Global emergent time, how does Tomita flow work?


by marcus
Tags: emergent, flow, global, time, tomita, work
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Dec26-12, 12:04 PM
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Tomita time is an intrinsic observer-independent time variable available to us for fully general relativistic analysis. As far as I know it is the only such time available for things like
general covariant statistical mechanics
GC statistical quantum mechanics
GC quantum field theory

Here I want to discuss some nuts&bolts of the Tomita flow construction.

In GR, "coordinate time" is not really a time---it's neither observable nor physically meaningful. On the other hand, observer time is not defined until one has already fixed on a particular curved space-time geometry. So observer time cannot be used if the geometry itself is included as part of the dynamics. Furthermore in a quantum theoretical treatment the problem is worse since a space-time cannot be determined any more than can a continuous particle trajectory.

But to do certain kinds of analysis we need an independent time variable. The Tomita flow is a one-parameter group of transformations of the observables algebra M that arises naturally as powers Ut of a distinguished unitary operator U. I want to discuss the natural way you get that operator U. Tomita time is the real number t that appears as the exponent. It is the parameter of the one-parameter group of changes that operate on the observable algebra M, mapping one element X of M to another.

X → UtXU-t

The construction is not all that complicated. It is described on page 517 of the Princeton Companion to Mathematics.
http://books.google.com/books?id=ZOf...20math&f=false

Thinking of M as the C* algebra generated by all possible measurements, this one-parameter group shifts and shuffles the observables among themselves. Obviously a key question is how do you construct the basic operator U whose powers Ut serve to represent the passage of time?

To start with, we represent what we think we know about the world (statistical correlations among measurements, levels of confidence, uncertainty, variances, expectation values) by a positive state function ω: M → ℂ.

Positive here means that for any X in the algebra ω(X* X) > 0, and equals zero only in case X itself is zero. The state ω defines an inner product on M which allows us to treat M as a hilbert space M.

Now the star mapping X → X* can be viewed as an operator S on the hilbert space M. The map S:MM is almost linear--the term for it is conjugate linear, because scalar multiplication carries over using the complex conjugate of the scalar.

I suppose the letter S is used to denote this operator version of the * because S stands for "star". Whatever the reason, in every treatment of Tomita flow I've seen, this notation has been used.

Now the first cleverness occurs. Since M is now a hilbert space we have a well-defined notion of the ADJOINT of an operator. That is also always denoted by a star. In matrix representations it's the conjugate transpose--you flip the matrix over and take conjugates when the entries are complex. And it's defined more generally using the inner product:
<T*X, Y> = <X, TY>.
So we can take the star of the star mapping S. It gets slightly non-trivial here simply because we are using * in two senses: the original C* algebra operation and the new adjoint available now that M is a hilbert space with an inner product. Now we can multiply S together with its adjoint S*.

That's it. S* S is a well-defined operator on the observables algebra M viewed as a hilbert space. It's easy to see that it is self-adjoint. Self adjoint operators are the analogs of real numbers, just as unitary operators are analogs of the numbers on the unit circle. To see that it is self-adjoint you just have to verify that
<S*S X, Y> = <X, S*S Y>

The unitary operator U, mentioned earlier, which is the seed of the Tomita flow, the seed of change, the seed of the passage of time in the world described by (M, ω) is simply given by
U = (S*S)i

It's a basic fact about hilbertspace that you can raise a self-adjoint operator to an imaginary power like i, and get a unitary. This is analog of raising a real number x to imaginary power and getting a number xi on the unit circle. Like those numbers e2πit we are always seeing.

So that is how U, the seed of the Tomita flow, is obtained.
The flow simply consists of bracketing an observable X with that unitary U raised to powers t and -t. That is what CHANGE is, how the passage of time works, in the star-algebra world defined by (M, ω)

Time is the logarithm of change, to the base U. And it is observer-independent.

Time is not a pseudo-spatial dimension, in this view of the world. It is the parameter of a flow on the observables.

X → Ut X U-t
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Dec26-12, 03:12 PM
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We need to keep handy a few links to discussions of Tomita flow, as a global emergent time, in the literature:
Here's a Vimeo video of part of a talk on Tomita time by Matteo Smerlak:
http://vimeo.com/33363491
from a March 2011 workshop at Nice.

Here's the article by Alain Connes and Carlo Rovelli:
http://arxiv.org/abs/gr-qc/9406019

and a seminal 1993 paper, The Statistical State of the Universe
http://siba.unipv.it/fisica/articoli....1567-1568.pdf
which shows how T-time recovers usual conventional time in several interesting contexts, notably standard Friedmann-model cosmology.

Wherever a local observer's time is defined, one can compare it with the global emergent T-time. The ratio between the two times is physically meaningful as shown in this paper by Smerlak and Rovelli.
http://arxiv.org/abs/1005.2985

Here's a recent paper showing how T-time could be used in approaches to general relativistic statistical mechanics and general covariant statistical QM.
http://arxiv.org/abs/1209.0065
===================

It is unlikely that QG can be formulated in terms of a 4D spacetime geometry, for the same reason that in quantumtheoretical treatments a particle does not have a definite trajectory. One can make a finite number of observations about where it goes, and they are correlated, but one cannot say that a continuous trajectory exists. Same with a 4D spacetime which is a trajectory of the geometry. This is explained in Chapter 1 of Approaches to Quantum Gravity (D. Oriti ed.)
http://arxiv.org/abs/gr-qc/0604045 (see page 4 of preprint)

This indicates that the set of all possible measurements M must replace the 4D manifold space-time, as a basis for GC QFT (general covariant quantum field theory). Each of the observables in M has its own uncertainty built in. Since the correlations are statistical we have no difficulty obtaining an indefinite causal structure, or an uncertain geometric evolution. Features that may be difficult to obtain with a 4D manifold seem to come automatically with the C* algebra M.

The state ω defined on M, represents what Bohr referred to as "what we can SAY" about nature or the world, as opposed to what IS.

Jeff Morton's blog on T-time (with John Baez comment):
http://theoreticalatlas.wordpress.co...d-tomita-flow/

A recent paper by Robert Oeckl on the Boundary Formulation of QT interestingly refers to the (M,ω) picture, suggesting that he, too, may be looking at it as a possible way to go.
http://arxiv.org/abs/1212.5571
A positive formalism for quantum theory in the general boundary formulation
Robert Oeckl (CCM-UNAM)
(Submitted on 21 Dec 2012)
We introduce a new "positive formalism" for encoding quantum theories in the general boundary formulation, somewhat analogous to the mixed state formalism of the standard formulation. This makes the probability interpretation more natural and elegant, eliminates operationally irrelevant structure and opens the general boundary formulation to quantum information theory.
28 pages

to clarify the relevance here is a quote from end of section 2 on page 4:
"...The time-evolution operator U ̃ restricted to self-adjoint operators produces self-adjoint operators. Moreover, it is positive, i.e., it maps positive operators to positive operators. It also conserves the trace so that it maps mixed states to mixed states. These considerations suggest that positivity and order structure should play a more prominent role at a foundational level than say the Hilbert space structure of H or the algebra structure of the operators on it from which they are usually derived.
Algebraic quantum field theory [7] is a great example of the fruitfulness of taking serious some of these issues. There, one abandons in fact the notion of Hilbert spaces in favor of more flexible structures built on C∗-algebras. Also, positivity plays a crucial role there in the concept of state."
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Dec26-12, 06:16 PM
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To get a little intuition about the form of the Tomita flow:
X → Ut X U-t
let's imagine a concrete example. Mozart takes time off from writing Don Giovanni and for his amusement he uses a crude spectrograph to measure the color of the sun. That is measurement X

Now we want to see what that measurement turns into when advanced by t = 200 years.
Ronald Reagan takes time off from golf, napping at staff meetings, and his busy schedule of public appearances to go out and use the same crude spectrograph to record the mix of wavelengths of the sun.

The recipe is UNDO 200 years of change. Do Mozart's measurement of sunlight. Then RESTORE 200 years of change

X → U+200 years X U-200 years

This is how we map the measurement Moz. made into the one Ron made.
I have to go, no time to edit back soon.

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Dec26-12, 10:29 PM
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Global emergent time, how does Tomita flow work?


Marcus, you always manage to make my brain explode! I think that's a good thing!
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Dec27-12, 12:37 AM
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Quote Quote by Drakkith View Post
Marcus, you always manage to make my brain explode! I think that's a good thing!
I'm delighted to hear this! I'm excited by this development in quantum gravity too. It must be what it feels like when a possibly major development comes along.
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Hi Marcus! Is this related to my question in the other thread?
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Dec27-12, 10:27 AM
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Quote Quote by MTd2 View Post
Hi Marcus! Is this related to my question in the other thread?
I replied in the other thread.
http://www.physicsforums.com/showthr...77#post4209977

EDIT to respond to following post:

Hi Arivero, KMS states are referred to quite a lot in the Connes-Rovelli paper, as I recall. Also in Rovelli's recent paper (on general relativistic statistical mechanics).
http://arxiv.org/abs/gr-qc/9406019
http://arxiv.org/abs/1209.0065

To give context for any readers not familiar with it, I'll simply quote the lead paragraph at WikiP.
http://en.wikipedia.org/wiki/KMS_state
"KMS state
In the statistical mechanics of quantum mechanical systems and quantum field theory, the properties of a system in thermal equilibrium can be described by a mathematical object called a Kubo-Martin-Schwinger state or, more commonly, a KMS state: a state satisfying the KMS condition. Kubo (1957) introduced the condition, Martin & Schwinger (1959) used it to define thermodynamic Greens functions, and Rudolf Haag, M. Winnink, and N. M. Hugenholtz (1967) used the condition to define equilibrium states and called it the KMS condition."
arivero
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First time I saw it, it was for some study of states (KMS states??) in C*-algebras.
marcus
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Dec27-12, 08:56 PM
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Hi Alejandro, I replied to you earlier in postscript to post#7, but wanted to ask what role you think the KMS condition plays? It gets mentioned (I gave some references earlier) but does not seem to be needed in defining Tomita flow or the associated time.

========quote Princeton Companion to Mathematics (2008) page 517 ==========
Modular theory exploits a version of the GNS construction (section 1.4). Let M be a self-adjoint algebra of operators. A linear functional φ: M → C is called a state if it is positive in the sense that φ(T*T) ≥ 0 for every T in M (this terminology is derived from the connection described earlier between Hilbert space theory and quantum mechanics). for the purposes of modular theory we restrict attention to faithful states, those for which φ(T*T) = 0 implies T = 0. If φ is a state, then the formula

<T1, T2> = φ(T1* T2)

defines an inner product on the vector space M. Applying the GNS procedure, we obtain a Hilbert space HM. The first important fact about HM is that every operator T in M determines an operator on HM. Indeed a vector V in HM is a limit V = limn→∞ Vn of elements in M, and we can apply an operator T in M to the vector V using the formula

TV = lim TVn

where on the right-hand side we use multiplication in the algebra M. Because of this observation, we can think of M as an algebra of operators on HM, rather than as an algebra of operators on whatever Hilbert space we began with.

Next, the adjoint operation equips the Hilbert space HM wtih a natural "anti linear" operator
S: HM → HM by the formula [see footnote]

S(V) = V*.

Since U*g = Ug-1 for the regular representations, this is indeed analogous to the operator S we encountered in our discussion of continuous groups. The important theorem of Minoru Tomita and Masamichi Takesaki asserts that, as long as the original state φ satisfies a continuity condition, the complex powers

Ut = (S*S)it

have the property that

Ut M U-t = M for all t.

The transformations of M given by the formula T → Ut T U-t are called the modular automorphisms of M.
Alain Connes proved that they depend only in a rather inessential way on the original faithful state φ. To be precise, changing φ changes the modular automorphisms only by inner automorphisms, that is, transformations of the form T → UTU-1 where U is a unitary operator in M itself. The remarkable conclusion is that every von Neumann algebra M has a canonical one-parameter group of "outer automorphisms," which is determined by M alone and not by the state φ that is used to define it.


[footnote] The interpretation of this formula on the completion HM of M is a delicate matter.

==endquote==
Nowhere here is the KMS condition invoked. I think KMS plays a role in one or more significant APPLICATIONS of the Tomita flow idea, but is not essential to its definition.

http://books.google.com/books?id=ZOf...20math&f=false

BTW the chapter on operator algebras this is quoted from is by Nigel Higson and John Roe (the PCM is edited by Timothy Gowers)
marcus
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Dec27-12, 10:00 PM
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I think the Tomita flow ("T-flow") and Tomita time ("T-time") are going to be important because physics will probably come to be formulated on a quantum geometric basis to handle certain extreme situations where the geometry is an important part of the dynamics. And QG is unlikely to involve spacetimes (= continuous trajectories). T-time is the only option I know of being proposed as an independent time variable for general covariant statistical mechanics or GC quantum theory. And it is intrinsic--arises naturally when we have a C*algebra M and a state ω representing the world and what we think we know about it.

Given that it is going to be of major importance, I would like to see if there's a good notation that will make it easy to grasp. What do you call (S* S)i a certain self-adjoint operator raised to the power i? It's a unitary operator on HM, the algebra M competed as hilbertspace. Earlier I denoted it simply by the letter U. Maybe that's not distinctive enough, what about Q? And let me try using A as a generic element of the algebra M.

Let's try that. S is the star operation on M carried over to HM. S* is the adjoint.
The product S* S is a positive self-adjoint operator on HM. A positive real number raised to the power i is just a number on the unit circle, and we can pick a basis that diagonalizes S* S and raise every eigenvalue to the power i. We get a unitary operator.
Let's try denoting that Q = (S* S)i

Then Tomita's theorem says we have a flow on M that is simply given by
A → Qt A Q-t
where t is a real number parameter of the flow, and corresponds to time (measured in Planck units in cases where it has been checked.)

So advancing a measurement or observation A, by some 200 years,
to get another measurement Q+200 years A Q-200 years
can be thought of intuitively as "undo 200 years of change, perform A, and restore the 200 years." Q is the basic Tomita unitary operator, Qt is how change is represented. Intuitively "T-time is the logarithm of change, to the base Q. And it is observer independent. And it is not a pseudospatial fake dimension"
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Dec27-12, 11:05 PM
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Quote Quote by arivero View Post
First time I saw it, it was for some study of states (KMS states??) in C*-algebras.
This is rather technical, readers may wish to skip this post. I found a source
http://arxiv.org/pdf/math-ph/0511034v1.pdf
that summarizes stuff about KMS (kubo martin schwinger) that Arivero was asking about.
This is from page 2, section 2.1:
==quote==
The modular automorphism group satisfies a condition which had already been used in mathematical physics to characterize equilibrium temperature states of quantum systems in statistical mechanics and field theory — the Kubo–Martin–Schwinger (KMS) condition. If M is a von Neumann algebra and {αt | t ∈ ℝ} is a σ-weakly continuous one-parameter group of automorphisms of M, then the state φ on M satisfies the KMS-condition at (inverse temperature) β (0 < β < ∞) with respect to {αt} if for any A,B ∈ M there exists a complex function FA,B(z) which is analytic on the strip {z ∈ C | 0 < Imz < β} and continuous on the closure of this strip such that
FA,B(t) = φ(αt(A)B) and FA,B(t + iβ) = φ(Bαt(A)) ,
for all t ∈ ℝ . In this case, φ(α(A)B) = φ(BA), for all A,B in a σ-weakly dense, α- invariant *-subalgebra of M. Such KMS-states are α-invariant, i.e. φ(αt(A)) = φ(A), for all A ∈ M, t ∈ ℝ, and are stable and passive (cf. Chapter 5 in [3] and [5]).
Every faithful normal state satisfies the KMS-condition at value β = 1 (henceforth called the modular condition) with respect to the corresponding modular automorphism group.
...
...
The modular automorphism group is therefore endowed with the analyticity associated with the KMS-condition, and this is a powerful tool in many applications of the modular theory to mathematical physics. In addition, the physical properties and interpretations of KMS-states are often invoked when applying modular theory to quantum physics.
Note that while the non-triviality of the modular automorphism group gives a measure of the non-tracial nature of the state, the KMS-condition for the modular automorphism group provides the missing link between the values ω(AB) and ω(BA), for all A, B ∈ M (hence the use of the term “modular”, as in the theory of integration on locally compact groups).
==endquote==
This article was written by Stephen Summers for the Elsevier Encyclopedia of Mathematical Physics.
So FWIW this summarizes the fact that the modular automorphism group (the Tomita flow we were talking about) has to do with the non-commutativity of the observables algebra.
And it also points out the usefulness in applications that comes from the modular group satisfying the KMS-condition.
Here's another possibly useful source: an open access article by Marc Rieffel and Alfons van Daele about Tomita's theorem and related, also called Tomita-Takesaki theory.
http://projecteuclid.org/DPubS?verb=...05&page=record
John86
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Marcus,
Not directly related links but Aspect and Connes talking about time in these talks !
Hope it is of your tatse..

http://noncommutativegeometry.blogspot.nl/

http://math.univ-lille1.fr/~cempi/co...programme.html
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Quote Quote by John86 View Post
... Connes talking about time ... !
Hope it is of your taste..

http://math.univ-lille1.fr/~cempi/co...programme.html
Connes in the first 5 minutes of the video:
"the origin of time is quantum mechanical and it is really coming from the non-commutativity of the variables.."

Very much to my taste! I hope others will watch the beginning of the Connes video. Thanks!
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Dec28-12, 09:18 PM
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One reason I think Tomita time is important is that it is so strongly motivated. It's not clear to me that the motivation to use the parameter of the Tomita flow as time, in the study of physical systems, was appreciated by Tomita, or by the younger mathematician Takesaki who helped bring his research to light in 1970. Tomita's interest was in abstract operator algebras. It was, I think, Connes and Rovelli who first realized that the Tomita theory applies to the "problem of time" in GR and the even more severe problems with time that people were faced with in QG. To review the motivation for T-time, I'll quote from page 2 of their 1994 paper.
== http://arxiv.org/abs/gr-qc/9406019 page 2 ==
In a general covariant theory there is no preferred time flow, and the dynamics of the theory cannot be formulated in terms of an evolution in a single external time parameter. One can still recover weaker notions of physical time: in GR, for instance, on any given solution of the Einstein equations one can distinguish timelike from spacelike directions and define proper time along timelike world lines. This notion of time is weaker in the sense that the full dynamics of the theory cannot be formulated as evolution in such a time. In particular, notice that this notion of time is state dependent.

Furthermore, this weaker notion of time is lost as soon as one tries to include either thermodynamics or quantum mechanics into the physical picture, because, in the presence of thermal or quantum “superpositions” of geometries, the spacetime causal structure is lost. This embarrassing situation of not knowing “what is time” in the context of quantum gravity has generated the debated issue of time of quantum gravity. As emphasized in [4], the very same problem appears already at the level of the classical statistical mechanics of gravity, namely as soon as we take into account the thermal fluctuations of the gravitational field. Thus, a basic open problem is to understand how the physical time flow that characterizes the world in which we live may emerge from the fundamental “timeless” general covariant quantum field theory [9].

In this paper, we consider a radical solution to this problem. This is based on the idea that one can extend the notion of time flow to general covariant theories, ...
==endquote==
martinbn
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Dec29-12, 10:33 AM
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Mathematically this is well known and straightforward given the algebra, but what exactly is M? Generated by all possible measurements is not clear enough (to me). What are observables in general relativity?

p.s. Sorry for the, I guess, stupid questions.
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Quote Quote by martinbn View Post
Mathematically this is well known and straightforward given the algebra, but what exactly is M? Generated by all possible measurements is not clear enough (to me). What are observables in general relativity?

p.s. Sorry for the, I guess, stupid questions.
Certainly not stupid questions! I can try to give my opinion. There has been a lot of discussion of this very thing: What are observables in general relativity? I think the question is intimately related to the challenge of combining GR with QM, and also with problems of time in GR. My opinion will necessarily be partly speculative.

You probably know that QM has evolved to a higher level of abstraction by throwing away the hilbert space (which can be recovered but is not essential to the analysis once one has the algebra).

Now what I think needs to happen and that is beginning to happen is that we can throw away the manifold that used to be necessary in GR.

Once GR has shed its manifold, and QM has discarded its Hilbertspace, the two will find they are together, they won't even have to TRY to unite---they will already be joined. This is how I think things are going.

So I think AdS/CFT is something of a distraction from the main evolutionary course, it is too elaborately manifold-based for it to be squarely in the right direction.
===here's a paraphrase something I wrote earlier which might help, or might be redundant===

People differ as to their interpretation of QM but what they are talking out mathematically is what John von Neumann presented in 1932 (Foundations of Quantum Mechanics) and subsequently in what is called von Neumann algebras. Originally these were algebras of operators on a Hilbertspace, but later they were axiomatized. It is the axiomatic stucture of the algebra, saying how its elements behave (the "operators" which no longer operate on anything) which gives meaning to the elements.
This is a familiar way mathematics develops.
The elements of a set in axiomatic set theory can be anything or nothing in particular, the structure gives them meaning. They are the primitives--one cannot say what they are. The "points" in a differential manifold are the primitives in diff geom. One cannot say what a "point" is, but the axiomatic structure saying how they behave gives them their meaning.

Originally von Neumann defined algebras of operators on a specific Hilbertspace. then that was axiomatized. what were operators on something are now primitives with no special meaning in and of themselves---their behavior and what we can do with them determined by the axioms and the C* algebra structure.

But we can THINK of M as an infinite menu of possible acts of measurement, whose results can be added, subtracted, multiplied with each other, having the * operation and the normed topology that von Neumann thought up etc.
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Dec29-12, 10:36 PM
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Martin, you might be interested in taking a look at the WikiP on John von Neumann:
http://en.wikipedia.org/wiki/John_von_Neumann
the man was phenomenal. Probably the greatest pure/applied mathematician born in the 20th century bar none.
Basic contributions in so many areas of mathematics and physics (including computer architecture and game theory). 1903-1957.
It helped me to get some perspective when I saw the comments by other mathematicians, about von Neumann and then saw what he stated he thought was his most essential accomplishment. This is from a short list of facts he submitted to the National Academy of Sciences: "The part of my work I consider most essential is that on quantum mechanics, which developed in Göttingen in 1926, and subsequently in Berlin in 1927–1929. Also, my work on various forms of operator theory, Berlin 1930 and Princeton 1935–1939; on the ergodic theorem, Princeton, 1931–1932."

He was the person who put quantum theory in the mathematical form it is today, and also axiomatized the operator *-algebra that evolved into the C* algebra.
Interpretation is another business---there can be lots of different interpretations of the same basic mathematical structures.
martinbn
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Dec30-12, 07:57 AM
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marcus (#16), yes, that helps me, and I think I understand your point. I do like the idea, as you said history shows that was the way in many areas, including algebraic geometry, Grothendieck's point of view, which can be paralleled to Gelfand's. The geometric object is determined by a certain algebra, so the properties of the algebra can be abstracted and the theory can be built without the geometric object and in greater generality. To come back to my question if we start with a Lorentzian manifold, what is the algebra? I suppose the generators have to correspond to measurements, but this needs a more formal description, and what are the relations? Now that I said this, I seem to remember seeing a paper by Geroch, where, if I am not mistaken, he was defining an algebra of observables, but I have to find it and look at it again.

#17 Oh, yes, I have looked into his life, he is indeed quite remarkable.


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