The DeWitt-Schwinger point-splitting procedure is known to be capable of yielding analytic expressions that are approximations to both $\langle\phi^{2}\rangle$ and $\langle T^{\mu\nu}\rangle$. The DeWitt-Schwinger point-splitting expansions in this thesis are carried out until they reach terms of order $m^{-2}$ for $\langle T^{\mu\nu}\rangle$, where $m$ is the mass of the quantized field. Presented here for the first time is an analytic approximation for $\langle T^{\mu\nu}\rangle$ for a massive, quantized charged scalar field in a general spacetime with a general electromagnetic field possessing $U(1)$ symmetry. After subtracting the infinites of the stress-energy tensor, which are contained in the terms in the expansion proportional to non-negative powers of $m$, the remaining expression, proportional to $m^{-2}$, serves as the ``DeWitt-Schwinger approximation'' to the actual renormalized value for $\langle T^{\mu\nu}\rangle$. This expression, along with an analytic expression for $\langle\phi^{2}\rangle$ of order $m^{-4}$, constitute the second of the two major results derived here.

The theory of DeWitt-Schwinger geodesic point-splitting began with the classic work of Schwinger \cite{Schwinger}. Schwinger calculated the Green function $G(x,x')$ associated with a fermion current produced by an external electromagnetic field, \begin{equation} \langle j^{\mu}(x)\rangle=\lim_{x'\rightarrow x} ie\mbox{ tr}[\gamma^{\mu}G(x,x')] \end{equation} where $e$ is the charge of the fermion field and the $\gamma^{\mu}$ are the Dirac matrices. He calculated the Green function by introducing a fictitious, non-quantum-mechanical Hilbert space in which calculations were performed \cite{bsd}. This Hilbert space was constructed in $4+1$-dimensions, with the $4$ familiar spacetime dimensions being supplemented by a fifth dimension identified as the proper time parameter $s$ in this fictitious space. Working within this fictitious Hilbert space, Schwinger was able to isolate the divergences that appeared in the quantum field integrals involved with the fermion field current, and use those divergences to renormalize the charge of the fermion current and the strength of the external field.

While Schwinger's original work was performed in flat spacetime, DeWitt recognized that Schwinger's method could provide a way to isolate the divergences that appeared in quantum field theory calculations in curved spacetime. Consider the semiclassical Einstein-Maxwell field equations, \begin{equation} G_{\mu\nu}=8\pi\langle T_{\mu\nu}\rangle, \label{EFE}\end{equation} and \begin{equation} F^{\mu\nu}{}_{;\nu}=4\pi\langle j^{\mu} \rangle. \label{MFE}\end{equation} Eqs.(\ref{EFE}) and (\ref{MFE}) treat the gravitational and electromagnetic fields classically, while treating the sources for these fields quantum mechanically. It is well known that, when the transition is made from classical to quantized fields, infinities appear in the expectation values on the right hand sides of Eqs.(\ref{EFE})--(\ref{MFE}) in both flat \cite{Schweber,BD1,BD2} and curved \cite{ndb2} spacetimes. These infinities can not be physical in the context of curved spacetime physics. For example, in non-gravitational physics, any infinities that appear in energy density calculations are considered to be ``zero-point energies'' and are summarily discarded. This rescaling of the zero point of energy in flat space does not change the physics and is allowed. However, in gravitational physics, all energy is a source of the gravitational field and thus a source of curvature. Zero-point energy, infinite or finite, may not be na\"{\i}vely thrown away since doing so throws away a source of curvature and thus change the physics. Yet there must be a way to correctly remove any unphysical infinities from the theory.

The geodesic point-splitting regularization scheme developed by DeWitt \cite{dtgf,bsd} is a fully covariant method whereby these unphysical infinities may be isolated, a process known as {\it regularization}. Then, in the process known as {\it renormalization}, the infinities are discarded by subtracting them from the unrenormalized field equations, leaving behind finite quantities that represent the physical universe. All of the quantities to be renormalized are VEVs constructed from products of the quantized field $\phi(x)$ and its derivatives. Products such as $\langle\phi(x)\phi(x')\rangle$ have their two constituent quantum fields evaluated at two spacetime points, $x$ and the nearby $x'$. This product is finite so long as the two points are separated in spacetime. The infinities will be shown to arise when these two spacetime points are brought together.

DeWitt based his scheme on the earlier proper-time method method used by Schwinger to calculate the Feynman Green function associated with a quantized fermion field. While there are other regularization methods, point-splitting has proven to be the most robust and trustworthy method of the lot \cite{ndb2}. This is because the point-splitting procedure is well-developed for a general spacetime of arbitrary curvature. Although algebraically quite complicated, point-splitting works in every case, fully isolating all infinite quantities.

Other regularization methods do exist \cite{ndb3}. Pauli-Villars regularization \cite{pv} requires the introduction of fictitious fields whose own divergences are chosen to exactly cancel the divergences of the physical field. The number of these fields introduced is chosen according to the number of divergences the physical fields possess, and these fields are allowed to either commute or anti-commute depending on whether they are required to add to or subtract from the divergences in the stress-energy tensor of the physical field.

Dimensional regularization involves the continuation of quantum field calculations to non-integral spacetime dimensions. Physical parameters requiring renormalization are shown to have bare values proportional to $(n-4)^{-1}$, where $n$ is the dimension of the spacetime. This fact requires the introduction into the original physical Lagrangian terms proportional to bare coefficient of adiabatic order $4$ that will serve to absorb the infinities. However, the renormalized quantities that result must have restrictions on their size in order to be consistent with observations \cite{stelle,hw}.

Adiabatic regularization has been used extensively in calculations of conformally flat spacetimes \cite{tsb,tsb2}. In this method, the subtraction of infinite quantities is based on the adiabatic expansion of the modes of the quantized field. However, the subtractions necessary to renormalize the physical parameters are often too difficult to evaluate. This means that, while the infinities are isolated, they can not serve to renormalize the VEVs of interest such as $\langle T^{\mu\nu}\rangle$.

The technique of $\zeta$-function regularization allows the effective Lagrangian to be written as a derivative of a $\zeta$ function {\it resembling} Riemann's $\zeta$ function on the curved space \cite{dc}. This formal technique for regularizing the effective action uses a generalized $\zeta$ function, $\zeta(\nu)$, whose argument $\nu$ must be analytically continued from regions where $\zeta(\nu)$ converges into regions where $\zeta(\nu)$ does not converge. This is necessary since $\nu=0$, a value for which the generalized $\zeta$ function does not converge, is the value of interest in quantum field calculations.

Point-splitting was chosen for the purposes of this thesis for two main reasons. First, the major advantage that point-splitting possesses over these and any other known regularization schemes is that it is the most efficient method to use when computing actual values for the quantized fields instead of working with purely formal manipulations of the effective action of the theory. Second, by using point-splitting for the present calculations of a complex scalar field in curved space, a connection may be made with the previous work of Christensen in using the point-splitting procedure for regularization of the real scalar field in curved space \cite{chrthesis,chr}. The results of this thesis will explicitly show that, if the charge of the complex scalar field is allowed to vanish, then the regularization counterterms presented here reduce exactly to those of Christensen.

After the point-splitting calculations have been performed and the infinities of the field equations are isolated then, in principle, they may be subtracted from the unrenormalized equations, leaving a finite remainder which contains the relevant physical information. In practice, the renormalized quantities are often too algebraically complicated to evaluate analytically in the case of general spacetimes. In the case of conformally invariant fields in conformally invariant spacetimes, the renormalized quantities been evaluated analytically \cite{ndbtext,bcf,tsb,dfcb}. These calculations have been performed by first renormalizing the Hadamard elementary function, $G^{(1)}(x,x')$, from which the quantities $\langle\phi^{2}\rangle$ and $\langle T_{\mu\nu}\rangle$ are constructed. The VEV $\langle\phi^{2}\rangle$ is directly proportional to $G^{(1)}(x,x')$ and is renormalized when $G^{(1)}(x,x')$ is renormalized. The stress-energy tensor $\langle T_{\mu\nu}\rangle$ is constructed from $G^{(1)}(x,x')$ and its covariant derivatives. The presence of these derivatives requires a slight modification of the renormalization of $G^{(1)}(x,x')$, yet these modifications in essence amount to carrying out certain power series expansions involving the two spacetime points $x$ and the nearby $x'$ to a higher order than when renormalizing $G^{(1)}(x,x')$.

Point-splitting renormalization calculations have been performed for specific spacetimes for both $\langle\phi^{2}\rangle$ and $\langle T_{\mu\nu}\rangle$ for the real scalar field. Candelas \cite{pc} computed a renormalized value for $\langle\phi^{2}\rangle$, or $\langle\phi^{2}\rangle_{ren}$, for the massless scalar field in Boulware, Hartle-Hawking, and Unruh vacuum states in the region exterior to the horizon of a Schwarzschild black hole. Some of the components of $\langle T_{\mu\nu}\rangle_{ren}$ were also renormalized. Frolov \cite{vpf1,vpf2} generalized the work of Candelas such that $\langle\phi^{2}\rangle_{ren}$ for the massless field could be calculated on the event horizon of a Reissner-Nordstr\"{o}m black hole and near the pole of the event horizon of a charged Kerr black hole. Candelas and Howard \cite{ch} calculated $\langle\phi^{2}\rangle_{ren}$ for the Hartle-Hawking vacuum in the Schwarzschild spacetime in the region exterior to the horizon. Candelas and Jensen \cite{cj} analytically continued the calculation of $\langle\phi^{2}\rangle_{ren}$ across the event horizon of a Schwarzschild black hole, giving an expression which is valid for a range of the Schwarzschild radial coordinate $r$. They numerically evaluated $\langle\phi^{2}\rangle_{ren}$ for the range of $r$ given by $0.5M Following the work of the investigators mentioned above, Anderson \cite{pra} has decribed a method whereby $\langle\phi^{2}\rangle_{ren}$ may be numerically computed for free scalar fields in a general static spherically symmetric spacetime. This method assumes the spacetime is asymptotically flat for the purpose of defining initial conditions for the modes sums to be computed for the fields. The fields may be massive or massless, at zero or non-zero temperature, and the spacetime may have arbitrary curvature coupling $\xi$. This scheme is fully renormalized, and the computations may be carried to arbitrary numerical precision.

Renormalization calculations for $\langle T_{\mu\nu}\rangle$ in specific spacetimes have also been performed. These calculations include the work by Howard and Candelas \cite{hch} wherein $\langle T_{\mu}{}^{\nu}\rangle_{ren}$ was calculated for a massless, conformally invariant scalar field in a Hartle-Hawking vacuum state in a Schwarzschild spacetime. Their work allowed numerical computations of $\langle T_{\mu}{}^{\nu}\rangle_{ren}$ in the region exterior to the event horizon and described methods for increasing the efficiency of the computations. Frolov and Zel'nikov \cite{vpf1,fz} calculated an approximate expression for $\langle T_{\mu}{}^{\nu}\rangle_{ren}$ for massive scalar, spinor, and vector fields in Ricci-flat spacetimes. This was done by calculating an approximation for the effective action using the generalization of the DeWitt-Schwinger technique developed by Barvinsky and Vilkovisky \cite{BV}. Brown, Ottewill, and Page derived an analytic approximation for $\langle T^{\mu\nu}\rangle$ in conformal spacetimes using the one-loop effective action for massless, conformally invariant scalar, spinor, and vector fields on static Einstein spaces \cite{bop}.

Building on the studies of $\langle T_{\mu\nu}\rangle$ above, Anderson, Hiscock, and Samuel \cite{AHS} have recently decribed a method whereby $\langle T_{\mu\nu}\rangle_{ren}$ may be calculated to arbitrary numerical precision for a general static spherically symmetric spacetime. The scalar field can be massless or massive, in a zero temperature or non-zero temperature state, and the coupling $\xi$ to the scalar curvature can be arbitrary. While calculationally quite intensive, this method is very powerful since it provides a way of using a fully renormalized stress-energy tensor in quantum field theory calculations in these static spherically symmetric spacetimes.

These renormalization calculations rely on the regularization counterterms previously derived by Christensen \cite{chr,chrthesis}. They use these counterterms to subtract infinities from the unrenormalized expressions for $\langle\phi^{2}\rangle$ and $\langle T_{\mu\nu}\rangle$, leaving behind finite remainders which correctly describe the physics. These subtractions have been performed for the specific spacetimes mentioned above and not for completely general spacetimes. Also, all of these studies have been performed using uncharged fields. The first of the two major results of this thesis are the point-splitting regularization counterterms that will be required for the study of quantized charged scalar fields interacting with the electromagnetic fields that may be present in curved spacetimes.

The quantities $\langle\phi^{2}\rangle$, $\langle j^{\mu}\rangle$, and $\langle T_{\mu\nu}\rangle$ are constructed from the Hadamard elementary function, $G^{(1)}(x,x')$, and its derivatives. The new feature present in this thesis is that these derivatives are now {\it gauge covariant} derivatives. As explained further at the end of this chapter, the coupling of the charged scalar field to the electromagnetic field of the curved spacetime will introduce new physics into the structure of $G^{(1)}$ and its derivatives. This new physics will not only modify the results of Christensen for a real scalar field, but will also be responsible for the generation of the current, $\langle j^{\mu}\rangle$, associated with this charged scalar field.

The Hadamard elementary function is constructed from $\langle\phi(x)\phi(x')\rangle$ and is finite so long as the points $x$ and $x'$ in the argument of $G^{(1)}(x,x')$ are not coincident. The point-splitting scheme is actually an asymptotic expansion of the biscalar $G^{(1)}(x,x')$ in powers of the mass $m$ of the quantized field, an expansion known as a ``DeWitt-Schwinger (DS) expansion.'' The expansion of $G^{(1)}(x,x')$ goes as, \begin{equation} G^{(1)}(x,x')= A_{+2}(x,x')m^{2}+A_{0}(x,x')m^{0}+A_{-2}(x,x')m^{-2}+\ldots, \label{g1morder}\end{equation} where the $A_{n}$ are coefficients constructed from curvature and electromagnetic tensors, and $m$ is the mass of the quantized field. Eq.(\ref{g1morder}) has been shown to contain infinities that arise when the points $x$ and $x'$ are brought together \cite{chr,chrthesis,ndb2,bsd}. These infinities have been shown to be contained in the first two terms of the right hand side of Eq.({\ref{g1morder}), or, in the terms proportional to nonnegative powers of $m$. These infinities may then be, in principle, subtracted from the unrenormalized expressions involving the charged quantized field, with the finite result containing real physical information. Unfortunately, these subtractions are difficult at best to evaluate analytically, as mentioned previously.

Fortunately, point-splitting provides a way out of this dilemma of difficult subtractions, at least in the case of massive quantized fields. The magnitudes of the $A_{n}$ depend directly on the strength of the curvature and electromagnetic fields of the spacetime since they are constructed from curvature and electromagnetic invariants. As the inverse power of $m$ becomes large, the $A_{n}$ contain higher and higher derivatives of both the spacetime metric and electromagnetic field functions. Yet, since Eq.(\ref{g1morder}) is an asymptotic series for $G^{(1)}(x,x')$, for some large value of $n$, the magnitude of the $A_{n}$ becomes large when compared with the magnitude of the $m^{-n}$. Beyond this value for $n$, the asymptotic expansion breaks down, and the series represention of $G^{(1)}(x,x')$ must be abandoned. Subtracting the infinite terms from the unrenormalized expression for $G^{(1)}(x,x')$ leaves a series finite in length containing finite terms. The first few, or even the first one, of these terms may be kept as a ``DeWitt-Schwinger approximation'' for the actual value of $G^{(1)}(x,x')$. The accuracy of this approximation will depend on how large the mass $m$ of the field is chosen when compared to the magnitude of the spacetime curvature invariants. Hereafter, the phrases ``DS expansion'' or ``DS approximation'' will refer to the large mass expansion of any quantity.

The point-splitting procedure is capable of yielding a DS approximation for both the vacuum polarization $\langle\phi^{2}\rangle$ (which is directly proportional to $G^{(1)}$) and the vacuum expectation value of the stress energy tensor $\langle T_{\mu\nu}\rangle$ (which is assembled from $G^{(1)}$ and its derivatives). Unfortunately, one major limitation of the point-splitting procedure is exposed whenever real particle production occurs. As will be shown in greater detail later, point-splitting is incapable of yielding either the imaginary part of the expansion of $G^{(1)}$ \cite{dtgf}, or odd powers of $m$ in Eq.(\ref{g1morder}). These two limitations mean that point-splitting is incapable of yielding a DS approximation for the current of the quantized charged scalar field $\langle j^{\mu}\rangle$.

This thesis calculates the regularization counterterms for the three VEVs $\langle\phi^{2}\rangle$, $\langle j^{\mu}\rangle$, and $\langle T_{\mu\nu}\rangle$. All of these will be derived for the case of a complex scalar field interacting with a classical background electromagnetic in a spacetime of arbitrary curvature. A scalar field is chosen in the present work in order to make a connection with the regularization work previously done by Christensen \cite{chr,chrthesis} and for simplicity. The added complication of spin need not be considered since essentially no new physics is expected to arise when spin effects are considered.

Chapter 2 contains a discussion of the degree of divergence in QED will be presented. The relationship between the momentum space representations of VEVs and the geodesic separation of spacetime points $x$ and $x'$ will be discussed. This discussion will show $G^{(1)}(x,x')$ (and equivalently $\langle\phi^{2}\rangle$) to have an expected quadratic divergence in the spacetime separation distance $|x-x'|$ as the two points $x$ and $x'$ are brought together. The quantities $\langle j^{\mu}\rangle$ and $\langle T_{\mu\nu}\rangle$ will be shown to potentially contain cubic and quartic divergences, respectively, in the separation distance $|x-x'|$.

Chapter 3 begins with a discussion of the local structure of spacetime upon which point-splitting regularization depends crucially. A Hamilton-Jacobi analysis of geodesic motion in the presence of gravitational and electromagnetic fields will lead to the biscalar of the geodetic interval, $\sigma(x,x')$. The biscalar $\sigma(x,x')$ will be shown to be equal to one-half the square of the geodesic distance between the points $x$ and $x'$, and will be shown to carry information about the structure of spacetime along the geodesic between the two points. The Van Vleck-Morette determinant, symbolized $D$, arises in a discussion of caustic surfaces in curved spacetimes, and will be shown to place constraints on how the points $x$ and $x'$ are separated. Differential equations that define $\sigma(x,x')$ and $D$ for the point-splitting procedure will be derived. The bivector of parallel transport, $g^{\mu}{}_{\nu'}$ will be shown as the means for conveying information from the nearby point $x'$ back to the stationary point $x$. A differential equation involving $g^{\mu}{}_{\nu'}$ that is required for the point-splitting procedure will be derived.

Chapter 4 presents a derivation of the geodesic point-splitting procedure using the quantities derived in Chapter 3. This chapter will start with Schwinger's original proper time method of calculating the Feynman Green function, and end with the presentation of the DS expansion of $G^{(1)}$. This will illustrate the need for $G^{(1)}$ and its derivatives to be rewritten in terms of local geometric quantities. The point-splitting ``recursion relations'' derived by Christensen \cite{chr} are a set of differential equations that are used to construct $G^{(1)}$ and its derivatives. In this chapter, a derivation of the recursion relations of the familiar form presented by Christensen \cite{chr} will follow. The crucial new feature in this thesis is the presence of the electromagnetic gauge field. The original recursion relations will be cast into the gauge-invariant form necessary for calculating effects due to {\it both} the gauge and gravitational fields. The gravitational field emerged in Christensen's work while solving the differential equations, or recursion relations, through the commutator of the covariant derivative \begin{equation} [\nabla_{\alpha},\nabla_{\beta}]V^{\mu}= R^{\mu}{}_{\nu\alpha\beta}V^{\nu}, \end{equation} where $\nabla_{\alpha}$ is the covariant derivative, $V^{\mu}$ is a purely geometric vector, and $R^{\mu}{}_{\nu\alpha\beta}$ is the Riemann tensor. In this thesis, the addition of the electromagnetic field requires the use of the gauge covariant derivative whose commutator is given by \begin{equation} [\nabla_{\alpha}-ieA_{\alpha},\nabla_{\beta}-ieA_{\beta}]W= ieF_{\beta\alpha}W, \end{equation} where $W$ is a scalar that explicitly depends on the gauge field, $A_{\alpha}$ is the vector potential for the gauge field, and $F_{\beta\alpha}$ is the Maxwell tensor. The effects of this gauge field in quantum field theory calculations in curved spacetime is the heart of the present work.

Chapter 5 is devoted to solving the recursion relations in order to construct the expansion of Eq.(\ref{g1morder}). The recursion relations can not be solved analytically and so they must be iteratively solved order by order. This iterative process is detailed and a listing of the geometric quantities that form the $A_{n}$ of Eq.(\ref{g1morder}) will be given.

In Chapter 6, the DS point-splitting expansions for $G^{(1)}$, $\langle j^{\mu}\rangle$, and $\langle T_{\mu\nu}\rangle$ will be given, with the divergent and finite parts of each clearly identified. These point-splitting expansions will be asymptotic series as in Eq.(\ref{g1morder}). The expansions for all three VEVs will explicitly show that the divergent parts are proportional to non-negative powers of $m$. The expansion for $G^{(1)}$ will include terms of order $m^{-2}$ and $m^{-4}$, and these terms may be used as a DS approximation to the true value for $G^{(1)}$ as discussed above. This chapter will discuss why the coefficients $B_{n}$ in the expansion for $\langle j^{\mu}\rangle$, \begin{equation} \langle j^{\mu}\rangle= B_{+2}(x,x')m^{2}+B_{0}(x,x')m^{0}+B_{-2}(x,x')m^{-2}+\ldots, \label{jmorder}\end{equation} vanish for all negative $n$. The DS approximations for $\langle\phi^{2}\rangle$ and $\langle T_{\mu\nu}\rangle$ will be presented in their most general form as combinations of the electromagnetic and curvature tensors. Previous workers derived expressions for $\langle\phi^{2}\rangle$ and $\langle T_{\mu\nu}\rangle$ for uncharged fields that contained only geometric terms. Here, the contributions of the electromagnetic field to these expressions will be shown.

Chapter 7 will discuss the possible future use of this work in studying the evolution of charged black hole interiors. The regularization counterterms presented here for $\langle j^{\mu}\rangle$ and $\langle T_{\mu\nu}\rangle$ are a first step towards studies of both gravitational and electromagnetic backreation effects. Also, the use of the general DeWitt-Schwinger approximation presented here for $\langle T_{\mu\nu}\rangle$ will be discussed.

The appendices following the Bibliography contain expressions which are too large to list in the body of this thesis. The equations in these appendices are quite long. So long, in fact, that there was no practical way to apply every simplification rule in order to write them in their most compact form. Rewriting or cancelling a few terms will make no difference in the physics contained in the longer expressions. These appendices are intended as an archive for the expressions derived in the DeWitt-Schwinger point-splitting procedure. Appendices A--E contain the coincidence limits of derivatives of the biscalars $\sigma(x,x')$, $\Delta^{1/2}(x,x')$, $a_{0}(x,x')$, $a_{1}(x,x')$, and $a_{2}(x,x')$. Appendix F contains the purely geometric regularization counterterms first derived by Christensen \cite{chr} that contribute to $\langle T_{\mu\nu}\rangle_{finite}$. In the interest of space, Chapter 6 contains only those terms in the DS expansion of $\langle T_{\mu\nu}\rangle$ which depend on the electromagnetic field. Appendix G contains the purely geometric terms of order $m^{-2}$ term of the DS approximation of $\langle T_{\mu\nu}\rangle$.