## Time, Parameter, Operator

<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\nArnold Neumaier wrote:\n\n&gt;\n&gt; Thus a real electron is no longer pointlike.\n&gt; See the entry \'\'Are electrons pointlike/structureless?\'\'\n&gt; in my theoretical physics FAQ\n&gt; http://www.mat.univie.ac.at/~neum/physics-faq.txt\n\nIn your FAQ you write\n\nBut physical, relativistic particles are not pointlike.\nAn intuitive argument for this is the fact that their localization\nto a region significantly smaller than the de Broglie wavelength\nwould need energies larger than that needed to create\nparticle-antiparticle pairs, which changes the nature of the system.\n\nThis statement can be found in many QFT textbooks, however\nI don\'t find it acceptable.\n\nTrue,\nthe electron localized in a small volume has uncertain energy,\nand the uncertainty can be larger that the energy 2mc^2\nrequired for the creation of an electron-positron pair.\nThis is because the operators of position and energy do not commute.\nHowever, the wave function of the localized electron may have\nnon-zero component in many-particle subspaces only if the electron\'s\nposition operator does not commute with the particle number operator(s).\nThis is not the case. The Newton-Wigner position operator of a single\nelectron in the Fock space commutes with number operators of all other\nparticles. Therefore, nothing forbids infinitely precise localization\nof electrons. No extra particles are created in this case.\n\nThe weakness of your argument can be also demonstrated by the following\nconsideration. Instead of "particle-antiparticle pairs" put there word\n"photons". Then your argument suggests that a particle cannot be\nlocalized even in regions larger than the de Broglie wavelength,\nbecause photons are massless and can be produced at huge quantities at\nany attempt of such localization. This is certainly not true.\n\nBy the way, Newton and Wigner themselves expressed doubts about the\nabove argument. See the discussion part of their famous paper\nNewton, T. D.; Wigner, E. P., Localized states for elementary systems,\nRev. Mod. Phys. 21, 400-406 (1949).\n\nEugene.\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">&nbsp;&nbsp;View this Usenet post in original ASCII form </a></div><P></jabberwocky>Arnold Neumaier wrote:

>
> Thus a real electron is no longer pointlike.
> See the entry ''Are electrons pointlike/structureless?''
> in my theoretical physics FAQ
> http://www.mat.univie.ac.at/~neum/physics-faq.txt

But physical, relativistic particles are not pointlike.
An intuitive argument for this is the fact that their localization
to a region significantly smaller than the de Broglie wavelength
would need energies larger than that needed to create
particle-antiparticle pairs, which changes the nature of the system.

This statement can be found in many QFT textbooks, however
I don't find it acceptable.

True,
the electron localized in a small volume has uncertain energy,
and the uncertainty can be larger that the energy $2mc^2$
required for the creation of an electron-positron pair.
This is because the operators of position and energy do not commute.
However, the wave function of the localized electron may have
non-zero component in many-particle subspaces only if the electron's
position operator does not commute with the particle number operator(s).
This is not the case. The Newton-Wigner position operator of a single
electron in the Fock space commutes with number operators of all other
particles. Therefore, nothing forbids infinitely precise localization
of electrons. No extra particles are created in this case.

The weakness of your argument can be also demonstrated by the following
consideration. Instead of "particle-antiparticle pairs" put there word
"photons". Then your argument suggests that a particle cannot be
localized even in regions larger than the de Broglie wavelength,
because photons are massless and can be produced at huge quantities at
any attempt of such localization. This is certainly not true.

By the way, Newton and Wigner themselves expressed doubts about the
above argument. See the discussion part of their famous paper
Newton, T. D.; Wigner, E. P., Localized states for elementary systems,
Rev. Mod. Phys. 21, $400-406$ (1949).

Eugene.



Eugene Stefanovich writes >The weakness of your argument can be also demonstrated by the following >consideration. Instead of "particle-antiparticle pairs" put there word >"photons". Then your argument suggests that a particle cannot be >localized even in regions larger than the de Broglie wavelength, >because photons are massless and can be produced at huge quantities at >any attempt of such localization. This is certainly not true. Isn't there a flaw here in that for photons there is no rest frame. As a consequence you can always find a frame where the energy is great enough to exceed pair formation. However, fortunately, isolated photons cannot spontaneously form a particle pair so the universe is safe. I think its true to say that any apparatus designed to localise a photon to greater resolution than a photon's wavelength (as seen by the apparatus) will result in the destruction of the photon. Whether the result shows the position of the photon or the result of a complex interplay between photon and detector is a moot point. -- Oz This post is worth absolutely nothing and is probably fallacious. Use oz@farmeroz.port995.com [ozacoohdb@despammed.com functions]. BTOPENWORLD address has ceased. DEMON address has ceased.