\int d\phi _{2}\cdots \int d\phi _{N-1}\,\langle \phi _{F}|e^{-iHT/N}|\phi _{N-1}\rangle \cdots \langle \phi _{2}|e^{-iHT/N}|\phi _{1}\rangle \langle \phi _{1}|e^{-iHT/N}|\phi _{I}\rangle .}$

Taking the limit N → ∞, the above product of integrals becomes the Feynman path integral:^{[1]:282[20]:12}

${\displaystyle \langle \phi _{F}|e^{-iHT}|\phi _{I}\rangle =\int {\mathcal {D}}\phi (t)\,\exp \left\{i\int _{0}^{T}dt\,L\right\},}$

where L is the Lagrangian involving ϕ and its derivatives with respect to spatial and time coordinates, obtained from the Hamiltonian H via Legendre transform. The initial and final conditions of the path integral are respectively

${\displaystyle \phi (0)=\phi _{I},\quad \phi (T)=\phi _{F}.}$

In other words, the overall amplitude is the sum over the amplitude of every possible path between the initial and final states, where the amplitude of a path is given by the exponential in the integrand.

Two-point correlation function[edit]

Main article: Correlation function (quantum field theory)

Now we assume that the theory contains interactions whose Lagrangian terms are a small perturbation from the free theory.

In calculations, one often encounters such expressions:

${\displaystyle \langle \Omega |T\{\phi (x)\phi (y)\}|\Omega \rangle ,}$

where x and y are position four-vectors, T is the time ordering operator (namely, it orders x and y according to their time-component, later time on the left and earlier time on the right), and ${\displaystyle |\Omega \rangle }$ is the ground state (vacuum state) of the interacting theory. This expression, known as the two-point correlation function or the two-point Green's function, represents the probability amplitude for the field to propagate from y to x.^[1]:82

In canonical quantisation, the two-point correlation function can be written as:^[1]:87

${\displaystyle \langle \Omega |T\{\phi (x)\phi (y)\}|\Omega \rangle =\lim _{T\to \infty (1-i\epsilon )}{\frac {\langle 0|T\left\{\phi _{I}(x)\phi _{I}(y)\exp \left[-i\int _{-T}^{T}dt\,H_{I}(t)\right]\right\}|0\rangle }{\langle 0|T\left\{\exp \left[-i\int _{-T}^{T}dt\,H_{I}(t)\right]\right\}|0\rangle }},}$

where ε is an infinitesimal number, ϕ_I is the field operator under the free theory, and H_I is the interaction Hamiltonian term. For the ϕ⁴ theory, it is^[1]:84

${\displaystyle H_{I}(t)=\int d^{3}x\,{\frac {\lambda }{4!}}\phi _{I}(x)^{4}.}$

Since λ is a small parameter, the exponential function exp can be expanded into a Taylor series in λ and computed term by term. This equation is useful in that it expresses the field operator and ground state in the interacting theory, which are difficult to define, in terms of their counterparts in the free theory, which are well defined.

In the path integral formulation, the two-point correlation function can be written as:^[1]:284

${\displaystyle \langle \Omega |T\{\phi (x)\phi (y)\}|\Omega \rangle =\lim _{T\to \infty (1-i\epsilon )}{\frac {\int {\mathcal {D}}\phi \,\phi (x)\phi (y)\exp \left[i\int _{-T}^{T}d^{4}z\,{\mathcal {L}}\right]}{\int {\mathcal {D}}\phi \,\exp \left[i\int _{-T}^{T}d^{4}z\,{\mathcal {L}}\right]}},}$

where ${\displaystyle {\mathcal {L}}}$ is the Lagrangian density. As in the previous paragraph, the exponential factor involving the interaction term can also be expanded as a series in λ.

According to Wick's theorem, any n-point correlation function in the free theory can be written as a sum of products of two-point correlation functions. For example,

${\displaystyle {\begin{aligned}\langle 0|T\{\phi (x_{1})\phi (x_{2})\phi (x_{3})\phi (x_{4})\}|0\rangle =&\langle 0|T\{\phi (x_{1})\phi (x_{2})\}|0\rangle \langle 0|T\{\phi (x_{3})\phi (x_{4})\}|0\rangle \\&+\langle 0|T\{\phi (x_{1})\phi (x_{3})\}|0\rangle \langle 0|T\{\phi (x_{2})\phi (x_{4})\}|0\rangle \\&+\langle 0|T\{\phi (x_{1})\phi (x_{4})\}|0\rangle \langle 0|T\{\phi (x_{2})\phi (x_{3})\}|0\rangle .\end{aligned}}}$

Since correlation functions in the interacting theory can be expressed in terms of those in the free theory, only the latter need to be evaluated in order to calculate all physical quantities in the (perturbative) interacting theory.^[1]:90

Either through canonical quantisation or path integrals, one can obtain:

${\displaystyle D_{F}(x-y)\equiv \langle 0|T\{\phi (x)\phi (y)\}|0\rangle =\lim _{\epsilon \to 0}\int {\frac {d^{4}p}{(2\pi )^{4}}}{\frac {i}{p_{\mu }p^{\mu }-m^{2}+i\epsilon }}e^{-ip_{\mu }(x^{\mu }-y^{\mu })}.}$

This is known as the Feynman propagator for the real scalar field.^{[1]:31,288[20]:23}

Feynman diagram[edit]

Main article: Feynman diagram

Correlation functions in the interacting theory can be written as a perturbation series. Each term in the series is a product of Feynman propagators in the free theory and can be represented visually by a Feynman diagram. For example, the λ¹ term in the two-point correlation function in the ϕ⁴ theory is

${\displaystyle {\frac {-i\lambda }{4!}}\langle 0|T\{\phi (x)\phi (y)\int d^{4}z\,\phi (z)\phi (z)\phi (z)\phi (z)\}|0\rangle .}$

After applying Wick's theorem, one of the terms is

${\displaystyle 12\cdot {\frac {-i\lambda }{4!}}\int d^{4}z\,D_{F}(x-z)D_{F}(y-z)D_{F}(z-z),}$

whose corresponding Feynman diagram is

Every point corresponds to a single ϕ field factor. Points labelled with x and y are called external points, while those in the interior are called internal points or vertices (there is one in this diagram). The value of the corresponding term can be obtained from the diagram by following "Feynman rules": assign ${\displaystyle -i\lambda \int d^{4}z}$ to every vertex and the Feynman propagator${\displaystyle D_{F}(x_{1}-x_{2})}$ to every line with end points x₁ and x₂. The product of factors corresponding to every element in the diagram, divided by the "symmetry factor" (2 for this diagram), gives the expression for the term in the perturbation series.^[1]:91-94

In order to compute the n-point correlation function to the k-th order, list all valid Feynman diagrams with n external points and k or fewer vertices, and then use Feynman rules to obtain the expression for each term. To be precise,

${\displaystyle \langle \Omega |T\{\phi (x_{1})\cdots \phi (x_{n})\}|\Omega \rangle }$

is equal to the sum of (expressions corresponding to) all connected diagrams with n external points. (Connected diagrams are those in which every vertex is connected to an external point through lines. Components that are totally disconnected from external lines are sometimes called "vacuum bubbles".) In the ϕ⁴ interaction theory discussed above, every vertex must have four legs.^[1]:98

In realistic applications, the scattering amplitude of a certain interaction or the decay rate of a particle can be computed from the S-matrix, which itself can be found using the Feynman diagram method.^[1]:102-115

Feynman diagrams devoid of "loops" are called tree-level diagrams, which describe the lowest-order interaction processes; those containing n loops are referred to as n-loop diagrams, which describe higher-order contributions, or radiative corrections, to the interaction.^[20]:44 Lines whose end points are vertices can be thought of as the propagation of virtual particles.^[1]:31

Renormalisation[edit]

Main article: Renormalisation

Feynman rules can be used to directly evaluate tree-level diagrams. However, naïve computation of loop diagrams such as the one shown above will result in divergent momentum integrals, which seems to imply that almost all terms in the perturbative expansion are infinite. The renormalisation procedure is a systematic process for removing such infinities.

Parameters appearing in the Lagrangian, such as the mass m and the coupling constant λ, have no physical meaning — m, λ, and the field strength ϕ are not experimentally measurable quantities and are referred to here as the bare mass, bare coupling constant, and bare field, respectively. The physical mass and coupling constant are measured in some interaction process and are generally different from the bare quantities. While computing physical quantities from this interaction process, one may limit the domain of divergent momentum integrals to be below some momentum cut-off Λ, obtain expressions for the physical quantities, and then take the limit Λ → ∞. This is an example of regularisation, a class of methods to treat divergences in QFT, with Λ being the regulator.

The approach illustrated above is called bare perturbation theory, as calculations involve only the bare quantities such as mass and coupling constant. A different approach, called renormalised perturbation theory, is to use physically meaningful quantities from the very beginning. In the case of ϕ⁴ theory, the field strength is first redefined:

${\displaystyle \phi =Z^{1/2}\phi _{r},}$

where ϕ is the bare field, ϕ_r is the renormalised field, and Z is a constant to be determined. The Lagrangian density becomes:

${\displaystyle {\mathcal {L}}={\frac {1}{2}}(\partial _{\mu }\phi _{r})(\partial ^{\mu }\phi _{r})-{\frac {1}{2}}m_{r}^{2}\phi _{r}^{2}-{\frac {\lambda _{r}}{4!}}\phi _{r}^{4}+{\frac {1}{2}}\delta _{Z}(\partial _{\mu }\phi _{r})(\partial ^{\mu }\phi _{r})-{\frac {1}{2}}\delta _{m}\phi _{r}^{2}-{\frac {\delta _{\lambda }}{4!}}\phi _{r}^{4},}$

where m_r and λ_r are the experimentally measurable, renormalised, mass and coupling constant, respectively, and

${\displaystyle \delta _{Z}=Z-1,\quad \delta _{m}=m^{2}Z-m_{r}^{2},\quad \delta _{\lambda }=\lambda Z^{2}-\lambda _{r}}$

are constants to be determined. The first three terms are the ϕ⁴ Lagrangian density written in terms of the renormalised quantities, while the latter three terms are referred to as "counterterms". As the Lagrangian now contains more terms, so the Feynman diagrams should include additional elements, each with their own Feynman rules. The procedure is outlined as follows. First select a regularisation scheme (such as the cut-off regularisation introduced above or dimensional regularization); call the regulator Λ. Compute Feynman diagrams, in which divergent terms will depend on Λ. Then, define δ_Z, δ_m, and δ_λ such that Feynman diagrams for the counterterms will exactly cancel the divergent terms in the normal Feynman diagrams when the limit Λ → ∞ is taken. In this way, meaningful finite quantities are obtained.^[1]:323-326

It is only possible to eliminate all infinities to obtain a finite result in renormalisable theories, whereas in non-renormalisable theories infinities cannot be removed by the redefinition of a small number of parameters. The Standard Model of elementary particles is a renormalisable QFT,^{[1]:719–727} while quantum gravity is non-renormalisable.^{[1]:798[20]:421}

Renormalisation group[edit]

Main article: Renormalization group

The renormalisation group, developed by Kenneth Wilson, is a mathematical apparatus used to study the changes in physical parameters (coefficients in the Lagrangian) as the system is viewed at different scales.^[1]:393 The way in which each parameter changes with scale is described by its β function.^[1]:417 Correlation functions, which underlie quantitative physical predictions, change with scale according to the Callan–Symanzik equation.^[1]:410-411

As an example, the coupling constant in QED, namely the elementary charge e, has the following β function:

${\displaystyle \beta (e)\equiv {\frac {1}{\Lambda }}{\frac {de}{d\Lambda }}={\frac {e^{3}}{12\pi ^{2}}}+O(e^{5}),}$

where Λ is the energy scale under which the measurement of e is performed. This differential equation implies that the observed elementary charge increases as the scale increases.^[23] The renormalized coupling constant, which changes with the energy scale, is also called the running coupling constant.^[1]:420

The coupling constant g in quantum chromodynamics, a non-Abelian gauge theory based on the symmetry group SU(3), has the following β function:

${\displaystyle \beta (g)\equiv {\frac {1}{\Lambda }}{\frac {dg}{d\Lambda }}={\frac {g^{3}}{16\pi ^{2}}}\left(-11+{\frac {2}{3}}N_{f}\right)+O(g^{5}),}$

where N_f is the number of quark flavours. In the case where N_f ≤ 16 (the Standard Model has N_f = 6), the coupling constant g decreases as the energy scale increases. Hence, while the strong interaction is strong at low energies, it becomes very weak in high-energy interactions, a phenomenon known as asymptotic freedom.^[1]:531

Conformal field theories (CFTs) are special QFTs that admit conformal symmetry. They are insensitive to changes in the scale, as all their coupling constants have vanishing β function. (The converse is not true, however — the vanishing of all β functions does not imply conformal symmetry of the theory.)^[24] Examples include string theory^[14] and N = 4 supersymmetric Yang–Mills theory.^[25]

According to Wilson's picture, every QFT is fundamentally accompanied by its energy cut-off Λ, i.e. that the theory is no longer valid at energies higher than Λ, and all degrees of freedom above the scale Λ are to be omitted. For example, the cut-off could be the inverse of the atomic spacing in a condensed matter system, and in elementary particle physics it could be associated with the fundamental "graininess" of spacetime caused by quantum fluctuations in gravity. The cut-off scale of theories of particle interactions lies far beyond current experiments. Even if the theory were very complicated at that scale, as long as its couplings are sufficiently weak, it must be described at low energies by a renormalisable effective field theory.^[1]:402-403 The difference between renormalisable and non-renormalisable theories is that the former are insensitive to details at high energies, whereas the latter do depend of them.^[8]:2 According to this view, non-renormalisable theories are to be seen as low-energy effective theories of a more fundamental theory. The failure to remove the cut-off Λ from calculations in such a theory merely indicates that new physical phenomena appear at scales above Λ, where a new theory is necessary.^[20]:156

Other theories[edit]

The quantisation and renormalisation procedures outlined in the preceding sections are performed for the free theory and ϕ⁴ theory of the real scalar field. A similar process can be done for other types of fields, including the complex scalar field, the vector field, and the Dirac field, as well as other types of interaction terms, including the electromagnetic interaction and the Yukawa interaction.

As an example, quantum electrodynamics contains a Dirac field ψ representing the electron field and a vector field A^μ representing the electromagnetic field (photon field). (Despite its name, the quantum electromagnetic "field" actually corresponds to the classical electromagnetic four-potential, rather than the classical electric and magnetic fields.) The full QED Lagrangian density is:

${\displaystyle {\mathcal {L}}={\bar {\psi }}(i\gamma ^{\mu }\partial _{\mu }-m)\psi -{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }-e{\bar {\psi }}\gamma ^{\mu }\psi A_{\mu },}$

where γ^μ are Dirac matrices, ${\displaystyle {\bar {\psi }}=\psi ^{\dagger }\gamma ^{0}}$, and ${\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }}$ is the electromagnetic field strength. The parameters in this theory are the (bare) electron mass m and the (bare) elementary charge e. The first and second terms in the Lagrangian density correspond to the free Dirac field and free vector fields, respectively. The last term describes the interaction between the electron and photon fields, which is treated as a perturbation from the free theories.^[1]:78

Shown above is an example of a tree-level Feynman diagram in QED. It describes an electron and a positron annihilating, creating an off-shell photon, and then decaying into a new pair of electron and positron. Time runs from left to right. Arrows pointing forward in time represent the propagation of positrons, while those pointing backward in time represent the propagation of electrons. A wavy line represents the propagation of a photon. Each vertex in QED Feynman diagrams must have an incoming and an outgoing fermion (positron/electron) leg as well as a photon leg.

Gauge symmetry[edit]

Main article: Gauge theory

If the following transformation to the fields is performed at every spacetime point x (a local transformation), then the QED Lagrangian remains unchanged, or invariant:

${\displaystyle \psi (x)\to e^{i\alpha (x)}\psi (x),\quad A_{\mu }(x)\to A_{\mu }(x)+ie^{-1}e^{-i\alpha (x)}\partial _{\mu }e^{i\alpha (x)},}$

where α(x) is any function of spacetime coordinates. If a theory's Lagrangian (or more precisely the action) is invariant under a certain local transformation, then the transformation is referred to as a gauge symmetry of the theory.^{[1]:482–483} Gauge symmetries form a group at every spacetime point. In the case of QED, the successive application of two different local symmetry transformations ${\displaystyle e^{i\alpha (x)}}$ and ${\displaystyle e^{i\alpha '(x)}}$ is yet another symmetry transformation ${\displaystyle e^{i[\alpha (x)+\alpha '(x)]}}$. For any α(x), ${\displaystyle e^{i\alpha (x)}}$ is an element of the U(1) group, thus QED is said to have U(1) gauge symmetry.^[1]:496 The photon field A_μ may be referred to as the U(1) gauge boson.

U(1) is an Abelian group, meaning that the result is the same regardless of the order in which its elements are applied. QFTs can also be built on non-Abelian groups, giving rise to non-Abelian gauge theories (also known as Yang–Mills theories).^[1]:489 Quantum chromodynamics, which describes the strong interaction, is a non-Abelian gauge theory with an SU(3) gauge symmetry. It contains three Dirac fields ψⁱ, i = 1,2,3 representing quark fields as well as eight vector fields A^a,μ, a = 1,...,8 representing gluon fields, which are the SU(3) gauge bosons.^[1]:547 The QCD Lagrangian density is:^[1]:490-491

${\displaystyle {\mathcal {L}}=i{\bar {\psi }}^{i}\gamma ^{\mu }(D_{\mu })^{ij}\psi ^{j}-{\frac {1}{4}}F_{\mu \nu }^{a}F^{a,\mu \nu }-m{\bar {\psi }}^{i}\psi ^{i},}$

where D_μ is the gauge covariant derivative:

${\displaystyle D_{\mu }=\partial _{\mu }-igA_{\mu }^{a}t^{a},}$

where g is the coupling constant, t^a are the eight generators of SU(3) in the fundamental representation (3×3 matrices),

${\displaystyle F_{\mu \nu }^{a}=\partial _{\mu }A_{\nu }^{a}-\partial _{\nu }A_{\mu }^{a}+gf^{abc}A_{\mu }^{b}A_{\nu }^{c},}$

and f^abc are the structure constants of SU(3). Repeated indices i,j,a are implicitly summed over following Einstein notation. This Lagrangian is invariant under the transformation:

${\displaystyle \psi ^{i}(x)\to U^{ij}(x)\psi ^{j}(x),\quad A_{\mu }^{a}(x)t^{a}\to U(x)\left[A_{\mu }^{a}(x)t^{a}+ig^{-1}\partial _{\mu }\right]U^{\dagger }(x),}$

where U(x) is an element of SU(3) at every spacetime point x:

${\displaystyle U(x)=e^{i\alpha (x)^{a}t^{a}}.}$

The preceding discussion of symmetries is on the level of the Lagrangian. In other words, these are "classical" symmetries. After quantisation, some theories will no longer exhibit their classical symmetries, a phenomenon called anomaly. For instance, in the path integral formulation, despite the invariance of the Lagrangian density ${\displaystyle {\mathcal {L}}[\phi ,\partial _{\mu }\phi ]}$ under a certain local transformation of the fields, the measure ${\displaystyle \int {\mathcal {D}}\phi }$ of the path integral may change.^[20]:243 For a theory describing nature to be consistent, it must not contain any anomaly in its gauge symmetry. The Standard Model of elementary particles is a gauge theory based on the group SU(3) × SU(2) × U(1), in which all anomalies exactly cancel.^[1]:705-707

The theoretical foundation of general relativity, the equivalence principle, can also be understood as a form of gauge symmetry, making general relativity a gauge theory based on the Lorentz group.^[26]

Noether's theorem states that every continuous symmetry, i.e. the parameter in the symmetry transformation being continuous rather than discrete, leads to a corresponding conservation law.^{[1]:17-18[20]:73} For example, the U(1) symmetry of QED implies charge conservation.^[27]

Gauge transformations do not relate distinct quantum states. Rather, it relates two equivalent mathematical descriptions of the same quantum state. As an example, the photon field A^μ, being a four-vector, has four apparent degrees of freedom, but the actual state of a photon is described by its two degrees of freedom corresponding to the polarisation. The remaining two degrees of freedom are said to be "redundant" — apparently different ways of writing A^μ can be related to each other by a gauge transformation and in fact describe the same state of the photon field. In this sense, gauge invariance is not a "real" symmetry, but a reflection of the "redundancy" of the chosen mathematical description.^[20]:168

To account for the gauge redundancy in the path integral formulation, one must perform the so-called Faddeev–Popov gauge fixing procedure. In non-Abelian gauge theories, such a procedure introduces new fields called "ghosts". Particles corresponding to the ghost fields are called ghost particles, which cannot be detected externally.^[1]:512-515 A more rigorous generalisation of the Faddeev–Popov procedure is given by BRST quantization.^[1]:517

Spontaneous symmetry breaking[edit]

Main article: Spontaneous symmetry breaking

Spontaneous symmetry breaking is a mechanism whereby the symmetry of the Lagrangian is violated by the system described by it.^[1]:347

To illustrate the mechanism, consider a linear sigma model containing N real scalar fields, described by the Lagrangian density:

${\displaystyle {\mathcal {L}}={\frac {1}{2}}(\partial _{\mu }\phi ^{i})(\partial ^{\mu }\phi ^{i})+{\frac {1}{2}}\mu ^{2}\phi ^{i}\phi ^{i}-{\frac {\lambda }{4}}(\phi ^{i}\phi ^{i})^{2},}$

where μ and λ are real parameters. The theory admits an O(N) global symmetry:

${\displaystyle \phi ^{i}\to R^{ij}\phi ^{j},\quad R\in \mathrm {O} (N).}$

The lowest energy state (ground state or vacuum state) of the classical theory is any uniform field ϕ₀ satisfying

${\displaystyle \phi _{0}^{i}\phi _{0}^{i}={\frac {\mu ^{2}}{\lambda }}.}$

Without loss of generality, let the ground state be in the N-th direction:

${\displaystyle \phi _{0}^{i}=\left(0,\cdots ,0,{\frac {\mu }{\sqrt {\lambda }}}\right).}$

The original N fields can be rewritten as:

${\displaystyle \phi ^{i}(x)=\left(\pi ^{1}(x),\cdots ,\pi ^{N-1}(x),{\frac {\mu }{\sqrt {\lambda }}}+\sigma (x)\right),}$

and the original Lagrangian density as:

${\displaystyle {\mathcal {L}}={\frac {1}{2}}(\partial _{\mu }\pi ^{k})(\partial ^{\mu }\pi ^{k})+{\frac {1}{2}}(\partial _{\mu }\sigma )(\partial ^{\mu }\sigma )-{\frac {1}{2}}(2\mu ^{2})\sigma ^{2}-{\sqrt {\lambda }}\mu \sigma ^{3}-{\sqrt {\lambda }}\mu \pi ^{k}\pi ^{k}\sigma -{\frac {\lambda }{2}}\pi ^{k}\pi ^{k}\sigma ^{2}-{\frac {\lambda }{4}}(\pi ^{k}\pi ^{k})^{2},}$

where k = 1,...,N-1. The original O(N) global symmetry is no longer manifest, leaving only the subgroup O(N-1). The larger symmetry before spontaneous symmetry breaking is said to be "hidden" or spontaneously broken.^[1]:349-350

Goldstone's theorem states that under spontaneous symmetry breaking, every broken continuous global symmetry leads to a massless field called the Goldstone boson. In the above example, O(N) has N(N-1)/2 continuous symmetries (the dimension of its Lie algebra), while O(N-1) has (N-1)(N-2)/2. The number of broken symmetries is their difference, N-1, which corresponds to the N-1 massless fields π^k.^[1]:351

On the other hand, when a gauge (as opposed to global) symmetry is spontaneously broken, the resulting Goldstone boson is "eaten" by the corresponding gauge boson by becoming an additional degree of freedom for the gauge boson. The Goldstone boson equivalence theorem states that at high energy, the amplitude for emission or absorption of a longitudinally polarised massive gauge boson becomes equal to the amplitude for emission or absorption of the Goldstone boson that was eaten by the gauge boson.^[1]:743-744

In the QFT of ferromagnetism, spontaneous symmetry breaking can explain the alignment of magnetic dipoles at low temperatures.^[20]:199 In the Standard Model of elementary particles, the W and Z bosons, which would otherwise be massless as a result of gauge symmetry, acquire mass through spontaneous symmetry breaking of the Higgs boson, a process called the Higgs mechanism.^[1]:690

Supersymmetry[edit]

Main article: Supersymmetry

All experimentally known symmetries in nature relate bosons to bosons and fermions to fermions. Theorists have hypothesised the existence of a type of symmetry, called supersymmetry, that relates bosons and fermions.^{[1]:795[20]:443}

The Standard Model obeys Poincaré symmetry, whose generators are spacetime translation P^μ and Lorentz transformation J_μν.^[28]:58–60 In addition to these generators, supersymmetry in (3+1)-dimensions includes additional generators Q_α, called supercharges, which themselves transform as Weyl fermions.^{[1]:795[20]:444} The symmetry group generated by all these generators is known as the super-Poincaré group. In general there can be more than one set of supersymmetry generators, Q_α^I, I = 1, ..., N, which generate the corresponding N = 1 supersymmetry, N = 2 supersymmetry, and so on.^{[1]:795[20]:450} Supersymmetry can also be constructed in other dimensions,^[29] most notably in (1+1) dimensions for its application in superstring theory.^[30]

The Lagrangian of a supersymmetric theory must be invariant under the action of the super-Poincaré group.^[20]:448 Examples of such theories include: Minimal Supersymmetric Standard Model (MSSM), N = 4 supersymmetric Yang–Mills theory,^[20]:450 and superstring theory. In a supersymmetric theory, every fermion has a bosonic superpartner and vice versa.^[20]:444

If supersymmetry is promoted to a local symmetry, then the resultant gauge theory is an extension of general relativity called supergravity.^[31]

Supersymmetry is a potential solution to many current problems in physics. For example, the hierarchy problem of the Standard Model — why the mass of the Higgs boson is not radiatively corrected (under renormalisation) to a very high scale such as the grand unified scale or the Planck scale — can be resolved by relating the Higgs field and its superpartner, the Higgsino. Radiative corrections due to Higgs boson loops in Feynman diagrams are cancelled by corresponding Higgsino loops. Supersymmetry also offers answers to the grand unification of all gauge coupling constants in the Standard Model as well as the nature of dark matter.^{[1]:796-797[32]}

Nevertheless, as of 2018, experiments have yet to provide evidence for the existence of supersymmetric particles. If supersymmetry were a true symmetry of nature, then it must be a broken symmetry, and the energy of symmetry breaking must be higher than those achievable by present-day experiments.^{[1]:797[20]:443}

Other spacetimes[edit]

The ϕ⁴ theory, QED, QCD, as well as the whole Standard Model all assume a (3+1)-dimensional Minkowski space (3 spatial and 1 time dimensions) as the background on which the quantum fields are defined. However, QFT a priori imposes no restriction on the number of dimensions nor the geometry of spacetime.

In condensed matter physics, QFT is used to describe (2+1)-dimensional electron gases.^[33] In high-energy physics, string theory is a type of (1+1)-dimensional QFT,^[20]:452[14] while Kaluza–Klein theory uses gravity in extra dimensions to produce gauge theories in lower dimensions.^[20]:428-429

In Minkowski space, the flat metric η_μν is used to raise and lower spacetime indices in the Lagrangian, e.g.

${\displaystyle A_{\mu }A^{\mu }=\eta _{\mu \nu }A^{\mu }A^{\nu },\quad \partial _{\mu }\phi \partial ^{\mu }\phi =\eta ^{\mu \nu }\partial _{\mu }\phi \partial _{\nu }\phi ,}$

where η^μν is the inverse of η_μν satisfying η^μρη^ρν = δ^μ_ν. For QFTs in curved spacetime on the other hand, a general metric (such as the Schwarzschild metric describing a black hole) is used:

${\displaystyle A_{\mu }A^{\mu }=g_{\mu \nu }A^{\mu }A^{\nu },\quad \partial _{\mu }\phi \partial ^{\mu }\phi =g^{\mu \nu }\partial _{\mu }\phi \partial _{\nu }\phi ,}$

where g^μν is the inverse of g_μν. For a real scalar field, the Lagrangian density in a general spacetime background is

${\displaystyle {\mathcal {L}}={\sqrt {|g|}}\left({\frac {1}{2}}g^{\mu \nu }\nabla _{\mu }\phi \nabla _{\nu }\phi -{\frac {1}{2}}m^{2}\phi ^{2}\right),}$

where g = det(g_μν), and ∇_μ denotes the covariant derivative.^[34] The Lagrangian of a QFT, hence its calculational results and physical predictions, depends on the geometry of the spacetime background.

In the physics of electromagnetism, the Abraham–Lorentz force (also Lorentz–Abraham force) is the recoil force on an accelerating charged particle caused by the particle emitting electromagnetic radiation. It is also called the radiation reaction force, radiation damping force^[1] or the self-force.^[2]

The formula predates the theory of special relativity and is not valid at velocities on the order of the speed of light. Its relativistic generalization is called the Abraham–Lorentz–Dirac force. Both of these are in the domain of classical physics, not quantum physics, and therefore may not be valid at distances of roughly the Compton wavelength or below.^[3] There is, however, an analogue of the formula that is both fully quantum and relativistic, called the "Abraham–Lorentz–Dirac–Langevin equation".^[4]

The force is proportional to the square of the object's charge, times the jerk (rate of change of acceleration) that it is experiencing. The force points in the direction of the jerk. For example, in a cyclotron, where the jerk points opposite to the velocity, the radiation reaction is directed opposite to the velocity of the particle, providing a braking action. The Abraham–Lorentz force is the source of the radiation resistance of a radio antenna radiating radio waves.

There are pathological solutions of the Abraham–Lorentz–Dirac equation in which a particle accelerates in advance of the application of a force, so-called pre-acceleration solutions. Since this would represent an effect occurring before its cause (retrocausality), some theories have speculated that the equation allows signals to travel backward in time, thus challenging the physical principle of causality. One resolution of this problem was discussed by Arthur D. Yaghjian^[5] and is further discussed by Fritz Rohrlich^[3] and Rodrigo Medina.^[6]

Definition and description[edit]

Mathematically, the Abraham–Lorentz force is given in SI units by

${\displaystyle \mathbf {F} _{\mathrm {rad} }={\frac {\mu _{0}q^{2}}{6\pi c}}\mathbf {\dot {a}} ={\frac {q^{2}}{6\pi \epsilon _{0}c^{3}}}\mathbf {\dot {a}} }$

or in Gaussian units by

${\displaystyle \mathbf {F} _{\mathrm {rad} }={2 \over 3}{\frac {q^{2}}{c^{3}}}\mathbf {\dot {a}} .}$

Here F_rad is the force, ${\displaystyle \mathbf {\dot {a}} }$ is the derivative of acceleration, or the third derivative of displacement), also called jerk, μ₀ is the magnetic constant, ε₀ is the electric constant, c is the speed of light in free space, and q is the electric charge of the particle.

Note that this formula is for non-relativistic velocities; Dirac simply renormalized the mass of the particle in the equation of motion, to find the relativistic version (below).

Physically, an accelerating charge emits radiation (according to the Larmor formula), which carries momentum away from the charge. Since momentum is conserved, the charge is pushed in the direction opposite the direction of the emitted radiation. In fact the formula above for radiation force can be derived from the Larmor formula, as shown below.

Background[edit]

In classical electrodynamics, problems are typically divided into two classes:

1. Problems in which the charge and current sources of fields are specified and the fields are calculated, and
2. The reverse situation, problems in which the fields are specified and the motion of particles are calculated.

In some fields of physics, such as plasma physics and the calculation of transport coefficients (conductivity, diffusivity, etc.), the fields generated by the sources and the motion of the sources are solved self-consistently. In such cases, however, the motion of a selected source is calculated in response to fields generated by all other sources. Rarely is the motion of a particle (source) due to the fields generated by that same particle calculated. The reason for this is twofold:

1. Neglect of the "self-fields" usually leads to answers that are accurate enough for many applications, and
2. Inclusion of self-fields leads to problems in physics such as renormalization, some of which are still unsolved, that relate to the very nature of matter and energy.

These conceptual problems created by self-fields are highlighted in a standard graduate text. [Jackson]

The difficulties presented by this problem touch one of the most fundamental aspects of physics, the nature of the elementary particle. Although partial solutions, workable within limited areas, can be given, the basic problem remains unsolved. One might hope that the transition from classical to quantum-mechanical treatments would remove the difficulties. While there is still hope that this may eventually occur, the present quantum-mechanical discussions are beset with even more elaborate troubles than the classical ones. It is one of the triumphs of comparatively recent years (~ 1948–1950) that the concepts of Lorentz covariance and gauge invariance were exploited sufficiently cleverly to circumvent these difficulties in quantum electrodynamics and so allow the calculation of very small radiative effects to extremely high precision, in full agreement with experiment. From a fundamental point of view, however, the difficulties remain.

The Abraham–Lorentz force is the result of the most fundamental calculation of the effect of self-generated fields. It arises from the observation that accelerating charges emit radiation. The Abraham–Lorentz force is the average force that an accelerating charged particle feels in the recoil from the emission of radiation. The introduction of quantum effects leads one to quantum electrodynamics. The self-fields in quantum electrodynamics generate a finite number of infinities in the calculations that can be removed by the process of renormalization. This has led to a theory that is able to make the most accurate predictions that humans have made to date. (See precision tests of QED.) The renormalization process fails, however, when applied to the gravitational force. The infinities in that case are infinite in number, which causes the failure of renormalization. Therefore, general relativity has an unsolved self-field problem. String theory and loop quantum gravity are current attempts to resolve this problem, formally called the problem of radiation reaction or the problem of self-force.

Derivation[edit]

The simplest derivation for the self-force is found for periodic motion from the Larmor formula for the power radiated from a point charge:

${\displaystyle P={\frac {\mu _{0}q^{2}}{6\pi c}}\mathbf {a} ^{2}}$.

If we assume the motion of a charged particle is periodic, then the average work done on the particle by the Abraham–Lorentz force is the negative of the Larmor power integrated over one period from ${\displaystyle \tau _{1}}$ to ${\displaystyle \tau _{2}}$:

${\displaystyle \int _{\tau _{1}}^{\tau _{2}}\mathbf {F} _{\mathrm {rad} }\cdot \mathbf {v} dt=\int _{\tau _{1}}^{\tau _{2}}-Pdt=-\int _{\tau _{1}}^{\tau _{2}}{\frac {\mu _{0}q^{2}}{6\pi c}}\mathbf {a} ^{2}dt=-\int _{\tau _{1}}^{\tau _{2}}{\frac {\mu _{0}q^{2}}{6\pi c}}{\frac {d\mathbf {v} }{dt}}\cdot {\frac {d\mathbf {v} }{dt}}dt}$.

The above expression can be integrated by parts. If we assume that there is periodic motion, the boundary term in the integral by parts disappears:

${\displaystyle \int _{\tau _{1}}^{\tau _{2}}\mathbf {F} _{\mathrm {rad} }\cdot \mathbf {v} dt=-{\frac {\mu _{0}q^{2}}{6\pi c}}{\frac {d\mathbf {v} }{dt}}\cdot \mathbf {v} {\bigg |}_{\tau _{1}}^{\tau _{2}}+\int _{\tau _{1}}^{\tau _{2}}{\frac {\mu _{0}q^{2}}{6\pi c}}{\frac {d^{2}\mathbf {v} }{dt^{2}}}\cdot \mathbf {v} dt=-0+\int _{\tau _{1}}^{\tau _{2}}{\frac {\mu _{0}q^{2}}{6\pi c}}\mathbf {\dot {a}} \cdot \mathbf {v} dt}$.

Clearly, we can identify

${\displaystyle \mathbf {F} _{\mathrm {rad} }={\frac {\mu _{0}q^{2}}{6\pi c}}\mathbf {\dot {a}} }$.

A more rigorous derivation, which does not require periodic motion, was found using an Effective Field Theory formulation.^[7][8] An alternative derivation, finding the fully relativistic expression, was found by Dirac.

Signals from the future[edit]

Below is an illustration of how a classical analysis can lead to surprising results. The classical theory can be seen to challenge standard pictures of causality, thus signaling either a breakdown or a need for extension of the theory. In this case the extension is to quantum mechanics and its relativistic counterpart quantum field theory. See the quote from Rohrlich ^[3] in the introduction concerning "the importance of obeying the validity limits of a physical theory".

For a particle in an external force ${\displaystyle \mathbf {F} _{\mathrm {ext} }}$, we have

${\displaystyle m{\dot {\mathbf {v} }}=\mathbf {F} _{\mathrm {rad} }+\mathbf {F} _{\mathrm {ext} }=mt_{0}{\ddot {\mathbf {v} }}+\mathbf {F} _{\mathrm {ext} }.}$

where

${\displaystyle t_{0}={\frac {\mu _{0}q^{2}}{6\pi mc}}.}$

This equation can be integrated once to obtain

${\displaystyle m{\dot {\mathbf {v} }}={1 \over t_{0}}\int _{t}^{\infty }\exp \left(-{t'-t \over t_{0}}\right)\,\mathbf {F} _{\mathrm {ext} }(t')\,dt'.}$

The integral extends from the present to infinitely far in the future. Thus future values of the force affect the acceleration of the particle in the present. The future values are weighted by the factor

${\displaystyle \exp \left(-{t'-t \over t_{0}}\right)}$

which falls off rapidly for times greater than ${\displaystyle t_{0}}$ in the future. Therefore, signals from an interval approximately ${\displaystyle t_{0}}$ into the future affect the acceleration in the present. For an electron, this time is approximately ${\displaystyle 10^{-24}}$ sec, which is the time it takes for a light wave to travel across the "size" of an electron, the classical electron radius. One way to define this "size" is as follows: it is (up to some constant factor) the distance ${\displaystyle r}$ such that two electrons placed at rest at a distance ${\displaystyle r}$ apart and allowed to fly apart, would have sufficient energy to reach half the speed of light. In other words, it forms the length (or time, or energy) scale where something as light as an electron would be fully relativistic. It is worth noting that this expression does not involve the Planck constant at all, so although it indicates something is wrong at this length scale, it does not directly relate to quantum uncertainty, or to the frequency–energy relation of a photon. Although it is common in quantum mechanics to treat ${\displaystyle \hbar \to 0}$ as a "classical limit", some speculate that even the classical theory needs renormalization, no matter how the Planck constant would be fixed.

Abraham–Lorentz–Dirac force[edit]

To find the relativistic generalization, Dirac renormalized the mass in the equation of motion with the Abraham–Lorentz force in 1938. This renormalized equation of motion is called the Abraham–Lorentz–Dirac equation of motion.^[9]

Definition[edit]

The expression derived by Dirac is given in signature (−, +, +, +) by

${\displaystyle F_{\mu }^{\mathrm {rad} }={\frac {\mu _{o}q^{2}}{6\pi mc}}\left[{\frac {d^{2}p_{\mu }}{d\tau ^{2}}}-{\frac {p_{\mu }}{m^{2}c^{2}}}\left({\frac {dp_{\nu }}{d\tau }}{\frac {dp^{\nu }}{d\tau }}\right)\right].}$

With Liénard's relativistic generalization of Larmor's formula in the co-moving frame,

${\displaystyle P={\frac {\mu _{o}q^{2}a^{2}\gamma ^{6}}{6\pi c}},}$

one can show this to be a valid force by manipulating the time average equation for power:

${\displaystyle {\frac {1}{\Delta t}}\int _{0}^{t}Pdt={\frac {1}{\Delta t}}\int _{0}^{t}{\textbf {F}}\cdot {\textbf {v}}\,dt.}$

Paradoxes[edit]

Similar to the non-relativistic case, there are pathological solutions using the Abraham–Lorentz–Dirac equation that anticipate a change in the external force and according to which the particle accelerates in advance of the application of a force, so-called preacceleration solutions. One resolution of this problem was discussed by Yaghjian,^[5] and is further discussed by Rohrlich^[3] and Medina.^[6]

COMO TAMBÉM  EM FUNÇÕES E EQUAÇÕES DE:

PRESENTES ABAIXO.

• Abraham–Lorentz force
• AdS/CFT correspondence
• Introduction to quantum mechanics
• Common integrals in quantum field theory
• Einstein–Maxwell–Dirac equations
• Form factor (quantum field theory)
• Green–Kubo relations
• Green's function (many-body theory)
• List of quantum field theories
• Quantization of a field
• Quantum electrodynamics
• Quantum field theory in curved spacetime
• Quantum flavordynamics
• Quantum hydrodynamics
• Quantum triviality
• Relation between Schrödinger's equation and the path integral formulation of quantum mechanics
• Relationship between string theory and quantum field theory
• Schwinger–Dyson equation
• Static forces and virtual-particle exchange
• Symmetry in quantum mechanics
• Theoretical and experimental justification for the Schrödinger equation
• Ward–Takahashi identity
• Wheeler–Feynman absorber theory
• Wigner's classification
• Wigner's theorem