Questions about BRST [Archive]

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P.Costello

Apr15-04, 11:37 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>I have recently begun a PhD in mathematical physics and am interested in\nstudying the mathematical aspects of the BRST method in quantum physics. I\nwas wondering if someone could answer a couple of basic questions for me.\n\nI have heard that BRST calculations have been used to make predictions that\nhave experimental verification. I was wondering what these calculations and\npredictions were and where I could find them?\n\nI am also aware that there are two forms of BRST, Lagrangean and Hamiltonian.\nI was wondering which form is used in experimental calculations, or indeed if\nboth are?\n\nAlso, I was wondering if there are any proofs of equivalence of the quantum\nversions of Lagrangean BRST and Hamiltonian BRST\n\nAny help would be greatly appreciated,\nPatrick Costello.\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 have recently begun a PhD in mathematical physics and am interested in
studying the mathematical aspects of the BRST method in quantum physics. I
was wondering if someone could answer a couple of basic questions for me.

I have heard that BRST calculations have been used to make predictions that
have experimental verification. I was wondering what these calculations and
predictions were and where I could find them?

I am also aware that there are two forms of BRST, Lagrangean and Hamiltonian.
I was wondering which form is used in experimental calculations, or indeed if
both are?

Also, I was wondering if there are any proofs of equivalence of the quantum
versions of Lagrangean BRST and Hamiltonian BRST

Any help would be greatly appreciated,
Patrick Costello.

Peter Woit

Apr16-04, 02:28 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>P.Costello wrote:\n\n>I have heard that BRST calculations have been used to make predictions that\n>have experimental verification. I was wondering what these calculations and\n>predictions were and where I could find them?\n>\n>\nBy now the standard way to treat the gauge symmetry of Yang-Mills theory\nin the derivation of the perturbative expansion is via BRST. This is\nstandard\nmaterial in many if not all QFT books. So any Feynman diagram calculation\nin the standard model, especially higher order ones, uses this in some\nsense or\nother.\n\n>I am also aware that there are two forms of BRST, Lagrangean and Hamiltonian.\n>I was wondering which form is used in experimental calculations, or indeed if\n>both are?\n>\n>\n>\nLagrangian BRST is what is used to get the correct Feynman diagram\nexpansion.\n\n>Also, I was wondering if there are any proofs of equivalence of the quantum\n>versions of Lagrangean BRST and Hamiltonian BRST\n>\n>\nHard enough to find anything like a sensible proof of the equivalence of the\nLagrangian and Hamiltonian formalisms in QFT, even before you throw in\nBRST....\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>P.Costello wrote:

>I have heard that BRST calculations have been used to make predictions that
>have experimental verification. I was wondering what these calculations and
>predictions were and where I could find them?
>
>
By now the standard way to treat the gauge symmetry of Yang-Mills theory
in the derivation of the perturbative expansion is via BRST. This is
standard
material in many if not all QFT books. So any Feynman diagram calculation
in the standard model, especially higher order ones, uses this in some
sense or
other.

>I am also aware that there are two forms of BRST, Lagrangean and Hamiltonian.
>I was wondering which form is used in experimental calculations, or indeed if
>both are?
>
>
>
Lagrangian BRST is what is used to get the correct Feynman diagram
expansion.

>Also, I was wondering if there are any proofs of equivalence of the quantum
>versions of Lagrangean BRST and Hamiltonian BRST
>
>
Hard enough to find anything like a sensible proof of the equivalence of the
Lagrangian and Hamiltonian formalisms in QFT, even before you throw in
BRST....

Buzurg Shagird

Apr17-04, 05:09 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>Peter Woit <woit@cpw.math.columbia.edu> wrote in message\nnews:<c5mi21\\$dc3\\$1@newsmaster.cc.colu mbia.edu>...\n\n> Hard enough to find anything like a sensible proof of the equivalence of the\n> Lagrangian and Hamiltonian formalisms in QFT, even before you throw in\n> BRST....\n\nIs it possible for the two formalisms to be equivalent if the\nHamiltonian\n(resp. Lagrangian) is no quadratic in momenta (resp. velocities)? I am\nthinking of Green functions being expressed as path integrals, and for\nquanum mechanics, if H is at most quadratic in momenta, we can\nintegrate\nover the momenta to get an alternative expression for the path\nintegral.\nWhich is the same as what we write as the Lagrangian form of the path\nintegral. So if a proof of equivalence of the two forms goes along the\nsame lines for field theory, won\'t we need H to be quadratic in\nmomenta?\n\n-S.\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>Peter Woit <woit@cpw.math.columbia.edu> wrote in message
news:<c5mi21$dc3$1@newsmaster.cc.columbia.edu>...

> Hard enough to find anything like a sensible proof of the equivalence of the
> Lagrangian and Hamiltonian formalisms in QFT, even before you throw in
> BRST....

Is it possible for the two formalisms to be equivalent if the
Hamiltonian
(resp. Lagrangian) is no quadratic in momenta (resp. velocities)? I am
thinking of Green functions being expressed as path integrals, and for
quanum mechanics, if H is at most quadratic in momenta, we can
integrate
over the momenta to get an alternative expression for the path
integral.
Which is the same as what we write as the Lagrangian form of the path
integral. So if a proof of equivalence of the two forms goes along the
same lines for field theory, won't we need H to be quadratic in
momenta?

-S.