Hillary Clinton has officially started her search for a running mate, and no potential candidate has evoked more speculation, and, depending on who is asked, wistfulness or dread, than Senator Elizabeth Warren of Massachusetts.

Last week, Ms. Warren had a secret meeting in her Washington condominium with James Hamilton, the lawyer leading the Clinton campaign’s vetting of potential vice-presidential candidates, according to two Democrats briefed on the conversation who requested anonymity to discuss a private conversation.

While the selection of Ms. Warren could help mollify liberal supporters of Senator Bernie Sanders of Vermont, there are ample reasons a Clinton-Warren ticket is unlikely to come to fruition.

Personal Chemistry

Although they have tried to display sisterly solidarity in recent days, Mrs. Clinton and Ms. Warren have had a frosty relationship in the past.