Take the Fermionic creation and destruction operators
ad (k) and a(k) where ad means a dagger.
One has a set of relationships including
ad(k) ad(k) =0 , a(k)a(k)=0
a(k) a(l) + a(l) a(k) =0.........(1)
also two more relations.
It is easy enough to solve the one state case, given in many sources,
giving specific 2 by 2 matrics.
However for the many state case do not know of a representation
It is easy also to propose nilpotent parametrized matrices, for example
x - xx
1 -x
where x is the parameter. But then one finds. if each different x is a different state, that (1) is not satisfied, the answer is proportional to
(x1-x2)(x1-x2) and not zero.
It is clear also that with constant matrices we will not have any success.
So this is a case that a proposed formality is give, without actual
math backing in concrete terms?
Ola
Thanks for the references.
These are very complete as to all the formal properties of the operators, but predictably
does not give an actual matrix form for each one, with required properties.
The one for one mode is x=0 in my matrix, the 1 in one
corner.
Best regards, Juan
Depends on what you mean by indefinite number of states, which is a rather indefinite statement. For sets of independent fermion operators that can be indexed by the natural numbers there is the Jordan-Wigner transformation, introduced in 1928: https://link.springer.com/chapter/10.1007%2F978-3-662-02781-3_9
The bosonization technique, from some 50 years later, relating fermion fields of a continuous variable x to boson fields of the same variable, has been used to prove Fermi-Bose duality in 1-dimensional systems (and used to solve some seemingly intractable problems). This is a more delicate method with respect to mathematical rigor.
Technical details are easily found by googling topics like "Jordan Wigner transformation", "Bosonization", "Fermi-Bose duality", "Klein factors", ...
Juan> predictably does not give an actual matrix form for each one, with required properties.
That is wrong. For any (small) number N of independent fermion operators, one can quite straightforwardly construct explicit (2N × 2N)-dimensional representations of all relevant matrices, directly from the formal properties.
Juan> So this is a case that a proposed formality is give, without actual math backing in concrete terms?
You should not so easily jump to premature conclusions, just because you cannot figure out something...
It particular when the topic has been investigated and applied for almost a century.
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