What is the most general mathematical framework for quantum

In summary, the Hilbert space approach, C* algebra approach, and rigged Hilbert space approach are all different mathematical frameworks for quantum mechanics. However, each approach has its limitations, such as not including distributions or mixed states, or not accounting for unbounded operators. There may not be a single, all-encompassing mathematical framework for quantum mechanics, as it could potentially violate Gödel's incompleteness theorem. In practice, most physicists use Dirac's formulation, which is made rigorous by the rigged Hilbert space approach.
  • #1
lfqm
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1.- The hilbert space approach does not include distributions (free particle, for example) nor mixed states.
2.- The C* algebra approach does not account for unbounded operators.
3.- Rigged Hilbert space approach does not include mixed states.

I'm not sure about path integral formulation... so I ask... What is the most general mathematical framework for quantum mechanics
 
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  • #2
Maybe such mathematical framework cannot exist because it would imply the existence of number operators, hereby violating Gödels incompleteness theorem?
 
  • #3
Lfqm, why do you think 1. and 3. don't include mixed states?
 
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  • #4
I'd say all three approaches together make up a consistent rigorous mathematical framework for non-relativistic QT.
 
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  • #5
I just started reading a bit on operator algebra's and apparently you need c*-algebra's to deal with situations where an infinite amount of particles are present, as in the thermodynamic limit. (http://www.springer.com/us/book/9783540170938)
 
  • #6
George Jones said:
Lfqm, why do you think 1. and 3. don't include mixed states?

Off course it does. That's because states are not elements of a vector space but are operators.

Study Ballentine.

This book gives our most powerful, complete and rigorous formulation:
https://www.amazon.com/dp/0387493859/?tag=pfamazon01-20

In practice most physicists use Diracs formulation which is made rigorous by the rigged hilbert space approch.

Thanks
Bill
 

Related to What is the most general mathematical framework for quantum

What is quantum mechanics?

Quantum mechanics is a branch of physics that studies the behavior of particles at a microscopic level, such as atoms and subatomic particles. It is a mathematical framework that describes the probabilistic nature of these particles and their interactions with each other.

What is the most general mathematical framework for quantum mechanics?

The most general mathematical framework for quantum mechanics is called the Hilbert space formalism. It uses linear algebra and complex numbers to represent quantum states and operators, and allows for the calculation of probabilities and expected values of measurements.

How does the Hilbert space formalism differ from classical mechanics?

In classical mechanics, the state of a system can be fully determined by specifying the position and momentum of all particles. In quantum mechanics, the state of a system is described by a wave function that contains all the information about the system. Additionally, in classical mechanics, measurements do not affect the state of a system, while in quantum mechanics, measurements can change the state of a system.

Are there other mathematical frameworks for quantum mechanics?

Yes, there are other mathematical frameworks for quantum mechanics, such as the path integral formalism and the operator algebra formalism. These frameworks use different mathematical tools and representations, but are ultimately equivalent to the Hilbert space formalism.

What are the practical applications of the mathematical frameworks for quantum mechanics?

The mathematical frameworks for quantum mechanics are used to understand and predict the behavior of particles at a microscopic level, which has numerous practical applications. These include the development of technologies such as transistors, lasers, and magnetic resonance imaging (MRI), as well as advancements in fields such as chemistry, material science, and cryptography.

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