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People often talk about "pure energy" in rather an informal way. In truth there is no proper notion of pure energy. Loosely one often means photons when the term pure energy is used. For example, you may come across statements like: when matter and antimatter collide they annihilate producing pure energy.



Energy is a property of "stuff"; that is a physical system. A configuration of a physical system will have a property that we can indirectly measure, which we call energy. One cannot have energy as some independent "thing".



As an analogy, you may talk about the colour of a car. Lets say a red car. Being red is a property of the car. One cannot talk about "red" as some notion independent of the objects we see as red in colour. Red does not exist by itself.



So, what is energy?



Informally energy is understood as a property of one physical system that allows it to preform work on another physical system. In essence this means that energy is the property associated with movement or change. It is the "doing" property.



Not that the above tell you what energy is. However, this is not really a problem as physics tends not to deal with the metaphysical notion of existence and what is. Physics deals with mathematically modelling nature. With that in mind, one should keep close the quote (I paraphrase) I believe is due to Feynman:

Energy is a number we can calculate at different points in time and find the same value.

That is energy is something we can calculate, given some configuration of a physical system and (given some technical stuff) we see that the energy does not change in time. That is, it is conserved.



A little more mathematically



We have a very powerful and beautiful theorem that relates symmetry and conservation laws.



Noether's first theorem: Any differentiable symmetry of the action of a physical system has a corresponding conserved charge.



This theorem is at the heart of modern physics and is based on the calculus of variations.



What does this theorem mean?



Theorems as theorems are by their very nature technical. But we can informally understand some consequences of this statement quite easily.



If the mathematical description of the physical system does not alter upon changes in time then there is a conserved quantity that we call energy.



This is as close to answering the question what is energy? as you can really get. Energy is the quantity that is associated with a physical system not explicitly depending on time.



The caveat here is that the physical system not depend explicitly on time. This is generally reasonable. From a physical perspective this seems natural, any experimental outcome should not depend on when you preform the experiment. Because you get the same result today as you will tomorrow, energy is conserved.



Back to pure energy



I hope I have explained that the notion of "pure energy" is not well founded. Energy is a number that is associated with physics not changing on when you preform your experiments.



Noether's first theorem makes this association with time and energy explicit. Other common conserved quantities exist:





Symmetry
Conserved Quantity




Translations in Time
Energy



Translations in Space
Linear Momentum



Rotations in Space
Angular Momentum




In the same way nobody talks about "pure angular momentum" as some thing in its own right, no one should use the term "pure energy".

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