# Gauge Transformations in Momentum Space?

by mikem@despammed.com
Tags: gauge, momentum, space, transformations
 P: n/a Eugene Stefanovich wrote in part: > [...] I don't think there is anything wrong with > regarding the Hilbert space of a single particle > as non-separable. After all, the number of points > in 3D space is not countable, and one can associate > a distinct basis vector (eigenvector of the position > operator) with each such point. That's because we can integrate over a 3D space. But, (at least with standard integration), we can't integrate over an infinite dimensional space in the same way. But about here, my detailed knowledge starts to dry up so I can't say much more. > Your arguments could be correct if your DEFINE > the Fock space as having not more than a finite > number of particles. Then, why I am not allowed > to DEFINE the Fock space as having any number > of particles from zero to infinity? Only because of the difficulty with performing standard integration over uncountably-infinite dimensional spaces. > [...] See, for example: A. Friedman, "Non-standard > extension of quantum logic and Dirac's bra-ket > formalism of quantum mechanics", Int. J. Theor. Phys. > 33 (1994), 307 [...] Is this paper on the archive, or somewhere else online? (It's a pain for me to travel to university libraries these days.) Regarding the other items in your post, I need to think about them for a while before replying.
 P: n/a Eugene Stefanovich wrote in part: > [...] I don't think there is anything wrong with > regarding the Hilbert space of a single particle > as non-separable. After all, the number of points > in 3D space is not countable, and one can associate > a distinct basis vector (eigenvector of the position > operator) with each such point. That's because we can integrate over a 3D space. But, (at least with standard integration), we can't integrate over an infinite dimensional space in the same way. But about here, my detailed knowledge starts to dry up so I can't say much more. > Your arguments could be correct if your DEFINE > the Fock space as having not more than a finite > number of particles. Then, why I am not allowed > to DEFINE the Fock space as having any number > of particles from zero to infinity? Only because of the difficulty with performing standard integration over uncountably-infinite dimensional spaces. > [...] See, for example: A. Friedman, "Non-standard > extension of quantum logic and Dirac's bra-ket > formalism of quantum mechanics", Int. J. Theor. Phys. > 33 (1994), 307 [...] Is this paper on the archive, or somewhere else online? (It's a pain for me to travel to university libraries these days.) Regarding the other items in your post, I need to think about them for a while before replying.
 P: n/a Eugene Stefanovich wrote in part: > [...] I don't think there is anything wrong with > regarding the Hilbert space of a single particle > as non-separable. After all, the number of points > in 3D space is not countable, and one can associate > a distinct basis vector (eigenvector of the position > operator) with each such point. That's because we can integrate over a 3D space. But, (at least with standard integration), we can't integrate over an infinite dimensional space in the same way. But about here, my detailed knowledge starts to dry up so I can't say much more. > Your arguments could be correct if your DEFINE > the Fock space as having not more than a finite > number of particles. Then, why I am not allowed > to DEFINE the Fock space as having any number > of particles from zero to infinity? Only because of the difficulty with performing standard integration over uncountably-infinite dimensional spaces. > [...] See, for example: A. Friedman, "Non-standard > extension of quantum logic and Dirac's bra-ket > formalism of quantum mechanics", Int. J. Theor. Phys. > 33 (1994), 307 [...] Is this paper on the archive, or somewhere else online? (It's a pain for me to travel to university libraries these days.) Regarding the other items in your post, I need to think about them for a while before replying.
 P: n/a Eugene Stefanovich wrote in part: > [...] I don't think there is anything wrong with > regarding the Hilbert space of a single particle > as non-separable. After all, the number of points > in 3D space is not countable, and one can associate > a distinct basis vector (eigenvector of the position > operator) with each such point. That's because we can integrate over a 3D space. But, (at least with standard integration), we can't integrate over an infinite dimensional space in the same way. But about here, my detailed knowledge starts to dry up so I can't say much more. > Your arguments could be correct if your DEFINE > the Fock space as having not more than a finite > number of particles. Then, why I am not allowed > to DEFINE the Fock space as having any number > of particles from zero to infinity? Only because of the difficulty with performing standard integration over uncountably-infinite dimensional spaces. > [...] See, for example: A. Friedman, "Non-standard > extension of quantum logic and Dirac's bra-ket > formalism of quantum mechanics", Int. J. Theor. Phys. > 33 (1994), 307 [...] Is this paper on the archive, or somewhere else online? (It's a pain for me to travel to university libraries these days.) Regarding the other items in your post, I need to think about them for a while before replying.
 P: n/a wrote in message news:1128407680.618420.49880@g43g2000cwa.googlegroups.com... > > Your arguments could be correct if your DEFINE > > the Fock space as having not more than a finite > > number of particles. Then, why I am not allowed > > to DEFINE the Fock space as having any number > > of particles from zero to infinity? > > Only because of the difficulty with performing > standard integration over uncountably-infinite > dimensional spaces. I would prefer to generalize the way we integrate things rather that stick to artificial separable spaces only to discover that they cannot accomodate the interacting systems we are most interested in. > > [...] See, for example: A. Friedman, "Non-standard > > extension of quantum logic and Dirac's bra-ket > > formalism of quantum mechanics", Int. J. Theor. Phys. > > 33 (1994), 307 [...] > > Is this paper on the archive, or somewhere else online? > (It's a pain for me to travel to university libraries these > days.) No, it's not on the Internet. I can send you a copy if you give me your address. Eugene.
 P: n/a wrote in message news:1128407680.618420.49880@g43g2000cwa.googlegroups.com... > > Your arguments could be correct if your DEFINE > > the Fock space as having not more than a finite > > number of particles. Then, why I am not allowed > > to DEFINE the Fock space as having any number > > of particles from zero to infinity? > > Only because of the difficulty with performing > standard integration over uncountably-infinite > dimensional spaces. I would prefer to generalize the way we integrate things rather that stick to artificial separable spaces only to discover that they cannot accomodate the interacting systems we are most interested in. > > [...] See, for example: A. Friedman, "Non-standard > > extension of quantum logic and Dirac's bra-ket > > formalism of quantum mechanics", Int. J. Theor. Phys. > > 33 (1994), 307 [...] > > Is this paper on the archive, or somewhere else online? > (It's a pain for me to travel to university libraries these > days.) No, it's not on the Internet. I can send you a copy if you give me your address. Eugene.
 P: n/a wrote in message news:1128407680.618420.49880@g43g2000cwa.googlegroups.com... > > Your arguments could be correct if your DEFINE > > the Fock space as having not more than a finite > > number of particles. Then, why I am not allowed > > to DEFINE the Fock space as having any number > > of particles from zero to infinity? > > Only because of the difficulty with performing > standard integration over uncountably-infinite > dimensional spaces. I would prefer to generalize the way we integrate things rather that stick to artificial separable spaces only to discover that they cannot accomodate the interacting systems we are most interested in. > > [...] See, for example: A. Friedman, "Non-standard > > extension of quantum logic and Dirac's bra-ket > > formalism of quantum mechanics", Int. J. Theor. Phys. > > 33 (1994), 307 [...] > > Is this paper on the archive, or somewhere else online? > (It's a pain for me to travel to university libraries these > days.) No, it's not on the Internet. I can send you a copy if you give me your address. Eugene.
 P: n/a wrote in message news:1128407680.618420.49880@g43g2000cwa.googlegroups.com... > > Your arguments could be correct if your DEFINE > > the Fock space as having not more than a finite > > number of particles. Then, why I am not allowed > > to DEFINE the Fock space as having any number > > of particles from zero to infinity? > > Only because of the difficulty with performing > standard integration over uncountably-infinite > dimensional spaces. I would prefer to generalize the way we integrate things rather that stick to artificial separable spaces only to discover that they cannot accomodate the interacting systems we are most interested in. > > [...] See, for example: A. Friedman, "Non-standard > > extension of quantum logic and Dirac's bra-ket > > formalism of quantum mechanics", Int. J. Theor. Phys. > > 33 (1994), 307 [...] > > Is this paper on the archive, or somewhere else online? > (It's a pain for me to travel to university libraries these > days.) No, it's not on the Internet. I can send you a copy if you give me your address. Eugene.
 P: n/a wrote in message news:1128407680.618420.49880@g43g2000cwa.googlegroups.com... > > Your arguments could be correct if your DEFINE > > the Fock space as having not more than a finite > > number of particles. Then, why I am not allowed > > to DEFINE the Fock space as having any number > > of particles from zero to infinity? > > Only because of the difficulty with performing > standard integration over uncountably-infinite > dimensional spaces. I would prefer to generalize the way we integrate things rather that stick to artificial separable spaces only to discover that they cannot accomodate the interacting systems we are most interested in. > > [...] See, for example: A. Friedman, "Non-standard > > extension of quantum logic and Dirac's bra-ket > > formalism of quantum mechanics", Int. J. Theor. Phys. > > 33 (1994), 307 [...] > > Is this paper on the archive, or somewhere else online? > (It's a pain for me to travel to university libraries these > days.) No, it's not on the Internet. I can send you a copy if you give me your address. Eugene.
 P: n/a wrote in message news:1128407680.618420.49880@g43g2000cwa.googlegroups.com... > > Your arguments could be correct if your DEFINE > > the Fock space as having not more than a finite > > number of particles. Then, why I am not allowed > > to DEFINE the Fock space as having any number > > of particles from zero to infinity? > > Only because of the difficulty with performing > standard integration over uncountably-infinite > dimensional spaces. I would prefer to generalize the way we integrate things rather that stick to artificial separable spaces only to discover that they cannot accomodate the interacting systems we are most interested in. > > [...] See, for example: A. Friedman, "Non-standard > > extension of quantum logic and Dirac's bra-ket > > formalism of quantum mechanics", Int. J. Theor. Phys. > > 33 (1994), 307 [...] > > Is this paper on the archive, or somewhere else online? > (It's a pain for me to travel to university libraries these > days.) No, it's not on the Internet. I can send you a copy if you give me your address. Eugene.
 P: n/a wrote in message news:1128407680.618420.49880@g43g2000cwa.googlegroups.com... > > Your arguments could be correct if your DEFINE > > the Fock space as having not more than a finite > > number of particles. Then, why I am not allowed > > to DEFINE the Fock space as having any number > > of particles from zero to infinity? > > Only because of the difficulty with performing > standard integration over uncountably-infinite > dimensional spaces. I would prefer to generalize the way we integrate things rather that stick to artificial separable spaces only to discover that they cannot accomodate the interacting systems we are most interested in. > > [...] See, for example: A. Friedman, "Non-standard > > extension of quantum logic and Dirac's bra-ket > > formalism of quantum mechanics", Int. J. Theor. Phys. > > 33 (1994), 307 [...] > > Is this paper on the archive, or somewhere else online? > (It's a pain for me to travel to university libraries these > days.) No, it's not on the Internet. I can send you a copy if you give me your address. Eugene.
 P: n/a wrote in message news:1128407680.618420.49880@g43g2000cwa.googlegroups.com... > > Your arguments could be correct if your DEFINE > > the Fock space as having not more than a finite > > number of particles. Then, why I am not allowed > > to DEFINE the Fock space as having any number > > of particles from zero to infinity? > > Only because of the difficulty with performing > standard integration over uncountably-infinite > dimensional spaces. I would prefer to generalize the way we integrate things rather that stick to artificial separable spaces only to discover that they cannot accomodate the interacting systems we are most interested in. > > [...] See, for example: A. Friedman, "Non-standard > > extension of quantum logic and Dirac's bra-ket > > formalism of quantum mechanics", Int. J. Theor. Phys. > > 33 (1994), 307 [...] > > Is this paper on the archive, or somewhere else online? > (It's a pain for me to travel to university libraries these > days.) No, it's not on the Internet. I can send you a copy if you give me your address. Eugene.
 P: n/a wrote in message news:1128407680.618420.49880@g43g2000cwa.googlegroups.com... > > Your arguments could be correct if your DEFINE > > the Fock space as having not more than a finite > > number of particles. Then, why I am not allowed > > to DEFINE the Fock space as having any number > > of particles from zero to infinity? > > Only because of the difficulty with performing > standard integration over uncountably-infinite > dimensional spaces. I would prefer to generalize the way we integrate things rather that stick to artificial separable spaces only to discover that they cannot accomodate the interacting systems we are most interested in. > > [...] See, for example: A. Friedman, "Non-standard > > extension of quantum logic and Dirac's bra-ket > > formalism of quantum mechanics", Int. J. Theor. Phys. > > 33 (1994), 307 [...] > > Is this paper on the archive, or somewhere else online? > (It's a pain for me to travel to university libraries these > days.) No, it's not on the Internet. I can send you a copy if you give me your address. Eugene.
 P: n/a Eugene Stefanovich wrote in part: > There is an infinite number of unitary transformations from > flavor eigenstates to mass eigenstates. Blasone-Vitiello's > transformation changes vacuum, which seems unphysical > to me. I would prefer to have a unique vacuum without > particles of any kind. This is achieved, for example, by > the following transformation: [.....] > > 1) U = 1 on the vacuum vector |0> > 2) U = a_v* a_1 + a_u* a_2 on one-particle > subspaces H_1 (+) H_2 = H_v (+) H_u > 3) U = whatever on the rest of the Fock space. I still can't make sense of this. It seems like you're defining U as a function of the vector on which it happens to be acting - which is not how one usually does transformations in Fock space. Normally one would write a' = U a U*, where U is independent of particle number. [BTW, since we seem to be the only 2 people interested in this sub-thread, perhaps we should move to email to avoid spr moderation delays?]
 P: n/a Eugene Stefanovich wrote in part: > There is an infinite number of unitary transformations from > flavor eigenstates to mass eigenstates. Blasone-Vitiello's > transformation changes vacuum, which seems unphysical > to me. I would prefer to have a unique vacuum without > particles of any kind. This is achieved, for example, by > the following transformation: [.....] > > 1) U = 1 on the vacuum vector |0> > 2) U = a_v* a_1 + a_u* a_2 on one-particle > subspaces H_1 (+) H_2 = H_v (+) H_u > 3) U = whatever on the rest of the Fock space. I still can't make sense of this. It seems like you're defining U as a function of the vector on which it happens to be acting - which is not how one usually does transformations in Fock space. Normally one would write a' = U a U*, where U is independent of particle number. [BTW, since we seem to be the only 2 people interested in this sub-thread, perhaps we should move to email to avoid spr moderation delays?]
 P: n/a Eugene Stefanovich wrote in part: > There is an infinite number of unitary transformations from > flavor eigenstates to mass eigenstates. Blasone-Vitiello's > transformation changes vacuum, which seems unphysical > to me. I would prefer to have a unique vacuum without > particles of any kind. This is achieved, for example, by > the following transformation: [.....] > > 1) U = 1 on the vacuum vector |0> > 2) U = a_v* a_1 + a_u* a_2 on one-particle > subspaces H_1 (+) H_2 = H_v (+) H_u > 3) U = whatever on the rest of the Fock space. I still can't make sense of this. It seems like you're defining U as a function of the vector on which it happens to be acting - which is not how one usually does transformations in Fock space. Normally one would write a' = U a U*, where U is independent of particle number. [BTW, since we seem to be the only 2 people interested in this sub-thread, perhaps we should move to email to avoid spr moderation delays?]
 P: n/a Eugene Stefanovich wrote in part: > There is an infinite number of unitary transformations from > flavor eigenstates to mass eigenstates. Blasone-Vitiello's > transformation changes vacuum, which seems unphysical > to me. I would prefer to have a unique vacuum without > particles of any kind. This is achieved, for example, by > the following transformation: [.....] > > 1) U = 1 on the vacuum vector |0> > 2) U = a_v* a_1 + a_u* a_2 on one-particle > subspaces H_1 (+) H_2 = H_v (+) H_u > 3) U = whatever on the rest of the Fock space. I still can't make sense of this. It seems like you're defining U as a function of the vector on which it happens to be acting - which is not how one usually does transformations in Fock space. Normally one would write a' = U a U*, where U is independent of particle number. [BTW, since we seem to be the only 2 people interested in this sub-thread, perhaps we should move to email to avoid spr moderation delays?]
 P: n/a Eugene Stefanovich wrote in part: > There is an infinite number of unitary transformations from > flavor eigenstates to mass eigenstates. Blasone-Vitiello's > transformation changes vacuum, which seems unphysical > to me. I would prefer to have a unique vacuum without > particles of any kind. This is achieved, for example, by > the following transformation: [.....] > > 1) U = 1 on the vacuum vector |0> > 2) U = a_v* a_1 + a_u* a_2 on one-particle > subspaces H_1 (+) H_2 = H_v (+) H_u > 3) U = whatever on the rest of the Fock space. I still can't make sense of this. It seems like you're defining U as a function of the vector on which it happens to be acting - which is not how one usually does transformations in Fock space. Normally one would write a' = U a U*, where U is independent of particle number. [BTW, since we seem to be the only 2 people interested in this sub-thread, perhaps we should move to email to avoid spr moderation delays?]