A Heuristic View of QM: Axioms & Gleason's Theorem

[bhobba]

Strictly speaking, this isn't about interpretations; it's about whether there is a way to justify QM formalism more intuitively. I put it here because it may help when discussing interpretations. I will let others decide on its implications or even if it is a valid way of viewing it.

Suppose two systems interact, and the result is several possible outcomes. We imagine that, at least conceptually, these outcomes can be displayed as a number on a digital readout. Such is an observation in QM. You may think all I need to know is the number. But I will be a bit more general than this and allow different outcomes to have the same number. To model this, we write the number from the digital readout of the ith outcome in position i of a vector. We arrange all the possible outcomes as a square matrix with the numbers on the diagonal. Those who know some linear algebra recognise this as a linear operator in diagonal matrix form. To be as general as possible, this is logically equivalent to a hermitian matrix in an assumed complex vector space where the eigenvalues are the possible outcomes. Why complex? That is a profound mystery of QM - it needs a complex vector space. Those that have calculated eigenvalues and eigenvectors of operators know they often have complex eigenvectors - so from an applied math viewpoint, it is only natural. But just because something is natural mathematically does not mean nature must oblige.

So we have the first Axiom of Quantum Mechanics:

To every observation, there exists a hermitian operator from a complex vector space such that its eigenvalues are the possible outcomes of the observation. This is called the Observable of the observation.

But nothing is mystical or strange about it, just a common sense way to model observations. The only actual assumption is it is from a complex vector space.

Believe it or not, that is all we need to develop Quantum Mechanics. This is because of Gleason's Theorem:

https://www.arxiv-vanity.com/papers/quant-ph/9909073/

This leads to the second axiom of QM.

The expected value of the outcome of any observable O, E(O), is E(O) = trace (OS), where S is a positive matrix of unit trace, called the state of a system.

These are the two axioms of QM from Ballantine.

Thanks
Bill

 

[CoolMint]

Mathemeatics is not invented but discovered. Evidently it underlies the relationships that eventually turn into observations and this is why everything is predictible. Until you hit the HUP, that is where the math forbids certain types of knowledge.
Math is neither a fluke nor a coincidence. Knowledge seems fundmental and math is the blueprint. Quantum fields, math and knowledge are the fundamental structure of reality.

Why complex vector space?

My guess is the essential indeterminancy of reality.

 

[bhobba]

Why complex vector space? My guess is the essential indeterminancy of reality.

I actually agree with the first bit, but can you explain the above in more detail?

Thanks
Bill

 

[CoolMint]

I actually agree with the first bit, but can you explain the above in more detail?

Thanks
Bill

Both real and imaginary numbers are required to describe quantum systems, hence complex-valued behavior can be predicted using complex valued numbers at each point at time t. Why is this so?
The fundamental indeterminancy of the world will likely never allow a classical description without imaginary numbers and complex vector space.

 

[A. Neumaier]

Believe it or not, that is all we need to develop Quantum Mechanics.

For most of quantum mechanics you need the canonical commutation relations. This was Heisenberg's starting point, and cannot be deduced from Gleason's theorem!

 

[malawi_glenn]

Mathemeatics is not invented but discovered.

That is a philosophical discussion.

 

[martinbn]

Both real and imaginary numbers are required to describe quantum systems, hence complex-valued behavior can be predicted using complex valued numbers at each point at time t. Why is this so?
The fundamental indeterminancy of the world will likely never allow a classical description without imaginary numbers and complex vector space.

This is not more detail. It is just an expression of your belief that it must be so.

 

[vanhees71]

For most of quantum mechanics you need the canonical commutation relations. This was Heisenberg's starting point, and cannot be deduced from Gleason's theorem!

Of course. For this you need the representation theory of the spacetime's symmetry group (Galilei or Poincare groups, depending on whether you want non-relativistic QM or relativistic QFT).

That's as in Newton's mechanics: You have a general valid set of postulates (including implicitly the space-time symmetry group btw), but there's no way to derive the concrete forces acting on the particles. This must come from empirical input, e.g., Newton's universal gravitational interaction force between point particles, which were discovered by analyzing which forces lead to the validity of Kepler's empirical law of planetary motion.

 

[CoolMint]

This is not more detail. It is just an expression of your belief that it must be so.

How is it my belief? This is textbook stuff.

 

[martinbn]

How is it my belief? This is textbook stuff.

You said this

Why complex vector space?

My guess is the essential indeterminancy of reality.

And he asked you for more detail. And you responded by

The fundamental indeterminancy of the world will likely never allow a classical description without imaginary numbers and complex vector space.

How is this more detail!!!

 

[Fra]

Strictly speaking, this isn't about interpretations; it's about whether there is a way to justify QM formalism more intuitively. I put it here because it may help when discussing interpretations. I will let others decide on its implications or even if it is a valid way of viewing it.
...

But nothing is mystical or strange about it, just a common sense way to model observations. The only actual assumption is it is from a complex vector space.

I've been giving this a lot of thought as awell, in the spirit of "physics from inference" from the perspective of an inside agent, and I agree the complexity is one weird thing, and another weird thing is what one needs to use uncountable numbers to index distinguishable events. But this is hard to discuss as it's always fuzzy.

But to address the reason for complexity, my intuitive understanding of this is that it has todo with datacompression and stability of agents. The reason for the complex state, is that an agents optimal state of information needs a more efficient encoding. And we know fourier transform is used in datacompression(having nothing todo with QM per see), and for a good reason. And the conjugate spaces are essentially defined by the fourier transfom. So if you want in your state of information, statistical information about Q and P and the same time, you either up with with a complex number - or one could also consider several possible spaces, that are defined by relations. But the former way makes for a more compact notation!

So I think the reason is both datacompression and stability of agents, and that the complex state space makes for an efficent notation for human physicists.

imo, a generalisation of this gets us into various dualities principles where different theories in different spaces, can give the the same predictions on a common boundary. The difference lies more in efficiency of representation and computational complexlitry.

Is my loose intutive view... pulled out of my "agent interpretation"

/Fredrik

 

[CoolMint]

You said this

And he asked you for more detail. And you responded by

How is this more detail!!!

I didn't. You cut out just 1 sentence from my reply and misrepresent it as my whole reply.Here is my full reply from post 4:

"Both real and imaginary numbers are required to describe quantum systems, hence complex-valued behavior can be predicted using complex valued numbers at each point at time t. Why is this so?
The fundamental indeterminancy of the world will likely never allow a classical description without imaginary numbers and complex vector space."

Nobody can say why things are as they are or why the laws of physics are what they are. What can be said is that the world appears to be fundamentally indeterminate.
I do not know why the world is indeterminate and neither I nor anyone else know more details why this is so.

 

[bhobba]

Of course. For this you need the representation theory of the spacetime's symmetry group (Galilei or Poincare groups, depending on whether you want non-relativistic QM or relativistic QFT).

I thought that was the purpose of Chapter 3 of Ballentine where Schrödinger's equation is derived. The physical assumption is the probabilities are frame independent ie the POR.

And yes, the Poisson Bracket approach is a very elegant way to derive the dynamics.

Thanks
Bill

 

[physika]

Mathemeatics is not invented but discovered

For me, made up.

That is a philosophical discussion

Right.

.......

 

[jim mcnamara]

Thread locked pending moderation

 

[berkeman]

After some cleanup and a Mentor discussion, the thread is reopened. A sockpuppet of a previously-banned member has been identified and banned now. Have a nice day.