California law (Government Code § 1029 and 1031) sets the minimum standards for selection as a peace officer by individual cities, counties and other governmental entities and jurisdictions. The California Commission on Peace Officer Standards and Training (POST) conducts the research that underlies the minimum standards. A GED is acceptable under POST standards. However, positions as a sworn officer (usually referred to as public safety officers -- "PSO's"-- in hiring literature) are very competitive and most successful candidates will not only have the high school diploma but also some college or other post-high school education.
There is a lot of good info for any potential PSO here: http://post.ca.gov/overview-selection-standards.aspx
My responses to questions on Avvo are never intended as legal advice and must not be relied upon as legal advice. I give legal advice only in the course of an attorney-client relationship. Exchange of information through Avvo's Questions forum does not establish an attorney-client relationship with me. That relationship is established only by individual consultation and execution of a written agreement for legal services.