There are several ways casinos handle “cash in the game”, depending on the casino and the locality / jurisdiction that the game is playing in. Traditionally, cash simply “plays” in the game. However, due to money laundering regulations, many casinos require players to exchange the cash for chips in order to play.
Most casinos allow for cash to play temporarily. That is, until the casino personnel change up the cash for chips. The dealer can announce “cash plays” for a player who has cash on the table, and “cash behind” to represent a player who has given his cash to the brush, and is awaiting the chips to come from the cage.
Reference
WSOP 2024 Live Action Rules
#55 – Cash does not play and is not permitted to remain on the table as any part of a participant’s stakes. If a participant is buying into a game, re-buying, or adding stakes to an existing buy-in, this participant must proceed to the WSOP Live-Action cage to purchase chips.