Why time is not an observable in quantum theory?

[rpt]

Why "time" is not an observable in quantum theory?

Why "time" is not considered as an observable as any other quantity in quantum theory?
Is it because we cannot imagine anything in the absence of time?

 

[Fredrik]

My view on this is that observables are equivalence classes of measuring devices, and that measuring devices are systems that have a specific type of interaction with the environment and with the measured system. Time isn't an observable of any system, because clocks don't need to interact with the system.

There are other threads about this, for example:

https://www.physicsforums.com/showthread.php?t=365383
https://www.physicsforums.com/showthread.php?t=361240

I think there are others, but these are the ones I could find easily.

 

[dextercioby]

Nonetheless, time is relative. Different observers can measure different values of time using similar (identincally built) instruments. I don't see an irefutable reason against the idea of time being one of the basic observables of a quantum system and consequently considering a linear, unbounded, self-adjoint operator with fully continuous spectrum in the mathematical formalism.

 

[Fredrik]

I don't have an irrefutable argument either (or at least I don't know for a fact that it's irrefutable), but it seems to me that we would have to change what we mean by an "observable" to be able to consider time an observable. A clock measuring "time" isn't measuring a property of another physical system. It's measuring a property of the curve in spacetime that represents its own motion.

 

[martinbn]

Isn't the standard argument (Pauli?) that if there is a time observable with the corresponding operator having the usual commutation relations with the Hamiltonian, then the Hamiltonian is not bounded from below?

 

[dextercioby]

Pauli's argument is basically flawed. He didn't (because he simply couldn't, in 1926 he didn't have all the mathematical tools for it) go full length with the mathematical justification of his statement.

@Fredrik, I thing not being able to uniquely specify what you mean by an observable (and even if you could, perhaps the notion you obtain is almost useless) is one of the features that makes quantum mechanics unique among the other commonly accepted physical theories. It has so many formulations, interpretations and mathematical approaches to it, that you'd have to spend at least a fully year trying to review them.

But hey, it's good the way it is. Else, some brilliant minds would have come up with a viable alternative so far.

And what's the definition of <property of a physical system> ?

 

[strangerep]

Pauli's argument is basically flawed. He didn't (because he simply couldn't, in 1926 he didn't have all the mathematical tools for it) go full length with the mathematical justification of his statement.

What is the flaw in Pauli's argument? (It seems to me that his argument still
goes through to the same conclusion if one formulates things carefully in
a rigged Hilbert space.)

And what's the definition of <property of a physical system> ?

I'll go with the SI definition of "quantity" which is:

A quantity in the general sense is a property
ascribed to phenomena, bodies, or substances that can be quantified for, or assigned to, a
particular phenomenon, body, or substance. [...] The value of a physical quantity is the
quantitative expression of a particular physical quantity as the product of a number and a
unit, the number being its numerical value.

I.e., a "quantity" is anything which (for a given system in a given state) can be associated
with a number by an act of measurement.

Kinda circular, one might think, but it actually allows an axiomatic
development of QM via quantity algebras.

 

[Fredrik]

Isn't the standard argument (Pauli?) that if there is a time observable with the corresponding operator having the usual commutation relations with the Hamiltonian, then the Hamiltonian is not bounded from below?

This sounds like an argument against the commutation relation, not against the observable.

@Fredrik, I thing not being able to uniquely specify what you mean by an observable (and even if you could, perhaps the notion you obtain is almost useless) is one of the features that makes quantum mechanics unique among the other commonly accepted physical theories.

I don't think that is a feature of QM. The definition of an observable as an equivalence class of measuring devices (see the first three pages of Araki) seems perfectly adequate to me. I know that most books don't include this definition, but I think that's just the authors being sloppy.

It has so many formulations, interpretations and mathematical approaches to it, that you'd have to spend at least a fully year trying to review them.

Now that's a feature of quantum mechanics. (A really annoying one).

And what's the definition of <property of a physical system> ?

When I say things like "there's no justification for assuming that a state vector represents all the properties of the system", I'm treating "property" as a primitive. It's not defined in terms of anything else. What my statement means is that there's no reason to think that all the "facts" about the system are contained in the state vector. Of course, now I have to explain that "fact" is another primitive, and that I don't just mean knowable facts, I mean all facts. (The things I said here define neither "property", nor "fact". My comments are just elucidations: a bunch of words that make the intended usage of an undefined term a bit clearer).

In the context of QM, a "property" can be defined (if we want to) as a statement about the results of future experiments on the system, that's true with probability 1.

 

[Demystifier]

Why "time" is not considered as an observable as any other quantity in quantum theory?

It is because a physical wave function cannot be localized in time. See e.g.
http://xxx.lanl.gov/abs/0811.1905 [Int. J. Quantum Inf. 7 (2009) 595]

 

[arkajad]

Because you have to specify: "time of what?". When you specify, then, with some little tweaks, time of events can be made into into an observable, though not represented by a usual linear Hermitian operator, like, for instance, "tunneling time". See, for instance Wigner's paper, "On the Time-Energy Uncertainty Relation", in "Aspects of Quantum Theory", Ed. Salam, E., and Wigner, E.P. , Cambridge University Press, Cambridge 1972 - though it contains a serious error.

 

[rpt]

Thinking behind my question was this,
If everything is emerging from emptiness. (Asuming Big-bang theory is correct)
Why should "time" be different from any other physical quatity observed in nature?

 

[MathematicalPhysicist]

If everything is emerging from emptiness. (Asuming Big-bang theory is correct)

Who says that?

Everything emerges from quantum fluctuations, whatever that means. (-:

 

[arkajad]

http://io9.com/5694701/does-cosmic-background-radiation-reveal-the-universe-before-the-big-bang" But I think that Roger Penrose is hesitating as regards to the issue of "quantum fluctuations".

 

[Fredrik]

If everything is emerging from emptiness. (Asuming Big-bang theory is correct)

There are several different big bang theories, but none of them says anything like that.

Everything emerges from quantum fluctuations, whatever that means. (-:

They also don't say that. (But I agree that quantum field theories suggest something like that).

 

[rpt]

What I meant meant by emptiness is not vacuum, but a condition that nothing exists including time and space.
That was my understanding of the condition before Big-bang (one of the many theories).
Is that not correct?

 

[arkajad]

There are several different big bang theories, but none of them says anything like that.

Yeah, they are silent on this issue.

 

[MathematicalPhysicist]

What I meant meant by emptiness is not vacuum, but a condition that nothing exists including time and space.
That was my understanding of the condition before Big-bang (one of the many theories).
Is that not correct?

A world with no space nor time is unphysical, so it's beyond physics, we may call it metaphysics.

 

[MathematicalPhysicist]

There are several different big bang theories, but none of them says anything like that.


They also don't say that. (But I agree that quantum field theories suggest something like that).

Well, I heard Yakir Ahoronov says that (though, it was in a popular TV show).

 

[Fredrik]

What I meant meant by emptiness is not vacuum, but a condition that nothing exists including time and space.
That was my understanding of the condition before Big-bang (one of the many theories).
Is that not correct?

The word "before" is a reference to an earlier time, so you seem to be talking about a time before time. That only makes sense if you think there's another kind of time, which has nothing to do with spacetime.

 

[arkajad]

That only makes sense if you think there's another kind of time, which has nothing to do with spacetime.

I am not the only one.

 

[rpt]

Fredrik,
I think there is a difficulty using words to describe ideas in this situation.
What is the mathematical description of this condition?
What happens to space-time at this point in accepted theories?

 

[Fredrik]

Fredrik,
I think there is a difficulty using words to describe ideas in this situation.
What is the mathematical description of this condition?
What happens to space-time at this point in accepted theories?

The big bang isn't an event in spacetime. Every event has a time coordinate t>0 in the cosmological (FLRW) coordinate system. The big bang is a property of the spacetime manifold that can be characterized in many different ways, one of them being that the proper distance between any two objects at constant spatial coordinates goes to zero as the time coordinate goes to zero.

To make sense of the phrase "before the big bang", we have to consider another theory, a theory in which the big bang is something that happens in spacetime (like a phase transition after a period of inflation) instead of a property of spacetime. It certainly doesn't make sense in the original big bang theory, which is just the claim that the large-scale behavior of the universe is described approximately by a FLRW solution.

I am not the only one.

What do you mean? Do you think there's another kind of time?

 

[dextercioby]

What is the flaw in Pauli's argument? (It seems to me that his argument still
goes through to the same conclusion if one formulates things carefully in
a rigged Hilbert space.)

I've based my assertion on the 2 articles one finds on arxiv written by Eric Galapon. Probably I may have not properly understood his conclusion, but what I got is what I wrote.

As for the <reformulation in RHS>, I didn't find any reference on it being ever made. Perhaps you can point me to some references.

Thanks

 

[strangerep]

As for the <reformulation in RHS>, I didn't find any reference on it being ever made.
Perhaps you can point me to some references.

No -- that's why I said "it seems to me...". :-)

I've based my assertion on the 2 articles one finds on arxiv written by Eric Galapon. Probably I may have not properly understood his conclusion, but what I got is what I wrote.

I looked at quant-ph/9908033 (assuming that's one of the papers you meant?),
but that was a while ago and I forget what I concluded about it.
I'll go take another look...

Cheers.

 

[rpt]

Fredrik,
Thanks for your explanation. I don't fully understand what you are saying because I do not have background in those theories you mention. I will have a read on these topics you have mentioned in your reply.

Forget about the "Big-bang" and that being an event in space-time.
Why does t=0 is excluded in the cosmological coordinate system?
Is it because it results a singularity in mathematical formulation of the problem?
Or because it converges to the situation where space cease to exist? (which is unphysical and therefore excluded).

Can a singularity in mathematics could mean something unphysical in nature?

Sorry that the questions I have are more fundamental in nature. The anwers you people give are really useful for my understanding.

 

[Careful]

First of all, time as a Hermitian self adjoint operator simply does not work even in the most elementary examples. For example, you may try to make such a type of quantization of the relativistic free particle and you will learn where the many difficulties reside. Second, I think it is obvious that time and space cannot be measured, since it would require an observer *outside* spacetime. The universe should in this view be regarded a single particle (such as in third quantization of the gravitational field) implying that observers should measure themselves inside the universe from the outside You will get into serious trouble with causality and locality if you insist upon such travesty. Moreover, one would get into logical contradictions such as: ''if I measure from outside the universe, why don't I *know* the content of the entire universe?''. Anyway, I think the idea of quantum geometry is safely dead.

 

[Fredrik]

Why does t=0 is excluded in the cosmological coordinate system?
Is it because it results a singularity in mathematical formulation of the problem?
Or because it converges to the situation where space cease to exist? (which is unphysical and therefore excluded).

The idea behind the FLRW solutions is to look for solutions that describe spacetime as a one-parameter family of spacelike hypersurfaces, each of which is homogeneous and isotropic (in a specific mathematical sense). There are three classes of such solutions: positive curvature, zero curvature, and negative curvature. Each solution in the positive curvature class describes spacetime as a one-parameter family of 3-spheres with radii that depend on the parameter that labels the 3-spheres. It's convenient to define the parameter so that the parameter difference between two arbitrary 3-spheres, let's call them Alice and Bob, is the proper time of a timelike geodesic that starts on Alice, ends on Bob, and is orthogonal to all the 3-spheres it passes through.

The requirement that this spacetime must satsify Einstein's equation tells us how the radius depends on the value of the parameter. One of the things we see is that the radius goes to zero as the parameter approaches some value from above. It's convenient to choose that value to be 0.

The "cosmological" coordinate system is defined by taking the time coordinate equal to the parameter. The spatial coordinates are chosen so that they're the same at all points that are intersected by a single timelike geodesic that's orthogonal to all the 3-spheres it passes through. (This really means that the tangent vector of the curve is orthogonal to the tangent space of the 3-sphere at the point of intersection).

These choices ensure that all the 3-spheres are labeled by a parameter value t>0. Each 3-sphere can be thought of as "space, at time t". We know that there can't be a 3-sphere in this family with parameter value 0, because its radius would have to be 0, which would make it a point, not a 3-sphere.

Even if we would add an additional point to spacetime just to be able to say that there's a t=0 in the theory, no coordinate system could cover a region that contains that point, so our spacetime wouldn't be a manifold.

Can a singularity in mathematics could mean something unphysical in nature?

Stuff in nature is physical by definition, isn't it? What this type of singularity means is that, according to the theory, if you specify a distance in meters, say L=10^-100, I can specify a time at which all the galaxies in the currently observable universe were contained in a region of volume L³. There's no reason to think that the theory is able to describe such extreme circumstances accurately. There are in fact good reasons to think it can't.

 

[haael]

It is because a physical wave function cannot be localized in time.

But what happens when we want to measure "time" of an event, not a wave. Physical waves surely don't have "time", but events do, i.e. a decay of a particle.

What do we do (mathematically) when we measure half-life time of some particle? What does Geiger counter output? Isn't is just good old "time"?

 

[arkajad]

But what happens when we want to measure "time" of an event, not a wave. Physical waves surely don't have "time", but events do, i.e. a decay of a particle.

Indeed. This kind of physics is discussed in specialized papers, not in in the textbooks. Why? Because it may lead beyond the existing quantum paradigm. Time of an event cannot be represented by a linear operator, it involves, for instance, non-linearity.

 

[rpt]

Fredrik,

What I understand from your reply is that the model you describe converges to a situation where space cease to exist when t=0.
I would consider a point as a limiting case of a sphere.
I think if we rely on mathematical model to describe a system, we should not let ourselves (I mean not being able to comprehend the idea that "space cease to exist") interfere with the mathematica model. You may argue that having that point t=0 has no significance to physical world. However it may give a clue to the reality of nature.

I wouldn't exclude the fact that reality of nature could be unphysical although everything in nature is physical.