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Linearity of momentum operator

  1. Jun 26, 2008 #1
    Is there a theory or any physical reasoning for the momentum operator being a linear, rather than an antilinear, operator?
  2. jcsd
  3. Jun 26, 2008 #2
    If I understand the basics correctly, all operators in quantum mechanics are linear. Not quite sure why, though.

    I guess that raises more questions than answers, though.
  4. Jun 26, 2008 #3
    The momentum operator is linear because the derivative operator is linear. The derivative operator is linear because addition is linear :smile:.
  5. Jun 27, 2008 #4
    An antilinear derivative operator will respect addition.
    In other words lets say the differential operator, D, obeys:

    D(f + g) = D(f) + D(g),
    D(cf) = cD(f), where c is a constant

    I define a differential operator B, and B obeys,
    B(f + g) = B(f) + B(g),
    B(cf) = c*B(f), where c* denotes complex conjugate of c

    My question is if there is a mathematical or physical reason why the operator P cannot be described by -iB.
  6. Jun 27, 2008 #5


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    The translation operator is [itex]e^{-ipr}[/itex] where p is the momentum operator and r the translation distance. The exponential is defined as a series, but the square of an anti-linear operator is linear, so the translation operator would be neither linear nor anti-linear. I think this might violate the theorem proved by Wigner that says that operators representing symmetry transformations must be either linear and unitary or anti-linear and anti-unitary, but I don't know if he proved that this must be the case, or just that it's possible to define the operators that way. (I think it's the latter actually).

    Also note that if we require the function T defined by [itex]T(r)=e^{-ipr}[/itex] to be continuous, then T(r) can't be anti-unitary for any r since it's unitary for r=0.

    If you want to know more, check out the appendix of chapter 2 of volume 1 of Weinberg's QFT book.
  7. Jun 27, 2008 #6
    Thanks Fredrik I had a quick look at Weinberg's version of Wigner's proof but it was a bit a long! I like you argument on continuity but you have already exponentiated a linear operator and then setting it equal to an anti-linear operator will obviously lead to some difficulties. You must begin earlier, considering the infintesimal unitary operator, just for translations for simplicity, and then choosing the P's to be anti-linear rather than linear.
    I have just began to look at this and so far it appears we get the constraint that the P's are antihermitian rather than hermitian but I will do some more work on this first.
  8. Jun 27, 2008 #7


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    The crucial point is that anti-linear transformations are discrete, since
    they involve a complex conjugation operation. Therefore, such
    transformations cannot be of the identity-connected type (i.e., they
    cannot be continuous with the identity transformation). This is
    similar to how a parity inversion is discrete, and not identity-connected.

    Examples of anti-linear transformations are charge conjugation
    and time-reversal for a Dirac particle (most RQM textbooks
    discuss this).

    The waters are muddied further, however, if you consider
    transformations that mix the annihilation and creation
    operators in QFT (since these implicit mix a field with
    its complex-conjugated counterpart). Look up
    "Bogoliubov transformations" (used in superconductivity
    and elsewhere) for examples of this. A subtle point here
    is that these transformations generally take you between
    inequivalent Hilbert spaces -- which is why the Wigner
    theorem mentioned by Fredrick may (superficially) seem
    to be in contradiction with such things.
  9. Jun 28, 2008 #8
    Its because we WANT operators to be linear, and that its meanigful. From the beginning starting from de Broglie and Schrödinger you describe a particle as a plane wave with [tex]k=p/\hbar, \omega=p/\hbar[/tex], and then you could easilly identify an operator that gives you p. Also Dirac made this wish of lineaity when he formulated the reativistic quantum mechanics.
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