MECÂNICA GRACELI GENERALIZADA - QUÂNTICA TENSORIAL DIMENSIONAL RELATIVISTA DE CAMPOS [1.207]

MECÂNICA GRACELI GENERALIZADA - QUÂNTICA TENSORIAL DIMENSIONAL RELATIVISTA DE CAMPOS.

MECÃNICA GRACELI GERAL - QTDRC.

$equação Graceli dimensional relativista tensorial quântica de campos$

$\psi ^{*}$

$[$ $\psi ^{*}$ $\psi ^{*}$ $G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$ $T_{\mu \nu }$ $/$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ $.$ = $\left|\psi ({\vec {r}},t)\right\rangle$ /

$G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$ $T_{\mu \nu }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ $.$ = $\left|\psi ({\vec {r}},t)\right\rangle$ +

$\psi ^{*}$

//////

[

$\psi ^{*}$

$TeoriaInteraçãomediadorMagnitude relativaComportamentoFaixaCromodinâmicaForça nuclear forteGlúon10411/r71,4 × 10-15 mEletrodinâmicaForça eletromagnéticaFóton10391/r2infinitoFlavordinâmicaForça nuclear fracaBósons W e Z10291/r5 até 1/r710-18 mGeometrodinâmicaForça gravitacionalgráviton101/r2infinito$

Teoria	Interação	mediador	Magnitude relativa	Comportamento	Faixa
Cromodinâmica	Força nuclear forte	Glúon	1041	1/r7	1,4 × 10-15 m
Eletrodinâmica	Força eletromagnética	Fóton	1039	1/r2	infinito
Flavordinâmica	Força nuclear fraca	Bósons W e Z	1029	1/r5 até 1/r7	10-18 m
Geometrodinâmica	Força gravitacional	gráviton	10	1/r2	infinito

$G* = OPERADOR DE DIMENSÕES DE GRACELI.$

DIMENSÕES DE GRACELI SÃO TODA FORMA DE TENSORES, ESTRUTURAS, ENERGIAS, ACOPLAMENTOS, , INTERAÇÕES DE CAMPOS E ENERGIAS, DISTRIBUIÇÕES ELETRÔNICAS, ESTADOS FÍSICOS, ESTADOS QUÂNTICOS, ESTADOS FÍSICOS DE ENERGIAS DE GRACELI, E OUTROS.

/ $\psi ^{*}$ [ $\psi ^{*}$ $T_{\mu \nu }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$

MECÂNICA GRACELI GENERALIZADA - QUÂNTICA TENSORIAL DIMENSIONAL RELATIVISTA DE INTERAÇÕES DE CAMPOS. EM ;

MECÂNICA GRACELI REPRESENTADA POR TRANSFORMADA.

dd = dd [G] = DERIVADA DE DIMENSÕES DE GRACELI.

- [ $G*$ $\hbar$ $.$ $\psi$

G { f [dd]} ´[d] $\psi ^{*}$ $\hbar$ $\psi$ $f [d] G*$ $T_{\mu \nu }$ $\psi$ $dd [G]$

O ESTADO QUÂNTICO DE GRACELI

- [ $G*$ $\hbar$ $.$ $\psi$ $\left|\psi \right\rangle ={\frac {1}{\sqrt {2}}}{\bigg (}\left|\uparrow \downarrow \right\rangle -\left|\downarrow \uparrow \right\rangle {\bigg )},$ ]

$G* = DIMENSÕES DE GRACELI TAMBÉM ESTÁ RELACIONADO COM INTERAÇÕES DE ENERGIAS, QUÂNTICAS, RELATIVÍSTICAS, , E INTERAÇÕES DE CAMPOS.$

o tensor energia-momento $T_{\mu \nu }$ é aquele de um campo eletromagnético,

In particle physics, Fermi's interaction (also the Fermi theory of beta decay or the Fermi four-fermion interaction) is an explanation of the beta decay, proposed by Enrico Fermi in 1933.[1] The theory posits four fermions directly interacting with one another (at one vertex of the associated Feynman diagram). This interaction explains beta decay of a neutron by direct coupling of a neutron with an electron, a neutrino (later determined to be an antineutrino) and a proton.[2]

Fermi first introduced this coupling in his description of beta decay in 1933.[3] The Fermi interaction was the precursor to the theory for the weak interaction where the interaction between the proton–neutron and electron–antineutrino is mediated by a virtual W− boson, of which the Fermi theory is the low-energy effective field theory.

According to Eugene Wigner, who together with Jordan introduced the Jordan–Wigner transformation, Fermi's paper on beta decay was his main contribution to the history of physics.[4]

History of initial rejection and later publication

[edit]

Fermi first submitted his "tentative" theory of beta decay to the prestigious science journal Nature, which rejected it "because it contained speculations too remote from reality to be of interest to the reader."[5][6] It has been argued that Nature later admitted the rejection to be one of the great editorial blunders in its history, but Fermi's biographer David N. Schwartz has objected that this is both unproven and unlikely.[7] Fermi then submitted revised versions of the paper to Italian and German publications, which accepted and published them in those languages in 1933 and 1934.[8][9][10][11] The paper did not appear at the time in a primary publication in English.[5] An English translation of the seminal paper was published in the American Journal of Physics in 1968.[11]

Fermi found the initial rejection of the paper so troubling that he decided to take some time off from theoretical physics, and do only experimental physics. This would lead shortly to his famous work with activation of nuclei with slow neutrons.

The "tentativo"

[edit]

Definitions

[edit]

The theory deals with three types of particles presumed to be in direct interaction: initially a “heavy particle” in the “neutron state” ( $\rho =+1$ ), which then transitions into its “proton state” ( $\rho =-1$ ) with the emission of an electron and a neutrino.

Electron state

[edit]

\psi =\sum _{s}\psi _{s}a_{s},

/ $\psi ^{*}$ [ $\psi ^{*}$

T_{\mu \nu }

$-i\hbar {\frac {\partial }{\partial x_{n}}}$

where $\psi$ is the single-electron wavefunction, $\psi _{s}$ are its stationary states.

$a_{s}$ is the operator which annihilates an electron in state $s$ which acts on the Fock space as

a_{s}\Psi (N_{1},N_{2},\ldots ,N_{s},\ldots )=(-1)^{N_{1}+N_{2}+\cdots +N_{s}-1}(1-N_{s})\Psi (N_{1},N_{2},\ldots ,1-N_{s},\ldots ).

$a_{s}^{*}$ is the creation operator for electron state $s:$

a_{s}^{*}\Psi (N_{1},N_{2},\ldots ,N_{s},\ldots )=(-1)^{N_{1}+N_{2}+\cdots +N_{s}-1}N_{s}\Psi (N_{1},N_{2},\ldots ,1-N_{s},\ldots ).

Neutrino state

[edit]

Similarly,

\phi =\sum _{\sigma }\phi _{\sigma }b_{\sigma },

/ $\psi ^{*}$ [ $\psi ^{*}$

T_{\mu \nu }

$-i\hbar {\frac {\partial }{\partial x_{n}}}$

where $\phi$ is the single-neutrino wavefunction, and $\phi _{\sigma }$ are its stationary states.

$b_{\sigma }$ is the operator which annihilates a neutrino in state $\sigma$ which acts on the Fock space as

b_{\sigma }\Phi (M_{1},M_{2},\ldots ,M_{\sigma },\ldots )=(-1)^{M_{1}+M_{2}+\cdots +M_{\sigma }-1}(1-M_{\sigma })\Phi (M_{1},M_{2},\ldots ,1-M_{\sigma },\ldots ).

$b_{\sigma }^{*}$ is the creation operator for neutrino state $\sigma$ .

Heavy particle state

[edit]

$\rho$ is the operator introduced by Heisenberg (later generalized into isospin) that acts on a heavy particle state, which has eigenvalue +1 when the particle is a neutron, and −1 if the particle is a proton. Therefore, heavy particle states will be represented by two-row column vectors, where

{\begin{pmatrix}1\\0\end{pmatrix}}

represents a neutron, and

{\begin{pmatrix}0\\1\end{pmatrix}}

represents a proton (in the representation where $\rho$ is the usual $\sigma _{z}$ spin matrix).

The operators that change a heavy particle from a proton into a neutron and vice versa are respectively represented by

Q=\sigma _{x}-i\sigma _{y}={\begin{pmatrix}0&1\\0&0\end{pmatrix}}

and

Q^{*}=\sigma _{x}+i\sigma _{y}={\begin{pmatrix}0&0\\1&0\end{pmatrix}}.

$u_{n}$ resp. $v_{n}$ is an eigenfunction for a neutron resp. proton in the state $n$ .

Hamiltonian

[edit]

The Hamiltonian is composed of three parts: $H_{\text{h.p.}}$ , representing the energy of the free heavy particles, $H_{\text{l.p.}}$ , representing the energy of the free light particles, and a part giving the interaction $H_{\text{int.}}$ .

H_{\text{h.p.}}={\frac {1}{2}}(1+\rho )N+{\frac {1}{2}}(1-\rho )P,

/ $\psi ^{*}$ [ $\psi ^{*}$

T_{\mu \nu }

$-i\hbar {\frac {\partial }{\partial x_{n}}}$

where $N$ and $P$ are the energy operators of the neutron and proton respectively, so that if $\rho =1$ , $H_{\text{h.p.}}=N$ , and if $\rho =-1$ , $H_{\text{h.p.}}=P$ .

H_{\text{l.p.}}=\sum _{s}H_{s}N_{s}+\sum _{\sigma }K_{\sigma }M_{\sigma },

/ $\psi ^{*}$ [ $\psi ^{*}$

T_{\mu \nu }

$-i\hbar {\frac {\partial }{\partial x_{n}}}$

where $H_{s}$ is the energy of the electron in the $s^{\text{th}}$ state in the nucleus's Coulomb field, and $N_{s}$ is the number of electrons in that state; $M_{\sigma }$ is the number of neutrinos in the $\sigma ^{\text{th}}$ state, and $K_{\sigma }$ energy of each such neutrino (assumed to be in a free, plane wave state).

The interaction part must contain a term representing the transformation of a proton into a neutron along with the emission of an electron and a neutrino (now known to be an antineutrino), as well as a term for the inverse process; the Coulomb force between the electron and proton is ignored as irrelevant to the $\beta$ -decay process.

Fermi proposes two possible values for $H_{\text{int.}}$ : first, a non-relativistic version which ignores spin:

H_{\text{int.}}=g\left[Q\psi (x)\phi (x)+Q^{*}\psi ^{*}(x)\phi ^{*}(x)\right],

/ $\psi ^{*}$ [ $\psi ^{*}$

T_{\mu \nu }

$-i\hbar {\frac {\partial }{\partial x_{n}}}$

and subsequently a version assuming that the light particles are four-component Dirac spinors, but that speed of the heavy particles is small relative to $c$ and that the interaction terms analogous to the electromagnetic vector potential can be ignored:

H_{\text{int.}}=g\left[Q{\tilde {\psi }}^{*}\delta \phi +Q^{*}{\tilde {\psi }}\delta \phi ^{*}\right],

/ $\psi ^{*}$ [ $\psi ^{*}$

T_{\mu \nu }

$-i\hbar {\frac {\partial }{\partial x_{n}}}$

where $\psi$ and $\phi$ are now four-component Dirac spinors, ${\tilde {\psi }}$ represents the Hermitian conjugate of $\psi$ , and $\delta$ is a matrix

{\begin{pmatrix}0&-1&0&0\\1&0&0&0\\0&0&0&1\\0&0&-1&0\end{pmatrix}}.

Matrix elements

[edit]

The state of the system is taken to be given by the tuple $\rho ,n,N_{1},N_{2},\ldots ,M_{1},M_{2},\ldots ,$ where $\rho =\pm 1$ specifies whether the heavy particle is a neutron or proton, $n$ is the quantum state of the heavy particle, $N_{s}$ is the number of electrons in state $s$ and $M_{\sigma }$ is the number of neutrinos in state $\sigma$ .

Using the relativistic version of $H_{\text{int.}}$ , Fermi gives the matrix element between the state with a neutron in state $n$ and no electrons resp. neutrinos present in state $s$ resp. $\sigma$ , and the state with a proton in state $m$ and an electron and a neutrino present in states $s$ and $\sigma$ as

H_{\rho =-1,m,N_{s}=1,M_{\sigma }=1}^{\rho =1,n,N_{s}=0,M_{\sigma }=0}=\pm g\int v_{m}^{*}u_{n}{\tilde {\psi }}_{s}\delta \phi _{\sigma }^{*}d\tau ,

/ $\psi ^{*}$ [ $\psi ^{*}$

T_{\mu \nu }

$-i\hbar {\frac {\partial }{\partial x_{n}}}$

where the integral is taken over the entire configuration space of the heavy particles (except for $\rho$ ). The $\pm$ is determined by whether the total number of light particles is odd (−) or even (+).

Transition probability

[edit]

To calculate the lifetime of a neutron in a state $n$ according to the usual quantum perturbation theory, the above matrix elements must be summed over all unoccupied electron and neutrino states. This is simplified by assuming that the electron and neutrino eigenfunctions $\psi _{s}$ and $\phi _{\sigma }$ are constant within the nucleus (i.e., their Compton wavelength is much larger than the size of the nucleus). This leads to

H_{\rho =-1,m,N_{s}=1,M_{\sigma }=1}^{\rho =1,n,N_{s}=0,M_{\sigma }=0}=\pm g{\tilde {\psi }}_{s}\delta \phi _{\sigma }^{*}\int v_{m}^{*}u_{n}d\tau ,

/ $\psi ^{*}$ [ $\psi ^{*}$

T_{\mu \nu }

$-i\hbar {\frac {\partial }{\partial x_{n}}}$

where $\psi _{s}$ and $\phi _{\sigma }$ are now evaluated at the position of the nucleus.

According to Fermi's golden rule[further explanation needed], the probability of this transition is

{\begin{aligned}\left|a_{\rho =-1,m,N_{s}=1,M_{\sigma }=1}^{\rho =1,n,N_{s}=0,M_{\sigma }=0}\right|^{2}&=\left|H_{\rho =-1,m,N_{s}=1,M_{\sigma }=1}^{\rho =1,n,N_{s}=0,M_{\sigma }=0}\times {\frac {\exp {{\frac {2\pi i}{h}}(-W+H_{s}+K_{\sigma })t}-1}{-W+H_{s}+K_{\sigma }}}\right|^{2}\\&=4\left|H_{\rho =-1,m,N_{s}=1,M_{\sigma }=1}^{\rho =1,n,N_{s}=0,M_{\sigma }=0}\right|^{2}\times {\frac {\sin ^{2}\left({\frac {\pi t}{h}}(-W+H_{s}+K_{\sigma })\right)}{(-W+H_{s}+K_{\sigma })^{2}}},\end{aligned}}

/ $\psi ^{*}$ [ $\psi ^{*}$ $T_{\mu \nu }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$

where $W$ is the difference in the energy of the proton and neutron states.

Averaging over all positive-energy neutrino spin / momentum directions (where $\Omega ^{-1}$ is the density of neutrino states, eventually taken to infinity), we obtain

\left\langle \left|H_{\rho =-1,m,N_{s}=1,M_{\sigma }=1}^{\rho =1,n,N_{s}=0,M_{\sigma }=0}\right|^{2}\right\rangle _{\text{avg}}={\frac {g^{2}}{4\Omega }}\left|\int v_{m}^{*}u_{n}d\tau \right|^{2}\left({\tilde {\psi }}_{s}\psi _{s}-{\frac {\mu c^{2}}{K_{\sigma }}}{\tilde {\psi }}_{s}\beta \psi _{s}\right),

where $\mu$ is the rest mass of the neutrino and $\beta$ is the Dirac matrix.

Noting that the transition probability has a sharp maximum for values of $p_{\sigma }$ for which $-W+H_{s}+K_{\sigma }=0$ , this simplifies to [further explanation needed]

t{\frac {8\pi ^{3}g^{2}}{h^{4}}}\times \left|\int v_{m}^{*}u_{n}d\tau \right|^{2}{\frac {p_{\sigma }^{2}}{v_{\sigma }}}\left({\tilde {\psi }}_{s}\psi _{s}-{\frac {\mu c^{2}}{K_{\sigma }}}{\tilde {\psi }}_{s}\beta \psi _{s}\right),

/ $\psi ^{*}$ [ $\psi ^{*}$

T_{\mu \nu }

$-i\hbar {\frac {\partial }{\partial x_{n}}}$

where $p_{\sigma }$ and $K_{\sigma }$ is the values for which $-W+H_{s}+K_{\sigma }=0$ .

Fermi makes three remarks about this function:

Since the neutrino states are considered to be free, $K_{\sigma }>\mu c^{2}$ and thus the upper limit on the continuous $\beta$ -spectrum is $H_{s}\leq W-\mu c^{2}$ .
Since for the electrons $H_{s}>mc^{2}$ , in order for $\beta$ -decay to occur, the proton–neutron energy difference must be $W\geq (m+\mu )c^{2}$
The factor

Q_{mn}^{*}=\int v_{m}^{*}u_{n}d\tau

/ $\psi ^{*}$ [ $\psi ^{*}$

T_{\mu \nu }

$-i\hbar {\frac {\partial }{\partial x_{n}}}$

in the transition probability is normally of magnitude 1, but in special circumstances it vanishes; this leads to (approximate) selection rules for

\beta

-decay.

Forbidden transitions

[edit]

As noted above, when the inner product $Q_{mn}^{*}$ between the heavy particle states $u_{n}$ and $v_{m}$ vanishes, the associated transition is "forbidden" (or, rather, much less likely than in cases where it is closer to 1).

If the description of the nucleus in terms of the individual quantum states of the protons and neutrons is accurate to a good approximation, $Q_{mn}^{*}$ vanishes unless the neutron state $u_{n}$ and the proton state $v_{m}$ have the same angular momentum; otherwise, the total angular momentum of the entire nucleus before and after the decay must be used.

Influence

[edit]

Shortly after Fermi's paper appeared, Werner Heisenberg noted in a letter to Wolfgang Pauli[12] that the emission and absorption of neutrinos and electrons in the nucleus should, at the second order of perturbation theory, lead to an attraction between protons and neutrons, analogously to how the emission and absorption of photons leads to the electromagnetic force. He found that the force would be of the form ${\frac {\text{Const.}}{r^{5}}}$ , but noted that contemporary experimental data led to a value that was too small by a factor of a million.[13]

The following year, Hideki Yukawa picked up on this idea,[14] but in his theory the neutrinos and electrons were replaced by a new hypothetical particle with a rest mass approximately 200 times heavier than the electron.[15]

Later developments

[edit]

Fermi's four-fermion theory describes the weak interaction remarkably well. Unfortunately, the calculated cross-section, or probability of interaction, grows as the square of the energy $\sigma \approx G_{\rm {F}}^{2}E^{2}$ . Since this cross section grows without bound, the theory is not valid at energies much higher than about 100 GeV. Here $G F$ is the Fermi constant, which denotes the strength of the interaction. This eventually led to the replacement of the four-fermion contact interaction by a more complete theory (UV completion)—an exchange of a W or Z boson as explained in the electroweak theory.

Fermi's interaction showing the 4-point fermion vector current, coupled under Fermi's Coupling Constant $G F$ . Fermi's Theory was the first theoretical effort in describing nuclear decay rates for β decay.

The interaction could also explain muon decay via a coupling of a muon, electron-antineutrino, muon-neutrino and electron, with the same fundamental strength of the interaction. This hypothesis was put forward by Gershtein and Zeldovich and is known as the Vector Current Conservation hypothesis.[16]

In the original theory, Fermi assumed that the form of interaction is a contact coupling of two vector currents. Subsequently, it was pointed out by Lee and Yang that nothing prevented the appearance of an axial, parity violating current, and this was confirmed by experiments carried out by Chien-Shiung Wu.[17][18]

The inclusion of parity violation in Fermi's interaction was done by George Gamow and Edward Teller in the so-called Gamow–Teller transitions which described Fermi's interaction in terms of parity-violating "allowed" decays and parity-conserving "superallowed" decays in terms of anti-parallel and parallel electron and neutrino spin states respectively. Before the advent of the electroweak theory and the Standard Model, George Sudarshan and Robert Marshak, and also independently Richard Feynman and Murray Gell-Mann, were able to determine the correct tensor structure (vector minus axial vector, $V - A$ ) of the four-fermion interaction.[19][20]

Fermi constant

[edit]

The most precise experimental determination of the Fermi constant comes from measurements of the muon lifetime, which is inversely proportional to the square of $G F$ (when neglecting the muon mass against the mass of the W boson).[21] In modern terms, the "reduced Fermi constant", that is, the constant in natural units is[3][22]

G_{\rm {F}}^{0}={\frac {G_{\rm {F}}}{(\hbar c)^{3}}}={\frac {\sqrt {2}}{8}}{\frac {g^{2}}{M_{\rm {W}}^{2}c^{4}}}=1.1663787(6)\times 10^{-5}\;{\textrm {GeV}}^{-2}\approx 4.5437957\times 10^{14}\;{\textrm {J}}^{-2}\ .

/ $\psi ^{*}$ [ $\psi ^{*}$ $T_{\mu \nu }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$

Here, $g$ is the coupling constant of the weak interaction, and $M W$ is the mass of the W boson, which mediates the decay in question.

In the Standard Model, the Fermi constant is related to the Higgs vacuum expectation value

v=\left({\sqrt {2}}\,G_{\rm {F}}^{0}\right)^{-1/2}\simeq 246.22\;{\textrm {GeV}}

.[23]

/ $\psi ^{*}$ [ $\psi ^{*}$

T_{\mu \nu }

$-i\hbar {\frac {\partial }{\partial x_{n}}}$

More directly, approximately (tree level for the standard model),

G_{\rm {F}}^{0}\simeq {\frac {\pi \alpha }{{\sqrt {2}}~M_{\rm {W}}^{2}(1-M_{\rm {W}}^{2}/M_{\rm {Z}}^{2})}}.

/ $\psi ^{*}$ [ $\psi ^{*}$

T_{\mu \nu }

$-i\hbar {\frac {\partial }{\partial x_{n}}}$

This can be further simplified in terms of the Weinberg angle using the relation between the W and Z bosons with $M_{\text{Z}}={\frac {M_{\text{W}}}{\cos \theta _{\text{W}}}}$ , / $\psi ^{*}$ [ $\psi ^{*}$ $T_{\mu \nu }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$

$so that$

G_{\rm {F}}^{0}\simeq {\frac {\pi \alpha }{{\sqrt {2}}~M_{\rm {Z}}^{2}\cos ^{2}\theta _{\rm {W}}\sin ^{2}\theta _{\rm {W}}}}.

/ $\psi ^{*}$ [ $\psi ^{*}$

T_{\mu \nu }

$-i\hbar {\frac {\partial }{\partial x_{n}}}$

In quantum field theory, a fermionic field is a quantum field whose quanta are fermions; that is, they obey Fermi–Dirac statistics. Fermionic fields obey canonical anticommutation relations rather than the canonical commutation relations of bosonic fields.

The most prominent example of a fermionic field is the Dirac field, which describes fermions with spin-1/2: electrons, protons, quarks, etc. The Dirac field can be described as either a 4-component spinor or as a pair of 2-component Weyl spinors. Spin-1/2 Majorana fermions, such as the hypothetical neutralino, can be described as either a dependent 4-component Majorana spinor or a single 2-component Weyl spinor. It is not known whether the neutrino is a Majorana fermion or a Dirac fermion; observing neutrinoless double-beta decay experimentally would settle this question.

Basic properties

[edit]

Free (non-interacting) fermionic fields obey canonical anticommutation relations; i.e., involve the anticommutators {a, b} = ab + ba, rather than the commutators [a, b] = ab − ba of bosonic or standard quantum mechanics. Those relations also hold for interacting fermionic fields in the interaction picture, where the fields evolve in time as if free and the effects of the interaction are encoded in the evolution of the states.

It is these anticommutation relations that imply Fermi–Dirac statistics for the field quanta. They also result in the Pauli exclusion principle: two fermionic particles cannot occupy the same state at the same time.

Dirac fields

[edit]

The prominent example of a spin-1/2 fermion field is the Dirac field (named after Paul Dirac), and denoted by $\psi (x)$ . The equation of motion for a free spin 1/2 particle is the Dirac equation,

\left(i\gamma ^{\mu }\partial _{\mu }-m\right)\psi (x)=0.\,

/ $\psi ^{*}$ [ $\psi ^{*}$

T_{\mu \nu }

$-i\hbar {\frac {\partial }{\partial x_{n}}}$

where $\gamma ^{\mu }$ are gamma matrices and $m$ is the mass. The simplest possible solutions $\psi (x)$ to this equation are plane wave solutions, $u(p)e^{-ip\cdot x}\,$ and $v(p)e^{ip\cdot x}\,$ . These plane wave solutions form a basis for the Fourier components of $\psi (x)$ , allowing for the general expansion of the wave function as follows,

\psi _{\alpha }(x)=\int {\frac {d^{3}p}{(2\pi )^{3}}}{\frac {1}{\sqrt {2E_{p}}}}\sum _{s}\left(a_{\mathbf {p} }^{s}u_{\alpha }^{s}(p)e^{-ip\cdot x}+b_{\mathbf {p} }^{s\dagger }v_{\alpha }^{s}(p)e^{ip\cdot x}\right).\,

/ $\psi ^{*}$ [ $\psi ^{*}$

T_{\mu \nu }

$-i\hbar {\frac {\partial }{\partial x_{n}}}$

u and v are spinors, labelled by spin, s and spinor indices $\alpha \in \{0,1,2,3\}$ . For the electron, a spin 1/2 particle, s = +1/2 or s = −1/2. The energy factor is the result of having a Lorentz invariant integration measure. In second quantization, $\psi (x)$ is promoted to an operator, so the coefficients of its Fourier modes must be operators too. Hence, $a_{\mathbf {p} }^{s}$ and $b_{\mathbf {p} }^{s\dagger }$ are operators. The properties of these operators can be discerned from the properties of the field. $\psi (x)$ and $\psi (y)^{\dagger }$ obey the anticommutation relations:

\left\{\psi _{\alpha }(\mathbf {x} ),\psi _{\beta }^{\dagger }(\mathbf {y} )\right\}=\delta ^{(3)}(\mathbf {x} -\mathbf {y} )\delta _{\alpha \beta }.

/ $\psi ^{*}$ [ $\psi ^{*}$

T_{\mu \nu }

$-i\hbar {\frac {\partial }{\partial x_{n}}}$

We impose an anticommutator relation (as opposed to a commutation relation as we do for the bosonic field) in order to make the operators compatible with Fermi–Dirac statistics. By putting in the expansions for $\psi (x)$ and $\psi (y)$ , the anticommutation relations for the coefficients can be computed.

\left\{a_{\mathbf {p} }^{r},a_{\mathbf {q} }^{s\dagger }\right\}=\left\{b_{\mathbf {p} }^{r},b_{\mathbf {q} }^{s\dagger }\right\}=(2\pi )^{3}\delta ^{3}(\mathbf {p} -\mathbf {q} )\delta ^{rs},\,

/ $\psi ^{*}$ [ $\psi ^{*}$

T_{\mu \nu }

$-i\hbar {\frac {\partial }{\partial x_{n}}}$

In a manner analogous to non-relativistic annihilation and creation operators and their commutators, these algebras lead to the physical interpretation that $a_{\mathbf {p} }^{s\dagger }$ creates a fermion of momentum p and spin s, and $b_{\mathbf {q} }^{r\dagger }$ creates an antifermion of momentum q and spin r. The general field $\psi (x)$ is now seen to be a weighted (by the energy factor) summation over all possible spins and momenta for creating fermions and antifermions. Its conjugate field, ${\overline {\psi }}\ {\stackrel {\mathrm {def} }{=}}\ \psi ^{\dagger }\gamma ^{0}$ , is the opposite, a weighted summation over all possible spins and momenta for annihilating fermions and antifermions.

With the field modes understood and the conjugate field defined, it is possible to construct Lorentz invariant quantities for fermionic fields. The simplest is the quantity ${\overline {\psi }}\psi \,$ . This makes the reason for the choice of ${\overline {\psi }}=\psi ^{\dagger }\gamma ^{0}$ clear. This is because the general Lorentz transform on $\psi$ is not unitary so the quantity $\psi ^{\dagger }\psi$ would not be invariant under such transforms, so the inclusion of $\gamma ^{0}\,$ is to correct for this. The other possible non-zero Lorentz invariant quantity, up to an overall conjugation, constructible from the fermionic fields is ${\overline {\psi }}\gamma ^{\mu }\partial _{\mu }\psi$ .

Since linear combinations of these quantities are also Lorentz invariant, this leads naturally to the Lagrangian density for the Dirac field by the requirement that the Euler–Lagrange equation of the system recover the Dirac equation.

{\mathcal {L}}_{D}={\overline {\psi }}\left(i\gamma ^{\mu }\partial _{\mu }-m\right)\psi \,

/ $\psi ^{*}$ [ $\psi ^{*}$

T_{\mu \nu }

$-i\hbar {\frac {\partial }{\partial x_{n}}}$

Such an expression has its indices suppressed. When reintroduced the full expression is

{\mathcal {L}}_{D}={\overline {\psi }}_{a}\left(i\gamma _{ab}^{\mu }\partial _{\mu }-m\mathbb {I} _{ab}\right)\psi _{b}\,

/ $\psi ^{*}$ [ $\psi ^{*}$

T_{\mu \nu }

$-i\hbar {\frac {\partial }{\partial x_{n}}}$

The Hamiltonian (energy) density can also be constructed by first defining the momentum canonically conjugate to $\psi (x)$ , called $\Pi (x):$

\Pi \ {\overset {\mathrm {def} }{=}}\ {\frac {\partial {\mathcal {L}}_{D}}{\partial (\partial _{0}\psi )}}=i\psi ^{\dagger }\,.

/ $\psi ^{*}$ [ $\psi ^{*}$

T_{\mu \nu }

$-i\hbar {\frac {\partial }{\partial x_{n}}}$

With that definition of $\Pi$ , the Hamiltonian density is:

{\mathcal {H}}_{D}={\overline {\psi }}\left[-i{\vec {\gamma }}\cdot {\vec {\nabla }}+m\right]\psi \,,

/ $\psi ^{*}$ [ $\psi ^{*}$

T_{\mu \nu }

$-i\hbar {\frac {\partial }{\partial x_{n}}}$

where ${\vec {\nabla }}$ is the standard gradient of the space-like coordinates, and ${\vec {\gamma }}$ is a vector of the space-like $\gamma$ matrices. It is surprising that the Hamiltonian density doesn't depend on the time derivative of $\psi$ , directly, but the expression is correct.

Given the expression for $\psi (x)$ we can construct the Feynman propagator for the fermion field:

D_{F}(x-y)=\left\langle 0\left|T(\psi (x){\overline {\psi }}(y))\right|0\right\rangle

/ $\psi ^{*}$ [ $\psi ^{*}$

T_{\mu \nu }

$-i\hbar {\frac {\partial }{\partial x_{n}}}$

we define the time-ordered product for fermions with a minus sign due to their anticommuting nature

T\left[\psi (x){\overline {\psi }}(y)\right]\ {\overset {\text{def}}{=}}\ \theta \left(x^{0}-y^{0}\right)\psi (x){\overline {\psi }}(y)-\theta \left(y^{0}-x^{0}\right){\overline {\psi }}(y)\psi (x).

/ $\psi ^{*}$ [ $\psi ^{*}$

T_{\mu \nu }

$-i\hbar {\frac {\partial }{\partial x_{n}}}$

Plugging our plane wave expansion for the fermion field into the above equation yields:

D_{F}(x-y)=\int {\frac {d^{4}p}{(2\pi )^{4}}}{\frac {i({p\!\!\!/}+m)}{p^{2}-m^{2}+i\epsilon }}e^{-ip\cdot (x-y)}

/ $\psi ^{*}$ [ $\psi ^{*}$

T_{\mu \nu }

$-i\hbar {\frac {\partial }{\partial x_{n}}}$

where we have employed the Feynman slash notation. This result makes sense since the factor

{\frac {i({p\!\!\!/}+m)}{p^{2}-m^{2}}}

/ $\psi ^{*}$ [ $\psi ^{*}$

T_{\mu \nu }

$-i\hbar {\frac {\partial }{\partial x_{n}}}$

is just the inverse of the operator acting on $\psi (x)$ in the Dirac equation. Note that the Feynman propagator for the Klein–Gordon field has this same property. Since all reasonable observables (such as energy, charge, particle number, etc.) are built out of an even number of fermion fields, the commutation relation vanishes between any two observables at spacetime points outside the light cone. As we know from elementary quantum mechanics two simultaneously commuting observables can be measured simultaneously. We have therefore correctly implemented Lorentz invariance for the Dirac field, and preserved causality.

More complicated field theories involving interactions (such as Yukawa theory, or quantum electrodynamics) can be analyzed too, by various perturbative and non-perturbative methods.

Dirac fields are an important ingredient of the Standard Model.

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MECÂNICA GRACELI GENERALIZADA - QUÂNTICA TENSORIAL DIMENSIONAL RELATIVISTA DE CAMPOS [1.207]

History of initial rejection and later publication

The "tentativo"

Definitions

Electron state

Neutrino state

Heavy particle state

Hamiltonian

Matrix elements

Transition probability

Forbidden transitions

Influence

Later developments

Fermi constant

Basic properties

Dirac fields

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