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Through unit analysis, one can identify the following relationship linking energy, action and power: $energy^2 = action \times power$

Alternatively, we rewrite this expression as:

$power = \frac{energy^2}{action}$

or;

$action = \frac{energy^2}{power}$

In light of this tight relationship, it seems odd that physicists prefer only to discuss energy and action when it comes to quantum field theory. It would seem that the inverse relationship between action and power would lead to an alternative formulation where the goal is to find extremum of power instead of action. So why is there very little discussion of power in physics?

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1 Answer 1

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we say that your formulae are "dimensionally correct", but that's it. They're dimensionally correct for a simple reason: energy is expressed in Joules which is the geometric average of Joule.seconds, the units of action, and Joules per second, the units of power.

But if you can construct an equation that is dimensionally correct, it doesn't yet mean that it is a correct identity - and it is very far from being a demonstrably useful one. I can't imagine any sensible context in which if would be true that "energy squared equals action times power". Moreover, even if such an identity existed, there could be a numerical coefficient, and it is pretty unlikely that it would be one.

So you're just playing with units - and yes, the left hand side has the same unit as the right hand side. That's a necessary condition but not a sufficient one for an equation to be OK. For such an equation to make sense, you must actually know "power of what", "energy of what", and "action of what" you are inserting into the equation. Otherwise you don't know what you're doing.

There is no important law of "extreme power". Why there should be one? Physical laws can't be discovered by randomly combining verbs and names of physical quantities. Science is not equivalent to the business of monkeys who randomly type letters and after some time, they inevitably write Hamlet just like Shakespeare. There is a law that systems that may get rid of some energy - or dissipate it into heat - will eventually do it. That means that they will minimize their energy. So energy minimization exists and is important for the description of the state of objects that become static.

Also, there is the law of least action which can be used to derive the differential equations governing all of classical physics. But there's no law of "extreme power". There's even no context that could be waiting for such a new law. So "extremized power" may be helpful for consumers of electricity who try to save power, and/or for energy companies who try to maximize their power production. But both groups usually have some other considerations aside from the power itself, too. So your "law" never works.

Power itself is simply not as fundamental as energy or action, much like the permittivity change per unit viscosity and per unit entropy per day is not. The simple term "power" shouldn't confuse you - linguistic simplicity of some terms doesn't imply anything for physics. The "power" is still some "energy per unit time" which is a derived, rather than fundamental, concept. Energy and action are simply more fundamental and primary than power.

Best wishes Lubos

P.S.: By the way, it's not difficult to produce as many dimensionally correct equations as many you want. In particular, every equation you write down is dimensionally OK in the Planck units because all quantities are dimensionless. Equivalently, one can always add the product of appropriate powers of universal constants - namely $c,\hbar,G,k_{Boltzmann},\epsilon_0$ - to all terms in an equation to transform a dimensionally wrong equation into a correct one. So the nontriviality or the "degree of shocking surprise" of the fact that you managed to construct a dimensionally correct equation involving energy, action, and power is exactly zero.

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Good answer. The point about dimensionally correct equations not implying validity is an important one. –  Noldorin Jan 17 '11 at 14:33
    
@lumo: Fair enough. –  Humble Jan 17 '11 at 15:15

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