As the Biden administration slowly coalesces, there have been many calls for its Justice Department to prosecute Donald Trump for any crimes he may have committed while in office. The hope, proponents of this view argue, is to establish that the president is subject to the rule of law and to deter future presidents from breaking the law.

The problem with this agenda is that there is little evidence that Mr. Trump did commit crimes as president. A conviction, given what we know now, is all but impossible. The calls to investigate him echo the president’s own calls to investigate Barack Obama, Hillary Clinton and Joe Biden based on mere speculation — calls that most people, especially liberals, rightly condemned.

The most plausible charge is that Mr. Trump obstructed justice by interfering with, and possibly lying to, Robert Mueller and his investigators. Critics also argue that Mr. Trump may have broken the law by threatening to withhold military aid to Ukraine unless the Ukrainian government announced the opening of an investigation into the Bidens. A third possible charge is that Mr. Trump corruptly mixed his financial affairs with government business.

All of these charges would face formidable difficulties in court. No former president has ever been prosecuted for crimes committed during his tenure. Courts tread cautiously when new legal ground is broken, worried about upsetting reasonable expectations about what the law is. And judges interpret criminal laws strictly because the defendant’s freedom is at stake.

Meanwhile, the constitutional powers of the president are extremely vague and have been interpreted expansively by generations of lawyers and judges on the left and right. This combination of strictly interpreted criminal law and expansively interpreted presidential powers all but dooms prosecution of the president except when the facts and law are exceptionally clear.

Consider, for example, the obstruction of justice charge. A private individual who lies to or misleads investigators obstructs justice and may go to jail. But our system gives prosecutors enormous discretion to drop a case, even a strong case, if they believe that prosecution does not advance justice.

The president is the prosecutor’s boss, and while his authority is not limitless, there is no recognized legal standard for distinguishing “obstruction” of a case by a president that is legitimate (like President Barack Obama’s refusal to allow investigations of George W. Bush-era torturers) from that which is illegal. So if Mr. Trump is prosecuted for obstruction of justice, a judge will be required to decide for the first time ever whether the president’s constitutional authority supersedes a statute that is not even clearly directed at the president (as opposed to private citizens). In such cases, courts would normally duck this question by interpreting the statute narrowly so as not to apply to a president.

Mr. Trump may have broken the law directing aid to Ukraine, but the law passed by Congress did not provide for criminal liability. Some people believe that Mr. Trump violated laws against bribery or extortion by refusing to release the money unless the Ukrainian government advanced his political interests. But any attempt to prove that Mr. Trump attempted to solicit a bribe would require overwhelming evidence that does not likely exist — especially because the aid was in fact finally released.

Prosecution would also hit a constitutional wall: The president’s constitutional powers over foreign affairs are by law and tradition nearly completely discretionary. And on top of this, the Supreme Court has expressed skepticism of efforts by prosecutors to use bribery statutes to criminalize the horse-trading, back-scratching and threats that are a normal part of politics and policymaking.

And that is also why Mr. Trump’s many financial entanglements were not criminal. Guests of Trump International Hotel in Washington who sought to curry favor with Mr. Trump may have transferred cash to his wallet through his organization. But this is not a bribe unless a quid pro quo can be proven, and that seems unlikely as well.

An investigation and potential indictment and trial of Mr. Trump would give the circus of the Trumpian presidency a central place in American politics for the next several years, sucking the air out of the Biden administration and feeding into Mr. Trump’s politically potent claims to martyrdom. Mr. Trump will portray the prosecution as revenge by the “deep state” and corrupt Democrats.

For many people, and not just his supporters, the charge will resonate. Prosecution based on technical statutes that have never before been applied to a president would convert this narrative into a powerful mythology that will haunt American politics for years to come.

The Democrats cannot win. If they push forward, prosecutors may drop the case or narrow it down to nothing in response to adverse judicial rulings. If they get to trial, Mr. Trump and his lawyers will turn the tables on the government and attack it for its partisan motivation.

Acquittal would be Mr. Trump’s crowning achievement, but even conviction would enhance his political standing with half the country by completing his martyrdom. If the goal of a trial of Donald Trump is to renew American faith in government, the effect will be the opposite.

Investigations into allegations that Mr. Trump was involved in credit and tax fraud and campaign finance violations before his presidency are another matter. But convictions based on such pre-presidential behavior will not reflect on his presidency nor hold lessons for future presidents.

And even these investigations run the risk of turning Mr. Trump into one of the last things we want him to be: a martyr.

Eric Posner, a professor at the University of Chicago Law School, is the author of, most recently, “The Demagogue’s Playbook: The Battle for American Democracy From the Founders to Trump.”

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