Miguel Carrion
Apr23-04, 03:22 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> [040421 08:32]:\n>\n> One more thing, what\'s the energy expression for a free klein gordon\n> field? Is it just\n>\n> E= int (dphi/dt)^2 dx + int(dphi/dx)^2 dx,\n\nThe energy of a Klein Gordon field configuration is\n\nE = {1/2} int [ (phi\')^2 + (d phi)^2 + m^2 phi^2 ] dx\n\nWhere the integral is extended over space, \' is time derivative and d\nis space derivative.\n\nThe kinetic energy is just\n\nT = {1/2} int (phi\')^2\n\nand the rest must be potential energy, mustn\'t it?, since it involves\nno time derivatives:\n\nV = {1/2} int [ (d phi)^2 + m^2 phi^2 ] dx\n\nThe first term is a sort of "elastic energy", and the second is a\npotential energy based solely on the value of phi at each point.\n\n> see, because in that case I\'m surprised that you\'d want to use phi\n> in L^2(R) as your config space since according to the energy\n> expression, the only condition is that dphi/dx be in L^2 not the phi\n> itself.\n\nActually, given what I said\n\nIn article <c622el\\$dfn\\$1@glue.ucr.edu>,\nMiguel Carrion <miguel@math-cl-n01.math.ucr.edu> wrote:\n>* 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>\n>as long as the kinetic energy is quadratic the configuration space of\n>a scalar field under the mexican hat potential is the same as the\n>configuration space of the Klein-Gordon field.\n\nIt turns out that the configuration space would have to be just L^2 of\nspace, with unrestricted derivatives! The kinetic energy not only\ndetermines the topology of the configuration space, but also the\n"canonically conjugate" variables, in this case phi\', and the poisson\nbracket or, equivalently, the symplectic structure\n\nomega(phi,psi)= int [ phi psi\' - phi\' psi ] dx\n\nSo the phase space would consist of functions phi in L^2 with time\nderivative phi\' also in L^2.\n\nWhat happens when you turn on a potential is that part of the\nconfiguration space gets cut out, because it consists of\ninfinite-energy configurations. In classical mechanics it seems that\nthis rarely or never happens, but the difference is that usually there\nis an open set of finite-energy points in configuration space. In the\nfield case, the finite-energy solutions tend to form a dense set with\nempty interior.\n\nFor the Klein-Gordon field, the finite-energy configuration space\nwould be the dense subspace consisting of phi with L^2 space\nderivative. Note, first, that the "mass" term is already finite,\ncourtesy of the kinetic energy. Also, the finite-energy configuration\nspace is a dense subspace of the "free" (zero potential) configuration\nspace. It has its own Hilbert space topology, but as a subspace of the\n"kinematical" configuration space it is a dense subspace and is not\ncomplete.\n\nTo sum up:\n\nFor scalar field theories with quadratic kinetic energy, the\n"kinematical configuration space" is L^2 on space.\n\nThe "dynamical configuration space" consists of the finite-energy\npoints of configuration space, which is a dense subset of the\nkinematical configuration space.\n\nIf the potential energy is a quadratic form, the dynamical\nconfiguration space is a Hilbert space. Otherwise, as in phi^4 theory\nor the mexican hat potential, the finite energy configuration space\nmay still be a vector space, but it won\'t be Hilbert.\n\nWhen one quantizes the Klein-Gordon field, an additional (this time\ncomplex) Hilbert space shows up.\n\nI suspect most of these nuances are unphysical, and physicists are\nseldom careful about them anyway.\n\nRegards,\n\nMiguel\n\n\n\n\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> [040421 08:32]:
>
> One more thing, what's the energy expression for a free klein gordon
> field? Is it just
>
> E= \int (dphi/dt)^2 dx + \int(dphi/dx)^2 dx,
The energy of a Klein Gordon field configuration is
E = {1/2} \int [ (\phi')^2 + (d \phi)^2 + m^2 \phi^2 ] dx
Where the integral is extended over space, ' is time derivative and d
is space derivative.
The kinetic energy is just
T = {1/2} \int (\phi')^2
and the rest must be potential energy, mustn't it?, since it involves
no time derivatives:
V = {1/2} \int [ (d \phi)^2 + m^2 \phi^2 ] dx
The first term is a sort of "elastic energy", and the second is a
potential energy based solely on the value of \phi at each point.
> see, because in that case I'm surprised that you'd want to use \phi
> in L^2(R) as your config space since according to the energy
> expression, the only condition is that dphi/dx be in L^2 not the \phi
> itself.
Actually, given what I said
In article <c622el$dfn$1@glue.ucr.edu>,
Miguel Carrion <miguel@math-cl-n01.math.ucr.edu> wrote:
>* 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?
>
>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.
It turns out that the configuration space would have to be just L^2 of
space, with unrestricted derivatives! The kinetic energy not only
determines the topology of the configuration space, but also the
"canonically conjugate" variables, in this case \phi', and the poisson
bracket or, equivalently, the symplectic structure
\omega(\phi,\psi)= \int [ \phi \psi' - \phi' \psi ] dx
So the phase space would consist of functions \phi in L^2 with time
derivative \phi' also in L^2.
What happens when you turn on a potential is that part of the
configuration space gets cut out, because it consists of
infinite-energy configurations. In classical mechanics it seems that
this rarely or never happens, but the difference is that usually there
is an open set of finite-energy points in configuration space. In the
field case, the finite-energy solutions tend to form a dense set with
empty interior.
For the Klein-Gordon field, the finite-energy configuration space
would be the dense subspace consisting of \phi with L^2 space
derivative. Note, first, that the "mass" term is already finite,
courtesy of the kinetic energy. Also, the finite-energy configuration
space is a dense subspace of the "free" (zero potential) configuration
space. It has its own Hilbert space topology, but as a subspace of the
"kinematical" configuration space it is a dense subspace and is not
complete.
To sum up:
For scalar field theories with quadratic kinetic energy, the
"kinematical configuration space" is L^2 on space.
The "dynamical configuration space" consists of the finite-energy
points of configuration space, which is a dense subset of the
kinematical configuration space.
If the potential energy is a quadratic form, the dynamical
configuration space is a Hilbert space. Otherwise, as in \phi^4 theory
or the mexican hat potential, the finite energy configuration space
may still be a vector space, but it won't be Hilbert.
When one quantizes the Klein-Gordon field, an additional (this time
complex) Hilbert space shows up.
I suspect most of these nuances are unphysical, and physicists are
seldom careful about them anyway.
Regards,
Miguel
>
> One more thing, what's the energy expression for a free klein gordon
> field? Is it just
>
> E= \int (dphi/dt)^2 dx + \int(dphi/dx)^2 dx,
The energy of a Klein Gordon field configuration is
E = {1/2} \int [ (\phi')^2 + (d \phi)^2 + m^2 \phi^2 ] dx
Where the integral is extended over space, ' is time derivative and d
is space derivative.
The kinetic energy is just
T = {1/2} \int (\phi')^2
and the rest must be potential energy, mustn't it?, since it involves
no time derivatives:
V = {1/2} \int [ (d \phi)^2 + m^2 \phi^2 ] dx
The first term is a sort of "elastic energy", and the second is a
potential energy based solely on the value of \phi at each point.
> see, because in that case I'm surprised that you'd want to use \phi
> in L^2(R) as your config space since according to the energy
> expression, the only condition is that dphi/dx be in L^2 not the \phi
> itself.
Actually, given what I said
In article <c622el$dfn$1@glue.ucr.edu>,
Miguel Carrion <miguel@math-cl-n01.math.ucr.edu> wrote:
>* 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?
>
>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.
It turns out that the configuration space would have to be just L^2 of
space, with unrestricted derivatives! The kinetic energy not only
determines the topology of the configuration space, but also the
"canonically conjugate" variables, in this case \phi', and the poisson
bracket or, equivalently, the symplectic structure
\omega(\phi,\psi)= \int [ \phi \psi' - \phi' \psi ] dx
So the phase space would consist of functions \phi in L^2 with time
derivative \phi' also in L^2.
What happens when you turn on a potential is that part of the
configuration space gets cut out, because it consists of
infinite-energy configurations. In classical mechanics it seems that
this rarely or never happens, but the difference is that usually there
is an open set of finite-energy points in configuration space. In the
field case, the finite-energy solutions tend to form a dense set with
empty interior.
For the Klein-Gordon field, the finite-energy configuration space
would be the dense subspace consisting of \phi with L^2 space
derivative. Note, first, that the "mass" term is already finite,
courtesy of the kinetic energy. Also, the finite-energy configuration
space is a dense subspace of the "free" (zero potential) configuration
space. It has its own Hilbert space topology, but as a subspace of the
"kinematical" configuration space it is a dense subspace and is not
complete.
To sum up:
For scalar field theories with quadratic kinetic energy, the
"kinematical configuration space" is L^2 on space.
The "dynamical configuration space" consists of the finite-energy
points of configuration space, which is a dense subset of the
kinematical configuration space.
If the potential energy is a quadratic form, the dynamical
configuration space is a Hilbert space. Otherwise, as in \phi^4 theory
or the mexican hat potential, the finite energy configuration space
may still be a vector space, but it won't be Hilbert.
When one quantizes the Klein-Gordon field, an additional (this time
complex) Hilbert space shows up.
I suspect most of these nuances are unphysical, and physicists are
seldom careful about them anyway.
Regards,
Miguel