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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|量子力学代写quantum mechanics代考|Galilean Gravitational Field

In einsteinian General Relativity one deals with a lorentzian spacetime metric $g$ and the associated Levi-Civita connection $K[g]$. Actually, the roles of the metric $g$ and of the Levi-Civita connection $K[g]$ are very different from an experimental viewpoint; in fact, the gravitational phenomena are described more directly by $K[g]$, rather than by $g$. The Levi-Civita connection $K[g]$ is fully determined by the lorentzian metric $g$, which plays, in a certain sense, the role of a potential. So, one is usually led to regard the metric $g$ as the gravitational field. We think that, in spite of this circumstance occurring in einsteinian General Relativity, it would be theoretically more appropriate to regard $K[g]$ as the gravitational field and $g$ as a metric field, suitable to describe different experiments and phenomena.

Actually, in the galilean framework, the galilean metric $g$ has signature $(0+$ $++$, hence it is unable to determine a spacetime connection via the Levi-Civita procedure. So, we cannot identify the gravitational field with $g$. Accordingly, in the galilean framework, we describe the gravitational field by means of an appropriate galilean connection $K^{\natural}$.

We define a “galilean spacetime connection” to be a torsion free linear connection $K: T \boldsymbol{E} \rightarrow T^* \boldsymbol{E} \otimes T T \boldsymbol{E}$ of the vector bundle $T \boldsymbol{E} \rightarrow \boldsymbol{E}$ (see Definitions 4.1.1, 4.2.3 and 4.3.1) which is time preserving and metric preserving, i.e. which fulfills the conditions $\nabla d t=0$ and $\nabla g=0$, along with the additional condition $A \underline{R}[g . K]=0$, with coordinate expression $R_{i \mu j v}=R_{j v i \mu}$.

Let us explain this additional condition. The curvature tensor of (pseudo-)riemannian connections fulfills certain well-known symmetry properties, which are direct consequence of the condition $\nabla g=0$. But, in the galilean case, the above symmetry property of the connection should be considered as an additional condition, because the spacelike metric is unable to yield it.

We stress that this symmetry condition is essential in our context because, later, it turns out to be equivalent to the closure of the cosymplectic phase 2-form $\Omega$ and this fact turns out to be equivalent to the Bianchi identity for the upper quantum connection $\mathcal{Y}^{\uparrow}$ (see Theorem 9.2.15 and Remark 15.2.3). Actually, the above condition concerning a symmetry property of the curvature tensor has been considered by several authors (see, for instance, $[259,260,263]$ ).

## 物理代写|量子力学代写quantum mechanics代考|Galilean Electromagnetic Field

Joining standard Quantum Mechanics with the Maxwell theory of electromagnetism is a delicate problem.

The Maxwell theory of electromagnetism is one of the most successful achievements of classical physics (see, for instance, $[164,166,189,322,333,343,346$, $347,418,419]$ ). It is very satisfactory from a phenomenological viewpoint, in complete agreement with einsteinian General Relativity and very elegant mathematically.

In standard literature, Quantum Mechanics and Maxwell theory are usually combined without specific care. However, these two theories are formally inconsistent. In fact, standard Quantum Mechanics is essentially a galilean theory, while the Maxwell theory is essentially a lorentzian theory. Actually, the above formal inconsistency does not manifestly appear when one deals with a fixed inertial observer: this fact explains why no serious troubles arise in the standard joining of the above two theories and why one does not pay too much attention to their formal inconsistency.

Now, as we are searching for a covariant formulation of Quantum Mechanics, with reference to any observer and possibly in a curved galilean spacetime, we have to overcome the above inconsistency. There are two ways: (1) to pass to a lorentzian formulation of Quantum Mechanics, (2) to search for a reduced galilean version of electromagnetism. Actually, as we have already pointed out, well-known successful lorentzian quantum theories have already been achieved, but they stand quite far from the original core of standard Quantum Mechanics and do not cover it fully. But the goal of the present book deals with a further understanding of Quantum Mechanics, within its original galilean framework. For this reason, despite ourselves, we are forced to deal with a reduced galilean version of electromagnetism, in order to get a formal consistency between the quantum theory and the electromagnetic theory.

# 量子力学代考

## 有限元方法代写

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## MATLAB代写

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