In this post, which is a continuation of Spin Chronicles Part 47, we'll get prepared for exploring the Tomita's "flow of time". It has very little, if anything at all, to do with "spin". But the two subjects both come under one umbrella - the umbrella of *-algebras, and it is somewhat instructive to discuss Tomita's flow on the simple example of the geometric Clifford algebra of space. Tomita's construction works for any von Neumann algebra with a cyclic and separating vector, whether finite or infinite-dimensional. The plan of this post is to discuss a general, though not the most general case. So we will deal with a possibly infinite-dimensional Hilbert space, and then see how things simplify (and become somewhat confusing at the same time) in the case of a finite-dimensional Clifford algebra A that was the subject of our study so far.
Let therefore H be a Hilbert space (complex, with a finite or countable orthonormal basis). For the spin 1/2 case we will take H = C2. Let A be the algebra A=B(H) of all bounded linear operators on H. This is our "algebra of observables" of some "quantum system". For spin 1/2 case A≃Mat(2,C). This is our von Neumann algebra. By "von Neumann" algebra we mean A=A''. So, it is a *-algebra of operators, equal to its double commutant.
It is here that we meet an example of the chicken and egg problem: what is more "primary", what should come first, the algebra, or the representation space on which the algebra acts? In geometrical algebra we usually start with the algebra, then we recover the representation space by examining minimal left ideals of the algebra. In quantum theory it is usually the other way around: we start with a Hilbert space and then decide which operators on this space represent "observables"? It would be more in tune with geometry if I would start with a possibly infinite-dimensional Clifford algebra. But that would take us too far from the problem at hand, and also would have to find a simple way how to present the concept, which I do not know at present. So let us stay as described above.
As I have mentioned above, we are not discussing the most general case. For instance it would be more general to take a direct sum of a finite or infinite number of Hilbert spaces ⨁iHi, and to take for A the direct sum ⨁i B(Hi) of corresponding B(Hi). This would let us to discuss "superselection sectors". We are not going to do it here, so let us stick with our simple case of just one H.
In previous posts we have discussed the GNS construction, which connects states of the algebra to its representations. We know that pure states lead to irreducible representations, mixed states to reducible ones. We know that a general state ω on B(H) is uniquely described by a "density" matrix:
ω(a) = Tr(ρa), a∈A,
where ρ is a non-negative operator on H, with Tr(ρ)=1.
Since we are including here the infinite-dimensional case, it is appropriate to recall that
Tr(ρa) = Σi (ei, ρa ei),
where ei is any orthonormal basis in H. The series is absolutely convergent, and its sum is independent of the choice of the orthonormal basis. Every density matrix has a purely discrete spectrum, so, in particular we can choose the orthonormal basis ei consisting of eigenvectors of ρ:
ρei = pi ei,
where pi, not necessarily all different (finite multplicities may occur - Why only finite for non-zero eigenvalues?), satisfy:
pi ≥ 0, Σi pi =1.
In this case we get (Why?)
Tr(ρa) = Σi pi (ei, a ei).
Thus, in general, ω is a mixture, a weighted sum (with probabilities pi) of pure states defined by the eigenvectors ei.
In an infinite-dimensional Hilbert space the landscape is much more interesting than in the finite-dimensional case. The algebra B(H) contains then several important non-trivial two-sided *-ideals (cf. Wikipedia Trace class):
{ finite rank} ⊂ { trace class } ⊂ { Hilbert-Schmidt } ⊂ { compact }.
For a finite-dimensional H, they all coincide and are equal to the whole B(H), but in the infinite-dimensional case all inclusions above are proper. For our purposes we dot need to worry now about finite rank and compact operators, but we need to know something about the trace class *-ideal, denoted B1(H), and the Hilbert-Schmidt *-ideal, denoted B2(H):
B1(H) = {T∈B(H): Tr(|T|) < ∞},
where |T| = √(T*T).
B2(H) = {T∈B(H): Tr(T*T) < ∞}.
If T,S∈B2(H), then TS∈B1(H). The ideal of of Hilbert-Schmidt operators B2(H) is a Hilbert space when equipped with the scalar product <S,T> defined as
<S,T> = Tr(S*T).
This Hilbert space plays the crucial role in the following. In the finite-dimensional case it coincides with the whole of B(H), but in the infinite-dimensional case it is a proper ideal of B(H).
Remark 1. It should be also noticed that while B2(H) is a complete Hilbert space when equipped with the scalar product <S,T>, it is not a closed subspace when considered as a subspace of B(H) equipped with the operator norm. The closure of B2(H) with respect to the operator norm is the ideal of compact operators.
Remark 2. This remark concerns our notation here. We were denoting the elements of the Clifford algebra A with letters like a,b,u,v etc. Now, when considering Hilbert spaces, we are using letters S,T, etc. for operators in B(H) - as it is done, for instance, in Wikipedia. This may be somewhat confusing, but once we know about it, we can live with it. Each symbol should always be considered within a context, since it is the context that gives meaning to the symbol.
Remark 3. I am assuming that the Reader knows the rudiments of quantum mechanics in a Hilbert space. Probably the best reference for that is Brian C. Hall, "Quantum Theory for Mathematicians, Springer 2013, in particular Ch. 19.2 Trace-Class and Hilbert-Schmidt operators, and Ch. 19.3 Density Matrices: The General Notion of the State of a Quantum System. But if you have any questions - don't hesitate to ask, and I will do my best to answer them.
The standard GNS construction
Now we need to return to the GNS construction. The door to this subject has been closed in Part 46, but now, dealing with "emergence of time", we need to reopen it, even if only a little bit to take a peek at it again, for the particular case when the state ω is a faithful state. For a general state ω we consider the left ideal Iω = {a∈A: ω(a*a)=0} and construct the quotient A/Iω. But for a faithful state ω, we have Iω={0}, and no quotient is needed in this case. Thus we define the inner-product (or "pre-Hilbert") space Hω = A, and equip it with the scalar product
<a,b>ω = ω(a*a).
For A=B(H) this becomes
<a,b>ω = Tr(ρa*b).
In the infinite-dimensional case Hω is not yet complete, so we define the Hilbert space Hω as the completion of Hω. So, we have
A = B(H) = Hω ⊂ Hω.
The scalar product <a,b>ω extends to Hω by continuity.
Notice that, for a faithful state ω, the space Hω, in fact, does not depend on ω, only Hω does. The same concerns the representation πω, which is defined on as Hω as
πω(a)b = ab,
and extended by continuity to Hω.
Similarly for the cyclic vector Ωω defined as
Ωω = 1∈A.
With these definitions we have
ω(a) = <Ωω, a Ωω>ω.
The advantage of the standard GNS construction is that the vector Ωω representing the state ω is independent of the state. The disadvantage is that the scalar product of the representation Hilbert space is ω-dependent. There is, however, an alternative construction in which the Hilbert space and the scalar product are fixed, ω-independent, but, instead, the cyclic vector representing ω is depends on ω. We will use this alternative construction in our discussion of the Tomita flow.
Alternative GNS construction for faithful states on B(H)
In the alternative construction of the representation unitarily equivalent to the standard construction we use one Hilbert space for all faithful states - the spaces B2(H) with the scalar product:
<a,b> = Tr(a*b).
The *-representation is the same as before
π(a)b = ab, (1)
but the cyclic vector, which we denote Ωρ, depends on the state:
Ωρ = ρ½. (2)
Here we notice two things. In (1) we have a∈B(H) and b∈B2(H). Since B2(H) is a two-sided ideal of B(H), the product ab is again in B2(H), therefore the representation is well defined. In (2) we have ρ½, ρ being a positive trace class operator. Thus ρ½, is a Hilbert-Schmidt operator (Why?). Therefore Ωρ is well defined vector of B2(H), with (Why?)
<Ωρ , Ωρ> = 1.
Moreover, we have (Why?)
ω(a) = <Ωρ , π(a)Ωρ>.
The vector Ωρ is cyclic for the representation π of B(H) on B2(H). (Why?)
Remark 4. According to Theorem 4.5.3 in Part 42 the representations πω on Hω and π on B2(H) are unitarily equivalent. But how can it be? In infinite dimension B2(H) is strictly smaller than B(H), and B(H) is a subspace of its completion Hω. How can a subset of some set be isomorphic to a superset of this set? Yet in infinitely dimensions this can easily happen. It certainly contradicts our intuitions based on finite-dimensional examples. We need to learn how to live with it, and how to make use of it.
Left and right representation
B2(H) is a two-sided ideal. But in (1) we have used only the fact that it is a left ideal. Our representation π defined in (1) is just the left regular representation, which previously we have denoted by L. So, instead of π we can as well use the notation L. But B2(H) is also a right ideal, therefore we can define another representation of B(H) on B2(H), let us denote it by π':
π'(a)b = ba.
Strictly speaking π' is the representation of the opposite algebra, since π'(ab)=π'(b)π'(a). We have then two subalgebras of B(B2(H)): π(A) and π'(A). Since left actions commute with right actions, the operators of π(A) commute with the operators of π'(A). In fact we have
π'(A) = π(A)', π(A) = π'(A)'.
Vector Ωρ is a cyclic and separating vector for both von Neumann algebras π(A) and π'(A). So, we have now a perfect symmetry. This symmetry will be used in the construction of the Tomita's flow that we are ready to discuss. This will be the subject of the following post. If you have any questions - please, ask.
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