MECÂNICA GRACELI GENERALIZADA - QUÂNTICA TENSORIAL DIMENSIONAL RELATIVISTA DE CAMPOS [1.212]

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,

Exchange symmetry or permutation symmetry

[edit]

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

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

T_{\mu \nu }

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

(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.

If the fields commute, meaning that the following holds:

\phi (x)\phi (y)=\phi (y)\phi (x),

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

T_{\mu \nu }

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

then only the symmetric part of $\psi$ contributes, so that $\psi (x,y)=\psi (y,x)$ , and the field will create bosonic particles.

On the other hand, if the fields anti-commute, meaning that $\phi$ has the property that

\phi (x)\phi (y)=-\phi (y)\phi (x),

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

T_{\mu \nu }

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

then only the antisymmetric part of $\psi$ contributes, so that $\psi (x,y)=-\psi (y,x)$ , and the particles will be fermionic.

In physics, and especially quantum field theory, an auxiliary field is one whose equations of motion admit a single solution. Therefore, the Lagrangian describing such a field $�$ contains an algebraic quadratic term and an arbitrary linear term, while it contains no kinetic terms (derivatives of the field):

{\mathcal {L}}_{\text{aux}}={\frac {1}{2}}(A,A)+(f(\varphi ),A).

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

T_{\mu \nu }

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

The equation of motion for $�$ is

A(\varphi )=-f(\varphi ),

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

T_{\mu \nu }

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

and the Lagrangian becomes

{\mathcal {L}}_{\text{aux}}=-{\frac {1}{2}}(f(\varphi ),f(\varphi )).

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

T_{\mu \nu }

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

Auxiliary fields generally do not propagate,^[1] and hence the content of any theory can remain unchanged in many circumstances by adding such fields by hand. If we have an initial Lagrangian ${\mathcal {L}}_{0}$ describing a field $\varphi$ , then the Lagrangian describing both fields is

{\mathcal {L}}={\mathcal {L}}_{0}(\varphi )+{\mathcal {L}}_{\text{aux}}={\mathcal {L}}_{0}(\varphi )-{\frac {1}{2}}{\big (}f(\varphi ),f(\varphi ){\big )}.

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

T_{\mu \nu }

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

Therefore, auxiliary fields can be employed to cancel quadratic terms in $\varphi$ in ${\mathcal {L}}_{0}$ and linearize the action ${\mathcal {S}}=\int {\mathcal {L}}\,d^{n}x$ . / $\psi ^{*}$ [ $\psi ^{*}$ $T_{\mu \nu }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$

Examples of auxiliary fields are the complex scalar field F in a chiral superfield,^[2] the real scalar field D in a vector superfield, the scalar field B in BRST and the field in the Hubbard–Stratonovich transformation.

The quantum mechanical effect of adding an auxiliary field is the same as the classical, since the path integral over such a field is Gaussian. To wit:

\int _{-\infty }^{\infty }dA\,e^{-{\frac {1}{2}}A^{2}+Af}={\sqrt {2\pi }}e^{\frac {f^{2}}{2}}.

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

T_{\mu \nu }

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

\left(\beta mc^{2}+c\sum _{n=1}^{3}\alpha _{n}p_{n}\right)\psi (x,t)=i\hbar {\frac {\partial \psi (x,t)}{\partial t}}

The left side represents the square of the momentum operator divided by twice the mass, which is the non-relativistic kinetic energy. Because relativity treats space and time as a whole, a relativistic generalization of this equation requires that space and time derivatives must enter symmetrically as they do in the Maxwell equations that govern the behavior of light — the equations must be differentially of the same order in space and time. In relativity, the momentum and the energies are the space and time parts of a spacetime vector, the four-momentum, and they are related by the relativistically invariant relation $E^{2}=m^{2}c^{4}+p^{2}c^{2}$

which says that the length of this four-vector is proportional to the rest mass $m$ . Substituting the operator equivalents of the energy and momentum from the Schrödinger theory produces the Klein–Gordon equation describing the propagation of waves, constructed from relativistically invariant objects, $\left(-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}+\nabla ^{2}\right)\phi ={\frac {m^{2}c^{2}}{\hbar ^{2}}}\phi$ / $\psi ^{*}$ [ $\psi ^{*}$ $T_{\mu \nu }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$

with the wave function $\phi$ being a relativistic scalar: a complex number which has the same numerical value in all frames of reference. Space and time derivatives both enter to second order. This has a telling consequence for the interpretation of the equation. Because the equation is second order in the time derivative, one must specify initial values both of the wave function itself and of its first time-derivative in order to solve definite problems. Since both may be specified more or less arbitrarily, the wave function cannot maintain its former role of determining the probability density of finding the electron in a given state of motion. In the Schrödinger theory, the probability density is given by the positive definite expression $\rho =\phi ^{*}\phi$ and this density is convected according to the probability current vector $J=-{\frac {i\hbar }{2m}}(\phi ^{*}\nabla \phi -\phi \nabla \phi ^{*})$ / $\psi ^{*}$ [ $\psi ^{*}$ $T_{\mu \nu }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$

with the conservation of probability current and density following from the continuity equation: $\nabla \cdot J+{\frac {\partial \rho }{\partial t}}=0~.$

The fact that the density is positive definite and convected according to this continuity equation implies that one may integrate the density over a certain domain and set the total to 1, and this condition will be maintained by the conservation law. A proper relativistic theory with a probability density current must also share this feature. To maintain the notion of a convected density, one must generalize the Schrödinger expression of the density and current so that space and time derivatives again enter symmetrically in relation to the scalar wave function. The Schrödinger expression can be kept for the current, but the probability density must be replaced by the symmetrically formed expression^{[further explanation needed]} $\rho ={\frac {i\hbar }{2mc^{2}}}\left(\psi ^{*}\partial _{t}\psi -\psi \partial _{t}\psi ^{*}\right).$ / $\psi ^{*}$ [ $\psi ^{*}$ $T_{\mu \nu }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$

which now becomes the 4th component of a spacetime vector, and the entire probability 4-current density has the relativistically covariant expression $J^{\mu }={\frac {i\hbar }{2m}}\left(\psi ^{*}\partial ^{\mu }\psi -\psi \partial ^{\mu }\psi ^{*}\right).$

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

The continuity equation is as before. Everything is compatible with relativity now, but the expression for the density is no longer positive definite; the initial values of both $ψ$ and $\partial t ψ$ may be freely chosen, and the density may thus become negative, something that is impossible for a legitimate probability density. Thus, one cannot get a simple generalization of the Schrödinger equation under the naive assumption that the wave function is a relativistic scalar, and the equation it satisfies, second order in time.

and they all mutually anti-commute: ${\begin{aligned}\alpha _{i}\alpha _{j}+\alpha _{j}\alpha _{i}&=0\quad (i\neq j)\\\alpha _{i}\beta +\beta \alpha _{i}&=0\end{aligned}}$ / $\psi ^{*}$ [ $\psi ^{*}$ $T_{\mu \nu }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$

In physics, the Majorana equation is a relativistic wave equation. It is named after the Italian physicist Ettore Majorana, who proposed it in 1937 as a means of describing fermions that are their own antiparticle.^[1] Particles corresponding to this equation are termed Majorana particles, although that term now has a more expansive meaning, referring to any (possibly non-relativistic) fermionic particle that is its own anti-particle (and is therefore electrically neutral).

There have been proposals that massive neutrinos are described by Majorana particles; there are various extensions to the Standard Model that enable this. The article on Majorana particles presents status for the experimental searches, including details about neutrinos. This article focuses primarily on the mathematical development of the theory, with attention to its discrete and continuous symmetries. The discrete symmetries are charge conjugation, parity transformation and time reversal; the continuous symmetry is Lorentz invariance.

Charge conjugation plays an outsize role, as it is the key symmetry that allows the Majorana particles to be described as electrically neutral. A particularly remarkable aspect is that electrical neutrality allows several global phases to be freely chosen, one each for the left and right chiral fields. This implies that, without explicit constraints on these phases, the Majorana fields are naturally CP violating. Another aspect of electric neutrality is that the left and right chiral fields can be given distinct masses. That is, electric charge is a Lorentz invariant, and also a constant of motion; whereas chirality is a Lorentz invariant, but is not a constant of motion for massive fields. Electrically neutral fields are thus less constrained than charged fields. Under charge conjugation, the two free global phases appear in the mass terms (as they are Lorentz invariant), and so the Majorana mass is described by a complex matrix, rather than a single number. In short, the discrete symmetries of the Majorana equation are considerably more complicated than those for the Dirac equation, where the electrical charge $U(1)$ symmetry constrains and removes these freedoms.

Definition

[edit]

The Majorana equation can be written in several distinct forms:

As the Dirac equation written so that the Dirac operator is purely Hermitian, thus giving purely real solutions.
As an operator that relates a four-component spinor to its charge conjugate.
As a 2×2 differential equation acting on a complex two-component spinor, resembling the Weyl equation with a properly Lorentz covariant mass term.^[2]^[3]^[4]^[5]

These three forms are equivalent, and can be derived from one-another. Each offers slightly different insight into the nature of the equation. The first form emphasises that purely real solutions can be found. The second form clarifies the role of charge conjugation. The third form provides the most direct contact with the representation theory of the Lorentz group.

Purely real four-component form

[edit]

The conventional starting point is to state that "the Dirac equation can be written in Hermitian form", when the gamma matrices are taken in the Majorana representation. The Dirac equation is then written as^[6]

\left(\,-i\,{\frac {\partial }{\partial t}}-i\,{\hat {\alpha }}\cdot \nabla +\beta \,m\,\right)\,\psi =0

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

T_{\mu \nu }

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

with ${\hat {\alpha }}$ being purely real 4×4 symmetric matrices, and $\beta$ being purely imaginary skew-symmetric; as required to ensure that the operator (that part inside the parentheses) is Hermitian. In this case, purely real 4‑spinor solutions to the equation can be found; these are the Majorana spinors.

Charge-conjugate four-component form

[edit]

The Majorana equation is

i\,{\partial \!\!\!{\big /}}\psi -m\,\psi _{c}=0~

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

T_{\mu \nu }

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

with the derivative operator ${\partial \!\!\!{\big /}}$ written in Feynman slash notation to include the gamma matrices as well as a summation over the spinor components. The spinor ${\textstyle \,\psi _{c}\,}$ is the charge conjugate of ${\textstyle \,\psi \,.}$ By construction, charge conjugates are necessarily given by

\psi _{c}=\eta _{c}\,C\,{\overline {\psi }}^{\mathsf {T}}~

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

T_{\mu \nu }

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

where $\,(\cdot )^{\mathsf {T}}\,$ denotes the transpose, $\,\eta _{c}\,$ is an arbitrary phase factor $\,|\eta _{c}|=1\,,$ conventionally taken as $\,\eta _{c}=1\,,$ and $\,C\,$ is a 4×4 matrix, the charge conjugation matrix. The matrix representation of $\,C\,$ depends on the choice of the representation of the gamma matrices. By convention, the conjugate spinor is written as