JIRA doesn't offer field level security. You need to look for a plugin that offers it. You may also be able to do it with scripting. The out of the box work around is to put the field on a transition screen only executable by one project role and the view screen. Then assign the project owner (do you mean project administrator?) to that role. I would think would put the transition on the Open status or in a follow on status after research has been done into the issue.
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