How to programmatically login/authenticate a user?

Yes, you can do this via something similar to the following:

use Symfony\Component\EventDispatcher\EventDispatcher,
    Symfony\Component\Security\Core\Authentication\Token\UsernamePasswordToken,
    Symfony\Component\Security\Http\Event\InteractiveLoginEvent;

public function registerAction()
{
    // ...
    if ($this->get("request")->getMethod() == "POST")
    {
        // ... Do any password setting here etc

        $em->persist($user);
        $em->flush();

        // Here, "public" is the name of the firewall in your security.yml
        $token = new UsernamePasswordToken($user, $user->getPassword(), "public", $user->getRoles());

        // For older versions of Symfony, use security.context here
        $this->get("security.token_storage")->setToken($token);

        // Fire the login event
        // Logging the user in above the way we do it doesn't do this automatically
        $event = new InteractiveLoginEvent($request, $token);
        $this->get("event_dispatcher")->dispatch("security.interactive_login", $event);

        // maybe redirect out here
    }
}

The event firing at the end isn’t automatically done when you set a token into the context, whereas it would be normally when using eg a login form or similar. Hence the reason for including it here. You may need to adjust the type of token used, depending on your use case – the UsernamePasswordToken shown above is a core token, but you can use others if required.

Edit: Adjusted the above code to explain the ‘public’ parameter and also add in the roles of the user into the token creation, based on Franco’s comment below.