# Poincaré and Galilei group - notation

On this slide

it just says that $$\mathcal{P}$$ and $$\mathcal{G}$$ are the Poincoré and Galilei groups, but I do not understand what they are made of.

What does $$\mathbb{R}^{1,3}$$ mean?

Why does $$\mathcal{G}$$ have the extra $$(\mathbb{R}^1 \times \mathbb{R}^3)$$?

$$\mathbf{R}^{1,3}$$ is Minkowski spacetime. It's a 4-dimensional real vector space with inner product \begin{align}x\cdot y = - x_0 y_0 + x_1 y_1 + x_2 y_2 + x_3 y_3.\end{align}

Note that in each case the decomposition is given in the form \begin{align} \text{Group = (Boosts and Rotations) } \times \text{(Spacetime translations)}. \end{align} (Strictly speaking, this should be a semidirect product rather than a direct product, but feel free to ignore this detail.)

For the case of the Poincaré group $$\mathcal{P}$$, the Lorentz subgroup $$SO(1,3)$$ includes both rotations and boosts — boosts by themselves aren't a subgroup. But for the Galilei group, boosts (parameterized by a velocity vector in $$\mathbf{R}^3$$) and rotations ($$SO(3)$$) are independent.

The "extra" $$(\mathbf{R}^1 \times \mathbf{R}^3)$$ appearing in $$\mathcal{G}$$ is the group of non-relativistic spacetime translations, the analogue of $$\mathbf{R}^{1,3}$$ for $$\mathcal{P}$$. Again, for $$\mathcal{G}$$, space and time translations decompose.

• so what does $\mathbb{R}^{1,3}$ mean? Translations in space(3)-time(1)? How is that different, in its represenatoin, from $(\mathbf{R}^1 \times \mathbf{R}^3)$? – SuperCiocia Dec 15 '18 at 12:38
• $\mathbb{R}^{1,3}$ is the vector space endowed with the Minkowski inner product that d_b wrote out; if you evaluate it for $x = y$, you simply get the Minkowski metric. Acting on it is the group SO(1,3) which preserves the Minkowski inner product — it is the analog of SO(3) which preserves the usual cartesian inner product (as well as the orientation). – Max Lein Dec 19 '18 at 1:12