[db] Add context propagation for PostgreSQL via SET application_name

.chloggen/postgresql-application-name-propagation.yaml

@@ -0,0 +1,5 @@
+change_type: enhancement
+component: db
+note: Add context propagation for PostgreSQL via `SET application_name`.
+issues: [2162]
+subtext:

docs/db/postgresql.md

@@ -9,6 +9,8 @@ linkTitle: PostgreSQL
 <!-- toc -->
 
 - [Spans](#spans)
+- [Context propagation](#context-propagation)
+  - [SET application\_name](#set-application_name)
 - [Metrics](#metrics)
 
 <!-- tocstop -->
@@ -165,6 +167,36 @@ and SHOULD be provided **at span creation time** (if provided at all):
 <!-- END AUTOGENERATED TEXT -->
 <!-- endsemconv -->
 
+## Context propagation
+
+**Status**: [Development][DocumentStatus]
+
+### SET application_name
+
+Instrumentations MAY propagate context using [`SET application_name`](https://www.postgresql.org/docs/current/runtime-config-logging.html#GUC-APPLICATION-NAME) by injecting fixed-length part of span context (trace-id, span-id, trace-flags, protocol version) before executing a query. For example, when using W3C Trace Context, only a string representation of [`traceparent`](https://www.w3.org/TR/trace-context/#traceparent-header) SHOULD be injected. Context injection SHOULD NOT be enabled by default, but instrumentation MAY allow users to opt into it.
+
+Variable context parts (`tracestate`, `baggage`) SHOULD NOT be injected since `application_name` value length is limited to less than [`NAMEDATALEN`](https://www.postgresql.org/docs/current/runtime-config-preset.html#GUC-MAX-IDENTIFIER-LENGTH) characters (63 characters in a standard build).
+
+Instrumentations that propagate context MUST execute `SET application_name` on the same physical connection as the SQL statement.
+
+The `application_name` value is visible in [`pg_stat_activity`](https://www.postgresql.org/docs/current/monitoring-stats.html#MONITORING-PG-STAT-ACTIVITY-VIEW) and can be included in PostgreSQL server logs via `%a` in [`log_line_prefix`](https://www.postgresql.org/docs/current/runtime-config-logging.html#GUC-LOG-LINE-PREFIX).
+
+Example:
+
+For a query `SELECT * FROM songs` where `traceparent` is `00-0af7651916cd43dd8448eb211c80319c-b7ad6b7169203331-01`:
+
+Run the following command on the same physical connection as the SQL statement:
+
+```sql
+SET application_name = '00-0af7651916cd43dd8448eb211c80319c-b7ad6b7169203331-01';
+```
+
+Then run the query:
+
+```sql
+SELECT * FROM songs;
+```
+
 ## Metrics
 
 PostgreSQL client instrumentations SHOULD collect metrics according to the general