Help with some notation in QFT

Question

I'm reading a paper on QFT and QEIs but i'm a little sketchy on some of the prerequisites. Can anyone tell me what this represents, $$C_{0}^{\infty}(M)$$

Where M is globally hyperbolic spacetime. I understand it has something to do with its topology but im not sure what.

Answer 1

This is the space of all smooth functions on $ M $ that vanish at $ \infty $. Let me make this more precise.

As $ M $ is a space-time manifold, there is an open cover $ \mathcal{U} $ of $ M $ and a $ \mathcal{U} $-sequence of embeddings $ \left( \phi_{U}: U \to \mathbb{R}^{4} \right)_{U \in \mathcal{U}} $ such that for all $ U,V \in \mathcal{U} $, $$ \phi_{U} \circ \phi_{V}^{-1}: \quad \mathbb{R}^{4} \supseteq {\phi_{V}}[U \cap V] \to {\phi_{U}}[U \cap V] \subseteq \mathbb{R}^{4} $$ is a smooth function between open subsets of $ \mathbb{R}^{4} $.

Now, to say that $ f \in {C_{0}}(M) $ means that

• $ f: M \to \mathbb{R} $;
• $ f \circ \phi_{U}^{-1}: \mathbb{R}^{4} \supseteq {\phi_{U}}[U] \to \mathbb{R} $ is a smooth function for each $ U \in \mathcal{U} $; and
• for any $ \epsilon > 0 $, there exists a compact subset $ K $ of $ M $ such that $ |f(x)| < \epsilon $ for all $ x \in M \setminus K $.

Answer 2

Generally speaking, $C^\infty_0(M)$ denotes the subset of $C^\infty(M)$ (smooth functions on $M$) consisting of compactly supported functions. [It is standard notation to use $C^\infty_0$ (as e.g. in the standard reference "Linear PDO's" by L. Hörmander), which is at least as common as $C^\infty_c$, whereas NO ONE uses $C^\infty_{00}$.]