Answer

The question whether operating something by means of water is like operating it by hand is one question. On that issue, the author of Even HaOzer and the Magen Avraham disagreed. Even HaOzer forbids operating a mill by means of water. Another question is what you are operating with the water. Igniting a match or grinding are forbidden labors.
I once thought about this analogy in another context: transferring a flame on a Jewish holiday. Why is turning on electricity not similar to transferring a flame? After all, the switch merely enables the transfer of electricity to the device operated by it.
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Questioner:
What about actions that today are not permitted because electricity is involved?
For example, an automatic faucet for washing hands, an elevator, an automatic door.
All these actions are not forbidden in themselves, yet seemingly I could imagine a system that uses water instead of electricity to perform them—and then electricity would not be involved here, only doors opening and elevators going up and down.
According to this electricity-water analogy, should actions like these be permitted?
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Rabbi:
You are conflating two different things. Electricity is forbidden because of itself (building, generating current, and the like), not only because of the actions it performs. That is not the case with water.

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Questioner:
Maybe this is a somewhat general question, but why is the analogy between electricity and an etrog and generating fragrance (from which the prohibition is derived) more fitting than the analogy between electricity and water (which seemingly would permit it)? Intuitively, at least, the water analogy is clearer.
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Rabbi:
I do not understand the question. If you examine these comparisons, you will immediately see the difference. Generating fragrance, like other cases of creating something new, deals with the prohibition against producing something new. This is a rabbinic prohibition and not an actual labor, but I do not see any problem with claiming that electricity too produces something new. This is not an analogy based on similarity of actions—which is the definition of labor—but on results, which were prohibited rabbinically.
By contrast, the analogy between electricity and water deals with the action, not the result. If you are talking about operating something by means of electricity or water, then as I already said, that is indeed similar (at least according to some views). But regarding activating the electricity itself, what connection does that have to water at all?
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Questioner:
I am trying to ask: what is there in electricity that is considered generating something new, which is not present in water? Why is the flow of water not considered generating something new, while the flow of electrons is?
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Rabbi:
The view of electricity as moving electrons (which leads to your analogy) is incorrect, both essentially and for our purposes. On the essential-physical level, that view too is classical. In quantum theory this is not the correct view (unless you see the electron as an abstract wave that moves, rather than as a drifting particle). And on the halakhic plane, even if the classical picture were correct, from our standpoint an electric current is something new (like fragrance), whereas water in motion is not essentially different from standing water. Therefore here there is generating something new, and there there is not. In your opinion, is the difference between a circuit with no current and one with current like the difference between water standing in the sink and water flowing out?
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Questioner:
You wrote to me:
“The view of electricity as moving electrons (which leads to your analogy) is incorrect, both essentially and for our purposes. On the essential-physical level, that view too is classical. In quantum theory this is not the correct view (unless you see the electron as an abstract wave that moves, rather than as a drifting particle). And on the halakhic plane, even if the classical picture were correct, from our standpoint an electric current is something new (like fragrance), whereas water in motion is not essentially different from standing water. Therefore here there is generating something new, and there there is not. In your opinion, is the difference between a circuit with no current and one with current like the difference between water standing in the sink and water flowing out?”
My answer—
Agreed, electricity is not only moving electrons, and to complete the classical description one must also refer to the electromagnetic waves that carry the information onward. The analogy between electricity and water includes that aspect as well, in the form of acoustic waves (the analogy also includes resistors, capacitors, and more). This is especially true for electric circuits that run on direct current, in which the slow movement of electrons (about 10 cm per hour) is essential to the operation of components connected to it. On the other hand, in my humble opinion, if we move to the quantum description we will never get out of it, because then we can already move on to field theory and straight to the Standard Model, and there we will not find useful differences between the phenomena.
In my opinion, the question about the difference between an electric circuit and a sink does not fit the analogy, because these are different levels of complexity. I would compare static electricity in a sweater to a sink with standing water; in such a comparison, in my opinion there is no difference between discharging the static electricity from the sweater and draining the water from the sink. If we look at a simple electric circuit containing only a battery and a switch, do you think the difference between when there is current and when there is not is different from the difference between two water tanks with a pipe between them and a valve in the middle?
I am having trouble understanding why electric current is something new in this picture.
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Rabbi:
In the classical description, you do not need waves. The electric current itself carries the information.
Your comment about the quantum description and the Standard Model is not relevant. Water is a classical phenomenon, and there is no reason to quantize it, and certainly not to apply second quantization. But electricity is essentially a quantum phenomenon.
In any case, this seems to me to be mere stubbornness. Every child can see that there is a big difference between a wire through which electric current flows (which is a completely different state of the wire) and a pipe through which water flows (which does not look like anything new at all. Nothing has been generated here). The current animates the wire, and that is an entirely new state, whereas the water does nothing to the pipe and does not appear to be a new state that has come into being. This seems very simple to me, but if you do not agree, then you do not. We are repeating ourselves.
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Questioner:
Forgive me if this comes across as nitpicking, but I still do not understand in what exact essential way a flow of water differs from a flow of electricity. My plain feeling is that in all matters of the prohibition of electricity on the Sabbath, they shot the arrow and then drew the target around it. There is no doubt that the prohibition is welcome, but in the end this is Jewish law and not folklore, and it is important to me to understand why it is halakhically forbidden. What I meant in my comment about the Standard Model is that the quantum description of electricity is not relevant to our discussion (which is what the Rabbi himself said about applying second quantization to water).
One could also write: “Every child can see that there is a big difference between a pipe through which water flows (which is a completely different state of the pipe) and a wire through which electric current flows (which does not look like anything new at all. Nothing has been generated here). The water animates the pipe, and that is an entirely new state, whereas the electric current does nothing to the wire and does not appear to be a new state that has come into being”—and it would be just as convincing.
I apologize if my words seem like mere stubbornness. I truly do not understand what is so essentially different between a simple electric circuit and a faucet in the bathtub (or even an optical fiber!) with respect to generating something new. In my opinion, there is a conceptual mistake here between end components connected to an electric circuit and the current itself.
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Rabbi:
I do not know how to explain it better. The difference seems self-evident to me, but apparently we do not agree on this.