The Boundary is Mixed

1. Jul 4, 2013

marcus

From time to time the Magic Grandmother of Philosophy flies down from the sky and touches physics with her magic wand and it the field becomes slightly different. Philosophy is the careful examination of the concepts we use to think with. It can spark revolutions in thought.

I've noticed it's not uncommon for people who have changed foundations in physics to have been educated in philosophy and even to have taught philosophy or held philosophy chairs at their universities for at least part of their careers. Wikipedia is wonderful for details like this. Maybe their experience in philosophy helped them, at a crucial moment, to ask a deeper question---which turned out (that time) to be the right one to ask.

I think of Ernst Mach asking what acceleration might be relative to, how could you tell, and what caused objects to resist acceleration. He held a philosophy chair at the U. of Vienna for part of his life. People who critically explore the concepts that other people take for granted. Emmy Noether wondering where conservation laws come from. Maybe she wasn't officially a philosopher but wouldn't you say she thought like one?

So I want to say the obvious thing, at least I guess it's obvious: the boundary of physics is not fixed, it's mixed. There is no clear impermeable demarcation between it and neighboring lines of investigation.

I was struck by the philosophical novelty of this paper:
http://arxiv.org/abs/1306.5206
The boundary is mixed
Eugenio Bianchi, Hal M. Haggard, Carlo Rovelli

In this case (of a revolutionary gambit that might or might not succeed) the philosophical opening moves, I think, go like this: with GENERAL COVARIANCE you can't have TIME until you have a solution to the equation, which is to say a geometric PROCESS. Time should hatch from the process.

And the word "STATE" has too much the favor of "state at a particular time". State is a cognate of "status" or "standing". So maybe the Hilbert space should not be a space of states but a space of processes.

And suddenly we have a new format for quantum field theory and quantum statistical mechanics---a new format that is general covariant.

Last edited: Jul 4, 2013
2. Jul 4, 2013

marcus

The new [general covariant] format for QFT and QSM is a triple (B, A, W) where B is a Hilbert space, A is an algebra of self-adjoint operators on B, and W is a linear functional defined on the hilbert B (a space of "processes" we could say, or of "boundary conditions"). W, by giving the amplitudes of elements of B, tells us the quantum DYNAMICS.

I posted about this earlier. What strikes me now is that this is just one instance of a general thing that happens from time to time in Physics. Somebody realizes that everybody has been thinking with concepts that are just a bit wrong---just a bit illogical or shallow. Much of physics can be described on the basis of QFT and QSM and these use special relativity. So you can say what are initial and final states. But in the real world, in nature, you cannot do this. The causality structure is not given, but remains to be determined.

So if you have a process in some bounded region, that you want to compute amplitudes for, you do not have initial and final states, all you have are the boundary plus the events recorded and measurements made on the boundary. I'll recall what was said previously.

Much
As a language note, we are habituated to refer to elements of a Hilbert space as "states". So even though it does not have quite the right connotation, since the usage in ingrained, the authors continue referring to elements ψ of the boundary Hilbert space B as "boundary states". So the function W gives you the amplitude W(Ψ) of a boundary state Ψ.

Last edited: Jul 4, 2013
3. Jul 4, 2013

atyy

Is the boundary concept here the same as in http://arxiv.org/abs/1209.4539 , except that since it is mixed, not pure, one should use a density matrix?

4. Jul 4, 2013

marcus

If you just look at the paper itself there no LQG formalism or any explicit reference to LQG. They are working at a different level of abstraction. In a paragraph in the last section they refer to Quantum Gravity in general terms. I can't relate this paper to the one you cite (by Dittrich, Hellmann,) which is entirely involved with LQG-type stuff, spinfoams etc.

In the Bianchi Haggard Rovelli paper some of their references e.g. [3,4,8,24, 28] are to explicitly LQG papers. But most references are to other stuff. I'm just don't see how to make the connection.

I see they refer to [22] work in progress by Bianchi Haggard "to appear 2013". this is mentioned 3 paragraphs before the end, on page 7, as having inspired the present paper and as containing some worked examples. Something puzzling is said there: That it might be better to treat the boundary hilbert space as non-separable.

==quote==
Up to this point we have emphasized the mixed state character of the boundary states in order to make a clear connection with the standard quantum formalism. However, note that from the perspective of the fully covariant general boundary formalism (see section IV) there is always a single boundary Hilbert space B that can be made bipartite in many different manners. From this point of view it is more natural to call these boundary states non-separable. Then, local gravitational states are entangled states. This was first appreciated in the context of the examples treated in [22], which was an inspiration for the present work.
==endquote==

In the *algebra description of a quantum system what is called a "state" is a positive linear functional defined on the algebra with certain properties, which I don't think necessarily comes from a vector in the hilbertspace or from a statistical combination of such. I'm trying to make sense of what they say in the abstract:
We observe that the distinction between pure and mixed states weakens in the general covariant context, and surmise that local gravitational processes are indivisibly statistical with no possible quantal versus probabilistic distinction.
I'm trying to recall particulars--studied this at one time. I'll have to leave this reply to you as provisional and go check details. You could be right about the representation of the state as a density matrix. But what if B does't factor into initial/final tensor form? Would a density matrix necessarily exist? It could be a case where when general covariance comes into it we are forced to go up a level of abstraction and take the C* algebra version of quantum theory. Sorry I'm still struggling with this and unable to answer more clearly.

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5. Jul 4, 2013

marcus

Atyy maybe since I don't understand how this distinction weakens between pure and mixed states I ought to put the relevant text out there for us all to look at. We may be able to help each other understand.
==quote page 4==
IV. GENERAL BOUNDARY
We now generalize the boundary formalism to genuinely (general) relativistic systems that do not have a non-relativistic formulation.

A quantum system is defined by the triple (B,A,W). The Hilbert space B is interpreted as the boundary state space, not necessarily of the tensor form. A is an algebra of self-adjoint operators on B. The elements A, B, ... ∈ A represent partial observables, namely quantities to which we can imagine associating measurement apparatuses, but whose outcome is not necessarily predictable (think for instance of a clock). The linear map W on B defines the dynamics.

Vectors Ψ ∈ B represent processes. If Ψ is an eigenstate of the operator A ∈ A with eigenvalue a, it represents a process where the corresponding boundary observable has value a. The quantity
W(Ψ) = ⟨W|Ψ⟩ (33)
is the amplitude of the process. Its modulus square (suitably normalized) determines the relative probability of distinct processes [8]. A physical process is a vector in B that has amplitude equal to one, namely satisfies
⟨W |Ψ⟩ = 1. (34)
The expectation value of an operator A ∈ A on a physical process Ψ is
⟨A⟩ = ⟨W |A|Ψ⟩. (35)

If a tensor structure in B is not given, then there is no a priori distinction between pure and mixed states. The distinction between quantum incertitude and statistical incertitude acquires meaning only if we can distinguish past and future parts of the boundary [12, 13].

So far, there is no notion of time flow in the theory. The theory predicts correlations between boundary observables. However, as pointed out in [14], a generic state Ψ on the algebra of local observables of a region defines a flow ατ on the observable algebra by the Tomita theorem [14], and the state Ψ satisfies the KMS condition for this flow...
==endquote==

I think what they mean by "of the tensor form" for the boundary Hilbert space B to split in a unique way into the tensor product of initial states and final states.

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6. Jul 4, 2013

atyy

marcus, I don't really have a comment, since I don't understand the same parts you mention. I only have a feeling the conclusion is right from my previous attempts to understand Oeckl's boundary proposal, and from AdS/CFT.

My feeling that the Dittrich work is related is because Hellmann's on that paper - is he the same as f-h?

Some time ago f-h posted favourably on the Oeckl-Rovelli boundary proposal, which seems the same as the Oeckl boundary formalism in the Bianch-Haggard-Rovelli paper. So I suspect the "boundary" is the same in the Dittrich-Hellmann-Kaminski paper.

Here's f-h's quote https://www.physicsforums.com/showpost.php?p=3431056&postcount=12

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7. Jul 5, 2013

atyy

I still haven't made any progress in understanding this. Could it be possible they mistyped, and intended to write that there is no distinction between entangled and unentangled states without dividing the system into two parts (which is what I usually associate with tensor structure)? Then if two parts of a system are entangled, the state (density matrix) of a subsystem is mixed, even though the full system is pure. So entanglement is a natural way of getting a mixed state, without giving up pure states.

They seem to say something close to this idea at the end, although it's perhaps not quite the same (p7, bottom left column): "Up to this point we have emphasized the mixed state character of the boundary states in order to make a clear connection with the standard quantum formalism. However, that from the perspective of the fully covariant boundary formalism (see section IV) there is always a single boundary Hilbert space B that can be made bipartite in many dierent manners. From this point of view it is more natural to call these boundary states nonseparable. Then, local gravitational states are entangled states."

8. Jul 5, 2013

martinbn

No, they didn't mean entagled/unentangled. Mathamatically it is the same, whether the tensor is pure or not, but the context here is different.

9. Jul 5, 2013

atyy

What I don't understand is why is a tensor structure needed for a mixed state. If I have just a single spin at one time, then I can write a mixed density matrix, can't I? Here I'm thinking a single spin has no tensor structure, since it's just "one object".

10. Jul 5, 2013

atyy

Ok, maybe I understand. By "state" they mean something like density matrix (Eq 6, 7). Normally the density matrix has things like |a><b|. But they're saying that in the most general case, it need not contain things like that, in which case "pure" and "mixed" aren't defined.

Last edited: Jul 5, 2013
11. Jul 14, 2013

atyy

Does their conclusion depend on the local field theory having a unique ground state (they mention the Wightman axioms somewhere)? If it does, is that assumption justifiable?