View Full Version : Where did the main (anti)commutation relation go?
<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\nIn the path integral formulation of QFT for bosons,\nthe canonical commutations relations are not\neven mentioned. (Ref: Peskin & Schroeder, ch9).\n\nFor fermions, P&S say that we must introduce\nanticommuting Grassman numbers. OK, but\nwhat about an anti-commutation relation corresponding\nto {a_k, a*_k\'} = i delta(k - k\') ? I don\'t see\nanything about that. Only the boring ones like\nab + ba = 0.\n\nSimilarly, for phi^4 theory (P&S pp284-289) I\ndon\'t see anything about how phi^dot doesn\'t\ncommute with phi, which is the case in the\ncanonical approach. Then, to derive the\nphi^4 Feynman rules on p289, P&S do the\nfollowing (see the 2nd unnumbered eqn on p289):\n\nexp[i Integral L] = exp[i Integral L_0] exp[i Integral L_int]\n\n~ exp[i Integral L_0] (1 - i Integral lambda/4! phi^4)\n\nBut L_0 contains phi^dot which doesn\'t commute with phi (at least,\nthat\'s what happens in the canonical formalism). The first step\nabove is not valid if L_0 doesn\'t commute with L_int. And\n[L_0, L_int] is O(lambda), so it should appear in the O(lambda)\nexpansion. I.e: there should be a delta fn at O(lamda). Actually,\nthere\'s probably more delta fns, because higher order\nBaker-Campbell-Hausdorf terms like [L_0, [L_0, L_int]] are\nalso O(lambda).\n\nWhat am I missing?\n\nTIA,\n\n- MikeM.\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>In the path integral formulation of QFT for bosons,
the canonical commutations relations are not
even mentioned. (Ref: Peskin & Schroeder, ch9).
For fermions, P&S say that we must introduce
anticommuting Grassman numbers. OK, but
what about an anti-commutation relation corresponding
to {a_k, a*_k'} = i \delta(k - k') ? I don't see
anything about that. Only the boring ones like
ab + ba = .
Similarly, for \phi^4 theory (P&S pp284-289) I
don't see anything about how \phi^dot doesn't
commute with \phi, which is the case in the
canonical approach. Then, to derive the
\phi^4 Feynman rules on p289, P&S do the
following (see the 2nd unnumbered eqn on p289):
\exp[i[/itex] Integral L] = \exp[i Integral L_0] \exp[i Integral [itex]L_{int}]
~ \exp[i Integral L_0] (1 - i Integral \lambda/4! \phi^4)
But L_0 contains \phi^dot which doesn't commute with \phi (at least,
that's what happens in the canonical formalism). The first step
above is not valid if L_0 doesn't commute with L_{int}. And
[L_0, L_{int}] is O(\lambda), so it should appear in the O(\lambda)
expansion. I.e: there should be a \delta fn at O(lamda). Actually,
there's probably more \delta fns, because higher order
Baker-Campbell-Hausdorf terms like [L_0, [L_0, L_{int}]] are
also O(\lambda).
What am I missing?
TIA,
- MikeM.
Lubos Motl
Sep25-04, 05:00 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>On Fri, 24 Sep 2004, MM wrote:\n\n> In the path integral formulation of QFT for bosons,\n> the canonical commutations relations are not\n> even mentioned. (Ref: Peskin & Schroeder, ch9).\n\nThe path integral formulation is a different formulation than the operator\nformulation. Because there are no real operators acting on the Hilbert\nspace in this approach, there are also no commutators as you know them.\nThe variables that you integrate over are thought of as purely classical,\ncommuting (or anticommuting, for Grassmann numbers between each other)\nconfigurations. The uncertainty principle is reflected by the\n"jitteriness" of the typical contributions to the path integral, and the\nclosest thing to the nonzero commutator can be derived from ordering\nambiguities, resulting from the ultraviolet behavior of correlators.\n\nYour multiply repeated question (of course that the answer to your\nquestion is identical in QED, lambda.phi^4, as well as for fermions) can\nalready be asked in quantum mechanics and has nothing special to do with\nquantum field theory. In quantum mechanics for a pointlike particle, you\neither write the path integral purely in terms of configurations of x(t) -\nand there is no "p" in this description - or a path integral involving\nboth x(t) and p(t), in which you can more transparently see the nonzero\ncommutator. Only the first case - the configurations of x(t) separately -\nis naturally generalized in relativistic quantum field theory - and the\nquestion "what is the commutator" is not really well-posed in Feynman\'s\napproach. The goal of Feynman\'s approach is to calculate the amplitudes of\nthe evolution operators between particular initial and final states.\n\nIt is these amplitudes that are equal like those derived from the operator\nformalism - which does not mean that Feynman\'s approach must copy the\nmachinery of operators. It does not.\n\n> But L_0 contains phi^dot which doesn\'t commute with phi (at least,\n> that\'s what happens in the canonical formalism).\n\nThe path integrals treats \\phi(x^\\mu), and consequently also its\nderivatives, as classical numbers (integration variables), and they do\ncommute with one another. You are not integrating over operators! The\nordering issues are reflected in the ultraviolet behavior of correlators\n(=path integral with insertions).\n\nSee e.g. Feynman-Hibbs\' book about the path integrals\n\nhttp://www.amazon.com/exec/obidos/tg/detail/-/0070206503/\n\nBest\nLubos\n_________________________________ _____________________________________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)\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>On Fri, 24 Sep 2004, MM wrote:
> In the path integral formulation of QFT for bosons,
> the canonical commutations relations are not
> even mentioned. (Ref: Peskin & Schroeder, ch9).
The path integral formulation is a different formulation than the operator
formulation. Because there are no real operators acting on the Hilbert
space in this approach, there are also no commutators as you know them.
The variables that you integrate over are thought of as purely classical,
commuting (or anticommuting, for Grassmann numbers between each other)
configurations. The uncertainty principle is reflected by the
"jitteriness" of the typical contributions to the path integral, and the
closest thing to the nonzero commutator can be derived from ordering
ambiguities, resulting from the ultraviolet behavior of correlators.
Your multiply repeated question (of course that the answer to your
question is identical in QED, \lambda.\phi^4, as well as for fermions) can
already be asked in quantum mechanics and has nothing special to do with
quantum field theory. In quantum mechanics for a pointlike particle, you
either write the path integral purely in terms of configurations of x(t) -
and there is no "p" in this description - or a path integral involving
both x(t) and p(t), in which you can more transparently see the nonzero
commutator. Only the first case - the configurations of x(t) separately -
is naturally generalized in relativistic quantum field theory - and the
question "what is the commutator" is not really well-posed in Feynman's
approach. The goal of Feynman's approach is to calculate the amplitudes of
the evolution operators between particular initial and final states.
It is these amplitudes that are equal like those derived from the operator
formalism - which does not mean that Feynman's approach must copy the
machinery of operators. It does not.
> But L_0 contains \phi^dot which doesn't commute with \phi (at least,
> that's what happens in the canonical formalism).
The path integrals treats \phi(x^\mu), and consequently also its
derivatives, as classical numbers (integration variables), and they do
commute with one another. You are not integrating over operators! The
ordering issues are reflected in the ultraviolet behavior of correlators
(=path integral with insertions).
See e.g. Feynman-Hibbs' book about the path integrals
http://www.amazon.com/exec/obidos/tg/detail/-/0070206503/
Best
Lubos
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Igor Khavkine
Sep27-04, 03:31 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>\nOn Fri, 24 Sep 2004 14:22:08 +0000, MM wrote:\n\n> In the path integral formulation of QFT for bosons, the canonical\n> commutations relations are not even mentioned. (Ref: Peskin & Schroeder,\n> ch9).\n\nThe fields that enter the action integral in the path integral calculation\nare not the quantum fields, but the classical ones. Except that in the\nfermion case you formally make multiplication anti-commutative.\n\nThe canonical commutation relations are not necessary in the path integral\nformalism. The path integral allows you to calculate directly what\nwould have been time ordered vacuum expectation values of various\noperators in the canonical picture.\n\nIf you want to recover the canonical commutation relations, you have to\nexpress them in terms of such expectation values. For instance, take\naa* - a*a = 1 and convert this expression into matrix form by taking its\nexpectation value for each state in the bosonic Fock space and inserting\na resolution of identity between each product of operators. Since you are\nonly allowed to use time-ordered expectation values, you have to fiddle\nwith the time labels for each operator and then take the limit in which\nall time labels are equal.\n\nAlso, to appease Arnold Neumaier, I should mention that none of this is\nrigorously defined mathematically. :-)\n\nHope this helps.\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>On Fri, 24 Sep 2004 14:22:08 +0000, MM wrote:
> In the path integral formulation of QFT for bosons, the canonical
> commutations relations are not even mentioned. (Ref: Peskin & Schroeder,
> ch9).
The fields that enter the action integral in the path integral calculation
are not the quantum fields, but the classical ones. Except that in the
fermion case you formally make multiplication anti-commutative.
The canonical commutation relations are not necessary in the path integral
formalism. The path integral allows you to calculate directly what
would have been time ordered vacuum expectation values of various
operators in the canonical picture.
If you want to recover the canonical commutation relations, you have to
express them in terms of such expectation values. For instance, take
aa* - a*a = 1 and convert this expression into matrix form by taking its
expectation value for each state in the bosonic Fock space and inserting
a resolution of identity between each product of operators. Since you are
only allowed to use time-ordered expectation values, you have to fiddle
with the time labels for each operator and then take the limit in which
all time labels are equal.
Also, to appease Arnold Neumaier, I should mention that none of this is
rigorously defined mathematically. :-)
Hope this helps.
Igor
Arnold Neumaier
Sep27-04, 03:32 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\nMM wrote:\n> In the path integral formulation of QFT for bosons,\n> the canonical commutations relations are not\n> even mentioned. (Ref: Peskin & Schroeder, ch9).\n\nThis is because the fields in the path integral are\nclassical fields that commute (or anticommute) _always_.\n\nIn the path integral formalism, QM arises by summing over\nclassical trajectories, so only classical objects enter\nthe stage. There are no operators at all in the formalism,\nany non-commutation in vacuum expectations is eliminated by\nnormal ordering requirement. And even the normal ordering\nis just notational, since one never orders anything...\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>MM wrote:
> In the path integral formulation of QFT for bosons,
> the canonical commutations relations are not
> even mentioned. (Ref: Peskin & Schroeder, ch9).
This is because the fields in the path integral are
classical fields that commute (or anticommute) _always_.
In the path integral formalism, QM arises by summing over
classical trajectories, so only classical objects enter
the stage. There are no operators at all in the formalism,
any non-commutation in vacuum expectations is eliminated by
normal ordering requirement. And even the normal ordering
is just notational, since one never orders anything...
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>Thanks Lubos, Igor & Arnold for your answers.\n\nLubos wrote:\n\n> [...] The ordering issues are reflected in the ultraviolet\n> behavior of correlators\n> (=path integral with insertions).\n>\n> See e.g. Feynman-Hibbs\' book about the path integrals\n>\n> http://www.amazon.com/exec/obidos/tg/detail/-/0070206503\n\nUnfortunately, that book\'s (used) price is upwards from\nUSD 340.00.\n\nI tried looking up some of your keywords in Weinberg,\nbut his index is quite poor. [Why can\'t physics authors\nprepare a decent index? :-( ]\n\nCan you (or anyone else) point me to a suitable chapter/page for\nthe insertion stuff in Weinberg? Alternatively, I also have P&S,\nZee, Itzykson, Bjorken & Drell, Glimme & Jaffe, Streater & Wightman.\n\n\nCheers & TIA,\n\n- MikeM.\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>Thanks Lubos, Igor & Arnold for your answers.
Lubos wrote:
> [...] The ordering issues are reflected in the ultraviolet
> behavior of correlators
> (=path integral with insertions).
>
> See e.g. Feynman-Hibbs' book about the path integrals
>
> http://www.amazon.com/exec/obidos/tg/detail/-/0070206503
Unfortunately, that book's (used) price is upwards from
USD 340.00.
I tried looking up some of your keywords in Weinberg,
but his index is quite poor. [Why can't physics authors
prepare a decent index? :-( ]
Can you (or anyone else) point me to a suitable chapter/page for
the insertion stuff in Weinberg? Alternatively, I also have P&S,
Zee, Itzykson, Bjorken & Drell, Glimme & Jaffe, Streater & Wightman.
Cheers & TIA,
- MikeM.
Arnold Neumaier
Sep28-04, 10:20 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\nMM wrote:\n> Thanks Lubos, Igor & Arnold for your answers.\n>\n> Lubos wrote:\n>\n> > [...] The ordering issues are reflected in the ultraviolet\n> > behavior of correlators\n> > (=path integral with insertions).\n> >\n> > See e.g. Feynman-Hibbs\' book about the path integrals\n> >\n> > http://www.amazon.com/exec/obidos/tg/detail/-/0070206503\n>\n> Unfortunately, that book\'s (used) price is upwards from\n> USD 340.00.\n>\n> I tried looking up some of your keywords in Weinberg,\n> but his index is quite poor. [Why can\'t physics authors\n> prepare a decent index? :-( ]\n>\n> Can you (or anyone else) point me to a suitable chapter/page for\n> the insertion stuff in Weinberg? Alternatively, I also have P&S,\n> Zee, Itzykson, Bjorken & Drell, Glimme & Jaffe, Streater & Wightman.\n\nI don\'t know what you refer to as \'insertion stuff\'.\nThe path integral is discussed in Weinberg I, Chapter 9, or\nPeskin/Schroeder, also Chapter 9. As you can see there, there are\nno operators at all, but the quantities obtained in the expansion\nof the path integral are time-ordered vacuum expectation values.\nSince the original ordering in a time-ordered vacuum expectation value\nis immaterial (apart from a sign for fermions), the same must be the\ncase for the path integral itself.\n\nTo answer Igor Khavkine, the path integral is ill-defined as a number,\nbut, after regularization, well-defined as a formal power series in\nhbar (the latter is often set to 1 to simplify typography,\nbut this make things more difficult to grasp). The Legendre transform\nof the logarithm is then also defined as a formal power series, and\nby letting the coupling constants depend on the regularization parameter\n(eps or Lambda), one can take the limit eps to 0 or Lambda to infty\nto get the effective action, again as a formal power series.\n\nFrom there, one can get the S-matrix, again as a formal power series.\nFOR QED, the first few terms give highly accurate approximations;\nfor other QFTs, partial resumming of these series give acceptable results\nin agreement with experiment.\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>MM wrote:
> Thanks Lubos, Igor & Arnold for your answers.
>
> Lubos wrote:
>
> > [...] The ordering issues are reflected in the ultraviolet
> > behavior of correlators
> > (=path integral with insertions).
> >
> > See e.g. Feynman-Hibbs' book about the path integrals
> >
> > http://www.amazon.com/exec/obidos/tg/detail/-/0070206503
>
> Unfortunately, that book's (used) price is upwards from
> USD 340.00.
>
> I tried looking up some of your keywords in Weinberg,
> but his index is quite poor. [Why can't physics authors
> prepare a decent index? :-( ]
>
> Can you (or anyone else) point me to a suitable chapter/page for
> the insertion stuff in Weinberg? Alternatively, I also have P&S,
> Zee, Itzykson, Bjorken & Drell, Glimme & Jaffe, Streater & Wightman.
I don't know what you refer to as 'insertion stuff'.
The path integral is discussed in Weinberg I, Chapter 9, or
Peskin/Schroeder, also Chapter 9. As you can see there, there are
no operators at all, but the quantities obtained in the expansion
of the path integral are time-ordered vacuum expectation values.
Since the original ordering in a time-ordered vacuum expectation value
is immaterial (apart from a sign for fermions), the same must be the
case for the path integral itself.
To answer Igor Khavkine, the path integral is ill-defined as a number,
but, after regularization, well-defined as a formal power series in
\hbar (the latter is often set to 1 to simplify typography,
but this make things more difficult to grasp). The Legendre transform
of the logarithm is then also defined as a formal power series, and
by letting the coupling constants depend on the regularization parameter
(eps or \Lambda), one can take the limit eps to or \Lambda to \infty
to get the effective action, again as a formal power series.
From there, one can get the S-matrix, again as a formal power series.
FOR QED, the first few terms give highly accurate approximations;
for other QFTs, partial resumming of these series give acceptable results
in agreement with experiment.
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>I wrote:\n\n>> [...]\n>> Can [Lubos] (or anyone else) point me to a suitable chapter/page for\n>> the insertion stuff in Weinberg? Alternatively, I also have P&S,\n>> Zee, Itzykson, Bjorken & Drell, Glimme & Jaffe, Streater & Wightman.\n\nArnold Neumaier replied:\n\n> I don\'t know what you refer to as \'insertion stuff\'.\n\nI was referring to what Lubos wrote, i.e:\n\n>>> [...] The ordering issues are reflected in\n>>> the ultraviolet behavior of correlators\n>>> [... =path integral with insertions]\n\n\n> As you can see [in Weinberg and P&S], there are\n> no operators at all, [...]\n\nYes, I had already realized that much from reading P&S.\nBut it just seemed a bit strange to me. Anyway, I obviously\nneed to read a lot more about path integrals to\nappreciate the finer points.\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>I wrote:
>> [...]
>> Can [Lubos] (or anyone else) point me to a suitable chapter/page for
>> the insertion stuff in Weinberg? Alternatively, I also have P&S,
>> Zee, Itzykson, Bjorken & Drell, Glimme & Jaffe, Streater & Wightman.
Arnold Neumaier replied:
> I don't know what you refer to as 'insertion stuff'.
I was referring to what Lubos wrote, i.e:
>>> [...] The ordering issues are reflected in
>>> the ultraviolet behavior of correlators
>>> [... =path integral with insertions]
> As you can see [in Weinberg and P&S], there are
> no operators at all, [...]
Yes, I had already realized that much from reading P&S.
But it just seemed a bit strange to me. Anyway, I obviously
need to read a lot more about path integrals to
appreciate the finer points.
Charles Torre
Oct26-04, 12:55 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>[Regarding path integrals]\n\n> I should mention that none of this is\n> rigorously defined mathematically. :-)\n>\n>\n\nI would say that there are some rigorous path integral results for\nmechanical systems and for free fields. See the book\n\n"Mathematical theory of Feynman Path Integrals",\nby Albeverio and Hoegh-Krohn.\n\ncharlie\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>[Regarding path integrals]
> I should mention that none of this is
> rigorously defined mathematically. :-)
>
>
I would say that there are some rigorous path integral results for
mechanical systems and for free fields. See the book
"Mathematical theory of Feynman Path Integrals",
by Albeverio and Hoegh-Krohn.
charlie
Arnold Neumaier
Oct26-04, 12:55 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>MM wrote:\n> I wrote:\n>\n> >> [...]\n> >> Can [Lubos] (or anyone else) point me to a suitable chapter/page for\n> >> the insertion stuff in Weinberg? Alternatively, I also have P&S,\n> >> Zee, Itzykson, Bjorken & Drell, Glimme & Jaffe, Streater & Wightman.\n>\n> Arnold Neumaier replied:\n>\n> > I don\'t know what you refer to as \'insertion stuff\'.\n>\n> I was referring to what Lubos wrote, i.e:\n>\n> >>> [...] The ordering issues are reflected in\n> >>> the ultraviolet behavior of correlators\n> >>> [... =path integral with insertions]\n>\n>\n> > As you can see [in Weinberg and P&S], there are\n> > no operators at all, [...]\n>\n> Yes, I had already realized that much from reading P&S.\n> But it just seemed a bit strange to me. Anyway, I obviously\n> need to read a lot more about path integrals to\n> appreciate the finer points.\n\nThe main strength of the path integral approach is precisely that\nit avoids quantum operators and replaces all operator arguments by\naverages over classical paths. (The main weakness is that this averaging\nprocess is logically ill-defined.)\n\nThe older canonical quantization approach was fraught with difficulties\nbecause of inconsistencies in the operator approach.\nFor example, the CCR that you wanted to see are\nvalid only in the free case, and no one knows how they should\nbe in the interacting case - though one knows that (anti)commutators\nmust still vanish at spacelike related arguments.\nMoreover, the renormalization program plays havoc with operators.\n\nUnfortunately, this means that dynamical isssues and bound states\nquestions, which are comparatively easy to handle in an operator\nframework, become almost intractable in the path integral approach.\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>MM wrote:
> I wrote:
>
> >> [...]
> >> Can [Lubos] (or anyone else) point me to a suitable chapter/page for
> >> the insertion stuff in Weinberg? Alternatively, I also have P&S,
> >> Zee, Itzykson, Bjorken & Drell, Glimme & Jaffe, Streater & Wightman.
>
> Arnold Neumaier replied:
>
> > I don't know what you refer to as 'insertion stuff'.
>
> I was referring to what Lubos wrote, i.e:
>
> >>> [...] The ordering issues are reflected in
> >>> the ultraviolet behavior of correlators
> >>> [... =path integral with insertions]
>
>
> > As you can see [in Weinberg and P&S], there are
> > no operators at all, [...]
>
> Yes, I had already realized that much from reading P&S.
> But it just seemed a bit strange to me. Anyway, I obviously
> need to read a lot more about path integrals to
> appreciate the finer points.
The main strength of the path integral approach is precisely that
it avoids quantum operators and replaces all operator arguments by
averages over classical paths. (The main weakness is that this averaging
process is logically ill-defined.)
The older canonical quantization approach was fraught with difficulties
because of inconsistencies in the operator approach.
For example, the CCR that you wanted to see are
valid only in the free case, and no one knows how they should
be in the interacting case - though one knows that (anti)commutators
must still vanish at spacelike related arguments.
Moreover, the renormalization program plays havoc with operators.
Unfortunately, this means that dynamical isssues and bound states
questions, which are comparatively easy to handle in an operator
framework, become almost intractable in the path integral approach.
Arnold Neumaier
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