Return to Answer

For a closed 2-form $\omega$ on a manifold $M$, the integrality of the closed 2-form, that is, $$ \int_\sigma \omega \in a{\bf Z}, \quad \mbox{for all} \quad \sigma \in H_2(M,{\bf Z}), $$ for some real number $a$, ensures the existence of a principal circle-bundle $Y$ (and its associated line bundle $L$) over $M$ and a connexion $\lambda$ with curvature $\omega$. Then, it is possible to lift some groups of automorphisms of $\omega$ (subgroups of ${\rm Diff}(M,\omega)$) as groups of automorphisms of $(Y,\lambda)$. This procedure is called prequantization because it is the first step of an answer to the Dirac program of quantization consisting in representing symmetries in classical mechanics by unitary transformations in some Hilbert space (that is supposed to have a physical meaning). I would not want to develop why one needs this bundle, at the first place, to answer Dirac's program, and are not contented just with the automorphisms of $\omega$, because it will lead us too far. If you are happy with this answer I'm fine, else I'll try to say a more few words(*).

P.S. The fact that the number $a$ is required to be a multiple of $\hbar$ comes just from physics consideration.

(*) Edited: I added a few words here

For a closed 2-form $\omega$ on a manifold $M$, the integrality of the closed 2-form, that is, $$ \int_\sigma \omega \in a{\bf Z}, \quad \mbox{for all} \quad \sigma \in H_2(M,{\bf Z}), $$ for some real number $a$, ensures the existence of a principal circle-bundle $Y$ (and its associated line bundle $L$) over $M$ and a connexion $\lambda$ with curvature $\omega$. Then, it is possible to lift some groups of automorphisms of $\omega$ (subgroups of ${\rm Diff}(M,\omega)$) as groups of automorphisms of $(Y,\lambda)$. This procedure is called prequantization because it is the first step of an answer to the Dirac program of quantization consisting in representing symmetries in classical mechanics by unitary transformations in some Hilbert space (that is supposed to have a physical meaning). I would not want to develop why one needs this bundle, at the first place, to answer Dirac's program, and are not contented just with the automorphisms of $\omega$, because it will lead us too far. If you are happy with this answer I'm fine, else I'll try to say a more few words(*).

P.S. The fact that the number $a$ is required to be a multiple of $\hbar$ comes just from physics consideration.

(*) Edited: I added a few words here

For a closed 2-form $\omega$ on a manifold $M$, the integrality of the closed 2-form, that is, $$ \int_\sigma \omega \in a{\bf Z}, \quad \mbox{for all} \quad \sigma \in H_2(M,{\bf Z}), $$ for some real number $a$, ensures the existence of a principal circle-bundle $Y$ (and its associated line bundle $L$) over $M$ and a connexion $\lambda$ with curvature $\omega$. Then, it is possible to lift some groups of automorphisms of $\omega$ (subgroups of ${\rm Diff}(M,\omega)$) as groups of automorphisms of $(Y,\lambda)$. This procedure is called prequantization because it is the first step of an answer to the Dirac program of quantization consisting in representing symmetries in classical mechanics by unitary transformations in some Hilbert space (that is supposed to have a physical meaning). I would not want to develop why one needs this bundle, at the first place, to answer Dirac's program, and are not contented just with the automorphisms of $\omega$, because it will lead us too far. If you are happy with this answer I'm fine, else I'll try to say a more few words.

P.S. The fact that the number $a$ is required to be a multiple of $\hbar$ comes just from physics consideration.

For a closed 2-form $\omega$ on a manifold $M$, the integrality of the closed 2-form, that is, $$ \int_\sigma \omega \in a{\bf Z}, \quad \mbox{for all} \quad \sigma \in H_2(M,{\bf Z}), $$ for some real number $a$, ensures the existence of a principal circle-bundle $Y$ (and its associated line bundle $L$) over $M$ and a connexion $\lambda$ with curvature $\omega$. Then, it is possible to lift some groups of automorphisms of $\omega$ (subgroups of ${\rm Diff}(M,\omega)$) as groups of automorphisms of $(Y,\lambda)$. This procedure is called prequantization because it is the first step of an answer to the Dirac program of quantization consisting in representing symmetries in classical mechanics by unitary transformations in some Hilbert space (that is supposed to have a physical meaning). I would not want to develop why one needs this bundle, at the first place, to answer Dirac's program, and are not contented just with the automorphisms of $\omega$, because it will lead us too far. If you are happy with this answer I'm fine, else I'll try to say a more few words(*).

P.S. The fact that the number $a$ is required to be a multiple of $\hbar$ comes just from physics consideration.

(*) Edited: I added a few words here

For a closed 2-form $\omega$ on a manifold $M$, the integrality of the closed 2-form, that is, for some real number $a$: $$ \int_\sigma \omega \in a{\bf Z}, \quad \mbox{for all} \quad \sigma \in H_2(M,{\bf Z}), $$ ensures the existence of a principal circle-bundle $Y$ (and its associated line bundle $L$) over $M$ and a connexion $\lambda$ with curvature $\omega$. Then, it is possible to lift some groups of automorphisms of $\omega$ (subgroups of ${\rm Diff}(M,\omega)$) as groups of automorphisms of $(Y,\lambda)$. This procedure is called prequantization because it is the first step of an answer to the Dirac program of quantization consisting in representing symmetries in classical mechanics by unitary transformations in some Hilbert space. I would not want to develop why one needs this bundle, at the first place, to answer Dirac's program because it will lead us too far. If you are happy with this answer I'm fine, else I'll try to say a more few words.

P.S. The fact that the number $a$ is required to be a multiple of $\hbar$ comes just from physics consideration.

For a closed 2-form $\omega$ on a manifold $M$, the integrality of the closed 2-form, that is, $$ \int_\sigma \omega \in a{\bf Z}, \quad \mbox{for all} \quad \sigma \in H_2(M,{\bf Z}), $$ for some real number $a$, ensures the existence of a principal circle-bundle $Y$ (and its associated line bundle $L$) over $M$ and a connexion $\lambda$ with curvature $\omega$. Then, it is possible to lift some groups of automorphisms of $\omega$ (subgroups of ${\rm Diff}(M,\omega)$) as groups of automorphisms of $(Y,\lambda)$. This procedure is called prequantization because it is the first step of an answer to the Dirac program of quantization consisting in representing symmetries in classical mechanics by unitary transformations in some Hilbert space (that is supposed to have a physical meaning). I would not want to develop why one needs this bundle, at the first place, to answer Dirac's program, and are not contented just with the automorphisms of $\omega$, because it will lead us too far. If you are happy with this answer I'm fine, else I'll try to say a more few words.

P.S. The fact that the number $a$ is required to be a multiple of $\hbar$ comes just from physics consideration.