# 物理代写|量子力学代写quantum mechanics代考|PHYS3034

## 物理代写|量子力学代写quantum mechanics代考|Criterion of Projectability

Our quantum bundle $\pi: \boldsymbol{Q} \rightarrow \boldsymbol{E}$ lives on the spacetime $\boldsymbol{E}$, but the cosymplectic phase 2-form $\Omega: J_1 \boldsymbol{E} \rightarrow \Lambda^2 T^* J_1 \boldsymbol{E}$ lives on the phase space $J_1 \boldsymbol{E}$. Therefore, in order to define a galilean upper quantum connection $Ч^{\uparrow}: \boldsymbol{Q}^{\uparrow} \rightarrow T^* J_1 \boldsymbol{E} \otimes T \boldsymbol{Q}^{\uparrow}$ suitable to be linked to $\Omega$, we have been forced to introduce, by pullback, the upper quantum bundle $\pi^{\uparrow}: \boldsymbol{Q}^{\uparrow} \rightarrow J_1 \boldsymbol{E}$, which lives on the phase space $J_1 \boldsymbol{E}$ (see Definition 9.1.3, 14.11.1, 15.1.6 and 15.1.5, Corollary 9.2.4, Theorems 9.2.8, 10.1.1, Postulate Q.1, Q.2, and Lemma 15.1.3).

In this way, we deal with an upper quantum bundle $Q^{\uparrow}$, whose base space $J_1 \boldsymbol{E}$ involves all observers $o$. Accordingly, also the upper quantum connection $\mathrm{Y}^{\uparrow}$, involves all observers $o$ implicitly, and, equivalently, the associated system of observed quantum connections ${\mathrm{Y}[o]}$, involves all observers $o$ explicitly.

Indeed, our quantum theory is aimed at deriving all dynamical quantum objects from the upper quantum connection $\mathrm{Y}^{\uparrow}$, getting rid of observers, in order to fulfill a covariance requirement. Accordingly, our procedure is based on the search of quantum objects which, on the one hand, are derived, in a natural way, from the upper quantum connection $\mathrm{U}^{\uparrow}$, but, on the other hand, are projectable on the quantum hundle $\boldsymbol{Q}$. We call this type of procedure “criterion of projectability”: actually it turns out to be a way to implement the covariance of the quantum theory (see Note 17.1.1).

## 物理代写|量子力学代写quantum mechanics代考|Dynamical Quantum Objects

According to the above covariant procedure based on the criterion of projectability, we exhibit the following dynamical quantum objects.
Actually, all objects below are achieved in two steps:
(1) we show two distinguished quantum objects, which live on the same source space $J_1 \boldsymbol{E}$ and have the same target space,

(2) for each of the above pairs, we show that their difference projects on the spacetime $\boldsymbol{E}$, so getting rid of the observers which are encoded in the phase space.
Indeed, this distinguished difference turns out to be our natural candidate for the searched dynamical quantum object.

Some of the following dynamical quantum objects are rather unusual and cannot be found in the literature, at least in our way of presentation.

We stress that all dynamical quantum objects above are global, observer independent and gauge independent. Actually, this property is related to the fact that they deal with the gravitational and electromagnetic fields effecting the quantum particle and do not involve other possible phenomenological fields. Of course, other additional phenomenological fields might be added by hand; but, so doing, we would possibly brake the covariance of the theory.

Moreover, we stress that the above procedure based on the criterion of projectability yields the dynamical quantum objects above without any relation to hamiltonian methods.

Indeed, these objects can be achieved by other differential geometric approaches related to the above criterion of projectability. In particular, one can prove that $S$ and $\mathrm{L}$ are determined by the only requirement of covariance (see [219]).

## 物理代写|量子力学代写quantum mechanics代考|Dynamical Quantum Objects

（1）我们展示了两个不同的量子对象，它们生活在同一个源空间上Ĵ1和并具有相同的目标空间，

(2) 对于上述每一对，我们表明它们的差异在时空上投影和，因此摆脱了在相空间中编码的观察者。

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