Just another point about negative probabilities. There can be a quantity that can be interpreted as a probability in many cases, but in extreme cases, the probability interpretation breaks down, and a different interpretation is needed.
Here's an example that isn't really physics, but I think it relates to the appearance of negative values for the "current" in solutions to Klein-Gordon equation:
Suppose I have a string in the x-y plane that runs basically in the x-direction, as shown in the top figure below. You can imagine that the exact shape of the string is nondeterministic. So instead of describing it by y = f(x), a function, we might instead have a probability density function P(y,x): the probability of the string passing through the point (x,y). If the shape of the string is well-behaved in a certain sense, you would expect the density function to satisfy:
\int_{-\infty}^{+\infty} P(y,x) dy = 1
That is, as long as the string is running basically in the x-direction, there will only be a single value of y for any fixed x.
If you knew something about how the probabilities for various shapes were computed, you could come up with a kind of "Green's function" G(x_1, y_1, x_2, y_2), which would be the probability of the string passing through (x_1, y_1) given that it passed through (x_2, y_2).
Now, imagine a wilder shape for the string, as shown in the middle figure. In this case, the string doubles-back at several places, so that y is no longer a single-valued function of x. At point \mathcal{A}, there is only one value of y that the string passes through. At point \mathcal{B}, there are three values of y. At point \mathcal{C}, there are five values. So it is no longer the case that P(y,x) must add up to 1 when integrated over all y (with x held fixed). However, what we can do is this: Let x(s), y(s) be a parametrization of the path of the string. In terms of this parametrization, define the direction of the string at a point, D(x,y) to be +1 if \frac{dx}{ds} > 0 and -1 if \frac{dx}{ds} < 0. Then we can define a new way of counting the string:
N(y,x) = D(x,y) P(y,x)
As shown in the bottom figure, the places where the string doubles back always results in some sections having a negative value for N(y,x) (the sections shown in red) and some sections having a positive value (the sections shown in black). For a fixed value of x, the red and black sections cancel, except for one black section. So we conclude that
In terms of this quantity, it will be true that \int_{-\infty}^{+\infty} N(y,x) dx = 1
If you had a formula for computing N(y,x), then in the simple cases (no doubling back), you can identify N with P and interpret it as a probability density. But in the more complex cases, it can be negative at points, and it has to be interpreted as some kind of "directed string density", which can be positive or negative.
