Igor Khavkine
May26-04, 05:12 AM
<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>\n\nI am trying to come to grips with bosonization. This is an equivalence\nbetween some fermionic and bosonic field theories in 1+1 dimensions.\nA precise example from relativistic field theory is the Thirring model,\nwhich is equivalent to the sine-Gordon model. Another example from\ncondensed matter theory is the Luttinger model (a fudged up 1D electron\ngas), which is equivalent to the 1D bosonic field.\n\nI think I\'ve got a fair understanding of the concepts and the\ntricks involved in the calculation, but I would like to clear up some\ntechnical issues.\n\nConceptually, paired fermions correspond to bosonic excitations, while\nkink-like solitons in the bosonic theory correspond to fermions.\n\nThe technically tricky part is constructing the soliton creation and\nannihilation operators from the boson field operators. The idea is\nto decompose the boson field (in the Heisenberg picture) into its\nleft and right moving components\n\nphi(x,t) = phi_L(x+vt) + phi_R(x-vt) .\n\nThe same can be done with fermion field operators psi_L and psi_R. Then,\nset psi_{L,R} ~ exp(i a phi_{L,R}), where the proportionality constant and\na are determined by requiring that psi_{L,R} satisfy the proper\nanti-commuration relations. However, to avoid infinities, the exponential\nmust be normal ordered (all creation operators are manually moved to the\nleft the annihilation operators to the right).\n\nIt is fairly easy to show that as constructed, the fermion operators\npsi(x) and psi*(x\') (* is read dagger) anticommute at equal times\nwhen x != x\'. However to show that\n\n{psi(x),psi*(x\')} = delta(x-x\'), (*)\n\nall the references I\'ve looked at claim that it is equivalent to show\nthat [n(x),psi(y)] = delta(x-y) psi(x), where n(x) = psi*(x)psi(x).\nWhy is the last equality equivalent to (*)? Is it possible to show that\n(*) holds directly?\n\nI\'m also confused about the issue of normal ordering. I see that it is\nnecessary to remove singularities that otherwise appear in products of\noperators at nearby spacetime points. I would be happy if normal\nordering appeared only in the definition of the constructed fermion\nfield operators. However, it seems that all operators built out of the\nfermions need to be further normal ordered. I would imagine that if I\ncould identify the ground states of the fermion and boson theories, as\nwell as build fermion operators with the right anticommutation relations,\nthen I would be able to construct the fermion Fock space and re-express\nall operators in the fermion theory using bosons. So how does normal\nordering fit into this picture?\n\nAlso, a good reference would also be appreciated. The references I\'ve\nfound were helpful to some degree, but their expositions are not entirely\nclear to me. But this may be because they assume some background knowledge\nthat I am not aware of.\n\nThanks in advance.\n\nIgor\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"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>I am trying to come to grips with bosonization. This is an equivalence
between some fermionic and bosonic field theories in 1+1 dimensions.
A precise example from relativistic field theory is the Thirring model,
which is equivalent to the sine-Gordon model. Another example from
condensed matter theory is the Luttinger model (a fudged up 1D electron
gas), which is equivalent to the 1D bosonic field.
I think I've got a fair understanding of the concepts and the
tricks involved in the calculation, but I would like to clear up some
technical issues.
Conceptually, paired fermions correspond to bosonic excitations, while
kink-like solitons in the bosonic theory correspond to fermions.
The technically tricky part is constructing the soliton creation and
annihilation operators from the boson field operators. The idea is
to decompose the boson field (in the Heisenberg picture) into its
left and right moving components
\phi(x,t) = \phi_L(x+vt) + \phi_R(x-vt) .
The same can be done with fermion field operators \psi_L and \psi_R. Then,
set \psi_{L,R} ~ \exp(i a \phi_{L,R}), where the proportionality constant and
a are determined by requiring that \psi_{L,R} satisfy the proper
anti-commuration relations. However, to avoid infinities, the exponential
must be normal ordered (all creation operators are manually moved to the
left the annihilation operators to the right).
It is fairly easy to show that as constructed, the fermion operators
\psi(x) and \psi*(x') (* is read dagger) anticommute at equal times
when x != x'. However to show that
{\psi(x),\psi*(x')} = \delta(x-x'), (*)
all the references I've looked at claim that it is equivalent to show
that [n(x),\psi(y)] = \delta(x-y) \psi(x), where n(x) = \psi*(x)\psi(x).
Why is the last equality equivalent to (*)? Is it possible to show that
(*) holds directly?
I'm also confused about the issue of normal ordering. I see that it is
necessary to remove singularities that otherwise appear in products of
operators at nearby spacetime points. I would be happy if normal
ordering appeared only in the definition of the constructed fermion
field operators. However, it seems that all operators built out of the
fermions need to be further normal ordered. I would imagine that if I
could identify the ground states of the fermion and boson theories, as
well as build fermion operators with the right anticommutation relations,
then I would be able to construct the fermion Fock space and re-express
all operators in the fermion theory using bosons. So how does normal
ordering fit into this picture?
Also, a good reference would also be appreciated. The references I've
found were helpful to some degree, but their expositions are not entirely
clear to me. But this may be because they assume some background knowledge
that I am not aware of.
Thanks in advance.
Igor
between some fermionic and bosonic field theories in 1+1 dimensions.
A precise example from relativistic field theory is the Thirring model,
which is equivalent to the sine-Gordon model. Another example from
condensed matter theory is the Luttinger model (a fudged up 1D electron
gas), which is equivalent to the 1D bosonic field.
I think I've got a fair understanding of the concepts and the
tricks involved in the calculation, but I would like to clear up some
technical issues.
Conceptually, paired fermions correspond to bosonic excitations, while
kink-like solitons in the bosonic theory correspond to fermions.
The technically tricky part is constructing the soliton creation and
annihilation operators from the boson field operators. The idea is
to decompose the boson field (in the Heisenberg picture) into its
left and right moving components
\phi(x,t) = \phi_L(x+vt) + \phi_R(x-vt) .
The same can be done with fermion field operators \psi_L and \psi_R. Then,
set \psi_{L,R} ~ \exp(i a \phi_{L,R}), where the proportionality constant and
a are determined by requiring that \psi_{L,R} satisfy the proper
anti-commuration relations. However, to avoid infinities, the exponential
must be normal ordered (all creation operators are manually moved to the
left the annihilation operators to the right).
It is fairly easy to show that as constructed, the fermion operators
\psi(x) and \psi*(x') (* is read dagger) anticommute at equal times
when x != x'. However to show that
{\psi(x),\psi*(x')} = \delta(x-x'), (*)
all the references I've looked at claim that it is equivalent to show
that [n(x),\psi(y)] = \delta(x-y) \psi(x), where n(x) = \psi*(x)\psi(x).
Why is the last equality equivalent to (*)? Is it possible to show that
(*) holds directly?
I'm also confused about the issue of normal ordering. I see that it is
necessary to remove singularities that otherwise appear in products of
operators at nearby spacetime points. I would be happy if normal
ordering appeared only in the definition of the constructed fermion
field operators. However, it seems that all operators built out of the
fermions need to be further normal ordered. I would imagine that if I
could identify the ground states of the fermion and boson theories, as
well as build fermion operators with the right anticommutation relations,
then I would be able to construct the fermion Fock space and re-express
all operators in the fermion theory using bosons. So how does normal
ordering fit into this picture?
Also, a good reference would also be appreciated. The references I've
found were helpful to some degree, but their expositions are not entirely
clear to me. But this may be because they assume some background knowledge
that I am not aware of.
Thanks in advance.
Igor