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All eigenvalues of a Hermitian matrix are real

  1. Mar 13, 2014 #1
    We know that all eigenvalues of a Hermitian matrix are real. How to explain this from the physics point of view?
     
  2. jcsd
  3. Mar 14, 2014 #2
    Explain that none of our instruments are capable of measuring imaginary quantities? Seriously? You imply that physics depends on the matrix, rather than the matrix is used to describe the physics. There are all sorts of mathematical objects and concepts, none of which need be "explained" from a physics point of view.
    Let me know when you measure a length, area, or count with a value of a+bi where a and b are real numbers and i = √-1. I am not familiar with any SI units using complex numbers, are you?
    One ought not to confuse different levels of abstraction, if one can avoid it...
     
  4. Mar 14, 2014 #3

    AlephZero

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    Hmm.... we measure quantities that we represent as complex numbers all the time, including distances, velocities, and accelerations.

    But then i do engineering. Maybe we have different thresholds of confusion and/or abstraction from physicists.

    You don't need any new units. if a+bi is a length, a and b both have units of meters. Otherwise, scaling a complex length by an arbitrary complex number wouldn't make any sense.

    But like you, I'm not sure exactly what the OP wants "explained" about this. If the eigenvalues of the math are always real numbers and the corresponding quantities in the physics are not, the explanation is that the math model doesn't match the real world.

    FWIW there are useful engineering math models where the matrices are not Hermitian, and the eigenvalues are physically meaningful but not real.
     
  5. Mar 14, 2014 #4
    The obvious answer that we measure only real quantities is in fact quite superficial.

    Measurement happens in an separate framework from the actual physical time evolution and encoding the possible results with hermitian operators is pure convenience. We could as well use antihermitian operators and take the imaginary eigenvalues as possible measurement outcomes. This would not change any of the physics involved.

    However, the much more important use of hermitian operators is as generators of unitary representations of Lie groups. Interestingly, antihermitian operators can be used too and would simplify the notation somewhat and are in fact the choice of many mathematicians. But because it is nice to be able to directly identify generators of a symmetry with possible observables, the convention is to use hermitian operators for group generation and for encoding of possible measurement outcomes with real measured quantities.

    Cheers,

    Jazz
     
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