Return to Question

Taking the usual definition of the propagator for a massless scalar field and taking the reciprocal:

$$f(x,y)\propto\left(\int \phi(x)\phi(y) e^{i\int \sqrt{-g}g^{\mu\nu}(z)\partial_\mu \phi(z)\partial_\nu\phi(z) d^4z}D\phi\right)^{-1}$$

When $g$ is the Minkowski metric we have that $f(x,y)\propto|x-y|^2$.

This suggests that the function $f(x,y)$ gives the proper time between two points traced by the shortest path. So if we let $g(x)$ become the metric of a general curved space-time, this should still hold true. We might write $f(x,y) = MinTime(x,y)^2$

However.... in such cases as when $g$ is the field of a gravitional mass, there may be more than one locally minimum path between two space-time points, in which case this couldn't be true. Hence either the integral cannot be valid for this metric or the result is something diffent such as the average of all the proper times-squared for locally minimum paths.

Also for the Minkowski case we can say that when $f(x,y)>0$ the points are space-like separated and $f(x,y)<0$ the points are time-like separated. Does $f$ give us similar information when $g$ is a general curved space (perhaps with singularities at gravitional sources?)

Edit: To clarify, I meant $x$ and $y$ to be points in 3+1 dimensional space-time. Which I think was understood.

Taking the usual definition of the propagator for a massless scalar field and taking the reciprocal:

$$f(x,y)\propto\left(\int \phi(x)\phi(y) e^{i\int \sqrt{-g}g^{\mu\nu}(z)\partial_\mu \phi(z)\partial_\nu\phi(z) d^4z}D\phi\right)^{-1}$$

When $g$ is the Minkowski metric we have that $f(x,y)\propto|x-y|^2$.

This suggests that the function $f(x,y)$ gives the proper time between two points traced by the shortest path. So if we let $g(x)$ become the metric of a general curved space-time, this should still hold true. We might write $f(x,y) = MaxTime(x,y)^2$

However.... in such cases as when $g$ is the field of a gravitional mass, there may be more than one locally minimum path between two space-time points, in which case this couldn't be true. Hence either the integral cannot be valid for this metric or the result is something diffent such as the average of all the proper times-squared for locally minimum paths.

Also for the Minkowski case we can say that when $f(x,y)>0$ the points are space-like separated and $f(x,y)<0$ the points are time-like separated. Does $f$ give us similar information when $g$ is a general curved space (perhaps with singularities at gravitional sources?)

Edit: To clarify, I meant $x$ and $y$ to be points in 3+1 dimensional space-time. Which I think was understood.

Taking the usual definition of the propagator for a massless scalar field and taking the reciprocal:

$$f(x,y)\propto\left(\int \phi(x)\phi(y) e^{i\int \sqrt{-g}g^{\mu\nu}(z)\partial_\mu \phi(z)\partial_\nu\phi(z) d^4z}D\phi\right)^{-1}$$

When $g$ is the Minkowski metric we have that $f(x,y)\propto|x-y|^2$.

This suggests that the function $f(x,y)$ gives the proper time between two points traced by the shortest path. So if we let $g(x)$ become the metric of a general curved space-time, this should still hold true. We might write $f(x,y) = MinTime(x,y)^2$

However.... in such cases as when $g$ is the field of a gravitional mass, there may be more than one locally minimum path between two space-time points, in which case this couldn't be true. Hence either the integral cannot be valid for this metric or the result is something diffent such as the average of all the proper times-squared for locally minimum paths.

Also for the Minkowski case we can say that when $f(x,y)>0$ the points are space-like separated and $f(x,y)<0$ the points are time-like separated. Does $f$ give us similar information when $g$ is a general curved space (perhaps with singularities at gravitional sources?)

Taking the usual definition of the propagator for a massless scalar field and taking the reciprocal:

$$f(x,y)\propto\left(\int \phi(x)\phi(y) e^{i\int \sqrt{-g}g^{\mu\nu}(z)\partial_\mu \phi(z)\partial_\nu\phi(z) d^4z}D\phi\right)^{-1}$$

When $g$ is the Minkowski metric we have that $f(x,y)\propto|x-y|^2$.

This suggests that the function $f(x,y)$ gives the proper time between two points traced by the shortest path. So if we let $g(x)$ become the metric of a general curved space-time, this should still hold true. We might write $f(x,y) = MinTime(x,y)^2$

However.... in such cases as when $g$ is the field of a gravitional mass, there may be more than one locally minimum path between two space-time points, in which case this couldn't be true. Hence either the integral cannot be valid for this metric or the result is something diffent such as the average of all the proper times-squared for locally minimum paths.

Also for the Minkowski case we can say that when $f(x,y)>0$ the points are space-like separated and $f(x,y)<0$ the points are time-like separated. Does $f$ give us similar information when $g$ is a general curved space (perhaps with singularities at gravitional sources?)

Edit: To clarify, I meant $x$ and $y$ to be points in 3+1 dimensional space-time. Which I think was understood.

Taking the usual definition of the propagator for a massless scalar field and taking the reciprocal:

$$f(x,y)\propto\left(\int \phi(x)\phi(y) e^{i\int \sqrt{-g}g^{\mu\nu}(z)\partial_\mu \phi(z)\partial_\nu\phi(z) dz^4}D\phi\right)^{-1}$$

When $g$ is the Minkowski metric we have that $f(x,y)\propto|x-y|^2$.

This suggests that the function $f(x,y)$ gives the proper time between two points traced by the shortest path. So if we let $g(x)$ become the metric of a general curved space-time, this should still hold true. We might write $f(x,y) = MinTime(x,y)^2$

However.... in such cases as when $g$ is the field of a gravitional mass, there may be more than one locally minimum path between two space-time points, in which case this couldn't be true. Hence either the integral cannot be valid for this metric or the result is something diffent such as the average of all the proper times-squared for locally minimum paths.

Also for the Minkowski case we can say that when $f(x,y)>0$ the points are space-like separated and $f(x,y)<0$ the points are time-like separated. Does $f$ give us similar information when $g$ is a general curved space (perhaps with singularities at gravitional sources?)

Taking the usual definition of the propagator for a massless scalar field and taking the reciprocal:

$$f(x,y)\propto\left(\int \phi(x)\phi(y) e^{i\int \sqrt{-g}g^{\mu\nu}(z)\partial_\mu \phi(z)\partial_\nu\phi(z) d^4z}D\phi\right)^{-1}$$

When $g$ is the Minkowski metric we have that $f(x,y)\propto|x-y|^2$.

This suggests that the function $f(x,y)$ gives the proper time between two points traced by the shortest path. So if we let $g(x)$ become the metric of a general curved space-time, this should still hold true. We might write $f(x,y) = MinTime(x,y)^2$

However.... in such cases as when $g$ is the field of a gravitional mass, there may be more than one locally minimum path between two space-time points, in which case this couldn't be true. Hence either the integral cannot be valid for this metric or the result is something diffent such as the average of all the proper times-squared for locally minimum paths.

Also for the Minkowski case we can say that when $f(x,y)>0$ the points are space-like separated and $f(x,y)<0$ the points are time-like separated. Does $f$ give us similar information when $g$ is a general curved space (perhaps with singularities at gravitional sources?)