The Foundational Questions Institute, FQXi, is running an essay contest, with the subject of the Nature of Time. Up to 21 prizes are to be awarded, with amounts ranging from $1000 to $10,000 per prize. But probably more important are the bragging rights. You have until December 1, 2008 to submit an essay. Your essay, assuming it’s “serious”, will be displayed on their website from soon after you turn it in until the contest concludes in mid December.
Right now, the contributions look fairly weak so who knows, maybe a little typing now will get you a little money, and some bragging rights, in early 2009. Of course I’ve typed up a contribution, and of course it has to do with the nature of time, as is suggested by the density matrix formulation of quantum mechanics.
I included some short comments on the subject of non Hermitian density matrices. These have something to do with raising and lowering operators and it’s worth typing up a quick blog post on the subject. Surely somebody is going to learn something.
First, rather than use bra-ket notation, which is slow and painful to format due to combined inadequacies of both wordpress and Roger Cortesi’s LaTex engine, (which last I use to create LaTex images), I will use operator notation. And for this subject, there’s no reason to have to worry about dependency on time, so let’s use wave functions with no time dependence.
Let f(x), g(x), and h(x) be three orthonormal wave functions. One converts these to density matrices by multiplication by the complex conjugate, with another copy of the space-time coordinate, i.e.:
This raises the issue of “what do you get when you replace the “g” in the above with “g”? That would be:
This is the raising operator that converts f to g and zeroes everything else.
This may not be obvious. Let w(x) be an arbitrary function of x. Define as a function of x as follows:
Then , and , so is the raising operator as defined.
More general raising operators can be defined similarly. For instance, if we wish to have an operator that raises f to g, and g to h, then we define it as:
In other words, we can make all sorts of raising and or lowering operators this way.
Non Hermitian Density Matrices
And what about “non Hermitian density matrices?” Let u(x) and v(x) be two normalized but not orthogonal wave functions. Define the “raising” operator from v to u as:
Density matrices are usually normalized so that . This requires that the above be divided by the constant k where
which is nonzero under the assumption that u and v are not orthogonal. Note that if u and v are orthogonal, as they would be for a raising operator, it is impossible to normalize the “non Hermitian density matrix” because this would require division by zero. But we can think of raising and lowering operators as the limits of non Hermitian density operators.
So can be thought of as a sort of normalized raising operator for non orthogonal states. But it meets all the requirements of a density matrix except one; it cannot be made from a wave function.
Note that the usual density matrices are Hermitian, replacing x with x’ changes the values to their complex conjugates. Making the same change with does not result in a simple complex conjugate so these objects are not Hermitian.
Wave functions from Density Matrices
Given a (pure) density matrix of the usual sort , it’s easy to define a wave function from the same object. Simply pick a point b where the density matrix is not identically zero, and then is a wave function that will create the density matrix . Alternatively, one could use instead . In this sense, the usual density matrices are two-sided. They give the same wave function, up to a complex multiple.
Translating this back into the matrix language, one obtains a state vector ket from a density matrix by taking any non-zero column. One can get a state vector bra by taking any non-zero row. For the usual Hermitian density matrix, the row and column one obtains these ways are complex multiples of each other.
So the generalization of Hermitian density matrices to non Hermitian density matrices amounts to allowing the left and right side wave functions to be different.
Density matrices and Complex Phases
When one converts a wave function / state vector to a density operator / matrix, the arbitrary complex phase of the wave function / state vector is lost. Density matrices still keep complex numbers, but now they’re expressed as the difference in phase between two points in space-time, the incoming and outgoing states.
When one moves from density matrices to non Hermitian density matrices, one ends up with differences in spin between incoming and outgoing states. In that sense, non Hermitian density operators are a natural generalization of density operators.