a student
Oct12-06, 04:13 AM
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 = 0.
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 ?
(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 = 0.
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 ?