  In this paper, we propose two new regulators for quantum field theories in
spacetimes with compactified extra dimensions. We refer to these regulators as
the ``extended hard cutoff'' (EHC) and ``extended dimensional regularization''
(EDR). Although based on traditional four-dimensional regulators, the key new
feature of these higher-dimensional regulators is that they are specifically
designed to handle mixed spacetimes in which some dimensions are infinitely
large and others are compactified. Moreover, unlike most other regulators which
have been used in the extra-dimension literature, these regulators are designed
to respect the original higher-dimensional Lorentz and gauge symmetries that
exist prior to compactification, and not merely the four-dimensional symmetries
which remain afterward. This distinction is particularly relevant for
calculations of the physics of the excited Kaluza-Klein modes themselves, and
not merely their radiative effects on zero modes. By respecting the full
higher-dimensional symmetries, our regulators avoid the introduction of
spurious terms which would not have been easy to disentangle from the physical
effects of compactification. As part of our work, we also derive a number of
ancillary results. For example, we demonstrate that in a gauge-invariant
theory, analogues of the Ward-Takahashi identity hold not only for the usual
zero-mode (four-dimensional) photons, but for all excited Kaluza-Klein photons
as well.
