Spin Chronicles Part 51: Tomita's thermal flow and Mirror symmetry

 We continue from Part 50.

Relation to "thermal time"

J - Mirror symmetry of  the time flow

To relate this purely mathematical construction to the concept of "thermal time" related to physics, we need to direct our attention to quantum dynamics. To not complicate our discussion we will use Planck units. In quantum dynamics, in the Schrodinger picture, time evolution of state vectors is described by unitary operators

U(t) = e^-itH,                     (1)

where H is the Hamiltonian (energy operator). Comparing U(t) with U_ρ(s)  given by Eq. (5) we see that both expressions agree if we set s=t and

H = -log ρ.                         (2)

This is a self-adjoint, and positive (Why?) operator, unbounded if H is infinite-dimensional (Why?), and we can interpret U_ρ(s) as the operators defining the time evolution of a quantum system.

In order to understand why it is called "thermal flow" let us calculate the expectation value <E>_ρ of the energy represented by the operator H. We have

<E>_ρ = Tr(ρ H) = -Tr(ρ log ρ).

But this is exactly the expression for von Neumann entropy of the state ρ! Therefore it is the entropy of the state that drives the evolution defined by Tomita's modular flow. Calling it a "thermal flow" is thus justified.

Remark. Usually, for instance in Ref. [1], there is a different explanation, based on the particular example of a Gibbs thermal equilibrium state. But, I think, the explanation above serve its purpose as well.

Let us consider now the induced evolution U_ρ(s) on the space of Hilbert-Schmidt operators B₂(H), defined by Eq. in Part 50

U_ρ(t)| a > = | ρ^itaρ^-it >, a∈B₂(H), t∈R.      (3)

We rewrite it using ρ^it =  U(t) = e^-itH:

U_ρ(t)| a > = | e^-itHaρ^itH >.                           (4)

Since U_ρ(t) is a group of unitary operators on B₂(H), representing the time evolution there, we can write

U_ρ(t) = e^-itℋ,

where ℋ is the Hamilton's operator on B₂(H). Differentiating (4) with respect to t at t=0 we then obtain

ℋ| a > = | Ha - aH > = | [H,a] >.

It is now easy to find eigenvectors and eigenvalues of  ℋ. From the spectral decomposition of :

ρ = ∑_n p_n P_n, p_n>0,  ∑_n p_n = 1,

and Eq. (1), we get

U(t) e_n = e^{it log p_n} e_n.

Therefore, with e_mn = |e_m)(e_n|, we have

U(t) e_mn = e^{it(log p_m-log p_n)} e_mn.

Differentiating with respect to t at t=0 we get

ℋ e_mn = -(log p_m - log p_n) e_mn.         (5)

Thus e_mn are eigenvectors of ℋ, and the corresponding eigenvalues of ℋ are differences of eigenvalues of of H. In particular the spectrum of ℋ is symmetric with respect to the eigenvalue 0.

The "ground state"  Ω_ρ

We have defined Ω_ρ as

 Ω_ρ = √ρ = ∑_n (p_n)^½ P_n.

But P_n = |e_n)(e_n| = e_nn. Therefore

 Ω_ρ = ∑_n (p_n)^½ e_nn.                     (6)

From (5) we see that ℋ vanishes not only on  Ω_ρ  (which is a stationary state for U_ρ(t)), but it vanishes also on each e_nn participating in the sum (6).

Mirror symmetry

The anti-unitary operator J provides a symmetry between the two "algebras of observables" π(A), acting on B₂(H) from the left, and π'(A) acting from the right. Left and right actions commute. In fact  π(A) and π'(A) are commutants of each other, while

π(A)∩π'(A) = C1

If the intersection of a von Neumann algebra with its commutant (the center of the algebra) is trivial, we call the algebra a factor. So π(A) and π'(A) are factors with Jπ(A)J=π'(A). We can easily get (How?)

JU_ρ(t)J = U_ρ(-t),

thus

JℋJ = -ℋ.

The operator J reverses the "flow of time". It reverses the "energy" sign.

In quantum theory commuting observables are interpreted as representing two "compatible measurements". Measuring one observable does not "disturb" the measurement of another observable. Here we have two quantum "worlds". One represented by the algebra π(A), the other represented by π'(A). Measurements of one world do not disturb measurements of the other world. The two worlds have opposite "time arrows" and opposite "energy spectra". The ground state  Ω_ρ is kind of situated perfectly in the middle. The "energy" can be added to this ground state, or subtracted from it.

The two worlds have opposite "time arrows" and opposite "energy spectra".

We have already seen a similar arrangement while studying the Clifford algebra of space Cl(V). The algebra acts on itself by left and by right actions. The two actions commute. The left regular representation is reducible. It has a commutant, which is the right regular representation. Elements of the algebra are multivectors. The spin group acts on multivectors simultaneously from the left and from the right, when spinor (here a vector in H) rotates by 180 degrees, a multivector (here an element of B₂(H)) rotates by 360 degrees. Going from H to B₂(H) some information is lost.

References

[1] A. Connes, C. Rovelli, "Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories", Class. Quantum Grav. 11 (1994) 2899 .