MECÂNICA GRACELI GENERALIZADA - QUÂNTICA DIMENSIONAL TENSORIAL RELATIVISTA DE CAMPOS [1.308]

MECÂNICA GRACELI GENERALIZADA - QUÂNTICA TENSORIAL DIMENSIONAL RELATIVISTA DE CAMPOS.

MECÃNICA GRACELI GERAL - QTDRC.

$equação Graceli dimensional relativista tensorial quântica de campos$

$\psi ^{*}$

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

$+$ $G$ * $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =

//////

[

$\psi ^{*}$

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

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.

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

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,

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

$G$ $E=mc^{2}$ ] $-i\hbar {\frac {\partial }{\partial x_{n}}}$

$/$ $G$ $E=mc^{2}$ ] $-i\hbar {\frac {\partial }{\partial x_{n}}}$

Em física quântica, a regra de ouro de Fermi expressa a taxa de transição (probabilidade por unidade de tempo) de um auto-estado de um Hamiltoniano $H_{0}$ para um contínuo de estados, devido a uma perturbação $H_{1}$ , que pode depender do tempo. Seu nome é uma homenagem ao físico italiano Enrico Fermi.

Dado um auto-estado $\scriptstyle |i\rangle$ do Hamiltoniano não perturbado $H_{0}$ , a probabilidade de transição para um estado $\scriptstyle |f\rangle$ é dado em primeira ordem de teoria de perturbação por

T_{i\rightarrow f}={\frac {2\pi }{\hbar }}\left|\langle f|H_{1}|i\rangle \right|^{2}\rho ,

$/$ $G$

E=mc^{2}

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

sendo $\scriptstyle \rho$ a densidade de estados finais.

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.\,

$/$ $G$

E=mc^{2}

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

where $\gamma ^{\mu }$ are gamma matrices and $�$ 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).\,

$/$ $G$

E=mc^{2}

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

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 }.

$/$ $G$

E=mc^{2}

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

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},\,

$/$ $G$

E=mc^{2}

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

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 \,

$/$ $G$

E=mc^{2}

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

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}\,

$/$ $G$

E=mc^{2}

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

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 }\,.

$/$ $G$

E=mc^{2}

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

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

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

$/$ $G$

E=mc^{2}

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

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

$/$ $G$

E=mc^{2}

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

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).

$/$ $G$

E=mc^{2}

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

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)}

$/$ $G$

E=mc^{2}

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

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

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

$/$ $G$

E=mc^{2}

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

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.

Bosons are particles whose wavefunction is symmetric under such an exchange or permutation, so if we swap the particles, the wavefunction does not change. Fermions are particles whose wavefunction is antisymmetric, so under such a swap the wavefunction gets a minus sign, meaning that the amplitude for two identical fermions to occupy the same state must be zero. This is the Pauli exclusion principle: two identical fermions cannot occupy the same state. This rule does not hold for bosons.

In quantum field theory, a state or a wavefunction is described by field operators operating on some basic state called the vacuum. In order for the operators to project out the symmetric or antisymmetric component of the creating wavefunction, they must have the appropriate commutation law. The operator

\iint \psi (x,y)\phi (x)\phi (y)\,dx\,dy

$/$ $G$

E=mc^{2}

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

(with $\phi$ an operator and $\psi (x,y)$ a numerical function with complex values) creates a two-particle state with wavefunction $\psi (x,y)$ , and depending on the commutation properties of the fields, either only the antisymmetric parts or the symmetric parts matter.

Let us assume that $x\neq y$ and the two operators take place at the same time; more generally, they may have spacelike separation, as is explained hereafter.