Is Quantum Time a Vector Instead of a Scalar?

[Jilang]

Why do we rule out the possibility that quantum time is a vector rather than a scalar? The necessity of the introduction of the number "i" would seem to indicate there are angles involved, with the angle being indeterminate until there is an interaction with the macroscopic world which determines the angle of time for the quantum system.

 

[bhobba]

Well there is a complication relativity introduces that space and time should be treated on the same footing. In standard QM position is an observable, time is a parameter. To get around this QFT has time and position both as parameters. Evidently treating time as an observable was also tried but led to extreme difficulties and was abandoned.

There are all sorts of reasons i is introduced, but the one I tend to focus on is Wigners Theorem - it only works for complex vector spaces.

The necessity of the introduction of the number "i" would seem to indicate there are angles involved, with the angle being indeterminate until there is an interaction with the macroscopic world which determines the angle of time for the quantum system.

I have zero idea what you mean. I think you will need to post some math, and if you can't that's probably a hint its not well fleshed out.

Thanks
Bill

 

[ChrisVer]

Well, without the i you wouldn't have stable states - the wavefunction would explode or die out exponentially in time...

 

[vanhees71]

That already answers your own question. The wave function (applicable in non-relativistic quantum theory for systems with a fixed number of particles) is a probability amplitude, i.e., it's modulus squared gives the probality distribution for the positions of the particles. Integrating all the position variables over all space must thus give 1,
$$\int_{\mathbb{R}^{3N}} \mathrm{d}^3 \vec{x}_1 \ldots \mathrm{d}^3 \vec{x}_N \; |\psi(t,\vec{x}_1,\ldots,\vec{x}_N|^2=1.$$
If now $\psi$ would grow or die out exponentially with time, this would violate this normalization constraint.

In quantum theory time evolution is a unitary mapping, and thus the total probability always stays conserved at the value 1 as it must be.

 

[bhobba]

In quantum theory time evolution is a unitary mapping, and thus the total probability always stays conserved at the value 1 as it must be.

That's where Wigners Theorem comes in - you need to go to complex vector spaces to guarantee a unitary evolution.

Thanks
Bill

 

[Jilang]

That's where Wigners Theorem comes in - you need to go to complex vector spaces to guarantee a unitary evolution.

Thanks
Bill

Yes, I'm happy with that and if we are to put space and time in an equal footing - Just as there are many different paths that can be taken through space to get from A to B would we not require many different time paths to get from t1 to t2?
A converging multiverse for want of a better description...

 

[Jilang]

Sorry Bill, I'm not the best at maths, but I've been looking into this a bit more and it does seem that quite a few folks are working in the formalism. Like Itzhak Bars and others
http://arxiv.org/abs/hep-th/0008164
I know discussions on interpretations are not encouraged on this site, but it does seem to offer a way forward to reconcile Many Worlds with the path integral formulation of QM.

 

[atyy]

Why do we rule out the possibility that quantum time is a vector rather than a scalar? The necessity of the introduction of the number "i" would seem to indicate there are angles involved, with the angle being indeterminate until there is an interaction with the macroscopic world which determines the angle of time for the quantum system.

Sorry Bill, I'm not the best at maths, but I've been looking into this a bit more and it does seem that quite a few folks are working in the formalism. Like Itzhak Bars and others
http://arxiv.org/abs/hep-th/0008164
I know discussions on interpretations are not encouraged on this site, but it does seem to offer a way forward to reconcile Many Worlds with the path integral formulation of QM.

It is not ruled out, as you yourself point out by referencing the work of Itzhak Bars and others.

Incidentally, the path integral formulation of quantum mechanics, when it exists, is not at odds with many-worlds as an approach to the interpretation of quantum mechanics.

 

[Jilang]

Thanks atyy, that's good to know.

 

[Jilang]

Just a last question that is bothering me. How do the paths in many world manage to interfere?

 

[atyy]

The path integral is only a tool to calculate the probability amplitude of going from one state to another. It's just a way of combining the Schroedinger equation and the Born rule for easy calculation. The probability amplitude is interpreted in exactly the same way as in normal quantum mechanics. In this sense, the path integral is not really an interpretation like Copenhagen or many-worlds, but just a calculational tool. This is why I said it is compatible with the many-worlds approach.

That said, I think David Wallace proposes a version of many-worlds where the worlds interfere. I don't know if any version of many-worlds really works, but you can take a look at Wallace's proposal in https://www.amazon.com/dp/0199546967/?tag=pfamazon01-20 .

 

[Jilang]

Many thanks.