David McCormick, the former chief executive of Bridgewater Associates, the world’s largest hedge fund, filed paperwork to run for the Senate in Pennsylvania as a Republican on Wednesday, entering a crowded but unsettled field in what is likely to be one of the most hotly contested midterm elections.

A former Treasury Department official and a former Army captain, Mr. McCormick, 56, joins a number of other major Republican and Democratic contenders vying to succeed Senator Pat Toomey, a Republican, who is retiring. His official announcement will come in the next day or two, according to a campaign adviser, Kristin Davison.

Mr. McCormick’s filing was made after the Pennsylvania Democratic Party asked federal election officials last week to investigate his spending on television ads in the Pittsburgh region without officially declaring himself a candidate.

The race is for the only open Senate seat in a state won by President Biden and is seen as a tossup, making it a critical battleground for control of the chamber, now divided 50-50 with Vice President Kamala Harris’s deciding vote giving Democrats a majority.

The early jockeying in the Republican field has been characterized by most candidates’ efforts to win the support of grass-roots voters who backed former President Donald J. Trump. They include Kathy Barnette, a conservative commentator who has fanned the false conspiracy that Mr. Trump won Pennsylvania in 2020, and Carla Sands, a wealthy former ambassador to Denmark under Mr. Trump, who has promised to “stand up to woke culture, censorship, and critical race theory.” Dr. Mehmet Oz, the heart surgeon and longtime television host, has framed his candidacy as a conservative response to the pandemic, criticizing mandates, shutdowns and actions by “elites” that restricted “our freedom.

Mr. McCormick has his own personal tie to Mr. Trump: His wife, Dina Powell McCormick, served on the National Security Council during the first year of the Trump Administration. The two were married in 2019. Hope Hicks, a former Trump aide, has been advising Mr. McCormick’s team, and other former Trump staffers, including Stephen Miller, are expected to do so, according to Politico.

Five months ahead of the May primary, the field is wide open, especially since the withdrawal in November of Sean Parnell, who was endorsed by Mr. Trump. Mr. Parnell suspended his campaign after losing a custody fight with his estranged wife, who accused him of spousal and child abuse.

Democrats also face a crowded primary contest. Unlike the Republicans, the leading Democrats in the race have experience in elected office. One theme that could animate the general election, depending on who emerges as the G.O.P. nominee, is the issue of who is an authentic Pennsylvanian. Dr. Oz, Ms. Sands and now Mr. McCormick all have roots in the state, but lived elsewhere in recent years and returned to run for Senate.

Ideologically, Republicans promoting Mr. McCormick’s bid have drawn comparisons between him and Glenn Youngkin, the former private equity executive who won the Virginia governor’s race in November by attracting the support of moderates as well as Trump devotees.

Largely unknown outside the financial world, Mr. McCormick grew up in Bloomsburg, Pa., near Wilkes-Barre, where his father was president of Bloomsburg University and later was named chancellor of the Pennsylvania State System of Higher Education. Mr. McCormick graduated from West Point and served five years in the Army, including as an officer in the 82nd Airborne Division during the Gulf War, then earned a Ph.D. in international relations at Princeton and wrote a book about the Army’s downsizing in the early 1990s.

A McKinsey consultant for several years, Mr. McCormick later ran the Pittsburgh-based internet auction company FreeMarkets, then sold it to the larger tech company Ariba in 2004. During the George W. Bush administration, he served in several roles, ultimately becoming under secretary of the Treasury for international affairs.

He joined Bridgewater in 2009 and in 2017, he was named co-C.E.O. of the Westport, Conn.-based hedge fund, which manages $150 billion in assets. His name was repeatedly floated to be the Defense Department deputy during the Trump administration.

In 2020, he became Bridgewater’s sole chief executive after his co-chief, Eileen Murray, left the firm. She later sued Bridgewater over a pay dispute that she said stemmed partly from gender discrimination. The suit was settled in 2020.

On Jan. 3, Mr. McCormick announced his resignation from Bridgewater, calling his potential Senate run “a way of devoting the next chapter of my life to public service” in a farewell email to employees.

Mr. McCormick bought a home recently in Pittsburgh’s East End to re-establish residency in the state, the Pittsburgh Post-Gazette reported. He had split his time between Connecticut and New York City in recent years, though since about 2010 he has owned the family Christmas tree farm where he was raised.

He highlighted the farm in a commercial introducing himself to Pennsylvania voters — as Dave McCormick of Bloomsburg, Pa. — that was broadcast before the holiday.