Degree of Divergence in Scalar QED

As mentioned in Chapter 1, the biscalar $\langle\phi(x)\phi(x')\rangle$ has its two constituent quantum fields evaluated at two spacetime points, $x$ and the nearby $x'$. This is finite so long as the two points are separated in spacetime, but diverges as the two points are brought together. The so-called {\it degree of divergence} indicates whether the quantities $\langle\phi^{2}\rangle$, $\langle j^{\mu}\rangle$, and $\langle T_{\mu\nu}\rangle$ diverge logarithmically, linearly, etc., when the points are brought together. Quantities that have a logarithmic, linear, etc., degree of divergence contain terms proportional to $\ln(|x-x'|)$, $|x-x'|^{-1}$, etc. The DeWitt-Schwinger point-splitting procedure isolates the infinities which appear in $\langle\phi(x)\phi(x')\rangle$ as $x'\rightarrow x$ and in the VEVs of $\langle\phi^{2}\rangle$, $\langle j^{\mu}\rangle$, and $\langle T_{\mu\nu}\rangle$. This chapter discusses the origin and structure of these infinities using the familiar tool of the Fourier expansion of quantum field operators.

Relativistic quantum field theory calculations of VEVs of operators may be performed by Fourier transforming to wavenumber space. It is known that infinities arise as the integration limits over the wavenumbers involved are extended to infinity. As an example of divergences appearing due to divergent integration over wavenumbers, consider the Wightman function, \begin{equation} G^{+}(x,x')\equiv\langle 0| \underline{\phi}(x)\underline{\phi}(x')|0\rangle, \label{wightmandef}\end{equation} in flat space. The Fourier transform of the operator $\underline{\phi}(x)$ is given by, \begin{equation} \underline{\phi}(x)=\int{d^{3}k\over \sqrt{(2\pi)^{3}2\omega_{k}}} \left[ \underline{a}(k)e^{-ik_{\alpha}x^{\alpha}}+ \underline{b}^{\dagger}(k)e^{ik_{\alpha}x^{\alpha}} \right], \label{phifourier}\end{equation} where $k^{\alpha}$ is the wavenumber of the basis functions $e^{\pm ik_{\alpha}x^{\alpha}}$, $\omega_{\vec{k}}\equiv k_{0}=\sqrt{\vec{k}\cdot\vec{k}+m^{2}}$, and the limits of integration extend from zero to infinity in each wavenumber integral. The action of the particle annihilation and creation operators on the vacuum state $|0\rangle$ is given by \begin{equation} \underline{a}(\vec{k})|0\rangle=0\hspace*{1.0cm},\hspace*{1.0cm} \underline{a}^{\dagger}(\vec{k})|0\rangle=|1(\vec{k})\rangle_{part}, \label{partbasisvac}\end{equation} where $|1(\vec{k})\rangle_{part}$ is a state with one particle of wavenumber $\vec{k}$. The action of the antiparticle annihilation and creation operators on the vacuum state $|0\rangle$ is given by \begin{equation} \underline{b}(\vec{k})|0\rangle=0\hspace*{1.0cm},\hspace*{1.0cm} \underline{b}^{\dagger}(\vec{k})|0\rangle=|1(\vec{k})\rangle_{anti}\ . \label{antibasisvac}\end{equation} where $|1(\vec{k})\rangle_{anti}$ is a state with one antiparticle of wavenumber $\vec{k}$. Substituting Eq.(\ref{phifourier}) into Eq.(\ref{wightmandef}), and using Eqs.(\ref{partbasisvac})--(\ref{antibasisvac}) along with the commutation relations, \begin{equation} [\underline{a}(\vec{k}),\underline{a}^{\dagger}(\vec{k'})]= [\underline{b}(\vec{k}),\underline{b}^{\dagger}(\vec{k'})]= \delta^{3}(\vec{k}-\vec{k'}), \end{equation} gives \begin{eqnarray} G^{+}(x,x')&=& \int{d^{3}k\ d^{3}k'\over (2\pi)^{3} \sqrt{2\omega_{\vec{k}}2\omega_{\vec{k}'}}} e^{-ik_{\alpha}x^{\alpha}+ik'_{\alpha}x'^{\alpha}} \langle0|\underline{a}(\vec{k})\underline{a}^{\dagger}(\vec{k}') |0\rangle \nonumber\\ &=& \int{d^{3}k\ d^{3}k'\over (2\pi)^{3} \sqrt{2\omega_{\vec{k}}2\omega_{\vec{k}'}}} e^{-ik_{\alpha}x^{\alpha}+ik'_{\alpha}x'^{\alpha}} \left[ \langle0|\underline{a}^{\dagger}(\vec{k}')\underline{a}(\vec{k}) |0\rangle+ \langle0|\delta^{3}(\vec{k}-\vec{k}')|0\rangle \right] \nonumber\\ &=& \int{d^{3}k\ d^{3}k'\over (2\pi)^{3} \sqrt{2\omega_{\vec{k}}2\omega_{\vec{k}'}}} e^{-ik_{\alpha}x^{\alpha}+ik'_{\alpha}x'^{\alpha}}\delta^{3}(k-k') \langle 0|0\rangle \nonumber\\ &=&\int{d^{3}k\over (2\pi)^{3}2\omega_{\vec{k}}} e^{-ik_{\alpha}(x^{\alpha}-x^{\alpha'})}. \label{wightmanfourier}\end{eqnarray}

The last line is finite so long as the points $x$ and $x'$ are not coincident. As the points $x$ and $x'$ are brought together, $G^{+}(x,x')$ diverges quadratically since $G^{+}(x,x')\sim\int k\ dk\sim k^{2}$ as $k\rightarrow\infty$. This is where a direct correlation exists between point-splitting and wavenumber space integrations. States with high wavenumbers, or high energy values, have short wavelengths. These short wavelength states are probing the spacetime at ever-decreasing length scales. As the wavenumbers diverge, the spacetime is probed to ever-decreasing length scales, and the vacuum self-energy diverges. Instead of investigating the behavior VEVs using divergent wavenumber integrations, point-splitting studies the behavior of VEVs explicitly in terms of small length scales.

It is straightforward to show that one and two derivatives of $G^{+}(x,x')$ give the cubic and quartic divergences \begin{eqnarray} {\partial\over \partial x}G^{+}(x,x')&\sim& \int k^{2}\ dk\sim k^{3},\quad k\rightarrow\infty , \nonumber\\ {\partial^{2}\over \partial x^{2}}G^{+}(x,x')&\sim& \int k^{3}\ dk\sim k^{4},\quad k\rightarrow\infty \ , \label{wightmanderivs}\end{eqnarray} where spacetime indices have been suppressed for simplicity. It should be noted that $G^{+}(x,x')$ is similar to the Hadamard elementary function $G^{(1)}(x,x')$ in that they are both constructed from the VEV of products of the fields $\phi(x)$ and $\phi(x')$.

While the above discussion of the divergences of the Wightman function is in flat space, it gives an intuitive picture of the physical origin of the divergences in quantum field theory. With $\langle\phi^{2}\rangle$, $\langle j^{\mu}\rangle$, and $\langle T_{\mu\nu}\rangle$ being constructed from $G^{(1)}(x,x')$ and its derivatives, it is possible to consider an analogy between the divergences of $G^{+}(x,x')$ and its derivatives and $G^{(1)}(x,x')$ and its derivatives. For example, the stress-energy tensor $\langle T_{\mu\nu}\rangle$ will be constructed from up to two derivatives of $G^{(1)}(x,x')$. With the result of Eq.(\ref{wightmanderivs}), the stress-energy tensor would potentially contain up to quartic divergences. This will be shown to be true. The current $\langle j^{\mu}\rangle$ will be constructed from up to one derivatives of $G^{(1)}(x,x')$ and would potentially contain up to cubic divergences. However, $\langle j^{\mu}\rangle$ will be shown to actually contain only a linear divergence.

While the above discussion provides a general framework within which to view the origin of the divergences in scalar QED, there exists a method for determining the specific degree of divergence for $\langle\phi^{2}\rangle$, $\langle j^{\mu}\rangle$, and $\langle T_{\mu\nu}\rangle$ individually. To calculate these degrees of divergence, it is necessary to consider the action functional for a complex scalar field coupled to the electromagnetic field in an arbitrary curved background \cite{MTW}; \begin{eqnarray} S[\phi,A_\mu,g_{\mu\nu}]&=&-{1\over 2}\int \mbox{\scri L}\ dV \\ &=&-{1\over 2}\int \mbox{\scri L}\ (-g)^{1/2}\ d^{4}x \\ &=& -{1\over 2}\int {(-g)^{1/2}\left[ (D_{\mu}\phi)(D^{\mu}\phi)^{*}+ (m^{2}+\xi R)\phi\phi^* - {1\over 4}F_{\mu\nu}F^{\mu\nu} \right] d^{4}x}, \label{totalaction} \end{eqnarray} where $\phi(x)={1\over \sqrt{2}}\left(\phi_{1}(x) + i\phi_{2}(x)\right)$ is the complex scalar field, $g$ is the determinant of the metric $g_{\mu\nu}$, $D_{\mu} \equiv (\nabla_\mu-ieA_\mu)$ is the gauge covariant derivative, $A^\mu$ is the classical electromagnetic vector potential, $e$ is the coupling between the complex scalar and the electromagnetic fields, $m$ is the mass of the complex scalar field, $\xi$ is the scalar curvature coupling, and $R$ is the scalar curvature. The Lagrangian density $\mbox{\scri L}$ may be rewritten in terms of interaction Lagrangians \begin{eqnarray} \mbox{\scri L}&=&\left[ (D_{\mu}\phi)(D^{\mu}\phi)^{*}+ (m^{2}+\xi R)\phi\phi^{*}-{1\over 4}F_{\mu\nu}F^{\mu\nu} \right] \nonumber\\ &=&\left[(\nabla_{\mu}-ieA_{\mu})\phi(\nabla^{\mu}+ieA^{\mu})\phi^{*}+ (m^{2}+\xi R)\phi\phi^{*}-{1\over 4}F_{\mu\nu}F^{\mu\nu} \right] \nonumber\\ &=& {1\over 2}\{\partial_{\mu}\phi,\partial^{\mu}\phi^{*}\}+ {1\over 2}(m^{2}+\xi R)\{\phi,\phi^{*}\}+ \nonumber\\ && {ie\over 2}\left[ \{D^{\mu}\phi,\phi^{*}\}-\{D^{\mu}\phi,\phi^{*}\}^{*}\right]A_{\mu}- {e^{2}\over 2}A_{\mu}A^{\mu}\{\phi,\phi^{*}\}- {1\over 4}F_{\mu\nu}F^{\mu\nu} \nonumber\\ &=& {1\over 2}\{\partial_{\mu}\phi,\partial^{\mu}\phi^{*}\}+ {1\over 2}(m^{2}+\xi R)\{\phi,\phi^{*}\} +j^{\mu}A_{\mu} -{e^{2}\over 2}A_{\mu}A^{\mu}\{\phi,\phi^{*}\} -{1\over 4}F_{\mu\nu}F^{\mu\nu} \label{totallagrangian}\end{eqnarray} where $j^{\mu}\equiv {ie\over 2}\left[ \{D^{\mu}\phi,\phi^{*}\}- \{D^{\mu}\phi,\phi^{*}\}^{*}\right]$ is the scalar field current, and the anti-commutators have arisen by symmetrizing with respect to the fields. Eq.(\ref{totallagrangian}) may be rewritten in terms of three interaction Lagrangian densities and the classical electromagnetic Lagrangian density; \begin{equation} \mbox{\scri L}\equiv \mbox{\scri L}_{I,1}+\mbox{\scri L}_{I,2}+ \mbox{\scri L}_{I,3}+\mbox{\scri L}_{EM}, \end{equation} where \begin{equation} \mbox{\scri L}_{I,1}\equiv {1\over 2}\{\partial_{\mu}\phi,\partial^{\mu}\phi^{*}\}+ {1\over 2}(m^{2}+\xi R)\{\phi,\phi^{*}\} \label{scalarinteract}\end{equation} is the interaction Lagrangian density for the scalar field in curved space studied by Christensen \cite{chr}, \begin{equation} \mbox{\scri L}_{I,2}\equiv j^{\mu}A_{\mu}={ie\over 2}\left[ \{D^{\mu}\phi,\phi^{*}\}-\{D^{\mu}\phi,\phi^{*}\}^{*}\right]A_{\mu} \label{jA}\end{equation} is the interaction Lagrangian density for the scalar field current $j^{\mu}$ and the classical background electromagnetic field $A_{\mu}$, \begin{equation} \mbox{\scri L}_{I,3}\equiv -{e^{2}\over 2}A_{\mu}A^{\mu}\{\phi,\phi^{*}\} \label{AAphiphi}\end{equation} is the interaction Lagrangian density for the scalar field and classical background electromagnetic field, and \begin{equation} \mbox{\scri L}_{EM}\equiv -{1\over 4}F_{\mu\nu}F^{\mu\nu} \label{FF}\end{equation} is the Lagrangian density for the classical background electromagnetic field. Note that Eq.(\ref{jA}) is an interaction first order in the coupling constant $e$, while Eq.(\ref{AAphiphi}) is second order in $e$.

The three VEVs $\langle\phi^{2}\rangle$, $\langle j^{\mu}\rangle$, and $\langle T_{\mu\nu}\rangle$ arise from functional variation of Eq.(\ref{totalaction}) for the total action with respect to the the fields $\phi^{*}$, $A^{\mu}$, and $g_{\mu\nu}$, respectively. The degree of divergence of each VEV may be predicted from analysis of the Feynman diagrams for the interaction Lagrangian density pertaining to each. For example, the total interaction Lagrangian density for the scalar field and the electromagnetic field is \begin{equation} \mbox{\scri L}_{\phi,A^{\mu}}\equiv \mbox{\scri L}_{I,2}+\mbox{\scri L}_{I,3}. \label{scalarphotontotal}\end{equation} The first term gives rise to the three-point Feynman graph of Figure \ref{scalarphotonfig}, while the second term is the four-point graph. \begin{figure}[h] \input{scalarphoton} \caption{Feynman diagrams for scalar field-photon interaction} \label{scalarphotonfig} \end{figure}

In the 3-point diagram, there are two external scalar (boson) lines, one external photon line, and one vertex. For scalar QED, the degree of divergence $D$ in four dimensions is given by \cite{Schweber} \begin{equation} D=4-P_{e}-Q_{e}, \end{equation} where $P_{e}$ and $Q_{e}$ are the number of external scalar and photon lines, respectively, in the Feynman diagrams for the interaction. The values $D=0,1,2,\ldots$ imply logarithmic, linear, quadratic, etc., divergences, while $D<0$ implies the interaction is not divergent. DeWitt has pointed out for the generalized Yang-Mills field in the presence of the gravitational field that the simplest possible diagram will be the most divergent \cite{dtgf}. With the total interaction given by Eq.(\ref{scalarphotontotal}), the three-point graph will be the most divergent. Thus, the expected degree of divergence is $D=1$, a linear divergence. The direct correspondence between the degree of divergence in the wavenumber Fourier transforms of the operators $\underline{\phi}(x)$ and the dependence upon the splitting of points in spacetime indicates that, when all calculations of the VEV of the current $\langle j^{\mu}\rangle$ are complete, the infinities should be proportional to $|x-x'|^{-1}$.

The vacuum polarization $\langle\phi^{2}\rangle$ for a real scalar field in curved space has well-known quadratic and logarithmic divergences as the split points $x$ and $x'$ are brought together \cite{chr}. The degree of this divergence should not change in the case of a complex scalar field since the complex field is constructed from the complex sum of two real scalar fields. Thus, $\langle\phi\phi^{*}\rangle\equiv\langle\phi^{2}\rangle$ should also have a quadratic divergence. The interaction Lagrangian density of Eq.(\ref{scalarinteract}) gives rise to a two-point graph with a degree of divergence $D=2$. This gives an expected quadratic divergence in full agreement with previous point-splitting results. With no gauge field contribution to this interaction, all of the divergences should be exactly the same as when no gauge field is present in the spacetime.

The VEV of the stress-energy tensor in curved space has well-known quartic, quadratic, linear, and logarithmic divergences in the case of a real scalar field with no gauge field present \cite{chr}. According to DeWitt \cite{dtgf}, the addition of the gauge field should contribute additional divergences within the existing logarithmic ones. These may be considered as arising from the Lagrangian density of Eq.(\ref{AAphiphi}) which has the logarithmic degree of divergence $D=0$.

Point-splitting will be shown capable of isolating all of the divergences of $\langle\phi^{2}\rangle$, $\langle j^{\mu}\rangle$, and $\langle T_{\mu\nu}\rangle$ predicted above.