Re: Normal ordering and VEV subtraction & simpler definition of normal

markwh04@yahoo.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>Igor Khavkine wrote:\n> I\'m looking at Wald\'s book on QFT in curved spacetime.\n[which works within the algebraic approach].\n\n> In a few simple cases, the normal ordering and the point splitting\n> prescriptin can be applied explicitly and the results agreee. But this\n> is still somewhat mysterious to me. Wald claims that "for physically\n> reasonable states in the standard Fock space, the singular behavior of\n> the bidistribution <phi(x)phi(y)> is the same as for\n> <0|phi(x)phi(y)|0>" (paraphrasing slightly).\n\nReasonable assumption is that the state space is\n{ A|0>: A from a class of well-behaved operators }\nThen for any state A|0>\n<phi(x)phi(y)> = <0| A* phi(x) phi(y) A |0>\nand the A\'s should have no effect on the singularity behavior.\n\nA simpler prescription which subsumes Wald and everyone else -- but\nwhich apparently eluded Wald and others -- is to just define\nN(phi(x)phi(y)) = unique (up to constant) operator Z\nsuch that Ad(Z) = Ad(phi(x)phi(y)).\nNote that even though phi(x)phi(y) may be singular, it\'s *adjoint* is\ngenerally not. Various authors try to state this point by pointing out\nthat the infinity is "only" a c-number, meaning that it isn\'t even\nthere under the Ad(). If ths spectrum of phi(x)phi(y) is bounded, the\nconstant is fixed by convention of setting the lower bound to 0.\n\n[Note: Ad(A) is the adjoint of A, defined as [A, ()]).\n\nIn an irreducible representation, the only operator A with Ad(A) = 0 is\na c-number. Therefore, the above prescription does indeed uniquely\nspecify the operator ... assuming that a solution exists.\n\nThe motivation for the foregoing is simply that nearly the only place\n(at least in QFT) the stress-energy tensor is used is in the context of\nthe Lie bracket and the (generalized) Heisenberg equations:\n[P_a, O] = i h-bar dO/dx^a.\n\nIn general terms, the concept of adjoint-equivalence allows you to\nconsider a larger class of operators which are all equal to the\noriginal class of bounded, well-behaved operators, modulo the\nequivalence. For these operators, normal ordering defined via\nadjoint-equivalence then specifies ... up to a constant ... a unique\nbounded operator which behaves the same as the original operator in all\ncontexts involving the commutators.\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>Igor Khavkine wrote:
> I'm looking at Wald's book on QFT in curved spacetime.
[which works within the algebraic approach].

> In a few simple cases, the normal ordering and the point splitting
> prescriptin can be applied explicitly and the results agreee. But this
> is still somewhat mysterious to me. Wald claims that "for physically
> reasonable states in the standard Fock space, the singular behavior of
> the bidistribution [itex]<\phi(x)\phi(y)>[/itex] is the same as for
> [itex]<0|\phi(x)\phi(y)|0>[/itex]" (paraphrasing slightly).

Reasonable assumption is that the state space is
[itex]{ A|0>: A[/itex] from a class of well-behaved operators }
Then for any state [itex]A|0><\phi(x)\phi(y)> = <0| A* \phi(x) \phi(y) A |0>[/itex]
and the A's should have no effect on the singularity behavior.

A simpler prescription which subsumes Wald and everyone else -- but
which apparently eluded Wald and others -- is to just define
[itex]N(\phi(x)\phi(y)) =[/itex] unique (up to constant) operator Z
such that Ad(Z) [itex]= Ad(\phi(x)\phi(y))[/itex].
Note that even though [itex]\phi(x)\phi(y)[/itex] may be singular, it's *adjoint* is
generally not. Various authors try to state this point by pointing out
that the infinity is "only" a c-number, meaning that it isn't even
there under the Ad(). If ths spectrum of [itex]\phi(x)\phi(y)[/itex] is bounded, the
constant is fixed by convention of setting the lower bound to .

[Note: Ad(A) is the adjoint of A, defined as [A, ()]).

In an irreducible representation, the only operator A with Ad(A) = is
a c-number. Therefore, the above prescription does indeed uniquely
specify the operator ... assuming that a solution exists.

The motivation for the foregoing is simply that nearly the only place
(at least in QFT) the stress-energy tensor is used is in the context of
the Lie bracket and the (generalized) Heisenberg equations:
[itex][P_a, O] = i[/itex] h-bar [itex]dO/dx^a[/itex].

In general terms, the concept of adjoint-equivalence allows you to
consider a larger class of operators which are all equal to the
original class of bounded, well-behaved operators, modulo the
equivalence. For these operators, normal ordering defined via
adjoint-equivalence then specifies ... up to a constant ... a unique
bounded operator which behaves the same as the original operator in all
contexts involving the commutators.

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