Wick rotation

xavatar@gmail.com

<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>Hi everyone,\n\nWhen doing some classic computations with Feynman diagrams, after\ncombining the denominators, one usually wants to Wick rotate the loop\nmomentum 0th component.\nBut this can be done in an obvious way only if some constraints are put\non the external momenta (for instance if all external momenta are space\nlike).\n\nSo how to obtain Green\'s functions at arbitrary external momenta? In\nsimple cases (eg the four point function), I understand this can be\nachieved with some analytic continuation but is there a more systematic\nway to do this?\n\nI have also heard of Osterwalder Schrader theorem but I don\'t see how\nit can be used to really compute something in Minkowski Green\'s\nfunctions from Euclidean correlation functions.\n\nIn fact all this message deal with the following fundamental (to me)\nproblem: in ordinary perturbative computation, Green\'s functions are\nmerely functions whereas, in mathematical physics, Green\'s functions\nshould be tempered distributions.\nI\'ve doing lot of bibliographical research but nothing came up...\nPlease, can anyone point me to something relevant about all this ?\n\nThanks for reading and sorry for my poor English :)\nXavatar\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>Hi everyone,

When doing some classic computations with Feynman diagrams, after
combining the denominators, one usually wants to Wick rotate the loop
momentum 0th component.
But this can be done in an obvious way only if some constraints are put
on the external momenta (for instance if all external momenta are space
like).

So how to obtain Green's functions at arbitrary external momenta? In
simple cases (eg the four point function), I understand this can be
achieved with some analytic continuation but is there a more systematic
way to do this?

I have also heard of Osterwalder Schrader theorem but I don't see how
it can be used to really compute something in Minkowski Green's
functions from Euclidean correlation functions.

In fact all this message deal with the following fundamental (to me)
problem: in ordinary perturbative computation, Green's functions are
merely functions whereas, in mathematical physics, Green's functions
should be tempered distributions.
I've doing lot of bibliographical research but nothing came up...
Please, can anyone point me to something relevant about all this ?

Thanks for reading and sorry for my poor English :)
Xavatar

Arnold Neumaier

<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>xavatar@gmail.com wrote:\n\n> I have also heard of Osterwalder Schrader theorem but I don\'t see how\n> it can be used to really compute something in Minkowski Green\'s\n> functions from Euclidean correlation functions.\n\nIf you know how to compute a function f_E(x_0,\\x) in Euclidean\nspacetime and know that it is analytic in x_0, you can\nanalytically continue to get the corresponding\nf_M(t,\\x) = f_E(it,\\x)\nin Minkowski spacetime. If you know f_E only numerically,\nyou can fit a rational function in t to the data and analytically\ncontinue the latter. This usually gives reasonable results.\n\nSo Osterwalder-Schrader is exactly the right thing to look for.\nType these two names into http://scholar.google.com/\nto get plenty of additional information!\n\n\n> In fact all this message deal with the following fundamental (to me)\n> problem: in ordinary perturbative computation, Green\'s functions are\n> merely functions whereas, in mathematical physics, Green\'s functions\n> should be tempered distributions.\n\nEven in ordinary perturbative computations, spacetime Green\'s functions\nare only distributions. It is just that physicists often don\'t care\nto restrict the usage of the term \'function\' to what mathematical\nphysicists call a function but allow things like the Delta function,\nwhich is not a function in the mathematical sense.\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>xavatar@gmail.com wrote:

> I have also heard of Osterwalder Schrader theorem but I don't see how
> it can be used to really compute something in Minkowski Green's
> functions from Euclidean correlation functions.

If you know how to compute a function [itex]f_E(x_0,\x)[/itex] in Euclidean
spacetime and know that it is analytic in [itex]x_0,[/itex] you can
analytically continue to get the corresponding
[itex]f_M(t,\x) = f_E(it,\x)[/itex]
in Minkowski spacetime. If you know [itex]f_E[/itex] only numerically,
you can fit a rational function in t to the data and analytically
continue the latter. This usually gives reasonable results.

So Osterwalder-Schrader is exactly the right thing to look for.
Type these two names into http://scholar.google.com/
to get plenty of additional information!

> In fact all this message deal with the following fundamental (to me)
> problem: in ordinary perturbative computation, Green's functions are
> merely functions whereas, in mathematical physics, Green's functions
> should be tempered distributions.

Even in ordinary perturbative computations, spacetime Green's functions
are only distributions. It is just that physicists often don't care
to restrict the usage of the term 'function' to what mathematical
physicists call a function but allow things like the [itex]\Delta[/itex] function,
which is not a function in the mathematical sense.

Arnold Neumaier

xavatar@gmail.com

<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> xavatar@gmail.com wrote:\n>\n> > I have also heard of Osterwalder Schrader theorem but I don\'t see\nhow\n> > it can be used to really compute something in Minkowski Green\'s\n> > functions from Euclidean correlation functions.\n>\n> If you know how to compute a function f_E(x_0,\\x) in Euclidean\n> spacetime and know that it is analytic in x_0, you can\n> analytically continue to get the corresponding\n> f_M(t,\\x) = f_E(it,\\x)\n> in Minkowski spacetime. If you know f_E only numerically,\n> you can fit a rational function in t to the data and analytically\n> continue the latter. This usually gives reasonable results.\n\nBut what about Green functions themselves? Continuation in n variables\nis not that simple...\n\nIn there a good approach to this in momentum space? When you perform\nthe analytic continuation to ix^0 what happens to the Fourier\ntransform?\n\n> > In fact all this message deal with the following fundamental (to\nme)\n> > problem: in ordinary perturbative computation, Green\'s functions\nare\n> > merely functions whereas, in mathematical physics, Green\'s\nfunctions\n> > should be tempered distributions.\n>\n> Even in ordinary perturbative computations, spacetime Green\'s\nfunctions\n> are only distributions. It is just that physicists often don\'t care\n> to restrict the usage of the term \'function\' to what mathematical\n> physicists call a function but allow things like the Delta function,\n> which is not a function in the mathematical sense.\n\nMy problem is that there are two points view:\n1- Green Functions are tempered distributions and that\'s it... Within\nthis approach I don\'t think one can go much further than Streater and\nWightmann\n2- Tempered distributions viewed as boundary values of analytic\nfunctions. These can me multiplied, integrated... Everything you need\nin order to compute Feynman diagrams...\n\nXavatar\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:
> xavatar@gmail.com wrote:
>
> > I have also heard of Osterwalder Schrader theorem but I don't see
how
> > it can be used to really compute something in Minkowski Green's
> > functions from Euclidean correlation functions.
>
> If you know how to compute a function [itex]f_E(x_0,\x)[/itex] in Euclidean
> spacetime and know that it is analytic in [itex]x_0,[/itex] you can
> analytically continue to get the corresponding
> [itex]f_M(t,\x) = f_E(it,\x)[/itex]
> in Minkowski spacetime. If you know [itex]f_E[/itex] only numerically,
> you can fit a rational function in t to the data and analytically
> continue the latter. This usually gives reasonable results.

But what about Green functions themselves? Continuation in n variables
is not that simple...

In there a good approach to this in momentum space? When you perform
the analytic continuation to [itex]ix^0[/itex] what happens to the Fourier
transform?

> > In fact all this message deal with the following fundamental (to
me)
> > problem: in ordinary perturbative computation, Green's functions
are
> > merely functions whereas, in mathematical physics, Green's
functions
> > should be tempered distributions.
>
> Even in ordinary perturbative computations, spacetime Green's
functions
> are only distributions. It is just that physicists often don't care
> to restrict the usage of the term 'function' to what mathematical
> physicists call a function but allow things like the [itex]\Delta[/itex] function,
> which is not a function in the mathematical sense.

My problem is that there are two points view:
1- Green Functions are tempered distributions and that's it... Within
this approach I don't think one can go much further than Streater and
Wightmann
2- Tempered distributions viewed as boundary values of analytic
functions. These can me multiplied, integrated... Everything you need
in order to compute Feynman diagrams...

Xavatar

Arnold Neumaier

<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>xavatar@gmail.com wrote:\n\n> Arnold Neumaier wrote:\n>\n>>xavatar@gmail.com wrote:\n>\n>>>I have also heard of Osterwalder Schrader theorem but I don\'t see how\n>>>it can be used to really compute something in Minkowski Green\'s\n>>>functions from Euclidean correlation functions.\n>>\n>>If you know how to compute a function f_E(x_0,\\x) in Euclidean\n>>spacetime and know that it is analytic in x_0, you can\n>>analytically continue to get the corresponding\n>> f_M(t,\\x) = f_E(it,\\x)\n>>in Minkowski spacetime. If you know f_E only numerically,\n>>you can fit a rational function in t to the data and analytically\n>>continue the latter. This usually gives reasonable results.\n>\n> But what about Green functions themselves? Continuation in n variables\n> is not that simple...\n\nThe idea is the same. Choose a functional form of some shape that\ncaptures what you think should happen in real space. Then\nfit the analytically continued functional form in Euclidean space\nto whatever you computed there, and substitute the resulting\nparameters into your original functional form.\n\n\n> In there a good approach to this in momentum space? When you perform\n> the analytic continuation to ix^0 what happens to the Fourier\n> transform?\n\nIt becomes a Laplace or inverse Laplace transform.\n\n\n> My problem is that there are two points view:\n> 1- Green Functions are tempered distributions and that\'s it... Within\n> this approach I don\'t think one can go much further than Streater and\n> Wightmann\n> 2- Tempered distributions viewed as boundary values of analytic\n> functions. These can me multiplied, integrated... Everything you need\n> in order to compute Feynman diagrams...\n\nAnd these are related. As long as you have anlaytic functions you\nare off real time, but you have functions with nice properties.\nOnce you go to the boundary values, you lose these nice properties\nand get only the distribution.\n\nThat\'s why one calculates in the analytic continuation and continues\nback only at the very end.\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>xavatar@gmail.com wrote:

> Arnold Neumaier wrote:
>
>>xavatar@gmail.com wrote:
>
>>>I have also heard of Osterwalder Schrader theorem but I don't see how
>>>it can be used to really compute something in Minkowski Green's
>>>functions from Euclidean correlation functions.
>>
>>If you know how to compute a function [itex]f_E(x_0,\x)[/itex] in Euclidean
>>spacetime and know that it is analytic in [itex]x_0,[/itex] you can
>>analytically continue to get the corresponding
>> [itex]f_M(t,\x) = f_E(it,\x)[/itex]
>>in Minkowski spacetime. If you know [itex]f_E[/itex] only numerically,
>>you can fit a rational function in t to the data and analytically
>>continue the latter. This usually gives reasonable results.
>
> But what about Green functions themselves? Continuation in n variables
> is not that simple...

The idea is the same. Choose a functional form of some shape that
captures what you think should happen in real space. Then
fit the analytically continued functional form in Euclidean space
to whatever you computed there, and substitute the resulting
parameters into your original functional form.

> In there a good approach to this in momentum space? When you perform
> the analytic continuation to [itex]ix^0[/itex] what happens to the Fourier
> transform?

It becomes a Laplace or inverse Laplace transform.

> My problem is that there are two points view:
> 1- Green Functions are tempered distributions and that's it... Within
> this approach I don't think one can go much further than Streater and
> Wightmann
> 2- Tempered distributions viewed as boundary values of analytic
> functions. These can me multiplied, integrated... Everything you need
> in order to compute Feynman diagrams...

And these are related. As long as you have anlaytic functions you
are off real time, but you have functions with nice properties.
Once you go to the boundary values, you lose these nice properties
and get only the distribution.

That's why one calculates in the analytic continuation and continues
back only at the very end.

Arnold Neumaier

xavatar@gmail.com

<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> > But what about Green functions themselves? Continuation in n\nvariables\n> > is not that simple...\n>\n> The idea is the same. Choose a functional form of some shape that\n> captures what you think should happen in real space. Then\n> fit the analytically continued functional form in Euclidean space\n> to whatever you computed there, and substitute the resulting\n> parameters into your original functional form.\n\nThis is more easy to say than to do as I have no explicit form for the\nfunctions I want to anatically continue :)\n\n> > In there a good approach to this in momentum space? When you\nperform\n> > the analytic continuation to ix^0 what happens to the Fourier\n> > transform?\n>\n> It becomes a Laplace or inverse Laplace transform.\n\nA Laplace transform is integrated from 0 to infinity whereas a Fourier\ntransform is integrated along R... To do this you need to have support\nproperty I suppose.\n\nXavatar\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:

> > But what about Green functions themselves? Continuation in n
variables
> > is not that simple...
>
> The idea is the same. Choose a functional form of some shape that
> captures what you think should happen in real space. Then
> fit the analytically continued functional form in Euclidean space
> to whatever you computed there, and substitute the resulting
> parameters into your original functional form.

This is more easy to say than to do as I have no explicit form for the
functions I want to anatically continue :)

> > In there a good approach to this in momentum space? When you
perform
> > the analytic continuation to [itex]ix^0[/itex] what happens to the Fourier
> > transform?
>
> It becomes a Laplace or inverse Laplace transform.

A Laplace transform is integrated from to infinity whereas a Fourier
transform is integrated along R... To do this you need to have support
property I suppose.

Xavatar

Arnold Neumaier

<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>xavatar@gmail.com wrote:\n> Arnold Neumaier wrote:\n>\n>\n>>>But what about Green functions themselves? Continuation in n\n>\n> variables\n>\n>>>is not that simple...\n>>\n>>The idea is the same. Choose a functional form of some shape that\n>>captures what you think should happen in real space. Then\n>>fit the analytically continued functional form in Euclidean space\n>>to whatever you computed there, and substitute the resulting\n>>parameters into your original functional form.\n>\n>\n> This is more easy to say than to do as I have no explicit form for the\n> functions I want to anatically continue :)\n\nOf course you have to guess one, and then check how well it fits.\nIf you don\'t even this much there is no hope to do the continuation!\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>xavatar@gmail.com wrote:
> Arnold Neumaier wrote:
>
>
>>>But what about Green functions themselves? Continuation in n
>
> variables
>
>>>is not that simple...
>>
>>The idea is the same. Choose a functional form of some shape that
>>captures what you think should happen in real space. Then
>>fit the analytically continued functional form in Euclidean space
>>to whatever you computed there, and substitute the resulting
>>parameters into your original functional form.
>
>
> This is more easy to say than to do as I have no explicit form for the
> functions I want to anatically continue :)

Of course you have to guess one, and then check how well it fits.
If you don't even this much there is no hope to do the continuation!

Arnold Neumaier

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