- #1
Heidi
- 418
- 40
- TL;DR Summary
- Do physicist and mathematicians agree with the définition of inequivalent ccr representations?
Fock spaces use lists of integers (0 and 1 in the fermionic case) to describe set of particles.
a list 0 1 1 0 1 0 0 0 ... for 1 fermion in the second third and fifth state may be associated to the real number 0.0110100000...
so this set of list is not countable. a Fock space select a countable subset of such lists
the simpler is found if we choose the lists with a finite number or 1. it contains 0 0 0 0 0 0 ...
we can do the same thing in the bosonic case.
i read this in a book:
Two representations in different subsets are unitarily inequivalent to each other.
If two different subsets can be used as base of representations for the operators;
{a i , a i † } and {α i , α i † } in the set {|n 1 , n 2 , . . . , n i , . . . }, these two representations are
unitarily inequivalent to each other, which means that a vector of one representation
for the operators {a i , a i † } is not expressed by a superposition of base vectors of
another representation for the operators {α i , α i † }. In quantum mechanics, we can
express a vector of one representation by a superposition of base vectors of another
representation, because all possible representations are unitary equivalent.This
situation disappears in the case of quantum field theory.
this the definition of unitarily inequivalence for a physicist.
a mathematician would have spoken of commutativity and intertwiners.
Are they talking about the same thing?
a list 0 1 1 0 1 0 0 0 ... for 1 fermion in the second third and fifth state may be associated to the real number 0.0110100000...
so this set of list is not countable. a Fock space select a countable subset of such lists
the simpler is found if we choose the lists with a finite number or 1. it contains 0 0 0 0 0 0 ...
we can do the same thing in the bosonic case.
i read this in a book:
Two representations in different subsets are unitarily inequivalent to each other.
If two different subsets can be used as base of representations for the operators;
{a i , a i † } and {α i , α i † } in the set {|n 1 , n 2 , . . . , n i , . . . }, these two representations are
unitarily inequivalent to each other, which means that a vector of one representation
for the operators {a i , a i † } is not expressed by a superposition of base vectors of
another representation for the operators {α i , α i † }. In quantum mechanics, we can
express a vector of one representation by a superposition of base vectors of another
representation, because all possible representations are unitary equivalent.This
situation disappears in the case of quantum field theory.
this the definition of unitarily inequivalence for a physicist.
a mathematician would have spoken of commutativity and intertwiners.
Are they talking about the same thing?