It is usually said that potential energy has an arbitrary additive
constant, and that it is a mere convention that we set that constant to
zero for electric potential energy, writing U = kq_1q_2/r. However,
relativity requires that the constant be zero. The energy of a system
consisting of an electron and a positron with both particles at rest is,
including an arbitrary additive constant C,
2mc^2 + U = 2mc^2 + (-ke^2/r + C)
As r -> infinity, the system energy approaches 2mc^2 + C, which shows that
C must be zero. A theorist colleague told me that there is some wiggle room
with respect to potential (kq/r) but not for potential energy.
When I pointed this out to my colleague Fred Reif at Carnegie Mellon, he
responded by pointing a similarity to the situation with entropy, which
pre-quantum mechanics had an arbitrary additive constant, but post-quantum
mechanics at absolute zero the entropy approaches zero.
It's interesting that two of the physics revolutions of the twentieth
century, relativity and QM, introduced absolute character for both energy
and entropy.