Heat kernel expansion in the background field formalism

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Andrei O. Barvinsky (2015), Scholarpedia, 10(6):31644. doi:10.4249/scholarpedia.31644 revision #150661 [link to/cite this article]
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Contents

Introduction

Heat kernel expansion and background field formalism represent the combination of two calculational methods within the functional approach to quantum field theory. This approach implies construction of generating functionals for matrix elements and expectation values of physical observables. These are functionals of arbitrary external sources or the mean field of a generic configuration -- the background field. Exact calculation of quantum effects on a generic background is impossible. However, a special integral (proper time) representation for the Green's function of the wave operator -- the propagator of the theory -- and its expansion in the ultraviolet and infrared limits of respectively short and late proper time parameter allow one to construct approximations which are valid on generic background fields. Current progress of quantum field theory, its renormalization properties, model building in unification of fundamental physical interactions and QFT applications in high energy physics, gravitation and cosmology critically rely on efficiency of the heat kernel expansion and background field formalism.

Background field formalism

Effective action in quantum field theory $\varGamma[\phi]$ is the functional of the mean field $\phi$, which contains all information about a quantum system, that is the set of correlation functions of its physical observables. For a system with the classical action $S[\varphi]$ these correlators can be obtained by functionally differentiating with respect to the auxiliary external source $J(x)$ the generating functional \[ \tag{1} e^{-W[J]/\hbar}= \int D\varphi \exp \frac1\hbar\left(-S[\varphi] -\int dx\,\varphi(x)J(x) \right), \] represented by the functional integral over the quantum field $\varphi(x)$. In particular, the lowest order correlation function -- the mean field $\phi(x)$, $\phi(x)\equiv\langle\varphi(x)\rangle=\delta W/\delta J(x)$, allows one to introduce the Legendre transform of $W[J]$, $\varGamma[\phi]=W[J]-\int dx\,J(x)\phi(x)$, and this effective action functional generates the equation \[ \tag{2} \frac{\delta\varGamma[\phi]}{\delta\phi(x)}=-J(x). \] Thus $\phi$ and $\varGamma[\,\phi\,]$ comprise quantum generalization of the classical field $\varphi$, its classical action $S[\,\phi\,]$ and classical equations of motion $\delta S/\delta\phi=-J$ in the presence of the external source (DeWitt B S, 1965).

Relations of the above type allow one to rewrite in a closed form the following functional integro-differential equation for $\varGamma[\phi]$ \[ \tag{3} e^{-\varGamma[\phi]/\hbar}= \int D\varphi \exp \frac1\hbar\left(-S[\varphi] +\int dx\,\big(\varphi(x) -\phi(x)\big)\frac{\delta\varGamma[\phi]}{\delta\phi(x)}\right), \] where the right hand side is the path integral over the quantum field $\varphi(x)$ parametrically depending on the mean field $\phi(x)$. This is the Euclidean QFT formulation, generating by Wick rotation IN-OUT scattering amplitudes (DeWitt B S, 1965) or by the Schwinger-Keldysh formalism IN-IN expectation values (Schwinger J, 1961; Keldysh L V, 1965).

Semiclassical expansion of $\varGamma[\phi]$ in powers of $\hbar$ starts with the classical action $S[\phi]$ followed by quantum terms -- contibutions of the saddle point of this integral at $\varphi=\phi$, \[ \tag{4} \varGamma[\phi]=S[\phi]+\hbar\varGamma_{\rm 1-loop}[\phi] +\hbar^2\varGamma_{\rm 2-loop}[\phi]+\ldots. \] Two lowest quantum orders read as the one-loop functional determinant of the operator of the second functional derivative of the classical action \[ \begin{eqnarray} &&\varGamma_{\rm 1-loop}=\frac12\ln\mathrm{Det} F(\nabla)= \frac12{\mathrm{Tr}}\ln F(\nabla), \tag{5}\\ &&F(\nabla)\delta(x,y)=\frac{\delta^2 S[\phi]} {\delta\phi(x)\delta\phi(y)}, \tag{6} \end{eqnarray} \] and the two two-loop terms, \[ \begin{eqnarray} &&\varGamma_{\rm 2-loop}=\frac18 \int dx_1 dx_2 dx_3 dx_4 G(x_1,x_2) \frac{\delta^4 S[\phi]} {\delta\phi(x_1)\delta\phi(x_2)\delta\phi(x_3)\delta\phi(x_4)} G(x_3,x_4) \\ &&\qquad\qquad+\frac1{12}\int dx_1 dx_2 dx_3dy_1 dy_2 dy_3\frac{\delta^3 S[\phi]} {\delta\phi(x_1)\delta\phi(x_2)\delta\phi(x_3)} G(x_1,y_1)G(x_2,y_2) G(x_3,y_3)\frac{\delta^3 S[\phi]} {\delta\phi(y_1)\delta\phi(y_2)\delta\phi(y_3)}, \tag{7} \end{eqnarray} \] built in terms of the Green's function $G(x,y)$ of $F(\nabla)$, \[ \begin{eqnarray} F(\nabla)G(x,y)=-\delta(x,y), \end{eqnarray} \] and multiple functional derivatives of the classical action $\delta^n S[\phi]/\delta\phi(y_1)...\delta\phi(y_n)$. Graphically these contributions to $\varGamma$ can be represented as the one-loop and two-loop Feynman diagrams with the lines corresponding to Green's functions and vertices determined by these functional derivatives,

Figure 1: Feynman diagrams of $\varGamma_{\rm 1-loop}$ and $\varGamma_{\rm 2-loop}$

It is important that the propagators and vertices of this diagrammatic technique are determined not on a trivial background of vanishing field or flat space metric but for a generic mean field configuration including all scalar, spinor, vector, metric, etc. fields of the model $\phi=\phi(x),\psi(x),A_\mu(x),g_{\mu\nu}(x),...$. Thus all these expressions are functionals of the mean field, rather than functions of particle momenta, coupling constants, etc. Functional differentiation of $\varGamma[\phi]$ with respect to $\phi$ gives the effective equations for the mean field and all needed correlation functions on its background. This method is called the background field formalism which generalizes the Feynman diagrammatic technique and signifies the use of functional methods in QFT.

Proper time method

Efficient tool for the calculation of the background field (or mean field) functionals is the Schwinger proper time method. It is based on the integral representation of the Green's function of the operator $F(\nabla)$ and its one-loop functional determinant in terms of the heat kernel $K(s)=e^{sF(\nabla)}$, \[ \begin{eqnarray} &&G\equiv -F^{-1}(\nabla)=\int_0^\infty dsK(s),\\ &&\frac12{\mathrm{Tr}} \ln F(\nabla)=-\frac{1}{2} \int_{0}^{\infty }\frac{ds}{s}{\rm Tr} K(s).\tag{8} \end{eqnarray} \] Here $G$ is the operator whose kernel is the Green's function $G(x,y)$. For a multicomponent field $\phi=\phi^A(x)$ whose components form a vector in the configuration space of the system, labeled by spin-tensor indices $A$, its operator $F(\nabla)$ and propagator $G(x,y)$ acquire the matrix structure denoted below by the hat, $\hat F(\nabla),\hat G(x,y)$. Correspondingly the kernel of $K(s)$ is a matrix $K^A_B(s|x,y)\equiv\hat K(s|x,y)$ acting in the representation space of $\phi^A(x)$ and the functional trace involves the trace over matrix indices, denoted below by ${\rm tr}$, \[ \begin{eqnarray} &&\hat K(s|x,y)=e^{s\hat F(\nabla)}\delta(x,y), \tag{9} \\ &&{\rm Tr} K(s)=\int dx\, {\rm tr}\hat K(s|x,x). \tag{10} \end{eqnarray} \]

The efficiency of this formalism is based on the heat equation for $\hat K(s|x,y)$ -- the Euclidean version of the evolutionary Schroedinger-type equation (explaining the name of proper time for the parameter $s$) \[ \begin{equation} \frac{\partial}{\partial s} \hat K(s|x,y)=\hat F(\nabla)\hat K(s|x,y), \quad \hat K(0|x,y)=\hat 1\delta(x,y), \tag{11} \end{equation} \] and powerful approximation methods for its solution for generic metric and matter fields entering the coefficients of the operator $F(\nabla)$.

Heat kernel technique and HAMIDEW coefficients

The coefficients of the minimal operators, whose second order derivatives form a covariant d'Alembertian acting in the representation space of $\phi^A$, \[ \begin{eqnarray} \hat F(\nabla)=\Box+\hat P- \frac{\hat 1}6R-m^2\hat 1, \quad \Box=g^{\mu\nu}\nabla_\mu\nabla_\nu. \tag{12} \end{eqnarray} \] are characterized by the set of "curvatures" \[ \begin{eqnarray} \Re=\hat P,\hat{\cal R}_{\mu\nu}, R_{\mu\nu\alpha\beta}, \tag{13} \end{eqnarray} \] the potential term $\hat P$ (the term $-\frac16 R\hat 1$ is singled out from it for convenience), fibre bundle curvature $\hat{\cal R}_{\mu\nu}$ -- the commutator of covariant derivatives acting on the vector $\phi$ or the matrix $\hat X$ -- and the Riemann tensor, \[ \begin{eqnarray} &&[\nabla_\mu,\nabla_\nu]V^\alpha=R^\alpha_{\beta\mu\nu}V^\beta,\\ &&[\nabla_\mu,\nabla_\nu]\phi=\hat{\cal R}_{\mu\nu}\phi,\quad [\nabla_\mu,\nabla_\nu]\hat X=[\hat{\cal R}_{\mu\nu},\hat X]. \end{eqnarray} \]

The heat kernel for (12) has a small (or early) time asymptotic expansion at $s\to 0$, \[ \begin{eqnarray} \hat K(s|x,y)=\frac{\Delta^{1/2}(x,y)}{(4\pi s)^{d/2}}g^{1/2}(y)e^{-\sigma(x,y)/2s-sm^2} \sum\limits_{n=0}^\infty s^n \hat a_n(x,y), \tag{14} \end{eqnarray} \] where $\sigma(x,y)$ is the Synge world function -- one half of the square of geodetic distance between the points $x$ and $y$, and $\Delta(x,y)=g^{-1/2}(x)|{\rm det}\partial_\mu^x\partial_\nu^y\sigma(x,y)|g^{-1/2}(y)$ is the (dedensitized) Pauli-Van Vleck-Morette determinant built of $\sigma(x,y)$ (both $\delta(x,y)$ and $\hat K(s|x,y)$ are defined above as zero weight densities with respect to $x$ and densities of weight one with respect to $y$, which explains the factor $g^{1/2}(y)$). The two-point matrix quantities $\hat a_n(x,y)$ bear the name of HAMIDEW (Gibbons G W, 1979) or Gilkey-Seeley coefficients praising the efforts of mathematicians and physicists in heat kernel theory (DeWitt B S , 1965; Gilkey P B, 1984) (see review of physics implications of this theory in (Barvinsky A O, Vilkovisky G A, 1985; Avramidi I G, 2000; Vassilevich D V, 2003)).

The equation for the world function $g^{\mu\nu}\nabla_\mu\sigma\nabla_\nu\sigma=2\sigma$ and the recurrent equations for $\hat a_n(x,y)$, which follow from (11), allow one to derive the coincidence limits for $\sigma(x,y)$, $\Delta^{1/2}(x,y)$ and $\hat a_n(x,y)$ to arbitrary order \[ \begin{eqnarray} &&\nabla_{\mu_1}\ldots\nabla_{\mu_k}\sigma(x,y) \big|_{y=x}, \quad \nabla_{\mu_1}\ldots\nabla_{\mu_k}\Delta^{1/2}(x,y)\big|_{y=x}, \tag{15} \\ &&\nabla_{\mu_1}\ldots\nabla_{\mu_k}\hat a_n(x,y)\big|_{y=x} \tag{16} \end{eqnarray} \] Their remarkable property is that they are local functions of the curvatures (13) and their covariant derivatives, starting with \[ \begin{eqnarray} &&\sigma\big|_{y=x}=0,\quad\nabla_{\mu}\nabla_{\nu}\sigma\big|_{y=x}=g_{\mu\nu},\quad \nabla_{\mu}\nabla_{\nu}\nabla_{\alpha}\sigma\big|_{y=x}=0,\\ &&\nabla_{\mu}\nabla_{\nu}\nabla_{\alpha}\nabla_{\beta}\sigma \big|_{y=x}=-\frac23R_{\mu(\alpha\nu\beta)},\\ &&\Delta^{1/2}\big|_{y=x}=1,\quad \nabla_{\mu}\Delta^{1/2}\big|_{y=x}=0,\quad \nabla_{\mu}\nabla_{\nu}\Delta^{1/2}\big|_{y=x}=\frac16R_{\mu\nu}, \end{eqnarray} \] and \[ \begin{eqnarray} \hat a_0(x,x) &=&\hat 1,\quad \nabla_\mu\hat a_0(x,y)\big|_{y=x}=0, \quad \nabla_\mu\nabla_\nu\hat a_0(x,y)\big|_{y=x}=\frac12\hat{\cal R}_{\mu\nu},\\ \hat a_1(x,x)&=&\hat P,\\ \hat{a}_2(x,x)&=&\frac1{180}(R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}-R_{\mu\nu} R^{\mu\nu})\hat{1} +\frac1{12}\hat{\cal R}_{\mu\nu}\hat{\cal R}^{\mu\nu}+\frac12\hat{P}^2+\frac16\Box\hat{P}+\frac1{180}\Box R\hat{1}. \end{eqnarray} \] The general structure of the coincidence limit $\hat a_n(x,x)$ is the sum of various covariant monomials of curvatures and their covariant derivatives of powers $m$ and $p$ with the total dimensionality $2n$, \[ \begin{eqnarray} \hat a_n(x,x)\propto \stackrel{2p}{\overbrace{\nabla\cdots\nabla}} \stackrel{m}{\overbrace{\Re\cdots\Re}},\quad m+p=n. \tag{17} \end{eqnarray} \]

UV divergences, anomalies and local expansion

In theories with a nonzero mass local nature of $\hat a_n(x,x)$ gives the local inverse mass expansion of the one-loop effective action (DeWitt B S, 1965; Barvinsky A O, Vilkovisky G A, 1985; Barvinsky A O, Mukhanov V F, 2002) \[ \begin{eqnarray} &&\frac12{\rm Tr}\ln F(\nabla)=\varGamma_{\mathrm{div}}+\varGamma_{\log }-\frac{1}{2} \left( \frac{m^{2}}{4\pi}\right)^{d/2} \sum\limits_{n=d/2+1}^{\infty} \frac{\Gamma(n-\frac{d}2)}{(m^{2})^{n}} \int dx\, g^{1/2}{\rm tr} \hat a_{n}(x,x). \tag{18} \end{eqnarray} \] For an even $d$ it begins with dimensionally regularized UV divergences, $\omega\to d/2$, and logarithmic in mass term determined by the first $d/2$ HAMIDEW coefficients \[ \begin{eqnarray} &&\varGamma_\mathrm{div}\!=\!\frac{1}{2(4\pi )^{d/2}}\sum\limits_{n=0}^{d/2} \left[-\frac{1}{\frac{d}2-\omega}-\psi\Big(\frac{d}{2}-n+1\Big)\right] \frac{(-m^2)^{\frac{d}2-n}}{(\frac{d}2-n)!} \int dx\, g^{1/2}{\rm tr} \hat a_{n}(x,x), \tag{19}\\ &&\varGamma_{\log }=\frac{1}{2(4\pi )^{d/2}}\sum\limits_{n=0}^{d/2} \frac{(-m^{2})^{\frac{d}2-n}}{\left(\frac{d}2-n\right) !} \ln \frac{m^{2}}{\mu^{2}}\int dxg^{1/2}{\rm tr} \hat a_{n}(x,x), \tag{20} \end{eqnarray} \] where $\mu^2$ is the mass parameter reflecting the renormalization ambiguity, and $\psi(x)$ is the logarithmic derivative of the Euler gamma function. Obviously, this asymptotic expansion makes sense only when $\hat a_n(x,x)/m^{2n}\ll 1$ which in virtue of (17) implies smallness of curvatures and their derivatives compared to the mass scale, \[ \tag{21} \begin{eqnarray} \frac{\cal R}{m^2}\ll 1,\quad \frac{\nabla\nabla}{m^2}\ll 1. \end{eqnarray} \]

Simplest example of this expansion is the calculation of the Coleman-Weinberg effective potential (Coleman S, Weinberg E, 1973) for the 4-dimensional scalar field with the $\lambda\varphi^4/12$ self-interaction. In the constant scalar field background playing the role of the mass parameter $m^2=\lambda\varphi^2$ the logarithmic part of (20) is contributed solely by the only nonzero HAMIDEW coefficient $a_0=1$ and represents the Coleman-Weinberg potential integrated over spacetime \[ \begin{eqnarray} \varGamma_\mathrm{CW} = \int d^4x \frac{\lambda^2\varphi^4}{64\pi^2} \ln \frac{\lambda\varphi^2}{\mu ^2}. \end{eqnarray} \]

Important application of local expansion is the calculation of stress tensor trace anomalies in conformally invariant theories, induced by renormalization of UV divergences which cause the breakdown of local Weyl invariance. For even dimensional spacetimes the UV divergences and trace anomaly are both defined by ${\rm tr}\hat a_{d/2}$ and in four dimensions read \[ \begin{eqnarray} &&\varGamma_{\rm div}=-\frac1{32\pi^2}\frac1{2-\omega} \int d^4x g^{1/2}{\rm tr}\hat a_2(x), \tag{22}\\ &&\langle T^\mu_\mu\rangle\equiv\frac2{g^{1/2}}g_{\mu\nu} \frac{\delta\varGamma}{\delta g_{\mu\nu}}=-\frac1{(4\pi)^2}{\rm tr}\hat a_2(x), \end{eqnarray} \] In particular, for conformally invariant fields of lowest spins ${\rm tr}\hat a_2(x)$ equals \[ \begin{eqnarray} \frac1{(4\pi)^2}{\rm tr}\hat a_2(x)= \mathbf{c} C_{\mu\nu\alpha\beta}^2-\mathbf{a} E -\mathbf{b} \Box R, \end{eqnarray} \] where $E=R_{\mu\nu\alpha\gamma}^2-4R_{\mu\nu}^2+R^2$ is the density of the Gauss-Bonnet invariant and the coefficients are contributed by $\mathbb{N}_0$ real scalars, $\mathbb{N}_{1/2}$ Dirac spinors and $\mathbb{N}_{1}$ vector multiplets (including relevant contributions of Faddeev-Popov ghosts subtracting temporal and longitudinal polarizations) \[ \begin{eqnarray} &&\mathbf{a}=\frac1{360(4\pi)^2} \big(\mathbb{N}_0+11 \mathbb{N}_{1/2}+ 62 \mathbb{N}_{1}\big), \tag{23}\\ &&\mathbf{c}=\frac1{120 (4\pi)^2} \big(\mathbb{N}_0+6\mathbb{N}_{1/2}+ 12 \mathbb{N}_{1}\big), \\ &&\mathbf{b}=-\frac1{180 (4\pi)^2} \big(\mathbb{N}_0+6\mathbb{N}_{1/2}+ 12 \mathbb{N}_{1}\big). \end{eqnarray} \] In the dimensional regularization the coefficient of $\Box R$, $\mathbf{b}$, is related to $\mathbf{c}$ by the equation $\mathbf{b}=-\frac23\mathbf{c}$, but in the zeta-function regularization this relation does not hold for a vector multiplet, $s=1$, and should be replaced by $\mathbf{b}=-\mathbf{c}$.

Generalizations: non-minimal operators and universal functional traces

For the operators of higher order derivatives and operators whose second derivatives do not form a d'Alembertian $\square$ -- the so called non-minimal ones -- the heat kernel technique is not directly applicable. However, calculations can be reduced to the application of (14) for the class of operators subject to the causality condition (Barvinsky A O, Vilkovisky G A, 1985). These operators have a special property of their principal symbol -- the higher derivative term with the derivatives replaced by a numerical vector $p_\mu$, modified by the mass term. The matrix determinant of this symbol is a polynomial function of $p^2\equiv g^{\mu\nu}p_\mu p_\nu$, \[ \begin{eqnarray} &&F(\nabla)=\hat G^{\mu\nu}\nabla_\mu\nabla_\nu+\hat P-\frac{\hat 1}6R-m^2\hat 1, \quad \hat G^{\mu\nu}\neq g^{\mu\nu}\hat 1 \tag{24} \\ &&{\rm det} \big(\hat G^{\mu\nu}p_\mu p_\nu-m^2\hat 1\big) ={\rm const}\times\prod\limits_{n=1}^N\big(p^2-M_n^2(m^2)\big). \end{eqnarray} \] The causality interpretation here follows from the fact that the characteristic surface for these operators, defined by the equation ${\rm det}(\hat G^{\mu\nu}p_\mu p_\nu)=0$, coincides with the light cone in the momentum space of $p_\mu$. As a consequence, the inverse of $\hat G^{\mu\nu}p_\mu p_\nu-m^2\hat 1$ has the form \[ \big(\hat G^{\mu\nu}p_\mu p_\nu-m^2\hat 1\big)^{-1}= \frac{\hat D(p)}{\prod\limits_{n=1}^N\big(p^2-M_n^2(m^2)\big)}, \] where $\hat D(p)$ is a polynomial in $p_\mu$. Therefore, the leading order for the Green's function of $F(\nabla)$ can be obtained from this expression by a formal replacement $p_\mu\to\nabla_\mu$ \[ \hat F^{-1}(\nabla)\delta(x,y)=\hat D(\nabla)\frac{\hat 1}{\prod\limits_{n=1}^N \big(\Box-M_n^2(m^2)\big)} \delta(x,y)+O(\Re), \] where the corrections $O(\Re)$ arising in view of the noncommutativity of $\nabla_\mu$ and the presence of the potential term in $F(\nabla)$ can be systematically expanded in powers of the curvatures.

The Mellin transform of this Green's function with respect to $m^2$ recovers the heat kernel of the nonminimal operator (24) and, in particular, local expansion of its functional trace. Since $\hat D(\nabla)$ is a differential operator all coefficients of this expansion have the form of the universal functional traces \[ \begin{eqnarray} &&\nabla_{\mu_1}\ldots\nabla_{\mu_k}\frac{\hat 1}{\prod\limits_{n=1}^N\big(\Box-M_n^2\big)} \delta(x,y)\Big|_{y=x}\nonumber\\ &&\qquad =\int\limits_0^\infty dss^{N-1} \int\limits_{\alpha_i\geq 0} d^N\alpha\delta\Big(1-\sum_{i=1}^N\alpha_i\Big) e^{-s\sum_{i=1}^N\alpha_iM_i^2}\nabla_{\mu_1}\ldots\nabla_{\mu_k}\hat K_\Box(s|x,y) \Big|_{y=x}, \tag{25} \end{eqnarray} \] with $\hat K_\Box(s|x,y)$ -- the heat kernel of the simplest minimal operator $\Box$. These traces can be directly expanded in local quantities of growing power in curvatures by using the heat kernel expansion (14), because the calculation of $\nabla_{\mu_1}\ldots\nabla_{\mu_k}\hat K_\Box(s|x,y)|_{y=x}$ reduces to taking the coincidence limits (15)-(16). This method allows one to recover also the heat kernel $\hat K(s|x,y)$ of the nonminimal operator (24) with separate arguments $x$ and $y$ and its nontrivial leading asymptotics for $s\to 0$ at $\sigma(x,y)\neq 0$ (Moss I G, Toms D J, 2014).

Calculation of universal functional traces is especially simplified when all $M_n^2$ in (25) coincide \[ \nabla_{\mu_1}\ldots\nabla_{\mu_k}\frac{\hat 1}{\big(\Box-M^2\big)^N} \delta(x,y)\Big|_{y=x}=\frac1{N!}\int\limits_0^\infty dss^{N-1} e^{-sM^2}\nabla_{\mu_1}\ldots\nabla_{\mu_k}\hat K_\Box(s|x,y)\Big|_{y=x}. \] Such structures are especially useful for finding the one-loop divergences for the massless minimal higher-derivative operators of the form $\hat F(\nabla)=\Box^N+\hat U(\nabla)$, because a formal expansion in powers of the lower derivative part of the operator $\hat U(\nabla)$, ${\rm Tr}\ln\hat F=N{\rm Tr}\ln\Box+{\rm Tr}\hat U[\hat 1/\Box^N]+O(\hat U^2)$, contains these expressions in a particularly simple case of $M^2=0$.

It is useful to generalize (25) by replacing the powers of $\Box$ in the denominators with the powers of the full minimal operator (12). In the massless case the logarithmic divergences of several such universal functional traces read, $\varepsilon\equiv 2-\omega\to 0$ (Jack I, Osborn H, 1984; Barvinsky A O, Vilkovisky G A, 1985) \[ \begin{eqnarray} &&\frac{\hat 1}{F(\nabla)}\delta(x,y)\Big|_{y=x}^{\rm div}=-\frac1{16\pi^2\varepsilon}g^{1/2}\hat P,\\ &&\nabla_\mu\frac{\hat 1}{F(\nabla)}\delta(x,y)\Big|_{y=x}^{\rm div} =-\frac1{16\pi^2\varepsilon}g^{1/2}\Big(\frac12\nabla_\mu\hat P-\frac16\nabla^\nu\hat{\cal R}_{\nu\mu}\Big),\\ &&\frac{\hat 1}{F^2(\nabla)}\delta(x,y)\Big|_{y=x}^{\rm div} =\frac1{16\pi^2\varepsilon}g^{1/2}\hat 1,\\ &&\nabla_\mu\frac{\hat 1}{F^2(\nabla)}\delta(x,y) \Big|_{y=x}^{\rm div}=0, \\ &&\nabla_\mu\nabla_\nu\frac{\hat 1}{F^2(\nabla)}\delta(x,y) \Big|_{y=x}^{\rm div}=\frac1{16\pi^2\varepsilon}g^{1/2} \Big(\frac16R_{\mu\nu}\hat 1-\frac12g_{\mu\nu}\hat P+\frac12\hat{\cal R}_{\mu\nu}\Big). \end{eqnarray} \]

From the viewpoint of the diagrammatic technique these universal functional traces represent tadpoles $\nabla_x\ldots\nabla_x G(x,y)|_{y=x}$ with the Green's function of the operator $F(\nabla)$ (or $F^2(\nabla)$, etc.), containing derivatives in the $x$-vertex

X-vertex.png

Moreover, their functional differentiation with respect to $P(y)$ generates other elements of Feynman diagrammatic technique -- polarization (or self energy) operators $\varPi(x,y)$ with the same set of derivatives in one of their vertices

Polarization-operators.png

Similarly, the functional differentiation with respect to the fibre bundle connection $\hat\varGamma_\mu$ implicit in $\nabla_\mu=\partial_\mu+\hat\varGamma_\mu$ generates the first order derivative in the second (labelled by $y$) vertex of $\varPi(x,y)$, the functional differentiation with respect to metric $g^{\alpha\beta}(y)$ gives two derivatives $\nabla_\alpha\nabla_\beta$ at $y$, etc. Application of these simple operations to the above table of universal functional traces gives UV divergences of various tadpoles and polarization operators which can also be used beyond one-loop order for multi-loop renormalization (Barvinsky A O, Vilkovisky G A, 1987).

Other expansions: nonlocal and nonanalytic effective action

Beyond the range of validity of the local expansion (21) the theory becomes nonanalytic either in the curvature $\Re$ or in spacetime derivatives $\nabla$ and requires alternative calculational methods. In the first case the effective action cannot be expanded in powers of small curvatures and field strengths, but its expansion can still be local. In the second case the effective action becomes essentially nonlocal but can be expanded in curvatures. Calculational methods for $\varGamma$ in these two cases rely on the approximation for the heat kernel with a generic non-small value of the proper time parameter and include the nonlocal covariant perturbation theory (Barvinsky A O, Vilkovisky G A, 1987; Barvinsky A O, Vilkovisky G A, 1990; Barvinsky A O, Gusev Yu V, Vilkovisky G A and Zhytnikov V V, 1993) and late time heat kernel asymptotics (Barvinsky A O, Mukhanov V F, 2002).

Covariant perturbation theory

Massless limit can be achieved within the covariant perturbation theory when the heat kernel and effective action are found as series in powers of the set of curvatures $(\hat P,\hat{\cal R}_{\mu\nu},R^\mu_{\nu\alpha\beta})$ with nonlocal covariant coefficients. From the viewpoint of the Schwinger-DeWitt expansion it corresponds to an infinite resummation of all terms (17) with a given power of the curvature and arbitrary number of derivatives, after which the result has a regular massless limit, \[ \begin{eqnarray} &&{\rm Tr}K(s)=\sum\limits_{n=0}^\infty{\rm Tr}K_n(s)= \sum\limits_{n=0}^\infty\int dx_{1}dx_{2}\ldots dx_{n} {\rm tr} K_{n}(s|x_{1},x_{2},\ldots x_{n}) \Re(x_{1})\Re(x_{2})\ldots \Re(x_{n}), \tag{26}\\ &&\varGamma=\sum\limits_{n=0}^{\infty }\int dx_{1}dx_{2}\ldots dx_{n}{\rm tr} \varGamma_{n}(x_{1},x_{2},\ldots x_{n}) \Re(x_{1})\Re(x_{2})\ldots \Re(x_{n}). \tag{27} \end{eqnarray} \] In each order the structure $\Re(x_{1})\Re(x_{2})\ldots \Re(x_{n})$ represents the set of monomials in powers of the curvatures $(\hat P,\hat{\cal R}_{\mu\nu},R_{\alpha\beta})$ acted upon by covariant derivatives of the finite order (generally bounded from above by a maximal number of possible contractions between the indices of derivatives and curvatures). For example, in the quadratic order these monomials reduce to five structures, $i=1,2,\ldots 5$, \begin{equation} \Re_1\Re_2({i})\equiv R_{\mu\nu}(x_1)R^{\mu\nu}(x_2)\hat 1, R(x_1)R(x_2) \hat 1,\hat P(x_1)R(x_2),\hat P(x_1)\hat P(x_2),\hat{\cal R}_{\mu\nu}(x_1)\hat{\cal R}^{\mu\nu}(x_2), \tag{28} \end{equation} whereas in the cubic order they form the set of twenty nine structures ranging from $\hat P(x_1)\hat P(x_2)\hat P(x_3)$ to $\nabla^\alpha\nabla^\beta R_{\mu\nu}(x_1)\nabla^\mu\nabla^\nu R_{\lambda\sigma}(x_2)\nabla^\lambda\nabla^\sigma R_{\alpha\beta}(x_3)\hat 1$.

The nonlocal coefficient of each such structure -- the form factor $K_{n}(s|x_{1},\ldots x_{n})$ and $\varGamma_{n}(x_{1},\ldots x_{n})=-\frac12\int_0^\infty ds K_{n}(s|x_{1},\ldots x_{n})/s$ -- can be represented as a function of $n$ covariant derivatives $\nabla_k$, $k=1,\ldots n$, acting on relevant arguments in the product of delta functions \[ K_{n}(s|x_{1},\ldots x_{n})=g^{1/2}(x_1)F_n(s|\nabla_1,\nabla_2,\ldots \nabla_n) \delta(x_1,x_2)\delta(x_1,x_3)\ldots \delta(x_1,x_n), \] and each term of (26) then reads as \[ \int dxg^{1/2}(x){\rm tr}F_n(s|\nabla_1,\ldots \nabla_n) \Re_1\Re_2\ldots \Re_n\big|_{x_1=x_2\ldots =x_n=x},\quad \Re_k\equiv\Re(x_k). \] Vanishing of total derivative terms in asymptotically flat and empty spacetime implies that these covariant derivatives are subject to $\nabla_1+\cdots+\nabla_n=0$ in the sense that \[ \int dx\,g^{1/2}(x){\rm tr}(\nabla_1+\cdots+\nabla_n)\Re_1\ldots\Re_n\,\big|_{x_1=x_2\ldots=x_n=x}\equiv \int dx\,g^{1/2}(x){\rm tr}\nabla\big(\Re_1(x)\ldots\Re_n(x)\big)=0 \] for any set of curvatures, which can be interpreted as the analogue of the conservation law for the total momentum (general covariance of this relation is guaranteed, of course, by the fact that the product $\Re_1\Re_2\ldots\Re_n$ here forms a vector whose index is contracted with the index of $\nabla$ -- see details of this formalism in (Barvinsky A O, Vilkovisky G A, 1987; Barvinsky A O, Vilkovisky G A, 1990; Barvinsky A O, Gusev Yu V, Vilkovisky G A and Zhytnikov V V, 1993). $F_n(s|\nabla_1,\ldots \nabla_n)$ in their turn can be expressed as functions of scalar invariants $g_{\mu\nu}\nabla_k^\mu\nabla_k^\nu\equiv\Box_k$ and $g_{\mu\nu}\nabla_k^\mu\nabla_m^\nu$ -- coordinate (and covariant) version of Mandelstam variables. It is important that the operator arguments of $F_n(s|\nabla_1,\ldots \nabla_n)$ are full covariant derivatves with respect to metric and fibre bundle connections, and the covariantly constant metric $g_{\mu\nu}=g_{\mu\nu}(x)$ is treated in formfactors as a c-number commuting with all $\nabla_k$.

These form factors were explicitly obtained in (Barvinsky A O, Vilkovisky G A, 1987; Barvinsky A O, Vilkovisky G A, 1990; Barvinsky A O, Gusev Yu V, Vilkovisky G A and Zhytnikov V V, 1993) up to $n=3$ inclusive. The first two orders are concise enough \[ {\rm Tr}K(s) = \frac1{(4\pi s)^\omega}\int dx g^{1/2} {\rm tr} \Big\{\hat{1}+s\hat{P}+s^2\sum^{5}_{i=1}F_2^{(i)}(-s\Box_2) \Re_1\Re_2({i})+{\rm O}[\Re^3]\Big\}. \tag{29} \] where the five curvature structures $\Re_1\Re_2(i),i=1,2,\ldots 5$, are listed in (28) and their formfactors $F_2^{(i)}(\xi)$, $\xi=-s\Box$, are expressed through the basic second-order form factor, \begin{equation} f(\xi)=\int^1_0\!d\alpha{\rm e}^{-\alpha(1-\alpha)\xi}, \end{equation} as follows \[ \begin{eqnarray} F_2^{(1)}(\xi) &=& \frac{f(\xi)-1+\frac16\xi}{\xi^2},\quad F_2^{(2)}(\xi) = \frac18\left[ \frac1{36}f(\xi)+\frac13\frac{f(\xi)-1}{\xi}-\frac{f(\xi)-1 +\frac16\xi}{\xi^2}\right], \\ F_2^{(3)}(\xi) &=& \frac1{12}f(\xi)+\frac12\frac{f(\xi)-1}{\xi},\quad F_2^{(4)}(\xi) = \frac12f(\xi),\quad F_2^{(5)}(\xi) = -\frac12\frac{f(\xi)-1}{\xi}. \end{eqnarray} \] They generate the quadratic part of the effective action for any dimension $2\omega$ (continued to the complex plane in dimensional regulariation) \[ \begin{eqnarray} &&\varGamma=-\frac{\Gamma(2-\omega)\Gamma(\omega+1)\Gamma(\omega-1)} {2(4\pi)^\omega\Gamma(2\omega+2)}\int dxg^{1/2}(x){\rm tr}\Big\{R_{\mu\nu}(-\Box)^{\omega-2}R^{\mu\nu} \hat 1 \\ &&\qquad\qquad\qquad\qquad -\frac{(4-\omega)(\omega+1)}{18} R(-\Box)^{\omega-2}R \hat 1 -\frac{2(2-\omega)(2\omega+1)}3\hat P(-\Box)^{\omega-2}R \\ &&\qquad\qquad\qquad\qquad +2(4\omega^2-1)\hat P(-\Box)^{\omega-2}\hat P +(2\omega+1) \hat{\cal R}_{\mu\nu}(-\Box)^{\omega-2}\hat{\cal R}^{\mu\nu}\Big\}+O[\Re^3] \end{eqnarray} \] For a conformally invariant scalar field, $\hat P=\hat1 R/6$, in two dimensions this expression generates the Polyakov action $\varGamma=\frac1{96\pi}{\rm tr} \hat 1\int d^2xR\frac1\Box R$. In four dimensions, $\omega\to 2$, this gives UV divergences (22) and the finite part with a nonlocal form factor $\sim \ln(-\Box/\mu^2)$.

Cubic order takes pages (Barvinsky A O, Gusev Yu V, Vilkovisky G A and Zhytnikov V V, 1993). In four dimensions the simplest contribution cubic in $\hat P$ reads \[ \begin{eqnarray} &&{\rm Tr}K_{3}(s)= \frac1{16\pi^2}\int d^4xg^{1/2}(x){\rm tr}\!\!\int\limits_{\alpha_i\geq 0}d^3\alpha \delta\big(1-\sum_{i=1}^3\alpha_i\big)\nonumber\\ &&\qquad\qquad\qquad\qquad\quad\times e^{s(\alpha_2\alpha_3\Box_1+\alpha_3\alpha_1\Box_2 +\alpha_1\alpha_2\Box_3)} \hat P_1\hat P_2\hat P_3 \Big|_{x_1=x_2=x_3=x}, \tag{30} \end{eqnarray} \] \begin{equation} \varGamma_{3}= \frac1{32\pi^2}\int d^4xg^{1/2}(x){\rm tr}\int\limits_{\alpha_i\geq 0}\frac{d^3\alpha \delta\big(1-\sum_{i=1}^3\alpha_i\big)} {\alpha_2\alpha_3\Box_1+\alpha_3\alpha_1\Box_2 +\alpha_1\alpha_2\Box_3} \hat P_1\hat P_2\hat P_3 \Big|_{x_1=x_2=x_3=x}. \end{equation}

In contrast to local Schwinger-DeWitt expansion the nonlocal covariant perturbation theory requires nontrivial analytic continuation rules from the Euclidean space setup to physical problems in spacetime with the Lorentzian signature. Along with the standard Wick rotation method for the IN-OUT matrix elements of physical observables these rules include the Euclidean version of the Schwinger-Keldysh formalism for IN-IN expectation values, which was developed in context of asymptotically-flat setup in (Barvinsky A O, Vilkovisky G A, 1987) and extended to asymptotically de Sitter backgrounds in (Higuchi A, Marolf D, Morrison I A, 2011; Korai Y, Tanaka T, (2013).

There exist numerous applications of this covariant perturbation theory to the particle creation phenomena (Mirzabekian A G, Vilkovisky G A, 1995; Mirzabekian A G, Vilkovisky G A, 1998), to vacuum backreaction of rapidly moving sources in QED (Vilkovisky G A, 1999a; Vilkovisky G A, 1999b) and in the theory of evolving quantum black holes (Vilkovisky G A, 2006a; Vilkovisky G A, 2006b).

Late time asymptotic expansion

Late time asymptotics of $K(s|x,y)$ is important for infrared properties of massless theories with covariantly constant background fields, $\nabla=0$. Very little is known about it in the background field method. The universal statement is that all orders of the expansion (26) except $n=0$ behave like $O(s^{1-d/2})$, $n\geq 1$, at $s\rightarrow \infty$ and \[ {\rm Tr}K(s)=\frac1{(4\pi s)^{d/2}}\left\{sW_0+W_1+O\left(\frac1s\right)\right\}. \tag{31} \] Therefore, in spacetime dimension $d\geq 3$ the effective action integral (8) is infrared convergent. In one and two dimensions this expansion for $\varGamma$ does not exist except for the special case of the massless theory in curved two-dimensional spacetime, when it reproduces the Polyakov effective action.

For the operator $F(\nabla)=\Box+\hat P$ in flat spacetime with vanishing fibre bundle curvature $\hat{\cal R}_{\mu\nu}=0$ resummation of the leading in $s\to\infty$ terms can be done explicitly and gives (Barvinsky A O, Mukhanov V F, 2002) \[ \begin{eqnarray} &&W_0=\int dxg^{1/2}{\rm tr}\hat P\hat\Phi,\quad W_1=\int dxg^{1/2}{\rm tr} \left(\hat 1 -2\nabla_\mu\hat\Phi\frac1{\Box+\hat P} \nabla^\mu\hat\Phi\right), \tag{32}\\ &&\hat\Phi=\hat 1- \frac{1}{\Box+\hat P}\hat P, \tag{33} \end{eqnarray} \] where $\hat\varPhi$ is the matrix-valued zero mode of $F(\nabla)$ with the unit boundary condition at infinity. In Cartesian coordinates (on spacetime and in fibre bundle with zero connection for $\nabla_\mu$) the heat kernel asymptotics is also available for separate points $x$ and $y$ \[ \begin{eqnarray} &&K(s|x,y)=\frac1{(4\pi s)^{d/2}} e^{-|x-y|^2/4s}\left[\hat\Phi (x)\hat\Phi (y) +\frac1s\hat\Omega_1(x,y)+ O\left(\frac1{s^2}\right) \right], \\ &&\hat\Omega_1(x,y)=\frac1{\Box_x+\hat P_x}(x-y)^\mu \nabla_\mu\hat\Phi(x)\hat\Phi(y)+\frac1{\Box_x+\hat P_x}\nabla_\mu\hat\Phi(x)\frac1{\Box_y+\hat P_y} \nabla^\mu\hat\Phi(y)+(x\leftrightarrow y). \end{eqnarray} \] The expression for $W_0$ can be directly generalized to curved spacetime by the price of additional Gibbons-Hawking surface integral at asymptotically flat infinity (Barvinsky A O, Gusev Yu V, Mukhanov V F, Nesterov D V, 2003). Expressions for $K(s|x,y)$ and $\hat\Omega_1(x,y)$ can also be cast into a covariant form in terms of the world function and its derivative, but thus far do not stand generalization to nonvanishing curvature because of infrared divergent spacetime integrals (Barvinsky A O, Gusev Yu V, Mukhanov V F, Nesterov D V, 2003).

This late time asymptotics gives a nonlocal and nonperturbative effective action which describes the transition between the compact domain of nearly constant field to its zero value at spacetime infinity. This is known for a class of scalar (one-component) operators $F(\nabla)=\Box-V$ with the potentials $V(x)\geq 0$ having a compact support in the spacetime domain $\cal D$ of size $L$, $V(x)=0,\quad |x|\geq L$. If this potential and its derivatives are bounded as $V(x)\leq V_0,\quad |\nabla V(x)|\leq V_0/L$, then the UV finite part of this effective action can be approximated for two opposite limits of the dimensionless combination $V_0 L^2$ (Barvinsky A O, Gusev Yu V, Mukhanov V F, Nesterov D V, 2003), \[ \varGamma\simeq \frac{1}{64\pi ^{2}} \int\limits_{\cal D} d^4xV^{2}\ln \left[\frac{\int_{\cal D} d^4xV^{2}}{\int_{\cal D} d^4x V\frac{\mu ^{2}}{V-\Box}V}\right], \quad V_0 L^2\ll 1, \tag{34} \] \[ \varGamma\simeq\varGamma_\mathrm{CW}+\frac{1}{64\pi^2}\frac{\left[\int\limits_{\cal D} d^4x \left(V-V\frac1{V-\Box}V\right)\right]^2}{\int\limits_{\cal D}d^4x}, \quad V_0 L^2\gg 1, \tag{35} \] where $\varGamma_\mathrm{CW}=\frac1{64\pi^2}\int d^4xV^2(x)\ln(V(x)/\mu^2)$ is the Coleman-Weinberg action. Both expressions reproduce the Coleman-Weinberg action in the limit of constant $V$, $\Box V=0$, but for spacetime gradients dominating over the magnitude of the potential get large nonlocal corrections. These expressions have not yet been utilized in physical applications, but might be useful within the cosmological constant problem (Barvinsky A O, Gusev Yu V, Mukhanov V F, Nesterov D V, 2003).

Heat kernel in spacetimes with boundaries

Important application field of the heat kernel is quantum theory of systems with spacetime boundaries. They include the Casimir effect, open and closed strings, fundamental branes and brane world physics in gravity and cosmology, etc. In the presence of boundaries the basic quantity of interest -- heat kernel trace and its short time expansion for the second order operator (12) is modified by the boundary terms of integer and half-integer power in $s$ (McKean H P, Singer I M, 1967), \[ \begin{eqnarray} &&{\rm Tr}K(s)=\frac1{(4\pi s)^{d/2}}\sum\limits_{k=0}^\infty s^{k/2}B_k,\\ &&B_{2n}=\int_{\cal M} d^dxg^{1/2}{\rm tr}\hat a_n(x,x) +\int_{\partial\cal M} d^{d-1}\sigma h^{1/2} b_{2n}(\sigma),\\ &&B_{2n+1}=\int_{\partial\cal M} d^{d-1}\sigma h^{1/2} b_{2n+1}(\sigma). \end{eqnarray} \] Here the volume (bulk) part of ${\rm Tr}K(s)$ is determined by the HAMIDEW coefficients considered above, and the surface terms are built in terms of geometric and field quantities induced on the boundary $\partial\cal M$ of the spacetime domain $\cal M$. With $\sigma=\sigma^a$, $a=1,2,\ldots,d-1$, denoting internal coordinates on $\partial\cal M$, the geometric quantities include induced metric on the boundary $h_{ab}$, $h\equiv\det h_{ab}$, extrinsic curvature $K_{ab}$, $K=h^{ab}K_{ab}$, and all possible $(d-1)$-dimensional and $d$-dimensional curvature invariants taken at the location of the boundary.

These surface terms essentially depend on the boundary conditions for $\hat K(s|x,y)$ (and correspondingly $\phi(x)$) at $\partial\cal M$. For the Dirichlet and generalized Neumann (Robin) boundary conditions \[ \begin{eqnarray} &&D:\quad \phi\big|_{\partial\cal M}=0,\\ &&N:\quad (\nabla_n-\hat S)\phi\big|_{\partial\cal M}=0, \end{eqnarray} \] ($\nabla_n$ denotes the derivative normal to the boundary) several lowest order boundary term integrands read \[ \begin{eqnarray} &&b_0^{D,N}(\sigma)=0, \\ &&b_1^D(\sigma)= -\frac{\sqrt\pi}2{\rm tr}\hat1, b_1^N(x)=\frac{\sqrt\pi}2{\rm tr}\hat1, \\ &&b_2^D(\sigma)=\frac13K{\rm tr}\hat1,\quad b_2^N(\sigma)={\rm tr}\Big(2\hat S+\frac13K\hat1\Big). \end{eqnarray} \] Boundary conditions may include $(d-1)$-dimensional covariant derivatives $D_a$ tangential to the boundary -- the so-called oblique boundary conditions (McAvity D M, Osborn H, 1991) \[ \Big(\nabla_n-\hat\varGamma^a D_a-\frac12(D_a \hat\varGamma^a)- \hat S\Big)\phi\Big|_{\partial\cal M}=0. \tag{36} \] For generic matrix-valued vector coefficients $\hat\varGamma^a$ their contribution to $b_n$ is not known, but in the case of commuting matrices $[\hat\varGamma^a,\hat\varGamma^b]=0$ the lowest order boundary terms read (McAvity D M, Osborn H, 1991; Dowker J S, Kirsten K, 1997; Dowker J S, Kirsten K, 1999; Avramidi I G, Esposito G, 1998a; Avramidi I G, Esposito G, 1998b) \[ \begin{eqnarray} &&b_1^O(x)=\frac{\sqrt\pi}2{\rm tr}\left[\frac2{\sqrt{1 +\hat\varGamma^2}}- \hat1\right], \tag{37}\\ &&b_2^O(x)={\rm tr}\left[\frac2{1+\hat\varGamma^2}\hat S+\frac13K\hat1 +\Big(\frac1{1+\hat\varGamma^2}-\frac{{\rm arctanh} \sqrt{-\hat\varGamma^2}}{\sqrt{-\hat\varGamma^2}}\Big) \Big(K-K^{ab}\frac{\hat\varGamma_a \hat\varGamma_b}{\hat\varGamma^2}\Big) \right], \tag{38} \end{eqnarray} \] where $\hat\varGamma^2=\hat\varGamma^a\hat\varGamma_a$. More details on mathematical aspects of boundary conditions in the heat kernel theory can be found in (Kirsten K, 2002;Fulling S A, 2003;Gilkey P B, 2004)

Acknowledgements

I am deeply grateful to G.A.Vilkovisky in collaboration with whom were obtained original results of this review and want to thank S.A.Fulling for helpful criticism that allowed me to improve this work. Also I wish to thank the hospitality of Theory Division of CERN where the last part of this work was accomplished. This work was partly supported by the RFBR grant No. 14-02-01173 and by the Tomsk State University Competitiveness Improvement Program.

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