## Gauge Transformations in Momentum Space?

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Most textbook treatments of gauge transformations do it in\nposition space. So far, I haven\'t found any that discuss\nin detail what they look like in momentum space, and what\nissues arise in the QFT Fock space.\n\nFor example, consider the U(1) group for electromagnetism\nacting on Dirac electrons in QED. Textbooks write it as\nsomething like:\n\nPsi(x) -&gt; exp(i theta(x)) Psi(x)\n\nwhere theta(x) is a real scalar function of x (in 3+1\nspacetime of course).\n\nTo pass to momentum space, we need to assume that\nexp(i theta(x)) has a reasonable Fourier transform,\nsuch that the transformation can be represented\nin momentum space as an integral operator whose\nkernel is a distribution.\n\nTake a simple case: theta(x) = wt, where \'w\' is a real constant.\nI.e: in position space we have\n\nPsi(x) -&gt; exp(iwt) Psi(x)\n\nIn momentum space, this just shifts the energy by an amount \'w\'\nI.e: E -&gt; E - w.\n\nSo old positive-energy modes in the energy range 0 to w get\ntransformed into negative-energy modes. In the 2nd-quantized Fock\nspace this means we\'re mixing some of the annihilation and creation\noperators - because they were defined in terms of the original +ve\nand -ve energy modes. Such mixing usually means that we\'re mapping\nbetween unitarily inequivalent representations, i.e: between\northogonal Fock spaces.\n\nI\'m interested in finding explicit operators which are form-invariant\nin both representations. I tried Google-Scholar but didn\'t have much\nsuccess.\n\nSo my question is:\n\nDo any textbooks or review papers discuss this stuff at length?\n(I don\'t mean just the usual Bogoliubov transformations from\ncondensed matter physics which map between inequivalent reps,\nbut specifically for standard model gauge transformations\nin momentum space, and hence Fock space(s).)\n\nTIA.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">&nbsp;&nbsp;View this Usenet post in original ASCII form </a></div><P></jabberwocky>Most textbook treatments of gauge transformations do it in
position space. So far, I haven't found any that discuss
in detail what they look like in momentum space, and what
issues arise in the QFT Fock space.

For example, consider the U(1) group for electromagnetism
acting on Dirac electrons in QED. Textbooks write it as
something like:

$$\Psi(x) -> \exp(i \theta(x)) \Psi(x)$$

where $\theta(x)$ is a real scalar function of x (in 3+1
spacetime of course).

To pass to momentum space, we need to assume that
$\exp(i \theta(x))$ has a reasonable Fourier transform,
such that the transformation can be represented
in momentum space as an integral operator whose
kernel is a distribution.

Take a simple case: $\theta(x) =$ wt, where 'w' is a real constant.
I.e: in position space we have

$$\Psi(x) -> \exp(iwt) \Psi(x)$$

In momentum space, this just shifts the energy by an amount 'w'
I.e: $E -> E - w$.

So old positive-energy modes in the energy range to w get
transformed into negative-energy modes. In the 2nd-quantized Fock
space this means we're mixing some of the annihilation and creation
operators - because they were defined in terms of the original +ve
and -ve energy modes. Such mixing usually means that we're mapping
between unitarily inequivalent representations, i.e: between
orthogonal Fock spaces.

I'm interested in finding explicit operators which are form-invariant
in both representations. I tried Google-Scholar but didn't have much
success.

So my question is:

Do any textbooks or review papers discuss this stuff at length?
(I don't mean just the usual Bogoliubov transformations from
condensed matter physics which map between inequivalent reps,
but specifically for standard model gauge transformations
in momentum space, and hence Fock space(s).)

TIA.

 PhysOrg.com physics news on PhysOrg.com >> Promising doped zirconia>> New X-ray method shows how frog embryos could help thwart disease>> Bringing life into focus


mikem@despammed.com wrote: > Most textbook treatments of gauge transformations do it in > position space. So far, I haven't found any that discuss > in detail what they look like in momentum space, and what > issues arise in the QFT Fock space. > > For example, consider the U(1) group for electromagnetism > acting on Dirac electrons in QED. Textbooks write it as > something like: > > $\Psi(x) -> \exp(i \theta(x)) \Psi(x)$ > > where $\theta(x)$ is a real scalar function of x (in 3+1 > spacetime of course). > > To pass to momentum space, we need to assume that > $\exp(i \theta(x))$ has a reasonable Fourier transform, > such that the transformation can be represented > in momentum space as an integral operator whose > kernel is a distribution. > > Take a simple case: $\theta(x) =$ wt, where 'w' is a real constant. > I.e: in position space we have > > $\Psi(x) -> \exp(iwt) \Psi(x)$ > > In momentum space, this just shifts the energy by an amount 'w' > I.e: $E -> E - w$. > > So old positive-energy modes in the energy range to w get > transformed into negative-energy modes. In the 2nd-quantized Fock > space this means we're mixing some of the annihilation and creation > operators - because they were defined in terms of the original +ve > and -ve energy modes. Such mixing usually means that we're mapping > between unitarily inequivalent representations, i.e: between > orthogonal Fock spaces. > > I'm interested in finding explicit operators which are form-invariant > in both representations. I tried Google-Scholar but didn't have much > success. > > So my question is: > > Do any textbooks or review papers discuss this stuff at length? > (I don't mean just the usual Bogoliubov transformations from > condensed matter physics which map between inequivalent reps, > but specifically for standard model gauge transformations > in momentum space, and hence Fock space(s).) > > TIA. I think quantum fields and their gauge transformations in the "momentum" representation have no meaning at all. You can switch between position and momentum representations of wave functions, but quantum fields are completely different beasts (see recent thread "Why no tensors in quantum mechanics?"). The only point to introduce quantum fields $\Psi(x,t)$ in QFT is to have convenient "building blocks" for the interacting Hamiltonian. The gauge invariance of $\Psi(x,t)$ is a heuristic aid in this construction. All this works only when (x,t) are coordinates on an abstract Minkowski space. Then you can explicitly ensure that $\Psi(x,t)$ transform by linear Lorentz formulas wrt the non-interacting representation of the Poincare group and that $\Psi(x,t)$ (anti)commute at "space-like" separation. This, in turn, guarantees (see Weinberg, vol. 1) that the interaction operator in the Hamiltonian constructed from $\Psi(x,t)$ is relativistically invariant. I have no idea how and why would you want to use "momentum-space" $\Psi(p,t)$ for this purpose. Eugene.



Eugene Stefanovich wrote: > I think quantum fields and their gauge transformations > in the "momentum" representation have no meaning at all. > [...] Certainly they're not of much use in standard treatments of QFT and the standard model. But "no meaning at all" seems a bit strong. > [...] > I have no idea how and why would you want to use > "momentum-space$" \Psi(p,t)$ for this purpose. 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. The textbooks I'm aware of give various calculation techniques using momentum-space annihilation/creation operators exclusively. So in the hope of leveraging those techniques I need to learn more about gauge transformations in momentum space. techniques for investigating such Bogoliubov transformations

## Gauge Transformations in Momentum Space?

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Eugene Stefanovich wrote:\n\n&gt; Unitarily inequivalent representations and disjoint Fock spaces\n&gt; is something I couldn\'t understand for a long time. This looks\n&gt; like infamous "parallel universes" to me.\n\nNot at all. They arise because the Fock space construction\nmust arbitrarily restrict to finite particle numbers in order for\nan inner product to exist. This is explained in Umezawa (below).\n\n\n&gt; Could you give one good example where these things are\n&gt; necessary for understanding real physical phenomena.\n\nI\'ll give you several...\n\nThey arise in condensed matter physics, i.e: bose-einstein\ncondensation, superfluidity and superconductivity. In textbooks\non these subjects, look for "Bogoliubov transformation" and\nin most cases you\'ll find UIRs lurking, although the textbooks\ndon\'t always bring out this point explicitly. Umezawa\'s text on\n"Thermofield Dynamics and Condensed States" gives the\nmost understandable presentation I\'ve seen so far.\n\nA different, more recent, area is neutrino oscillations.\nBlasone et al have shown that the Fock space of\ndefinite flavour states is unitarily inequivalent to\nthat definite mass states. See, for example,\nhep-ph/9501263, and also the review article by\nCapolupo: hep-th/0408228. This means that to\nunderstand the QFT of neutrino oscillations fully,\nwe need to understand UIRs and disjoint Fock\nspaces.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">&nbsp;&nbsp;View this Usenet post in original ASCII form </a></div><P></jabberwocky>Eugene Stefanovich wrote:

> 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.

Not at all. They arise because the Fock space construction
must arbitrarily restrict to finite particle numbers in order for
an inner product to exist. This is explained in Umezawa (below).

> Could you give one good example where these things are
> necessary for understanding real physical phenomena.

I'll give you several...

They arise in condensed matter physics, i.e: bose-einstein
condensation, superfluidity and superconductivity. In textbooks
on these subjects, look for "Bogoliubov transformation" and
in most cases you'll find UIRs lurking, although the textbooks
don't always bring out this point explicitly. Umezawa's text on
"Thermofield Dynamics and Condensed States" gives the
most understandable presentation I've seen so far.

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,
http://www.arxiv.org/abs/hep-ph/9501263, and also the review article by
Capolupo: http://www.arxiv.org/abs/hep-th/0408228. This means that to
understand the QFT of neutrino oscillations fully,
we need to understand UIRs and disjoint Fock
spaces.



Eugene Stefanovich wrote: > wrote in message > news:1126839691.917142.163680@g43g20...egroups.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



Eugene Stefanovich wrote: > [...] > 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. I presume you meant Fock "space". :-) > This wouldn't be the case even if all components of $|0'>$ > in the old basis were "zero" in the limit of infinite volume. Actually, it would. I.e: it means that the new vacuum cannot be expressed as a mathematically meaningful superposition of vectors in the old Fock space. Unfortunately, I have to go away for a couple of days, so I can't respond more thoroughly right now. But I promise to do so when I return. For now, I'll just make a few general remarks: It is probably true that *all* the vectors in the new Fock space are orthogonal to *every* vector in the old Fock space, (though I haven't explicitly checked this for the Blasone-Vitiello - I should remedy that). However, Umezawa contains the essence of such a calculation, though he leaves it to the reader to fill in a fair bit of detail. Do you have access to a copy of Umezawa? It explains a lot of the theory of UIR much better than I can. The basic idea is that every new vector is orthogonal to every old vector. Therefore, none of the new vectors can be expressed as superpositions of the old vectors. That's essentially what defines a UIR. I'll follow up on this, and your 2nd question, more fully in a few days time.



mikem@despammed.com wrote: > Eugene Stefanovich wrote: > > > [...] > > 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. > > I presume you meant Fock "space". :-) That's right. Sorry. > > > This wouldn't be the case even if all components of $|0'>$ > > in the old basis were "zero" in the limit of infinite volume. > > Actually, it would. I.e: it means that the new vacuum > cannot be expressed as a mathematically meaningful > superposition of vectors in the old Fock space. I doubt that. > It is probably true that *all* the vectors in the new Fock space > are orthogonal to *every* vector in the old Fock space, > (though I haven't explicitly checked this for the > Blasone-Vitiello - I should remedy that). However, Umezawa > contains the essence of such a calculation, though he > leaves it to the reader to fill in a fair bit of detail. Do you > have access to a copy of Umezawa? I couldn't find this book in my usual library. I'll try another place later. > It explains a lot of > the theory of UIR much better than I can. The basic idea > is that every new vector is orthogonal to every old vector. > Therefore, none of the new vectors can be expressed as > superpositions of the old vectors. That's essentially what > defines a UIR. I have seen similar discussions in many places. I don't have these sources with me now. If I remember well, the idea was to 1) apply a unitary tranformation to the vacuum vector $|0'> = U|0>$ 2) Find components of $|0'>$ in a basis 3) Observe that in some limit all these components tend to zero 4) Conclude that the vector $|0'>$ goes outside the original Hilbert space H in this limit. This seems unfair to me. In these examples, even if all components of $|0'>$ tend to zero, their number tends to infinity, and the sum of squares of the components $|0'>$ in any basis in H should remain 1. The unitary operator U is explicitly defined within the Hilbert space H, so it is beyond me how it can bring any vector outside of H. In my opinion, mathematicians overemphasise the subtle differences between finite-dimensional and infinite-dimensional Hilbert spaces. I am not convinced that these differences have any physical meaning. Though, this could be just my ignorance. Eugene. > > I'll follow up on this, and your 2nd question, more fully > in a few days time. Have a nice trip. I look to hear from you soon. Eugene.



Eugene Stefanovich wrote: > 1. In my opinion [the Blasone-Vitiello] 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'm not quite sure what you mean here. My textbooks say that $= \exp(ipx),$ meaning that a position eigenstate |x> is _not_ orthogonal to a momentum eigenstate |p>. But perhaps you meant something else? > 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 stared at this for a while, but I'm still unsure what you mean. Did you omit an "$\exp" and/or$ an Integral $and/or a "\phi"$ in your definition of U above? >> [...] The basic idea is that every new vector is orthogonal to every >> old vector. Therefore, none of the new vectors can be expressed as >> superpositions of the old vectors. That's essentially what defines a >> UIR. > [...] If I remember well, the idea was to > > 1) apply a unitary tranformation to the vacuum vector $|0'> = U|0>$ > 2) Find components of $|0'>$ in a basis > 3) Observe that in some limit all these components tend to zero > 4) Conclude that the vector $|0'>$ goes outside the original > Hilbert space H in this limit. > > This seems unfair to me. In these examples, even if all components of > $|0'>$ tend to zero, their number tends to infinity, and the sum of > squares of the components $|0'>$ in any basis in H should remain 1. I look at it this way: if we have a (continuously-parametrized, infinite) basis |b> for H, then any other vector |v> can be expressed as an integral superposition, whose coefficients are given by taking the inner product between |v> and each respective $|b>, i$.e: $$|v> =[/itex] Integral $db |b>$$ So if [itex]$ is for every b , the Integral must be 0, showing that |v> cannot be expressed as a superposition of $|b>'s$. This is quite different from the position/momentum case where $= \exp(ipx)$ which is non-zero. > The unitary operator U is explicitly defined within the Hilbert space > H, so it is beyond me how it can bring any vector outside of H. If one examines the U carefully, it is not really correct to say that it is explicitly defined "within" the Hilbert (Fock) space. A crucial step in the construction of Fock space is to restrict it to have only state vectors whose total particle number is finite. Without this restriction, one cannot define an inner product on the space, because the usual Riemann-Lebesgue integral calculus doesn't work: we can't approximate an arbitrary vector therein by a countable sum arbitrarily closely, as is necessary when defining integrals rigorously. Umezawa explains (some of) this. But if you can't get a copy, part of it appears in a summary I posted to spr back on 15-Dec-2004 in a thread titled "Degenerate vacua in QFT": http://www.lns.cornell.edu/spr/2004-12/msg0065860.html modulo some followup corrections by Arnold Neumaier. :-) The "U" maps vectors in the Fock space into other vectors in the larger non-separable space, of which the Fock space is but a subspace. The total particle number of those "other vectors" turns out to be infinite, proving that they lie outside Fock space, which by construction contains only vectors of *finite* total particle number.



mikem@despammed.com wrote: > > 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'm not quite sure what you mean here. My textbooks say that > $= \exp(ipx),$ meaning that a position eigenstate |x> is _not_ > orthogonal to a momentum eigenstate |p>. But perhaps you meant > something else? If you require that |p> is a normalized vector, then $$= N \exp(ipx)$$ where the normalization factor N is basically "zero". One can formally say that $N = 1/\sqrt(V)$ where V is the "volume of space", i.e., infinity. The probability of finding definite momentum particle in each finite volume W is $W/V,$ which is "zero". However, this does not mean that this state is "outside" of the Hilbert space. The probability of finding the particle "somewhere" (i.e. the intergal over V) is $V/V = 1$. > > > 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 stared at this for a while, but I'm still unsure what you mean. > Did you omit an "$\exp" and/or$ an Integral $and/or a "\phi"$ in > your definition of U above? Sorry, I should have been more specific. The full definition of U is: 1) $U = 1$ on the vacuum vector |0> $2) U = a_v* a_1 + a_u* a_2$ on one-particle subspaces $H_1 (+) H_2 = H_v (+) H_u3) U =$ whatever on the rest of the Fock space. > > >> [...] The basic idea is that every new vector is orthogonal to > every > >> old vector. Therefore, none of the new vectors can be expressed as > >> superpositions of the old vectors. That's essentially what defines > a > >> UIR. > > > [...] If I remember well, the idea was to > > > > 1) apply a unitary tranformation to the vacuum vector $|0'> = U|0>$ > > 2) Find components of $|0'>$ in a basis > > 3) Observe that in some limit all these components tend to zero > > 4) Conclude that the vector $|0'>$ goes outside the original > > Hilbert space H in this limit. > > > > This seems unfair to me. In these examples, even if all components > of > > $|0'>$ tend to zero, their number tends to infinity, and the sum of > > squares of the components $|0'>$ in any basis in H should remain 1. > > I look at it this way: if we have a (continuously-parametrized, > infinite) basis |b> for H, then any other vector |v> can be expressed > as an integral superposition, whose coefficients are given by taking > the inner product between |v> and each respective $|b>, i$.e: > > $|v> =$ Integral $db |b>$ > > So if  is for every b , the Integral must be 0, showing that |v> > cannot be expressed as a superposition of $|b>'s$. This is quite > different from the position/momentum case where $= \exp(ipx)$ which > is non-zero. There is full analogy. If we use (as we should) normalized vectors for |x> and $|p>,$ then we obtain $$= 1/\sqrt(V) \exp(ipx)|p> =[/itex] Integral $dx |x>= 1/\sqrt(V)$ Integral $dx \exp(ipx) |x>$$ The wave function (the density of the probability amplitude) of |p> in the position representation must be $$1/\sqrt(V) \exp(ipx)$$ because the volume integral of its square is Integral [itex]dx |1/\sqrt(V) \exp(ipx)|^2 = 1/V$ Integral dx = 1 as it should. The function $\exp(ipx)$ without the normalization factor $1/\sqrt(V)$ does not have a probabilistic interpretation, because the volume integral of its square is infinite. > > The unitary operator U is explicitly defined within the Hilbert > space > > H, so it is beyond me how it can bring any vector outside of H. > > If one examines the U carefully, it is not really correct to say > that it is explicitly defined "within" the Hilbert (Fock) space. A > crucial step in the construction of Fock space is to restrict it > to have only state vectors whose total particle number is finite. > Without this restriction, one cannot define an inner product on the > space, because the usual Riemann-Lebesgue integral calculus doesn't > work: we can't approximate an arbitrary vector therein by a countable > sum arbitrarily closely, as is necessary when defining integrals > rigorously. Umezawa explains (some of) this. But if you can't get a > copy, part of it appears in a summary I posted to spr back on > 15-Dec-2004 in a thread titled "Degenerate vacua in QFT": > > http://www.lns.cornell.edu/spr/2004-12/msg0065860.html > > modulo some followup corrections by Arnold Neumaier. :-) > > The "U" maps vectors in the Fock space into other vectors in the > larger non-separable space, of which the Fock space is but a > subspace. The total particle number of those "other vectors" turns > out to be infinite, proving that they lie outside Fock space, which > by construction contains only vectors of *finite* total particle > number. Thank you for the reference. I've seen similar arguments in other places, but they do not make much sense to me. I have a strong feeling that the distinction between separable and non-separable Hilbert spaces was invented by mathematicians to make life of physicists miserable. 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. 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? I have thought about these issues and came to a conclusion that there could be some non-trivial math involved, but it has no significance to physics. Again, I am not talking about superconductivity and spontaneously broken vacuum symmetries - the issues I'm not so familiar with. Maybe there is deep physical truth concerning UIR in these fields, I just don't know. For myself, I found another more comfortable attitude to these issues. This attitude is not frequently discussed in physics literature, but I found it rather illuminating. This is based on the so-called "non-standard analysis" first developed by A. Robinson in 1960. This is too vast a field to be described in one post, but the basic idea is to treat finite, "infinitely small" and "infinitely large" quantities on the same footing. In this approach, the numbers like $1/\sqrt(V),$ where V is the volume of the entire space, make perfect sense, and there is no trouble to calculate the integral Integral $dx |1/\sqrt(V) \exp(ipx)|^2 = 1$ even if the integrand is "zero" everywhere. There are few papers which try to apply this approach to quantum mechanics. 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 (he was my student back in old times). The non-standard analysis is now a well-established branch of mathematics. I think, its use in QM may clarify some conceptual issues, but I don't expect any physical discoveries there. Eugene.



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.