Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

Suppose that a matrix

$$A ~=~ x_1 B + x_2 C$$

is a linear combination of two self-adjoint matrices $B$ and $C$.

I'm interested in when $A$ represents a physical quantity.

When the linear combination is a complex combination, then $B$ and $C$ have to be commutable for $A$ to represent any physical quantity, cf. this Phys.SE post.

Now suppose that $x_1$ and $x_2$ are real. What happens in this case? If $B$ and $C$ are noncommutable, does $A$ still represent physical quantity?

share|improve this question
add comment

1 Answer

Yes, if $B$ and $C$ are self-adjoint $n\times n$ matrices (and even if they are mutually non-commuting), and if $x_1$ and $x_2$ are real numbers, then

$$A= x_1 B + x_2 C $$

is also a self-adjoint matrix $A\!=\!A^{\dagger}$, and hence a physical observable.

share|improve this answer
1  
In infinite dimensions, $C$ is Hermitian but not necessarily self-adjoint. One needs some domain compatibility condition for the selfadjointness of $C$. –  Arnold Neumaier Sep 27 '12 at 17:33
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.