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- 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?