Functorial properties of Schwinger-DeWitt expansion and Mellin-Barnes representation

We consider integral kernels for functions <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>f</a:mi> <a:mo stretchy="false">(</a:mo> <a:mover accent="true"> <a:mi>F</a:mi> <a:mo stretchy="false">^</a:mo> </a:mover> <a:mo stretchy="false">)</a:mo> </a:math> of a minimal second-order differential operator <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mover accent="true"> <g:mi>F</g:mi> <g:mo stretchy="false">^</g:mo> </g:mover> <g:mo stretchy="false">(</g:mo> <g:mo>∇</g:mo> <g:mo stretchy="false">)</g:mo> </g:math> on a curved spacetime. We show that they can be expanded in a functional series, analogous to the DeWitt expansion for the heat kernel, by integrating the latter term-by-term. This procedure leads to a separation of two types of data: all information about the bundle geometry and the operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mover accent="true"> <m:mi>F</m:mi> <m:mo stretchy="false">^</m:mo> </m:mover> <m:mo stretchy="false">(</m:mo> <m:mo>∇</m:mo> <m:mo stretchy="false">)</m:mo> </m:math> is still contained in the standard heat kernel coefficients <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:msub> <s:mover accent="true"> <s:mi>a</s:mi> <s:mo stretchy="false">^</s:mo> </s:mover> <s:mi>k</s:mi> </s:msub> <s:mo stretchy="false">[</s:mo> <s:mi>F</s:mi> <s:mo stretchy="false">|</s:mo> <s:mi>x</s:mi> <s:mo>,</s:mo> <s:msup> <s:mi>x</s:mi> <s:mo>′</s:mo> </s:msup> <s:mo stretchy="false">]</s:mo> </s:math> (we call this property “off-diagonal functoriality”), while information about the function <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"> <z:mi>f</z:mi> </z:math> is encoded in some new scalar functions <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"> <bb:msub> <bb:mi mathvariant="double-struck">B</bb:mi> <bb:mi>α</bb:mi> </bb:msub> <bb:mo stretchy="false">[</bb:mo> <bb:mi>f</bb:mi> <bb:mo stretchy="false">|</bb:mo> <bb:mi>σ</bb:mi> <bb:mo stretchy="false">]</bb:mo> </bb:math> and <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"> <hb:msub> <hb:mi mathvariant="double-struck">W</hb:mi> <hb:mi>α</hb:mi> </hb:msub> <hb:mo stretchy="false">[</hb:mo> <hb:mi>f</hb:mi> <hb:mo stretchy="false">|</hb:mo> <hb:mi>σ</hb:mi> <hb:mo>,</hb:mo> <hb:msup> <hb:mi>m</hb:mi> <hb:mn>2</hb:mn> </hb:msup> <hb:mo stretchy="false">]</hb:mo> </hb:math> , which we call basis and complete massive kernels, respectively. These objects are calculated for operator functions of the form <nb:math xmlns:nb="http://www.w3.org/1998/Math/MathML" display="inline"> <nb:mi>exp</nb:mi> <nb:mo stretchy="false">(</nb:mo> <nb:mo>−</nb:mo> <nb:mi>τ</nb:mi> <nb:msup> <nb:mover accent="true"> <nb:mi>F</nb:mi> <nb:mo stretchy="false">^</nb:mo> </nb:mover> <nb:mi>ν</nb:mi> </nb:msup> <nb:mo stretchy="false">)</nb:mo> <nb:mo>/</nb:mo> <nb:mo stretchy="false">(</nb:mo> <nb:msup> <nb:mover accent="true"> <nb:mi>F</nb:mi> <nb:mo stretchy="false">^</nb:mo> </nb:mover> <nb:mi>μ</nb:mi> </nb:msup> <nb:mo>+</nb:mo> <nb:mi>λ</nb:mi> <nb:mo stretchy="false">)</nb:mo> </nb:math> as multiple Mellin-Barnes integrals. The article also discusses subtle issues such as the validity of the term-by-term integration, the regularization of IR divergent integrals, and the physical interpretation of the resulting expansions.