What is the physical interpretation of imaginary lengths?

Question

My question is about the meaning of imaginary lengths, which occur often in the solution to various numerical problems in Physics. Generally imaginary quantities are discarded as nothing but mathematical tools, but this also means our narrowmindedness, for in such a case we are escaping curiosity. The following is only an example of such a question.

A boat moves relative to water with a velocity $n$ times less than the river flow velocity. At what angle to the stream direction must the boat move to minimize drifting? Take the river width to be $d$. (Adopted from Problems in General Physics, I.E. Irodov, Mir Publishers, Moscow, 1979)

Let $\vec{v_0}$ be the stream velocity and $\vec{v'}$ be the velocity of the boat w.r.t water. Let $\frac {v_0}{v'}=n>0$ s.t. the boat is drifted. Let $\vec{v'}$ make an angle $\theta$ with $\vec{v_0}$. Then time taken to cross the river, $$ t=\frac {d}{v'\sin \theta} $$ In this time interval, drifting of the boat, $$ x=(v'\cos \theta+v_0)t $$ $$ =(\cot \theta + n\csc \theta)d $$ For minimum, $$ \frac {dx}{d\theta}=-d \csc\theta(\csc\theta +n\cot \theta) = 0 $$ Or, $$ \csc\theta +n\cot \theta=0 $$ Or, $$ \theta = \pi - \sec^{-1} (n) $$ In this case, drifting, $$ x_0 = d\sqrt {n^2 -1} $$

My Problem

I dared to investigate the other case, i.e. when $\csc \theta =0$. Mathematically (as well physically, for there is no drift possible) this is impossible for $\theta \in \mathbb {R}$. But for $\theta \in \mathbb {C}$, this implies that the boat does have a drift, not a real and observable drift, but an imaginary drift $x_0 = id$. The problem begins here. As far as I know, physics is the subject of the observed and observable things. But then this definition of physics looks very naive when we come to acknowledge that electrons do exist, although we can never directly observe them in strict sense (or in other words, Heisenberg's principle wouldn't allow us to do so!). But there are always indirect evidences. In the same spirit, does the above solution point towards the existence of an alternate dimension, in which the boat does drift a distance $d$? This is not my personal theory, rather I have been motivated by the fact that time is taken as an imaginary dimension in Minkowski spacetime. So, should I take it to be the traversal of the body (in this case the boat) from one point of time to another, or a dilation or contraction of time? Any help would be appreciated.

Answer 1

Physics deals with finding models that fit reality well, and then using them as explanations for further observations. Many models have limited domains of applicability and do not make sense outside them because their core assumptions do not work outside. Sometimes a model does produce unexpected predictions outside the normal range and they actually fit patterns in reality because the model and its assumptions did catch some underlying pattern better than expected (consider the use of complex indices of refraction).

Would an imaginary angle be of the first or second case here? As far as I can see, it is a case of the first: the angle is introduced to denote an actual real angle the boat is taking relative to the water. One could extend the dynamics and say positions and velocities are complex, but it seems that only the real case has a direct physical meaning unless one wants to invent a complex Newtonian mechanics (which doesn't make much sense since energy can become arbitrarily negative by having imaginary velocities: it is very much a second case extension of our model).

Answer 2

Unless I misunderstand the problem, I think you might be approaching it from the wrong direction. There is a given velocity magnitude available to move the boat toward the opposite shore, $\vec{v}_{max} = \frac {|\vec{v}_1|}{n} \vec {\epsilon}_x$. $\vec {\epsilon}_x$ is a unit vector directed straight across the water toward the shore. The time $t$ required to reach the opposite shore will be minimized if $v_{max}$ is directed straight toward the opposite shore. So now you can calculate the angle of motion of the boat (relative to $\vec {v}_1$) as simply the angle of $\frac {v_1}{n}\vec{\epsilon}_x + |\vec{v}_1|\vec{\epsilon}_y$.

Answer 3

Whenever an imaginary component shows up in a calculation of an observable quantity, in the case of a calculation involving a law of physics, you have either done the calculation wrong, or one or more of the inputs you used were not physically possible inputs.

I can't think of any instance in which an imaginary quantity ever shows up as a value for anything other than an intermediate step in a calculation.

For example, even though some CKM matrix elements used to determine the probability of weak force quark flavor transitions have imaginary components, all observables derived from calculations involving the complex elements have real number answers between 0 and 100%.

There are times in quantum physics where, for example, a sub-part of a calculation used to determine the probability of something happening has a negative probability due to complex numbers in intermediate calculations. But, that never causes the actual probability of happening to fall below 0% because there are always other probability that have to be considered before considering the overall probability of something happening that get added in that end up counterbalancing that contribution, so there is not a negative probability observable.

Some of the more common ways that you can mistakenly end up with an imaginary valued answer (or a negative answer for quantities that are always either zero or have positive real values (e.g. rest mass)), are (1) to fail to use an absolute value when one is required, as illustrated by the answer from @S.McGrew, (2) to use scalar values for variables, when you should be using variables that are vectors or tensors, (3) to treat modulo quantities as real number quantities, (4) to put a wrong sign on an input because you are using inconsistent coordinates for your inputs, or (5) to use hypothetical inputs that cannot be physical in the context of the particular problem you are working with, often for some subtle reason that that you hadn't considered originally when you posed the hypothetical. Also note that $e{^i}{^\theta}=cos\theta+isin\theta$, so it can produce real number valued outputs at select values of $\theta$.