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Quantities like position and time allow us to place our origin anywhere, but can the same be said for energy? I was thinking about the way we have defined the quantity gravitational potential, and for any finite distance from a 'planet' or a body, an abject would be said to have negative gravitational potential, conveying that a negative amount of work is done (in bringing the object from 'infinity' to that position, but this seems impossible as negative energy does not seem to physically exist (as far as I know). So, does negative energy really exist, or does it not (implying that negative work done is just another mathematical argument)?

Edit: The primary reason I ask this question is because I was wondering whether negative work done is fundamentally different from negative debts (when talking about money), or are they just the same - a mathematical convenience.

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To be honest, signs don't exist. Energies don't exist. They are merely methods or models or descriptions invented for us to describe forces, tendencies, absorption of radiation, impacts, vibrational motions, directions etc. So, does negative energy exist? We can just invent it, like you just described for the potential energy, so yes.

  • In some situations a sign describes nothing but the size of values (when it doesn't matter where the origin is, such as with potential energies with arbitrarily chosen reference points),
  • in other situations the sign describes the mathematical act of adding or subtracting (like heat absorbed or expelled),
  • while in yet other situations signs describe directions on a predefined axis (your axis defines forward, so a speed moving you backwards is negative).
  • Etc.

A sign is a mathematical invention that means different things in different contexts, depending on what we need to do.

conveying that a negative amount of work is done

A "negative amount of work" belongs to the 2nd bullet point above. Work being negative just means that energy is leaving (mathematically subtracted) the object. The negative sign is here nothing more than mathematical.

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    $\begingroup$ What you are saying is just that mathematics is a means to describe reality, and not real itself. That's true. So obviously, the question cannot be if energy has the same status of reality as this chair, this is clear, so it's quite an empty statement to say that "energy does not exist". The question is about our description: Does it play a fundamental role in this description which sign we have? $\endgroup$
    – Luke
    Jul 13, 2018 at 10:25
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    $\begingroup$ @Luke Which is answered in the last part of this answer. Apart form that the OP specifically asks: "Does negative energy really exist?", thus the above approach to an answer. $\endgroup$
    – Steeven
    Jul 13, 2018 at 12:11
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    $\begingroup$ This is wrong. Energy is just as physical as a chair is, you can define it positivistically and measure it in the same way a chair is also measured by your eyes by sensing the light and other things. It's useless, as well as meaningless, to call something "just a mathematical construct", everything in physics is a mathematical construct, that also exists physically. $\endgroup$
    – Abhimanyu Pallavi Sudhir
    Jul 13, 2018 at 17:04
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    $\begingroup$ @Steeven Energy is just another component of "momentum". If the point of view for energy you presented here is true, then it should also be true for other momentum components. $\endgroup$
    – Oktay Doğangün
    Jul 13, 2018 at 18:45
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    $\begingroup$ @Steeven Energy and three-momentum are the components of the (four-)momentum which is the conserved quantity under the local translation symmetry on space-time. Just like time being a component of the (four-)position. When you switch between reference frames, actually you kind of rotate (technically called boost) between energy and three-momentum components, e.g., $E' = E \cosh \xi + p \sinh \xi$ where $\xi$ is the rapidity. There is no physical distinction between space & time or momentum & energy, it is just our perception of time foliates the space-time into time and 3 bulk components. $\endgroup$
    – Oktay Doğangün
    Jul 13, 2018 at 19:13
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The question about "existence" is philosophical one, not a physics one. Some people would argue that you can't be sure about the existence of anything, including yourself.

Physics is concerned with creating mathematical models that can make predictions about the outcome of a experiment or situation. All physical models have limited applicability, i.e. the predictions are only accurate if certain conditions are met. So far there aren't any models that are truly universal and applicable in every possible situation.

Negative, or positive, energy are just that: concepts that are useful to make quantitative predictions in certain situations (but not in others).

For example, negative energy is a useful concept is energy is moving around inside a system. If it's moving one way you call it "positive" , if it's moving the other way it's negative.

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    $\begingroup$ If you can question the existence of anything, then what is physics about? Personally, I believe that something that I call "objective reality" must be postulated. $\endgroup$
    – my2cts
    Jul 13, 2018 at 15:51
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Of course it is very important in some calculations if the energy change in some system is positive or negative. If you are building a power plant, rather try to get a positive amount of energy out!

But that's only differences in energy. The point of energy zero is defined arbitrarily and you can always redefine it, so that the absolute sign of $E$ is unimportant.

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  • $\begingroup$ Well I would put the zero of energy at the vacuum. Unfortunately that has an energy of plus $10^{123} $ unless there is a mistake somewhere 😀. $\endgroup$
    – my2cts
    Jul 13, 2018 at 15:48
  • $\begingroup$ @Luke That does make sense. One question though; for position, we can 'set' an origin simply by declaring where (whatever that may mean) the origin is, but how does one go about 'setting' a zero for energy? $\endgroup$
    – user154527
    Jul 14, 2018 at 5:49
  • $\begingroup$ @Supernova As I said, the vacuum sets the lower limit to energy. $\endgroup$
    – my2cts
    Jul 14, 2018 at 8:36
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There's no negative energy, and the origin for energy is absolute, not arbitrary. To measure the energy of a system, you can simply measure its mass, then use $E = mc^2$ to find its energy (in a frame where its net momentum is zero). The energy is always positive because the mass is always positive. Maybe you can find some really exotic counterexamples with the Casimir effect or something, but basically it's positive.

When we say that the gravitational energy for the Earth-moon system is negative, it just means that the energy is lower than it would be if the moon were further out. The Earth-moon system has a certain mass. It's less than the mass of the Earth plus the mass of the moon, each considered separately. This is called the "mass defect".

So we say there's negative energy in the gravitational binding of them to each other, but that's just so that the total sum of energy comes out right. The total amount of energy is still positive because the mass of the system is positive.

When people say that the origin of potential energy doesn't matter, they mean that forces are derivatives of the potential energy. If you add a constant to the potential energy, that doesn't change any derivatives, so if all you care about is finding the forces, it doesn't matter if you add a constant to the potential energy. However, there is still only one choice of zero for the energy of an entire system that accounts for all the energy, and that's one that makes $E = mc^2$ true.

Note: this presumes we're dealing with systems that have a rest frame; massless systems don't have a rest frame.

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Energy is a concept which can be positive or negative. However, it is interesting in that, in many situations, it is meaningful to define a "minimum" energy. This is different than other things, like positions. There is rarely a meaningful "minimum position," unless there is a symmetry which makes such a concept meaningful.

Because there is a minimum, it can be meaningful to assign it to the value 0. We could have picked any value to assign it to, but zero is convenient. If we make such an assignment, then it is trivial to show that energy must always be non-zero.

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The energy density equations for an electric field for a capacitor with a dielectric contain a negative component. This energy has to be supplied to pull the dielectric of a capacitor. Unlike most negative energy components this one has neither a floating base (like most potential energies) nor an obvious rule that keeps the total energy positive. So far, no attempts have succeeded to make a dialectic low enough mass and charge the capacitor to a high enough voltage so that the negative energy component exceeds the mass energy of the dielectric. If this ever succeeded (and I know of no law that absolutely forbids it though it seems unlikely to work) then negative energy components would be realized and we could hold wormholes open.

The consequence for constructed negative energy in physics is backwards time travel.

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The answers so far to this question have been both irrelevant and wrong. The question isn't some philosophical nonsense about whether negative energy really exists or whether energy really exists. It is a straightforward question, about whether there is a shift-invariance in energy, i.e. if the energy of the universe increased by a fixed quantity, would there be an impact on the physics?

The answer is clearly no, there is no impact, you do have "energy translational" invariance. The reason this is not talked about is that it's completely useless. It makes no sense to increase the energy of every object by the same amount (because then the system "A + B" will have to increase its energy by 2E, not by E), so what you're really talking about is a shift in your unit system, where you call "0 J" as "20 J" instead, and then energy is not easily additive anymore (you have to subtract 20 every time you add things). This is like using Celsius instead of Kelvin, or pH instead of proton concentration, there is no physical importance to this.

Note that it is impossible to have any unit system in which negative energy doesn't exist, as you can get arbitrarily negative gravitational potential energies near a black hole, and to translate -∞ to 0, you need to map everything else to +∞.

Edit -- Global shifts in local energy density are meaningful to speak of, since it's a local phenomenon and A + B doesn't need to shift by the same amount as A and B. In this case, the physics obviously isn't invariant, since you get more gravity.

Answering the question that wasn't asked, but everyone else is answering --

Yes, energy does exist, it means nothing to say that something is "just a mathematical construct". You can define energy positivistically, and measure it, this is fundamentally no different from measuring a chair, which also means you're measuring the light from the chair incident on your eyes, etc.

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    $\begingroup$ Sorry but this is incorrect. General relativity is sensitive to global shifts in the energy. There is no "energy translational" invariance in the universe. $\endgroup$
    – AccidentalFourierTransform
    Jul 13, 2018 at 17:13
  • $\begingroup$ @AccidentalFourierTransform What exactly are you referring to by "global shifts in energy"? Like I mentioned, a simplistic "increase everything's energy" transformation is inconsistent -- the only possible interpretation of such a thing is an alternate unit system where energy isn't additive (and all physical theories are clearly invariant to these). Shifts in energy density are a different issue altogether. $\endgroup$
    – Abhimanyu Pallavi Sudhir
    Jul 13, 2018 at 17:39
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    $\begingroup$ "The question isn't some philosophical nonsense about whether negative energy really exists" Quote from the OP: "does negative energy really exist, or does it not". I don't think the question goes as deep as you interpret it. $\endgroup$
    – Steeven
    Jul 13, 2018 at 18:17
  • $\begingroup$ @Steeven Read the first line of the question. I'm not interpreting anything, he's asking about whether shifts in energy leave physics invariant. It's not all that deep, but it's a physics question. $\endgroup$
    – Abhimanyu Pallavi Sudhir
    Jul 13, 2018 at 19:52
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    $\begingroup$ I think you've answered your interpretation of the question. This is fine. The problem is claiming "the answers so far to this question have been both irrelevant and wrong" because they answered their own interpretations. Dial it back. Keep it to your interpretation of the question and your answer. Use votes and comments to comment on other answers. Ask for clarity from the OP rather than declaring The One True Interpretation. $\endgroup$
    – Schwern
    Jul 13, 2018 at 23:45