I have watched only one Arnold Schwarzenegger film, and rather enjoyed the absurdity of it. But I have never met anyone, male or female, who likes the huge, steroid-inspired muscles that are Schwarzenegger's trademark. Which creates an abiding mystery about Arnie: how on earth does he do it? How does the Austrian son of a Nazi storm trooper, who can hardly speak English, become an American hero worth, at a conservative estimate, $200m? I am told his movie career is in decline - despite an original fee of around $30m for Terminator 3, his latest artistic effort - so the new question is: can Arnie really become the next governor of California?

The answer is maybe. Arnie now needs police protection wherever he goes; he is being mobbed by his fans; and is being advised by a former governor of California, a zillionaire financier, and a former US secretary of state. Brits once guffawed at Ronald Reagan's entry into politics; anyone underestimating Arnie and his chances of achieving political office is mistaken.

First, though, a word about how California got itself into this mess. It has the fifth- or sixth-largest economy in the world - the figures are contradictory. But it also has an obscure 1911 constitutional amendment under which, if more than 12 per cent of the number of people who voted in the previous election for a statewide politician sign a petition asking for that politician to be "recalled" - that is, sacked - a new election can be held.

And that is exactly what has happened. A Republican state senator (who has already tearfully withdrawn his own candidature for the governorship) organised a recall of the extremely unpopular Democrat governor, Gray Davis; he needed 897,156 signatures, but gathered around 1.5 million to make sure.

Davis - whose physical stature is the precise converse of Arnie's - was first elected governor of California in 1998. But he became deeply unpopular when power bills quadrupled. Like all states, California has also been hit by the let-them-eat-bread policies of Dubbya and, as the largest, California has been hit the most. So Davis is now campaigning not on his record but on the line that the only way Republicans can win major political office is to undo properly run elections - and his tactic shows some signs of resonating with heavily Democratic Californians.

The Republican-inspired rerun of the election will be held on 7 October. Voters will first be asked whether they agree with the recall. And a recent poll in the Los Angeles Times showed that just 50 per cent currently agree with it. If the vote is no, Davis stays as governor. But if it is yes, then the field is open to any one of a ludicrous 134 candidates, ranging from Arnie to the publicity-fond Arianna Huffington to Gary Coleman (a man who has the stature and voice of a child and has starred as such in long-forgotten television comedies) to Larry Flynt, a wheelchair-bound pornographer.

The dark horse in all this is the hitherto obscure Cruz Bustamante. He is currently Davis's deputy. The two have never got on personally, let alone politically, and, much to Davis's fury, Bustamante has elected to stand as a Democrat. This means it is quite possible that the electorate will agree to a recall but then vote in Bustamante as a Democrat alternative to Davis, rather than let the governorship go to a Republican. Indeed, the Los Angeles Times poll showed 35 per cent support for Bustamante, compared to Arnie's 22 per cent.

The little we know about the real Arnie is that he shrewdly invested his takings from winning seven Mr Olympia titles in property deals and then somehow seemed to captivate Hollywood with his muscles and his Austro-American gruntings. The reason he is taken seriously by people such as Pete Wilson (the former governor of California), Warren Buffett (the financier) and George Shultz (Reagan's secretary of state) is that, under the first President Bush, he was appointed a sports proponent to encourage fitness in children. He visited all 50 states, using his own money, and gained credibility in the Republican circles he assiduously cultivated. It helped for stature, too, to be married to Maria Shriver of the Kennedy clan.

Arnie is supposedly pursuing the middle ground in Californian politics, being pro-abortion, pro-environment, and so on; there are two serious Republican candidates who will take votes from him unless they, too, withdraw. His biggest following, the pollsters tell us, is among young Latino males - and they tend not to vote. We have heard little of the wit and wisdom of Arnie, except perhaps such pearls as "I will be a governor for the people for a change because, because I want to represent the people because the only thing that counts for me is the people". Quite, Arnie, quite: but at least, because you were born in Austria, the US constitution forbids you to become president. Sorry about that.