Introduction to Mathematical Physics/Relativity/Introduction

In this chapter we focus on the ideas of symmetry and transformations. More precisely, we study the consequences on the physical laws of transformation invariance. The reading of the appendices Tensors and Groups is thus recommended for those who are not familiar with tensorial calculus and group theory. In classical mechanics, a material point of mass $m$ is referenced by its position $r$ and its momentum $p$ at each time $t$ . Time does not depend on the reference frame used to evaluate position and momentum. The Newton's law of motion is invariant under Galileean transformations. In special and general relativity, time depends on the considered reference frame. This yields to modify classical notions of position and momentum. Historicaly, special relativity was proposed to describe the invariance of the light speed. The group of transformations that leaves the new form of the dynamics equations is the Lorentz group. Quantum, kinetic, and continuous description of matter will be presented later in the book.^[1].

↑ In quantum mechanics, physical space considered is a functional space, and the state of a system is represented by a "wave" function $\phi (r,t)$ of space and time. Quantity $|\phi (r,t)|^{2}dr$ can be interpreted as the probability to have at time $t$ a particle in volume $dr$ . Wave function notion can be generalized to systems more complex than those constituted by one particle. A kinetic description of system constituted by an large number of particles consists in representing the state of considered system by a function $f(r,p,t)$ called "repartition" function, that represents the probability density to encounter a particle at position $r$ with momentum $p$ . Continuous description of matter refers to several functions of position and time to describe the state of the physical system considered.

[1] In quantum mechanics, physical space considered is a functional space, and the state of a system is represented by a "wave" function $\phi (r,t)$ of space and time. Quantity $|\phi (r,t)|^{2}dr$ can be interpreted as the probability to have at time $t$ a particle in volume $dr$ . Wave function notion can be generalized to systems more complex than those constituted by one particle. A kinetic description of system constituted by an large number of particles consists in representing the state of considered system by a function $f(r,p,t)$ called "repartition" function, that represents the probability density to encounter a particle at position $r$ with momentum $p$ . Continuous description of matter refers to several functions of position and time to describe the state of the physical system considered.

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