Commutator technique for the heat kernel of minimal higher derivative operators

We suggest a new technique of the asymptotic heat kernel expansion for minimal higher derivative operators of a generic <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mn>2</a:mn><a:mi>M</a:mi></a:math>th order, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>F</c:mi><c:mo stretchy="false">(</c:mo><c:mo>∇</c:mo><c:mo stretchy="false">)</c:mo><c:mo>=</c:mo><c:mo stretchy="false">(</c:mo><c:mo>−</c:mo><c:mo>□</c:mo><c:msup><c:mo stretchy="false">)</c:mo><c:mi>M</c:mi></c:msup><c:mo>+</c:mo><c:mo>⋯</c:mo></c:math>, in the background field formalism of gauge theories and quantum gravity. This technique represents the conversion of the recently suggested Fourier integral method of generalized exponential functions [A. O. Barvinsky and W. Wachowski, Heat kernel expansion for higher order minimal and nonminimal operators, ] into the commutator algebra of special differential operators, which allows one to express expansion coefficients for <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>F</i:mi><i:mo stretchy="false">(</i:mo><i:mo>∇</i:mo><i:mo stretchy="false">)</i:mo></i:math> in terms of the Schwinger-DeWitt coefficients of a minimal second-order operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>H</m:mi><m:mo stretchy="false">(</m:mo><m:mo>∇</m:mo><m:mo stretchy="false">)</m:mo></m:math>. This procedure is based on special functorial properties of the formalism including the Mellin-Barnes representation of the complex operator power <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msup><q:mi>H</q:mi><q:mi>M</q:mi></q:msup><q:mo stretchy="false">(</q:mo><q:mo>∇</q:mo><q:mo stretchy="false">)</q:mo></q:math> and naturally leads to the origin of generalized exponential functions without directly appealing to the Fourier integral method. The algorithm is essentially more straightforward than the Fourier method and consists of three steps ready for a computer codification by symbolic manipulation programs. They begin with the decomposition of the operator into a power of some minimal second-order operator <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>H</u:mi><u:mo stretchy="false">(</u:mo><u:mo>∇</u:mo><u:mo stretchy="false">)</u:mo></u:math> and its lower derivative perturbation part <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mi>W</y:mi><y:mo stretchy="false">(</y:mo><y:mo>∇</y:mo><y:mo stretchy="false">)</y:mo></y:math>, <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:mi>F</cb:mi><cb:mo stretchy="false">(</cb:mo><cb:mo>∇</cb:mo><cb:mo stretchy="false">)</cb:mo><cb:mo>=</cb:mo><cb:msup><cb:mi>H</cb:mi><cb:mi>M</cb:mi></cb:msup><cb:mo stretchy="false">(</cb:mo><cb:mo>∇</cb:mo><cb:mo stretchy="false">)</cb:mo><cb:mo>+</cb:mo><cb:mi>W</cb:mi><cb:mo stretchy="false">(</cb:mo><cb:mo>∇</cb:mo><cb:mo stretchy="false">)</cb:mo></cb:math>, followed by considering their multiple nested commutators. The second step is the construction of special local differential operators—the perturbation theory in powers of the lower derivative part <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"><kb:mi>W</kb:mi><kb:mo stretchy="false">(</kb:mo><kb:mo>∇</kb:mo><kb:mo stretchy="false">)</kb:mo></kb:math>. The final step is the so-called procedure of their “Syngification,” consisting of a special modification of the covariant derivative monomials in these operators by the Synge world function <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"><ob:mi>σ</ob:mi><ob:mo stretchy="false">(</ob:mo><ob:mi>x</ob:mi><ob:mo>,</ob:mo><ob:msup><ob:mi>x</ob:mi><ob:mo>′</ob:mo></ob:msup><ob:mo stretchy="false">)</ob:mo></ob:math> with their subsequent action on the Hadamard-Minakshisundaram-DeWitt coefficients of <sb:math xmlns:sb="http://www.w3.org/1998/Math/MathML" display="inline"><sb:mi>H</sb:mi><sb:mo stretchy="false">(</sb:mo><sb:mo>∇</sb:mo><sb:mo stretchy="false">)</sb:mo></sb:math>. Published by the American Physical Society 2024