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Imagine the following path integral

$$\mathcal{Z}= \int_{x(0)=x_i}^{x(T)=x_f} \mathcal{D}x \, e^{\frac{i}{\hbar}S[x]}.$$$$\int_{x(0)=x_i}^{x(T)=x_f} \mathcal{D}x \, e^{\frac{i}{\hbar}S[x]}.$$

This integral is defined over the space of all paths that satisfy the boundary conditions $$x(0)=x_i$$ and $$x(T)=x_f$$. I am interested in defining a metric on this space in order to be able to quantify how similar two different paths are. I appreciate any suggestions on how to proceed.

Imagine the following path integral

$$\mathcal{Z}= \int_{x(0)=x_i}^{x(T)=x_f} \mathcal{D}x \, e^{\frac{i}{\hbar}S[x]}.$$

This integral is defined over the space of all paths that satisfy the boundary conditions $$x(0)=x_i$$ and $$x(T)=x_f$$. I am interested in defining a metric on this space in order to be able to quantify how similar two different paths are. I appreciate any suggestions on how to proceed.

Imagine the following path integral

$$\int_{x(0)=x_i}^{x(T)=x_f} \mathcal{D}x \, e^{\frac{i}{\hbar}S[x]}.$$

This integral is defined over the space of all paths that satisfy the boundary conditions $$x(0)=x_i$$ and $$x(T)=x_f$$. I am interested in defining a metric on this space in order to be able to quantify how similar two different paths are. I appreciate any suggestions on how to proceed.

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# Defining a metric on the space of paths

Imagine the following path integral

$$\mathcal{Z}= \int_{x(0)=x_i}^{x(T)=x_f} \mathcal{D}x \, e^{\frac{i}{\hbar}S[x]}.$$

This integral is defined over the space of all paths that satisfy the boundary conditions $$x(0)=x_i$$ and $$x(T)=x_f$$. I am interested in defining a metric on this space in order to be able to quantify how similar two different paths are. I appreciate any suggestions on how to proceed.