Arnold Schwarzenegger is in good hands ... 'cause his new chick is a very experienced, high-profile physical therapist ... TMZ has learned.

Her name is Heather ... and she made her first public appearance with Arnold during a romantic dinner date in Santa Monica over the weekend.

The next day, the two met up for a brunch ... and everybody knows there's no such thing as a platonic brunch.

Sources tell TMZ ... Heather is not your average physical therapist ... she's worked with a myriad of professional athletes including UFC badass Georges St-Pierre.

It's unclear if Heather is actually treating Arnold ... but one source close to the Governator tells us, "She is not in the normal group of people who treat him."

Plus, THEY BRUNCHED TOGETHER.

We smell romance ... and huevos rancheros.