How does abstract algebra relate to canonically conjugate in physics?

  • Thread starter nateHI
  • Start date
  • Tags
    Conjugate
In summary, in physics, momentum and position are canonically conjugate because they are the Fourier transforms of each other. In abstract algebra, this means that for a group G and its natural homomorphism ψ:G->G/H with H as the kernel, the elements p and x would be in the same coset in the set of cosets G/H. In quantum mechanics, this is connected to the Heisenberg group, while in classical mechanics, it is related to the Poisson bracket and symplectic structure on phase space.
  • #1
nateHI
146
4
Momentum and position are canonically conjugate in physics because they are the Fourier transforms of each other.

In the context of abstract algebra what would that mean. More precisely, Let G be the group they both (p and x) belong to and let ψ:G->G/H be the natural homomorphism where H is the kernel of ψ. Would p and x be in the same coset in the set of cosets G/H?

Dang, I lost my train of thought and I'm not sure where I'm going with this now. I guess my question now is, please relate canonically conjugate in group theory to Fourier transforms.

Thanks, Nate
 
Last edited:
Physics news on Phys.org

What is the meaning of canonically conjugate?

Canonically conjugate refers to a pair of variables in physics that are related through a mathematical transformation known as a canonical transformation. These variables have a special relationship where their corresponding equations of motion are described by Hamilton's equations.

How are canonically conjugate variables related to each other?

Canonically conjugate variables are related through a specific type of mathematical transformation called a canonical transformation. This transformation preserves the fundamental equations of motion, known as Hamilton's equations, and allows for a different perspective to be taken on the system.

What are some examples of canonically conjugate variables?

Some common examples of canonically conjugate variables include position and momentum, energy and time, and angular position and angular momentum. These pairs of variables are related through a canonical transformation and have a special relationship in terms of their equations of motion.

What is the importance of canonically conjugate variables in physics?

Canonically conjugate variables play a crucial role in the study of classical mechanics and quantum mechanics. They allow for a deeper understanding of the dynamics of a system and provide a different perspective on the behavior of physical systems. They also have applications in fields such as statistical mechanics and thermodynamics.

How are canonically conjugate variables used in practical applications?

Canonically conjugate variables have practical applications in various fields, including physics, chemistry, and engineering. They are used in the development of mathematical models to describe the behavior of physical systems and can help predict the future state of a system. They are also useful in the design and optimization of physical systems and processes.

Similar threads

A Conjugate variables in the Fourier and Legendre transforms
    • Quantum Physics
Replies
1
Views
2K
A Unitary representations of Lie group from Lie algebra
    • Quantum Physics
Replies
5
Views
1K
A How to derive Quantum Mechanics in curved physical space?
    • Quantum Physics
Replies
6
Views
918
I Uncertainty between position and angular frequency - is this correct?
    • Quantum Physics
Replies
2
Views
1K
A Are there two different adjoint representations used in the SM?
    • Quantum Physics
Replies
9
Views
1K
Courses Is proof based Linear Algebra be similar to Abstract Algebra
    • STEM Academic Advising
Replies
6
Views
4K
A Does Antony Valentini's "sub-quantum measurement" really work?
    • Quantum Physics
Replies
1
Views
1K
Courses Which Abstract Algebra Sequence is Best for Aspiring High Energy Physicists?
    • STEM Academic Advising
Replies
3
Views
1K
I Bracket VS wavefunction notation in QM
    • Quantum Physics
3
Replies
82
Views
8K
Counting Cosets in Abstract Algebra | Pinter's Self Study
    • Linear and Abstract Algebra
Replies
4
Views
1K

Hot Threads

Recent Insights

Back
Top