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I'll start by sketching the specific model I'm looking at, a quantum spin chain.

This is defined as N spins (2 basis states) on a 1D lattice i.e. the sites are a subset of ##\mathbb{Z}##.

Then we defined the state-space as

[tex]

\{ \psi\in\mathcal{H}_N \text{ with } ||\psi|| = 1\}\cong S^d \subset \mathbb{R}^{d+1}

[/tex]

Some clarifications are needed here first ##\mathcal{H}_N = \otimes^N \mathcal{H}_s## where ##\mathcal{H}_s\cong \mathcal{C}^2## is the single-spin state space.

The isomorphism above with the d-sphere is clear (to me) with ##d\equiv 2\cdot 2^N##.

Now I can get to the crucial part, we put a measure on the d-sphere naturally we pick the surface measure ##d\sigma(\psi)\text{ on }S^d##. The defining property given too us is

[tex]

\mathbb{E}(f) = \int_{S^d} d\sigma(\psi) f(\psi) = \int_{\mathbb{R}^{d+1}} d\sigma(\psi)\, \delta( ||\psi||-1 ) \, f\left(\frac{\psi}{||\psi||}\right)

[/tex]

Where f is a function on the d-sphere or the (d+1) dimensional real space.

This all seems okay so far.

As an example the following function was defined ##f(\psi) = \langle\psi|O|\psi\rangle = \langle O \rangle_\psi = \text{Tr}\left(O|\psi\rangle\langle\psi|\right)##.

Then we can apply the definition above

[tex]

\mathbb{E}(\langle O\rangle_\psi) = \text{Tr}\left(O|\psi\rangle\langle\psi|\right) = \text{Tr}\left(O\int d\sigma(\psi)|\psi\rangle\langle\psi|\right)

[/tex]

Now the claim is made that this is equal to ##\text{tr}(O)## which uses the normalized trace ##\text{tr}(\cdot) \equiv d^{-1}\text{Tr}(\cdot)##.

I don't get how they arrive at this claim. It is mentioned that we need to use that the measure ##d\sigma## is invariant under all unitary ##U##.

The lecturer made the remark that ##\int d\sigma(\psi)|\psi\rangle\langle\psi| ={1\!\!1}_d##.

This seems logical but where did the normalization of the trace come from if that's true?

I've been thinking about this stuff for hours now but I can't resolve my difficulties with the idea.

Thanks,

Joris

P.S.: I'm not very familiar with measure theory

P.P.S.:

The nice thing is that if the above holds, we can interpret the expectation as that of a thermal ensemble in the limit ##\beta\downarrow 0##.

Which in turn will be used when talking about the "Eigenstate Thermalization Hypothesis".

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# Expectation for many-body system

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