I was looking at reviews for Sakurai's Quantum Mechanics textbook, and some mentioned it being outdated, specifically mentioning his use of imaginary time. Is this idea deliberately avoided in modern treatments?
I can't see why a simple parameter change $t\to it$, would be or not be an outdated concept. It doesn't make things significantly prettier, but it doesn't hurt anything either.
With that said I've never before heard the specific phrase imaginary time so maybe it is outdated.
I can't see why a simple parameter change t->it, would be or not be an outdated concept. It doesn't make things significantly prettier, but it doesn't hurt anything either.
The idea of imaginary time was introduced in this context in an effort to shoehorn Euclidean geometry into special relativity. The minus sign in the spacetime interval
$$\mathrm ds^2 = \color{red}{-} \mathrm dt^2 + \mathrm dx^2 + \mathrm dy^2 + \mathrm dz^2$$
was unsettling (the Euclidean norm is positive-definite), and so it was resolved by simply using $it$ as the time coordinate, in which case $$\mathrm ds^2 = \mathrm d(it)^2 + \mathrm dx^2 + \mathrm dy^2 + \mathrm dz^2$$
While this does clear up that minus sign, it also begs the question of why the time coordinate should be imaginary, so this is a zero-sum game.
In the modern understanding of special relativity, we realize that spacetime is not Euclidean, but rather Minkowski. The fact that the Minkowski (pseudo)norm is not positive definite is physically significant, and is tied to the fact that light-like trajectories must remain light-like under arbitrary coordinate transformations (which is in turn tied to the fact that $c$ has a fundamental significance to the geometry of spacetime). In that way, the minus sign which we were trying to hide by introducing imaginary time is in fact essential and physical.
Additionally, this geometrical understanding of special relativity can be straightforwardly generalized to a curved Lorenzian manifold for General Relativity, in a way which is impossible for the Euclidean geometry + $it$ prescription.
Let me try to clear up the comments and answers already given by simply stating: imaginary time is not an outdated concept at all! In fact, if you ever take a course on QFT (which I suppose depends upon your professor and text), you'll want to use the imaginary time transformation namely $t\rightarrow -i\tau$ when doing momentum integrals in order to make them analytically solvable (the Minkowski integrals can be solved as well, but require a more careful process). In this context (and in the literature), this is process is called a Wick rotation. The Wick rotation is still widely used in almost all aspects of physics that uses aspects of QFT.
As an example of how the Wick rotation is used in research, consider de Sitter spacetime given as \begin{equation} ds^2 = \frac{1}{(\eta H)^2}\left(-d\eta^2 +d\mathbf{x}^2\right) \end{equation} where $\eta$ is conformal time and $H$ is the Hubble parameter. We can perform a conformal Wick rotation namely $\eta\rightarrow i\tau$ giving us an all plus signature on the metric. This is now Euclidean de Sitter or abbreviated typically as EdS space. I can also perform the following transformation giving $H\rightarrow i\ell$ where $\ell$ is some new inverse length parameter, and I get now an overall minus sign, which is actually Euclidean Anti-de Sitter space (EAdS).
So no, imaginary time (or Wick rotations are not outdated); it is still a handy math trick, but there are caveats when using it. For example, the results in EAdS compared to EdS can disagree in the IR, and you have to be extremely careful when performing the analytic continuation (which is all that a Wick rotation is) since you could hit poles (of any kind) that prevents such a transformation. This is sometimes why it is not used in certain cases.
You can call it imaginary time or you can call it negative spatial distance. It doesn't matter. The second is just a Wick rotation of the first.
You can also flip the signature and use the Einstein representation, +---, in which time is positive.
But no matter what you want to call it, time "acts" like negative distance.
BTW, if you take a vector extending into both time and space, rotating it into the space direction (to the right) is acceleration. That gives the interval more extent in space and less in time. That's time dilation. But that's Minkowski space.
If you rotate a vector in Lorentzian space (where one dimension is negative, i.e., reality) the length of the interval vector will be negative if an object exceeds the speed of light.
Elapsed time is negative distance.
