Classical vs quantum grassmann variables [Archive]

a student

Sep7-05, 01:24 AM

Igor Khavkine

Sep11-05, 03:34 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 2005-09-07, a student <of_1001_nights@hotmail.com> wrote:\n> Grassmann numbers correspond to objects with squares equal to zero\n> (similarly, imaginary numbers correspond to objects with squares less\n> than zero). Assuming they form a (distributive) algebra, it follows\n> that all elements must anticommute:\n> gh + hg = (g+h)^2 - g^2 - h^2 = 0.\n>\n> When one quantises (and it is easier to consider discrete modes rather\n> than continuous fields here), one assumes that each g has a conjugate\n> h, such that they are mapped to operators G and H satisfying\n> GH + HG = constant times hbar,\n> and one goes on to derive Fermi-Dirac statistics, etc. This is all\n> perfectly analogous to the \'bosonic\' case, where anticommutation is\n> replaced by commutation.\n>\n> What interests me is the classical limit hbar->0, where everything\n> anticommutes as per the first paragraph. Are there some nice models\n> which throw light on this limit ?\n\nThere are two classical limits that are of importance. One is for states\nwith low uncertainty in the number of particles/excitations (this is\nordinary classical particle mechanics) and the other is for states with\nlarge uncertainty in the number of particles/excitations (this is\nclassical field theory).\n\nFor fermions, the first kind of limit gives classical particles that\nidentical and obey the Pauli exclusion principle (although the latter\nshows up only when they are packed *really* closely together and quantum\neffects can no longer be ignored).\n\nThe second kind of classical limit is more mysterious and is what I\nthink you are asking about. The classical field obtained in this case is\nGrassmann valued. I don\'t know how to make this limit less mysterious.\nIf we look at real valued classical fields, we can always fall back on\nintuition gained from observing real world examples like fluids,\nelectromagnetism, vibrating membranes, etc. Unfortunately, there isn\'t\na Grassmann valued field theory that falls within our usual real of\nexperience. The two conditions of large excitation number and large\nuncertainty in that number, which imply this classical limit, are hard\nto meet for fermions, since they tend to be massive (first condition)\nand have conserved charge (second condition).\n\nThe formal theory of Grassmann valued fields can be straightforwardly\nworked out, but it is unclear how to relate it to something we can\neasily recognize.\n\nTo meet the first condition for the relevance of this classical limit,\none needs high density. To meet the second condition, one needs high\ntemperatures. This way, charge conservation can be circumvented by\nhaving a large number of particles and antiparticles participating in\nconstant creation/annihilation events, thus creating large uncertainty\nin the total particle number. I don\'t know exactly where such conditions\nwill be met. Some possibilities are stellar interiors, nuclear\ncollisions, or early universe. All of which are difficult to observe.\n\nI don\'t know if I\'ve shed some light on your question, but perhaps I\'ve\nat least pointed to a place where to look, even if still in the dark.\n\nIgor\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 2005-09-07, a student <of_1001_nights@hotmail.com> wrote:
> Grassmann numbers correspond to objects with squares equal to zero
> (similarly, imaginary numbers correspond to objects with squares less
> than zero). Assuming they form a (distributive) algebra, it follows
> that all elements must anticommute:
> gh + hg = (g+h)^2 - g^2 - h^2 = .
>
> When one quantises (and it is easier to consider discrete modes rather
> than continuous fields here), one assumes that each g has a conjugate
> h, such that they are mapped to operators G and H satisfying
> GH + HG = constant times \hbar,
> and one goes on to derive Fermi-Dirac statistics, etc. This is all
> perfectly analogous to the 'bosonic' case, where anticommutation is
> replaced by commutation.
>
> What interests me is the classical limit \hbar->0, where everything
> anticommutes as per the first paragraph. Are there some nice models
> which throw light on this limit ?

There are two classical limits that are of importance. One is for states
with low uncertainty in the number of particles/excitations (this is
ordinary classical particle mechanics) and the other is for states with
large uncertainty in the number of particles/excitations (this is
classical field theory).

For fermions, the first kind of limit gives classical particles that
identical and obey the Pauli exclusion principle (although the latter
shows up only when they are packed *really* closely together and quantum
effects can no longer be ignored).

The second kind of classical limit is more mysterious and is what I
think you are asking about. The classical field obtained in this case is
Grassmann valued. I don't know how to make this limit less mysterious.
If we look at real valued classical fields, we can always fall back on
intuition gained from observing real world examples like fluids,
electromagnetism, vibrating membranes, etc. Unfortunately, there isn't
a Grassmann valued field theory that falls within our usual real of
experience. The two conditions of large excitation number and large
uncertainty in that number, which imply this classical limit, are hard
to meet for fermions, since they tend to be massive (first condition)
and have conserved charge (second condition).

The formal theory of Grassmann valued fields can be straightforwardly
worked out, but it is unclear how to relate it to something we can
easily recognize.

To meet the first condition for the relevance of this classical limit,
one needs high density. To meet the second condition, one needs high
temperatures. This way, charge conservation can be circumvented by
having a large number of particles and antiparticles participating in
constant creation/annihilation events, thus creating large uncertainty
in the total particle number. I don't know exactly where such conditions
will be met. Some possibilities are stellar interiors, nuclear
collisions, or early universe. All of which are difficult to observe.

I don't know if I've shed some light on your question, but perhaps I've
at least pointed to a place where to look, even if still in the dark.

Igor