There is a new act touring California this summer: The Pete and Jerry Show.

"Pete" is San Diego Mayor Pete Wilson, the GOP nominee for the Senate seat now held by retiring Republican S. I. Hayakawa. "Jerry" is Gov. Edmund G. (Jerry) Brown Jr., Wilson's Democratic opponent.

The Pete and Jerry Show opened the day after the June 8 primary with a live, nine-minute joint appearance on KABC-TV's local news show here. The candidates, calling one another "Jerry" and "Pete," have made subsequent appearances on KABC-TV and KNXT-TV, the CBS affiliate here. And they promise return engagements.

In his election night victory speech, Brown challenged Wilson to a series of 12 debates, which Wilson accepted. So far two debates, sponsored by the League of Women Voters, have been scheduled for the fall, one in Los Angeles, one in San Francisco.

In California, the primary season is usually followed by a summer lull as candidates prepare for the traditional Labor Day start of campaigning for the general election. Then, rested, the combatants open their campaigns with smiles and slogans.

Later, they talk seriously and earnestly about issues. And finally, they do whatever they can to annoy each other. This year, the Senate horses are out of the gate and trying to annoy each other early.

Wilson started by buying 10-second television commercials, aired statewide, in which he said: "This is Pete Wilson. And I just want to say, 'Thanks California, for your vote of confidence. I'm not going to let you down.' "

The "mini-debates" feature local news anchors asking the two candidates about the issues.

About the only thing they agree on is their support for Israel in its invasion of Lebanon, which both say is necessary for Israel's security. But even on that question they are not in complete accord: Brown would like to cancel the Airborne Warning and Control Systems (AWACS) sale to Saudi Arabia, which supports the Palestine Liberation Organization. Wilson said he would not yet support a cancellation.

Wilson, claiming that an imbalance of nuclear weapons favors the Soviet Union over the United States, said he could not support the nuclear arms freeze. Brown, a strong supporter of the movement to freeze "these crazy, insane weapons," denied that an imbalance exists.

Brown, who hopes to capitalize on the electorate's economic woes, made specific suggestions for improving the national economy, all of which run counter to Reaganomics. Wilson criticized Congress for its lack of action on a new budget.

Wilson, who hopes to capitalize on Brown's current low standing in the polls, accused Brown of being a "taxer and spender." Brown countered with the contention that he is known for being "frugal" and "cheap."

When Wilson repeated the charge, Brown responded that, "I have cranked down spending in California more than any governor in a long, long time," and added: "Pete, you're going to have to get some other ammunition because the facts won't support that little statement you just made."

Wilson and Brown have been seated near each other during these appearances. In one appearance, Wilson kept patting Brown's arm with regularity as he addressed him: "Jerry, . . . " Pat, pat.

Brown responded to the patting by casting an incredulous glance at the hand on his arm.

It is safe to say that if Brown gets upset by the patronizing pat, it won't ruin Wilson's day. And Labor Day is still 10 weeks away.