Evaluation of the QFT generating functional [Archive]

Martin Lohmann

Apr24-04, 08:16 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>The functional approach to quantum field theory often uses a\ngenerating functional from which one can obtain all the interesting\nquantities such as n-point-functions and, through the reduction\nformula, S-matrix-elements. The generating functional of an\ninteracting field with source J is given by the formula (which, I\nhope, is not too unreadable):\n\n/ 4\ni| L (-i d_J) d x\n/ int\nZ[J] = e Z [J]\n0\n\nwhere d_J is the functional derivative with respect to J, L_int is the\ninteraction Lagrangian and Z0[J] is the generating functional for the\nnoninteracting field, which is a path integral and can often be\nevaluated exactly because the kinetic terms in Lagrangian densities\nare usually of Gaussian type.\nNow, one mostly evaluates the generating functional for interacting\nfields by power-expanding the operator on the right-hand side of the\nabove formula, but I heared that there are also situations where one\ncan evaluate the functional exactly. These situations are where the\ninteraction lagrangian is of the form\n\nL = psi A(x,y) psi\nint\n\nwhere psi is the field associated to the source J and A is a function\nsymmetric in x and y.\n\nUnfortunately, I was not able to find much literature about the\nevaluation of the generating functional, because most textbooks on QFT\ntreat functional methods only very briefly. Can anyone provide me with\ngood references about the evaluation of the generating functional? I\nam particularly interested in exact evaluations and, consequently, in\na theory of the operator on the right-hand side of the above formula\n(if such a theory exists).\n\nThanks in advance, Martin Lohmann\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>The functional approach to quantum field theory often uses a
generating functional from which one can obtain all the interesting
quantities such as n-point-functions and, through the reduction
formula, S-matrix-elements. The generating functional of an
interacting field with source J is given by the formula (which, I
hope, is not too unreadable):

/ 4i| L (-i d_J) d x/ \int[/itex]
Z[J] = e Z [J]

where d_J is the functional derivative with respect to J, L_{int} is the
interaction Lagrangian and Z0[J] is the generating functional for the
noninteracting field, which is a path integral and can often be
evaluated exactly because the kinetic terms in Lagrangian densities
are usually of Gaussian type.
Now, one mostly evaluates the generating functional for interacting
fields by power-expanding the operator on the right-hand side of the
above formula, but I heared that there are also situations where one
can evaluate the functional exactly. These situations are where the
interaction lagrangian is of the form

L [itex]= \psi A(x,y) \psi\int

where \psi is the field associated to the source J and A is a function
symmetric in x and y.

Unfortunately, I was not able to find much literature about the
evaluation of the generating functional, because most textbooks on QFT
treat functional methods only very briefly. Can anyone provide me with
good references about the evaluation of the generating functional? I
am particularly interested in exact evaluations and, consequently, in
a theory of the operator on the right-hand side of the above formula
(if such a theory exists).

Thanks in advance, Martin Lohmann

Arnold Neumaier

Apr28-04, 01:46 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>Martin Lohmann wrote:\n> The functional approach to quantum field theory often uses a\n> generating functional from which one can obtain all the interesting\n> quantities such as n-point-functions and, through the reduction\n> formula, S-matrix-elements. The generating functional of an\n> interacting field with source J is given by the formula (which, I\n> hope, is not too unreadable):\n>\n> / 4\n> i| L (-i d_J) d x\n> / int\n> Z[J] = e Z [J]\n> 0\n\nIt is almost unreadable with viewers such as mine that have proportional\nfonts. It is far better to write integrals and indices in a simplified\nlatex style, such as\nintegral L_int dx^4\n\n\n> Now, one mostly evaluates the generating functional for interacting\n> fields by power-expanding the operator on the right-hand side of the\n> above formula, but I heared that there are also situations where one\n> can evaluate the functional exactly.\n\nOne can evaluate the integrals exactly if the exponent is at most\nquadratic in all fields, and the quadratic form is nondegenerate.\nThen one can complete ther square and apply a gaussian formula.\n\n\nArnold Neumaier\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>Martin Lohmann wrote:
> The functional approach to quantum field theory often uses a
> generating functional from which one can obtain all the interesting
> quantities such as n-point-functions and, through the reduction
> formula, S-matrix-elements. The generating functional of an
> interacting field with source J is given by the formula (which, I
> hope, is not too unreadable):
>
> / 4
> i| L (-i d_J) d x
> / \int
> Z[J] = e Z [J]
>

It is almost unreadable with viewers such as mine that have proportional
fonts. It is far better to write integrals and indices in a simplified
latex style, such as
integral L_{int} dx^4

> Now, one mostly evaluates the generating functional for interacting
> fields by power-expanding the operator on the right-hand side of the
> above formula, but I heared that there are also situations where one
> can evaluate the functional exactly.

One can evaluate the integrals exactly if the exponent is at most
quadratic in all fields, and the quadratic form is nondegenerate.
Then one can complete ther square and apply a gaussian formula.

Arnold Neumaier

Danny Ross Lunsford

Apr28-04, 02:27 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>Arnold Neumaier wrote:\n\n> It is almost unreadable with viewers such as mine that have proportional\n> fonts. It is far better to write integrals and indices in a simplified\n> latex style, such as\n> integral L_int dx^4\n\nFine, but you can use Mozilla 1.5 or better and get mono spacing for\nemail and news. For Linux, there are builds that include font\nanti-aliasing which makes beautiful reading. Windows XP has this\nbuilt-in. You can use any monospaced font. Good ones are Andale Mono and\nCrystal. Even Courier looks fine when anti-aliased.\n\nIf you have a laptop, a recent Linux distribution or Windows XP will\nallow you to use sub-pixel color antialiasing, which produces\ntype-quality fonts even on 1024x768. At higher resolutions the display\nis spectacular.\n\nI don\'t know Macs, but the "Quartz" UI has these features as well.\n\nI went through a lot of newsreaders before settling on Mozilla\'s\nbuilt-in. It\'s the right combination of features and simplicity.\n\nIt wouldn\'t be *that* hard to have a MathML enabled group. That would be\nvery nice!\n\n-danny\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>Arnold Neumaier wrote:

> It is almost unreadable with viewers such as mine that have proportional
> fonts. It is far better to write integrals and indices in a simplified
> latex style, such as
> integral L_{int} dx^4

Fine, but you can use Mozilla 1.5 or better and get mono spacing for
email and news. For Linux, there are builds that include font
anti-aliasing which makes beautiful reading. Windows XP has this
built-in. You can use any monospaced font. Good ones are Andale Mono and
Crystal. Even Courier looks fine when anti-aliased.

If you have a laptop, a recent Linux distribution or Windows XP will
allow you to use sub-pixel color antialiasing, which produces
type-quality fonts even on 1024x768. At higher resolutions the display
is spectacular.

I don't know Macs, but the "Quartz" UI has these features as well.

I went through a lot of newsreaders before settling on Mozilla's
built-in. It's the right combination of features and simplicity.

It wouldn't be *that* hard to have a MathML enabled group. That would be
very nice!

-danny

Martin Lohmann

Apr28-04, 02:28 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>> It is almost unreadable with viewers such as mine that have proportional\n> fonts. It is far better to write integrals and indices in a simplified\n> latex style, such as\n> integral L_int dx^4\n\nI am sorry about this. It reads:\n\nZ[J] = exp[i integral L_int(i d_J) d4x] Z0[J]\n\n\n\n\n> One can evaluate the integrals exactly if the exponent is at most\n> quadratic in all fields, and the quadratic form is nondegenerate.\n> Then one can complete ther square and apply a gaussian formula.\n\nThis is for the path integral. I know that one can evaluate the\ncorresponding path integral exactly if it is gaussian. But I wondered\nif there is a theory of operators like the exp[i integral L_int(i\nd_J) d4x] in the above formula. I did never come across mathematical\ntreatments of such functional differential operators, so I ask for\nreference.\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>> It is almost unreadable with viewers such as mine that have proportional
> fonts. It is far better to write integrals and indices in a simplified
> latex style, such as
> integral L_{int} dx^4

I am sorry about this. It reads:

Z[J] = \exp[i integral L_{int}(i d_J) d4x] Z0[J]

> One can evaluate the integrals exactly if the exponent is at most
> quadratic in all fields, and the quadratic form is nondegenerate.
> Then one can complete ther square and apply a gaussian formula.

This is for the path integral. I know that one can evaluate the
corresponding path integral exactly if it is gaussian. But I wondered
if there is a theory of operators like the \exp[i integral L_{int}(id_J) d4x] in the above formula. I did never come across mathematical
treatments of such functional differential operators, so I ask for
reference.

Arnold Neumaier

Apr29-04, 05:38 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>\nMartin Lohmann wrote:\n>>It is almost unreadable with viewers such as mine that have proportional\n>>fonts. It is far better to write integrals and indices in a simplified\n>>latex style, such as\n>> integral L_int dx^4\n>\n>\n> I am sorry about this. It reads:\n>\n> Z[J] = exp[i integral L_int(i d_J) d4x] Z0[J]\n>\n>\n>\n>\n>\n>>One can evaluate the integrals exactly if the exponent is at most\n>>quadratic in all fields, and the quadratic form is nondegenerate.\n>>Then one can complete ther square and apply a gaussian formula.\n>\n>\n> This is for the path integral. I know that one can evaluate the\n> corresponding path integral exactly if it is gaussian. But I wondered\n> if there is a theory of operators like the exp[i integral L_int(i\n> d_J) d4x] in the above formula. I did never come across mathematical\n> treatments of such functional differential operators, so I ask for\n> reference.\n\nI don\'t know if these things are well-defined mathematically,\nlittle about path integrals is. But on a formal level, an operator\nexp(X) where X is an operator is just the value U(1) of the solution\nof the differential equation Udot(t)=X U(t) with U(0)=1 (the identity).\nSo the mathematical problem would be to show that his initial value\nproblem has a unique solution for your choice of argument of the exp.\nThe conditions for this are given by the Hille-Yoshida theorem\n(see books on functional analysis or Vol 3 of Thirring\'s Math. Phys.\ntreatise); but I didn\'t check underwhich conditions on L these apply.\n\n\nArnold Neumaier\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>Martin Lohmann wrote:
>>It is almost unreadable with viewers such as mine that have proportional
>>fonts. It is far better to write integrals and indices in a simplified
>>latex style, such as
>> integral L_{int} dx^4
>
>
> I am sorry about this. It reads:
>
> Z[J] = \exp[i integral L_{int}(i d_J) d4x] Z0[J]
>
>
>
>
>
>>One can evaluate the integrals exactly if the exponent is at most
>>quadratic in all fields, and the quadratic form is nondegenerate.
>>Then one can complete ther square and apply a gaussian formula.
>
>
> This is for the path integral. I know that one can evaluate the
> corresponding path integral exactly if it is gaussian. But I wondered
> if there is a theory of operators like the \exp[i integral L_{int}(i
> d_J) d4x] in the above formula. I did never come across mathematical
> treatments of such functional differential operators, so I ask for
> reference.

I don't know if these things are well-defined mathematically,
little about path integrals is. But on a formal level, an operator
\exp(X) where X is an operator is just the value U(1) of the solution
of the differential equation Udot(t)=X U(t) with U(0)=1 (the identity).
So the mathematical problem would be to show that his initial value
problem has a unique solution for your choice of argument of the \exp.
The conditions for this are given by the Hille-Yoshida theorem
(see books on functional analysis or Vol 3 of Thirring's Math. Phys.
treatise); but I didn't check underwhich conditions on L these apply.

Arnold Neumaier

Martin Lohmann

Apr29-04, 12:47 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>\n\n> I don\'t know if these things are well-defined mathematically,\n> little about path integrals is. But on a formal level, an operator\n> exp(X) where X is an operator is just the value U(1) of the solution\n> of the differential equation Udot(t)=X U(t) with U(0)=1 (the identity).\n> So the mathematical problem would be to show that his initial value\n> problem has a unique solution for your choice of argument of the exp.\n> The conditions for this are given by the Hille-Yoshida theorem\n> (see books on functional analysis or Vol 3 of Thirring\'s Math. Phys.\n> treatise); but I didn\'t check underwhich conditions on L these apply.\n>\n>\n> Arnold Neumaier\n\nThank you for advise. The problem about books on functional analysis\nis that you get to know alot about such things like Banach-spaces\nwhich are not of much importance to theoretical physics because you\nnearly always work with Hilbert-spaces, so most treatments are much to\ngeneral. What would be very helpful is a volume that is concentrated\non differential or functional differential operators (the function\nspace L^2 is a Hilbert-space). I´ll look for Thirrings work, and,\nhopefully, I will find some reference to a more precise treatment of\nsuch operators later.\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 don't know if these things are well-defined mathematically,
> little about path integrals is. But on a formal level, an operator
> \exp(X) where X is an operator is just the value U(1) of the solution
> of the differential equation Udot(t)=X U(t) with U(0)=1 (the identity).
> So the mathematical problem would be to show that his initial value
> problem has a unique solution for your choice of argument of the \exp.
> The conditions for this are given by the Hille-Yoshida theorem
> (see books on functional analysis or Vol 3 of Thirring's Math. Phys.
> treatise); but I didn't check underwhich conditions on L these apply.
>
>
> Arnold Neumaier

Thank you for advise. The problem about books on functional analysis
is that you get to know alot about such things like Banach-spaces
which are not of much importance to theoretical physics because you
nearly always work with Hilbert-spaces, so most treatments are much to
general. What would be very helpful is a volume that is concentrated
on differential or functional differential operators (the function
space L^2 is a Hilbert-space). I´ll look for Thirrings work, and,
hopefully, I will find some reference to a more precise treatment of
such operators later.

Arnold Neumaier

Apr30-04, 10:36 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>Martin Lohmann wrote:\n>>I don\'t know if these things are well-defined mathematically,\n>>little about path integrals is. But on a formal level, an operator\n>>exp(X) where X is an operator is just the value U(1) of the solution\n>>of the differential equation Udot(t)=3DX U(t) with U(0)=3D1 (the identi=\nty).\n>>So the mathematical problem would be to show that his initial value\n>>problem has a unique solution for your choice of argument of the exp.\n>>The conditions for this are given by the Hille-Yoshida theorem\n>>(see books on functional analysis or Vol 3 of Thirring\'s Math. Phys.\n>>treatise); but I didn\'t check underwhich conditions on L these apply.\n>=20\n> Thank you for advise. The problem about books on functional analysis\n> is that you get to know alot about such things like Banach-spaces\n> which are not of much importance to theoretical physics because you\n> nearly always work with Hilbert-spaces, so most treatments are much to\n> general.=20\n\nBut still useful, since every Hilbert space is also a Banach space.\nYou can therefore read everything about Banach spaces as applying to\nHilbert spaces as well.\n\n\n> What would be very helpful is a volume that is concentrated\n> on differential or functional differential operators (the function\n> space L^2 is a Hilbert-space). I=B4ll look for Thirrings work, and,\n> hopefully, I will find some reference to a more precise treatment of\n> such operators later.\n\nIn the above case, you just need to look up Hille-Yoshida in the index;\nnot read everything about Banach spaces.\n\n\nMost of what is known rigorously about path integrals is in the\nbook by Glimm and Jaffe; but it takes time to get used to it\nif you come from physics...\n\n\nArnold Neumaier\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>Martin Lohmann wrote:
>>I don't know if these things are well-defined mathematically,
>>little about path integrals is. But on a formal level, an operator
>>\exp(X) where X is an operator is just the value U(1) of the solution
>>of the differential equation Udot(t)=3DX U(t) with U(0)=3D1 (the identi=
ty).
>>So the mathematical problem would be to show that his initial value
>>problem has a unique solution for your choice of argument of the \exp.
>>The conditions for this are given by the Hille-Yoshida theorem
>>(see books on functional analysis or Vol 3 of Thirring's Math. Phys.
>>treatise); but I didn't check underwhich conditions on L these apply.
>=20
> Thank you for advise. The problem about books on functional analysis
> is that you get to know alot about such things like Banach-spaces
> which are not of much importance to theoretical physics because you
> nearly always work with Hilbert-spaces, so most treatments are much to
> general.=20

But still useful, since every Hilbert space is also a Banach space.
You can therefore read everything about Banach spaces as applying to
Hilbert spaces as well.

> What would be very helpful is a volume that is concentrated
> on differential or functional differential operators (the function
> space L^2 is a Hilbert-space). I=B4ll look for Thirrings work, and,
> hopefully, I will find some reference to a more precise treatment of
> such operators later.

In the above case, you just need to look up Hille-Yoshida in the index;
not read everything about Banach spaces.

Most of what is known rigorously about path integrals is in the
book by Glimm and Jaffe; but it takes time to get used to it
if you come from physics...

Arnold Neumaier

Martin Lohmann

May3-04, 05:37 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>> But still useful, since every Hilbert space is also a Banach space.\n> You can therefore read everything about Banach spaces as applying to\n> Hilbert spaces as well.\n>\n>\nThats, of course, right, but the problem is, as I said, that things\nget to general. It would of course be a possibility to go that way,\nbut this means a lot of extra work, because banach spaces are not that\nwell-behaved as hilbert spaces. I dont really need generality because\nthe problem is very special: an exponential functional differential\noperator acting on some functional which is also not too special (also\nan exponential quadratic in the sources). The only difficulties will\narise from the fact that there is a differential operator in the\nexponent which needs a careful treatment. I am looking for this\ncareful treatment.\n\n\n\n> In the above case, you just need to look up Hille-Yoshida in the index;\n> not read everything about Banach spaces.\n\nI\'ll look for that, thanks.\n\n>\n>\n> Most of what is known rigorously about path integrals is in the\n> book by Glimm and Jaffe; but it takes time to get used to it\n> if you come from physics...\n>\nYes, I heared about this book, but they told me that it is neither\nsuitable for physicists (which I suppose I am) nor for mathematicians.\nBut lets see, maybe it is helpful.\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>> But still useful, since every Hilbert space is also a Banach space.
> You can therefore read everything about Banach spaces as applying to
> Hilbert spaces as well.
>
>
Thats, of course, right, but the problem is, as I said, that things
get to general. It would of course be a possibility to go that way,
but this means a lot of extra work, because banach spaces are not that
well-behaved as hilbert spaces. I dont really need generality because
the problem is very special: an exponential functional differential
operator acting on some functional which is also not too special (also
an exponential quadratic in the sources). The only difficulties will
arise from the fact that there is a differential operator in the
exponent which needs a careful treatment. I am looking for this
careful treatment.

> In the above case, you just need to look up Hille-Yoshida in the index;
> not read everything about Banach spaces.

I'll look for that, thanks.

>
>
> Most of what is known rigorously about path integrals is in the
> book by Glimm and Jaffe; but it takes time to get used to it
> if you come from physics...
>
Yes, I heared about this book, but they told me that it is neither
suitable for physicists (which I suppose I am) nor for mathematicians.
But lets see, maybe it is helpful.