I (extended) Page-Wootters formalism

[Heidi]

Hi Pfs,
I am beginning to read this short paper
https://arxiv.org/abs/1504.04215
(quantum time)
i read things like that on first page:
Ht is the space of a system T (we call it the clock system) isomorphic to a Hilbrt space of a particle on a line. this space is equipped with
coordinates T and Omega (with [T,Omega] = i. they represent position and momentum
they say that according to certain restrictions (which ones?) they can be interpreted as time and energy indicators.
I would like to read your comments on these restrictions.

 

[LittleSchwinger]

they say that according to certain restrictions (which ones?) they can be interpreted as time and energy indicators.

The projector onto the physical state subspace given in equation (1).

 

[Heidi]

thanks for the answer. other questions will follow....

 

[Heidi]

I do not understand what is (mathematically) the norm in equation 26.
How is it defined?
thanks.

 

[LittleSchwinger]

I do not understand what is (mathematically) the norm in equation 26.
How is it defined?
thanks.

It's the typical Hilbert space norm induced by the inner product.

 

[Heidi]

I see that you like short answers
could you give details about the inner product which is used here?

 

[PeterDonis]

I see that you like short answers

The answer he gave was perfectly appropriate for an "A" level thread. (Indeed, more than should have been needed--see below.)

could you give details about the inner product which is used here?

If you don't know what a Hilbert space inner product is, you do not have the background knowledge for an "A" level thread. (Indeed, even your question about the norm does not show that level of background knowledge.)

I am changing the thread level to "I".

 

[LittleSchwinger]

Heidi just to say, in a friendly way, if you don't immediately know what a Hilbert space product is it will be very difficult to learn the Page-Wooter formalism. What textbooks have you used to learn Quantum Theory?

 

[Heidi]

Zee,Robert Wald....

 

[LittleSchwinger]

Do you mean Wald's "Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics"?

 

[Heidi]

I read these books a long time ago. i am 76 years old now...

 

[LittleSchwinger]

No worries!

So if I said the norm is
$||\psi|| = |(\psi,\psi)|^{1/2}$
would that make more sense to you?

With $(\psi,\psi)$ being the usual inner product between vectors in a Hilbert space.

 

[Demystifier]

Zee,Robert Wald....

That explains why you don't know what is Hilbert space.

 

[PeterDonis]

That explains why you don't know what is Hilbert space.

Wald's monograph discusses those in some detail so I don't think he can be the reason.

(Zee is another matter, though...)

 

[Demystifier]

Wald's monograph discusses those in some detail so I don't think he can be the reason.

Wald discusses it at a rather abstract and advanced level, which is hard to understand for someone who haven't seen it before at a more concrete and elementary level. If Wald and Zee are the only quantum books that he has seen in his life, it's very hard to learn basics of QM just from that.