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Let $M$ be a Riemannian manifold and $\sigma$ the world function. The Van-Vleck-Morette determinant $D$ is defined by $$D(x,x')=\det(-\sigma_{;\mu\nu{}'})$$ Regarding the semi-colon: In chapter $4.1$ of [K] it is claimed that it "denotes differentiation with respect to the Levi-Civita connection", but in other references it is simply interpreted as partial differentiation$^1$: $$\sigma_{;\mu\nu{}'}=\frac{\partial^2\sigma}{\partial x^\mu\partial(x')^\nu}$$ In any case, it is not clear to me why $D$ is supposedly coordinate-independent (even if we restrict ourselves to normal coordinates), as it is claimed here.

$^1$

From chapter $4.1$ of [K]:

From page $38$ of [B]:

From [C]:

References

[K] Klaus Kirsten, Spectral Functions in Mathematics and Physics

[B] Barvinsky and Vilkovisky, The generalized Schwinger-Dewitt technique in gauge theories and quantum gravity

[C] Steven Christensen, Vacuum expectation value of the stress tensor in an arbitrary curved background: The covariant point-separation method

Let $M$ be a Riemannian manifold and $\sigma$ the world function. The Van-Vleck-Morette determinant $D$ is defined by $$D(x,x')=\det(-\sigma_{;\mu\nu{}'})$$ Regarding the semi-colon: In chapter $4.1$ of [K] it is claimed that it "denotes differentiation with respect to the Levi-Civita connection", but in other references it is simply interpreted as partial differentiation$^1$: $$\sigma_{;\mu\nu{}'}=\frac{\partial^2\sigma}{\partial x^\mu\partial(x')^\nu}$$ In any case, it is not clear to me why $D$ is supposedly coordinate-independent (even if we restrict ourselves to normal coordinates), as it is claimed here.

$^1$

From chapter $4.1$ of [K]:

From page $38$ of [B]:

From [C]:

References

[K] Klaus Kirsten, Spectral Functions in Mathematics and Physics

[B] Barvinsky and Vilkovisky, The generalized Schwinger-Dewitt technique in gauge theories and quantum gravity

[C] Steven Christensen, Vacuum expectation value of the stress tensor in an arbitrary curved background: The covariant point-separation method

Let $M$ be a Riemannian manifold and $\sigma$ the world function. The Van-Vleck-Morette determinant $D$ is defined by $$D(x,x')=\det(-\sigma_{;\mu\nu{}'})$$ Regarding the semi-colon: In chapter $4.1$ of [K] it is claimed that it "denotes differentiation with respect to the Levi-Civita connection", but in other references it is simply interpreted as partial differentiation$^1$: $$\sigma_{;\mu\nu{}'}=\frac{\partial^2\sigma}{\partial x^\mu\partial(x')^\nu}$$ In any case, it is not clear to me why $D$ is supposedly coordinate-independent (even if we restrict ourselves to normal coordinates), as it is claimed here.

$^1$

From chapter $4.1$ of [K]:

From page $38$ of [B]:

From [C]:

References

[K] Klaus Kirsten, Spectral Functions in Mathematics and Physics

[S] Barvinsky and Vilkovisky, The generalized Schwinger-Dewitt technique in gauge theories and quantum gravity

[C] Steven Christensen, Vacuum expectation value of the stress tensor in an arbitrary curved background: The covariant point-separation method

Let $M$ be a Riemannian manifold and $\sigma$ the world function. The Van-Vleck-Morette determinant $D$ is defined by $$D(x,x')=\det(-\sigma_{;\mu\nu{}'})$$ Regarding the semi-colon: In chapter $4.1$ of [K] it is claimed that it "denotes differentiation with respect to the Levi-Civita connection", but in other references it is simply interpreted as partial differentiation$^1$: $$\sigma_{;\mu\nu{}'}=\frac{\partial^2\sigma}{\partial x^\mu\partial(x')^\nu}$$ In any case, it is not clear to me why $D$ is supposedly coordinate-independent (even if we restrict ourselves to normal coordinates), as it is claimed here.

$^1$

From chapter $4.1$ of [K]:

From page $38$ of [B]:

From [C]:

References

[K] Klaus Kirsten, Spectral Functions in Mathematics and Physics

[B] Barvinsky and Vilkovisky, The generalized Schwinger-Dewitt technique in gauge theories and quantum gravity

[C] Steven Christensen, Vacuum expectation value of the stress tensor in an arbitrary curved background: The covariant point-separation method

Let $M$ be a Riemannian manifold and $\sigma$ the world function. The Van-Vleck-Morette determinant $D$ is defined by $$D(x,x')=\det(-\sigma_{;\mu\nu{}'})$$ Regarding the semi-colon: In chapter $4.1$ of [K] it is claimed that it "denotes differentiation with respect to the Levi-Civita connection" (I don't know what this means), but in other references it is simply interpreted as partial differentiation: $$\sigma_{;\mu\nu{}'}=\frac{\partial^2\sigma}{\partial x^\mu\partial(x')^\nu}$$ In any case, it is not clear to me why $D$ is supposedly coordinate-independent (even if we restrict ourselves to normal coordinates), as it is claimed here.

Edit: In view of the comments, I need to provide more evidence to support my claims:

From chapter $4.1$ of [K]:

From page $38$ of [B]:

From [C]:

References

[K] Klaus Kirsten, Spectral Functions in Mathematics and Physics

[S] Barvinsky and Vilkovisky, The generalized Schwinger-Dewitt technique in gauge theories and quantum gravity

[C] Steven Christensen, Vacuum expectation value of the stress tensor in an arbitrary curved background: The covariant point-separation method

Let $M$ be a Riemannian manifold and $\sigma$ the world function. The Van-Vleck-Morette determinant $D$ is defined by $$D(x,x')=\det(-\sigma_{;\mu\nu{}'})$$ Regarding the semi-colon: In chapter $4.1$ of [K] it is claimed that it "denotes differentiation with respect to the Levi-Civita connection", but in other references it is simply interpreted as partial differentiation$^1$: $$\sigma_{;\mu\nu{}'}=\frac{\partial^2\sigma}{\partial x^\mu\partial(x')^\nu}$$ In any case, it is not clear to me why $D$ is supposedly coordinate-independent (even if we restrict ourselves to normal coordinates), as it is claimed here.

$^1$

From chapter $4.1$ of [K]:

From page $38$ of [B]:

From [C]:

References

[K] Klaus Kirsten, Spectral Functions in Mathematics and Physics

[S] Barvinsky and Vilkovisky, The generalized Schwinger-Dewitt technique in gauge theories and quantum gravity

[C] Steven Christensen, Vacuum expectation value of the stress tensor in an arbitrary curved background: The covariant point-separation method