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In electromagnetism, we often define a field propagating in the $z$ direction over time $t$, with wavenumber $k$ and angular frequency $\omega$ as $\text{cos}(-kz+\omega t + \phi)$. Choosing $z = 0$ for convenience and using the phasor (complex) notation, we can express the field as $\text{exp}[i(\omega t + \phi)]$. We can view this as a vector rotating in the complex plane (see figure). By convention, the vector rotates counterclockwise with increasing $t$, and $\phi$ is defined as the phase at $t=0$ measured counterclockwise from the Real axis.

Is the counterclockwise rotation direction implicit in the definition of the field, or can we simply say that the phasor rotates clockwise instead? In that case, can I interpret a field defined as $\text{cos}(kz - \omega t - \phi)$ as being no different from $\text{cos}(-kz+\omega t + \phi)$?

Phasor rotates counter clockwise.

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I'll answer by first comparing with something else where similar issues arise.

Your question is like asking whether I can interpret the angular momentum vector $\bf L$ as being no different from $-{\bf L}$. The answer is no. However, one can admit that the rule whereby the direction of $\bf L$ is obtained from the sense of rotation contains an arbitrary human convention. The standard convention is to use a right hand rather than left hand rule. Once one has adopted that rule one should stick to it. But then someone may come along and note that we could have adopted a left hand rule in the first place, and then all angular momenta and all torques would switch direction, but no physically observable movement, whether displacement or rotation, would change.

Coming now to phasors, the general background is that whenever one has a result using $i = \sqrt{-1}$, one could write down another correct formula by replacing $i$ with $-i$ throughout ones system of equations. However, it is not a correct mathematical operation to introduce such a sign change to some terms and not others in any given equation or deduction. In your example, there is an arbitrary human convention in saying which sign of $i$ to use for which sense of rotation of the field, but once one has made a choice, one must then forever after consistently stick to that choice.

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