What are the fundamental concepts and axioms of quantum mechanics?

[rpt]

What are the axioms of quantum Physics?
Does time/space considered fundamental quantities that
considered to exist in nature when formulating quantum theory?

 

[Fredrik]

What are the axioms of quantum Physics?

This is a good summary. I've had problems linking to that page before. If the link doesn't work, do a Google search for "dirac-von neumann axioms" (without the quotes) and click the link to Strocchi's book at Google Books. It should take you directly to page 72.

Strocchi's book isn't easy to read (unless you know a lot of math) so as an alternative, I suggest Isham.

Does time/space considered fundamental quantities that
considered to exist in nature when formulating quantum theory?

The axioms I linked to only treat time as fundamental. To replace time with spacetime, you should drop the axiom about the Schrödinger equation, and replace it with an axiom that says that there's a group homomorphism from the symmetry group of spacetime (which can be either Galilei spacetime or Minkowski spacetime, so the group is either the Galilei group or the Poincaré group) into the group of symmetries of the quantum system (bijections on the set of states). This recovers the Schrödinger equation, because the time translation group is a subgroup of the symmetry group of spacetime.

Chapter 2 of Weinberg ("The quantum theory of fields", vol. 1) is a pretty good place to start reading about these things. The only book I know that seems to have all the details is Varadarajan ("Geometry of quantum theory"). Varadarajan's book is really cool, but it's very difficult.

 

[rpt]

Thanks Fredrik,

I am not a person from a strong mathematical background. I am an engineer (I know some maths). Please forgive me if my comments sounds unreasonable (please correct me).

I read the google book reference you linked in your reply. To my understanding, two basic axioms are state, and observables.

I feel that accepting the existence of state as fundamantal concept implies existence of time as fundamental. State has no meaning in absence of time.

Next paragraph you say, time is replaced with spacetime in more general theory.
Does that mean existence of time and space as fundamental quantities come into picture,
not in isolation but as a unified fundamental quantity called space-time.
So is it correct to summerize as "Space-time is a fundamental quantity that exists in nature"
when formulating modern quantum theory?

 

[Fredrik]

I feel that accepting the existence of state as fundamantal concept implies existence of time as fundamental. State has no meaning in absence of time.

Yes, the concept of state only makes sense if it's possible to prepare a system in different ways, and then perform a measurement on it. So a theory that involves states must include some concept of "time".

Next paragraph you say, time is replaced with spacetime in more general theory.
Does that mean existence of time and space as fundamental quantities come into picture,
not in isolation but as a unified fundamental quantity called space-time.

Not really. The concept of "spacetime" becomes useful the moment we need to start talking about events and their coordinates. Spacetime is just the set of all events, and a coordinate system is a function that assigns a 4-tuple (t,x,y,z) to each event. Galilei spacetime is the spacetime of pre-relativistic physics. Minkowski spacetime is the spacetime of special relativity. Special relativity brings space and time together in the sense that it says that two events that you would describe as being separated only in space, I would describe as separated in space and time (assuming that you would also describe me as having a non-zero velocity component in the direction from the location of one of the events to the location of the other). Non-relativistic classical mechanics doesn't do anything like that, but the concept of spacetime is still useful there.

So is it correct to summerize as "Space-time is a fundamental quantity that exists in nature"
when formulating modern quantum theory?

I don't like to say things like that, because the theory doesn't. It doesn't talk about what "exists". It just tells you how to calculate probabilities of possibilities. However, the axioms of the theory are motivated by such ideas. For example, if there exists a mathematical "thing" that an observer S can use as a represent of the state of the system when he calculates the theory's predictions about results of experiments on it, then there must exist another such "thing" that an observer who's just like S but rotated clockwise by an angle θ can use to make those predictions. It's when we translate such ideas to mathematical language that we end up with the axiom that there's a group homomorphism from the symmetry group of spacetime into the group of symmetry transformations on the set of states.

By the way, the Schrödinger equation is obtained from a special case of that. It's what we end up with when we consider the group of translations in time instead of the full group of symmetries of spacetime (i.e. the group of functions that represent a change of coordinates from one inertial frame to another).

 

[rpt]

Thanks Fredrik,

Your ideas getting clearer to me now. So is it correct to say that,

Quantum theory does not explain the reality of nature but can predict results of expreiments accurately in an environment where space-time is intutive.

To my understanding according to the Big-bang theory, there was no space-time before Bigbang. Can that state of universe be described by quantum theory? (asumming Big-bang theory was correct). Furthermore it implies that all the physical quantities observed (space time, etc.) are not fundamental. This is a separate question to axioms of quantum theory, but it was a part of the question had in my mind when I did the original post.

I appreciate your comments, but I do not expect a perfect answer.

 

[Fredrik]

So is it correct to say that,

Quantum theory does not explain the reality of nature but can predict results of expreiments accurately in an environment where space-time is intutive.

Yes, I would say that's accurate. At least the part before "in an environment". It's not clear what that part of the statement means, at least for someone who hasn't been reading the posts you're responding to (i.e. my posts).

To my understanding according to the Big-bang theory, there was no space-time before Bigbang. Can that state of universe be described by quantum theory? (asumming Big-bang theory was correct).

If you mean the state a very short time after the big bang, no one really knows. I doubt that a theory that describes things in terms of what can be measured, makes predictions in the form of probabilities of possibilities, and is sophisticated enough to incorporate the local symetries of spacetime, can be a different theory than QM. But since spacetime is likely to be something entirely different in a theory that can handle the really early times, it's unclear if quantum mechanics as we know it would survive.

If you meant "the time before the big bang", what I just said is still relevant, but I also have to point out that your intuition that tells you that such a time must exist is wrong. See my comments below.

Furthermore it implies that all the physical quantities observed (space time, etc.) are not fundamental.

Space and time, with the properties your experiences tell you they have, are described by Galilei spacetime. That spacetime is simply the mathematical "thing" that makes our intuitive ideas about space and time precise. Space and time in the real world have different properties than that. We know this for sure because of experiments that show that special relativity (which uses Minkowski spacetime instead) make better predictions about results of experiments than theories based on Galilei spacetime. This is all we need to know that "space and time", as defined by our intuition, don't really exist in the real world.

We also know that space and time in the real world have different properties than Minkowski spacetime, because of experiments that show that general relativity (which uses a smooth Lorentzian manifold with properties determined by an equation) makes better predictions than special relativity. So we know that the space and time described by Minkowski spacetime doesn't exist in the real world either.

And neither does the spacetime of general relativity, because the equation that determines the specific properties of space and time involves the properties of matter, described classically. And we know that on small scales, it's just wrong to describe matter classically, because experiments are telling us that quantum theories make better predictions than classical theories.

That last part is how we know that some sort of union between QM and GR is necessary if a theory is to make predictions about times very close to the big bang. Hm, I've never quite thought of this before, but since there's no evidence to support the idea that QM actually describes reality (i.e. that it tells us what actually happens to the system even at times between measurements), it's unclear what a union of QM and GR would really accomplish. What I just said suggests that such a theory wouldn't describe spacetime at the earliest times, and the whole thing about state preparation and measurement is meaningless too, since measuring devices couldn't exist even in principle. The theory could still make better predictions than GR about cosmology, and possibly about particle physics too, so it wouldn't be useless, but it probably wouldn't make sense to say that it describes the early times.

It's also unclear if such a theory would make sense of the concept of "before" the big bang.

(There's already more than one big bang theory, and in some of them, in particular the versions involving inflation, the concept of "before" already makes sense).

 

[rpt]

Thanks Fredrik,

I am really greatful to you as you are the only person replied to my post and what you are saying is perfectly make sense to me. I assume most/all of others agree with you by keeping quite. Earlier I had the wrong understanding that quantum mechanics can reveal the reality of nature.

Although its is a slightly different topic I would like to extend the discussion on this thread, further as it is directly related to original question I had.

Does time and space (or spacetime) is considered a quantized quantity in quantum Physics?
I have heard about the "Planks time" but do not know whether its is really a concept in advanced quantum theories.

 

[nismaratwork]

Thanks Fredrik,

I am really greatful to you as you are the only person replied to my post and what you are saying is perfectly make sense to me. I assume most/all of others agree with you by keeping quite. Earlier I had the wrong understanding that quantum mechanics can reveal the reality of nature.

Although its is a slightly different topic I would like to extend the discussion on this thread, further as it is directly related to original question I had.

Does time and space (or spacetime) is considered a quantized quantity in quantum Physics?
I have heard about the "Planks time" but do not know whether its is really a concept in advanced quantum theories.

No one else responded because Fredrik had this one beautifully, but a bit of advice... if you really want to get into your last question, I'd make it a new thread, not just a new post here.

 

[rpt]

Sorry nimsara,
I am new here.
I will post a new thread.
I thought it would be best to discuss on the same thread.

People, please do not reply to this thread on the question "Is time/space quantized"
Please reply to original question on "axioms on quantum physics" if you have any comments.

 

[nismaratwork]

Sorry nimsara,
I am new here.
I will post a new thread.
I thought it would be best to discuss on the same thread.

People, please do not reply to this thread on the question "Is time/space quantized"
Please reply to original question on "axioms on quantum physics" if you have any comments.

No need to apologize! It's just more likely that you'll start the conversation you want in a new thread. Sorry if I came across as upset, I enjoyed the links you that you got in post #2, so I consider this thread a winner.

 

[rpt]

Fredrik,

What I meant by "in an environment where space time is intiutive" is this.
When we interpret the results of an experiment, it is understood by relating these abstract mathematical objects to quantities that are familiar to us by intution.

 

[andrebourbaki]

ok vi -c "source Doc*/init" answer

Many textbooks have altered significantly the original axioms of
Quantum Mechanics, of Dirac and von Neumann. (Weyl has
priority but only had five axioms, the famous 'reduction/collapse of
the wave packet' did not occur to him or Schr\"odinger.)
But in their classic books they did not list them separately
or all in the same place. You can find them in my paper on

The Axiomatisation of Physics, at

http://arxiv.org/abs/0705.2554
on p. 10.

I still think Dirac is the best introductory text on QM, but
he pays zero attention to the philosophical issues that
interest people. He also never mentioned parity invariance,
and when parity was found to be broken, and he was asked
what he thought about it, he replied "I said nothing about it
in my book."
So after reading the list of the axioms, the best place to
go to understand them is the first three chapters of Dirac.
You can learn all sorts of things from Dirac...what an integral
is, what diagonalisation of a matrix is, the spectral theorem...

To give you my own thoughts on the answer to the second half of
your question, and they might be controversial, it is clear
that time is a fundamental concept in these axioms, as Fredrik
said clearly (this is not controversial). Also, time plays
a seemingly different role from space (this seems to many
to be a blemish since the QM is not manifestly relativistic,
even when you formulate a relativistic QM it doesn't "look"
relativistic when you use the familiar kind of axioms).
But the space position and momentum operators are no
more fundamental than the spin
operators...or any of the many other dynamical variables
(I am thinking of occupation numbers)
we now see are important parts of Nature. So I wonder if,
after all, space is not fundamental in QM. Certainly QM is
non-local.
When one looks at combined systems,
the most natural place for them to live is a mixture of
position coordinates, momentum coordinates, and spin variables,
to say the least. And perhaps more will be necessary in
the future.

Now, Einstein thought this was a flaw in QM and it does stand
in the way of general relativity, not to have space-time be
a fundamental notion (really, it is the group of all diffeomorphisms
of space-time which is fundamental). Only the future will tell
whether QM has to be formulated more geometrically to be compatible
with General Relativity, or whether General Relativity needs to
be made more abstract in order to define an appropriate group
action on QM. This is a question Bell was worried about, too.