WASHINGTON (CNN) -- On the heels of signing a measure that officially moves the California presidential primary up to February 5, Gov. Arnold Schwarzenegger told CNN Thursday he hopes Golden State voters will play an important role in selecting each party's nominee.

"All we want to do is, if it is a Democrat or Republican that is in the White House, I think it is advantageous to California if those candidates come out to California, know what our issues are, know that they care about infrastructure, about health care reform, about the environment, about political reform, all of those things," Schwarzenegger told CNN's Wolf Blitzer.(Watch Video: Schwarzenegger makes a visit to "The Situation Room")

While many political analysts have suggested California's GOP primary will help the fields more moderate candidates -- former New York City Mayor Rudy Giuliani and Arizona Sen. John McCain -- Schwarzenegger said that was not a factor in the decision to change the primary's date.(Related Video: Begala and Bennett weigh in on how the California primary will affect the race)

The California governor also said he would likely not endorse a candidate before the primary process, saying he welcomes "all candidates to come in here and talk about the issues that are important to California."

Regarding his own political future, Schwarzenegger said he hadn't given any thought to the possibility of running for president even if an amendment was passed that removed the restriction on people not born in the United States.

"For me, I don't even want to think about all of this, running for president, not running for president," he said "I'm happy, the job that I have. And I'm not even thinking about, OK, this is something I can't do because I'm -- I wasn't born here. I only think about the things that I can do."

He also said he hasn't given any thought to challenging California Sen. Barbara Boxer in 2010.

-- CNN Ticker Producer Alexander Mooney