B Are subspaces of Hilbert space real?

[Jaycurious]

When orthogonal states of a quantum system is projected into subspaces A and B are A and B real spaces?

 

[anuttarasammyak]

Hi. Hilbert space for QM is a complex Hilbert space. The subspaces are also complex, I think.

 

[PeroK]

When orthogonal states of a quantum system is projected into subspaces A and B are A and B real spaces?

By "real" do you mean a vector space over the field of real numbers? If so, then no. A subspace of a vector space is, by definition, a vector space over the same field of scalars: in this case the complex numbers.

 

[Jaycurious]

By "real" do you mean a vector space over the field of real numbers? If so, then no. A subspace of a vector space is, by definition, a vector space over the same field of scalars: in this case the complex numbers.

Thanks, this is what I was asking. Another question. Are observers associated with subspaces A and B?

The way I understand it is that the states of an electron in Hilbert space like spin up and spin down are projected into subspaces A and B.

These subspaces can be in a pure state with zero entropy or if subspace B is in a mixed state then the entropy of entanglement increases.

So if subspaces are complex valued like Hilbert space then what's the difference between subspaces and Hilbert space?

 

[PeroK]

Thanks, this is what I was asking. Another question. Are observers associated with subspaces A and B?

The way I understand it is that the states of an electron in Hilbert space like spin up and spin down are projected into subspaces A and B.

These subspaces can be in a pure state with zero entropy or if subspace B is in a mixed state then the entropy of entanglement increases.

I'm sorry, but I can make no sense of that except your last question.

So if subspaces are complex valued like Hilbert space then what's the difference between subspaces and Hilbert space?

Look up the defintion of vector space and subspace. The short answer is that a subspace is only part of a vector space.

Technically, of course, a subspace of a Hilbert space is also a Hilbert space. I don't know why some physicists talk about "Hilbert Space", as though there is only one. Mathematically, there are many Hilbert spaces. See:

https://en.wikipedia.org/wiki/Hilbert_space

 

[sysprog]

Yeah, @PeroK, and who knows how many spaces Dr. Hilbert could show you if you were to check into his hotel . . .

 

[Bruzote]

Yeah, @PeroK, and who knows how many spaces Dr. Hilbert could show you if you were to check into his hotel . . .

That hotel is impossible to sleep in! First, they overbook the rooms. Then they keep waking you up all night, asking you to move one room over in order to make room for the new arrivals. It feels like never-ending exhaustion.

 

[waternohitter]

That hotel is impossible to sleep in! First, they overbook the rooms. Then they keep waking you up all night, asking you to move one room over in order to make room for the new arrivals. It feels like never-ending exhaustion.

Yup, I agree with you because I experienced this.