- #1

- 24,775

- 792

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

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 U

X → U

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=ZO...6AEwAw#v=onepage&q=minoru tomita math&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 U

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

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

Now the star mapping X → X* can be viewed as an operator S on the hilbert space

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

<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)

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 x

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

X → U

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 U

^{t}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 → U

^{t}XU^{-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=ZO...6AEwAw#v=onepage&q=minoru tomita math&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 U

^{t}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:__M__→__M__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 x

^{i}on the unit circle. Like those numbers e^{2πit}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 → U

^{t}X U^{-t}
Last edited: