Fock spaces and inequivalent ccr representations

In summary, Fock spaces are used to describe sets of particles by using lists of integers, and in the fermionic case, the lists consist of 0s and 1s. These lists can be associated with real numbers and a Fock space selects a countable subset of such lists. In the simpler case, the Fock space chooses lists with a finite number of 1s. This concept can also be applied to the bosonic case. In quantum mechanics, two representations using different subsets are unitarily inequivalent, meaning that a vector in one representation cannot be expressed as a superposition of base vectors in another representation. This distinction is also recognized by mathematicians, who refer to it as commutativity and intertwin
  • #1
Heidi
411
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?
 
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  • #3
It seems that physicists and mathematicians are talking about the same thing. But when a mathematician sees a bijective intertertwiner A between H1 and H2 with their own represention of a group or an algebra, the physicist sees only one hilbert space H (with its scalar product), A becomes the identity and there are two representations on H.
if all can be well computed they say that the representations are unitarily equivalent. But if the existence of the two representations leads to contradictions or ill defined things they say that there are unitatily inequivalent.
 
  • #4
  • #5
Heidi said:
i read this in a book

Which book?

Heidi said:
the physicist sees only one hilbert space H (with its scalar product)

I'm not sure this is true; I think physicists consider unitarily inequivalent representations to lie in different Hilbert spaces. In the quote you give from a book, it says the basis vectors of one representation cannot be expressed as superpositions of the basis vectors of another representation, if the two representations are unitarily inequivalent. But if both sets of basis vectors belong to the same Hilbert space, each set must be expressible as superpositions of the other set, since it's a vector space axiom that any vector in the space can be expressed as a linear combination of a set of basis vectors.
 
  • #6
PeterDonis said:
Which book?

the book is "liquid glass transition" written by Toyosuki Kitamura (chapter 3).
 

1. What is a Fock space?

A Fock space is a mathematical concept used in quantum mechanics to describe the state of a system with an infinite number of particles. It is a Hilbert space that contains all possible states of the system, including those with varying numbers of particles.

2. What is the significance of Fock spaces in quantum mechanics?

Fock spaces are important in quantum mechanics because they allow us to describe and analyze systems with an infinite number of particles, such as the electromagnetic field. They also provide a framework for understanding the creation and annihilation of particles in quantum systems.

3. What are CCR representations?

CCR stands for "canonical commutation relations," which are a set of mathematical equations that describe the fundamental behavior of quantum systems. CCR representations are different mathematical representations of these relations, which can be used to study different aspects of a quantum system.

4. What does it mean for two CCR representations to be inequivalent?

Two CCR representations are considered inequivalent if they cannot be transformed into each other through a unitary transformation. This means that they describe fundamentally different aspects of a quantum system and cannot be used interchangeably.

5. How are Fock spaces and CCR representations related?

Fock spaces and CCR representations are closely related because Fock spaces are often used to construct CCR representations. In addition, the creation and annihilation operators in Fock spaces satisfy the CCR relations, making them a useful tool for studying quantum systems described by CCR representations.

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