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# 物理代写|粒子物理代写Particle Physics代考|PHYS159 The Lorentz and Poincaré Groups

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## 物理代写|粒子物理代写Particle Physics代考|The Lorentz and Poincaré Groups

The Poincaré transformations consist of the spatial rotations, the boosts and the global translations of space-time. They are of the general form:
$$x^{\prime \mu}(x)=\Lambda_\nu^\mu x^\nu+a^\mu .$$
and leave the Minkowski metric $\eta_{\mu \nu}$ invariant. The invariance of the physical laws under these transformations has been tested with impressive accuracy. In this section we will study the group structure of Lorentz and Poincaré transformations and their applications in the physics of elementary particles.

The physical theories we will consider in this book are formulated in terms of a local Lagrangian density $\mathcal{L}$, which is a function of the dynamical fields $\Phi(x)$ and their first derivatives. Out of $\mathcal{L}$ we form the action $S[\Phi]$ as
$$S[\Phi]=\int \mathrm{d}^4 x \mathcal{L}(\Phi(x), \partial \Phi(x))$$
We see that, by construction, the action, if it exists, is translationally invariant. Therefore, if we enforce the invariance of $\mathcal{L}$ under Lorentz transformations, the invariance of the action under the full Poincaré symmetry is guaranteed. This leads us to study first the representation theory of the Lorentz group. We will consider the fields $\Phi(x)$ belonging to irreducible representations and build out of them invariant Lagrangian densities. As a second step we shall study the representation theory of the full Poincaré group.

## 物理代写|粒子物理代写Particle Physics代考|The Lorentz group

We have already seen that the Lorentz group preserves the Minkowski metric which, in our notation, is $\eta=\operatorname{diag}(1,-1,-1,-1)$, with one $+$ and three $-$ signs. It is often denoted by $O(1,3)$. We have also seen that it is a non-compact Lie group. In order to build invariant Lagrangian densities we must know how the fields $\Phi(x)$ transform. We are only interested in fields with a finite number of components, so we will study the finite-dimensional representations of the Lorentz group. There is a general theorem according to which all non-trivial unitary representations of a non-compact group are infinite-dimensional. Therefore, contrary to what we did with the symmetry groups we have considered so far, we will study non-unitary representations of the Lorentz group.
A coordinate $x^\mu$ is transformed linearly as
$$x^{\prime \mu}(x)=\Lambda_\nu^\mu x^\nu$$

## 物理代写|粒子物理代写粒子物理学代考|洛伦兹群和Poincaré群

Poincaré转换由时空的空间旋转、推进和全局平移组成。它们具有一般形式:
$$x^{\prime \mu}(x)=\Lambda_\nu^\mu x^\nu+a^\mu .$$

$$S[\Phi]=\int \mathrm{d}^4 x \mathcal{L}(\Phi(x), \partial \Phi(x))$$

## 物理代写|粒子物理代写粒子物理学代考|洛伦兹群

$$x^{\prime \mu}(x)=\Lambda_\nu^\mu x^\nu$$

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