Miguel Carrion
Apr22-04, 02:39 PM
<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>* collin collin <ultraman2002@hotmail.com> [040415 19:18]:\n>\n> Do you know for a field under the mexican hat potential\n> what is the space of fields used as a configuration space\n> and what is the topology on it?\n\nWhat follows is just a long-winded and pedantic way to say I have no\nidea, so feel free to ignore it ;-)\n\nIf we were dealing with a system with finitely many degrees of\nfreedom, the configuration space would be essentially independent of\nthe potential, and its geometry and topology would be determined by\n(or would determine) the kinetic energy. The naive treatment of\nquantum fields assumes that the techniques that work for ordinary\nmechanical systems also work for continuum systems, and so as long as\nthe kinetic energy is quadratic the configuration space of a scalar\nfield under the mexican hat potential is the same as the configuration\nspace of the Klein-Gordon field. For this I would probably use the\nHilbert space of square-integrable scalar functions (on space) with\nsquare-integrable gradients. On a general spacetime you can basically\nchoose any spacelike surface as your "space".\n\nThe problem is that this approach leads to perturbative quantization,\nwhich is based on the assumption that the ground state(s) of the\ninteracting (nonlinear) theory are in the Fock space of the\nnon-interacting (linear) theory with a quadratic potential, which is\nknown to be false.\n\nAlready in classical mechanics there is a view of the Phase space\nalternative to "the cotangent bundle of the configuration space", and\nthat is that the Phase space is the solution manifold of the equations\nof motion. The cotangent bundle of the "configuration space" is just\none convenient coordinatization of Phase space. The problem with this\nview is that, for nonlinear equations of motion, it leads to phase\nspaces which are not vector spaces. In the case of the scalar field,\nthis means that we should expect the phase space of the classical\nmexican-hat potential to be a nonlinear infinite-dimensional manifold,\nand the relation of this to Fock space, if there is any, is at best\nobscure to me.\n\nRegards,\n\nMiguel\n\n\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"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>* collin collin <ultraman2002@hotmail.com> [040415 19:18]:
>
> Do you know for a field under the mexican hat potential
> what is the space of fields used as a configuration space
> and what is the topology on it?
What follows is just a long-winded and pedantic way to say I have no
idea, so feel free to ignore it ;-)
If we were dealing with a system with finitely many degrees of
freedom, the configuration space would be essentially independent of
the potential, and its geometry and topology would be determined by
(or would determine) the kinetic energy. The naive treatment of
quantum fields assumes that the techniques that work for ordinary
mechanical systems also work for continuum systems, and so as long as
the kinetic energy is quadratic the configuration space of a scalar
field under the mexican hat potential is the same as the configuration
space of the Klein-Gordon field. For this I would probably use the
Hilbert space of square-integrable scalar functions (on space) with
square-integrable gradients. On a general spacetime you can basically
choose any spacelike surface as your "space".
The problem is that this approach leads to perturbative quantization,
which is based on the assumption that the ground state(s) of the
interacting (nonlinear) theory are in the Fock space of the
non-interacting (linear) theory with a quadratic potential, which is
known to be false.
Already in classical mechanics there is a view of the Phase space
alternative to "the cotangent bundle of the configuration space", and
that is that the Phase space is the solution manifold of the equations
of motion. The cotangent bundle of the "configuration space" is just
one convenient coordinatization of Phase space. The problem with this
view is that, for nonlinear equations of motion, it leads to phase
spaces which are not vector spaces. In the case of the scalar field,
this means that we should expect the phase space of the classical
mexican-hat potential to be a nonlinear infinite-dimensional manifold,
and the relation of this to Fock space, if there is any, is at best
obscure to me.
Regards,
Miguel
>
> Do you know for a field under the mexican hat potential
> what is the space of fields used as a configuration space
> and what is the topology on it?
What follows is just a long-winded and pedantic way to say I have no
idea, so feel free to ignore it ;-)
If we were dealing with a system with finitely many degrees of
freedom, the configuration space would be essentially independent of
the potential, and its geometry and topology would be determined by
(or would determine) the kinetic energy. The naive treatment of
quantum fields assumes that the techniques that work for ordinary
mechanical systems also work for continuum systems, and so as long as
the kinetic energy is quadratic the configuration space of a scalar
field under the mexican hat potential is the same as the configuration
space of the Klein-Gordon field. For this I would probably use the
Hilbert space of square-integrable scalar functions (on space) with
square-integrable gradients. On a general spacetime you can basically
choose any spacelike surface as your "space".
The problem is that this approach leads to perturbative quantization,
which is based on the assumption that the ground state(s) of the
interacting (nonlinear) theory are in the Fock space of the
non-interacting (linear) theory with a quadratic potential, which is
known to be false.
Already in classical mechanics there is a view of the Phase space
alternative to "the cotangent bundle of the configuration space", and
that is that the Phase space is the solution manifold of the equations
of motion. The cotangent bundle of the "configuration space" is just
one convenient coordinatization of Phase space. The problem with this
view is that, for nonlinear equations of motion, it leads to phase
spaces which are not vector spaces. In the case of the scalar field,
this means that we should expect the phase space of the classical
mexican-hat potential to be a nonlinear infinite-dimensional manifold,
and the relation of this to Fock space, if there is any, is at best
obscure to me.
Regards,
Miguel