MECÂNICA GRACELI GENERALIZADA - QUÂNTICA TENSORIAL DIMENSIONAL RELATIVISTA DE CAMPOS [911]

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

$G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$

$\psi ^{*}$

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[

$\psi ^{*}$

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

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 E CAMPOS, DISTRIBUIÇÕES ELETRÔNICAS, ESTADOS FÍSICOS, ESTADOS QUÂNTICOS, ESTADOS FÍSICOS DE ENERGIAS DE GRACELI, E OUTROS.

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

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

The Feynman–Kac formula, named after Richard Feynman and Mark Kac, establishes a link between parabolic partial differential equations (PDEs) and stochastic processes. In 1947, when Kac and Feynman were both Cornell faculty, Kac attended a presentation of Feynman's and remarked that the two of them were working on the same thing from different directions.^[1] The Feynman–Kac formula resulted, which proves rigorously the real-valued case of Feynman's path integrals. The complex case, which occurs when a particle's spin is included, is still an open question.^[2]

It offers a method of solving certain partial differential equations by simulating random paths of a stochastic process. Conversely, an important class of expectations of random processes can be computed by deterministic methods.

Theorem[edit source]

Consider the partial differential equation

{\frac {\partial u}{\partial t}}(x,t)+\mu (x,t){\frac {\partial u}{\partial x}}(x,t)+{\tfrac {1}{2}}\sigma ^{2}(x,t){\frac {\partial ^{2}u}{\partial x^{2}}}(x,t)-V(x,t)u(x,t)+f(x,t)=0,

defined for all

x\in \mathbb {R}

and

t\in [0,T]

, subject to the terminal condition

u(x,T)=\psi (x),

where

\mu ,\sigma ,\psi ,V,f

are known functions,

�

is a parameter, and

u:\mathbb {R} \times [0,T]\to \mathbb {R}

is the unknown. Then the Feynman–Kac formula tells us that the solution can be written as a conditional expectation

$u(x,t)=E^{Q}\left[\int _{t}^{T}e^{-\int _{t}^{\tau }V(X_{s},s)\,ds}f(X_{\tau },\tau )d\tau +e^{-\int _{t}^{T}V(X_{\tau },\tau )\,d\tau }\psi (X_{T})\,{\Bigg |}\,X_{t}=x\right]$

/ $\psi ^{*}$ [ $\psi ^{*}$ under the probability measure $�$ such that $�$ is an Itô process driven by the equation

dX_{t}=\mu (X,t)\,dt+\sigma (X,t)\,dW_{t}^{Q},

with

W^{Q}(t)

is a Wiener process (also called Brownian motion) under

�

, and the initial condition for

X(t)

X(t)=x

Intuitive interpretation[edit source]

Suppose we have a particle moving according to the diffusion process

dX_{t}=\mu (X,t)\,dt+\sigma (X,t)\,dW_{t}^{Q},

Let the particle incur "cost" at a rate of

f(X_{s},s)

at location

X_{s}

at time

�

. Let it incur a final cost at

\psi (X_{T})

Also, allow the particle to decay. If the particle is at location $X_{s}$ at time $�$ , then it decays with rate $V(X_{s},s)$ . After the particle has decayed, all future cost is zero.

Then, $u(x,t)$ is the expected cost-to-go, if the particle starts at $(t,X_{t}=x)$ .

Partial proof[edit source]

A proof that the above formula is a solution of the differential equation is long, difficult and not presented here. It is however reasonably straightforward to show that, if a solution exists, it must have the above form. The proof of that lesser result is as follows:

Let $u(x,t)$ be the solution to the above partial differential equation. Applying the product rule for Itô processes to the process

Y(s)=\exp \left(-\int _{t}^{s}V(X_{\tau },\tau )\,d\tau \right)u(X_{s},s)+\int _{t}^{s}\exp \left(-\int _{t}^{r}V(X_{\tau },\tau )\,d\tau \right)f(X_{r},r)\,dr

one gets: / $\psi ^{*}$ [ $\psi ^{*}$

{\begin{aligned}dY_{s}={}&d\left(\exp \left(-\int _{t}^{s}V(X_{\tau },\tau )\,d\tau \right)\right)u(X_{s},s)+\exp \left(-\int _{t}^{s}V(X_{\tau },\tau )\,d\tau \right)\,du(X_{s},s)\\[6pt]&{}+d\left(\exp \left(-\int _{t}^{s}V(X_{\tau },\tau )\,d\tau \right)\right)du(X_{s},s)+d\left(\int _{t}^{s}\exp \left(-\int _{t}^{r}V(X_{\tau },\tau )\,d\tau \right)f(X_{r},r)\,dr\right)\end{aligned}}

Since

d\left(\exp \left(-\int _{t}^{s}V(X_{\tau },\tau )\,d\tau \right)\right)=-V(X_{s},s)\exp \left(-\int _{t}^{s}V(X_{\tau },\tau )\,d\tau \right)\,ds,

the third term is

O(dt\,du)

and can be dropped. We also have that

d\left(\int _{t}^{s}\exp \left(-\int _{t}^{r}V(X_{\tau },\tau )\,d\tau \right)f(X_{r},r)dr\right)=\exp \left(-\int _{t}^{s}V(X_{\tau },\tau )\,d\tau \right)f(X_{s},s)ds.

Applying Itô's lemma to $du(X_{s},s)$ , it follows that

{\begin{aligned}dY_{s}={}&\exp \left(-\int _{t}^{s}V(X_{\tau },\tau )\,d\tau \right)\,\left(-V(X_{s},s)u(X_{s},s)+f(X_{s},s)+\mu (X_{s},s){\frac {\partial u}{\partial X}}+{\frac {\partial u}{\partial s}}+{\tfrac {1}{2}}\sigma ^{2}(X_{s},s){\frac {\partial ^{2}u}{\partial X^{2}}}\right)\,ds\\[6pt]&{}+\exp \left(-\int _{t}^{s}V(X_{\tau },\tau )\,d\tau \right)\sigma (X,s){\frac {\partial u}{\partial X}}\,dW.\end{aligned}}

/ $\psi ^{*}$ [ $\psi ^{*}$ The first term contains, in parentheses, the above partial differential equation and is therefore zero. What remains is:

dY_{s}=\exp \left(-\int _{t}^{s}V(X_{\tau },\tau )\,d\tau \right)\sigma (X,s){\frac {\partial u}{\partial X}}\,dW.

Integrating this equation from $�$ to $�$ , one concludes that:

Y(T)-Y(t)=\int _{t}^{T}\exp \left(-\int _{t}^{s}V(X_{\tau },\tau )\,d\tau \right)\sigma (X,s){\frac {\partial u}{\partial X}}\,dW.

Upon taking expectations, conditioned on $X_{t}=x$ , and observing that the right side is an Itô integral, which has expectation zero,^[3] it follows that:

E[Y(T)\mid X_{t}=x]=E[Y(t)\mid X_{t}=x]=u(x,t).

The desired result is obtained by observing that:

E[Y(T)\mid X_{t}=x]=E\left[\exp \left(-\int _{t}^{T}V(X_{\tau },\tau )\,d\tau \right)u(X_{T},T)+\int _{t}^{T}\exp \left(-\int _{t}^{r}V(X_{\tau },\tau )\,d\tau \right)f(X_{r},r)\,dr\,{\Bigg |}\,X_{t}=x\right]

and finally

u(x,t)=E\left[\exp \left(-\int _{t}^{T}V(X_{\tau },\tau )\,d\tau \right)\psi (X_{T})+\int _{t}^{T}\exp \left(-\int _{t}^{s}V(X_{\tau },\tau )\,d\tau \right)f(X_{s},s)\,ds\,{\Bigg |}\,X_{t}=x\right]

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MECÂNICA GRACELI GENERALIZADA - QUÂNTICA TENSORIAL DIMENSIONAL RELATIVISTA DE CAMPOS [911]

Theorem[edit source]

Intuitive interpretation[edit source]

Partial proof[edit source]

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

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Theorem[edit source]

Intuitive interpretation[edit source]

Partial proof[edit source]

/ G* = = [ ] ω , , .=

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