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

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|Principle of Covariance

There is another word which is strictly interrelated with relativity: covariance. In a few words, we can define this notion in the following way.

Let us consider a physical theory which is formalised by a well defined fundamental geometric framework. Then, we shall define the (local) covariance group to be the (local) group of automorphisms of this geometric framework. Accordingly, we say that the theory is covariant if its fundamental laws turn out to be equivariant with respect to the action of the covariance (local) group.

For instance, in the einsteinian Special Relativity, we deal with a lorentzian affine space (Minkowski space); hence, the covariance group is the group of affine isometries (Lorentz transformations). Analogously, in the einsteinian General Relativity, we deal with a lorentzian manifold; hence the covariance group is the group of isometric diffeomorphisms.

Our classical galilean theory is achieved by postulating a geometric structure of spacetime in three steps. Accordingly, the group of automorphisms and the induced covariance can be expressed in three steps.
(1) We start by postulating a spacetime manifold fibred over absolute time. So, $a t$ this step, the covariance group turns out to be the group of fibred automorphisms of spacetime over affine automorphisms of the base space. Accordingly, at this step, the physical laws are covariant if they are equivariant with respect to this group of automorphisms.

Indeed, the above fibred geometric structure can be fully represented by a suitable atlas of adapted charts. Hence, at this step, the covariance of the theory can be read, in coordinates, as coordinate free expression of physical laws.
(2) Then, we postulate a riemannian metric on the fibres of the spacetime fibred space. So, at this step, the covariance group turns out to be the group of fibred automorphisms of spacetime as in step 1, which further yield isometric diffeomorphisms of the fibres. Accordingly, at this step, the physical laws are covariant if they are equivariant with respect to this group of automorphisms.

In order to read the covariance of physical laws in coordinates, it would not be sufficient to refer to charts adapted to the fibring, but it would be necessary to refer also to suitable adapted frames.

## 物理代写|量子力学代写quantum mechanics代考|Intrinsic, Observed and Coordinate Languages

It is worth discussing three kinds of possible languages used in the formulation of a physical theory: the intrinsic language, the language of coordinates and the language based on observers.

In Geometry, some “original basic” concepts are unavoidably defined, through coordinates, by means of an explicit equivariance property with respect to a certain transition rule of charts. This is the case, for instance, of the concepts of manifolds and jet spaces. But, once these basic objects have been introduced in coordinates, one can proceed by means of formal “intrinsic methods”, which do not require, at each step, the explicit mention of coordinates and their equivariance properties. In fact, such an equivariance is ensure a priory by those intrinsic methods. This is the case, just as an example, of the concepts of exterior differential and Lie derivatives of forms on manifolds.

So, there are at least two ways to deal with the covariance of a physical theory. Namely, if the mathematical language of the theory is systematically expressed in coordinates, then the covariance of the theory needs to be explicitly checked at any step. Conversely, if the physical concepts and laws of the theory are expressed in terms of an intrinsic geometric language, then the covariance is ensured a priori.
Most physical theories in standard literature are usually formulated in coordinates.

In the presesent hook, in general, we first present the hasic concepts and laws hy means of an intrinsic genmetric language. However, we systematically add a further description in coordinates. as well. Indeed, both languages turn out to be useful: the first one is basically convenient for its concise character, the second one is useful for emphasising further very useful features.

But, besides the intrinsic and coordinate formulations of a physical theory, it is also worth considering an intermediate approach which stands in between the intrinsic and the coordinate languages. Namely, this approach deals with observers.

## 物理代写|量子力学代写quantum mechanics代考|Principle of Covariance

(1) 我们首先假设一个在绝对时间上纤维化的时空流形。所以，一个吨这一步，协方差群变成了时空的纤维自同构在基空间的仿射自同构上的群。因此，在这一步，如果物理定律对于这组自同构是等变的，则它们是协变的。

(2) 然后，我们假设时空纤维空间的纤维有黎曼度量。因此，在这一步，协方差群变成了与步骤 1 一样的时空纤维自同胚群，这进一步产生了纤维的等距微分同胚。因此，在这一步，如果物理定律对于这组自同构是等变的，则它们是协变的。

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## MATLAB代写

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