## 物理代写|量子力学代写quantum mechanics代考|PHYSICS3544

2023年3月29日

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## 物理代写|量子力学代写quantum mechanics代考|Phase Musical Morphisms

We consider a galilean spacetime connection $K$ and the associated phase fields $\Gamma, \gamma, \Omega, \Lambda$. Then, we discuss the natural linear musical phase morphisms $\Omega^b$ and $\Lambda^{\sharp}$; indeed, they are not isomorphisms. But, after having chosen a phase time scale $\tau$, these morphisms yield natural isomorphism by means of the additional help of $\gamma$. The above geometric construction will be used later for the definition of hamiltonian lift of phase functions and consequently for the definition of Poisson Lie bracket (see Definition 11.3.6 and Theorem 11.4.6).

Actually, in this context, we shall see that every phase function provides a distinguished time scale, which will be used for the above definitions.

Let us consider a galilean spacetime connection $K: T \boldsymbol{E} \rightarrow T^* \boldsymbol{E} \otimes T T \boldsymbol{E}$ (see Definition 4.3.1) and the associated phase objects (see Theorem 9.2.1 and Corollary 9.2.4)
\begin{aligned} & \Gamma[K]: J_1 \boldsymbol{E} \rightarrow T^* \boldsymbol{E} \otimes T J_1 \boldsymbol{E}, \quad \gamma[K]: J_1 \boldsymbol{E} \rightarrow \mathbb{T}^* \otimes T J_1 \boldsymbol{E}, \ & \Omega[G, K]: J_1 \boldsymbol{E} \rightarrow \Lambda^2 T^* J_1 \boldsymbol{E}, \quad \Lambda[G, K]: J_1 \boldsymbol{E} \rightarrow \Lambda^2 V J_1 \boldsymbol{E} \end{aligned}

We recall that $\Omega[G, K]$ turns out to be a cosymplectic phase 2 -form; hence, we can derive from it the phase 1 -forms (see Theorem 10.1.8)
$$\mathcal{L}[G, K]: J_1 \boldsymbol{E} \rightarrow T^* \boldsymbol{E} \text { and } \mathcal{M}[G, K]: J_1 \boldsymbol{E} \rightarrow T^* \boldsymbol{E} .$$

## 物理代写|量子力学代写quantum mechanics代考|Hamiltonian Phase Lift of Phase Functions

Let us consider a galilean spacetime connection $K$ and the associated phase fields $\Gamma, \gamma, \Omega, \Lambda$. Now, we are in a position to discuss the phase lifts of phase functions $f$. We start by defining the scaled hamiltonian lift of phase functions with respect to an arbitrary phase time scale $\tau$. Further, we show that every phase function $f$ yields a distinguished phase time scale $f^{\prime \prime}$. Hence, we obtain a (natural) hamiltonian lift of phase functions.

Indeed, the hamiltonian lift of phase functions fulfills the following useful property $i_{X_{\gamma_{\operatorname{Lim}}}[f]} \Omega=d f-\gamma \cdot f$ and yields the equivalence $i_{X{ }{t{\operatorname{lum}}}[f]} \Omega=d f \Leftrightarrow \gamma \cdot f=0$.
The hamiltonian lift of phase functions will be largely used throughout to book, in several contexts.

Let us consider a galilean spacetime connection $K$ and the associated phase objects $\Gamma, \gamma, \Omega, \Lambda$. By recalling the natural splitting of the tangent space of phase space (see Proposition 11.1.1), we introduce the scaled hamiltonian lift of phase functions, with respect to a phase time scale $\tau$. Then, we prove some technical statements, which will be frequently used throughout the book.

At a first insight, this definition of scaled hamiltonian phase lift might appear to be rather arbitrary. However, this concept arises, in a natural way, in several steps of this book.

More generally, this concept arises, in a natural way, in a general theorem which classifies the infinitesimal symmetries of a pair $(\omega, \Omega)$ consisting of a 1-form $\omega$ and closed 2-form $\Omega$ (see [205]).

In the next section, we shall exhibit a distinguished choice of this time scale associated with each phase function.

# 量子力学代考

## 物理代写|量子力学代写quantum mechanics代考|Phase Musical Morphisms

$K: T \boldsymbol{E} \rightarrow T^* \boldsymbol{E} \otimes T T \boldsymbol{E}$ (参见定义 4.3.1) 和相关 的相对象（参见定理 9.2.1 和推论 9.2.4）
$$\Gamma[K]: J_1 \boldsymbol{E} \rightarrow T^* \boldsymbol{E} \otimes T J_1 \boldsymbol{E}, \quad \gamma[K]: J_1 \boldsymbol{E} \rightarrow \mathbb{T}^* \otimes$$

$\mathcal{L}[G, K]: J_1 \boldsymbol{E} \rightarrow T^* \boldsymbol{E}$ and $\mathcal{M}[G, K]: J_1 \boldsymbol{E} \rightarrow T^* \boldsymbol{E}$

## 物理代写|量子力学代写quantum mechanics代考|Hamiltonian Phase Lift of Phase Functions

$\Gamma, \gamma, \Omega, \Lambda$. 现在，我们可以讨论相位函数的相位提升 $f$. 我们首先定义相对于任意相位时间尺度的相位函数的缩 放哈密尔顿提升 $\tau$. 此外，我们表明每个相位函数 $f$ 产生 一个显着的阶段时间尺度 $f^{\prime \prime}$. 因此，我们获得了相函数 的 (自然) 哈密顿提升。

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