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物理代写|粒子物理代写Particle Physics代考|PHYS159 Elements of Classical Field Theory

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物理代写|粒子物理代写Particle Physics代考|Elements of Classical Field Theory

Although the natural framework to describe the interactions among elementary particles is the quantum theory, we will start here by recalling some elements of the classical theory of fields. There is a good reason for that. As we alluded to in the first chapter, in order to obtain a quantum theory we start from the corresponding classical theory to which we apply the quantisation prescription. This applies to any physical system, no matter whether it has a finite or an infinite number of degrees of freedom. It follows that the knowledge of the classical system is essential in the formulation of the corresponding quantum system. Since the quantum theory of fields will be the language of elementary particle physics, it is essential to understand the corresponding classical field theory.

物理代写|粒子物理代写Particle Physics代考|Lagrangian and Hamiltonian Mechanics

The high level of conceptualisation of classical mechanics has, since the 19 th century, played an essential part in the development of physical theories. We shall give here a very brief review of the main results with no proofs, essentially in order to fix terminology and notations.

The Lagrangian. Let us consider a system with $N$ degrees of freedom and let $q_{a}(t), a=1, \ldots, N$, denote the corresponding generalised coordinates. We will assume that they determine a point $\boldsymbol{q}$ in an $N$-dimensional differentiable manifold $\mathcal{M}$, for example the $N$-dimensional real space $\mathbb{R}^{N} .{ }^{1}$ We shall call $\mathcal{M}$ the configuration space of the system. Since $\mathcal{M}$ is differentiable, we can consider at every point $\boldsymbol{q}$ the set of $N$ tangent vectors $\dot{q}{a}(t)=\mathrm{d} q{a}(t) / \mathrm{d} t$ of curves passing through $\boldsymbol{q}$. Together with $q_{a}$ they span a $2 N$-dimensional space, which we shall call $\mathcal{T}(\mathcal{M}) .{ }^{2}$

A Lagrangian $L$ is a real function of the $2 N$ variables $q_{a}$ and $\dot{q}{a}$ and, possibly, the time $t$, i.e. $L\left(q{a}, \dot{q}_{a}, t\right): \mathcal{T}(\mathcal{M}) \times \mathbb{R} \rightarrow \mathbb{R}$. An important mathematical tool, which was developed for functional analysis problems, of the kind we shall deal with in this book, is the calculus of variations. For the simple case of $\mathcal{M}=\mathbb{R}^{N}$ it derives the following well-known theorem:

• Consider $\boldsymbol{q} \in \mathbb{R}^{N}$ and let $\gamma=\left{t, \boldsymbol{q} \mid \boldsymbol{q}=\boldsymbol{q}(t), t_{0} \leq t \leq t_{1}\right}$ be a curve in $\mathbb{R}^{N} \times \mathbb{R}$ such that $\boldsymbol{q}\left(t_{0}\right)=\boldsymbol{q}{0}$ and $\boldsymbol{q}\left(t{1}\right)=\boldsymbol{q}{1}$, and let the Lagrangian $L: \mathbb{R}^{N} \times \mathbb{R}^{N} \times \mathbb{R} \rightarrow \mathbb{R}$ be a sufficiently regular function of $2 N+1$ variables. We can prove that the curve $\gamma$ is extremal for the action functional defined by $S[\gamma]=\int{t_{0}}^{t_{1}} L(\boldsymbol{q}, \dot{\boldsymbol{q}}, t) \mathrm{d} t$ in the space of the curves joining $\left(t_{0}, \boldsymbol{q}{0}\right)$ to $\left(t{1}, \boldsymbol{q}_{1}\right)$ if and only if the Euler-Lagrange equations
$$\frac{\mathrm{d}}{\mathrm{d} t}\left(\frac{\partial L}{\partial \dot{q}}\right)-\frac{\partial L}{\partial \boldsymbol{q}}=0$$
are satisfied along $\gamma$.

物理代写|粒子物理代写Particle Physics代考|Lagrangian and Hamiltonian Mechanics

$$\frac{\mathrm{d}}{\mathrm{d} t}\left(\frac{\partial L}{\partial \dot{q}}\right)-\frac{\partial L}{\partial \boldsymbol{q}}=0$$

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