Complex Hilbert Space as a Symplectic Space?

In summary, the conversation is about the symplectic form in complex Hilbert space and how it relates to the standard inner-product. It is noted that the imaginary part of the inner-product is symplectic and antisymmetric, with an example provided to illustrate this. The conversation ends with one participant thanking another for pointing this out.
  • #1
Bacle
662
1
Hi All: in the page:

http://mathworld.wolfram.com/SymplecticForm.html,

Complex Hilbert space, with "the inner-product" I<x,y> , where <.,.> is the inner-product

Does this refer to taking the imaginary part of the standard inner-product ? If so, is

I<x,y> symplectic in Complex Hilbert Space? It is obviously bilinear, but I don't see

how it is antisymmetric , i.e., I don't see that I<x,y>=-I<y,x>

Am I missing something?

Thanks.
 
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  • #2
For any complex number c=a+ib (with a,b real), we have Im(c*) = Im(a-ib) = -b = -Im(a+ib) = -Im c, so

Im<x,y>=Im(<y,x>*)=-Im<y,x>
 
Last edited:
  • #3
Yes, how dumb of me. Thanks, Fredrik.
 

1. What is a complex Hilbert space?

A complex Hilbert space is a mathematical concept that is used to describe infinite-dimensional vector spaces over the complex numbers. It is a generalization of Euclidean space, which is finite-dimensional, and it is often used in fields such as quantum mechanics and signal processing.

2. How is a complex Hilbert space related to symplectic spaces?

A complex Hilbert space can be viewed as a special case of a symplectic space, where the symplectic form is given by the imaginary part of the inner product. This means that a complex Hilbert space can be seen as a symplectic space with additional structure.

3. What is the significance of using a complex Hilbert space as a symplectic space?

Using a complex Hilbert space as a symplectic space allows for a more elegant and efficient mathematical formulation of physical theories, especially in the context of quantum mechanics. It also allows for the use of powerful mathematical tools, such as the theory of symplectic geometry, to study and understand complex systems.

4. How is a symplectic space different from a complex Hilbert space?

While a complex Hilbert space is a special case of a symplectic space, there are some key differences between the two. One major difference is that a symplectic space is defined over a real vector space, while a complex Hilbert space is defined over a complex vector space. Additionally, a symplectic space does not have a notion of distance, unlike a complex Hilbert space where the inner product can be used to define a metric.

5. What are some applications of using a complex Hilbert space as a symplectic space?

There are many applications of using a complex Hilbert space as a symplectic space, especially in the fields of quantum mechanics and control theory. For example, in quantum mechanics, the wave function of a system can be represented as a vector in a complex Hilbert space, allowing for the use of symplectic geometry to study the evolution of the system. In control theory, complex Hilbert spaces are used to model and analyze complex systems with multiple inputs and outputs.

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