## Gauge Transformations in Momentum Space?

Eugene Stefanovich wrote:
> <mikem@despammed.com> wrote in message
>
>>I'm trying to find out whether standard model gauge
>>groups (acting on fermions) correspond to
>>Bogoliubov transformations mapping between
>>disjoint Fock spaces, i.e: between unitarily
>>inequivalent representations.

>
>
> Unitarily inequivalent representations and disjoint Fock spaces
> is something I couldn't understand for a long time. This looks like
> infamous "parallel universes" to me. Could you give one
> good example where these things are
> necessary for understanding real physical phenomena.

Superconductivity is the most conspicuous example.

Arnold Neumaier

 Eugene Stefanovich wrote: > wrote in message > news:1126839691.917142.163680@g43g2000cwa.googlegroups.com... > >>I'm trying to find out whether standard model gauge >>groups (acting on fermions) correspond to >>Bogoliubov transformations mapping between >>disjoint Fock spaces, i.e: between unitarily >>inequivalent representations. > > > Unitarily inequivalent representations and disjoint Fock spaces > is something I couldn't understand for a long time. This looks like > infamous "parallel universes" to me. Could you give one > good example where these things are > necessary for understanding real physical phenomena. Superconductivity is the most conspicuous example. Arnold Neumaier
 mikem@despammed.com wrote: > A different, more recent, area is neutrino oscillations. > Blasone et al have shown that the Fock space of > definite flavour states is unitarily inequivalent to > that definite mass states. See, for example, > hep-ph/9501263, and also the review article by > Capolupo: hep-th/0408228. This means that to > understand the QFT of neutrino oscillations fully, > we need to understand UIRs and disjoint Fock > spaces. OK, let's skip superconductivity and talk about neutrinos. I looked at Blasone-Vitiello paper. This is a good example of what seems confusing about UIR for me. They find a unitary transformation which makes flavor eigenstates (or creation-annihilation operators) from mass eigenstates (or creation-annihilation operators). This transformation also changes the vacuum vector. In particular, it makes the new vacuum |0'> orthogonal to the old vacuum |0>. I have two questions: 1. In my opinion this construction does not mean that the new vacuum lies in a different Fock state. This wouldn't be the case even if all components of |0'> in the old basis were "zero" in the limit of infinite volume. Each of the components may tend to zero, but the number of components tends to infinity. So that if you correctly sum up the infinite number of "zeros" you should still get a vector of unit norm. In my view, this is not dissimilar to the normalized plane wave. The wavefunction of the state with definite momentum is "zero" everywhere in the position space. However, if you integrate its square over the entire universe you should get 1. You wouldn't say that momentum eigenstates lie in a separate Hilbert space, wouldn't you? I think that in order to evaluate correctly the expressions like "zero probability density" x "infinite volume" one should be careful with limits. The "nonstandard analysis" may be useful there. 2. 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: U = a_v* a_1 + a_u* a_2 where a_1, a_2 are annihilation operators of the mass eigenstates a_v* and a_u* are creation operators of the flavor eigenstates (e.g., a_v* = cos(phi) a_1* + sin(phi) a_2*) It is 1) unitary in the 0-particle and 1-particle sectors 2) transforms a_1, a_2 to a_v and a_u, respectively 3) does not change vacuum. I am sure I am missing some important point regarding UIR. Could you please let me know what this point is? Eugene.
 mikem@despammed.com wrote: > A different, more recent, area is neutrino oscillations. > Blasone et al have shown that the Fock space of > definite flavour states is unitarily inequivalent to > that definite mass states. See, for example, > hep-ph/9501263, and also the review article by > Capolupo: hep-th/0408228. This means that to > understand the QFT of neutrino oscillations fully, > we need to understand UIRs and disjoint Fock > spaces. OK, let's skip superconductivity and talk about neutrinos. I looked at Blasone-Vitiello paper. This is a good example of what seems confusing about UIR for me. They find a unitary transformation which makes flavor eigenstates (or creation-annihilation operators) from mass eigenstates (or creation-annihilation operators). This transformation also changes the vacuum vector. In particular, it makes the new vacuum |0'> orthogonal to the old vacuum |0>. I have two questions: 1. In my opinion this construction does not mean that the new vacuum lies in a different Fock state. This wouldn't be the case even if all components of |0'> in the old basis were "zero" in the limit of infinite volume. Each of the components may tend to zero, but the number of components tends to infinity. So that if you correctly sum up the infinite number of "zeros" you should still get a vector of unit norm. In my view, this is not dissimilar to the normalized plane wave. The wavefunction of the state with definite momentum is "zero" everywhere in the position space. However, if you integrate its square over the entire universe you should get 1. You wouldn't say that momentum eigenstates lie in a separate Hilbert space, wouldn't you? I think that in order to evaluate correctly the expressions like "zero probability density" x "infinite volume" one should be careful with limits. The "nonstandard analysis" may be useful there. 2. 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: U = a_v* a_1 + a_u* a_2 where a_1, a_2 are annihilation operators of the mass eigenstates a_v* and a_u* are creation operators of the flavor eigenstates (e.g., a_v* = cos(phi) a_1* + sin(phi) a_2*) It is 1) unitary in the 0-particle and 1-particle sectors 2) transforms a_1, a_2 to a_v and a_u, respectively 3) does not change vacuum. I am sure I am missing some important point regarding UIR. Could you please let me know what this point is? Eugene.
 mikem@despammed.com wrote: > A different, more recent, area is neutrino oscillations. > Blasone et al have shown that the Fock space of > definite flavour states is unitarily inequivalent to > that definite mass states. See, for example, > hep-ph/9501263, and also the review article by > Capolupo: hep-th/0408228. This means that to > understand the QFT of neutrino oscillations fully, > we need to understand UIRs and disjoint Fock > spaces. OK, let's skip superconductivity and talk about neutrinos. I looked at Blasone-Vitiello paper. This is a good example of what seems confusing about UIR for me. They find a unitary transformation which makes flavor eigenstates (or creation-annihilation operators) from mass eigenstates (or creation-annihilation operators). This transformation also changes the vacuum vector. In particular, it makes the new vacuum |0'> orthogonal to the old vacuum |0>. I have two questions: 1. In my opinion this construction does not mean that the new vacuum lies in a different Fock state. This wouldn't be the case even if all components of |0'> in the old basis were "zero" in the limit of infinite volume. Each of the components may tend to zero, but the number of components tends to infinity. So that if you correctly sum up the infinite number of "zeros" you should still get a vector of unit norm. In my view, this is not dissimilar to the normalized plane wave. The wavefunction of the state with definite momentum is "zero" everywhere in the position space. However, if you integrate its square over the entire universe you should get 1. You wouldn't say that momentum eigenstates lie in a separate Hilbert space, wouldn't you? I think that in order to evaluate correctly the expressions like "zero probability density" x "infinite volume" one should be careful with limits. The "nonstandard analysis" may be useful there. 2. 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: U = a_v* a_1 + a_u* a_2 where a_1, a_2 are annihilation operators of the mass eigenstates a_v* and a_u* are creation operators of the flavor eigenstates (e.g., a_v* = cos(phi) a_1* + sin(phi) a_2*) It is 1) unitary in the 0-particle and 1-particle sectors 2) transforms a_1, a_2 to a_v and a_u, respectively 3) does not change vacuum. I am sure I am missing some important point regarding UIR. Could you please let me know what this point is? Eugene.
 mikem@despammed.com wrote: > A different, more recent, area is neutrino oscillations. > Blasone et al have shown that the Fock space of > definite flavour states is unitarily inequivalent to > that definite mass states. See, for example, > hep-ph/9501263, and also the review article by > Capolupo: hep-th/0408228. This means that to > understand the QFT of neutrino oscillations fully, > we need to understand UIRs and disjoint Fock > spaces. OK, let's skip superconductivity and talk about neutrinos. I looked at Blasone-Vitiello paper. This is a good example of what seems confusing about UIR for me. They find a unitary transformation which makes flavor eigenstates (or creation-annihilation operators) from mass eigenstates (or creation-annihilation operators). This transformation also changes the vacuum vector. In particular, it makes the new vacuum |0'> orthogonal to the old vacuum |0>. I have two questions: 1. In my opinion this construction does not mean that the new vacuum lies in a different Fock state. This wouldn't be the case even if all components of |0'> in the old basis were "zero" in the limit of infinite volume. Each of the components may tend to zero, but the number of components tends to infinity. So that if you correctly sum up the infinite number of "zeros" you should still get a vector of unit norm. In my view, this is not dissimilar to the normalized plane wave. The wavefunction of the state with definite momentum is "zero" everywhere in the position space. However, if you integrate its square over the entire universe you should get 1. You wouldn't say that momentum eigenstates lie in a separate Hilbert space, wouldn't you? I think that in order to evaluate correctly the expressions like "zero probability density" x "infinite volume" one should be careful with limits. The "nonstandard analysis" may be useful there. 2. 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: U = a_v* a_1 + a_u* a_2 where a_1, a_2 are annihilation operators of the mass eigenstates a_v* and a_u* are creation operators of the flavor eigenstates (e.g., a_v* = cos(phi) a_1* + sin(phi) a_2*) It is 1) unitary in the 0-particle and 1-particle sectors 2) transforms a_1, a_2 to a_v and a_u, respectively 3) does not change vacuum. I am sure I am missing some important point regarding UIR. Could you please let me know what this point is? Eugene.
 mikem@despammed.com wrote: > A different, more recent, area is neutrino oscillations. > Blasone et al have shown that the Fock space of > definite flavour states is unitarily inequivalent to > that definite mass states. See, for example, > hep-ph/9501263, and also the review article by > Capolupo: hep-th/0408228. This means that to > understand the QFT of neutrino oscillations fully, > we need to understand UIRs and disjoint Fock > spaces. OK, let's skip superconductivity and talk about neutrinos. I looked at Blasone-Vitiello paper. This is a good example of what seems confusing about UIR for me. They find a unitary transformation which makes flavor eigenstates (or creation-annihilation operators) from mass eigenstates (or creation-annihilation operators). This transformation also changes the vacuum vector. In particular, it makes the new vacuum |0'> orthogonal to the old vacuum |0>. I have two questions: 1. In my opinion this construction does not mean that the new vacuum lies in a different Fock state. This wouldn't be the case even if all components of |0'> in the old basis were "zero" in the limit of infinite volume. Each of the components may tend to zero, but the number of components tends to infinity. So that if you correctly sum up the infinite number of "zeros" you should still get a vector of unit norm. In my view, this is not dissimilar to the normalized plane wave. The wavefunction of the state with definite momentum is "zero" everywhere in the position space. However, if you integrate its square over the entire universe you should get 1. You wouldn't say that momentum eigenstates lie in a separate Hilbert space, wouldn't you? I think that in order to evaluate correctly the expressions like "zero probability density" x "infinite volume" one should be careful with limits. The "nonstandard analysis" may be useful there. 2. 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: U = a_v* a_1 + a_u* a_2 where a_1, a_2 are annihilation operators of the mass eigenstates a_v* and a_u* are creation operators of the flavor eigenstates (e.g., a_v* = cos(phi) a_1* + sin(phi) a_2*) It is 1) unitary in the 0-particle and 1-particle sectors 2) transforms a_1, a_2 to a_v and a_u, respectively 3) does not change vacuum. I am sure I am missing some important point regarding UIR. Could you please let me know what this point is? Eugene.
 mikem@despammed.com wrote: > A different, more recent, area is neutrino oscillations. > Blasone et al have shown that the Fock space of > definite flavour states is unitarily inequivalent to > that definite mass states. See, for example, > hep-ph/9501263, and also the review article by > Capolupo: hep-th/0408228. This means that to > understand the QFT of neutrino oscillations fully, > we need to understand UIRs and disjoint Fock > spaces. OK, let's skip superconductivity and talk about neutrinos. I looked at Blasone-Vitiello paper. This is a good example of what seems confusing about UIR for me. They find a unitary transformation which makes flavor eigenstates (or creation-annihilation operators) from mass eigenstates (or creation-annihilation operators). This transformation also changes the vacuum vector. In particular, it makes the new vacuum |0'> orthogonal to the old vacuum |0>. I have two questions: 1. In my opinion this construction does not mean that the new vacuum lies in a different Fock state. This wouldn't be the case even if all components of |0'> in the old basis were "zero" in the limit of infinite volume. Each of the components may tend to zero, but the number of components tends to infinity. So that if you correctly sum up the infinite number of "zeros" you should still get a vector of unit norm. In my view, this is not dissimilar to the normalized plane wave. The wavefunction of the state with definite momentum is "zero" everywhere in the position space. However, if you integrate its square over the entire universe you should get 1. You wouldn't say that momentum eigenstates lie in a separate Hilbert space, wouldn't you? I think that in order to evaluate correctly the expressions like "zero probability density" x "infinite volume" one should be careful with limits. The "nonstandard analysis" may be useful there. 2. 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: U = a_v* a_1 + a_u* a_2 where a_1, a_2 are annihilation operators of the mass eigenstates a_v* and a_u* are creation operators of the flavor eigenstates (e.g., a_v* = cos(phi) a_1* + sin(phi) a_2*) It is 1) unitary in the 0-particle and 1-particle sectors 2) transforms a_1, a_2 to a_v and a_u, respectively 3) does not change vacuum. I am sure I am missing some important point regarding UIR. Could you please let me know what this point is? Eugene.
 mikem@despammed.com wrote: > A different, more recent, area is neutrino oscillations. > Blasone et al have shown that the Fock space of > definite flavour states is unitarily inequivalent to > that definite mass states. See, for example, > hep-ph/9501263, and also the review article by > Capolupo: hep-th/0408228. This means that to > understand the QFT of neutrino oscillations fully, > we need to understand UIRs and disjoint Fock > spaces. OK, let's skip superconductivity and talk about neutrinos. I looked at Blasone-Vitiello paper. This is a good example of what seems confusing about UIR for me. They find a unitary transformation which makes flavor eigenstates (or creation-annihilation operators) from mass eigenstates (or creation-annihilation operators). This transformation also changes the vacuum vector. In particular, it makes the new vacuum |0'> orthogonal to the old vacuum |0>. I have two questions: 1. In my opinion this construction does not mean that the new vacuum lies in a different Fock state. This wouldn't be the case even if all components of |0'> in the old basis were "zero" in the limit of infinite volume. Each of the components may tend to zero, but the number of components tends to infinity. So that if you correctly sum up the infinite number of "zeros" you should still get a vector of unit norm. In my view, this is not dissimilar to the normalized plane wave. The wavefunction of the state with definite momentum is "zero" everywhere in the position space. However, if you integrate its square over the entire universe you should get 1. You wouldn't say that momentum eigenstates lie in a separate Hilbert space, wouldn't you? I think that in order to evaluate correctly the expressions like "zero probability density" x "infinite volume" one should be careful with limits. The "nonstandard analysis" may be useful there. 2. 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: U = a_v* a_1 + a_u* a_2 where a_1, a_2 are annihilation operators of the mass eigenstates a_v* and a_u* are creation operators of the flavor eigenstates (e.g., a_v* = cos(phi) a_1* + sin(phi) a_2*) It is 1) unitary in the 0-particle and 1-particle sectors 2) transforms a_1, a_2 to a_v and a_u, respectively 3) does not change vacuum. I am sure I am missing some important point regarding UIR. Could you please let me know what this point is? Eugene.
 mikem@despammed.com wrote: > A different, more recent, area is neutrino oscillations. > Blasone et al have shown that the Fock space of > definite flavour states is unitarily inequivalent to > that definite mass states. See, for example, > hep-ph/9501263, and also the review article by > Capolupo: hep-th/0408228. This means that to > understand the QFT of neutrino oscillations fully, > we need to understand UIRs and disjoint Fock > spaces. OK, let's skip superconductivity and talk about neutrinos. I looked at Blasone-Vitiello paper. This is a good example of what seems confusing about UIR for me. They find a unitary transformation which makes flavor eigenstates (or creation-annihilation operators) from mass eigenstates (or creation-annihilation operators). This transformation also changes the vacuum vector. In particular, it makes the new vacuum |0'> orthogonal to the old vacuum |0>. I have two questions: 1. In my opinion this construction does not mean that the new vacuum lies in a different Fock state. This wouldn't be the case even if all components of |0'> in the old basis were "zero" in the limit of infinite volume. Each of the components may tend to zero, but the number of components tends to infinity. So that if you correctly sum up the infinite number of "zeros" you should still get a vector of unit norm. In my view, this is not dissimilar to the normalized plane wave. The wavefunction of the state with definite momentum is "zero" everywhere in the position space. However, if you integrate its square over the entire universe you should get 1. You wouldn't say that momentum eigenstates lie in a separate Hilbert space, wouldn't you? I think that in order to evaluate correctly the expressions like "zero probability density" x "infinite volume" one should be careful with limits. The "nonstandard analysis" may be useful there. 2. 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: U = a_v* a_1 + a_u* a_2 where a_1, a_2 are annihilation operators of the mass eigenstates a_v* and a_u* are creation operators of the flavor eigenstates (e.g., a_v* = cos(phi) a_1* + sin(phi) a_2*) It is 1) unitary in the 0-particle and 1-particle sectors 2) transforms a_1, a_2 to a_v and a_u, respectively 3) does not change vacuum. I am sure I am missing some important point regarding UIR. Could you please let me know what this point is? Eugene.
 mikem@despammed.com wrote: > A different, more recent, area is neutrino oscillations. > Blasone et al have shown that the Fock space of > definite flavour states is unitarily inequivalent to > that definite mass states. See, for example, > hep-ph/9501263, and also the review article by > Capolupo: hep-th/0408228. This means that to > understand the QFT of neutrino oscillations fully, > we need to understand UIRs and disjoint Fock > spaces. OK, let's skip superconductivity and talk about neutrinos. I looked at Blasone-Vitiello paper. This is a good example of what seems confusing about UIR for me. They find a unitary transformation which makes flavor eigenstates (or creation-annihilation operators) from mass eigenstates (or creation-annihilation operators). This transformation also changes the vacuum vector. In particular, it makes the new vacuum |0'> orthogonal to the old vacuum |0>. I have two questions: 1. In my opinion this construction does not mean that the new vacuum lies in a different Fock state. This wouldn't be the case even if all components of |0'> in the old basis were "zero" in the limit of infinite volume. Each of the components may tend to zero, but the number of components tends to infinity. So that if you correctly sum up the infinite number of "zeros" you should still get a vector of unit norm. In my view, this is not dissimilar to the normalized plane wave. The wavefunction of the state with definite momentum is "zero" everywhere in the position space. However, if you integrate its square over the entire universe you should get 1. You wouldn't say that momentum eigenstates lie in a separate Hilbert space, wouldn't you? I think that in order to evaluate correctly the expressions like "zero probability density" x "infinite volume" one should be careful with limits. The "nonstandard analysis" may be useful there. 2. 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: U = a_v* a_1 + a_u* a_2 where a_1, a_2 are annihilation operators of the mass eigenstates a_v* and a_u* are creation operators of the flavor eigenstates (e.g., a_v* = cos(phi) a_1* + sin(phi) a_2*) It is 1) unitary in the 0-particle and 1-particle sectors 2) transforms a_1, a_2 to a_v and a_u, respectively 3) does not change vacuum. I am sure I am missing some important point regarding UIR. Could you please let me know what this point is? Eugene.
 mikem@despammed.com wrote: > A different, more recent, area is neutrino oscillations. > Blasone et al have shown that the Fock space of > definite flavour states is unitarily inequivalent to > that definite mass states. See, for example, > hep-ph/9501263, and also the review article by > Capolupo: hep-th/0408228. This means that to > understand the QFT of neutrino oscillations fully, > we need to understand UIRs and disjoint Fock > spaces. OK, let's skip superconductivity and talk about neutrinos. I looked at Blasone-Vitiello paper. This is a good example of what seems confusing about UIR for me. They find a unitary transformation which makes flavor eigenstates (or creation-annihilation operators) from mass eigenstates (or creation-annihilation operators). This transformation also changes the vacuum vector. In particular, it makes the new vacuum |0'> orthogonal to the old vacuum |0>. I have two questions: 1. In my opinion this construction does not mean that the new vacuum lies in a different Fock state. This wouldn't be the case even if all components of |0'> in the old basis were "zero" in the limit of infinite volume. Each of the components may tend to zero, but the number of components tends to infinity. So that if you correctly sum up the infinite number of "zeros" you should still get a vector of unit norm. In my view, this is not dissimilar to the normalized plane wave. The wavefunction of the state with definite momentum is "zero" everywhere in the position space. However, if you integrate its square over the entire universe you should get 1. You wouldn't say that momentum eigenstates lie in a separate Hilbert space, wouldn't you? I think that in order to evaluate correctly the expressions like "zero probability density" x "infinite volume" one should be careful with limits. The "nonstandard analysis" may be useful there. 2. 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: U = a_v* a_1 + a_u* a_2 where a_1, a_2 are annihilation operators of the mass eigenstates a_v* and a_u* are creation operators of the flavor eigenstates (e.g., a_v* = cos(phi) a_1* + sin(phi) a_2*) It is 1) unitary in the 0-particle and 1-particle sectors 2) transforms a_1, a_2 to a_v and a_u, respectively 3) does not change vacuum. I am sure I am missing some important point regarding UIR. Could you please let me know what this point is? Eugene.
 mikem@despammed.com wrote: > A different, more recent, area is neutrino oscillations. > Blasone et al have shown that the Fock space of > definite flavour states is unitarily inequivalent to > that definite mass states. See, for example, > hep-ph/9501263, and also the review article by > Capolupo: hep-th/0408228. This means that to > understand the QFT of neutrino oscillations fully, > we need to understand UIRs and disjoint Fock > spaces. OK, let's skip superconductivity and talk about neutrinos. I looked at Blasone-Vitiello paper. This is a good example of what seems confusing about UIR for me. They find a unitary transformation which makes flavor eigenstates (or creation-annihilation operators) from mass eigenstates (or creation-annihilation operators). This transformation also changes the vacuum vector. In particular, it makes the new vacuum |0'> orthogonal to the old vacuum |0>. I have two questions: 1. In my opinion this construction does not mean that the new vacuum lies in a different Fock state. This wouldn't be the case even if all components of |0'> in the old basis were "zero" in the limit of infinite volume. Each of the components may tend to zero, but the number of components tends to infinity. So that if you correctly sum up the infinite number of "zeros" you should still get a vector of unit norm. In my view, this is not dissimilar to the normalized plane wave. The wavefunction of the state with definite momentum is "zero" everywhere in the position space. However, if you integrate its square over the entire universe you should get 1. You wouldn't say that momentum eigenstates lie in a separate Hilbert space, wouldn't you? I think that in order to evaluate correctly the expressions like "zero probability density" x "infinite volume" one should be careful with limits. The "nonstandard analysis" may be useful there. 2. 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: U = a_v* a_1 + a_u* a_2 where a_1, a_2 are annihilation operators of the mass eigenstates a_v* and a_u* are creation operators of the flavor eigenstates (e.g., a_v* = cos(phi) a_1* + sin(phi) a_2*) It is 1) unitary in the 0-particle and 1-particle sectors 2) transforms a_1, a_2 to a_v and a_u, respectively 3) does not change vacuum. I am sure I am missing some important point regarding UIR. Could you please let me know what this point is? Eugene.
 mikem@despammed.com wrote: > A different, more recent, area is neutrino oscillations. > Blasone et al have shown that the Fock space of > definite flavour states is unitarily inequivalent to > that definite mass states. See, for example, > hep-ph/9501263, and also the review article by > Capolupo: hep-th/0408228. This means that to > understand the QFT of neutrino oscillations fully, > we need to understand UIRs and disjoint Fock > spaces. OK, let's skip superconductivity and talk about neutrinos. I looked at Blasone-Vitiello paper. This is a good example of what seems confusing about UIR for me. They find a unitary transformation which makes flavor eigenstates (or creation-annihilation operators) from mass eigenstates (or creation-annihilation operators). This transformation also changes the vacuum vector. In particular, it makes the new vacuum |0'> orthogonal to the old vacuum |0>. I have two questions: 1. In my opinion this construction does not mean that the new vacuum lies in a different Fock state. This wouldn't be the case even if all components of |0'> in the old basis were "zero" in the limit of infinite volume. Each of the components may tend to zero, but the number of components tends to infinity. So that if you correctly sum up the infinite number of "zeros" you should still get a vector of unit norm. In my view, this is not dissimilar to the normalized plane wave. The wavefunction of the state with definite momentum is "zero" everywhere in the position space. However, if you integrate its square over the entire universe you should get 1. You wouldn't say that momentum eigenstates lie in a separate Hilbert space, wouldn't you? I think that in order to evaluate correctly the expressions like "zero probability density" x "infinite volume" one should be careful with limits. The "nonstandard analysis" may be useful there. 2. 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: U = a_v* a_1 + a_u* a_2 where a_1, a_2 are annihilation operators of the mass eigenstates a_v* and a_u* are creation operators of the flavor eigenstates (e.g., a_v* = cos(phi) a_1* + sin(phi) a_2*) It is 1) unitary in the 0-particle and 1-particle sectors 2) transforms a_1, a_2 to a_v and a_u, respectively 3) does not change vacuum. I am sure I am missing some important point regarding UIR. Could you please let me know what this point is? Eugene.
 mikem@despammed.com wrote: > A different, more recent, area is neutrino oscillations. > Blasone et al have shown that the Fock space of > definite flavour states is unitarily inequivalent to > that definite mass states. See, for example, > hep-ph/9501263, and also the review article by > Capolupo: hep-th/0408228. This means that to > understand the QFT of neutrino oscillations fully, > we need to understand UIRs and disjoint Fock > spaces. OK, let's skip superconductivity and talk about neutrinos. I looked at Blasone-Vitiello paper. This is a good example of what seems confusing about UIR for me. They find a unitary transformation which makes flavor eigenstates (or creation-annihilation operators) from mass eigenstates (or creation-annihilation operators). This transformation also changes the vacuum vector. In particular, it makes the new vacuum |0'> orthogonal to the old vacuum |0>. I have two questions: 1. In my opinion this construction does not mean that the new vacuum lies in a different Fock state. This wouldn't be the case even if all components of |0'> in the old basis were "zero" in the limit of infinite volume. Each of the components may tend to zero, but the number of components tends to infinity. So that if you correctly sum up the infinite number of "zeros" you should still get a vector of unit norm. In my view, this is not dissimilar to the normalized plane wave. The wavefunction of the state with definite momentum is "zero" everywhere in the position space. However, if you integrate its square over the entire universe you should get 1. You wouldn't say that momentum eigenstates lie in a separate Hilbert space, wouldn't you? I think that in order to evaluate correctly the expressions like "zero probability density" x "infinite volume" one should be careful with limits. The "nonstandard analysis" may be useful there. 2. 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: U = a_v* a_1 + a_u* a_2 where a_1, a_2 are annihilation operators of the mass eigenstates a_v* and a_u* are creation operators of the flavor eigenstates (e.g., a_v* = cos(phi) a_1* + sin(phi) a_2*) It is 1) unitary in the 0-particle and 1-particle sectors 2) transforms a_1, a_2 to a_v and a_u, respectively 3) does not change vacuum. I am sure I am missing some important point regarding UIR. Could you please let me know what this point is? Eugene.
 mikem@despammed.com wrote: > A different, more recent, area is neutrino oscillations. > Blasone et al have shown that the Fock space of > definite flavour states is unitarily inequivalent to > that definite mass states. See, for example, > hep-ph/9501263, and also the review article by > Capolupo: hep-th/0408228. This means that to > understand the QFT of neutrino oscillations fully, > we need to understand UIRs and disjoint Fock > spaces. OK, let's skip superconductivity and talk about neutrinos. I looked at Blasone-Vitiello paper. This is a good example of what seems confusing about UIR for me. They find a unitary transformation which makes flavor eigenstates (or creation-annihilation operators) from mass eigenstates (or creation-annihilation operators). This transformation also changes the vacuum vector. In particular, it makes the new vacuum |0'> orthogonal to the old vacuum |0>. I have two questions: 1. In my opinion this construction does not mean that the new vacuum lies in a different Fock state. This wouldn't be the case even if all components of |0'> in the old basis were "zero" in the limit of infinite volume. Each of the components may tend to zero, but the number of components tends to infinity. So that if you correctly sum up the infinite number of "zeros" you should still get a vector of unit norm. In my view, this is not dissimilar to the normalized plane wave. The wavefunction of the state with definite momentum is "zero" everywhere in the position space. However, if you integrate its square over the entire universe you should get 1. You wouldn't say that momentum eigenstates lie in a separate Hilbert space, wouldn't you? I think that in order to evaluate correctly the expressions like "zero probability density" x "infinite volume" one should be careful with limits. The "nonstandard analysis" may be useful there. 2. 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: U = a_v* a_1 + a_u* a_2 where a_1, a_2 are annihilation operators of the mass eigenstates a_v* and a_u* are creation operators of the flavor eigenstates (e.g., a_v* = cos(phi) a_1* + sin(phi) a_2*) It is 1) unitary in the 0-particle and 1-particle sectors 2) transforms a_1, a_2 to a_v and a_u, respectively 3) does not change vacuum. I am sure I am missing some important point regarding UIR. Could you please let me know what this point is? Eugene.
 mikem@despammed.com wrote: > A different, more recent, area is neutrino oscillations. > Blasone et al have shown that the Fock space of > definite flavour states is unitarily inequivalent to > that definite mass states. See, for example, > hep-ph/9501263, and also the review article by > Capolupo: hep-th/0408228. This means that to > understand the QFT of neutrino oscillations fully, > we need to understand UIRs and disjoint Fock > spaces. OK, let's skip superconductivity and talk about neutrinos. I looked at Blasone-Vitiello paper. This is a good example of what seems confusing about UIR for me. They find a unitary transformation which makes flavor eigenstates (or creation-annihilation operators) from mass eigenstates (or creation-annihilation operators). This transformation also changes the vacuum vector. In particular, it makes the new vacuum |0'> orthogonal to the old vacuum |0>. I have two questions: 1. In my opinion this construction does not mean that the new vacuum lies in a different Fock state. This wouldn't be the case even if all components of |0'> in the old basis were "zero" in the limit of infinite volume. Each of the components may tend to zero, but the number of components tends to infinity. So that if you correctly sum up the infinite number of "zeros" you should still get a vector of unit norm. In my view, this is not dissimilar to the normalized plane wave. The wavefunction of the state with definite momentum is "zero" everywhere in the position space. However, if you integrate its square over the entire universe you should get 1. You wouldn't say that momentum eigenstates lie in a separate Hilbert space, wouldn't you? I think that in order to evaluate correctly the expressions like "zero probability density" x "infinite volume" one should be careful with limits. The "nonstandard analysis" may be useful there. 2. 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: U = a_v* a_1 + a_u* a_2 where a_1, a_2 are annihilation operators of the mass eigenstates a_v* and a_u* are creation operators of the flavor eigenstates (e.g., a_v* = cos(phi) a_1* + sin(phi) a_2*) It is 1) unitary in the 0-particle and 1-particle sectors 2) transforms a_1, a_2 to a_v and a_u, respectively 3) does not change vacuum. I am sure I am missing some important point regarding UIR. Could you please let me know what this point is? Eugene.