# Is renormalization associated with a volume scale or with an energy-momentum and length scale?

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Given that real-space renormalization blocks together small volume elements to construct larger volume elements, is it more appropriate/helpful to consider the renormalization scale to be a volume scale instead of an energy scale?

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retagged Mar 7, 2014
In a quantum mechanical theory momenta and inverse length scales are related, organizing degrees of freedom by length or momentum scales is equivalent, which one to use is a matter of convenience (and I think both are used in practice). I think the relation to energy (rather than momentum) scale is specific to relativistic theories. Those are general purpose comments, maybe you are asking a more specific question?

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Thanks, Moshe. It's a Question that comes out of my current computations, in which it appears to be much more natural to think of renormalization in terms of measure instead of in terms of lengths. I hope, I suppose, that someone will be struck enough by a similar aspect in their own research that they feel the urge to Comment or Answer. Long shot, of course.

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The distinction is not clear in the question, at least not to me, more detail will probably increase the likelihood of helpful answers.

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The question doesn't make sense to me at all. In hbar = c = 1 units length = 1 / energy and of course volume = length ^ 3. So energy scale defines length scale defines volume scale. It is the same thing.

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For a volume scale, one only needs the tensor $\epsilon^{\mu\nu\alpha\beta}$, one does not need any metric. This is splitting hairs at some level, of course, but I take it that conceptual progress is about making a personal choice of what distinctions to take seriously and which to put aside for another day.

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You mean you want to consider a QFT which depends on a volume form but no metric? Do you have an example in 4D?

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I have a theory, better to say a class of toy models, ... which I hope to arXiv soon, that I think casts a curious light on renormalization. It's not that there is no metric, however, it's just that the roles of the metric and of the volume form are distinct enough to make my Question make sense -- I would not have asked it and it would not have made sense to me a month ago.

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Still a scale is just a quantity with dimension. If you have a volume scale, doesnt matter how you got it, you can take its root and get a length scale and vice versa.

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Squark, I think you're right. One might be interested in other structure, but a scale is just a scale. Thanks.

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OK, so question can be closed?

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@Squark, you can vote to close, presumably as "not a real question". I tried to tease out a question here, but to my mind unsuccessfully. On the other hand, I don't want to make all decisions for the community.

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I'm not sure what harm having this question here does, nor am I sure what benefit there is in closing it. The point of the question is to ask whether anyone else is doing research in which this kind of distinction is of interest. I don't take it to be a strong argument unless you have yourselves done calculations in which such a distinction makes sense, but if one *is* interested in other structure than just scale, a volume scale is not the same as an energy scale.

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