In Quantum Mechanis (QM), the dynamical variables are the (quantized) coordinates $x_j$ and their canonical conjugate $p_j = -i\partial_j$ with the commutation relation $[x_j,p_k]=i\delta_{jk}$ acting as operators on the quantum state space.

What exactly happens to that state space when we change the underlying geometry or topology of "physical" space (the spacetime manifold that serves as the background for the quantum system)? How does this change in geometry/topology reflect on the Hilbert space?

In Quantum Field Theory (QFT), the dynamical objects are the (quantized) fields $\phi (x^\mu)$ and the coordinates $x_i$ are demoted to mere labels. What happens in this case? How does a change in geometry/topology alter the resulting Fock space?

I am new to this area, so what I need would be a basic explanation (for QM and QFT) how to make the connection between the two concepts geometry/topology of physical space and resulting properties of the quantum state space - if such a wish makes sense at all.

In Quantum Mechanis (QM), the dynamical variables are the (quantized) coordinates $x_j$ and their canonical conjugate $p_j = -i\partial_j$ with the commutation relation $[x_j,p_k]=i\delta_{jk}$ acting as operators on the quantum state space.

What exactly happens to that state space when we change the underlying geometry or topology of "physical" space - the (spacetime) manifold that serves as the background for the quantum system? How does this change in geometry/topology reflect on the Hilbert space?

In Quantum Field Theory (QFT), the dynamical objects are the (quantized) fields $\phi (x^\mu)$ and the coordinates $x_i$ are demoted to mere labels. What happens in this case? How does a change in geometry/topology alter the resulting Fock space?

I am new to this area, so what I need would be a basic explanation (for QM and QFT) how to make the connection between the two concepts geometry/topology of physical space and resulting properties of the quantum state space - if such a wish makes sense at all.

In Quantum Mechanis (QM), the dynamical variables are the (quantized) coordinates $x_j$ and their canonical conjugate $p_j = -i\partial_j$ with the commutation relation $[x_j,p_k]=i\delta_{jk}$ acting as operators on the quantum state space.

What exactly happens to that state space when we change the underlying geometry or topology of "physical" space, that is the coordinates $x_j$ ? How does this change in geometry/topology reflect on the Hilbert space?

In Quantum Field Theory (QFT), the dynamical objects are the (quantized) fields $\phi (x^\mu)$ and the coordinates $x_i$ are demoted to mere labels. What happens in this case? How does a change in geometry/topology alter the resulting Fock space?

I am new to this area, so what I need would be a basic explanation (for QM and QFT) how to make the connection between the two concepts geometry/topology of physical space and resulting properties of the quantum state space - if such a wish makes sense at all.

In Quantum Mechanis (QM), the dynamical variables are the (quantized) coordinates $x_j$ and their canonical conjugate $p_j = -i\partial_j$ with the commutation relation $[x_j,p_k]=i\delta_{jk}$ acting as operators on the quantum state space.

What exactly happens to that state space when we change the underlying geometry or topology of "physical" space (the spacetime manifold that serves as the background for the quantum system)? How does this change in geometry/topology reflect on the Hilbert space?

In Quantum Field Theory (QFT), the dynamical objects are the (quantized) fields $\phi (x^\mu)$ and the coordinates $x_i$ are demoted to mere labels. What happens in this case? How does a change in geometry/topology alter the resulting Fock space?

I am new to this area, so what I need would be a basic explanation (for QM and QFT) how to make the connection between the two concepts geometry/topology of physical space and resulting properties of the quantum state space - if such a wish makes sense at all.